Civil Engineering Reference
In-Depth Information
strength (Figure 6.24b) and the shear strength (Figure 6.24c) of the concrete are exhausted.
Looking at the steel bars, the peak point is accompanied by large strains of the transverse
steel bars in Figure 6.24(f) and (g), but not accompanied by the yielding of the longitudinal
steel bars in Figure 6.24(d) and (e). In other words, failure is caused by the crushing of
concrete and the yielding of transverse steel, but without the yielding of the longitudinal
steel.
4. The strains in the longitudinal steel behave in a very interesting way after reaching point
2 (less than the yield point). While the uniaxial strain decreases elastically after point 2,
Figure 6.24(e), the biaxial strain increases significantly along a descending branch (Figure
6.24d) due to the Poisson effect. The incorporation of Poisson effect allows SMM to predict
the descending branches shown in Figure 6.24 (a), (b) and (c).
5. While the loading history of shear stress versus shear strain curve in Figure 6.24(a) moves
from point 2 to a typical point 3 in the descending branch, both the stresses and the strains
in the transverse steel increase rapidly into the strain-hardening region, as shown in Figure
6.24(f) and (g). Beyond point 3, the transverse steel stresses eventually decrease. However,
this decrease of stresses is accompanied by an elastic decrease of uniaxial strain in Figure
6.24 (g), but is also accompanied by an increase of biaxial strain in Figure 6.24 (f).
6.1.12.4 Comparison of Predicted and Experimental
γ
t
Curves
Figure 6.25 compares the SMM predicted shear stress versus shear strain (
τ
t
-
τ
t
−
γ
t
) curves of
16 panels with the experimental curves. The agreement is very good. The 16 panels include
6 panels from series
B
of Pang and Hsu (1995), 4 panels from series VB of Zhang and Hsu
(1998) and 6 panels from series M of Chintrakarn (2001). Series B has a concrete strength
of 42 MPa and
ρ
/ρ
t
ratios ranging from 1.5 to 5. Series VB has a concrete strength of
100 MPa and
ρ
/ρ
t
ratios ranging from 2 to 5. Series M has a concrete strength of 42 MPa
and
ρ
/ρ
t
ratios ranging from 4 to infinity. It can be concluded that SMM is applicable to RC
2-D elements with concrete strengths up to 100 MPa and
ρ
/ρ
t
ratios up to infinity.
6.2 Fixed Angle Softened Truss Model (FA-STM)
6.2.1 Basic Principles of FA-STM
In Section 6.2, we will study the fixed angle softened truss model (FA-STM). FA-STM was
developed before the Poisson effect was understood and before the establishment of the
softened membrane model (SMM), as described in Section 6.1.
The essence of FA-STM is to neglect the Poisson effect in SMM. This has two implications.
First, both the Hsu/Zhu ratios
ν
12
and
ν
21
are assumed to be zero in SMM and second, the
biaxial strains (
ε
t
). As a result,
FA-STM can be considered a special case of SMM, and is simpler than SMM. However,
FA-STM can not correctly predict the descending branch of the load-deformation curves.
FA-STM is useful when we are interested only in the peak shear strength and the behavior
before the peak, and when we would like to have a more accurate prediction than RA-STM
given in Section 5.4.
In FA-STM, the three equilibrium equations are the same as those in SMM, i.e Equations
(6.1)-(6.3). The three compatibility equations, however, are different from those in SMM, i.e.
ε
1
,
ε
2
,
ε
,
ε
t
) are identical to the uniaxial strains (¯
ε
1
,¯
ε
2
,¯
ε
,¯
