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
:
926
?
d
s1
0
:
926
?
140
129
:
60 mm
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):
9
:
25
?
10
6
129
m
0
;
k
z
s1
60
71
:
34 kN
=
m
Following on from that it is possible to determine the strain in the reinforcing steel from
the area of the reinforcing bars and the modulus of elasticity of the reinforcement:
F
s1
:
71
:
34
?
10
3
4
:
43
?
10
2
F
s1
a
s1
?
E
s
ε
s1
?
200
0
:
75 mm
=
m
Assuming a compressive strain in the concrete
ε
c
>
2mm/m, 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
c
ε
c
ε
c
ε
s
1
ε
12
ε
c
1000
?
?
140
?
20
?
2
Equilibrium of the internal forces results in an equation for calculating the compressive
strain in the concrete:
F
s1
F
c
71
2
c
ε
c
ε
c
ε
12
ε
c
:
34 kN
=
m
1000
?
?
140
?
20
?
0
:
75
2
Solving the equation in the permissible range of values results in
ϵ
c
=
0.21mm/m. As
this value is
>
2mm/m, the above assumption was justi
ed. The relative depth of the
compression zone can now be determined with the help of the strains:
ε
c
ε
c
ε
s
0
:
21
ξ
75
0
:
22
0
:
21
0
:
Using the coef
cient
k
a
(for
ε
c
>
2mm/m), i.e. the result according to Section 3.2, it is
now possible to determine the internal lever arm:
ε
c
24
4
?
ε
c
8
21
24
4
?
0
:
21
8
0
:
k
a
0
:
34
a
k
a
?
ξ
?
d
s1
0
:
34
?
0
:
22
?
140
10
:
41 mm
z
s1
d
s1
a
140
10
:
41
129
:
59 mm
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