Reinforced concrete

1. 1. A rectangular cross-section reinforced concrete element with dimensions b × h = 20 × 45 cm
is reinforced with 2∅12 reinforcement of class A 400 in the compression zone, and 3∅25
reinforcement of class A400 in the tension zone, Rs = Rsc = 350 MPa, 𝐴𝑠
′ = 2,26 cm2, As =
14.72 cm2. Concrete is B 25 and its compressive strength is Rb= 14.5 MPa. Determine the load-
bearing capacity of the section and the relative depth of the compression zone of the section.
Take as= 6 cm and 𝑎𝑠
′ = 3 cm for the tension and compression zones during the calculations.
SOLUTİON: First find the load-bearing height of the cross-section:
ℎ0 = h - 𝑎𝑠= 45-6= 39 cm
Then from the moment equation we calculate the moment coefficient:
𝑅𝑠 ⋅ 𝐴𝑠= 𝜉 ⋅ 𝑅𝑏 ⋅ 𝑏 ⋅ ℎ0 + 𝑅𝑠𝑐 ⋅ 𝐴𝑠
′
=> 𝜉=
𝑅𝑠⋅𝐴𝑠−𝑅𝑠𝑐⋅𝐴𝑠
′
𝑅𝑏⋅𝑏⋅ℎ0
Based on the given:
𝜉=
𝑅𝑠⋅𝐴𝑠−𝑅𝑠𝑐⋅𝐴𝑠
′
𝑅𝑏⋅ 𝑏 ⋅ ℎ0
=
350 ⋅ 14.72 −350 ⋅2.26
14.5 ⋅ 20 ⋅ 39
= 0.385 < 𝜉𝑅 = 0,5333
The we obtained for the moment coefficient:
𝛼𝑚 = 𝜉 ⋅ (1 −
𝜉
2
)= 0.3856 ⋅ (1-
0.3856
2
) = 0.311 < 𝛼𝑚𝑟=0,3910
Finally, for the load-bearing capacity of a double-layer reinforced section given by the moment
equation, we obtain:
𝑀𝑢𝑙𝑡 = 𝛼𝑚 ⋅ 𝑅𝑏 ⋅ 𝑏 ⋅ ℎ0
2
+ 𝑅𝑠𝑐 ⋅ 𝐴𝑠
′
⋅(ℎ0 − 𝑎𝑠
′
)= 0.3113 ⋅ 14.5 ⋅ 103
⋅ 0,2 ⋅ 0,392
+
+ 350 ∙ 103
∙ 2.26 ∙ 10−4
∙ (0.39 − 0,03) = 165,7 𝑘𝑁 ∙ 𝑚

2. 2. Rectangular cross-sectional element with dimensions b × h = 30 × 40 cm is made of B 15
concrete with compressive strength Rb = 8.5 MPa in the first group of limit states and class A400
with total area As = 19.63 cm2 in the tensile zone. Reinforced with 5∅25 reinforcement, Rs = Rsc=
350 MPa. Determine the load-bearing capacity of the section and the relative depth of the
compression zone. Take as = 5 cm during calculations
3. Rectangular cross-sectional element with dimensions b × h = 30 × 85 cm is made of B20
concrete with strength Rb= 11.5 MPa in the first group of limit cases and class A 400 with total
area As = 2.27 cm2 in the tension zone. It is reinforced with 2∅12 periodic profile reinforcement,
and in the tension zone by 4 ∅ 25 of class A400 reinforcement with total area As= 19.625 cm2,
Rs = Rsc = 350 MPa. Determine the load-bearing capacity of the section and the relative depth of
the compression zone. Take 𝑎𝑠
′ = 3 cm and as= 4 cm during the calculations.
SOLUTİON: First find the load-bearing height of the cross-section:
ℎ0 = h - 𝑎𝑠= 85-4= 81 cm
Then from the moment equation we calculate the moment coefficient:
𝑅𝑠 ⋅ 𝐴𝑠= 𝜉 ⋅ 𝑅𝑏 ⋅ 𝑏 ⋅ ℎ0 + 𝑅𝑠𝑐 ⋅ 𝐴𝑠
′
=> 𝜉=
𝑅𝑠⋅𝐴𝑠−𝑅𝑠𝑐⋅𝐴𝑠
′
𝑅𝑏⋅𝑏⋅ℎ0
Based on the given:
𝜉=
𝑅𝑠⋅𝐴𝑠−𝑅𝑠𝑐⋅𝐴𝑠
′
𝑅𝑏⋅ 𝑏 ⋅ ℎ0
=
350 ⋅ 19.625 −350 ⋅2.27
11.5 ⋅ 30 ⋅ 81
= 0.217 < 𝜉𝑅 = 0,5333
The we obtained for the moment coefficient:
𝛼𝑚 = 𝜉 ⋅ (1 −
𝜉
2
)= 0.2174 ⋅ (1-
0.2174
2
) = 0.193 < 𝛼𝑚𝑟=0,3910

3. Finally, for the load-bearing capacity of a double-layer reinforced section given by the moment
equation, we obtain:
𝑀𝑢𝑙𝑡 = 𝛼𝑚 ⋅ 𝑅𝑏 ⋅ 𝑏 ⋅ ℎ0
2
+ 𝑅𝑠𝑐 ⋅ 𝐴𝑠
′
⋅(ℎ0 − 𝑎𝑠
′
)= 0.1938 ⋅ 11.5 ⋅ 103
⋅ 0,3 ⋅ 0,812
+
+ 350 ∙ 103
∙ 2.27 ∙ 10−4
∙ (0.81 − 0,03) = 500,7 𝑘𝑁 ∙ 𝑚
4. Rectangular cross-sectional element with dimensions sm
h
b 80
40

