复合受力下钢筋混凝土构件承载力的统一表达
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摘要
在各种灾害作用下，钢筋混凝土构件的复杂受力实际是由轴力、弯矩、剪力、扭矩的不同组合形成，其破坏机理十分复杂。目前，除轴力、弯矩单一受力情况国内外有比较统一的计算方法外，对剪、扭加入引起的破坏理论并不统一。长期以来，国内外研究学者对钢筋混凝土构件在复合受力作用下的研究已取得许多成果。一些靠试验数据回归得到的承载力计算公式，如我国现行的钢筋混凝土结构设计规范，缺少统一的理论基础，各回归公式之间也不存在内在联系。多年来，众学者一直致力于钢筋混凝土构件组合受力统一破坏理论的研究,利用协调，平衡和本构关系对构件受复合力作用进行全过程分析，取得了一些领先的成果。目前仅有这种理论可以分析钢筋混凝土构件受拉-压弯剪扭作用的破坏机理，但由于在计算中引入协调条件使计算变得复杂，让工程界难以接受。一个成熟的理论，它应该在理论上是统一的，在表达上是简明的。总之，与相对完善和成熟的钢结构计算理论相比，钢筋混凝土构件在复合力作用下的破坏理论还不够成熟。
为方便工程应用，尤其是分析构件在灾难荷载作用下的破坏，本文针对量大面广的矩形断面的钢筋混凝土构件在轴力、弯矩、剪力、扭矩共同作用下的破坏提出了一个形式简单并且能展示构件破坏机理的统一理论。本文的理论利用了钢筋和混凝土的破坏准则，寻找到承载力极限状态下不违反边界条件且满足平衡条件的应力分布，从而得到构件承受复合力破坏时各个外力的相互关系。依据极限分析中下限解定理可知，本文得到的结果是一个偏于安全的极限下限解，本文寻找的极限应力状态越接近实际情况，这个下限解应该越接近真实解。
根据大量试验现象可知，构件在承受的轴压力相对较小时，构件的混凝土随着荷载的增大，首先出现裂缝，之后部分混凝土退出工作，剩余的混凝土和钢筋继续承担的荷载越来越大，直到构件破坏，此类破坏为延性破坏；而在构件承受轴压力相对较大时，在混凝土先被压碎的同时构件也随之完全破坏，几乎没有像延性破坏那样的破坏发展阶段，此类破坏为脆性破坏。本文在利用极限下限解定理建立统一破坏方程时，根据两类破坏模式的不同机理，分别建立了破坏方程。
在研究延性破坏时本文建立了如下的分析思路：首先需要寻找到构件在极限状态下的三维翘曲破坏面，其次描述出破坏面上的应力分布，最后根据平衡条件可以得到各个外力之间的相互关系。在研究极限状态下的破坏面时，本文提出了一种全新的组成破坏面各边角的计算方法，根据这个方法找到的破坏面不仅能够描述构件在受复合力作用时的三维翘曲破坏面，还可以在构件承受单一荷载时较为准确地退化成简单的破坏面。在寻找破坏面上的应力状态时，本文对钢筋和混凝土的贡献分别进行了详细的分析。通过对理论和试验研究的分析，本文总结出破坏面上的钢筋的贡献，由于不需借助协调条件进行计算，表达简单便于应用。破坏面上的混凝土分为受拉区和受压区。受拉区的混凝土由于骨料咬合的作用在开裂后可继续承担部分外力，本文提出了一种简便的计算方法，仅需一个可以简单计算的折减系数kT，便可量化受拉开裂区混凝土的贡献。对处于复杂应力状态且受压破坏的混凝土，可以用主应力空间混凝土三维破坏准则计算应力状态。这样，破坏面上的混凝土的应力分布得到了简单并较为准确的量化。由于构件承受复合力的不同，受压区不同，破坏面形态也不同，在分析不同破坏形态下的极限平衡条件后，构件受拉-压弯剪扭作用下的破坏方程可以用非常简明的方式表达出来，同时破坏方程还清晰的表现了各个外力间的相互关系。
脆性破坏的最明显的特征是，随着混凝土先被压坏构件也破坏，此时混凝土的应力状态直接决定着构件的状态。根据破坏机理，本文寻找到了构件受复合力作用时的正应力和剪应力的分布，当混凝土的应力状态满足主应力空间内三维破坏准则时，认为构件破坏。得到的分析方法计算简单，便于应用，并反映了构件破坏的最主要特征。
经和试验结果对比后发现，本文得到的破坏理论与实际破坏情况吻合较好。
本文用相同的分析方法对更复杂的受力情况，即矩形截面钢筋混凝土构件受双向拉-压弯剪扭作用时的破坏情况，进行了初步探索。和试验对比发现，本文理论取得了比较理想的结果。
与现行混凝土结构设计规范相比，统一破坏理论不仅计算出的极限承载力更为准确，还能展示出破坏时各个外力之间的相互作用关系。
与其他分析方法相比，能够分析拉-压弯剪扭共同作用的理论与数值方法都比十分繁琐，不适于工程中推广，而本文得到的统一破坏理论恰好弥补了这个工程应用中的空缺。
总之，本论文提出的矩形截面钢筋混凝土构件受复合力作用下的统一破坏理论不仅能够清晰合理的阐释构件的破坏机理，并能快速便捷的分析出构件的极限承载力和破坏模式，对于推广到工程实际应用中有重要的参考意义。
Failures of reinforced concrete (RC) structures in hazards are mainly caused bydifferent combinations of axial loads, bending, shear force and torsion, and failuremechanisms are very complex. While RC elements under axial loads and bending areunderstood uniquely, different theories have been developed to demonstrate failuremechanisms of RC elements subjected to shear and torsion. During decades, dozens ofmethods have been achieved to analyze structures under one or several external loads.Some methods provide equations obtained by regression analyses of experimental results,such as "Code for design of concrete structures"(GB50010-2010). Apparently, equationsbased on empirical regressions have no theoretical basis, and few connections amongequations can be observed. For many years, researchers such as Professor Hsu have beenworking on a unified theory using compatibility conditions, equilibrium conditions, andconstitutive laws. Currently, only this approach can analyze structures subjected to axialloads, bending, shear force, and torsion at any loading stage and give out accurate results,but in practice it seems too complex and lengthy to be used. A mature theory inengineering should be very rigorous and unified in theory, convenient and simple inexpression, and also related parameters should be easily determined by traditional testmethods. Comparing with the theory used in steel structures, existing theories for RCmembers under complex external loads are still immature.
