Liu et al 2017a - Revision history

Rimni at 11:05, 30 August 2021

2021-08-30T11:05:53Z

Rimni: /* 3. Finite element calculation model */

2021-08-30T10:14:26Z

‎3. Finite element calculation model

Rimni at 12:11, 14 May 2020

2020-05-14T12:11:20Z

Rimni at 12:11, 14 May 2020

2020-05-14T12:11:00Z

Rimni at 11:08, 14 May 2020

2020-05-14T11:08:03Z

Rimni at 10:59, 14 May 2020

2020-05-14T10:59:54Z

Rimni at 10:49, 14 May 2020

2020-05-14T10:49:53Z

Rimni at 10:47, 14 May 2020

2020-05-14T10:47:23Z

Rimni at 10:35, 14 May 2020

2020-05-14T10:35:05Z

Rimni at 10:34, 14 May 2020

2020-05-14T10:34:36Z

@@ Line 346: / Line 346: @@
 ===4.1 Low-cycle repeated loading===
-Constrain the finite element nodes at the top and bottom of the column, and apply a uniform downward load of 858 KN on the top of the column. Apply a concentrated load in the vertical direction to the overhanging beam end, sequentially changing the direction of the load (up or down) in each load step, and gradually increase the concentrated load in the steps [19,20].
+Constrain the finite element nodes at the top and bottom of the column, and apply a uniform downward load of 858 KN on the top of the column. Apply a concentrated load in the vertical direction to the overhanging beam end, sequentially changing the direction of the load (up or down) in each load step, and gradually increase the concentrated load in the steps [19,20] ([[#tab-5|Table 5]]).
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
 '''Table 5. '''Loading situation</div>
+<div id='tab-5'></div>
 {| style="width: 55%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
 |-
@@ Line 376: / Line 376: @@
-After the five load steps, the formation and development of cracks in the joint model can be got (Figure 8).
+After the five load steps, the formation and development of cracks in the joint model can be got ([[#img-8|Figure 8]]).
-−
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 403: / Line 403: @@
 It can be seen from the figure that after the first load step has been loaded, the cracks at the upper part of the beam head first appear due to the tensile stress. After the second load step is applied, there are cracks in the lower part of the beam head. In the third downward concentrated load step, the cracks develop to the column limb, the column part above the beam begins to crack because of the pulling force. The fourth load step is upward, the damage of the beam is aggravated, and the lower part of the column limb emerges the cracking phenomenon. With the fifth load step, fractures occur in the middle of the beam, the junction of beam and column limb is the most devastated, and the top of the column also fractures due to the uniform force.
-After the five load steps were loaded, the nephogram of major principal stress which is along the x-axis direction can be obtained (Figure 9).
+After the five load steps were loaded, the nephogram of major principal stress which is along the x-axis direction can be obtained ([[#img-9|Figure 9]]).
+<div id='img-9'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 425: / Line 425: @@
 In monotonic loading situation''', '''a vertically downward concentrated load is applied at the beam end and gradually increases until some elements of the model are destroyed [19,21].
-Figure 10 shows the development of joint cracks with increasing load.
+[[#img-10|Figure 10]] shows the development of joint cracks with increasing load.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 453: / Line 454: @@
 When the load is added to 50KN, a element at beam head is completely damaged.
-Figure 11 is enlarged views of the joint when the load force is 50KN, the rhombus appearing in a element means the element is completely destroyed. The rhombus is marked out by a red circle.
+[[#img-11|Figure 11]] is enlarged views of the joint when the load force is 50KN, the rhombus appearing in a element means the element is completely destroyed. The rhombus is marked out by a red circle.
+<div id='img-11'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 467: / Line 468: @@
-The relationship between the displacement of the beam end and the force under monotonic loading can be got as shown in Figure 12.
+The relationship between the displacement of the beam end and the force under monotonic loading can be got as shown in [[#img-12|Figure 12]].
+<div id='img-12'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 477: / Line 478: @@
-When the beam end is loaded to 50KN, the joint nephogram of major principal stress which is also along the ''x''-axis direction can be acquired as shown in Figure 13.
+When the beam end is loaded to 50KN, the joint nephogram of major principal stress which is also along the ''x''-axis direction can be acquired as shown in [[#img-13|Figure 13]].
+<div id='img-13'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-

