Civil Engineering Reference
In-Depth Information
Strain compatibility equations
α
1
−
γ
12
2
ε
1
cos
2
ε
2
sin
2
ε
=
α
1
+
α
1
cos
α
1
¯
¯
¯
2sin
(6.93)
α
1
+
γ
12
2
ε
1
sin
2
ε
2
cos
2
ε
t
¯
=
¯
α
1
+
¯
2sin
α
1
cos
α
1
(6.94)
γ
t
2
=
α
1
+
γ
12
2
(cos
2
sin
2
(¯
ε
1
−
¯
ε
2
)sin
α
1
cos
α
1
−
α
1
)
(6.95)
Notice in Equations (6.93)-(6.95) that the symbols ¯
ε
1
,¯
ε
2
,¯
ε
and ¯
ε
t
are the
uniaxial
strains,
not the biaxial strains (
ε
1
,
ε
2
,
ε
and
ε
t
) in SMM. The advantage of using uniaxial strains
1
,
2
,
(¯
ε
1
,¯
ε
2
,¯
ε
and ¯
ε
t
) in FA-STM is that they can be directly related to the stresses (
σ
σ
12
,
f
and
f
t
) in the equilibrium equations by uniaxial stress-strain relationships obtained
directly from the uniaxial tests of RC 2-D elements. The uniaxial stress-strain relationships of
smeared concrete and smeared steel bars are given in Sections 6.1.6-6.1.10 and are summarized
as follows.
τ
Uniaxial constitutive relationships
The solution of the three stress equilibrium equations and the three strain compatibility equa-
tions requires two constitutive matrices, one for concrete and one for steel. The concrete
constitutive matrix is:
⎡
⎤
⎡
⎤
⎡
⎤
ε
1
¯
¯
E
1
00
0
E
2
0
00
G
12
c
1
σ
⎣
⎦
⎣
⎦
=
⎣
⎦
=
ε
2
γ
12
2
c
2
σ
(6.96)
c
12
τ
E
1
,
E
2
and
The 3
×
3 matrix in Equation (6.96) contains three diagonal nonlinear moduli
E
1
represents the
E
2
represents the
G
12
.
1
2
σ
−
ε
1
curve given in Section 6.1.8.
¯
σ
−
ε
2
curve
¯
given in Section 6.1.6 and 6.1.7; and
G
12
gives the
12
τ
−
γ
12
curve in section 6.1.10.
The steel constitutive matrix is:
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
E
s
ρ
f
ρ
t
f
t
0
ρ
0
0
ε
¯
¯
E
t
ρ
t
ε
t
0
0
0
(6.97)
0
0
0
E
s
E
t
are for mild steel bars in
The two diagonal elements in the 3
×
3matrix
ρ
and
ρ
t
E
s
E
t
represent the
the
- and
t
-directions, respectively.
ρ
and
ρ
t
are the steel ratios.
and
ε
t
curves, respectively, for the smeared stress-strain curves of mild steel bars
embedded in concrete. These curves are given in Section 6.1.9.
f
−
¯
ε
and
f
t
−
¯
6.2.2 Solution Algorithm
6.2.2.1 Summary of Governing Equations
The twelve governing equations for RC 2-D elements are summarized below:
