ULTIMATE STRENGTH OF R.C. BEAMS (LIMIT STATE OF COLLAPSE BY FLEXURE)

The following assumptions are made for calculating the ultimate moment of resistance or the strength at limit state of flexural collapse of reinforced concrete beams (IS: 456, Clause 38.1): 

1. Plane sections remain plane in bending up to the point of failure i.e. strains are proportional to distance from the neutral axis. 

Figure 1. Strain diagram and stress blocks: (a) Section; (b) Strain diagram (plane sections remain plane); (c) Stress block with partial safety factors, and (d) Simple rectangular stress block (BS).

2. Ultimate limit state of bending failure is deemed to have been reached when the strain in concrete at the extreme bending compression fibre εcu reaches 0.0035. 

3. The stress distribution across the compression face will correspond to the stress-strain diagram for concrete in compression. Any suitable shape like parabolic, rectangular or any combinations of shapes that give results which are in substantial agreement with tests may be assumed for this compression block. For design purpose, the maximum compressive strength in the structure is assumed as 0.67 times the characteristic laboratory cube strength i.e. 2/3fck. With an additional partial factor of γm = 1.5 applied to concrete strength, the values of the maximum concrete stress in a beam will be 0.446fck which can be taken as equal to 0.45fck for all practical purposes. In figure 2, it should be noted that γm = 1.5 is applied over the whole stress-strain curve to obtain the design stress-strain curve for concrete. 

Figure 2. Design stress-strain curves for concrete in compression: (a) Laboratory test curves; and (b) Idealized curves

4. The tensile strength of concrete is neglected as the section is assumed to be cracked up to the neutral axis. 

5. The stress in steel will correspond to the corresponding strain in the steel εs, and can be read off from the stress-strain diagram of the steel. For design purposes, a partial safety factor of 1.15 is used for strength of steel so that maximum stress in steel is limited to fy/1.15 = 0.87fy. It should be noted that the design stress-strain curve for cold worked steel is obtained by applying partial safety factor γm = 1.15 over the region starting from 0.8fy of the actual stress-strain curve for steel. 

6. In order to avoid sudden and brittle compression failure in singly reinforced beams, the limiting value of the depth of compression block is to be obtained according to IS: 456 by assuming the strain of tension steel at failure (εsu) to be not less than the following: 

Where 

εsu = strain in steel at ultimate failure 

fy = characteristic strength of steel 

Es = modulus of elasticity of steel = 200 x 10^3 N/mm^2