A simple unified theory is proposed in this thesis for widely used rectangularreinforced concrete elements subjected to axial loads, bending, shear force and torsion.This simple-formed theory is able to demonstrate failure mechanisms of structures subjected to complex loads. By using failure criteria of steel and concrete, stressdistributions at ultimate limit state satisfying equilibrium conditions can be describedwithout violating boundary conditions. Therefore, the interaction relationship amongexternal loads can be achieved. According to the Lower-bound Theorem, the obtainedsolution is a lower-bound limit analysis. If described stress distributions are rather close tothe realist limit state, the solution would be fairly accurate.
As observed in experiments, if the applied axial compression is relatively small, afterthe appearing of cracks, part of concrete can not contribute any more, reinforcing steelsand remaining concrete continue to carry more external loads after cracking until failureoccurs, and the failure mechanism is ductile failure. However, if the axial compression israther large, elements would fail instantly when stresses of concrete reach the failurecriterion and concrete is crushed, the stress development is different from the developmentof ductile failure, and the stress state at failure is called the brittle failure limit state. Whileestablishing a unified theory based on the Lower-bound Theorem, each failure mechanismare studied separately, and different interaction equations are derived.
The strategy of analyzing ductile failure is established as follows. At first, the warped3-dimensional failure surface is required, then stress distributions on the failure surfaceshould be described, and finally equilibrium conditions would lead to the interactionrelationship among external loads. While analyzing the failure surface, this thesis providesa new method to calculate inclinations of significant cracks. With this new method, notonly the warped failure surface of a reinforced concrete element subjected to complexexternal loads can be described, it can also rather accurately describe the failure surfacewhen an element is only subjected to only one kind of external load. During finding stressdistributions on the failure surface, contributions of concrete and steels are analyzedseparately. Based on theories and experiments, contributions of steels exposed on the failure surface can be presumed. Since compatibility conditions are not considered duringthe process, the expression of steel contribution is simple and practical. At failure thenormal-section of concrete will be divided into two parts, a tension zone and acompression zone. Concrete in tension zone reaches the failure criterion first, cracks duringloading, and still contributes after cracking because of aggregates’ interlock. A newapproach to calculate the contribution of concrete in tension zone is proposed. Byintroducing a reduction coefficient kT, which can be easily obtained, the contribution ofconcrete in tension zone can be quantified. The complicated stress state of concrete incompression zone can be determined using the3-dimensional failure criterion. Therefore,the stress distribution of concrete at failure can be easily quantified. Different failuremechanisms are distinguished by the location of compression zone caused by differentcombinations of external loads, and after analyzing equilibrium conditions of all failuremechanisms, the interaction equation can be simply expressed and also can demonstratethe relationship among different external loads.
