Working Stress & Limit State Method

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

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

Permissible stresses (IS 456 Annex B)

Maximum shear stress with shear reinforcement, ${\tau }_{c,max}$ (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 $A_{st}$ is replaced by an equivalent concrete area $m{A}_{st}$ at the same level (strain compatibility with $E_s/E_c=m$). Compression steel $A_{sc}$ is replaced by $(1.5m-1)A_{sc}$ 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 $kd$.

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

Doubly reinforced beams (WSM)

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

where $M_1=Rb{d}^2$, and the stress in compression steel is ${\sigma }_{sc}=1.5m{\sigma }_c^{\prime }$ with ${\sigma }_c^{\prime }$ the concrete stress at the level of the steel.

What a limit state is

A limit state is a condition beyond which a structure (or part of it) no longer fulfils its design purpose. Limit state design makes the structure safe against all relevant limit states with an acceptable probability, instead of only keeping elastic stresses below permissible values (as working stress design does).

The structure is designed for the collapse limit states and checked for serviceability.

Characteristic values

• Characteristic strength of a material — the value below which not more than 5% of test results are expected to fall ($f_{ck}$ for concrete, $f_y$ for steel).
• Characteristic load — the load that has only a 95% probability of not being exceeded during the life of the structure. In practice, loads from IS 875 (dead, imposed, wind) and IS 1893 (earthquake) are used as characteristic loads.

Partial safety factors

For materials (${\gamma }_m$)

Concrete gets the larger factor because its strength is more variable (site mixing, compaction, curing).

For loads (${\gamma }_f$)

Assumptions for flexure (limit state of collapse)

1. Plane sections remain plane — strain varies linearly with depth.
2. Maximum strain in concrete at the outermost compression fibre is 0.0035 in bending.
3. The design stress–strain curve of concrete is parabolic–rectangular, with maximum design stress $0.67f_{ck}/1.5=0.446f_{ck}$.
4. Tensile strength of concrete is ignored.
5. Stresses in reinforcement follow the design stress–strain curve of steel, with ${\gamma }_m$ = 1.15 (design yield $0.87f_y$).
6. The maximum strain in the tension steel at failure must not be less than $\frac{f_y}{1.15E_s}+0.002$ (that is, $\frac{0.87f_y}{E_s}+0.002$), so that steel yields before concrete crushes (ductile failure).

The stress block

Integrating the parabolic–rectangular stress diagram over the compression depth $x_u$ of a rectangular section of width $b$:

Where the numbers come from: the average stress of the block is $0.8095\times 0.446f_{ck}=0.36f_{ck}$, and its centroid lies at $0.416x_u\approx 0.42x_u$.