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

Doubly-reinforced Sections When the applied design bending moment exceeds the concrete capacity (i.e. 0.156 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:

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.

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:

At the yield with Fy =460 N/mm2, the stress strain ζsc = ζst = 0.00219 as specified in the code

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 :

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

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:

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

Area of tension reinforcement

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..

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

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

Area of compression reinforcement Area of tension reinforcement