Civil Engineering Reference
In-Depth Information
5. Beyond the peak point 3, the long descending branch predicted by SMM in Figure 6.27(a)
is accompanied by the long descending branches of biaxial steel strains (
ε
t
) predicted
by SMM in Figure 6.27(d) and (f). These large biaxial steel strains in the longitudi-
nal and transverse steel are the direct results of considering the large Poisson effect
(Hsu/Zhu ratio
ε
,
ν
12
=
1.9). It should be noted, however, that the concrete compressive
strain, ¯
ε
2
=
ε
2
, is not affected by Poisson effect, because Hsu/Zhu ratio
ν
21
=
0. Even
so, the uniaxial concrete compressive strain, ¯
ε
2
, in Figure 6.27(b) shows long descending
branches.
6. The influence of Poisson effect in the descending branches of longitudinal steel can also
be observed by comparing the biaxial steel strains
ε
t
in Figure 6.27(d) and (f),
respectively, calculated by FA-STM and by SMM. The biaxial steel strains calculated by
SMM, which considers the Hsu/Zhu ratios, provides a long and gently descending curve.
In contrast, the biaxial steel strains calculated by FA-STM, which neglects the Hsu/Zhu
ratios, follows an elastic, unloading straight line.
7. The uniaxial steel strains ¯
ε
and
ε
t
calculated by FA-STM and SMM and shown in Figure
6.27(e) and (g), respectively, are essentially the same. The descending branches always
exhibit elastic, unloading straight line.
ε
and ¯
In summary, in the design of concrete structures subjected to monotonic, static loading, the
FA-STM can be very useful in calculating the peak shear strength. However, in the design of
concrete structures subjected to earthquake loading, the SMM must be used to calculate the
hysteretic loops, because ductility and energy dissipation play a vital role.
6.3 Cyclic Softened Membrane Model (CSMM)
6.3.1 Basic Principles of CSMM
Sections 6.1 and 6.2 have presented the softened membrane model (SMM) and the fixed
angle softened truss model (FA-STM), respectively, for predicting the behavior of RC 2-D
elements subjected to monotonic static loading. In Section 6.3, we will study the cyclic
softened membrane model (CSMM) which is an extension of the softened membrane model
(SMM) to predict the behavior of RC 2-D elements subjected to reversed cyclic loadings.
Since cyclic loadings are encountered in dynamic and earthquake actions, CSMM will be
applied to structures discussed in Chapters 9 and 10.
Similar to SMM, the CSMM is based on the same equilibrium and compatibility equations
(6.1)-(6-6), and the same constitutive matrices for concrete and steel given by Equations (6.7)
and (6.8). However, the constitutive relationships of concrete and steel are generalized for
application to reversed cyclic loading as follows:
1. The constitutive relationships of smeared concrete must be modified in three aspects: (a)
the tensile and compressive envelope curves are expressed by equations applicable to both
the positive and negative directions of the cyclic loading; (b) the compression envelope
curves should take into account the damaging effect of perpendicular compressive stress
in previous loading cycles; and (c) the unloading and reloading curves in both the tension
