Limit state design in IS 456: partial safety factors made simple

Older working stress design kept stresses below a fraction of the material strength. Limit state design, the basis of IS 456:2000, instead checks that a structure does not reach any state that makes it unfit for use.

Two families of limit states

• Limit state of collapse: flexure, shear, compression, torsion, overturning. The structure must not fail.
• Limit state of serviceability: deflection and cracking. The structure must stay usable and durable.

Partial safety factors

Loads are increased and strengths reduced, so the margin is placed where the uncertainty is.

Load factors (γ_f) for the limit state of collapse:

Combination	DL	LL	WL/EL
DL + LL	1.5	1.5	—
DL + WL	1.5 or 0.9	—	1.5
DL + LL + WL	1.2	1.2	1.2

The 0.9 on dead load is used where dead load helps stability, as in checking overturning.

Material factors (γ_m): 1.5 for concrete and 1.15 for steel. Concrete gets the larger factor because its quality on site varies more.

Design strengths

Steel:

f_d = f_y1.15 = 0.87 f_y

Concrete in flexure: the cube strength f_ck is first reduced by 0.67 to allow for the difference between a cube and concrete in a real member, and then divided by 1.5:

0.67 f_ck1.5 = 0.446 f_ck

The stress block

IS 456 uses a parabolic–rectangular stress block. Its total compressive force and lever arm work out to:

C = 0.36 f_ck b x_u acting at 0.42 x_u from the top

These two numbers appear in every beam calculation that follows.

Serviceability in one line

Deflection is usually controlled by keeping span/effective depth within the basic values: 7 for cantilevers, 20 for simply supported and 26 for continuous spans (spans up to 10 m), adjusted for steel stress and percentage.