@@ Line 107: / Line 107: @@
 ===3.1 Joint model===
-The calculation model in this paper is the frame joint of T-shaped column which is the side column of special-shaped column frame structure [2], as follows.
+The calculation model in this paper is the frame joint of T-shaped column which is the side column of special-shaped column frame structure [2], as follows ([[#img-1|Figures 1]] to [[#img-4|4]]).
+<div id='img-1'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 116: / Line 116: @@
 |}
+<div id='img-2'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 124: / Line 124: @@
 |}
+<div id='img-3'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 132: / Line 132: @@
 |}
+<div id='img-4'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 145: / Line 145: @@
 ===3.2 Mechanical properties of materials===
-Mechanical properties of C30 concrete are shown in Table 1 [16].
+Mechanical properties of C30 concrete are shown in [[#tab-1|Table 1]] [16].
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
 '''Table 1.''' Concrete mechanical properties</div>
+<div id='tab-1'></div>
 {| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
 |-
@@ Line 166: / Line 166: @@
-The mechanical properties of HRB335 and HPB300 bars are as follows (Table 2).
+The mechanical properties of HRB335 and HPB300 bars are as follows ([[#tab-2|Table 2]]).
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
 '''Table 2. '''Steel mechanical properties</div>
+<div id='tab-2div>
 {| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
 |-
@@ Line 249: / Line 249: @@
 |}
-In the formulas above, <math>{d_c} </math> is the uniaxial compressive damage evolution parameter of concrete; <math>\alpha _{c}</math> is the descending section reference value of uniaxial compressive stress-strain curve of concrete, taking 0.74 for C30 concrete in this paper; <math>f_{c,r}</math> is the representative value of uniaxial compressive strength of concrete, taking 20.1N / mm<sup>2</sup>; <math>\varepsilon _{c,r}</math> is peak compressive strain of concrete corresponding to <math>f_{c,r}</math>, taking <math>1.47\times 10^3</math>. When the above values are put into the formula operation, the stress corresponding to a certain strain can be obtained, as shown in the following table:
+In the formulas above, <math>{d_c} </math> is the uniaxial compressive damage evolution parameter of concrete; <math>\alpha _{c}</math> is the descending section reference value of uniaxial compressive stress-strain curve of concrete, taking 0.74 for C30 concrete in this paper; <math>f_{c,r}</math> is the representative value of uniaxial compressive strength of concrete, taking 20.1N / mm<sup>2</sup>; <math>\varepsilon _{c,r}</math> is peak compressive strain of concrete corresponding to <math>f_{c,r}</math>, taking <math>1.47\times 10^3</math>. When the above values are put into the formula operation, the stress corresponding to a certain strain can be obtained, as shown in the [[#tab-3|Table 3]].
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
@@ Line 270: / Line 270: @@
-Drawing concrete four-fold line stress-strain relationship, as follows:
+Drawing concrete four-fold line stress-strain relationship, as follows ([[#img-5|Figure 5]]).
+<div id='img-5'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 288: / Line 288: @@
 The non-linear simulation in this paper obeys the von Mises yield criterion and multilinear isotropic hardening model [14,15].
-Model the joint and mesh it with tetrahedrons as below:
+Model the joint and mesh it with tetrahedrons as below ([[#img-6|Figures 6]] and [[#img-7|7]]).
+<div id='img-6'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 297: / Line 297: @@
 |}
+<div id='img-7'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 306: / Line 306: @@
-The reinforcement ratios of each part of the joint are as follows (Table 4).
+The reinforcement ratios of each part of the joint are as follows ([[#tab-4|Table 4]]).
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
 '''Table 4. '''Joint reinforcement ratios</div>
+<div id='tab-4'></div>
 {| style="width: 51%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;"
 |-

@@ Line 493: / Line 493: @@
 As shown, due to the stress concentration effect, the stress in the junction of beam and column limb is greater. And the stress in the compression side of beam head is the largest, which exceeds the compressive strength standard value of the concrete.
-==5 Conclusions==
+==5. Conclusions==
 In this paper, by the elasto-plastic finite element analysis of a T-shaped column joint model, the crack development and the stress distribution of the joint under repeated and monotonic loading were observed， its failure mode was studied. The following conclusions can be drawn:

