Latest revision as of 09:57, 1 February 2021

4 VIGAS TRIDIMENSIONALES DE MATERIAL COMPUESTO

4.1 INTRODUCCIÓN

Una viga tridimensional (3D), también llamada pieza, es un sólido prismático alargado sometido a esfuerzos axiles, de flexión, de cortante y de torsión. Se pueden encontrar estructuras con vigas 3D en edificios y construcciones industriales, arcos, placas rigidizadas, partes estructurales de vehículos de transporte terrestre, fuselajes de aviones y naves espaciales, cascos de barcos, piezas mecánicas, etc. La Figura 4.1 muestra ejemplos esquemáticos de estructuras formadas por piezas rectas.

En este capítulo estudiaremos el cálculo por el MEF de vigas 3D de sección arbitraria y material compuesto. Muchos de los conceptos son extensiones de los estudiados para vigas planas en los Capítulos 1–3 y se añaden algunos temas avanzados. En aras de la exhaustividad, se explican los principales conceptos teóricos de la teoría de vigas 3D, aunque de forma concisa. Se recomienda a los lectores que no estén familiarizados con el análisis de vigas 3D, el estudio de libros clásicos de Resistencia de Materiales y Análisis Estructural [Li,OR,SJ,Ti2,3].

En la primera parte del capítulo se aborda el estudio de una viga 3D de material compuesto mediante una extensión de las teorías de vigas de Euler-Bernoulli y Timoshenko estudiadas en los Capítulos 1 y 2. Se describe la formulación de elementos de viga 3D de dos nodos rectos y curvos. Se supone que la relación constitutiva entre las tres tensiones y deformaciones significativas tiene forma diagonal. Esto restringe la aplicabilidad de la formulación a un rango específico (aunque amplio) de materiales compuestos. La torsión libre (torsión de Saint-Venant) se estudia primero. Esta teoría supone que las tensiones y deformaciones inducidas por el alabeo de la sección son nulas. Esta hipótesis es exacta cuando el momento torsor es constante a lo largo de la longitud de la viga y el alabeo no está restringido en ningún punto. La teoría de Saint-Venant es también una buena aproximación cuando el torsor no es uniforme en vigas de secciones macizas (rectangular, cuadrada, circular, etc.), en secciones formadas por piezas rectangulares delgadas (sección angular, sección en T, etc) y en secciones huecas celulares (tubos, cajones con ancho/largo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \le 4} , etc.). Se explica también una particularización del elemento de viga 3D de dos nodos basado en la teoría de Saint-Venant para emparrillados planos.

Para otros tipos de secciones bajo torsión no uniforme, o torsión uniforme restringida, deben tenerse en cuenta las tensiones y deformaciones de alabeo [MB,OR,Vl]. Se presenta con detalle una teoría de torsión para vigas de pared delgada abierta que exhiben fuertes efectos de alabeo. También se describe brevemente una versión refinada de esta teoría que tiene en cuenta las deformaciones tangenciales inducidas por la torsión.

Al final del capítulo, presentamos un procedimiento para desarrollar elementos de viga 3D mediante una degeneración de elementos de sólido 3D. Esta formulación es aplicable a vigas de material compuesto y es una alternativa a los métodos tradicionales para desarrollar elementos de viga usando teorías ``clásicas de vigas. En el Capítulo 10 se aplican los elementos de viga 3D para su uso como rigidizadores de placas.

4.2 DEFINICIONES BÁSICAS PARA UNA VIGA 3D DE MATERIAL COMPUESTO

4.2.1 Ejes locales y globales

Una viga 3D es un sólido prismático de longitud Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L}

y área transversal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A}

, orientado en la dirección longitudinal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'} , cuyas dimensiones en el plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'z'} , ortogonal a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'} , son relativamente pequeñas comparadas con la longitud Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L} . El punto O obtenido por intersección del eje de la viga Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

con la sección transversal se supone   situado en el eje neutro. Por lo tanto, el eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}
se denomina en adelante eje de la viga o eje neutro indistintamente.

La geometría de la viga se define en un sistema de coordenadas global Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x,y,z}

(Figura 4.2).

En lo que sigue supondremos que el eje de la viga es recto y las propiedades de los materiales y las geométricas constantes a lo largo del eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'} .

La definición del eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

de la viga como el eje neutro desacopla los efectos de flexión y axial (Apartado 3.6). Los efectos de flexión y de torsión también están desacoplados cuando las fuerzas externas en las direcciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}
actúan sobre el centro de cortantes (denominado también centro de esfuerzos cortantes o centro de   torsión) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

. La línea recta que conecta los puntos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

de todas las secciones es el eje elástico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x'}

. En nuestros desarrollos supondremos que los ejes neutro y elástico son paralelos, aunque no coincidentes (Figura 4.2).

El punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}

sobre el eje neutro coincide con el centro de gravedad de la sección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G}
para secciones homogéneas (Figura  4.2 y Apartado 3.6).

Para secciones macizas homogéneas, secciones de pared delgada cerradas y secciones de pared delgada abiertas con simetría doble, los puntos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}
coinciden (Figuras 4.2 y 4.6).

Para vigas de material compuesto, el centro de gravedad (G), el centro de cortantes (C) y el eje neutro (O) no suelen coincidir (Figura 4.3).

4.2.2 Comportamiento constitutivo

Las tres tensiones locales en una sección (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'},\tau _{x'y'},\tau _{x'z'}} ) (Figura 4.4) se relacionan con las deformaciones conjugadas (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varepsilon _{x'},\gamma _{x'y'},\gamma _{x'z'}} ) mediante la ecuación constitutiva

La matriz constitutiva Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol D}'}

de dimensión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 3\times 3}
se deduce de la ecuación constitutiva general de la elasticidad 3D [On4] haciendo cero las deformaciones que se desprecian en la teoría de vigas 3D (es decir, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varepsilon _{y'} = \varepsilon _{z'}=\gamma _{y'z'}=0}

). Para un material heterogéneo, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol D}'}

es una matriz llena (simétrica). Para las teorías de vigas estudiadas en este capítulo supondremos que el material es ortótropo con la orientación de uno de los ejes principales del material  coincidente con el eje de la viga. Bajo esta hipótesis, la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol D}'}
tiene una simple (y estándar) forma diagonal

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E}

es el módulo de Young longitudinal (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E=E_{x'}}

) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{z'}}
son los módulos de rigidez transversal (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{y'}=G_{x'y'}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{z'}=G_{x'z'}}

). Para material isótropo

Los parámetros del material pueden variar en cada punto de la sección, siempre que la anterior hipótesis se mantenga.

La forma simple de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol D}'}

de la Ec.(4.2) nos permitirá hacer uso de conceptos de la teoría clásica de vigas que probablemente sean familiares para muchos lectores.

Sección de materiales compuesto laminados

Para una sección de material compuesto laminado (Figura 4.5a) supondremos que los parámetros constitutivos son constantes dentro de cada capa. Los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{z'}}
para la capa se pueden calcular rotando las ecuaciones constitutivas para el material de la capa (supuesto en estado de tensión plana, es decir, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _n=0}

) de los ejes laminares (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L,T,n} ) a los ejes locales de la viga (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',y',z'} ), siguiendo un procedimiento similar al explicado en el Apartado 8.3.3. La matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol D}'}

resultante es una matriz llena que puede  diagonalizarse eliminando en primer lugar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varepsilon _{y'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle  \gamma _{y'z'}}

, que se suponen nulos, e imponiendo después que las relaciones constitutivas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{y'z'}}
estén desacopladas.

Un procedimiento alternativo es mantener la expresión completa de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol D}'}

y después simplificar su forma generalizada (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat{\boldsymbol    D}'}

) que relaciona los esfuerzos y las deformaciones generalizadas [Va,VOO].

En nuestros desarrollos aceptaremos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol D}'}

es diagonal, lo cual ocurre invariablemente cuando el ángulo entre el eje longitudinal laminar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L}
y el eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}
es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0^\circ }
o Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 90^\circ }
(Figura 4.5a). A pesar de esta simplificación, este modelo es aplicable a un gran número de vigas de material compuesto laminado.

Bajo las hipótesis anteriores es útil definir Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

como el eje neutro, e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}
como los ejes principales de inercia de la sección.

4.2.3 Parámetros constitutivos resultantes y eje neutro

Elijamos un sistema de coordenadas ortogonal arbitrario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar x, \bar y,\bar z}

vinculado al centro de gravedad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G}

, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar x}

paralelo a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

. Una simple traslación de este sistema al eje neutro O (cuya posición es aún desconocida) da el sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', \bar y',\bar z'}

vinculado al punto O (Figura 4.5b). Los parámetros constitutivos resultantes (generalizados) axiales y flectores se definen como

Para vigas 3D de materiales compuestos e isótropos, el eje local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', \bar y',\bar z'}
se define como el eje neutro si

El eje neutro en el sistema de coordenadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',\bar y',\bar z'}

satisface

Las Ecs.(4.7) dan la posición del punto neutro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O(\bar y_0, \bar z_0)}

en el sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', \bar y,\bar z}
como (Figura 4.5b)

La posición del eje neutro no cambia si el sistema local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', \bar y',\bar z'}

se gira alrededor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}
para dar el sistema de coordenadas local final Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', y', z'}
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y' ,z'}
son los ejes principales de inercia.

4.2.4 Ejes principales de inercia

Los ejes principales de inercia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y', z'}

que definen el sistema de coordenadas local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',y' ,z'}
(en el punto sobre el eje neutro O) se obtienen en función de los parámetros constitutivos  en el sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', \bar  y',\bar z'}
como sigue.

Las coordenadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y', z'}

de un punto arbitrario de la sección se expresan en función de las coordenadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar y', \bar z'}

, es decir

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha }

es el ángulo entre el eje principal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y el Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar  y'}
(Figura 4.5b). Usando las Ecs.(4.4) obtenemos

Los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): y'

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): z'
son ejes principales de inercia si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \hat    D_{b_{y'z'}}= 0

, es decir, si

Los parámetros constitutivos de flexión principales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_{b_{z'}}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_{b_{y'}}}
se obtienen en función de los parámetros constitutivos en el sistema de coordenadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', \bar y',\bar z'}
como

Usando la Ec.(4.10) se obtiene

4.2.5 Resumen de los pasos para definir el sistema de coordenadas local

Los pasos para definir el sistema de coordenadas local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', y', z'} , vinculado al punto O sobre el eje neutro (Figura 4.5b), para una viga 3D son:

1. Encontrar el centro de gravedad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G} de la sección y definir el sistema ortogonal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar x ,\bar y , \bar z} vinculado al punto G, en el que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar x} es paralelo a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'} .
2. Encontrar la posición Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar y_0, \bar z_0} del punto O sobre el eje neutro que define los sistemas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x', \bar y', \bar z'} vinculados a dicho punto (Ecs.(4.7)). Calcular los parámetros constitutivos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_a} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_{b_{\bar y'}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_{b_{\bar z'}}} y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_{b_{\bar y'\bar z'}}} mediante las Ecs.(4.4).
3. Calcular el ángulo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha } que define la posición del sistema de coordenadas local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x' ,y', z'} vinculado al punto O (Ec.(4.11)).
4. Calcular los parámetros de flexión en los ejes principales de inercia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_{b_{y'}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_{b_{ z'}}} mediante las Ecs.(4.12).

Para material homogéneo el eje neutro coincide con el centro de gravedad y

Las direcciones principales de inercia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y',z'}

se definen en este caso por

y los momentos principales de inercia son

con

Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar y'}

o Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar z'}
son ejes de simetría, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_{\bar    y'\bar z'}=0}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_{y'}=I_{\bar y'}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_{z'} =I_{\bar z'}} .

4.2.6 Cálculo del centro de esfuerzos cortantes

Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

el eje neutro y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y',z'}
los ejes principales de inercia de una sección. El campo de tensiones está definido por las tensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'z'}}
(Figura 4.4). Las tensiones tangenciales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'y'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'z'}}
definen los esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{y'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q _{z'}}
y el momento torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{x'}}
(denominado “el torsor” por brevedad)

El torsor con respecto al eje elástico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x'}

que pasa por el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}
de coordenadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {y'}_c,z'_c}
es

El punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

es el centro de esfuerzos cortantes si los esfuerzos debidos a los efectos de flexión satisfacen

lo que da

El procedimiento para calcular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {y'}_c}

en una viga de material compuesto es el siguiente:

• Calcular la distribución de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'z'}}
  sobre la sección para un esfuerzo cortante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{z'}}
siguiendo el   procedimiento explicado en el Apéndice D para el caso general, o el   del Apartado 3.7 para flexión cilíndrica.

• Calcular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{x'}}

mediante la Ec.(4.16) y después Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {y'}_c}

Se repiten los mismos pasos para calcular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {z'}_c} empleando el cortante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{y'}} . Claramente, el método también es aplicable a vigas homogéneas.

La Figura 4.6 muestra la posición del centro de gravedad O y del centro de esfuerzos cortantes C para algunas secciones. Para otras secciones ver [PCh,Yo].

Por conveniencia, se define un nuevo sistema de referencia local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x',\hat y',\hat z'}

en el centro de esfuerzos cortantes C tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat   x',\hat y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat z'}
son paralelos a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}

, respectivamente (Figura 4.7).

4.2.7 Propiedades del centro de esfuerzos cortantes

Las propiedades del centro de esfuerzos cortantes son las siguientes:

• Las fuerzas externas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{\hat y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{\hat z'}}
aplicadas   en el centro de esfuerzos cortantes no producen un giro de la sección. Esto   quiere decir que las tensiones tangenciales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'y'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'z'}}
debidas a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{y'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{z'}}
se asocian a una flexión   únicamente.

