The Hysteretic Model
The smooth hysteretic model presented herein is a variation of the model originally proposed by Bouc (1967) and modified by several others (Wen 1976; Baber Noori 1985). The model was developed in the context of moment-curvature relationships of beam-columns. Therefore the stress variable is here
referred to as “moment” (M) and the strain variable as “curvature” (ф). In the case of shear-walls the hysteretic loop can be described in terms of a shear force-shear deformation relationship.
The use of such a hysteretic constitutive law is necessary for the effective simulation of the behavior of R/C structures under cyclic loading, since often structures that undergo inelastic deformations and cyclic behavior weaken and lose some of their stiffness and strength. Moreover, gaps tend to develop due to cracking causing the material to become discontinuous. The Bouc-Wen Hysteretic Model is capable of simulating stiffness degradation, strength deterioration and progressive pinching effects.
The model can be visualized as a linear and a nonlinear element in parallel, as shown in Figure 3. The relation between generalized moments and curvatures is given by:
where My is the yield moment; фу is the yield curvature; a is the ratio of the post-yield to the initial elastic stiffness and z(t) is the hysteretic component defined below.
The nondimensional hysteretic function z(t) is the solution of the following non-linear differential equation:
In the above expression A, B, C, D & E are constants which control the shape of the hysteretic loop for each direction of loading, while the exponents nB, nC, nD & nE govern the transition from the elastic to the plastic state. Small values of ni lead to a smooth transition, however as ni increases the transition becomes sharper tending to a perfectly bilinear behavior in the limit (n^^).
The program defaults are:
1 1 – MA = 1, C = D=0 & B = , E = where e = —, b=1 and nB =nE =n————————————- (3)
bnB enE M+ B E
The parameters C, D control the gradient of the hysteretic loop after unloading occurs. The assignment of null values for both, results to unloading stiffness equal to that of the elastic branch. Also, the model is capable of simulating non symmetrical yielding, so if the positive yield moment is regarded as a reference point, the resulting values for B and E are those presented in equation (3). The hysteretic parameter Kz is then limited in the range of 0 to 1, while the hysteretic function z varies from -|My/My| to 1.
“Plastique ” – A Computer Program for Analysis of Multi-Storey Buildings
Finally, the flexural stiffness can be expressed as:
K = EI = = M.
a- +(1 – a)-
a— + (1 – a)Kz 1
6.1 Hysteretic behavior Variations a) Stiffness Degradation
The stiffness degradation that occurs due to cyclic loading is taken into account by introducing the parameter ^ into the differential equation:
The parameter ^ depends on the current, р = ф/ фу, and maximum achieved plasticity, pmax = фтях / фу. Sk is a constant which controls the rate of stiffness decay. Common values for Sk are 0.1 and 0.05.
b) Strength Deterioration
The strength deterioration is simulated by multiplying the yield moment My with a degrading parameter Sp:
In the above expression Sd, Sp1, Sp2 are constants controlling the amount of strength deterioration; pc is the maximum plasticity that can be reached, pc =фи /фу; |dEdiss is the energy dissipated before
unloading occurs and finally Emon is the amount of energy absorbed during a monotonic loading until failure as shown in Figure 4.
c) Pinching or Slip
Pinching of hysteretic loops due to shear cracking and bond slip of the reinforcement is commonly observed in reinforced concrete structures during cyclic loading. This phenomenon is taken into account by introducing in the expression of Kz the “slip length” parameter a, and a function f (z):
In the above equation Kis the hysteretic parameter affected by pinching, Kz is the original expression of the hysteretic parameter obtained from (3) and parameters a, f(z) are given by:
where As is a control parameter which may be linked to the size of crack opening or reinforcement slip or both; p/ is the normalized curvature attained at the load reversal prior the current loading circle; zs is the range where slip occurs. A non zero value of the parameter zm will shift the effective slip region so that it is symmetric about z=zm.