Some problems in the inelastic response of structures subjects to earth quakes

Abstract

In this chapter, bilinear-hysteretic response of a one-storeyed building structure to several real earthquakes is discussed. The structure, consisting of a rectangular slab supported on a series of portal frames, is assumed to have two degrees of freedom. The several real earthquake accelerograms chosen for the study are: El Centro, May 1940, N-S component; Taft, July 1952, S69°S component; and El Centro Borrego Mountain, April 1968, N-S component. It was intended to study the influence of various parameters like eccentricity, yield strength, linear elastic period, type of ground motion, and non-uniformity in the strength of the frames on the response. P- effect was also included during the course of this investigation. A nominal 1% viscous damping was assumed for the structure. The structure is assumed to deform in the direction of the input ground motion, and therefore the frames were supposed to yield when the resistance force along the direction of displacement reached its yield value. The structure responded in the inelastic range for El Centro and Taft ground motions. The response to Borrego Mountain quake was, however, mostly in the elastic range. For response situations which involve inelastic yielding, small eccentricities (up to 5%) do not lead to a significant change in the ductilities from the zero-torsion case. The effect of torsion becomes appreciable only for large eccentricities. However, for responses in the elastic range, even small eccentricities can lead to significant torsional behavior. Exterior frames, in general, suffered large ductility demands; the ratio of the ductilities of exterior frames was seen to increase with eccentricity when the response was inelastic. This ratio, however, remained more or less constant with eccentricity (from e = 0.05b onwards) when the response was elastic. It is noted that when the yield strength of the frames decreased by half, the ductilities increased by nearly 75% for e/b = 0 and by 200% for e/b = 0.30, for structures with a frame period of 0.25 sec subjected to El Centro quake. However, structures with longer frame periods, like 1.0 sec, suffered only marginal increase in ductility due to the strength reduction. For Taft ground motion, the increase in the ductility due to a similar strength reduction was less compared to that caused by El Centro motion. It may be noted that a reduction in strength, in general, leads to increased ductilities, which is, however, influenced by eccentricity, fundamental period, and the type of ground motion. The influence of P- effect was also studied and it was seen that it had negligible influence on the ductility requirements for the type of structures and ground motion considered. Structures with shorter periods seem to have been affected more by P- effect. The effect of doubling the strength of only the exterior frames was seen to decrease the ductility demands of the exterior frames by nearly 60% at eccentricities other than zero. However, a slight reduction in the interior frame ductilities was also observed at these eccentricities. When e/b = 0, the strengthening of the exterior frames caused an increase in the interior frame ductilities, although there was a marginal decrease in the ductility demands of the exterior frames. It was also noted that the contribution made by the two modes of the structure was conditioned by the fundamental period, ratio of the two frequencies of the structure, and the frequency content of the quake. In situations where the fundamental mode response was dominant, the exterior frame closer to the mass center experienced maximum ductility. When both modes contributed significantly to the response, any of the two exterior frames could experience the maximum ductility. Observations made on hysteretic energy dissipation revealed that no simple correlation can exist between the maximum ductility of a particular frame and the hysteretic energy dissipated by it. It was found that the total hysteretic energy dissipated by a frame is strongly influenced by the number of times it crosses the yield level. In the particular case of the structure with 0.25 sec frame period, the exterior frame close to the mass center dissipated more than 60% of the total energy dissipated by the structure as a whole. Thus, it appears that the severity of the effect of eccentricity is revealed more by the hysteretic energy dissipation rather than the maximum ductility. The inelastic response of a single-storeyed structure, square in plan and supported on four columns, subjected to the El Centro, May 1940 earthquake has been discussed. The model is assumed to possess three degrees of freedom. Interaction between forces in the two orthogonal directions has been considered. The responses of the structure to simultaneous action of the two orthogonal, horizontal components of the ground motion and to one of the two components have been compared. The influence of eccentricity, yield strength, and the period of the structure on the response has been studied. The investigation showed that torsion does not seem to significantly influence the results for small eccentricities. However, for large eccentricities like e/a = 0.30, although the maximum ductilities of some columns are reduced due to torsion, the largest of the maximum ductilities is increased. This increase ranges from 5% to 50% when compared with the zero-eccentricity case, depending on the period and the yield strength of the structure. It is seen that disparity between the largest of the maximum ductilities and the smallest of the maximum ductilities increases with eccentricity for the two-component input, while it decreases with eccentricity for the one-component input. The columns nearer the mass center generally experience larger ductilities. This conclusion is valid only for the El Centro, 1940 quake discussed in this chapter. Its validity for other earthquakes needs to be investigated. Offhand, it might appear that if the four columns were identical, one would expect no torsion even in the case of two-component excitation. However, results indicate that the structure has torsional instability under biaxial yielding, and small errors in computation could be magnified because of dynamic instability. This possibility has also been noted by Housner and Newmark and Rosenblueth. This phenomenon has been brought to light by the present study and it provides scope for further investigation.