• Las fuerzas externas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{z'}}
aplicadas en un   punto arbitrario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D}
se equilibran por tensiones tangenciales debidas   a los esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{z'}= F_{\hat z'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{y'}=F_{\hat y'}}
y a un   torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_ {\hat x'}= F_{z'}{(y'}_D -c'_c)-F_{y'}{(z'}_z'_c'_c)}

.

• Por lo anterior, cualquier carga contenida en el plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'z'}

se   puede reducir a dos cargas (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{\hat y'}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{\hat z'}} ) actuando en el centro de esfuerzos cortantes y un torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}}

alrededor del eje   elástico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x'}

. Las dos fuerzas originan desplazamientos en los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}
y los correspondientes estados de flexión (en los   planos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'z'}

, respectivamente), mientras que el torsor induce un giro de la sección (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}} ) alrededor del eje elástico (Figura 4.7).

y torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): M_{\hat x'}
     actuando en el centro de esfuerzos cortantes C

• La tensiones tangenciales debidas a la flexión en el plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'y'}

satisfacen

las debidas a la flexión en el plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'z'}

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'z'}}

) satisfacen

y las debidas al torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}}

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'y'}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'z'}} ) satisfacen

• Para torsión uniforme no coaccionada, un torsor actuando en el centro de esfuerzos cortantes induce únicamente tensiones tangenciales en la sección. Para un torsor no uniforme o una torsión uniforme coaccionada, se inducen también tensiones axiales de alabeo. En todos los casos, el esfuerzo axil y los momentos flectores inducidos por un torsor actuando en el centro de esfuerzos cortantes son cero (Apéndice F).

• El centro de esfuerzos cortantes no se desplaza cuando la sección está sometida a un torsor actuando en el punto neutro O. Esta propiedad se deduce del teorema de reciprocidad de Maxwell-Betti (Figura 4.8), es decir

Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{\hat z'}}

pasa por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}
entonces el giro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta ^F_{z'}}
debido a la fuerza Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{\hat z'}}
es cero. Por consiguiente, el desplazamiento vertical del centro de esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle  w_c^{'M}}
debido a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{x'}}
es también cero. Lo mismo ocurre para una fuerza Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta ^F_{y'}}
actuando en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

. Esto explica porqué el centro de esfuerzos cortantes también se llama en ocasiones centro de giro.

Figura 4.8: El centro de esfuerzos cortantes no se desplaza bajo un torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): M_{x'} actuando en el punto O sobre el eje neutro

• Si la sección es simétrica respecto a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}

, el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

se   ubica sobre ese eje y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}= 0}

. También si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}

es un punto   de doble simetría, los puntos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}
coinciden. Finalmente si la   sección se define como un ensamblaje de paredes delgadas que   intersectan en un único punto, el centro de esfuerzos cortantes prácticamente   coincide con dicho punto (Figura 4.6).

• Para material homogéneo la posición de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

es normalmente una   propiedad geométrica de la sección. La posición del centro de   esfuerzos cortantes para distintas secciones homogéneas se puede encontrar en   muchas publicaciones [PCh,Yo].

• Una excepción de la anterior afirmación son las vigas de pared delgada muy deformables y con sección abierta. En este caso el centro de esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

ya no es una propiedad de la sección y   depende de las condiciones de contorno y las fuerzas externas (ya   que las tensiones tangenciales debidas a la torsión no son nulas en   la línea media) [Hy].

• Para una viga de material compuesto laminado la posición del centro de esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

depende de la geometría y de las propiedades  de cada   capa. El Apéndice I describe un procedimiento para calcular la   posición de C en secciones abiertas de pared delgada y material compuesto laminado.

4.3 VIGAS 3D DE MATERIAL COMPUESTO DE SAINT-VENANT

Estudiamos en primer lugar el cálculo por el MEF de vigas 3D de material compuesto que satisfacen la hipótesis de torsión libre de Saint-Venant, es decir, la sección de la viga se puede deformar libremente debido al torsor. Como consecuencia, la tensiones (y las deformaciones) axiles debidas a la torsión (efectos de alabeo) son cero, o de poca importancia. Como ya se ha mencionado anteriormente, esta hipótesis la cumplen las secciones macizas, las secciones en T y las secciones cerradas de pared delgada (incluyendo las secciones multicelulares). La formulación también es aplicable a secciones abiertas de pared delgada, si el torsor es uniforme y el alabeo no está coaccionado a lo largo de la viga. En el Apartado 4.10 se presenta una formulación refinada para secciones abiertas de pared delgada que tiene en cuenta los efectos de alabeo. En todos los casos, se admiten las hipótesis de Timoshenko para la flexión, es decir, la secciones planas permanecen planas pero no necesariamente ortogonales al eje de la viga. Esto introduce efectos de deformación de cortante en los modos de flexión, como se describe en el Capítulo 2 para vigas planas.

En el Apartado 4.7 se explica brevemente la formulación de vigas 3D de Saint-Venant siguiendo las hipótesis de Euler-Bernoulli.

4.3.1 Campos de desplazamientos y de deformaciones. Teoría de Timoshenko

Asumiremos que el centro de esfuerzos cortantes C y el eje neutro O no coinciden. Por consiguiente, existirá un acoplamiento entre los efectos axiles, de flexión/cortante y torsor si las variables cinemáticas se eligen en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}

o en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

.

Se puede, sin embargo, conseguir un desacoplamiento de dichos efectos eligiendo las siguientes variables cinemáticas [BD5]:

• el desplazamiento axial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u'}

en el punto neutro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {u'}_0}

),

• los desplazamiento Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w'}
en la dirección de los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat z'}

, respectivamente, en el centro de esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {v'}_c,w'_c}

),

• el giro de torsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}

, y

• los giros Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{z'}}
alrededor de los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}
en O (Figura 4.9).

Nótese que, al contrario que en capítulos anteriores, hemos elegido la representación vectorial para los giros (Figura 4.9).

Se destaca también que como el eje elástico es paralelo al eje neutro, los giros torsores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{x'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}
tienen el mismo valor (Figura 4.9). En lo que sigue mantendremos el giro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}
como variable cinemática, por conveniencia.

Las variables cinemáticas en el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

se expresan en función de sus valores en el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}

, antes de su transformación a ejes globales.

El campo de desplazamientos sobre la sección de la viga debido a los efectos flectores, según la teoría de Timoshenko (Capítulo 2), se escribe como

donde los subíndices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c}
denotan los puntos de muestreo de cada desplazamiento.

El campo de desplazamientos se completa con el movimiento debido a la torsión. En la teoría de Saint-Venant se supone que las secciones de la viga giran libremente alrededor del eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

por efecto de un  giro torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}
que varía linealmente a lo largo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x'}
(y también de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

). Los desplazamientos en las direcciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}
son Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {-(z'-z'}_c)\theta _{\hat x'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {(y'-y'}_c)\theta _{\hat x'}}

, respectivamente, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {y'}_c}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {z'}_c}
son las coordenadas del centro de esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

.

El desplazamiento axial introducido por el giro torsor es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega \phi _\omega }

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }
es la función de   alabeo y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega }
es la tasa de torsión. Se supone también que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega }
coincide con la variación del giro torsor a lo largo del eje de la viga, es decir

La Figura 4.10 muestra los desplazamientos de torsión libre de Saint-Venant de un punto arbitrario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P}

de la sección.

en los eje locales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): x',y,z'

La hipótesis (4.24) equivale a despreciar en el análisis los efectos de las deformaciones de cortante inducidas por la torsión. En el Apartado 4.11 se estudia cómo tener en cuenta esas deformaciones en vigas de sección abierta de pared delgada.

El desplazamiento de torsión se superpone al inducido por los movimientos axial y de flexión. El campo de desplazamientos resultante se puede escribir como

El vector de movimientos locales es

El campo de deformaciones (local) se puede deducir de las Ecs.(4.1) y (4.24) como

Las Ecs.(4.25) y (4.27) muestran la superposición de los movimientos y las deformaciones debidos a los efectos axil, flector y torsor.

Como se supone que el giro torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _ { \hat x' }}

varía linealmente a lo largo del eje elástico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x' }

, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle { \partial ^2 \theta _{\hat x'} \over {\partial { x'}^2}}=0} , y el ángulo de torsión no contribuye a la deformación axil Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varepsilon _{x'}} . Este no ocurre así en vigas con secciones de pared delgada abierta (Apartado 4.10).

La Ec.(4.27) se puede reescribir como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol \varepsilon }'}

es el vector de deformaciones generalizadas locales dado por

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol S}_1}

es la matriz de transformación de deformaciones

Recordemos que si el centro de esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

y el eje neutro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}
coinciden entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {y'}_z'_c=c=0}

.

4.3.2 Tensiones, esfuerzos y matriz constitutiva generalizada

El vector de esfuerzos se define como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N}

es el esfuerzo axil, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{y'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{z'}}
son los esfuerzos cortantes según los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}

, respectivamente, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{z'}}
son los momentos flectores alrededor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}

, respectivamente y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}}

es el torsor alrededor del eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x'}
(Figura 4.9).

La Ec.(4.30) se puede reescribir como

Nótese que

Recuérdese que en un estado de flexión, los esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{z'}}
no son nulos mientras que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}=0}

. Por el contrario, en un estado de torsión pura Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{y'} = Q_{z'}=0} , mientras que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}}

 no es nulo (Ecs.(4.20) y (4.21)).

4.3.3 Matriz constitutiva generalizada

El desacoplamiento entre los efectos de flexión y torsión implica que los esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{z'}}
se calculan a partir de las tensiones tangenciales inducidas por la flexión únicamente, mientras que el torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}}
se calcula a partir de las tensiones tangenciales debidas a la torsión. Teniendo esto en cuenta, la relación entre esfuerzos  y las deformaciones generalizadas se puede obtener  a partir de la Ec.(4.30), usando las Ecs.(4.1) y (4.27), como

donde

Sustituyendo en (4.33) la expresión de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varepsilon _{x'}}

de la Ec.(4.27) y recordando que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y',z'}
son los ejes principales de inercia se obtiene

La matriz constitutiva generalizada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat{\boldsymbol D}'}

tiene la siguiente forma diagonal

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {D}_a, \hat {\boldsymbol D'}_f}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t}
denotan las contribuciones axil, de flexión y de torsión en la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol    D}'}
con

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_{z'}}
son los coeficientes de corrección del cortante que tienen en cuenta una distribución no uniforme de las tensiones tangenciales. Estos coeficientes se pueden calcular como se describe en el Apartado 3.8 y en el Apéndice D

La forma diagonal de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \hat {\boldsymbol D}'

en la Ec.(4.36.a)   se obtiene sólo si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): x'
es el eje neutro e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): y'
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): z'
son ejes   principales de inercia. De lo contrario, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \hat {\boldsymbol D}'
es una   matriz llena (Ejemplo 1).

Las integrales de la Ec.(4.36.b) para vigas de material compuesto se llevan a cabo teniendo en cuenta la distribución de las propiedades del material sobre la sección. Para una viga de material heterogéneo (que satisfaga la Ec.(4.1)) es conveniente dividir la sección en celdas, cada una con propiedades del material diferentes. La integración se lleva a cabo evaluando las integrales de la Ec.(4.36.b) para cada celda y efectuando tras ello la suma correspondiente.

Para vigas de material compuesto laminado con una orientación del material como la mostrada en la Figura 3.3, los parámetros constitutivos generalizados axiles, flectores y de cortante se pueden calcular a partir de las Ecs.(3.12) como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {h_k = z'}_{k+1}{-z'}_k}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b_k}
son el espesor y el ancho de la capa Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_l}

es el número de capas y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (\cdot )^k}
denota valores para la capa Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k}

.

El cálculo de la rigidez torsional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t}

depende de la función de alabeo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

. Esta función se puede obtener como se explica en el Apartado 4.3.4.

La Tabla 4.1 muestra el valor promedio de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t}

para dos secciones de material compuesto laminado obtenido resolviendo las Ecs.(4.45) para obtener la función de alabeo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w}
y luego calculando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t}
por la Ec.(4.35b). Para ambos cálculos se utilizó el MEF con mallas de triángulos de tres nodos para discretizar la sección [DB5,Bo].

Las expresiones (4.37) se simplifican para material homogéneo a

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_{z'}}
son los momentos principales de inercia, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_{y'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_{z'}}
dependen de la geometría de la sección (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k_{y'}=k_{z'}=5/6}
para sección rectangular) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle J}
es la inercia torsional dada por

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {y = y' - y'}_c} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {z = z'-z'}_c} .

La Figura 4.11 muestra los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle J}

para algunas secciones homogéneas [BD5,PCh,Yo].

y tensión tangencial máxima Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \tau _{i}
para algunas secciones homogéneas. El punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): i
muestra     la posición de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \tau _{i}

Las tres tensiones y deformaciones locales en un punto de una sección se pueden calcular a partir de los esfuerzos. Teniendo en cuenta el desacoplamiento entre las tensiones de flexión y torsión, deducimos de las Ecs.(4.28a), (4.35) y (4.1)

donde

para las tensiones y deformaciones inducidas por los efectos axil y flector, y

para las tensiones y deformaciones de cortante inducidas por la torsión libre.

