Civil Engineering Reference
In-Depth Information
E
1
and ¯
E
2
into Equations (6.13) and (6.14), respectively,
1
2
Substituting ¯
ε
1
=
σ
/
ε
2
=
σ
/
gives:
⎡
⎣
⎤
⎦
E
1
E
1
ν
12
0
⎡
⎣
⎤
⎦
⎡
⎤
1
−
ν
12
ν
21
1
−
ν
12
ν
21
ε
1
ε
2
γ
12
2
1
σ
E
2
E
2
⎣
⎦
=
2
ν
21
σ
(6.15)
0
12
τ
1
−
ν
12
ν
21
1
−
ν
12
ν
21
G
12
0
0
c
c
where
σ
1
,
σ
2
=
smeared (average) stresses of concrete in the 1- and 2-directions, respectively.
2
, represent both the
biaxial
stresses and the
uniaxial
stresses.
Comparing Equation (6.7) with (6.15) gives the relationships between the biaxial moduli
and the uniaxial moduli:
1
and
The symbols,
σ
σ
E
1
E
1
=
(6.16)
(1
−
ν
12
ν
21
)
E
2
E
2
=
(6.17)
(1
−
ν
12
ν
21
)
Equations (6.16) and (6.17) state that the biaxial moduli of concrete are simply the uniaxial
moduli of concrete divided by (1
−
ν
12
ν
21
). This is applicable in both directions of principal
compression and principal tension.
Constitutive matrix of smeared steel bars
In the SMM, the constitutive matrix of steel bars is derived from the compatibility equations
(6.4) and (6.5) for biaxial strains
ε
t
. Substituting Equations (6.11) and (6.12)
into Equations (6.4) and (6.5) results in a set of two long equations, each with five terms on
the right-hand side:
ε
1
,
ε
2
,
ε
and
α
1
−
γ
12
2
ε
1
cos
2
ε
2
sin
2
ε
2
cos
2
ε
1
sin
2
ε
=
¯
α
1
+
¯
2sin
α
1
cos
α
1
−
ν
12
¯
α
1
−
ν
21
¯
α
1
(6.18)
α
1
+
γ
12
2
ε
1
sin
2
ε
2
cos
2
ε
2
sin
2
ε
1
cos
2
ε
t
=
¯
α
1
+
¯
2sin
α
1
cos
α
1
−
ν
12
¯
α
1
−
ν
21
¯
α
1
(6.19)
Notice that the first three terms on the right-hand side of Equations (6.18) and (6.19) are the
transformation forms of unaxial strains ¯
ε
and ¯
ε
t
:
α
1
−
γ
12
2
ε
1
cos
2
ε
2
sin
2
ε
=
¯
¯
α
1
+
¯
2sin
α
1
cos
α
1
(6.20)
α
1
+
γ
12
2
ε
1
sin
2
ε
2
cos
2
¯
ε
t
=
¯
α
1
+
¯
2sin
α
1
cos
α
1
(6.21)
Substituting Equations (6.20) and (6.21) into Equations (6.18) and (6.19), respectively,
gives:
ε
2
cos
2
ε
1
sin
2
ε
=
ε
−
ν
12
¯
¯
α
1
−
ν
21
¯
α
1
(6.22)
ε
2
sin
2
ε
1
cos
2
ε
t
=
ε
t
−
ν
12
¯
¯
α
1
−
ν
21
¯
α
1
(6.23)
