Working Stress Method Notes

Last reviewed 16 Sept 2026 · 6 min read

Philosophy

The working stress method (WSM) is an elastic method. Under service (working) loads, stresses in concrete and steel computed by elastic theory must not exceed permissible stresses, obtained by dividing material strengths by factors of safety. IS 456:2000 now treats limit state design as the main method and keeps WSM in Annex B; WSM remains in use for water-retaining structures (with IS 3370) and is widely asked in exams.

Assumptions

At any cross-section, plane sections remain plane after bending.
All tensile stress is taken by the reinforcement; concrete in tension is ignored.
Stress–strain relationship of both materials is linear (elastic) at working loads.
The modular ratio $m = \frac{280}{3 σ _{c b c}}$ is used (it allows for long-term creep).
Perfect bond between steel and concrete.

Permissible stresses (IS 456 Annex B)

Grade	$σ_{c b c}$ bending compression (N/mm²)	$σ_{cc}$ direct compression (N/mm²)	$m$
M15	5.0	4.0	18.67
M20	7.0	5.0	13.33
M25	8.5	6.0	10.98
M30	10.0	8.0	9.33
M35	11.5	9.0	8.11
M40	13.0	10.0	7.18

Steel	$σ_{s t}$ in tension (N/mm²)	In compression in columns (N/mm²)
Fe 250 (mild)	140 (130 for bars over 20 mm)	130
Fe 415	230	190
Fe 500	275	190

Maximum shear stress with shear reinforcement, $τ_{c, ma x}$ (WSM): M15 1.6, M20 1.8, M25 1.9, M30 2.2, M35 2.3, M40 2.5 N/mm².

Transformed (equivalent) section

Steel of area $A_{s t}$ is replaced by an equivalent concrete area $m A_{s t}$ at the same level (strain compatibility with $E_{s} / E_{c} = m$ ). Compression steel $A_{sc}$ is replaced by $(1.5 m - 1) A_{sc}$ in IS practice (the factor 1.5 allows for creep of the surrounding concrete).

Singly reinforced rectangular beam

Width $b$ , effective depth $d$ , neutral axis depth $k d$ .

FormulaNeutral axis by equating moments of areas

\frac{b ( k d ) ^{2}}{2} = m A_{s t} (d - k d)

Critical (balanced) neutral axis — both materials reach permissible stresses together:

k_{c} = \frac{m σ _{c b c}}{m σ _{c b c} + σ _{s t}}

Lever arm $j d$ with $j = 1 - \frac{k}{3}$ .

Moment of resistance:

M_{r} = \frac{1}{2} σ_{c b c} k j b d^{2} = R b d^{2} (concrete), M_{r} = σ_{s t} A_{s t} j d (steel)

Section	Condition	Behaviour
Balanced	$k = k_{c}$	Both reach permissible stresses together
Under-reinforced	$k < k_{c}$ ( $A_{s t}$ less than balanced)	Steel reaches permissible stress first — gives warning; preferred
Over-reinforced	$k > k_{c}$	Concrete reaches permissible stress first — uneconomical; brittle

For an actual section: find $k$ from the steel provided, then $M_{r}$ is the smaller of the concrete and steel values.

Design constants for common combinations

Concrete / steel	$k_{c}$	$j$	$R$ (N/mm²)
M20 / Fe 250	0.400	0.867	1.21
M20 / Fe 415	0.289	0.904	0.91
M25 / Fe 415	0.288	0.904	1.11

Doubly reinforced beams (WSM)

When $M > R b d^{2}$ and the depth is restricted, compression steel is added. With the critical neutral axis:

M - M_{1} = σ_{sc} A_{sc} (d - d^{'})

where $M_{1} = R b d^{2}$ , and the stress in compression steel is $σ_{sc} = 1.5 m σ_{c}^{'}$ with $σ_{c}^{'}$ the concrete stress at the level of the steel.

Downloaded from civilgyantests.com · Not for resale. Verify every code clause and date against the official source.

Working Stress Method