El cálculo de las tensiones tangenciales debidas a la torsión presenta algunas características particulares, ya que requiere el conocimiento de la función de alabeo. Este tema se trata en los próximos apartados para el caso general y para secciones cerradas de pared delgada. Las secciones abiertas de pared delgada se tratan en el Apartado 4.10.

Ejemplo 4.1:

Obtener la matriz constitutiva generalizada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat{\boldsymbol D}'}

para una posición arbitraria del eje de la viga.

Supongamos que el eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

no coincide con el eje neutro y que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y',z'}
no son ejes principales de inercia. Sustituyendo las expresión de la tensión axil Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}}
(Ec.(4.27)) en la integral para el vector de esfuerzos (Ec.(4.33)) da

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_t}

se define en la Ec.(4.34). Con un poco de álgebra se obtiene

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat{\boldsymbol \varepsilon }'}

dado por la Ec.(4.28b).

Vemos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol D}'}

es ahora una matriz llena (simétrica). Esto implica que los efectos axiles y flectores están acoplados; es decir, que una fuerza axial induce efectos de flexión y viceversa. Los términos de fuera de la diagonal en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol D}'}
desaparecen si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}
es el eje neutro y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y',z'}
son los ejes principales de inercia (Apartado 4.2.3).

En la práctica, se puede emplear tanto la forma llena de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol D}'}

o la diagonal dada en la Ec.(4.36a) obteniéndose resultados idénticos. La más sencilla forma diagonal requiere del cálculo “a priori” de la posición del eje neutro y los ejes principales de inercia.

Estas consideraciones no afectan al valor de la rigidez torsional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {D}_t} .

4.3.4 Cálculo de las tensiones tangenciales debidas a la torsión y de la función de alabeo

Las tensiones tangenciales debidas a la torsión libre de Saint-Venant se expresan en función de los desplazamientos como (ver Ecs.(4.1) y (4.27))

La ecuación de equilibrio en la dirección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

(teniendo en cuenta que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}}
es cero bajo torsión pura) es (Apéndice B)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A}

es la sección de la viga. Sustituyendo las Ecs.(4.40) en (4.41) da

La Ec.(4.42) es una ecuación de Laplace que debe satisfacer la siguiente condición en la frontera de la sección transversal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Gamma }

[ZTZ]

Teniendo en cuenta que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_{y'}=-{\partial z'\over \partial s}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_{z'}={\partial    y'\over \partial s}}
y usando las Ecs.(4.40) tenemos

Las Ecs.(4.42) y (4.44) se simplifican para material homogéneo a

La solución de estas ecuaciones diferenciales proporciona la distribución de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

para secciones de forma geométrica sencilla y material homogéneo. Para el caso general, las Ecs.(4.45) se resuelven típicamente empleando el MEF o el método de diferencias finitas  [BD4,Bo,OR,PCh,Yo,ZTZ].

Las Ecs.(4.45) proporcionan una distribución de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

sobre la sección que no es única, ya que su solución no se ve afectada por añadir una constante a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

. Esto no es un problema ya que las tensiones tangenciales dependen de las derivadas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

(Ec.(4.40)) y, por consiguiente, los resultados no están influenciados por el valor de dicha constante.

Una vez se ha encontrado la distribución de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

sobre la sección, se puede calcular la rigidez torsional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\hat D}_t}
mediante la Ec.(4.36b). Esto se suele hacer normalmente con la misma malla de elementos finitos, o de diferencias finitas, utilizada para resolver las Ecs.(4.45) [BD4,Bo]. La Tabla 4.1 muestra un ejemplo de este procedimiento para calcular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\hat D}_t}
en dos vigas de material compuesto laminado empleando el MEF.

La Figura 4.11 muestra la posición de la tensión tangencial máxima en algunas secciones homogéneas [BD5,PCh,Yo].

4.3.5 Secciones de pared delgada cerradas

Dada la geometría de las secciones de pared delgada cerradas es conveniente separar las tensiones tangenciales en sus componentes normal y tangencial a la línea media de la pared Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x's}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'n}}

, denominadas en adelante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _s}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _n}
por simplicidad (Figura 4.12). Es también usual suponer que la tensión tangencial normal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _n}
es nula en todo el   espesor de la pared, ya que las caras externas de la viga no tienen tensiones y el espesor de la pared es pequeño. El flujo de las tensiones tangenciales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _s}
(en adelante flujo de cortante) se define como

De la ecuación de equilibrio local para un elemento diferencial de la viga de pared delgada (Ec.(4.41)) deducimos (suponiendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _n =0} ) [BC]

Esta distribución de flujo de cortante constante genera un torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}}

alrededor del punto O sobre el eje neutro  (coincidente con el centro de esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}

) dado por

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A}

es el área encerrada por la linea central de la pared con perímetro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {L_s}}

. De las Ecs.(4.46) y (4.48) la tensión tangencial resultante del torsor se encuentra como

Esta ecuación se completa con la variación del giro torsor a la largo de la viga (Ec.(4.24))

La rigidez torsional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t}

se puede obtener igualando el trabajo virtual externo complementario [ZTZ,BC] realizado por la tensión tangencial y usando la Ec.(4.49), es decir

De la Ec.(4.51) encontramos, después de simplificar y tener en cuenta que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}}

es constante a lo largo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {L_s}}

,

Para una sección cerrada arbitraria de espesor de pared constante

La Ec.(4.53) muestra que la sección de rigidez torsional máxima es el tubo circular de pared delgada. La tensión tangencial se deduce de la Ec.(4.49) como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R_m}

es el radio medio del tubo.

4.3.6 Torsión de secciones multicelulares

En secciones multicelulares, por razones de equilibrio se requiere que el flujo de cortante sea constante en cada pared. Además, en cada nudo de conexión, la suma de flujos que concurren en él debe ser cero [AMR,BC,OR].

Estos requerimientos de continuidad se satisfacen automáticamente si los flujos de cortante se suponen constantes en cada celda. Para la sección de cuatro celdas de la Figura 4.13 los flujos de cortante que circulan alrededor de cada celda se denominan Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{(1)}, f^{(2)},f^{(3)}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{(4)}}

, y se indica su sentido positivo [BC]. La Figura 4.13 también ilustra los flujos de cortante que concurren en el nudo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {E}} . La condición de continuidad en el nudo se satisface porque Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (f^{(4)}) + (f^{(3)}-f^{(4)}) + (f^{(2)}-f^{(3)})+ (-f^{(2)})=0} .

La solución del problema requiere el cálculo de los flujos de cortante constantes, uno alrededor de cada celda. El torsor total de la sección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}}

 es la suma de los torsores de las celdas individuales, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}^{(i)}}

, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}

indica el número de celda, es decir

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_{cel}}

es el número de celdas y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {A}^{(i)}}
el área encerrada por la celda Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}
de perímetro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^{(i)}}

. Esta ecuación por si sola no permite la determinación de los flujos de cortante de las Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_{cel}}

celdas.

Se pueden encontrar ecuaciones adicionales expresando las condiciones de compatibilidad que requieren que la tasa de torsión de las distintas celdas sea idéntica. En respuesta al flujo de cortante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{(i)}}

que actúa dentro de una celda, se desarrolla en la celda una tasa de torsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega ^{(i)}}

. La compatibilidad de las deformaciones de todas las celdas proporciona Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_{cel} - 1}

ecuaciones adicionales

La relación entre la tasa de torsión y el torsor de cada celda se deduce de la Ec.(4.51) como

Las Ecs.(4.56) y (4.57) proporcionan las Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_{cells}}

ecuaciones necesarias para encontrar los Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_{cel}}
flujos de cortante en las celdas de una sección multicelular sometida a torsión.

Ejemplo 4.2:

Sección transversal de dos celdas.

La sección de pared delgada mostrada en la Figura 4.14 (tomada de [BC]) representa una sección muy idealizada de una estructura de perfil aerodinámico. La parte frontal (curva) se llama borde de ataque, el pilar vertical se llama larguero y la parte trasera borde de salida. La Ec.(4.55) da

Figura 4.14: Sección de pared delgada de dos celdas sometida a torsión [BC]

La condición de compatibilidad requiere tasas de torsión idénticas para las dos celdas. La Ec.(4.57) proporciona la tasa de torsión para la celda frontal como

y la tasa de torsión para la celda del borde de salida es

Igualando las dos tasas de torsión se llega a la segunda ecuación de los flujos de cortante

que se simplifica a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{(1)} =1.04 f^{(2)}} .

Este resultado, junto con la Ec.(4.58), se usan para obtener Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{(1)}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{(2)}}
como

Nótese que el flujo de cortante en la celda frontal, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{(1)}} , es sólo un 4% mayor que en el borde de salida, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f^{(2)}} . Por consiguiente, el flujo de cortante en el larguero, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^2 (f^{(1)}-f^{(2)}) = 0.04 M_1/(6+1.04\pi )} , que es un valor muy pequeño.

Como la rigidez torsional de la sección cerrada es proporcional al cuadrado del área encerrada, la mayor contribución a la rigidez torsional es de la sección cerrada exterior, que es la unión de las dos celdas. Así, el mayor flujo de cortante circula por el exterior, dejando el larguero prácticamente sin carga.

La rigidez torsional se calcula como el cociente entre el torsor y la tasa de torsión (Ec.(4.50)). Como las tasas de torsión de las dos celdas son iguales, se puede emplear tanto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega ^{(1)}}

como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega ^{(2)}}

. Utilizando, por ejemplo, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega ^{(1)}}

da

4.3.7 Principio de trabajos virtuales

El PTV se escribe como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol t}'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol p}' _i}
son fuerzas distribuidas en el eje de la viga y cargas puntuales, respectivamente, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle V}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L}
son el volumen y la la longitud de la viga.

Las componentes del vector de desplazamientos virtuales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta {\boldsymbol u}'}

y los vectores de fuerzas externas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol t}'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol p'}_i}
se expresan en el sistema de coordenadas local como (Figura 4.15)

Nótese que las fuerzas distribuidas y puntuales que actúan a lo largo de los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat y'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat z'}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{y'},f_{z'},F_{{y'}_i},~F_{{z'}_i}}

) así como el torsor (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m_{\hat x'},M_{\hat {x'}_i}} ) actúan en el eje elástico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x'} , mientras que las fuerzas axiales (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{x'}, F_{{x'}_i}} ) y los momentos flectores (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m_{y'},m_{z'},M_{{y'}_i},~M_{{z'}_i}} ) actúan en el eje neutro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

(Figura 4.15).

El trabajo virtual interno se puede escribir en función de los esfuerzos y las deformaciones virtuales generalizadas. Empleando notación matricial podemos escribir el primer miembro de la Ec.(4.63) usando las Ecs.(4.28a) y (4.32) como

La primera integral del primer miembro de la ecuación anterior se puede expresar, usando la Ec.(4.31), como

La última integral del segundo miembro de la Ec.(4.65) vale cero como se muestra a continuación. Recordando que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \displaystyle {\partial \theta _{\hat x'}\over \partial x'} }

es constante e integrando por partes tenemos

La integral de superficie del segundo miembro de la Ec.(4.67) es cero ya que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'y'}n_{y'} + \tau _{x'z'} n_{z'}=0}

en los contornos de la viga, en ausencia de otras fuerzas de superficie (Apéndice B). Por otra parte,  la tensión axil Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}}
originada por la de torsión es nula, y de las ecuaciones de equilibrio de la elasticidad 3D (Apéndice B) se tiene

y, por tanto, se deduce de la Ec.(4.67) que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I=0}

en un estado de torsión.

Para las tensiones tangenciales originadas por la flexión, de la anterior ecuación de equilibrio se deduce

y, por tanto

De las Ecs.(4.69) y (4.27) se llega a

Sustituyendo (4.70) en (4.69b) se obtiene

Esta integral se anula si

Estas condiciones se cumplen siempre y pueden demostrarse teniendo en cuenta que el esfuerzo axil y los momentos flectores inducidos por un torsor actuando en el centro de esfuerzos cortantes son cero (Apartado 4.2.7). En conclusión, el PTV se puede escribir como

4.4 DISCRETIZACIÓN POR ELEMENTOS FINITOS RECTOS. ELEMENTO DE VIGA DE TIMOSHENKO 3D DE DOS NODOS

4.4.1 Definición del eje neutro y la obtención de las matrices del elemento

La línea de referencia de la viga (eje neutro) se discretiza en elementos finitos rectos 1D de continuidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^0}

y longitud Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle l^{(e)}}

, como el elemento de viga de dos nodos de la Figura 4.16. Las coordenadas de un punto en la línea de referencia se obtienen por interpolación isoparamétrica [On4]

siendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol x}_i}

el vector de coordenadas del nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol N}_i=N_i(\xi ){\boldsymbol I}_3}

, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol I}_3}

es la matriz unidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 3\times 3}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_i(\xi )}
es la función de forma 1D del nodo  (Figura 2.4) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}
es el número de nodos del elemento. El vector unitario tangente  al eje neutro en un nodo se puede obtener de manera sencilla por

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol x}_ {\scriptscriptstyle i+1/2}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol x}_ {\scriptscriptstyle i-1/2}}
son las coordenadas de los puntos medios de los elementos adyacentes al nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}

.

Los vectores unitarios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{\scriptscriptstyle 2_i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol  e}_{\scriptscriptstyle 3_i}}
se definen en las direcciones principales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}
en cada nodo, respectivamente. Para vigas con sección no uniforme las direcciones principales pueden variar en cada punto. La posición de los vectores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e } _ { \scriptscriptstyle 1 }}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{\scriptscriptstyle 2}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{\scriptscriptstyle 3}}
se interpolan dentro de cada elemento a partir de los valores nodales como

Las expresiones anteriores son particularmente útiles para elementos 1D curvos (Apartado 4.6).

