Nonlinear Finite Element Analysis of Reinforced Concrete L-Shaped Column Frame Joint



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<big>Nonlinear Finite Element Analysis of Reinforced Concrete L-Shaped Column Frame Joint</big></div>

==Abstract==

Based on the concrete finite element model of nonlinear constitutive relation, the mechanical properties of L-shaped frame joint under low cyclic loading and monotonic loading were studied. The process of formation and development of joint cracks and the joint stress distributions under loading were analyzed. The failure mechanism of L-shaped column joint was further verified. Which provides a reference for the popularization and application of L-shaped columns.

==Key words  ==

L-Shaped Column Joint, Nonlinear Analysis, Finite Element, Cyclic Loading, Monotonic Loading

=1. Introduction=

Over the past 30 years, many scholars have carried out a large number of experimental and theoretical research on special-shaped columns<sup>[1-6]</sup>.With the development of special-shaped column structure practical application around the world, the seismic performance of special-shaped column joints has been paid more attention to<sup>[7]</sup>.

<span id='translation_sen_id-2'></span><span id='translation_sen_id-5'></span>The joint is the beam and column intersecting area of the frame, which is an important part of the frame and is in a very complicated stress state. And for the special-shaped column frame structure joints, there are orthogonally stretched limbs affecting the force situation of column. The stress distribution is more intricate <sup>[8]</sup>.

In this paper, the mechanical properties of L-shaped column frame joint under low cyclic loading and monotonic loading were analyzed with the finite element software ANSYS to simulate it’s performance under earthquake. And provide some research material for seismic performance of special-shaped column frame structure.

=2. Joint model=

The calculation model selected in this paper is the corner post joint of special-shaped column structure  (ie, L-shaped column joint) . The cross-section views of joint members are as follows:

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''' 
{|
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| [[Image:draft_177_463205977-image1.jpeg|246px]]
| [[Image:draft_177_463205977-image2.jpeg|center|204px]]
|}
'''</div>

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'''     Figure 1'''. L-shaped column cross-section.              '''Figure 2'''. Rectangular beam section.</div>

The 3D view and the side view of the beam-column joint:

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{|
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| [[Image:draft_177_463205977-image3.jpeg|240px]]
| [[Image:draft_177_463205977-image4.jpeg|center|318px]]
|}
</div>

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'''Figure 3'''. 3D view of joint.                  '''Figure 4'''. Lateral view of joint.</div>

The column height is 2.4m, beam length is 1.5m. The limb length of L-shaped column is 600mm, limb thickness is 200mm, limb length-thickness ratio is 3. The beam section is 400 × 200mm, aspect ratio of the beam is 2. The longitudinal bars of the L-shaped column and the rectangular beam are of HRB335 and the stirrups are made of HPB300 rebars. The longitudinal reinforcement diameter of column and beam is 16mm. Column stirrup diameter is 10mm and stirrup spacing is 150mm. Beam stirrup diameter is 6mm and stirrup spacing is also 150mm. Reinforcement arrangement is shown in the figures above. The concrete strength grade of column and beam is both C30.

=3. Finite element model=

''3.1. Mechanical properties of materials''

The mechanical parameters of C30 concrete used in this paper<sup>[9]</sup> are shown in the following table:

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
'''Table 1. '''Concrete mechanical properties.</div>

{| style="width: 73%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Strength level
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|C30
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Elastic modulus（MPa）
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|30000
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Poisson's ratio
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.2
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Standard value of axial compressive strength（MPa）
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|20.1
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Standard value of axial tensile strength（MPa）
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|2.01
|}


The mechanical properties of HRB335 and HPB300 rebars<sup>[9]</sup> used in the joint are as follows:

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'''Table 2.'''Rebar mechanical properties.</div>

{| style="width: 63%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Rebar grades
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|HRB335
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|HPB300
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Elastic modulus（MPa）
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|200000
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|210000
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Poisson's ratio
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.27
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.27
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Standard value of yield strength（MPa）
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|335
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|300
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Tangent modulus after yield（MPa）
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|2000
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|2000
|}


''3.2. Constitutive relationship of C30 concrete''