@@ Line 508: / Line 508: @@
 [1] JGJ 149-2006. Technical specification for concrete structures with specially shaped columns[S]. Beijing: China Architecture & Building Press, 2006.
-[2] S.C. YAN, et al. Comprehension and application of technical specification of concrete special-shaped column [M]. Beijing: China Architecture & Building Press, 2007.
+[2] Yan S.C., et al. Comprehension and application of technical specification of concrete special-shaped column. Beijing: China Architecture & Building Press, 2007.
-[3] L.N. Lowes, N. Mitra. A beam-column joint model for simulating the earthquake response of reinforced concrete frames[R]. Pacific Earthquake Engineering Research Center, University of California at Berkeley, 2004.
+[3]  Lowes L.N., Mitra N. A beam-column joint model for simulating the earthquake response of reinforced concrete frames. Pacific Earthquake Engineering Research Center, University of California at Berkeley, 2004.
-[4]L. Li, J.B. Mander, R.P. Dhakal. Bidirectional cyclic loading experiment on a 3D beam-column joint designed for damage avoidance[J]. Journal of Structural Engineering, 134(11):1733-1742, 2008.
+[4] Li L.,  Mander J.B.,  Dhakal R.P. Bidirectional cyclic loading experiment on a 3D beam-column joint designed for damage avoidance. Journal of Structural Engineering, 134(11):1733-1742, 2008.
-[5] N. Hooda, J. Narwal, et al. An experimental investigation on structural behaviour of beam column joint[J]. International Journal of Innovative Technology & Exploring Engineering,  3(3):84-88, 2013.
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-[8] L.N. Ramamurthy, T.A.H. Khan. L-shaped column design for biaxial eccentric[J]. Journal of Structural Engineering, 109(8):1903-1917, 1983.
+[8]  Ramamurthy L.N.,  Khan T.A.H. L-shaped column design for biaxial eccentric. Journal of Structural Engineering, 109(8):1903-1917, 1983.
-[9] C.T.T. Hsu. T-shaped reinforced concrete members under biaxial bending and axial compression [J]. ACI Structural Journal，86(4):460-468, 1989．
+[9]  Hsu C.T.T. T-shaped reinforced concrete members under biaxial bending and axial compression. ACI Structural Journal, 86(4):460-468, 1989．
-[10] Mallikarjuna, P. Mahadevappa. Computer-aided analysis of reinforced concrete columns subjected to axial compression and bending. Part II: T-shaped sections [J]. Computers & Structures, 53(6):1317-1356, 1994.
+[10] Mallikarjuna, Mahadevappa P.  Computer-aided analysis of reinforced concrete columns subjected to axial compression and bending. Part II: T-shaped sections. Computers & Structures, 53(6):1317-1356, 1994.
-[11] W.L. Cao, S.L. Gao, et al. Nonlinear analysis of T-shaped reinforced concrete columns[J]. World Information On Earthquake Engineering, (1):32-37, 1996.
+[11]  Cao W.L.,  Gao S.L., et al. Nonlinear analysis of T-shaped reinforced concrete columns. World Information On Earthquake Engineering, (1):32-37, 1996.
-[12] D. Wang, C.K. Huang, et al. Research on experiment and design method of specially-shaped columns under eccentric loading [J]. Journal of Building Structures, 22(5):37-42, 2001.
+[12]  Wang D.,  Huang C.K., et al. Research on experiment and design method of specially-shaped columns under eccentric loading. Journal of Building Structures, 22(5):37-42, 2001.
-[13]C. H. Chen, H. Gong, et al. Experimental study and nonlinear finite element analysis of concrete frame joint with T-shaped columns[J]. Industrial Construction, 47(3):76-82,88, 2017.
+[13] Chen C.H., Gong H., et al. Experimental study and nonlinear finite element analysis of concrete frame joint with T-shaped columns. Industrial Construction, 47(3):76-82,88, 2017.
-[14] ANSYS, Inc. ANSYS structural analysis guide release 8.0[S]. SAP, IP Inc, 2000.
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+[15]  Jiang J.J.,  Lu X.Z. Finite element analysis of concrete structures (Second edition). Beijing: Tsinghua University Press, 2013.
-[16] GB50010-2010. Code for design of concrete structures [S]. Beijing: China Architecture & Building Press, 2010.
+[16] GB50010-2010. Code for design of concrete structures. Beijing: China Architecture & Building Press, 2010.
-[17] Z. He, J.P. OU. Nonlinear analysis of reinforced concrete structures [M]. Harbin: Harbin Institute of Technology Press, 2007.
+[17]  He Z.,   Ou J.P. Nonlinear analysis of reinforced concrete structures. Harbin: Harbin Institute of Technology Press, 2007.
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-[19] P. Jun, V. Mechtcherine. Behaviour of strain-hardening cement-based composites (SHCC) under monotonic and cyclic tensile loading: Part 1- Experimental investigations[J]. Cement & Concrete Composites, 32(10):801-809, 2010.
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-[20] H. Ma, X.Y. Sun, et al. Finite Element Analysis of Reinforced Concrete Beam-Column Joint under Low Cyclic Loading with Different Load Systems[J]. Applied Mechanics & Materials, 351-352:351-354, 2013.
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-[21] I.N. Doudoumis. Finite element modelling and investigation of the behaviour of elastic infilled frames under monotonic loading[J]. Engineering Structures, 29(6):1004-1024, 2007.
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 </div>