Los desplazamientos locales se interpolan como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol I}_{\scriptscriptstyle {6}}}

es la matriz unidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 6\times  6}

. Sustituyendo la Ec.(4.77) en el vector de deformaciones generalizadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol \varepsilon } '}

de (4.28b) se obtiene

con

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B}_{a_i}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B}_{f_i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B}_{t_i}}
son las contribuciones axil, de flexión y de torsión a la matriz de deformaciones generalizadas del nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}

.

Sustituyendo la expresión de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol \varepsilon }'}

de la Ec.(4.78) en el PTV (Ec.(4.73)) y usando las Ecs.(4.35) y (4.77) se obtiene la matriz de rigidez del elemento y el vector de fuerzas nodales equivalentes para cargas distribuidas expresadas en ejes locales como

Introduciendo las Ecs.(4.36a) y (4.79) en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K }_ { ij } ^{\prime (e)}}

da

Las matrices del elemento se evalúan normalmente mediante cuadraturas numéricas. El bloqueo por cortante se puede evitar subintegrando las contribuciones del cortante en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K } _ { ij } ^{\prime (e)}} . El elemento de viga 3D más simple es el elemento de dos nodos lineal con una cuadratura uniforme de un punto. Su matriz de rigidez local se puede obtener explícitamente como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (\cdot )_c}

indica valores en el centro del elemento. La expresión de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [ {\boldsymbol B }_i^\prime ]_c}
es

El Cuadro 1 muestra la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K } _ { ij } ^{\prime (e)}}

para el elemento de viga 3D de dos nodos. Nótese que es una extensión de la matriz de rigidez para el elemento de viga plana de Timoshenko de dos nodos con una cuadratura de un punto (Ec.(2.39)).

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ~

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol K}_{11}^{\prime (e)}=\left[\begin{matrix} \displaystyle {\hat D_a\over l^{(e)}}  & 0 & 0 & 0 & 0&0\\ 0 & \displaystyle {\hat D_{s_{y'}}\over l^{(e)}}  & 0 & 0 & 0 & \displaystyle {\hat D_{s_{y'}}\over 2}\\ 0 & 0 & \displaystyle {\hat D_{s_{z'}}\over l^{(e)}} & 0& -\displaystyle {\hat D_{s_{z'}}\over 2} & 0\\ 0 & 0 & 0 & \displaystyle {\hat D_t \over l^{(e)}} & 0 & 0\\ 0 & 0 & -\displaystyle {\hat D_{s_{z'}}\over 2}  & 0& \left( \displaystyle {\hat D_{s_{z'}}\over 4}l^{(e)}+ \displaystyle {\hat  D_{b_{y'}} \over l^{(e)}} \right)&0\\ 0 & \displaystyle {\hat D_{s_{y'}}\over 2} & 0 & 0 &0&\left(\displaystyle {\hat D_{s_{y'}}\over  4}l^{(e)}+\displaystyle {\hat  D_{b_{z'}} \over l^{(e)}}\right)&\\ \end{matrix}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol K}_{12}^{\prime (e)}=\left[\begin{matrix} -\displaystyle {\hat D_a\over l^{(e)}}  & 0 & 0 & 0 & 0& 0\\ 0 & -\displaystyle {\hat D_{s_{y'}}\over l^{(e)}} &  0 & 0 & 0& \displaystyle {\hat D_{s_{y'}}\over 2} \\ 0 & 0 & -\displaystyle {\hat D_{s_{z'}}\over l^{(e)}}  & 0& -\displaystyle {\hat D_{s_{z'}}\over 2} & 0\\ 0 & 0 & 0 & -\displaystyle {\hat D_t \over l^{(e)}} & 0 & 0\\ 0 & 0 & \displaystyle {\hat D_{s_{z'}}\over 2}  & 0& \left( \displaystyle {\hat D_{s_{z'}}\over 4}l^{(e)}-\displaystyle {\hat  D_{b_{y'}} \over l^{(e)}} \right)&0\\ 0 & -\displaystyle {\hat D_{s_{y'}}\over 2} &  0 &0&0&\left(\displaystyle {\hat D_{s_{y'}}\over  4}l^{(e)}-\displaystyle {\hat  D_{bz'} \over l^{(e)}} \right) \end{matrix}\right]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol K}_{22}^{\prime (e)}=\left[\begin{matrix} \displaystyle {\hat {D}_a\over l^{(e)}} & 0  & 0 & 0 & 0&0\\ 0 & \displaystyle {\hat D_{s_{y'}}\over l^{(e)}} & 0 &  0 &0& -\displaystyle {\hat D_{s_{y'}}\over 2} &\\ 0 & 0 & \displaystyle {\hat D_{s_{z'}}\over l^{(e)}}  & 0& \displaystyle {\hat D_{s_{z'}}\over 2} &  0\\ 0 & 0 & 0 & \displaystyle {\hat D_t \over l^{(e)}} & 0 & 0\\ 0 & 0 & \displaystyle {\hat D_{s_{z'}}\over 2}  & 0& \left( \displaystyle {\hat D_{s_{z'}}\over 4}l^{(e)}+\displaystyle \hat  D_{b_{y'}}  \right)& 0\\ 0 & -\displaystyle {\hat D_{s_{y'}}\over 2} & 0  & 0 &0&\left(\displaystyle {\hat D_{s_{y'}}\over  4}l^{(e)}+\displaystyle {\hat  D_{b_{z'}} \over l^{(e)}}\right)\\ \end{matrix}\right]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol K}_{21}^{\prime (e)}=[{\boldsymbol K}_{12}^{\prime (e)}]^T
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ~

Cuadro 4.1: Matrices de rigidez locales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K}_{ij}^{\prime (e)}}

de un elemento de viga 3D de Timoshenko de   dos nodos con una cuadratura uniforme de un punto

4.4.2 Transformaciones de rigidez y de fuerzas

Antes del ensamblaje es necesario referir todas las variables nodales al punto O sobre el eje neutro. De las Ecs.(4.25) obtenemos (para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'=z'=0} )

La relación entre el vector de desplazamientos nodales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol a'}_{i}}

y el vector que contiene los GDLs en el punto O (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar {\boldsymbol a'}_{i}}

) es

donde

En la Ec.(4.84b) hemos tomado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'} = \theta _{x'}} , como se menciona en el Apartado 4.3.1.

Las fuerzas puntuales nodales se transforman como

donde

La Ec.(4.85a) se deduce fácilmente si tenemos en cuenta que todas las componentes de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol p'}_{i}}

coinciden con las de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol p'}_{i}}

, excepto los torsores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{{x'}_i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat {x'}_i}}
que se relacionan mediante (Figura 4.17)

Figura 4.17: Componentes de los vectores de fuerzas nodales puntuales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol p'}_{i} y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \bar {\boldsymbol p'}_{i}

La transformación de la matriz de rigidez local del elemento a ejes globales es como sigue. La matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K }_{ij}^{\prime (e)}}

se transforma primero al sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',y',z'}
situado  en el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}
sobre el eje neutro como

Para elementos rectos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol L}_i = {\boldsymbol L}_j} .

La matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{\boldsymbol K}_{ij}^{\prime (e)}}

se transforma finalmente al sistema cartesiano global Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x,y,z}
como

donde los vectores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{1_i}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{2_i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{3_i}}
se definen como se explica en el Apartado 4.4.1.

Las dos transformaciones son equivalentes a usar la siguiente matriz de deformaciones generalizadas

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B'}_i}

dada por la Ec.(4.79). La matriz de rigidez global se calcula por

La transformación de fuerzas nodales equivalentes a ejes globales sigue pasos similares. Primero se transforman las componente del vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol f}^{'(e)}}

de la Ec.(4.80) a los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',y',z'}
por

Las fuerzas nodales equivalentes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar {\boldsymbol f}^{(e)}_i}

se transforman seguidamente al sistema de coordenadas global por

4.5 ELEMENTO DE VIGA 3D DE TIMOSHENKO DE DOS NODOS CUASI-EXACTO

El elemento de viga exacto de Timoshenko de dos nodos del Apartado 2.9 se puede ampliar al caso 3D. El elemento resultante proporciona resultados nodales exactos en el análisis de la flexión de vigas planas rectas 3D bajo cargas actuando en los planos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'z'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'z'}

. Para cargas arbitrarias (incluyendo torsores) y geometría curva de la viga (discretizada con elementos rectos de dos nodos) los resultados dejan de ser “exactos”. Sin embargo, el elemento tiene un comportamiento excelente para vigas 3D gruesas y esbeltas y estructuras reticulares, y su precisión es normalmente mejor (para un mismo número de elementos) que la del elemento de viga 3D de Timoshenko de dos nodos del apartado anterior.

El Cuadro 2 muestra la matriz de rigidez del elemento de viga 3D de Timoshenko de dos nodos “cuasi-exacto”.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ~

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \!\!\!\!{\boldsymbol K}_{11}^{\prime (e)}\!\!=\!\!\left[\begin{matrix} \displaystyle {\hat D_a\over l^{(e)}}  & 0 & 0 & 0 & 0& 0\\ 0 &12\phi _{y'} \hat D_{b_{z'}} & 0  & 0 & 0&\displaystyle 6 \phi _{y'}\hat D_{b_{z'}}l^{(e)} \\ 0 & 0 & 0& \displaystyle 12\phi _{z'} \hat D_{b_{y'}}  & -\displaystyle 6 \phi _{z'}\hat D_{b_{y'}}l^{(e)} & 0\\ 0 & 0 & 0 & \displaystyle {\hat D_t \over l^{(e)}}& 0 & 0\\ 0 & 0 & -\displaystyle 6 \phi _{z'}\hat D_{b_{y'}}l^{(e)} &0& \displaystyle (4+\beta _{z'}) \hat D_{b_{y'}}(l^{(e)})^2 &0\\ 0 & \displaystyle 6 \phi _{y'}\hat D_{b_{z'}}l^{(e)}  & 0  & 0 &0&(4+\beta _{y'}) \hat D_{b_{z'}}(l^{(e)})^2\\ \end{matrix}\right]\!\!\!\!\!\!
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \!\!\!{\boldsymbol K}_{12}^{\prime (e)}\!=\!\left[\begin{matrix} -\displaystyle {\hat D_a\over l^{(e)}} & 0 & 0  & 0 & 0&0\\ 0 &-12\phi _{y'} \hat D_{b_{z'}} & 0  & 0 & 0&\displaystyle 6 \phi _{y'}\hat D_{b_{z'}}l^{(e)} \\ 0 & 0 & \displaystyle -12\phi _{z'} \hat D_{b_{y'}}  & 0& -\displaystyle 6 \phi _{z'}\hat D_{b_{y'}}l^{(e)}  & 0\\ 0 & 0 & 0 & -\displaystyle {\hat D_t \over l^{(e)}}& 0 & 0\\ 0 & 0 & \displaystyle 6 \phi _{z'}\hat D_{b_{y'}}l^{(e)}  &0& \displaystyle (2-\beta _{z'}) \hat D_{b_{y'}}(l^{(e)})^2 &0\\ 0 & -\displaystyle 6 \phi _{y'}\hat D_{b_{z'}}l^{(e)}  & 0 & 0 &0&(2-\beta _{y'}) \hat D_{b_{z'}}(l^{(e)})^2\\ \end{matrix}\right]\!\!\!
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":):  {\boldsymbol K}_{22}^{\prime (e)}=\left[\begin{matrix} \displaystyle {\hat D_a\over l^{(e)}} & 0 & 0  & 0 & 0&0\\ 0 &12\phi _{y'} \hat D_{b_{z'}} &  0 & 0 & 0& -\displaystyle 6 \phi _{y'}\hat D_{b_{z'}}l^{(e)} \\ 0 & 0 & \displaystyle 12\phi _{z'} \hat D_{b_{y'}}  & 0& \displaystyle 6 \phi _{z'}\hat D_{b_{y'}}l^{(e)} & 0\\ 0 & 0 & 0 & \displaystyle {\hat D_t \over l^{(e)}}& 0 & 0\\ 0 & 0 & \displaystyle 6 \phi _{z'}\hat D_{b_{y'}}l^{(e)}  &0& \displaystyle (4+\beta _{z'}) \hat D_{b_{y'}}(l^{(e)})^2 &0\\ 0 & -\displaystyle 6 \phi _{y'}\hat D_{b_{z'}}l^{(e)}  & 0 & 0 &0&(4+\beta _{y'}) \hat D_{b_{z'}}(l^{(e)})^2\\ \end{matrix}\right]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ~

Cuadro 4.2: Elemento de viga 3D de Timoshenko de dos nodos cuasi-exacto. Matrices de rigidez locales

El vector de fuerzas nodales equivalentes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol f}^{\prime (e)}}

para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{\hat x'}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{\hat y'}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{\hat z'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m_{\hat x'}}
constantes y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m_{y'}= m_{z'}=0}
es

En la obtención de las Ecs.(4.91) hemos supuesto que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{x'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m_{\hat x'}}
son constantes a lo largo del elemento. También

En las Ecs.(4.91b,c), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{\hat {y'}_1},f_{\hat {z'}_1}}

se deducen de las Ecs.(2.94b) como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{\hat y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{\hat z'}}
se deducen de la Ec.(2.87), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_m = \left(1-\frac{x'}{l^{(e)}}  \right)\frac{x'}{l^{(e)}}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta _{y'},\beta _{z'}}
se dan en la Figura 4.2.