In brittle failure analysis, concrete fails first, and failures of RC elements happen atthe same time. The stress state of concrete decides the condition of element. According tothe failure mechanism, stress distributions at this moment can be presumed based onexperimental results. When stresses of concrete reach the3-dimensional failure criterion ofconcrete, failure occurs, and then the interaction relationship can be expressed. Throughthe proposed simple and practical analysis, the main characteristic of failure mechanisms isreflected.
Then, reinforced concrete elements subjected to axial loads, biaxial bending, biaxialshear, and torsion are analyzed using the same strategy. After the comparison with testresults, the preliminary theory can rather accurately predict ultimate bearing capacities.
After comparing theoretical results with experimental results, a relatively good agreement has been found. After the comparison with "Code for design of concretestructures"(GB50010-2010), the proposed theory is found to be more accurate and morecapable of explaining failure mechanisms.
The problem of current approaches that can explain failure mechanisms of RCelements subjected to axial load, bending, shear force, and torsion and numerical analysesis that they all require complex computation and they are not suitable for engineeringanalyses. Comparing to other methods, the proposed unified theory can be identified as thefilling of this vacancy in engineering.
All in all, the proposed unified theory of rectangular reinforced concrete elementssubjected to axial loads, bending, shear, and torsion not only can explicitly explain failuremechanisms, ultimate bearing capacities can be easily calculated and the failure mode canbe quickly determined. Overall, this theory is significative for the analysis of reinforcedconcrete structures in engineering.
为方便工程应用，尤其是分析构件在灾难荷载作用下的破坏，本文针对量大面广的矩形断面的钢筋混凝土构件在轴力、弯矩、剪力、扭矩共同作用下的破坏提出了一个形式简单并且能展示构件破坏机理的统一理论。本文的理论利用了钢筋和混凝土的破坏准则，寻找到承载力极限状态下不违反边界条件且满足平衡条件的应力分布，从而得到构件承受复合力破坏时各个外力的相互关系。依据极限分析中下限解定理可知，本文得到的结果是一个偏于安全的极限下限解，本文寻找的极限应力状态越接近实际情况，这个下限解应该越接近真实解。
根据大量试验现象可知，构件在承受的轴压力相对较小时，构件的混凝土随着荷载的增大，首先出现裂缝，之后部分混凝土退出工作，剩余的混凝土和钢筋继续承担的荷载越来越大，直到构件破坏，此类破坏为延性破坏；而在构件承受轴压力相对较大时，在混凝土先被压碎的同时构件也随之完全破坏，几乎没有像延性破坏那样的破坏发展阶段，此类破坏为脆性破坏。本文在利用极限下限解定理建立统一破坏方程时，根据两类破坏模式的不同机理，分别建立了破坏方程。
在研究延性破坏时本文建立了如下的分析思路：首先需要寻找到构件在极限状态下的三维翘曲破坏面，其次描述出破坏面上的应力分布，最后根据平衡条件可以得到各个外力之间的相互关系。在研究极限状态下的破坏面时，本文提出了一种全新的组成破坏面各边角的计算方法，根据这个方法找到的破坏面不仅能够描述构件在受复合力作用时的三维翘曲破坏面，还可以在构件承受单一荷载时较为准确地退化成简单的破坏面。在寻找破坏面上的应力状态时，本文对钢筋和混凝土的贡献分别进行了详细的分析。通过对理论和试验研究的分析，本文总结出破坏面上的钢筋的贡献，由于不需借助协调条件进行计算，表达简单便于应用。破坏面上的混凝土分为受拉区和受压区。受拉区的混凝土由于骨料咬合的作用在开裂后可继续承担部分外力，本文提出了一种简便的计算方法，仅需一个可以简单计算的折减系数kT，便可量化受拉开裂区混凝土的贡献。对处于复杂应力状态且受压破坏的混凝土，可以用主应力空间混凝土三维破坏准则计算应力状态。这样，破坏面上的混凝土的应力分布得到了简单并较为准确的量化。由于构件承受复合力的不同，受压区不同，破坏面形态也不同，在分析不同破坏形态下的极限平衡条件后，构件受拉-压弯剪扭作用下的破坏方程可以用非常简明的方式表达出来，同时破坏方程还清晰的表现了各个外力间的相互关系。
脆性破坏的最明显的特征是，随着混凝土先被压坏构件也破坏，此时混凝土的应力状态直接决定着构件的状态。根据破坏机理，本文寻找到了构件受复合力作用时的正应力和剪应力的分布，当混凝土的应力状态满足主应力空间内三维破坏准则时，认为构件破坏。得到的分析方法计算简单，便于应用，并反映了构件破坏的最主要特征。
经和试验结果对比后发现，本文得到的破坏理论与实际破坏情况吻合较好。
本文用相同的分析方法对更复杂的受力情况，即矩形截面钢筋混凝土构件受双向拉-压弯剪扭作用时的破坏情况，进行了初步探索。和试验对比发现，本文理论取得了比较理想的结果。
与现行混凝土结构设计规范相比，统一破坏理论不仅计算出的极限承载力更为准确，还能展示出破坏时各个外力之间的相互作用关系。
