1. By :Dr. Aeid A. Abdulrazeg
Structural Design I
Course Code: CIVL312
By :Dr. Aeid A. Abdulrazeg
By :Dr. Aeid A. Abdulrazeg

2. Doubly-reinforced Sections
When the applied design bending moment exceeds the concrete capacity (i.e bd2fcu) compression reinforcement is required.
Consider the rectangular beam shown in Figure in which the neutral axis depth is equal to d/2 and both tension (As) and compression (As′) reinforcement are present:

3. As is the area of tension reinforcement,
As′ is the area of compression reinforcement,
Ft is the force in the tensile reinforcement,
Fc is the compression force in the concrete,
Fcs is the force in the compression reinforcement,
ζcs is the strain in the compression reinforcement,
b, d, fcu and ζs are as before.

4. The force in the compression reinforcement is dependent on the stress and consequently the strain ζcs. In order to ensure that this is greater than or equal to the yield stress, the ratio of (d′ / x) is limited to 0.37 as shown in Figure below:

5. At the yield with Fy =460 N/mm2, the stress strain
ζsc = ζst =
as specified in the code

6. The moment of resistance of the section is developed by the action of the combined compressive forces (Fc and Fcs) and the tensile force (Ft), separated by the lever arms z and (d − d′ ) respectively.
Consider the moment of the compressive forces about the line of action of Ft :

7. When considering singly-reinforced sections the symbol K is defined in terms of the applied moment as

8. and hence
This value of M can be substituted in the equation for As′ and rewritten as:
The limiting value of K when defining singly-reinforced sections is K ′ = 0.156, giving:

9. Tensile force = Compressive force
Area of compression reinforcement
The required area of tension reinforcement can be determined by equating the compressive and the tensile forces acting on the cross-section.
Tensile force = Compressive force

10. Area of tension reinforcement

11. Example 1
The rectangular beam shown in Figure is required to resist an ultimate design bending moment of 340 kN.m. Using the data given, determine the required areas of main reinforcing steel..

15. Example 2
The rectangular beam shown in Figure is required to resist an ultimate design bending moment of 160 kN.m. Using the data given, determine the required areas of main reinforcing steel.
Given that:
fcu = 30 MPa
fy = 460 MPa
200 mm
300 mm
70 mm
R.C. section for Example 2
By :Dr. Aeid A. Abdulrazeg

16. 0.18 > 0.156. Therefore, the section is doubly reinforced
Since εcs< εyield the compression steel will not yielded, the stress fcs must be calculated and used in the modified equations for areas of reinforcement.
70.00 mm
150.0 mm

17. Area of compression reinforcement
Area of tension reinforcement