La expresión de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol f'}^{(e)}_1}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol f'}^{(e)}_2}
para cargas distribuidas uniformes y triangulares y para cargas puntuales se puede deducir de la Tabla 2.2.

Si se desprecian los efectos de deformación de cortante, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta _{y'}=\beta _{z'}=0}

y los vectores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{f}^{\prime (e)}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol f'}^{(e)}_2}
de la Ec.(4.91) son

4.6 ELEMENTOS DE VIGA DE TIMOSHENKO CURVOS

La formulación anterior es aplicable a vigas 3D moderadamente curvas, es decir, cuando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {a\over R}\ll 1} , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R}

es el radio de curvatura de la línea de referencia de la viga y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a}
es una dimensión característica de la sección.

La linea curva se aproxima por una interpolación isoparamétrica usando elementos 1D curvos de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}

nodos [On4]. El vector tangente unitario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_1 }
en un punto se obtiene como

Para los nodos extremos compartidos por dos elementos, el vector tangente nodal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{1_i}}

se obtiene por un simple promedio.

Los vectores unitarios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{2_i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{3_i}}
se definen nodalmente en las direcciones principales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}
para cada sección nodal respectivamente. La posición de los vectores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_1, {\boldsymbol e}_2}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_3}
dentro del elemento se obtiene mediante la Ec.(4.76).

Las expresiones del elemento se deducen de las de vigas rectas sustituyendo la derivada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{\partial N_i}\over \partial x'}}

por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{\partial N_i}\over \partial s}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle dx'}
por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ds}

, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s}

es la coordenada curvilínea. Las derivadas curvilíneas se calculan como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle J={{\partial s}\over \partial \xi }}

se obtiene de la descripción isoparamétrica empleando la Ec.(4.74) (ver también el Apartado 9.8.2). Las integrales del elemento se evalúan numéricamente teniendo en cuenta que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ds=Jd\xi }

.

La formulación curvilínea de vigas curvas coincide con la desarrollada en el Capítulo 9 como un caso particular de elementos de lámina de revolución curvos (Apartado 9.8). En el Apartado 4.12 se presenta una formulación general para vigas 3D curvas obtenida por degeneración de elementos de sólido.

4.7 VIGAS 3D DE EULER-BERNOULLI. TEORÍA DE SAINT-VENANT

Se pueden desarrollar elementos de viga 3D de Sain-Venant según la teoría de Euler-Bernoulli a partir de la formulación del apartado anterior, imponiendo simplemente que los giros locales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{z'}}
coincidan con las   pendientes del eje neutro. Con el criterio de signos de la Figura 4.9,

Sustituyendo estas expresiones en las Ecs.(4.27) se obtiene el campo de deformaciones locales como

es decir, las deformaciones de cortante son debidas únicamente a la torsión (un estado de flexión no induce deformaciones de cortante). Los vectores de deformaciones generalizadas locales y de esfuerzos son

La nueva forma de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol S}_1}

de la Ec.(4.29) es

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A = {{\partial \omega }\over \partial y'}{-(z'-z'}_c)}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B={\partial \omega \over \partial z'}{+(y'-y'}_c)}

. Se introducen cambios similares en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol S}_2}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol S}_1}
de las Ecs.(4.31) y (4.39b).

Los cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{y'}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_{z'}}
se calculan a posteriori a partir de la distribución de tensiones tangenciales debidas a la flexión siguiendo procedimientos de  Resistencia de Materiales [Ti2,3]. El PTV viene dado por la Ec.(4.73). Se requieren aproximaciones por elementos finitos de continuidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^1}
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {v'}_c}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {w'}_c}
debido a la presencia de sus segundas derivadas en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat{\boldsymbol \varepsilon } '}

.

El elemento de viga 3D de Euler-Bernoulli más sencillo tiene dos nodos y emplea una interpolación lineal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^0}

para el desplazamiento axial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {u'}_0}
y el giro torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{x'}}

, y una aproximación de Hermite de continuidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^1}

para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {v'}_c}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {w'}_c} . La interpolación del campo de desplazamientos se escribe como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_i={1\over 2}(1+\xi \xi _i)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_i^{ \scriptscriptstyle H }}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\bar N}_i^{ \scriptscriptstyle H }}
son las funciones de forma cúbicas Hermíticas (Ec.(1.11a)). El signo negativo en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\bar N}_i^{ \scriptscriptstyle H }}
en la tercera fila de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol N } _i}
es consecuencia de la definición de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{y'}}
(Ec.(4.97)). Sustituyendo la Ec.(4.101) en (4.99) se obtiene

La matriz de rigidez local del elemento se da en la Ec.(4.80). La integración explícita es posible y la expresión resultante coincide con la del análisis estructural matricial clásico (Cuadro 3). La matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K}_{ij}^{\prime (e)}}

se puede deducir despreciando los efectos de la deformación de cortante (es decir, haciendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta _{y'}=\beta _{z'}=0}

) en las expresiones del Cuadro 4.2. La transformación a ejes globales sigue los mismos pasos del Apartado 4.4.2.

El vector de fuerzas nodales equivalentes se presenta en la Ec.(4.80). Las cargas distribuidas inducen ahora momentos flectores debido a la interpolación Hermítica de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v^\prime _c}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w^\prime _c}

, similarmente a los elementos de viga de Euler-Bernoulli. El vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol f}^{\prime (e)}_i}

para una carga uniformemente distribuida coincide con las Ecs.(4.93).

La formulación de elementos de viga curvos de Euler-Bernoulli usando una descripción curvilínea sigue lo explicado en el Apartado 4.6.

El elemento de viga de Euler-Bernoulli de dos nodos se puede mejorar empleando una interpolación Hermítica de la geometría. Esto es una aproximación mejor para vigas curvas y normalmente se combina con una interpolación de Hermite para el desplazamiento axial. La aproximación de mayor orden hace más difícil obtener una forma explícita de la matriz de rigidez del elemento.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol K}_{11}^{\prime (e)}= \displaystyle \frac{1}{(l^{(e)})^3}\left[\begin{matrix} \displaystyle \hat D_a (l^{(e)})^2 & 0 & 0 & 0 & 0&0\\ 0 &12 \hat D_{b_{z'}} & 0 & 0 & 0 &\displaystyle 6 \hat D_{b_{z'}}l^{(e)} \\ 0 & 0 & \displaystyle 12 \hat D_{b_{y'}} & 0& -\displaystyle 6 \hat D_{b_{y'}}l^{(e)} & 0\\ 0&0&0&\displaystyle \hat D_t (l^{(e)})^2 & 0 & 0\\ 0 & 0 & -\displaystyle 6 \hat D_{b_{y'}}l^{(e)} &0& \displaystyle 4\hat D_{b_{y'}}(l^{(e)})^2 &0\\ 0 & \displaystyle 6 \hat D_{b_{z'}}l^{(e)} & 0 & 0 &0& 4 \hat D_{b_{z'}}(l^{(e)})^2\\ \end{matrix}\right]

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol K}_{12}^{\prime (e)}= \displaystyle \frac{1}{(l^{(e)})^3}\left[\begin{matrix} -\displaystyle \hat D_a (l^{(e)})^2 & 0  & 0 & 0 & 0& 0\\ 0 &-12\hat D_{b_{z'}} & 0  & 0 &0&  \displaystyle 6 \hat D_{b_{z'}}l^{(e)}\\ 0 & 0 & \displaystyle -12 \hat D_{b_{y'}}  & 0& -\displaystyle 6 \hat D_{b_{y'}}l^{(e)}  & 0\\  0&0&0&-\displaystyle \hat D_t (l^{(e)})^2&0&0\\ 0 & 0 & \displaystyle 6 \hat D_{b_{y'}}l^{(e)}  &0& \displaystyle 2 \hat D_{b_{y'}}(l^{(e)})^2 &0\\ 0 & -\displaystyle 6 \hat D_{b_{z'}}l^{(e)}  & 0 & 0 &0&2\hat D_{b_{z'}}(l^{(e)})^2\\ \end{matrix}\right]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":):  {\boldsymbol K}_{22}^{\prime (e)}=\displaystyle \frac{1}{(l^{(e)})^3}\left[\begin{matrix} \displaystyle \hat D_a (l^{(e)})^2 &  0 & 0 & 0 & 0& 0\\ 0 &12\hat D_{b_{z'}} & 0  & 0 & 0& -\displaystyle 6 \hat D_{b_{z'}}l^{(e)} \\ 0 & 0 & \displaystyle 12 \hat D_{b_{y'}}  & 0& \displaystyle 6 \hat D_{b_{y'}}l^{(e)}  & 0\\  0&0&0&\displaystyle \hat D_t (l^{(e)})^2&0&0\\ 0 & 0 & \displaystyle 6 \hat D_{b_{y'}}l^{(e)} &0& \displaystyle 4\hat D_{b_{y'}}(l^{(e)})^2 &0\\ 0 & -\displaystyle 6 \hat D_{b_{z'}}l^{(e)}  & 0  & 0 &0&4\hat D_{b_{z'}}(l^{(e)})^2\\ \end{matrix}\right]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\boldsymbol K}_{21}^{'(e)}=[{\boldsymbol K}_{12}^{'(e)}]^T\quad ,\quad  \bar {\boldsymbol K}_{ij}^{'(e)}= {\boldsymbol L}^T  {\boldsymbol K}_{ij}^{\prime }{\boldsymbol L} \quad ,\quad {\boldsymbol L}={\boldsymbol L}_i = {\boldsymbol L}_j
(Ec.(4.84b))

Cuadro 4.3: Matriz de rigidez local de un elemento de viga 3D de Euler-Bernoulli de dos nodos

Los elementos de viga de Euler-Bernoulli, por su propia naturaleza, están libres del efecto de bloqueo por cortante, aunque en su versión curva pueden sufrir del bloqueo de membrana inducido por los efectos axiales. El remedio es usar una aproximación de mayor orden para el desplazamiento axial e integración uniforme reducida para toda la matriz de rigidez (Apartado 9.15).

Los elementos de viga 3D de Euler-Bernoulli coinciden en su versión plana con los de arco plano que se estudian en el Apartado 9.9.3.2 como un caso particular de los elementos de de revolución curvos delgados.

4.8 EMPARRILLADOS PLANOS

Es habitual encontrar estructuras formadas por un ensamblaje de vigas tal que:

• las vigas están unidas por sus centros de gravedad situados en el mismo plano global Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle xy}

• las vigas están orientadas de tal manera que el plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle xy}

  coincide con el plano de simetría Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'y'}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {z'}_= =0}

) de todas las vigas. El eje principal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}

es paralelo al eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z}
global.

Este tipo de estructura se llama emparrillado plano. Los modelos de emparrillado plano se usan habitualmente para el análisis de ensamblajes tipo losa-viga en puentes y forjados de edificios, entre otras aplicaciones. La Figura 4.18 muestra un ejemplo esquemático de un emparrillado plano esviado.

La Figura 4.19 muestra los ejes globales para una viga en un emparrillado plano. Cada viga está sometida a fuerzas concentradas o distribuidas en la dirección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z}

actuando sobre el eje elástico (incluida la flexión en el plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'z'}

) y a torsores alrededor del eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x'} . Por lo tanto, las variables cinemáticas satisfacen

El campo de desplazamientos se expresa como

El vector de deformaciones generalizadas locales y el vector de esfuerzos son

La matriz constitutiva generalizada es

Figura 4.19: Ejes locales y globales de una viga en un emparrillado plano

La expresión del PTV se da en la Ec.(4.73). La interpolación de los movimientos en el elemento de emparrillado plano de dos nodos se escribe por la Ec.(4.77) con

La matriz de deformaciones generalizadas es

La matriz local de rigidez Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K }_{ ij }^{\prime (e)}}

de un elemento de viga de emparrillado plano de Timoshenko de dos nodos se deduce del Cuadro 1 como

con

El Cuadro 4 muestra la matriz de rigidez local de los elementos de viga de Timoshenko y Euler-Bernoulli de dos nodos cuasi-exactos para emparrillados planos. Estas matrices se pueden deducir de las expresiones de los Cuadros 2 y 3, respectivamente.

Elemento de viga de Timoshenko de dos nodos cuasi-exacto para emparrillado plano

En ambos casos

Cuadro 4.4: Emparrillado plano. Matrices de rigidez de los elementos de viga de Timoshenko y Euler-Bernoulli de dos nodos cuasi-exactos

La matriz de rigidez global viene dada por las Ecs.(4.86)–(4.87) con

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C = \cos \alpha , S=\operatorname{sen}\alpha }

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha }
es el ángulo que el eje neutro (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

) forma con el eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}

global (Figura 4.18).

El vector de fuerzas nodales equivalentes para cargas distribuidas se calcula en ejes globales como

Para carga uniformemente distribuida, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol f}_i^{\prime (e)}= {l^{(e)}\over 2} [f_{\hat z'},m_{\hat x'},m_{y'}]^T} .

Recuérdese que las fuerzas verticales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{\hat z'}}

y el torsor distribuido Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m_{\hat x'}}
actúan en el eje elástico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat x'}
(Figura 4.15).

4.9 EJEMPLOS DEL COMPORTAMIENTO DE LOS ELEMENTOS DE VIGA 3D DE TIMOSHENKO

El primer ejemplo es el análisis de una viga en voladizo circular de sección rectangular con una carga puntual actuando en el extremo libre. La Figura 4.20 muestra la geometría y las diferentes soluciones para la flecha en el extremo con las siguientes mallas de elementos de viga 3D de Timoshenko: diez elementos de dos nodos (lineales) y un solo elemento de seis nodos (quíntico). La solución numérica en el segundo caso coincide prácticamente con la analítica [Ji,Ti3].

