Civil Engineering Reference
In-Depth Information
In the fixed angle theory, the solution of these three stress equilibrium equations and the
three strain compatibility equations requires stress-strain relationships for concrete and for
mild steel bars. The three sets of stress-strain curves for concrete are the
1
σ
−
ε
1
curve, the
2
12
σ
−
γ
12
curve. In the matrix form, we can summarize these concrete
constitutive relationships as:
−
ε
2
curve, and the
τ
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎤
ε
1
ε
2
γ
12
2
1
σ
E
1
00
0
E
2
0
00
G
12
⎣
⎦
2
σ
(4.58)
12
τ
where the 3
3 matrix in Equation. (4.58) is called the constitutive matrix for fixed angle
theories, and the three nonlinear moduli
E
1
,
E
2
and
G
12
represent the
×
1
2
σ
−
ε
1
curve, the
σ
−
ε
2
12
curve, and the
τ
−
γ
12
curve, respectively.
4.3.3 Rotating Angle Theory
In the rotating angle theory the direction of cracks is defined by the rotating angle
α
r
in the
principal
r
d
coordinate of concrete, as shown previously in Figure 4.11(g). Similar to the
derivation of equilibrium equations for the fixed angle theories, we can derive the reverse
matrix [
T
]
−
1
that transforms a set of concrete stresses in the
r
−
−
d
coordinate (
σ
r
and
σ
d
)into
c
t
and
c
a set of concrete stresses in the
−
t
coordinate (
σ
,
σ
τ
t
). Notice that the shear stress in
concrete
d
coordinate is a principal coordinate.
As a result, the three transformation equations for concrete element are:
⎡
τ
rd
must vanish (i.e.
τ
rd
=
0), because the
r
−
⎤
⎡
⎤
⎡
⎤
c
cos
2
sin
2
σ
α
r
α
r
−
2sin
α
r
cos
α
r
σ
r
σ
d
0
⎣
⎦
=
⎣
⎦
⎣
⎦
c
t
sin
2
cos
2
σ
α
r
α
r
2sin
α
r
cos
α
r
(4.59)
c
t
(cos
2
sin
2
τ
sin
α
r
cos
α
r
−
sin
α
r
cos
α
r
α
r
−
α
r
)
Substituting Equation (4.59) for concrete elements into Equations (4.46)-(4.48) for RC
elements results in three equilibrium equations for RC elements as follows:
σ
=
σ
r
cos
2
α
r
+
σ
d
sin
2
α
r
+
ρ
f
(4.60)
=
σ
r
sin
2
α
r
+
σ
d
cos
2
σ
t
α
r
+
ρ
t
f
t
(4.61)
τ
t
=
(
σ
r
−
σ
d
)sin
α
r
cos
α
r
(4.62)
Similarly, the three compatibility equations in the rotating angle theories are expressed as:
ε
=
ε
r
cos
2
α
r
+
ε
d
sin
2
α
r
(4.63)
=
ε
r
sin
2
α
r
+
ε
d
cos
2
ε
t
α
r
(4.64)
γ
t
2
=
(
ε
r
−
ε
d
)sin
α
r
cos
α
r
(4.65)
Equations (4.63)-(4.65) represent the transformation relationship between the strains (
ε
,
ε
t
and
γ
t
)inthe
−
t
coordinate of the steel grid element and the principal strains (
ε
r
and
ε
d
)inthe
r
d
coordinate of the concrete element.
In the rotating angle theory, the solution of the three equilibrium equations and the three
compatibility equations also requires the stress-strain relationships of concrete and mild steel
−