In this paper, nonlinear simulation is based on the multi-linear isotropic strengthening model (MISO) provided by ANSYS. This model is subject to the von Mises yield criterion<sup>[10-12]</sup>. For determining the stress-strain relationship of concrete, equations are selected as follows<sup>[9]</sup>:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| [[Image:draft_177_463205977-image5.png|600px]]
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| [[Image:draft_177_463205977-image6.png|600px]]
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}


among them，

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| [[Image:draft_177_463205977-image7.png|600px]]
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| [[Image:draft_177_463205977-image8.png|600px]]
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| [[Image:draft_177_463205977-image9.png|600px]]
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}


In the equations,  [[Image:draft_177_463205977-image10.png|600px]] is the uniaxial compression damage evolution parameter of concrete;  [[Image:draft_177_463205977-image11.png|600px]] is the reference value of the stress-strain curve descending section of concrete under uniaxial compression, 0.74 is taken in this paper;  [[Image:draft_177_463205977-image12.png|600px]] is the representative value of concrete uniaxial compressive strength , taking 20.1N/mm<sup>2</sup> ;  [[Image:draft_177_463205977-image13.png|600px]] is the concrete peak compressive strain corresponding to the uniaxial compressive strength [[Image:draft_177_463205977-image12.png|600px]] , taking 1.47×10<sup>-3</sup>. Replace the above values into the formulas, a stress corresponding to the certain strain can be obtained , as follows:

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'''Table 3. '''Stress and strain relationship of C30 concrete.</div>

{| style="width: 63%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|strainε
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|stressσ（MPa）
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.0005
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|15
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.001
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|19
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.0015
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|20
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.002
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|19
|}


The four-fold line stress-strain relationship of concrete is shown below:

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 [[Image:draft_177_463205977-image14.jpeg|348px]] </div>

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'''Figure 5. '''Stress-strain relationship of C30 concrete.</div>

''3.3. Establishment of the model''

In order to build the model, three-dimensional space entity unit SOLID65 is used. The reinforced bars are dispersed in concrete elements in the real constants manner (ie, smeared model)<sup>[13]</sup>, so the element material is considered as continuous and uniform. The elements adopt the W-W five-parameter failure criterion and the crack distribution model to check the cracking and crushing of concrete<sup>[14]</sup>.

Build the model and meshing it with tetrahedral grids as follows:

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{|
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| [[Image:draft_177_463205977-image15.jpeg|288px]]
| [[Image:draft_177_463205977-image16.jpeg|center|276px]]
|}
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'''            Figure 6.  '''Finite element model of joint.        '''Figure 7. '''Finite element model of joint after meshing.</div>

The reinforcement ratios of all four parts of the joint are as follows:

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'''Table 4. '''Reinforcement ratios of the joint.</div>

{| style="width: 66%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Direction
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Reinforcement ratio（%）
|-
|  rowspan='1' style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Part 1
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|X
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.2615
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Y
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.523
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Z
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|1.0048
|-
|  rowspan='1' style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Part 2
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|X
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.523
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Y
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.523
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Z
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|2.0096
|-
|  rowspan='1' style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Part 3
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|X
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.523
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Y
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.2615
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Z
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|1.0048
|-
|  rowspan='1' style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Part 4
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|X
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|1.0048
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Y
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.0942
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Z
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.1884
|}

=4. Calculation results=

''4.1. Low Cyclic Loading''

Constrain the top and bottom nodes of the column model and apply a 850KN vertical downward load on the top surface of the column(ie, the column axial compression ratio is 0.3). Applying a concentrated load in the vertical direction at the end of the beam, changing the direction of the load in each load step (up or down), and gradually increasing the concentrated load. By this means, the low cycle repetitive loading to the model is simulated<sup>[15]</sup>.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
'''Table 5. '''Load situation.</div>

{| style="width: 67%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Load step
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Value（KN）
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Direction
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Load step 1
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|10
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Vertical up
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Load step 2
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|10
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Vertical down
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Load step 3
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|20
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Vertical up
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Load step 4
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|20
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Vertical down
|-
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Load step 5
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|30
|  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Vertical up
|}


