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State space of base shear-torque relationship
In fact, the degradation type of bi- and tri-linear force-displacement relationship should be more suitable
than the idealized elasto-plastic one for most cases of reinforced concrete building structures in seismic
ductility design. This conclusion can be verified directly from the comparison of the abovementioned
experimental response and the idealized BST surface shown in fig.3a)~c). Then, correspondingly, the
single space surrounded by the BST surface can be developed to 2 or 3 state spaces signifying
respectively the elastic, inelastic even collapsed state of the related structural system. In other words, for
evaluating the seismic performance of asymmetric RC structures realistically, the ultimate BST surface
in the idealized model should be developed to nonlinear space(s) of the base shear- torque responses. The
first space signifies the elastic response while the second space represents the inelastic characteristics
when yield damage occurred in structure, and the third space means that the shear-torque response
exceeds the ultimate capacity of the system. The boundary between the first and the second space
describes a kind of critical state that the yielding damage initiates in system. Whilst, the boundary
between the second and the third space should theoretically signify that the system begins to collapse.
As example, abovementioned specimen2 (bi-directional asymmetry with 3 lateral resistant planes along
the y-direction of ground excitations) here is used to show how the 3 state-spaces model for evaluation of
the seismic response of the base shear-torque is established. From the experimental data for the yielding
strength f y of the reinforcement used in the model system shown in table 1 respectively, the yield
boundary dividing the elastic and the inelastic response spaces can be determined according to the
method that De La Llera and A.K. Chopra established, i.e., its* location is completely same to that of the
BST surface in the idealized model but they have quiet different meaning. Using the similar way and the
ultimate strength/, shown in table 1 , the eight vertices of the collapse boundary can be easily determined
based on the following formulas* 31 :
5 = — | 7 6 ~ — 2 i -j — —-} i % — — ^
y^ = -y\ ' y 6 = -^ > ^ = -^ > ^ = -^
( 1 )
where, in a more general sense, the symbol "*" signifies two kinds of critical state: the onset of yielding
and reaching at the ultimate strength. If the structure system just starts to be damaged (reaches its yield
strength), parameters in the above formulas have the same meanings with those of in the idealized BST
surface. If the structural response reaches the maximum capacity of the system, these parameters can be
determined based on the ultimate strength of structural members. As shown in fig.4, structural
parameters controlling the response state space of the studied system are explained as,
1). y* =VJV* Q is the normalized base shear along x-direction, while F r
v
0 = ^^ / vn and
^/o = X- /«' are ^e yi e^m§ an d ultimate lateral resistant capacities along the x-direction of the
studied system respectively, in which, f yxl and f llxl are the corresponding yield and ultimate strength of
the zth lateral resistant plane, and Mis the number of the lateral resistant planes along x-direction. If the
earthquake actions along the x-direction (besides slong the y-direction) should be taken into account, i.e.,
j7 r =£0, the shape of the state space would be changed with the change of the value of V x . Otherwise, if
only single-directional earthquake excitation is studied, i.e., F r =0, the effects of this item should be
ignored.
2). J/, V 0 = Y^ f m and V" Q = ^N f uyl are the lateral yield and ultimate capacity of the system along
y-direction, f yyi and f liy , are the lateral yield and ultimate strength of the lateral resistant planes,
respectively. N is the number of the lateral resistant planes along y-direction. The value of V^ and V^