El segundo ejemplo es la viga helicoidal biempotrada mostrada en la Figura 4.21. Se considera la carga por peso propio. La Tabla 4.2 muestra la convergencia de la flecha central y los máximos y mínimos de algunos esfuerzos para diferentes mallas de elementos de viga 3D de Timoshenko: lineal recto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (n=2)}

y cuadrático Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (n=3)}

, cúbico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (n=4)} , cuártico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (n=5)}

y quíntico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (n=6)}
curvos. Nótese la mayor precisión de los elementos curvos para mallas gruesas (en particular para predecir el torsor máximo).

La eficiencia computacional del elemento curvo se puede mejorar condensando los GDLs internos antes de proceder a la solución global.

 Kg/cmFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): ^2

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \hbox{peso especifico} = 2.5

4.10 VIGAS CON SECCIÓN DE PARED DELGADA ABIERTA

Cuando una viga de pared delgada se somete a un torsor, se generan tensiones tangenciales. A su vez, estas tensiones causan una deformación de la sección fuera del plano denominada alabeo. Aunque la magnitud de los desplazamientos de alabeo es normalmente pequeña, pueden tener influencia en el comportamiento torsional de la estructura.

Los efectos de alabeo son particularmente relevantes en secciones de pared delgada abierta sometidas a torsión no uniforme, o a torsión uniforme coaccionada. En ambos casos, la tasa de torsión varía a lo largo del eje de la viga. Esto contrasta con la teoría de Saint-Venant estudiada en el Apartado 4.3 que supone que la tasa de torsión es constante a lo largo de la viga.

El alabeo también puede aparecer en secciones de pared delgada cerrada sometidas a torsión no uniforme. Estos efectos son menos relevantes que para las secciones abiertas y este problema no se considerará aquí. El lector interesado puede consultar a la literatura sobre el tema [BB,BC,BLD,BT2,OR,Pi,Vl].

A continuación desarrollaremos una formulación de elementos finitos para análisis de vigas con sección de pared delgada abierta sometidas a efectos axiles y flectores y que tenga en cuenta el alabeo. Se supone que la cinemática sigue la teoría de vigas de Timoshenko (Capítulo 2).

4.10.1 Descripción geométrica

Consideremos una viga recta de sección de pared delgada abierta de longitud Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L}

con propiedades geométricas y del material independientes de la coordenada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

. Supondremos que el eje neutro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}

y los ejes principales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z'}
son conocidos (origen en el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O}

), así como la posición del centro de esfuerzos cortantes y las coordenadas del eje elástico local (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {y'},0,0} ). La pared delgada se define por una superficie media parametrizada por la coordenada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s}

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0\le s\le  L_s}
y por el espesor de la pared Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t}
que se supone constante por simplicidad. Los puntos extremos en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s=0}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s=L_s}
son Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F}

, respectivamente (Figura 4.22).

La posición de un punto arbitrario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p}

en la línea media se define en el sistema cartesiano fijo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol i},{\boldsymbol j},{\boldsymbol k}}
por

y

El vector tangente a la línea media en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p}

es (por componentes)

El vector normal unitario en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p}

se define como

con

La posición de un punto arbitrario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q}

en el espesor se define por la coordenada de espesor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta }
(Figura 4.22b). En el sistema cartesiano fijo

Las coordenadas del punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q}

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {y'}_qz'_q_q) \equiv (y',z')}
se pueden expresar en el sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec{i},\vec{j},\vec{k}}
como

Finalmente, definimos el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol c}

que une el centro de esfuerzos cortantes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}
y el punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p}
sobre la línea media (Figura 4.22b). El vector tangente Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol t}
también se puede obtener en función del vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \boldsymbol c}
como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol t}= \frac{d{\boldsymbol c}}{ds}}

.

4.10.2 Hipótesis cinemáticas. Teoría de Timoshenko

Como ya se ha mencionado, un torsor actuando en una sección de paredes delgadas abierta puede producir un desplazamiento axial significativo debido al alabeo. Si el alabeo se coacciona debido, por ejemplo, a la presencia de rigidizadores o extremos empotrados la torsión inducirá una tensión axil Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'} (x',s,\zeta )}

que se deberá añadir a las tensiones axiles producidas por la flexión.

Los desplazamientos del punto arbitrario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q}

debidos a la   torsión se pueden expresar como

donde, como es habitual, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}

es la tasa de torsión; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u',u_t}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_n}
indican los desplazamiento de torsión en los ejes locales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',t,n}

, respectivamente; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_n}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_t}
son las proyecciones del vector c en los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}

, respectivamente y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta }

es la coordenada del punto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q}
en la dirección normal (Figura 4.22b).

Las Ecs.(4.119) muestran claramente que la torsión induce una variación lineal del desplazamiento tangencial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_t

en el espesor de la pared.

La relación entre los desplazamientos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v',w'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_t,u_n}
es

Los desplazamientos totales se obtienen sumando los desplazamientos debidos a los efectos axiles y flectores de la teoría de vigas de Timoshenko, a los desplazamientos de torsión. De las Ecs.(4.25), (4.119) y (4.120) se obtiene el vector de desplazamientos locales como

Una diferencia clave con la formulación de Saint-Venant es que el giro torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}

no es una función lineal y, por consiguiente, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\partial ^2\theta _{\hat x'}\over \partial x^{\prime 2}} \not = 0 }

. Por lo tanto, la torsión origina deformaciones y tensiones axiles no nulas como se muestra a continuación.

4.10.3 Función de alabeo y deformaciones y tensiones debidas a la torsión

Consideremos una viga de sección de pared delgada abierta bajo torsión. Los desplazamientos inducidos por la torsión se dan en las Ecs.(4.119).

La función de alabeo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

se define de  manera que en un estado de torsión, la deformación de cortante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma _{x'\zeta }}
es cero en toda la sección, y la deformación de cortante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma _{x's}}
es cero en la línea central, es decir

La hipótesis (4.122) y la Ec.(4.119) llevan a

La Ec.(4.124) implica que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\partial \omega \over \partial \zeta } +c_t=0} . Integrando en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta }

da

La hipótesis (4.123) y las Ecs.(4.119) llevan a

Por lo que,

Combinando las Ecs.(4.125) y (4.127) se deduce

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega _D}

es una constante (típicamente Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega _D =\omega  (0,0)}

). La función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega _s}

en la Ec.(4.128) se denomina coordenada de   área sectorial (Figura 4.23).  El cálculo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega _s}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega _D}
se detalla en el Apéndice F.

Las Ecs.(4.125) y (4.128) llevan a la siguiente expresión de la función de alabeo

lo que conduce a (teniendo en cuenta que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\partial \omega _s\over \partial s}=c_n}

de la Ec.(4.127))

Figura 4.23: Coordenada de área sectorial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \omega _s

El campo de deformaciones debido a la torsión se obtiene de las Ecs.(4.119) y (4.130) como

La expresión de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma _{x's}}

se puede reescribir usando las siguientes relaciones

En la obtención de la Ec.(4.132) hemos usado las identidades Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol t}= {\partial \mathbf{c}\over \partial s}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\partial    \mathbf{t}\over \partial s}={1\over R} { \boldsymbol n}}
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R}
es el radio de curvatura de la pared (Figura 4.23) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol c}^T \cdot {\boldsymbol n}=c_n}

.

Las deformaciones no nulas debidas a la torsión se expresan finalmente como

Para una sección de paredes delgadas formada por un ensamblaje de segmentos rectos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R=\infty }

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma _{x's}=-2\zeta {\partial \theta _{\hat x'}\over \partial x'}}

.

Las deformaciones de torsión se añaden a las deformaciones axil y de flexión como en la torsión de Saint-Venant. Esto se detalla en el próximo apartado.

Las tensiones inducidas por la torsión se deducen de las Ecs.(4.133) y (4.1) como

En la Ec.(4.134b) hemos supuesto que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{y'}= G_{z'}=G} .

El índice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}^\omega }
denota que esta tensión axil se debe a los efectos de alabeo. Recuérdese que esta tensión es cero en la teoría de Saint-Venant. La tensión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}^\omega }
varía linealmente con el espesor mediante la dependencia de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta }
con la función de alabeo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }
(Ec.(4.129)). En la práctica Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }
se supone generalmente constante en el espesor y también Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}^\omega }
(Apartado 4.10.7 y [BC,OR,Pi,Ti3]).

La Ec.(4.134b) muestra que la tensión tangencial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x's}}

(o Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _s}
por simplicidad) varía linealmente sobre el espesor de la pared. Esto es una diferencia clave con las secciones   de pared delgada cerradas, en donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \tau _s
es constante sobre el espesor (Apartado 4.3.5).

La relación entre las deformaciones y las tensiones de cortante en los ejes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',y',z'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',t,n}
es

También

En el Apartado 4.10.7 se dan más detalles del cálculo de las tensiones debidas a la torsión en secciones de pared delgada abiertas.

4.10.4 Esfuerzos y ecuación constitutiva generalizada

Las tensiones no nulas son la suma de las tensiones inducidas por los efectos axiles, de flexión y de torsión, es decir

La relación tensión-deformación coincide con la Ec.(4.1) con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol D}'}

dada por la Ec.(4.2). Los esfuerzos se definen como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\omega }}

se denomina bimomento (Figura 4.24) [BC,OR,Ti3].

en una sección en T

Sustituyendo la tensión tangencial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x's}}

en función de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'y'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x'z'}}
mediante la Ec.(4.136) y usando la relación tensión-deformación (Ec.(4.1)) en (4.138) se obtiene

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol \varepsilon }'}

es el vector de deformaciones generalizadas y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol D}'}
es la matriz  constitutiva generalizada dados por

y

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {D}_a}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol D}_f}
se obtienen por las Ecs.(4.36) y la matriz de torsión constitutiva es

con

En la expresión de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {D}_t}

hemos supuesto que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{y'} = G_{z'} = G}

.

Para material homogéneo

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle J=\iint _A \zeta ^2 \left(2 + {c_n^2\over R}\right)dA}

es la inercia torsional y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_\omega = \int \!\!\int _A  \omega ^2 dA}
es el módulo de inercia de alabeo. La Figura 4.27 muestra los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle J}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_w}
de varias secciones.

Para una sección curva de espesor uniforme

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_s}

es la longitud de la línea media (Figura 4.22). Estas expresiones son exactas para una sección circular [OR,Ti3].

Para secciones abiertas de pared delgada homogéneas formadas por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}

segmentos rectos de espesor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_i}
y longitud Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle l_i}
[OR,Ti3],

La tensión tangencial máxima en cada segmento ocurrirá en los bordes situados a una distancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \pm t/2}

de la línea media. Su valor se deduce de la Ec.(4.134b) para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta =\pm \frac{t}{2}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R=\infty }
y la Ec.(4.139) como

Claramente, la tensión tangencial máxima se encuentra en el segmento de mayor espesor.

Ejemplo 4.3:

Comparación de secciones de pared delgada abiertas y cerradas.

El comportamiento torsional de secciones de pared delgada cerradas es bastante diferente del de las secciones abiertas. Para secciones cerradas, la tensión tangencial se distribuye uniformemente en el espesor de la pared (Figura 4.25), mientras que en las secciones abiertas se tiene una distribución lineal. La rigidez torsional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t}

es proporcional al cuadrado del área encerrada para secciones cerradas (Ec.(4.53)) mientras que es proporcional al cubo del espesor en secciones abiertas (Ec.(4.143a) [BC].

Figura 4.25: Tubos de pared delgada cerrados y abiertos

Considérense, por ejemplo, dos anillos de radio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R_m}

y espesor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t}
idénticos pero uno abierto y el otro cerrado (Figura 4.25). La rigidez torsional de las secciones abierta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat  D_t^{cerrado}}
y cerrada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t^{abierto}}
se dan en las Ecs.(4.53) y (4.143a), respectivamente, como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t^{cerrado}=2\pi GR_m^3 t}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t^{abierto}=2\pi GR_m t^3/3}

. Su cociente es

Si las dos secciones están sometidas al mismo torsor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\hat x'}} , las tensiones tangenciales máximas en la sección abierta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{\max }^{abierto}}

y cerrada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{\max }^{cerrado}}
se obtienen de las Ecs.(4.144) y (4.54), respectivamente como

Su cociente se puede expresar como

Para una sección típica de pared delgada con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R_m =20t}

la rigidez torsional de la sección cerrada será 1200 veces más grande que la de la sección abierta. Bajo el mismo torsor, la tensión tangencial máxima en la sección abierta será 60 veces mayor que en la sección cerrada. En otras palabras, la sección cerrada puede soportar 60 veces más torsor para alcanzar un nivel de tensiones igual al de la abierta [BC].

4.10.5 Principio de trabajos virtuales

El PTV para una carga distribuida es

El trabajo virtual interno se puede escribir usando las Ecs.(4.137)–(4.139) como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol \sigma }'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {\boldsymbol \varepsilon }'}
se definen en las Ecs.(4.138) y (4.140a).

Los términos subrayados en la Ec.(4.146) se deben a efectos torsionales.