@@ Line 445: / Line 445: @@
 |colspan="2" style="text-align: center;font-size:75%;"|(e) 50KN
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" style="padding:10px;"| '''Figure 10.''' Crack growth pattern under monotonic loading
+| colspan="2" style="padding:10px;"| '''Figure 10'''. Crack growth pattern under monotonic loading
 |}
@@ Line 458: / Line 458: @@
 |-
 |  style="text-align: center;padding:10px;"| [[Image:review_Liu_2017c-image32.png|300px]]
-|style="text-align: center;padding:10px; [[Image:review_Liu_2017c-image33.jpeg|center|300px]]
+|style="text-align: center;padding:10px; |[[Image:review_Liu_2017c-image33.jpeg|center|300px]]
 |-
 |style="text-align: center;font-size:75%;"|(a) Destroyed part
 | style="text-align: center;font-size:75%;"|(b) Partial enlargement of the destroyed part
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" style="padding:10px;"| '''Figure 11.''' Completely destroyed part in the joint
+| colspan="2" style="padding:10px;"| '''Figure 11'''. Completely destroyed part in the joint
 |}
-The relationship between the displacement of the beam end and the force under monotonic loading can be got as as shown in Figure 12.
+The relationship between the displacement of the beam end and the force under monotonic loading can be got as shown in Figure 12.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
@@ Line 473: / Line 473: @@
 |style="padding:10px;"| [[Image:review_Liu_2017c-image34.jpeg|600px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 12.''' Curve of beam end displacement and load force under monotonic loading
+| colspan="1" style="padding-bottom:10px;"| '''Figure 12'''. Curve of beam end displacement and load force under monotonic loading
 |}
-When the beam end is loaded to 50KN, the joint nephogram of major principal stress which is also along the ''x''-axis direction can be acquired as follows:
+When the beam end is loaded to 50KN, the joint nephogram of major principal stress which is also along the ''x''-axis direction can be acquired as shown in Figure 13.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
@@ Line 487: / Line 487: @@
 |  style="text-align: center;font-size: 75%;"|(b) Side view
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" style="padding:10px;"| '''Figure 13. '''Major principal stress nephogram under the monotonic loading
+| colspan="2" style="padding:10px;"| '''Figure 13'''. Major principal stress nephogram under the monotonic loading
 |}

