NON-LINEAR ANALYSIS OF REINFORCED CONCRETE FRAMED STRUCTURES

Abstract

Prediction of behaviour of reinforced concrete framed structures for entire range of loading upto failure assumes a special significance in view of greatly increasing use of ultimate load design methods. Although these methods incorporate effects of material nonlinearity to some extent, they generally present the preassumed behaviour at ultimate stage only. Moreover, effects of axial forces on displacement are also not considered rationally in the above methods so that the other modes of failure remain undetected in the analysis. Hence a method of analysis to predict response of reinforced concrete frame structures at various stages of loading with sufficient accuracy and rapidity becomes necessary to develop as an aid to rational method of design. Such an analy sis, however, is complex and it requires voluminous computational effort because of influences of material and geometric nonlinear!- ties. Rational development of these methods has only now become possible because of greater understanding of behaviour of such structures, greater research inputs, improved solution techniques and easy accessibility to faster and larger computers. Two major approaches developed in the past for nonlinear analysis of rein forced concrete frames consist of an elastic-plastic analysis using hinge concept (73,8>+) and the finite element approach employing unidimensional elements to discretise the frame members (5,55). Both these methods have their own limitations discussed at appropriate places in the text. An attempt has been made in the present study to develop a rapid but realistic method of -11- nonlinear analysis to predict the behaviour of reinforced concrete framed structures including frame-shear wall systems. The proposed analytical model considers both material and geometric nonlinearities by taking into account nonlinear material constitutive laws, spread of plasticity, axial deformations shear deformations, beam column effects and changes of geometry. The method, based on displacement approach of analysis, employs a solution algorithm of incremental and iterative type using modi fied Newton-Raphson procedure. Corrective load vectors based on plastic curvatures to satisfy material laws and on equilibrium of member forces under deformed geometry are applied during itera tions. The stiffness is kept constant during each iteration. Material inelastic properties are accounted for by gene rating axial load-moment-curvature data for every typical rein forced concrete section in the frame based on specified stressstrain relation of concrete and steel. This data for a section is stored after fitting it to mathematical expressions resulting in economy of both computer storage and computation time. Tangent stiffness of members is calculated by considering the spread of plasticity along the length of members depending upon current axial load and curvature levels. The effect of axial force on member stiffness (beam-column effect) is evaluated for a member assigning varying flexural rigidities along its length. The expressions for fixed end forces due to span loads on beams have also been developed considering the effect of vary ing rigidities in such members. Numerical integration using Gauss quadrature has been employed to evaluate above member -iiistiffnesses and fixed end forces. Lengths of beams and columns have been subdivided into four and three zones respectively to take care of reinforcement variation and expected plastic zones. This subdivision enables a more realistic evaluation of member stiffness. To test the accuracy of proposed method of analysis, ten test examples comprising of beams, columns, frames and shear wallframe structure as reported by other investigators have been ana lysed. These structures display marked material, geometrical or both types of nonlinearities. The analytical load-deflection diagrams have been compared with experimental values and with theoretical results as predicted by other investigators, wherever available. The proposed method has been used for predicting the behav iour of twenty storeyed coupled shear walls in elastic and postelastic ranges under action of lateral loads. Load-deflection and load-force diagrams are presented for three coupled shear walls having different relative stiffnesses of component walls and appropriate conclusions have been drawn. The method has also been applied to predict the behaviour of four single bay five storey frames and a twenty storey shear wall framed structure designed by prevalent design approaches- Their load deflection curves have been drawn and their performance against design load has been evaluated. Results obtained by the present method have been compared with published data, wherever feasible. -iv- It has been established that full range analysis of larger frames using proposed method of analysis is economically possible and it can be used as an aid to a rational design.