| Research Abstract |
The maximum potential energy of a linear system, W_E, is assumed to be proportional to that of the non-linear system, W_P, like W_E= _<alpha>W_P. alpha is the constant related with period or force-deflection relationship of the system. This realation is able to be reduced to the following equation, using more concrete imaged parameters. mu=f (C_y, E, beta, gamma) ・・・ (1) where, mu is ductility ratio. Ductility ratio obtained by maximum response displacement diveded by the yield displacement of the system, should be a function of yield strength ratio C_y, destructive power of ground motion E, constant beta determined from the shape of force-deflection relationship and constant gamma determined from period of teh system. As parameters C_y, E, beta are given a priori, constant gamma should be determined. Here, we assumed that gamma is a s hape of function as gamma=g (C_1/T+C_2) by the experimental work. If constants C_1, C_2 are determined, we can get the maximum response ductility ratio
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from Eq. 1. Non-linear response analysis for single-degree of freedom system having various periods were conducted to more than 10 earthquake records that are commonly used in earthquake resistant design in Japan. Appling the least square method to the reponse ductility ratios vs periods relations, constants C_1, C_2 were determined. Howver, Eq. 1 is available in the period range of 0.2 <less than or equal> T <less than or equal> 1.2 sec., and also, of course, mu <greater than or equal> 1. Maximum acceleration, maximum velocity and spectrum intensity are taken as the scale which indicates the destructive power of ground motion. Although maximum acceleration is not necessarily the adequate scale to evaluate the destructive power to structure, maximum velocity or spectrum internsity was found to be the fairly proper scale to evaluate the destructive power of ground motion to structure. This is because when earthquake records were normalized by maximum velocity or spectrum intensity, the non-linear responses were very closely scattered around the values obtained from Eq. 1, otherwise normalizing by maximum acceleraton, responses were widely scattered around the values from Eq. 1. In case of extending this relation (Eq. 1) to the problem of multi-degree of freedom system, the lst mode shape of the system was assumed not to change through linear stage and non-linear stage. From this assumption ductility ratio of each story mu_i was derived as mu_< : >=phi_i+mu・・・ (2) Here, phi_i is the constant determined from the mode shape and story yield displacement, and mu corresponds to ductility factor obtained from Eq. 1. Response analyses were conducted to the muiti-degree of freedom models, having yield strength distributions alongside story height that were given by Japanese code and also inverse triangular force distribution mode. Compared the real response values to the estimated values from Eq. 2, both were coincident well. Less
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