与其他分析方法相比，能够分析拉-压弯剪扭共同作用的理论与数值方法都比十分繁琐，不适于工程中推广，而本文得到的统一破坏理论恰好弥补了这个工程应用中的空缺。
总之，本论文提出的矩形截面钢筋混凝土构件受复合力作用下的统一破坏理论不仅能够清晰合理的阐释构件的破坏机理，并能快速便捷的分析出构件的极限承载力和破坏模式，对于推广到工程实际应用中有重要的参考意义。
Failures of reinforced concrete (RC) structures in hazards are mainly caused bydifferent combinations of axial loads, bending, shear force and torsion, and failuremechanisms are very complex. While RC elements under axial loads and bending areunderstood uniquely, different theories have been developed to demonstrate failuremechanisms of RC elements subjected to shear and torsion. During decades, dozens ofmethods have been achieved to analyze structures under one or several external loads.Some methods provide equations obtained by regression analyses of experimental results,such as "Code for design of concrete structures"(GB50010-2010). Apparently, equationsbased on empirical regressions have no theoretical basis, and few connections amongequations can be observed. For many years, researchers such as Professor Hsu have beenworking on a unified theory using compatibility conditions, equilibrium conditions, andconstitutive laws. Currently, only this approach can analyze structures subjected to axialloads, bending, shear force, and torsion at any loading stage and give out accurate results,but in practice it seems too complex and lengthy to be used. A mature theory inengineering should be very rigorous and unified in theory, convenient and simple inexpression, and also related parameters should be easily determined by traditional testmethods. Comparing with the theory used in steel structures, existing theories for RCmembers under complex external loads are still immature.
A simple unified theory is proposed in this thesis for widely used rectangularreinforced concrete elements subjected to axial loads, bending, shear force and torsion.This simple-formed theory is able to demonstrate failure mechanisms of structures subjected to complex loads. By using failure criteria of steel and concrete, stressdistributions at ultimate limit state satisfying equilibrium conditions can be describedwithout violating boundary conditions. Therefore, the interaction relationship amongexternal loads can be achieved. According to the Lower-bound Theorem, the obtainedsolution is a lower-bound limit analysis. If described stress distributions are rather close tothe realist limit state, the solution would be fairly accurate.
As observed in experiments, if the applied axial compression is relatively small, afterthe appearing of cracks, part of concrete can not contribute any more, reinforcing steelsand remaining concrete continue to carry more external loads after cracking until failureoccurs, and the failure mechanism is ductile failure. However, if the axial compression israther large, elements would fail instantly when stresses of concrete reach the failurecriterion and concrete is crushed, the stress development is different from the developmentof ductile failure, and the stress state at failure is called the brittle failure limit state. Whileestablishing a unified theory based on the Lower-bound Theorem, each failure mechanismare studied separately, and different interaction equations are derived.