El trabajo externo para cargas distribuidas se escribe como

con los vectores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta {\boldsymbol u}'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol t}'}
definidos como en la Ec.(4.64) y

La expresión de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta W}

para cargas puntuales y momentos concentrados es

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol p'}_i}

está definido en la Ec.(4.64) y

El subíndice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}

en las anteriores ecuaciones denota el punto del eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}
donde se aplica la carga axial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F_{{x'}_i}}

.

4.10.6 Elemento de viga de Timoshenko de dos nodos con sección de pared delgada abierta

La formulación de elementos de viga 3D de Timoshenko con sección de pared delgada abierta se puede obtener superponiendo los efectos axil, de flexión (con y sin deformación de cortante) y de torsión. Se requiere continuidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^1}

para aproximar el giro de torsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}

, ya que en el PTV aparecen sus segundas derivadas (Ec.(4.146)). Esto se puede implementar con elementos rectos de dos nodos escogiendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{\partial \theta _{\hat x'}}/{\partial x'}}}

como el séptimo GDL nodal y una interpolación Hermítica cúbica estándar para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}

.

La interpolación del campo de desplazamientos se escribe como

con

y

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_i}

es la función de forma lineal estándar y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_i^H, \bar  N_i^H}
son las funciones de forma Hermíticas cúbicas (Ecs.(1.11a)).

La relación entre las deformaciones generalizadas y los movimientos nodales se escribe como

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B}_{a_i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B}_{f_i}}
dado por la Ec.(4.82b) y

La matriz de rigidez del elemento se obtiene como se explica en el Apartado 4.4.1. Sus diferentes términos coinciden con los del Cuadro 1, con la excepción de los siguientes términos que completan la matriz de rigidez Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 14\times 14}

del elemento en cuestión (7 GDLs por nodo)

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {k'}_{ij}{=k'}_{ji}} .

Se puede escoger una interpolación diferente para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}}

que conduce a resultados nodales “exactos” usando funciones de forma hiperbólicas como se describe en [BD5].

La transformación de los primeros seis GDLs nodales a ejes globales es idéntica a lo descrito en el Apartado 4.4.2. La transformación de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{\partial \theta _{\hat x'}}\over {\partial x'}}}

es más enrevesada para estructuras reticuladas y barras rectas de sección variable [GP,Gu,Sh]. El grado de compatibilidad nodal entre dos elementos depende del cumplimiento de la siguiente ecuación [BD5]

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c\le 1}

y los subíndices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b}
denotan valores en cada uno de los dos elementos adyacentes. Se han propuesto diferentes valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c}
sobre la base de análisis por el MEF locales para vigas reticuladas con secciones en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H}
y en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U}
[Sh]. Una alternativa es suponer que las cantidades nodales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {{\partial \theta _{\hat x'}} \over \partial x'}}
son discontinuas entre elementos y que hay tantas variables de alabeo en un nodo como elementos conectados a él [Ak,BD5].

4.10.7 Cálculo de tensiones debidas a la torsión en secciones de pared delgada abiertas

Las tensiones axil y tangencial inducidas por la torsión en vigas de pared delgada abiertas se pueden calcular mediante la Ec.(4.134). Para paredes rectas (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R\to \infty } ) y

Los valores máximos ocurren en los bordes de las paredes (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta = \pm \frac{t}{2}} ), es decir

Esta tensión tangencial se llama a veces tensión tangencial de Saint-Venant (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x's}^{sv}} ), indicando que no incluye la tensión tangencial inducida por la tensión axil debida al alabeo.

La tensión axil inducida por el alabeo se puede calcular usando las Ecs.(4.134a) y (4.138) como

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega (s,\zeta )= g(s) - c_t (s)\zeta }

(Ec.(4.125)).

y (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \tau _{x's}^\omega (s)
para tres secciones      de pared delgada abiertas

Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g(s)\not =0} , la contribución del término Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_t \xi }

es generalmente despreciable. La Figura 4.26 muestra la distribución de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g(s)}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}^\omega (s,0)}
para tres secciones. El cálculo completo para una de las secciones se presenta en el Ejemplo 4.4. Los valores máximos de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}^\omega }
se encuentran usualmente en los extremos de la sección (puntos D en la Figura 4.27) y son independientes de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta }

. Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g(s)=0}

estos valores se encuentran en los puntos extremos y en las esquinas. En general podemos escribir

La Figura 4.27 muestra el valor y la posición de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta _1}

(y por consiguiente de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \max \vert \sigma _{x'}^\omega \vert }

) para diferentes secciones.

Tensión tangencial debida a la tensión axil de alabeo σ_x'^ω

La tensión axil Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{x'}^\omega }

induce un campo de tensiones tangenciales adicional. La llamada tensión tangencial de alabeo (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x's}^\omega }

) se puede calcular integrando las ecuaciones de equilibrio locales a posteriori. El método sigue los razonamientos usados para calcular la distribución de tensiones tangenciales en vigas planas bajo cargas de flexión (Apartado 3.7 y Apéndice D). Las ecuaciones de equilibrio son

Introduciendo la Ec.(4.156) en (4.158) da

Es usual despreciar la variación sobre el espesor de las tensiones tangenciales de alabeo. En ese caso, la Ec.(4.159) se reescribe (usando la Ec.(4.125)) para dar

Consecuentemente, las tensiones tangenciales debidas al alabeo son nulas si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g(s)=0} . Los valores máximos se encuentran en los puntos en los que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_\omega }

toma un valor máximo, es decir

La Figura 4.27 muestra el valor y la posición de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta _2}

(y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \max \vert \tau _{x's}^\omega \vert }

) para diferentes secciones.

La tensión tangencial total debida a la torsión es la suma de la tensión tangencial de Saint-Venant (Ec.(4.154)) y la tensión tangencial de alabeo (Ec.(4.160)).

La Figura 4.28 muestra la distribución de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{x's}}

en una sección abierta de pared delgada.

Ejemplo 4.4: Cálculo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_\omega }

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_\omega }
para una sección en doble L.

- Solución

Consideremos una sección de pared delgada en forma de doble Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L}

como la que se muestra en la figura siguiente:

Distribución de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \omega _s(s)

Segmento DA: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0\le s\le b}

; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -b\le y \le 0}
; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z = -\frac{h}{2}}

Segmento AB: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b\le s\le b+h}

; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y=0}
; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle  -\frac{h}{2}\le z \le \frac{h}{2}}

Segmento BF: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b+h\le s\le b+2h}

; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0\le y \le b}
; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z = \frac{h}{2}}

Valor medio: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega _m=-\omega _D}

Área sectorial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_\omega }

Despreciando la variación en el espesor, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega (s)=g(s)}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_\omega (s,0) = \int _0^s g(s) ds}

.

Módulo de inercia sectorial: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_\omega = \int _A \omega ^2 dA = \frac{th^2b^3}{12}\left(\frac{b+2h}{2b+h}\right)}

La contribución del término Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle -c_t\zeta }

de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega (s,\zeta )}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_\omega }
es despreciable. Su valor es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_\omega = \frac{t^3}{12} \left(\frac{2b^3}{3}+\frac{h^3}{12}\right)}

Las siguientes figuras muestran la distribución de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega _s (s), g(s)}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_\omega (s)}
en la sección en doble L considerada.

4.11 ELEMENTOS DE VIGA DE PARED DELGADA ABIERTA DE TIMOSHENKO QUE CONSIDERAN LA TENSIÓN TANGENCIAL DEBIDA A LA TORSIÓN

4.11.1 Ecuaciones básicas

Las tensiones tangenciales inducidas por la torsión pueden ser importantes en vigas cortas empotradas y en vigas de pared delgada abierta con material compuesto. Estos términos se pueden tener en cuenta en la teoría presentada anteriormente siguiendo razonamientos similares a los usados para introducir el efecto de la deformación de cortante en la teoría clásica de vigas. La tasa de torsión (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega } ) se define ahora como suma del cambio del giro de torsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(\frac{\partial \theta _{\hat x'}}{\partial x'} \right)}

y una tasa adicional (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _s}

) inducida por las tensiones tangenciales debidas a la torsión, es decir

La tasa Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _s}

se puede interpretar como (el valor negativo de) la deformación de cortante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma _t}
introducida por los efectos de torsión (Figura 4.29). Claramente, si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _s=0}

, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega = \frac{\partial \theta _{\hat x'}}{\partial x'}}

y se recupera la definición clásica de la tasa de torsión de la Ec.(4.25) [Va,VOO].

El campo de desplazamientos inducido por los efectos de torsión se escribe en ejes locales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',t,n}

en la nueva teoría como

La única diferencia con la Ec.(4.119) es la definición del desplazamiento axial. Nótese que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega }

se toma ahora como una variable independiente.

Las deformaciones inducidas por la torsión (denominadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol \varepsilon }^\prime _t} ) son

Usando la Ec.(4.130) y suponiendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\partial \omega \over \partial \zeta } = -c_t}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle  {\partial c_t \over \partial s}=1}
se obtiene

con

Los esfuerzos debidos a la torsión son (aceptando que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{c_n}{R}=0} )

La ecuación constitutiva se supone de la forma

Sustituyendo esta ecuación en (4.166) se obtiene

Una simple multiplicación da

Una comparación de las Ecs.(4.141a) y (4.169) muestra los términos introducidos por la deformación de cortante en la matriz constitutiva de torsión. Para material homogéneo

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_\omega }

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat D_t }
coinciden con las expresiones de la Ec.(141b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \displaystyle \left(\hbox{para } \frac{c_n}{R}=0\right)}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \displaystyle \hat D_{s_t} = \iint _A  |c|^2 GdA}

. Usando las Ecs.(4.167), (4.165b) y (4.168) se deduce que los efectos de torsión contribuyen los siguientes términos al PTV

Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _s =0}

entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma _t =0}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \kappa _s = {{ \partial \theta _ { \hat x' } }\over \partial x' } } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \kappa _\omega = {{ \partial ^2 \theta _ { \hat x' } }\over \partial x^{\prime 2} } }

y el PTV recupera la expresión de la Ec.(4.146) para los términos de torsión.

Nótese que en el PTV sólo aparece la derivada primera del ángulo de torsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{\hat x'}} . Esto permite emplear una interpolación de continuidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): C^0

para todas las variables de   movimiento. Estas variables incluyen la tasa de torsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _\omega }
como un GDL adicional.

4.11.2 Discretización por elementos finitos

La interpolación de los movimientos para un elemento de viga de dos nodos se escribe como

En la Ec.(4.172) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_i}

son las funciones de forma 1D lineales estándar (Figura 2.4).

Las deformaciones generalizadas se expresan en función de los GDLs nodales como

con

y

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B}_{a_i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B}_{f_i}^\prime }
se obtienen ampliando las expresiones de la Ec.(4.79) con una columna de ceros y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol    B}^\prime _{t_i}}
son las contribuciones de la torsión a la matriz de de deformaciones generalizadas dadas por

La matriz de rigidez del elemento tiene la forma estándar

donde

Introduciendo las Ecs.(4.173c) y (4.175b) en (4.175a) da

El bloqueo de cortante inducido por los efectos de flexión y torsión se puede eliminar en el elemento de viga de dos nodos usando una cuadratura reducida de un punto para integrar todos los términos de la matriz de rigidez del elemento.

Se puede seguir el mismo procedimiento para desarrollar elementos de viga 3D de pared delgada abierta de tres y cuatro nodos que tengan en cuenta los efectos de la deformación de cortante debidos a la torsión. El comportamiento del elemento cuadrático mejora usando una cuadratura reducida de dos puntos. El elemento cúbico tiene un comportamiento excelente usando una cuadratura completa de cuatro puntos [Va,VOS].

Se pueden encontrar otros procedimientos numéricos y analíticos para el análisis de vigas de pared delgada abierta que tengan en cuenta la deformación de cortante inducida por la torsión en [BT2,BW2,FM2, Ko2,KP,KSK,Le,LL3,PK,ST].

4.11.3 Ejemplos

Viga de material compuesto laminado en voladizo sometida a torsor en el extremo

La Figura 4.30 muestra una viga en voladizo de material compuesto laminado de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L=1000}

mm con sección en doble T. Tanto la pared central como las dos alas tienen siete capas con orientaciones simétricas [0,90,0,90,0,90,0] con respecto al eje de la viga. Hemos considerado dos casos de carga: (a) un torsor de 1 KNFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \times }

mm actuando en el extremo libre, y (b) una carga vertical de 1KN actuando también en el extremo libre. Ambos problemas se han resuelto con diferentes mallas de elementos de viga lineales de dos nodos y cuadráticos de tres nodos con integración completa y reducida y con elementos cúbicos de cuatro nodos con integración completa. Se han analizado los mismos problemas con una malla de 550 cuadriláteros de lámina plana DKQ (Apartado 8.12.3) mostrada en la Figura 4.30 para obtener una solución de referencia a efectos de comparación.

bajo un torsor en el extremo (a) y     cociente de flechas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): w_A^b/w_A^s
para  carga puntual en el extremo     (b) en función del coeficiente de esbeltez  (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lambda 

La Figura 4.31 muestra el cociente entre los resultados de viga y lámina plana para el ángulo de torsión (para la carga de torsión) y la flecha (para la carga vertical) en el extremo, en función del coeficiente de esbeltez Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =L/h}

para la malla de un   solo elemento. El gráfico muestra que

• El elemento de viga lineal de dos nodos con integración completa de dos puntos (L-F) se bloquea para vigas esbeltas. La cuadratura reducida de un punto (L-R) elimina el bloqueo de cortante y proporciona resultados excelentes para vigas cortas y esbeltas.
• El elemento de viga cuadrático de tres nodos con integración completa de tres puntos (Q-F) presenta un ligero bloqueo para vigas esbeltas. Se obtienen resultados excelentes para todos los casos empleando la cuadratura reducida de dos puntos uniforme (Q-R).
• El elemento de viga cúbico de cuatro nodos con integración completa de cuatro puntos (C-F) no bloquea y da resultados precisos para vigas cortas y esbeltas.