After the five load steps were loaded, the development of the joint model cracks was obtained, as Figure 8:

==a)                               b)                             c)==

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{|
|-
| [[Image:draft_177_463205977-image17.jpeg|180px]]
| [[Image:draft_177_463205977-image18.jpeg|center|180px]]
|}
 [[Image:draft_177_463205977-image19.jpeg|180px]] </div>

Load step 1                   Load step2                     Load step 3

==d)                             e)==

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{|
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| [[Image:draft_177_463205977-image20.jpeg|180px]]
| [[Image:draft_177_463205977-image21.jpeg|center|174px]]
|}
</div>

Load step 4                   Load step 5

'''              Figure 8.''' Crack development of joint model under cyclic loading.

It can be seen from the figure, after loading the first load step, the lower part of beam head begins to crack. After the second load step, the upper part of the beam head also cracks. And the top of the column due to uniform force, there is the case of crushed. The third upward load step, the cracks in the core area of the joint develops, some part of column below the beam also begins to crack. And then apply the concentrated load step four, damage of beam head increases, part of column above the beam shows the phenomenon of cracking. The fifth loading step causes cracks in the middle of the beam, and the crevices at the beam end is because that the concentrated load is applied directly to the nodes of the beam end.

After the fifth load step is loaded, the stress nephograms of the model along the beam extension direction (the absolute value of the stress in this direction is the maximum) are as follows:

 
{|
|-
| [[Image:draft_177_463205977-image22.jpeg|294px]]
| [[Image:draft_177_463205977-image23.jpeg|center|294px]]
|}


'''Figure 9.''' Stress nephogram of isometric view        '''Figure 10.''' Stress nephogram of lateral view under cyclic loading.

under cyclic loading.

As shown in the figures, the stress at the compression side of the beam is greater, and the stress at the beam head is the largest. The stress at the column part of the joint increases due to the squeezing action, and the stress contour is obliquely developed. In contrast to the stress distribution, there are cracks on the stretch side of beam and column , which is due to the fact that the tensile strength of the concrete is much smaller than the compressive strength.

''4.2. Monotonic loading''

As in the case of cyclic loading, a vertical downward load of 850KN is applied to the top surface of the column. Then apply a concentrated load in the vertical downward direction at the end of the beam, and gradually increase the load until the joint is destroyed.

After the finite element simulation, it can be seen that when the load reaches 55KN, some of the elements at the beam head are completely destroyed, as shown below:

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:draft_177_463205977-image24.jpeg|288px]] </div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
'''Figure 11.''' The most severely damaged part of the joint under monotonic loading.</div>

In the figure above, the appearance of the rhombus represents the complete destruction of the element.

And figure 12 shows the development of joint cracks as the load increases:

==a)                             b)                              c)==

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{|
|-
| [[Image:draft_177_463205977-image25.jpeg|180px]]
| [[Image:draft_177_463205977-image26.jpeg|center|180px]]
|}
 [[Image:draft_177_463205977-image27.jpeg|180px]] </div>

=10KN                           20KN                          30KN=

==   d)                              e)                              f)==

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
{|
|-
| [[Image:draft_177_463205977-image28.jpeg|180px]]
| [[Image:draft_177_463205977-image29.jpeg|center|180px]]
|}
 [[Image:draft_177_463205977-image30.jpeg|186px]] </div>

=40KN                           50KN                           60KN=

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
'''Figure 12.''' Crack development joint model under monotonic loading.</div>

<span id='translation_sen_id-1'></span>It can be seen from the figure that the cracks first appeared in the beam head, and then developed to the column side and beam end. As the load increases, the completely damaged elements also first appeared at the beam head. Due to the stretch effect, the column above the beam appears more cracks than the below part. The complete damage mode of the lower part of beam head is crushed.