@@ Line 425: / Line 425: @@
 In monotonic loading situation''', '''a vertically downward concentrated load is applied at the beam end and gradually increases until some elements of the model are destroyed [19,21].
-Figure 10 shows the development of joint cracks with increasing load:
+Figure 10 shows the development of joint cracks with increasing load.
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
-−
-{|
 |-
-| style="text-align: center;"|[[Image:review_Liu_2017c-image27.png|center|270px]]
+|  style="text-align: center;padding:10px;"|[[Image:review_Liu_2017c-image27.png|center|270px]]
-| style="text-align: center;"|[[Image:review_Liu_2017c-image28.png|center|270px]]
+| style="text-align: center;padding:10px;"|[[Image:review_Liu_2017c-image28.png|center|270px]]
 |-
 | style="text-align: center;font-size:75%;"|(a) 10KN
 | style="text-align: center;font-size:75%;"|(b) 20KN
 |-
-| style="text-align: center;"|[[Image:review_Liu_2017c-image29.png|center|270px]]
+| style="text-align: center;padding:10px;"|[[Image:review_Liu_2017c-image29.png|center|270px]]
-| style="text-align: center;"|[[Image:review_Liu_2017c-image30.png|center|270px]]
+| style="text-align: center;padding:10px;"|[[Image:review_Liu_2017c-image30.png|center|270px]]
 |-
 | style="text-align: center;font-size:75%;"|(c) 30KN
 | style="text-align: center;font-size:75%;"|(d) 40KN
 |-
-|colspan="2" style="text-align: center;"|[[Image:review_Liu_2017c-image31.png|center|270px]]
+|colspan="2" style="text-align: center;padding:10px;"|[[Image:review_Liu_2017c-image31.png|center|270px]]
 |-
 |colspan="2" style="text-align: center;font-size:75%;"|(e) 50KN
-|}</div>
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 10.''' Crack growth pattern under monotonic loading
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
+|}
-'''Figure 10.''' Crack growth pattern under monotonic loading.</div>
@@ Line 456: / Line 453: @@
 When the load is added to 50KN, a element at beam head is completely damaged.
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+Figure 11 is enlarged views of the joint when the load force is 50KN, the rhombus appearing in a element means the element is completely destroyed. The rhombus is marked out by a red circle.
-{|
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-| style="text-align: center;"|[[Image:review_Liu_2017c-image32.png|300px]]
+|  style="text-align: center;padding:10px;"| [[Image:review_Liu_2017c-image32.png|300px]]
-|style="text-align: center;"| [[Image:review_Liu_2017c-image33.jpeg|center|300px]]
+|style="text-align: center;padding:10px; [[Image:review_Liu_2017c-image33.jpeg|center|300px]]
 |-
 |style="text-align: center;font-size:75%;"|(a) Destroyed part
 | style="text-align: center;font-size:75%;"|(b) Partial enlargement of the destroyed part
-|}</div>
+|- style="text-align: center; font-size: 75%;"
-Figure 11 is enlarged views of the joint when the load force is 50KN, the rhombus appearing in a element means the element is completely destroyed. The rhombus is marked out by a red circle.
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+When the beam end is loaded to 50KN, the joint nephogram of major principal stress which is also along the ''x''-axis direction can be acquired as follows:
-[[Image:review_Liu_2017c-image34.jpeg|600px]] </div>
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;>
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
-'''Figure 12.''' Curve of beam end displacement and load force under monotonic loading.</div>
-−
-−
-When the beam end is loaded to 50KN, the joint nephogram of major principal stress which is also along the x-axis direction can be acquired as follows:
-−
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-−
-{|
 |-
-| style="text-align: center;"|[[Image:review_Liu_2017c-image35.jpeg|288px]]
+|  style="text-align: center;padding:10px;"| [[Image:review_Liu_2017c-image35.jpeg|288px]]
-|style="text-align: center;"| [[Image:review_Liu_2017c-image36.jpeg|center|288px]]
+|  style="text-align: center;padding:10px;"| [[Image:review_Liu_2017c-image36.jpeg|center|288px]]
 |-
-|style="text-align: center;font-size:75%;"|(a) Isometric view
+|  style="text-align: center;font-size: 75%;"|(a) Isometric view
-| style="text-align: center;font-size:75%;"|(b) Side view
+|  style="text-align: center;font-size: 75%;"|(b) Side view
-|}</div>
+|- style="text-align: center; font-size: 75%;"
+| colspan="2" style="padding:10px;"| '''Figure 13. '''Major principal stress nephogram under the monotonic loading
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;>
+|}
-'''Figure 13. '''Major principal stress nephogram under the monotonic loading</div>

@@ Line 244: / Line 244: @@
 {| style="text-align: center; margin:auto;"
 |-
-| style="text-align: center;" | <math> x=\frac{\varepsilon }{\varepsilon _{c,r} }</math>
+| style="text-align: center;" | <math> x=\displaystyle\frac{\varepsilon }{\varepsilon _{c,r} }</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)
@@ Line 280: / Line 280: @@
-It can be seen that the elastic modulus of the concrete is 30000 MPa at the beginning. When the strain reaches <math>0.5\times 10^{-3}</math>, the elastic modulus reduces to 8000 MPa. And with the strain reaching <math>1.0\times 10^{-3}</math>, the elastic modulus decreases to 2000 MPa. Once the strain reaches <math>1.5\times 10^{3}</math>, the stress begins to drop. The constitutive relationship of the concrete has a "softening" decline.
+It can be seen that the elastic modulus of the concrete is 30000 MPa at the beginning. When the strain reaches <math>0.5\times 10^{-3}</math>, the elastic modulus reduces to 8000 MPa. And with the strain reaching <math>1.0\times 10^{-3}</math>, the elastic modulus decreases to 2000 MPa. Once the strain reaches <math>1.5\times 10^{-3}</math>, the stress begins to drop. The constitutive relationship of the concrete has a "softening" decline.
 ===3.4 The establishment of the joint model===