La Figura 4.32 muestra la convergencia de los cocientes de flechas y giros en el extremo para una viga esbelta (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda =20} ) con el número de elementos para todos los casos estudiados. Todas las soluciones convergen a los valores de referencia.

Nótese que se ha usado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta \equiv \theta _{\hat x'}}

en las Figuras 4.31 y 4.32 por simplicidad.

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (\cdot )^s
denotan las soluciones de viga     y lámina

Vigas en U en voladizo y empotradas bajo cargas puntuales en el extremo

El siguiente ejemplo es el análisis de una viga en voladizo con sección en U de material compuesto laminado sometida a dos cargas puntuales actuando en el extremo libre. La Figura 4.33 muestra la geometría de la viga y las propiedades del material. El problema se ha resuelto con los siguientes tres elementos:

• Malla de 20 elementos de viga de pared delgada abierta de Timoshenko de dos nodos teniendo en cuenta las tensiones tangenciales debidas a la torsión con integración reducida uniforme de un punto (Apartado 4.11).

• Malla de 20 elementos de viga de pared delgada abierta de Euler-Bernoulli de tres nodos basada en la teoría de Saint-Venant (Apartado 4.7).

• Malla de 550 cuadriláteros de lámina DKQ de nueve nodos (Apartado 8.12.3).

El problema se ha resuelto para los dos materiales y cargas siguientes:

Material homogéneo isótropo (acero).

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E=210000}

MPa  , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu =0,30}

Cargas: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_{y'}=25000}

N , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_{z'}=-25000}
N

Material compuesto laminado. Laminado de 10 capas [90,0Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle _{4}]_s}

de matriz de cristal epoxi con las siguientes propiedades:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): E_1=53780 MPa   ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E_2 =17930} MPa  ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu _{12} =0,25}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): G_{12}=G_{11}=8960 MPa   ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G_{23}=3450} MPa
Cargas: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_{y'}=250} N,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_{z'}=-250} N

La Figura 4.34 muestra la distribución de la flecha del centro de esfuerzos cortantes y del ángulo de torsión para material homogéneo para los tres elementos considerados. Los resultados están normalizados con la flecha y el giro máximos obtenidos con la malla de elementos de lámina DKQ. Los resultados para material compuesto laminado son prácticamente coincidentes con los de la Figura 4.34.

El elemento de viga de Timoshenko de dos nodos proporciona resultados muy precisos. Nótese la discrepancia en los resultados del ángulo de torsión para el elemento de viga de Euler-Bernoulli de dos nodos basado en la teoría de Saint-Venant.

La Tabla 4.3 muestra los valores máximos de los desplazamientos laterales y verticales del centro de esfuerzos cortantes y el giro de torsión para los tres elementos considerados con secciones de material homogéneo y material compuesto laminado. Las distribuciones son prácticamente coincidentes para los tres elementos.

La Figura 4.35 y la Tabla 4.4 muestran una serie de resultados similares para una viga en U biempotrada de las mismas dimensiones. Las conclusiones son las mismas que para la viga en voladizo.

Como conclusión general, el sencillo elemento de barra 3D de Timoshenko de dos nodos con integración reducida de un punto tiene un comportamiento excelente para el análisis de vigas abiertas de pared delgada.

4.12 ELEMENTOS DE VIGA 3D DEGENERADOS

Los elementos de viga 3D también se pueden obtener imponiendo las siguientes restricciones a los elementos de sólido 3D [On4]:

1. Variación lineal de los desplazamientos en cada sección (hipótesis de sección plana de Saint-Venant),
2. Las dimensiones de la sección no cambian (lo que limita los GDLs nodales),
3. Hipótesis de tensión plana en ejes locales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (\sigma _{y'}=\sigma _{z'}=\gamma _{y'z'}=0)} .

El proceso es análogo al explicado para obtener elementos de lámina degenerados en el Capítulo 10.

El punto de partida es un elemento prismático. Por simplicidad sólo consideraremos hexaedros (Figura 4.36). Esto limita la formulación a vigas con sección rectangular. Se pueden modelar otras secciones empleando una sección rectangular de igual área y las mismas propiedades de inercia.

4.12.1 Descripción de la geometría y el campo de desplazamientos

El eje de referencia de la viga se define como la línea que une los centros de gravedad de las secciones. Se define en cada nodo un sistema de coordenadas curvilíneo local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x',y',z'}

de tal manera que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x'}
es tangente al eje de la viga e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y',z'}
son las direcciones principales de inercia (Figura 4.36). La geometría se expresa en forma isoparamétrica como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}

es el número de elementos, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_i(\xi )}
es la función de forma 1D Lagrangiana del nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}
[On4], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol    x}_i=[x_i,y_i,z_i]^{\scriptscriptstyle T }}
contiene las coordenadas cartesianas del nodo, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a_i}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b_i}
son las dimensiones de la sección en el nodo y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \eta }

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \zeta }

son las coordenadas naturales transversales (Figura 4.36).

El campo de desplazamientos se define siguiendo la hipótesis de la teoría de vigas de Timoshenko para el giro de la sección como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol a}^{\prime (e)}_i=[u_i, v_i, w_i, \theta _{{x'}_i}, \theta _{{y'}_i}, \theta _{{z'}_i}]^{ \scriptscriptstyle T}} , y

con

Nótese que las componentes de los vectores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{1i}, {\boldsymbol e}_{2i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_{3i}}
se expresan en el sistema de coordenadas global.

El vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol a}^{\prime (e)}_i}

contiene los tres desplazamientos globales del nodo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_i}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_i} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_i}

y los tres giros locales: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{{ x'}_i}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{{ y'}_i}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{{ z'}_i}}

(definidos en forma vectorial).

4.12.2 Campo de deformaciones

Teniendo en cuenta que los desplazamientos se expresan en ejes diferentes, las deformaciones locales (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol \varepsilon }'} ) y globales (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol \varepsilon }} ) en un punto se relacionan por

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol \varepsilon }}

es el vector de deformaciones estándar de la elasticidad 3D [On4,ZTZ]

y

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle e^x_{\scriptscriptstyle {1}},e^y_{\scriptscriptstyle {1}},e^z_{\scriptscriptstyle {1}}}

son las componentes en ejes globales de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_1}
en el punto en el que se calculan las deformaciones. Estas componentes se pueden obtener por interpolación de los valores nodales. Lo mismo aplica para las componentes de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_2}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol e}_3}

.

Las componentes de deformación global se obtienen en función de los movimientos nodales como sigue.

Primero, se obtienen las derivadas de los desplazamientos globales con respecto a las coordenadas naturales como

Las derivadas de los desplazamientos globales con respecto a las coordenadas cartesianas y naturales se relacionan por la inversa de la matriz jacobiana 3D Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol J}}

como

donde

Los elementos de la matriz jacobiana Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol J}}

se calculan de la descripción isoparamétrica (4.177) como

La matriz de deformaciones globales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol B}}

se obtiene sustituyendo la interpolación de los desplazamientos (4.178a) en (4.179b). La expresión completa se muestra en el Cuadro 5. Usando este resultado y la Ec.(4.179a) se obtiene la relación entre las deformaciones locales y los movimientos locales como

con

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\bar{\boldsymbol B}'}_{ab_i}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\bar{\boldsymbol B}'}_{s_i}}
contienen las contribuciones axil-flector y de cortante a la matriz de deformaciones locales y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol S}_1}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol S}_2}
son la primera y últimas dos filas de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol S}}
de la Ec.(4.179c), respectivamente.

Cuadro 4.5: Matriz de deformaciones globales de un elemento de viga 3D degenerado

Los giros nodales se expresan a continuación en ejes globales, obteniéndose la relación final entre las deformaciones locales y los movimientos globales como

donde

con

En lo anterior Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol a}^{(e)}_i=[u_i, v_i, w_i, \theta _{x_i}, \theta _{y_i}, \theta _{z_i}]^{ \scriptscriptstyle T}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol  T}_i}
es la matriz de transformación de la Ec.(4.178c). La Ec.(4.184a) relaciona las deformaciones locales y los movimientos nodales globales.

La ecuación constitutiva se expresa en ejes locales mediante la Ec.(4.1). Esta formulación permite considerar secciones de la viga de material heterogéneo, tal y como se explica en el apartado siguiente.

4.12.3 Matriz de rigidez y vector de fuerzas nodales equivalentes del elemento

Sustituyendo las Ec.(4.178a) y (4.183a) y la ecuación constitutiva (4.1) en la expresión de PTV (Ec.(4.63)) se obtiene la matriz de rigidez y el vector de fuerzas nodales equivalentes del elemento en ejes globales como

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle V^{(e)}}

es el volumen del elemento sólido original. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K}^{(e)}_{ab}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K}^{(e)}_{s}}
son las contribuciones de los efectos de axil-flexión y de cortante a la matriz de rigidez global, respectivamente, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol b}}
son las fuerzas de volumen (peso propio), t son las fuerzas distribuidas actuando en alguna de las caras del elemento (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \eta =\pm{1}}
o Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \xi{-\pm}1}

) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol p}^{(e)}_i}

son vectores de fuerzas puntuales nodales. Todas las componentes se definen en ejes globales como

Si las fuerzas distribuidas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol t}}

actúan en el eje de la viga, entonces la integral de área de la Ec.(4.185b) se sustituye por una integral de línea sobre la longitud del elemento y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \eta =\zeta=0}
en la expresión de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N_i}
de la Ec.(4.178b).

La integración explícita sobre la sección para elementos curvos tiene algunas dificultades. Es sin embargo posible siguiendo procedimientos similares a los explicados en el Capítulo 10 para elementos de lámina degenerados. Para vigas rectas, puede calcularse de forma analítica la matriz de rigidez del elemento. Para un elemento de viga de dos nodos la matriz de rigidez y el vector de fuerzas nodales equivalentes tiene expresiones idénticas a las obtenidas en el Apartado 4.4 partiendo de la teoría de vigas 3D. En la práctica, se emplea una cuadratura de Gauss 3D para la integración de las matrices y vectores del elemento; es decir

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_\xi ,~n_\eta ~ n_\zeta }

son los puntos de integración en las direcciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \xi ,\eta ,\zeta }

, respectivamente y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle W_p, W_q,W_r}

son los pesos correspondientes. Para material homogéneo es habitual escoger Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_\eta =n_\zeta=2}

. Para vigas de material heterogéneo es necesaria una cuadratura de mayor orden (o incluso integración por celdas). Para una sección de material compuesto laminado es suficiente una integración por capas.

El bloqueo de cortante se evita usando una cuadratura reducida para integrar la matriz de rigidez de cortante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol K}_s^{(e)}}

Típicamente se escoge Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_\xi=1}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_\xi=2}
para los elementos de viga degenerados de dos y tres nodos, como en vigas planas. Para elementos rectos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\boldsymbol    K}_{ab}^{(e)}}
es usual emplear una cuadratura completa (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \eta _\xi=2}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \eta _\xi=3}
para elementos de viga de dos y tres nodos, respectivamente). Se recomienda la integración uniforme reducida para todos los términos de la matriz de rigidez en el caso curvo para aliviar el bloqueo debido a los efectos axiles (Apartados 9.5 y 10.11.1).

Esta formulación se puede adaptar a la teoría de Euler-Bernoulli haciendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta _{ z'}={\partial v'\over \partial x'},\quad \theta _{x'}= -{\partial w'\over \partial x'}} , satisfaciendo así la condición de ortogonalidad del giro de la normal. Esto lleva a la desaparición de las deformaciones de cortante debidas a la flexión, e introduce la necesidad de aproximaciones de continuidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^1}

para los desplazamientos locales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v'}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w'}

, ya que aparecen sus segundas derivadas en la expresión de la deformación axil. Se puede seguir utilizando una interpolación de continuidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^0}

para el desplazamiento axial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u'}

. Esto se puede implementar definiendo la interpolación de los desplazamientos en los ejes locales seguida de su transformación a ejes globales. El elemento de viga de Euler-Bernoulli degenerado de dos nodos más simple emplea una aproximación cúbica Hermítica para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w'}
y un campo lineal para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u'}

. Para material homogéneo, sección constante y carga uniforme, la matriz de rigidez y el vector de fuerzas nodales equivalentes coinciden con las expresiones obtenidas mediante la teoría de vigas 3D clásica (Apartado 4.7).

4.13 CONCLUSIONES

En este capítulo hemos estudiado la formulación de elementos de viga 3D, válidos para el análisis de vigas de material compuesto, usando las teorías de Timoshenko y Euler-Bernoulli. Se ha presentado la teoría de torsión libre de Saint-Venant y la más sofisticada teoría de torsión que tiene en cuenta los efectos de alabeo en secciones de pared delgada abiertas. El elemento de viga de Timoshenko de dos nodos con un único punto de integración es probablemente la opción más útil para el análisis de todo tipo de vigas 3D.

Los elementos de viga 3D obtenidos por degeneración de elementos sólidos 3D pueden ser una opción interesante en algunos casos.

Es posible acoplar los elementos de viga 3D con elementos de placa o de lámina. Esto es de utilidad para el análisis de estructuras de placas y láminas rigidizadas con vigas. Este tema se estudia en el Apartado 10.21.