Civil Engineering Reference
In-Depth Information
If the RC element has a thickness
h
of 305 mm., the steel area per unit length
A
/
s
and
A
t
/
s
,
is:
A
s
=
A
t
s
=
ρ
14 mm
2
/mm
h
=
0
.
0103(305)
=
3
.
We could use two layers of No. 7 bars at 245 mm spacing in both directions.
A
/
s
=
3.16 mm
2
/mm
3.14 mm
2
/mm, OK.
A
t
/
s
=
2(387)/245
=
>
The smeared steel stresses
ρ
f
y
and
ρ
t
f
ty
, are shown in Figure 5.3(d), in which
ρ
f
y
=
ρ
t
f
ty
=
4.26 MPa. The stresses in the concrete element are shown by the Mohr circle in Figure
5.3(c) using the following additional values:
σ
−
ρ
f
y
=
2
.
13
−
4
.
26
=−
2
.
13 MPa
σ
t
−
ρ
t
f
ty
=−
2
.
13
−
4
.
26
=−
6
.
39 MPa
Equation (5.35) gives:
−
τ
t
−
3
.
69
σ
d
=
α
r
=
=−
8
.
52 MPa
α
r
cos
.
.
sin
(0
500)(0
866)
In conclusion, Figure 5.3(b), (c) and (d) clearly illustrate that the rotating angle
α
r
is equal to
the fixed angle
ρ
t
f
ty
are equal.
This interesting case is a direct consequence of Equation (5.32), which relates the double
angles of the steel grid element, the concrete element and the RC element. When cot2
α
1
when the smeared steel stresses in both directions
ρ
f
y
and
α
s
for
90
◦
), Equation (5.32) requires that cot2
the steel grid is zero (i.e. 2
α
s
=
α
r
for the concrete
element is equal to cot2
α
1
for the whole RC element.
5.2 Strain Compatibility of RC 2-D Elements
5.2.1 Transformation Type of Compatibility Equations
In Chapter 4, Section 4.3.3, we derived the three compatibility equations, (4.63)-(4.65), for
the rotating angle theory of a reinforced concrete 2-D element as follows:
ε
=
ε
r
cos
2
α
r
+
ε
d
sin
2
α
r
(5.41)
=
ε
r
sin
2
α
r
+
ε
d
cos
2
ε
t
α
r
(5.42)
γ
t
2
=
(
ε
r
−
ε
d
)sin
α
r
cos
α
r
(5.43)
The compatibility condition, defined by Equations (5.41)-(5.43), contains six variables,
namely,
α
r
. When any three of the six variables are given, the other three
can be solved by these three compatibility equations.
It is interesting to note that the four normal strains
ε
,
ε
t
,
γ
t
,
ε
r
,
ε
d
and
ε
,
ε
t
,
ε
r
and
ε
d
have a simple relationship.
Adding Equations (5.41) and (5.42) gives
ε
+
ε
t
=
ε
r
+
ε
d
(5.44)
Equation (5.44) is the principle of first invariance for strains.
From Eq. (5.44), the cracking strain
ε
r
can be expressed as:
ε
r
=
ε
+
ε
t
−
ε
d
(5.45)