@@ Line 224: / Line 224: @@
 {| style="text-align: center; margin:auto;"
 |-
-| [[Image:review_Liu_2017c-image14.png|100px]]
+| style="text-align: center;" | <math> \rho _{c} =\displaystyle\frac{f_{c,r} }{E_{c} \varepsilon _{c,r} } </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (10)
@@ Line 234: / Line 234: @@
 {| style="text-align: center; margin:auto;"
 |-
-| [[Image:review_Liu_2017c-image15.png|150px]]
+| style="text-align: center;" | <math>n=\displaystyle\frac{E_c\varepsilon _{c,r} }{E_c\varepsilon _{c,r} -f_{c,r} }</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (11)
@@ Line 244: / Line 244: @@
 {| style="text-align: center; margin:auto;"
 |-
-| [[Image:review_Liu_2017c-image16.png|70px]]
+| style="text-align: center;" | <math> x=\frac{\varepsilon }{\varepsilon _{c,r} }</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)
 |}
+In the formulas above, <math>{d_c} </math> is the uniaxial compressive damage evolution parameter of concrete; <math>\alpha _{c}</math> is the descending section reference value of uniaxial compressive stress-strain curve of concrete, taking 0.74 for C30 concrete in this paper; <math>f_{c,r}</math> is the representative value of uniaxial compressive strength of concrete, taking 20.1N / mm<sup>2</sup>; <math>\varepsilon _{c,r}</math> is peak compressive strain of concrete corresponding to <math>f_{c,r}</math>, taking <math>1.47\times 10^3</math>. When the above values are put into the formula operation, the stress corresponding to a certain strain can be obtained, as shown in the following table:
-In the formulas above, ''d<sub>c</sub>'' is the uniaxial compressive damage evolution parameter of concrete; ''α<sub>c</sub>'' is the descending section reference value of uniaxial compressive stress-strain curve of concrete, taking 0.74 for C30 concrete in this paper; ''f<sub>c,r  </sub>''is the representative value of uniaxial compressive strength of concrete, taking 20.1N / mm<sup>2</sup>; ''ε<sub>c,r</sub>'' is peak compressive strain of concrete corresponding to ''f<sub>c,r</sub>'', taking 1.47×10<sup>-3</sup>. When the above values are put into the formula operation, the stress corresponding to a certain strain can be obtained, as shown in the following table:
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
@@ Line 281: / Line 280: @@
-It can be seen that the elastic modulus of the concrete is 30000 MPa at the beginning. When the strain reaches 0.5×10<sup>-3</sup>, the elastic modulus reduces to 8000 MPa. And with the strain reaching 1.0×10<sup>-3</sup>, the elastic modulus decreases to 2000 MPa. Once the strain reaches 1.5×10<sup>-3</sup>, the stress begins to drop. The constitutive relationship of the concrete has a "softening" decline.
+It can be seen that the elastic modulus of the concrete is 30000 MPa at the beginning. When the strain reaches <math>0.5\times 10^{-3}</math>, the elastic modulus reduces to 8000 MPa. And with the strain reaching <math>1.0\times 10^{-3}</math>, the elastic modulus decreases to 2000 MPa. Once the strain reaches <math>1.5\times 10^{3}</math>, the stress begins to drop. The constitutive relationship of the concrete has a "softening" decline.
 ===3.4 The establishment of the joint model===

@@ Line 211: / Line 211: @@
 {| style="text-align: center; margin:auto;"
 |-
-| style="text-align: center;" | <math> d_{c} =\left\{\begin{array}{ll} 1-\frac{\rho _{c} n}{n-1+x^{n} } \; \; \;  &  x\le 1 \\
+| style="text-align: center;" | <math> d_{c} =\left\{\begin{array}{ll} 1-\displaystyle\frac{\rho _{c} n}{n-1+x^{n} } \; \; \;  &  x\le 1 \\
--\frac{\rho _{c} }{\alpha _{c} (x-1)^{2} +x} \; \; \; & x>1 \end{array}\right. </math>
+-\displaystyle\frac{\rho _{c} }{\alpha _{c} (x-1)^{2} +x} \; \; \; & x>1 \end{array}\right. </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (9)