RESPONSE OF ARCHES UNDER EARTHQUAKE EXCITATION

Abstract

This investigation concerns with the computation of response of arches under earthquake excitations. Methods of analysis are presented for computing linear elastic response and nonlinear response considering large deflections and material inelasticity. Many problems have been solved for a wide range of parameters in order to arrive at significant conclusions and a practical design approach. Two methods are presented for computing free vibration characteristics of arches. In the first method an equivalent lumped mass mathematical model with distributed elasticity is used to compute frequencies and mode shapes by influence coefficient method. This method is applicable to arches of arbitrary shape ani distribution of mass and stiff ness. A range of governing parameters of the structure are considered to determine its influence on the free vibration characteristics of fixed circular, parabolic and Whitney arches. This study highlights the significance of various parameters in the free vibration characteristics. In the second method closed-form expressions are derived for natural frequencies and mode shapes. This method is applicable to only symmetri cal and uniform circular arches with fixed as well as hinged ends. The results of closed-form expressions compare well with the first method. ii The elastic earthquake response considering small deflections is found by timewise superposition of natural modes of vibration. Two actually recorded accelerograms con sidered are (i) Longitudinal component of Koyna earthquake of December 11,1967, (ii) modified version of SICentro Earth quake of May 18,1940, N-S component. Modified SICentro 3arthnuake is obtained by toning down its acceleration ordinates so as to make its undamped spectral intensity equal to that of Koyna Earthquake. The horizontal ground motion alone and simultaneous action of horizontal and vertical ground motion are considered for determination of response. The ground motion is considered to be equal at the two supports. The practical range of variation of each of the arch parameters, namely, rise-span ratio, span to radius of gyration ratio, variation of moment of inertia along span, shape, fundamental period, mass distribution and damping, is considered to deter mine its influence on the earthquake response. The variations of bending moment, shear, thrust and deflection along the span and also the maximum strain energy stored during the ground motion on the arch are computed. Most of the responses are worked cut for hangar type mass distribution. Few problems with bridge type of mass distribution are also studied. The earthquake response is also computed by .vootmean- square combination of modes at various points en the arch for comparison with timewise response. The earthquake response Ill for uniform circular arches is also computed using closedform expressions. It is found that this approximate method is accurate enough for practical design purposes. The results of seismic coefficient approach are compared with those of dynamic analysis. It is seen that uniform seismic coefficient leads to unconservative distribution of forces on the arch. A numerical method of computing response of arches under ground motion considering the effect of large deflec tions and inelasticity of the material is also presented. The distributed mass as well as the elasticity of the struc ture are lumped at discrete points. For computation of the response in the inelastic range, the cross-sectional area of the arch is approximated by three lumped areas in the crosssection, two at the flanges and one at the centre of the web. The stress strain curve for each lumped area is considered to be elasto-plastic. The effect of vertical static loading is concurrently considered in the dynamic analysiso The damping is considered to be mass projjortional. The computation of response consists in solving a set of second order simul taneous nonlinear differential equations. Certain problems of circular arches are solved using this method. The elastic linear, elastic nonlinear and inelastic responses of circular arches are compared with each other to demonstrate the signi ficance of non-linear analysis. IV As a practical result of this study, the normalized distribution of forces on the fixed arches along the span is averaged and design curves are proposed. For use with these distributions, the support forces couli be computed either by the seismic coefficient approach or the root-mean-square combination of first three antisymmetric and first three symmetric modes for the probable earthquake for which suffi cient data is tabulated for ready use.