Under the monotonic loading, the relationship between the displacement of the beam end and the load is shown in the following figure:

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:draft_177_463205977-image31.jpeg|258px]] </div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
'''Figure 13. '''Curve of beam end displacement and load under monotonic loading.</div>

When the load is applied to 55kN，the stress nephogram of the model along the beam extension  direction (the absolute value of the stress in this direction is the maximum) are as follows:

 
{|
|-
| [[Image:draft_177_463205977-image32.jpeg|306px]]
| [[Image:draft_177_463205977-image33.jpeg|center|294px]]
|}


'''Figure 14.''' Stress nephogram of isometric view                  '''Figure 15.''' Stress nephograms of lateral view

under monotonic loading.                                   under monotonic loading.

As shown in the figures, the stress at the compression side of the beam is greater. And the stress at the beam head is the largest, which exceeds the standard value of axial compressive strength of C30 concrete. The stress in the column part below the beam is larger due to the squeezing action.

=5. Conclusions=

In this paper, the nonlinear finite element analysis of an L-shaped column joint model is carried out to study the development of cracks and stress distribution of the joint under cyclic loading and monotonic loading. The following conclusions can be drawn:

1. The L-shaped frame column satisfies the design principle of "strong column and weak beam". Whether the joint is loaded repeatedly or monotonically, its failure mode is beam failure, and the plastic hinge first appears at the beam head.

2.The constitutive relation of concrete used in this paper is feasible for nonlinear simulation.

<span id='translation_sen_id-3'></span>3.In order to ensure that the L-shaped column joints are not prematurely damaged, more steel bars should be set in the beam heads.

==References==

[1] J Marin．Design Aids for L-Shaped Reinforced Concrete Columns. Journal of the American Concrete Institute, 1979, 76( 11): 1197-1216.

[2] LN Ramamurthy，TAH Khan．L-Shaped Column Design for Biaxial Eccentric. Journal of STructural Engineering, 1983, 109( 8) : 1903-1917.

[3] Mallikarjuna, P Mahadevappa. Computer-aided Analysis of Reinforced-Concrete Columns Subjected to Axial-Compression and Bending. 1. L-Shaped Sections. Computers & Structures, 1992, 44(5): 1121-1138.

[4] J Feng, Q Chen, Z Li. Experiment Study on Shear Strength of L-Shaped Columns . Journal of South China University of Technology (Natural Science), 1995, 23(1): 68-75.

[5] W Cao , et al. Behavior of  “L” shape Column under Different Directional Cyclic Loading. Earthquake Engineering and Engineering Vibration, 1995, 15(1): 67-72.

[6] D Wang, C Huang, et al. Research on Experiment and Design Method of Specially-Shaped Columns Under Eccentric Loading. Journal of Building Structures, 2001, 22(5): 37-42.

[7] Z Wang , et al. Experimental Study on Seismic Performance of Prefabricated L-Shaped Column Frame Joints. Journal of Nanjing University (Natural Science Edition), 2016, 38 (3): 76-80.

[8] S Yan , et al. Comprehension and Application of Technical Specification of Concrete Special-Shaped Column. Beijing: China Architecture & Building Press, 2007.

[9] China Academy of Building Research. GB50010-2010. Code for Design of Concrete Structures. Beijing: China Architecture & Building Press, 2010.

[10] J Jiang, X Lu. Finite Element Analysis of Concrete Structures (Second Edition). Beijing: Tsinghua University Press, 2013.

[11] Z He , J Ou . Nonlinear analysis of reinforced concrete structures. Harbin: Harbin Institute of Technology Press, 2007.

[12] F Legeron , P Paultre , J Mazars . Damage Mechanics Modeling of Nonlinear Seismic Behavior of Concrete Structures. Journal of Structural Engineering - ASCE, 2005, 131(6):949-954.

[13] ANSYS, Inc. ANSYS structural Analysis Guide Release 8.0. SAP, IP Inc, 2000.

[14] KJ Willam, EP Warnke. Constitutive Model for the Triaxial Behavior of Concrete // Proceedings of International Association for Bridge and Structure Engineering. 1975, Vol 19. ISMES, Bergamo, Italy.

[15] BP Sinha , KH Gerstle, LG Tulin. Stress-Strain Relations for Concrete under Cyclic Loading. Journal of American Concrete Institute，1964, 62(2): 195-210.

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