Civil Engineering Reference
In-Depth Information
6
Fixed Angle Shear Theories
6.1 Softened Membrane Model (SMM)
6.1.1 Basic Principles of SMM
In this chapter we will study three models in the fixed angle theory: the fixed angle softened truss
model (FA-STM), the softened membrane model (SMM), and the cyclic softened membrane
model (CSMM). The second and third models, SMM and CSMM are also based on the fixed
angle concept, even without the special fixed angle (FA) label in the title. Since the FA-STM is
a special case of SMM, we will first study SMM in this section before reducing it to FA-STM
in Section 6.2. Then, the CSMM will be treated in Section 6.3.
From Section 4.3.2, the SMM satisfies Navier's three principles as follows:
6.1.1.1 Stress equilibrium equations
c
1
cos
2
c
2
sin
2
c
σ
=
σ
α
1
+
σ
α
1
−
τ
12
2sin
α
1
cos
α
1
+
ρ
f
(6.1)
c
1
sin
2
c
2
cos
2
c
σ
t
=
σ
α
1
+
σ
α
1
+
τ
12
2sin
α
1
cos
α
1
+
ρ
t
f
t
(6.2)
c
c
12
(cos
2
c
sin
2
τ
t
=
(
σ
1
−
σ
2
)sin
α
1
cos
α
1
+
τ
α
1
−
α
1
)
(6.3)
It should be emphasized that these three equilibrium equations for fixed angle theory exhibit
the following characteristics. The equations are established on the basis of the principal 1-2
coordinate of the applied stresses on the RC element, in which the shear stress
τ
12
=
0.
c
c
However, the concrete shear stress
τ
12
, is not equal to zero (see Figure 4.14a). These
τ
12
terms
are the sources of the 'contribution of concrete' (
V
c
).
Strain compatibility equations
α
1
−
γ
12
2
ε
=
ε
1
cos
2
α
1
+
ε
2
sin
2
2sin
α
1
cos
α
1
(6.4)
α
1
+
γ
12
2
=
ε
1
sin
2
α
1
+
ε
2
cos
2
ε
t
α
1
cos
α
1
2sin
(6.5)
cos
2
α
1
γ
t
2
=
α
1
+
γ
12
2
sin
2
(
ε
1
−
ε
2
)sin
α
1
cos
α
1
−
(6.6)
