POST-PEAK RESPONSE ANALYSIS USING THE FINITE ELEMENT METHOD

Abstract

With increasing load, a structure undergoes increasing deformation. Beyond a critical or peak load level, structure's inability to take any more loads causes failure. Failure can be distributed or localized. In general, the failure of civil engineering structures is localized and is caused by a series of densely populated cracks which coalesce in an extremely small region. The numerical simulation of crack formation and propagation has been a subject of considerable research. Although, post-peak states are usually not tolerated in the design of structures, the knowledge of post-peak behaviour can be of great help in understanding the strengths and weaknesses of structures. Further, the understanding of the failure modes is important to avoid brittle failure. It is for this reason that the capabilities to predict post-peak deformation behaviour is essential in addition to finding the ultimate load carrying capacity. The underlying aim of the thesis is to simulate post-peak behaviour of the structure using strain softening plasticity and the finite element method. In this regard the objectives of the thesis are outlined as follows. • To review the literature related to computational plasticity with emphasis on strain softening and localization. • To evolve benchmark tests in elastoplasticity particularly under strain softening conditions. • To study the post-yield behaviour, conditions of localization and mesh sensitivity issues using numerical samples with various yield criteria. • To develop algorithms for dynamic strain softening problems and to conduct studies on the possible use of strain softening under seismic forces. • To study post-peak response of some realistic structures. Some recent developments in computational elastoplasticity are discussed. Current literature in the area of strain softening plasticity and its use in simulating post-peak behaviour is reviewed. Issues related to the use of indirect displacement control and evolution of localization conditions are also reviewed. Emphasis is laid on the Hoffman yield criterion which is pressure sensitive and valid for anisotropic elastoplasticity. The present study, however, is limited to the isotropic form of the Hoffman criterion. A return mapping algorithm using the backward Euler (») scheme for this criterion is "discussed. The evolution of this criterion for strain softening plasticity wherein both equivalent compressive and tensile strengths are assumed to reduce as compared to when only the tensile strength is assumed to decline is considered. The possibility of using strain softening elastoplasticity for the prediction of post-peak seismic response is explored. Numerical implementation of strain softening has been known to cause problems of convergence, load step sensitivity and discretization sensitivity (or mesh sensitivity). Many of these difficulties have been surmounted for static analysis. Numerical problems associated with the use of strain softening in the solution of dynamic problems are highlighted and some methods of overcoming them discussed. Consideration is generally limited to one dimensional problems arising out of elastoplastic strain softening behaviour. The results indicate that dynamic response does not become unbounded due to strain softening. Strain softening, however, introduces a large zero frequency component as compared to strain hardening or perfect plasticity. The frequency content at frequencies other than zero is not significantly altered. These preliminary investigations indicate that strain softening in conjunction with an appropriate stress updating algorithm can be employed in the seismic analysis ofstructures. The analysis of industrial structures and substructures is often conducted using elastoplastic constitutive laws, in conjunction with the finite element method. The finite element codes may be used as a "black box" by personnel who may be inadequately trained in the method. In order to train analysts and to check the validity offinite element codes, the benchmarks can be of paramount importance. Further, exact solutions in computational elastoplasticity cannot be used directly as these often pertain to solutions that are valid only for particular cases. However, these solutions can be used as benchmark tests to check the validity of finite element codes and accuracy of numerical solution procedures. In the present study tests for three different yield criteria viz. von Mises, Mohr Coulomb and Hoffman are discussed. The perfectly plastic as well as strain hardening/softening cases are examined. The benchmark tests are based on prescribed displacement field format. Tests are evolved that can be used to verify the ability of finite element packages in accurately predicting first yield and flow in the post-elastic regime. The exact integration of constitutive equations for an isotropic plastic von Mises material that incorporate linear hardening/softening for some specific cases are developed. Illustrative tests are included. (///) Exact solutions for Mohr Coulomb criterion that include linear strain hardening or softening plasticity and the presence of singular regions are also developed. A number of biaxial and triaxial illustrative tests are included. The isotropic form of the Hoffman criterion is a cylindrical paraboloid in the principal stress space. As such it is not straightforward to evolve closed form solutions for this criterion. However, tests that can illustrate the stress changes in the principal stress space and can serve as tools for understanding, are studied. Simple tests under perfect plasticity and strain softening conditions are examined. Post-peak response can be described as the response of a structure that is incapable of sustaining any additional loads. The post-peak behaviour is associated with progressive failure of the structure, which in turn can be modelled using softening plasticity. Strain softening implies declining equivalent yield strength parameter in the yield criterion. This does not, necessarily imply post-peak (declining) load-deflection response and the load might actually increase. The behaviour of elastoplastic von Mises, Mohr Coulomb and Hoffman materials under simple load paths and considering perfect/strain softening plasticity are studied. In addition to the movement of the stress point in the principal stress space emphasis is laid on the load displacement behaviour. The study also examines the use of the accoustic tensor as a localization indicator. Uniaxial compression tests on single elements used in the study indicate that for associated von Mises plasticity the localization conditions are not necessarily satisfied immediately after first yield, even under strain softening conditions. Critical values of the softening parameter are evaluated such that the localization condition is satisfied immediately after first yield. It is seen that if the softening parameter is of greater magnitude than the evaluated critical magnitude, then the localization direction is not unique. Increasing the softening parameter beyond a certain magnitude may lead to instability. It is seen that this limit is more stringent than the local uniqueness requirements. Single element compressive tests indicate that a descending or constant load displacement response is obtained only after the satisfaction of the localization condition. Similar uniaxial tests indicate that it is far easier to satisfy the localization conditions with the Mohr Coulomb criterion. The satisfaction of the localization condition is accompanied by post-peak behaviour for simple uniaxial test. For these tests a flat or a (IV) descending load displacement response is observed depending on the assumption of perfect or strain softening plasticity. The study shows that post-peak behaviour using the Hoffman criterion is strongly influenced by the ratio of uniaxial tensile and compressive strengths. In case of softening plasticity the post-yield response is totally different when both equivalent tensile and compressive strengths are assumed to reduce as compared to when only the tensile strength is assumed to decline. Mesh sensitivity of the post-peak response is also studied for von Mises and Hoffman criterion. Good (mesh insensitive) results are obtained when nonlocal material laws are employed. The algorithms and ideas developed are applied to some engineering problems. Postpeak response of simple systems such as acantilever beam, plane strain tension specimen and a notched beam is studied. The failure patterns of a slope under varying post-yield conditions are examined. Strain softening plasticity is also applied to the seismic analysis of Koyna dam.