The strategy of analyzing ductile failure is established as follows. At first, the warped3-dimensional failure surface is required, then stress distributions on the failure surfaceshould be described, and finally equilibrium conditions would lead to the interactionrelationship among external loads. While analyzing the failure surface, this thesis providesa new method to calculate inclinations of significant cracks. With this new method, notonly the warped failure surface of a reinforced concrete element subjected to complexexternal loads can be described, it can also rather accurately describe the failure surfacewhen an element is only subjected to only one kind of external load. During finding stressdistributions on the failure surface, contributions of concrete and steels are analyzedseparately. Based on theories and experiments, contributions of steels exposed on the failure surface can be presumed. Since compatibility conditions are not considered duringthe process, the expression of steel contribution is simple and practical. At failure thenormal-section of concrete will be divided into two parts, a tension zone and acompression zone. Concrete in tension zone reaches the failure criterion first, cracks duringloading, and still contributes after cracking because of aggregates’ interlock. A newapproach to calculate the contribution of concrete in tension zone is proposed. Byintroducing a reduction coefficient kT, which can be easily obtained, the contribution ofconcrete in tension zone can be quantified. The complicated stress state of concrete incompression zone can be determined using the3-dimensional failure criterion. Therefore,the stress distribution of concrete at failure can be easily quantified. Different failuremechanisms are distinguished by the location of compression zone caused by differentcombinations of external loads, and after analyzing equilibrium conditions of all failuremechanisms, the interaction equation can be simply expressed and also can demonstratethe relationship among different external loads.
In brittle failure analysis, concrete fails first, and failures of RC elements happen atthe same time. The stress state of concrete decides the condition of element. According tothe failure mechanism, stress distributions at this moment can be presumed based onexperimental results. When stresses of concrete reach the3-dimensional failure criterion ofconcrete, failure occurs, and then the interaction relationship can be expressed. Throughthe proposed simple and practical analysis, the main characteristic of failure mechanisms isreflected.
Then, reinforced concrete elements subjected to axial loads, biaxial bending, biaxialshear, and torsion are analyzed using the same strategy. After the comparison with testresults, the preliminary theory can rather accurately predict ultimate bearing capacities.
After comparing theoretical results with experimental results, a relatively good agreement has been found. After the comparison with "Code for design of concretestructures"(GB50010-2010), the proposed theory is found to be more accurate and morecapable of explaining failure mechanisms.
The problem of current approaches that can explain failure mechanisms of RCelements subjected to axial load, bending, shear force, and torsion and numerical analysesis that they all require complex computation and they are not suitable for engineeringanalyses. Comparing to other methods, the proposed unified theory can be identified as thefilling of this vacancy in engineering.
All in all, the proposed unified theory of rectangular reinforced concrete elementssubjected to axial loads, bending, shear, and torsion not only can explicitly explain failuremechanisms, ultimate bearing capacities can be easily calculated and the failure mode canbe quickly determined. Overall, this theory is significative for the analysis of reinforcedconcrete structures in engineering.
引文
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[2] Y. Xie, S. H. Ahmad, T. Yu等. Shear ductility of reinforced concrete beams of normal andhigh-strength concrete[J]. ACI Structural Journal.1994.91(2):140-149.
[3] R. S. Pendyala, P. Mendis. Experimental study on shear strength of high-strength concrete beams[J].ACI Structural Journal.2000.97(4):564-571.
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[7] A. Bishara, J.-C. Peir. Reinforced Concrete Rectangular Columns in Torsion[J]. Journal of theStructural Division, Proceedings of ASCE.1968.94(ST12):2913-2933.
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[9]赵嘉康,张连德,卫云亭.钢筋混凝土压弯剪扭构件的抗扭强度研究[J].土木工程学报.1993.26(1):20-30.
[10] W. F. Chen. Plasticity in Reinforced Concrete[M]. Fort Lauderdale. J Ross Publishing.2007.
[1] X. Chen, X. Liu. Three dimensional failure surface of reinforced concrete element subjected tocomplex loads[C]. Advances in Civil Engineering and Building Materials. Hongkong.2012.
[2] V. Baikov, E. Sigalov. Reinforced Concrete Structures[M]. Moscow. Mir Publishers1981.
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[4] W. F. Chen. Plasticity in Reinforced Concrete[M]. Fort Lauderdale. J Ross Publishing.2007.
[5]陈惠发, A.F.萨里普.弹性与塑性力学[M].中国建筑工业出版社.2004.
[6] T. T. C. Hsu. Toward a unified nomenclature for reinforced-concrete theory[J]. Journal of StructuralEngineering.1996.122(3):275-283.
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[16] A. K. Sharma, G. S. Pandit. Torsion Tests on Concentrically Prestressed Beams[J]. Indian ConcreteJournal.1979.53(2):54-59.
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