 , is reinforced in the
compression zone by 12
3 MPa
Rsc 350
 reinforcement with area 2
39
,
3 sm
As 
 of A400 class
(𝑅𝑠𝑐 = 350 𝑀𝑃𝑎), but in the tension zone by 40
3 reinforcement ( MPa
Rs 350
 ) with area
2
68
,
37 sm
As  of class A 400 . The strength of concrete for calculations in the case of class
50
B and the limit-state of the first group is MPa
Rb 5
,
27
 . Determine the load-bearing capacity
of the section. During calculations, take 𝑎𝑠
′ = 3cm and as = 6 cm.
SOLUTİON: First find the load-bearing height of the cross-section:
ℎ0 = h - 𝑎𝑠= 80-6= 74 cm
Then from the moment equation we calculate the moment coefficient:
𝑅𝑠 ⋅ 𝐴𝑠= 𝜉 ⋅ 𝑅𝑏 ⋅ 𝑏 ⋅ ℎ0 + 𝑅𝑠𝑐 ⋅ 𝐴𝑠
′
=> 𝜉=
𝑅𝑠⋅𝐴𝑠−𝑅𝑠𝑐⋅𝐴𝑠
′
𝑅𝑏⋅𝑏⋅ℎ0
Based on the given:
𝜉=
𝑅𝑠⋅𝐴𝑠−𝑅𝑠𝑐⋅𝐴𝑠
′
𝑅𝑏⋅ 𝑏 ⋅ ℎ0
=
350 ⋅ 37.68 −350 ⋅3.39
27.65 ⋅ 40 ⋅ 74
= 0.146 < 𝜉𝑅 = 0,5333
The we obtained for the moment coefficient:

4. 𝛼𝑚 = 𝜉 ⋅ (1 −
𝜉
2
)= 0.1466⋅ (1-
0.1466
2
) = 0.135 < 𝛼𝑚𝑟=0,3910
Finally, for the load-bearing capacity of a double-layer reinforced section given by the moment
equation, we obtain:
𝑀𝑢𝑙𝑡 = 𝛼𝑚 ⋅ 𝑅𝑏 ⋅ 𝑏 ⋅ ℎ0
2
+ 𝑅𝑠𝑐 ⋅ 𝐴𝑠
′
⋅(ℎ0 − 𝑎𝑠
′
)= 0.1359 ⋅ 27.65 ⋅ 103
⋅ 0,4 ⋅ 0,742
+
+ 350 ∙ 103
∙ 3.39 ∙ 10−4
∙ (0.74 − 0,03) = 907.3 𝑘𝑁 ∙ 𝑚
5. Rectangular cross-sectional element with dimensions b × h = 30 × 60 cm, is reinforced in the
compression zone by 4∅10 reinforcement with area 𝐴𝑠
′ = 3,14 cm2 of A400 class
(𝑅𝑠𝑐 = 350 𝑀𝑃𝑎), but in the tension zone by 4∅28 reinforcement (𝑅𝑠 = 350 𝑀𝑃𝑎) with area
𝐴𝑠 = 24,63 𝑐𝑚2 of class A 400 . The strength of concrete for calculations in the case of class B
60 and the limit-state of the first group is 𝑅𝑏 = 33.0 𝑀𝑃𝑎. Determine the load-bearing capacity of
the section. During calculations, take 𝑎𝑠
′ = 3cm and as = 4.5 cm.
SOLUTİON: First find the load-bearing height of the cross-section:
ℎ0 = h - 𝑎𝑠= 60-4.5= 55.5 cm
Then from the moment equation we calculate the moment coefficient:
𝑅𝑠 ⋅ 𝐴𝑠= 𝜉 ⋅ 𝑅𝑏 ⋅ 𝑏 ⋅ ℎ0 + 𝑅𝑠𝑐 ⋅ 𝐴𝑠
′
=> 𝜉=
𝑅𝑠⋅𝐴𝑠−𝑅𝑠𝑐⋅𝐴𝑠
′
𝑅𝑏⋅𝑏⋅ℎ0
Based on the given:
𝜉=
𝑅𝑠⋅𝐴𝑠−𝑅𝑠𝑐⋅𝐴𝑠
′
𝑅𝑏⋅ 𝑏 ⋅ ℎ0
=
350 ⋅ 24.63 −350 ⋅3.14
33 ⋅ 30 ⋅ 55.5
= 0.136 < 𝜉𝑅 = 0,5333

5. The we obtained for the moment coefficient:
𝛼𝑚 = 𝜉 ⋅ (1 −
𝜉
2
)= 0.1369⋅ (1-
0.1369
2
) = 0.127 < 𝛼𝑚𝑟=0,3910
Finally, for the load-bearing capacity of a double-layer reinforced section given by the moment
equation, we obtain:
𝑀𝑢𝑙𝑡 = 𝛼𝑚 ⋅ 𝑅𝑏 ⋅ 𝑏 ⋅ ℎ0
2
+ 𝑅𝑠𝑐 ⋅ 𝐴𝑠
′
⋅(ℎ0 − 𝑎𝑠
′
)= 0.1275 ⋅ 33 ⋅ 103
⋅ 0,3 ⋅ 0,5552
+
+ 350 ∙ 103
∙ 3.14 ∙ 10−4
∙ (0.555 − 0,03) = 446,5 𝑘𝑁 ∙ 𝑚