Civil Engineering Reference
In-Depth Information
An iterative method is used to determine the prestrain condition in the cross-section. The
calculation below uses the internal lever arm of the reinforcing steel, determined
iteratively, in order to demonstrate the method briefly. The internal lever arm, which
represents the iteration variable, is
z
s1
0
:
905
?
d
s1
0
:
904
?
653
590
:
4mm
The tensile force in the steel at the time of strengthening for the maximummoment can be
calculated from the moment and the internal lever arm (see Section 3.2 and Figure 3.3):
240
?
10
6
590
M
0
;
k
z
s1
F
s1
4
406
:
5kN
:
Following on from that it is possible to determine the prestrain in the reinforcing steel
from the area of the reinforcing bars and the modulus of elasticity of the reinforcement:
406
:
5
?
10
3
30
:
79
?
10
2
F
s1
A
s1
?
E
s
ε
s1
?
200
0
:
66 mm
=
m
Assuming a compressive strain in the concrete
ε
c
>
2mm/m and a compression zone
contained completely within the slab, the compressive force in the concrete according to
Section 3.2 can be calculated approximately using the parabola-rectangle diagram for
concrete under compression as follows:
2
c
F
c
b
?
x
?
f
ck
?
α
R
b
?
ξ
?
d
s1
?
f
ck
?
ε
12
ε
c
2
2
c
ε
c
ε
c
ε
s
1
?
653
?
30
?
ε
12
ε
c
1000
?
2
Equilibrium of the internal forces results in an equation for calculating the compressive
strain in the concrete:
F
s1
F
c
406
:
5kN
1000
?
2
c
ε
c
ε
c
?
653
?
30
?
ε
12
ε
c
0
:
66
2
Solving the equation results in
ε
c
=
0.26mm/m. As this value is
>
2mm/m, the above
assumption was justi
ed. The relative depth of the compression zone
ξ
and the depth of
the compression zone
x
can now be determined with the help of the strains. As the depth
of the compression zone is less than the depth of the slab, the above assumption -
compression zone located fully within slab
-
was correct.
ε
0
:
26
c
ε
c
ε
s
ξ
66
0
:
28
0
:
26
0
:
x
ξ
?
d
s1
0
:
28
?
653
182
:
8mm
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