Limit State Design Method

Limit State Design Method (LSM) is now preferred (IS 800:2007) because it is more rational and economical than the Working Stress Method (WSM).

1. Realistic behaviour: WSM assumes linear elastic behaviour only. LSM considers the actual behaviour up to failure, including post-elastic strength, local and lateral buckling, and redistribution.
2. Separate partial safety factors: WSM uses one factor of safety for everything. LSM applies $\gamma_f$ to loads (different for DL, LL, WL) and $\gamma_m$ to material and connections, so each uncertainty is treated by its own level of variability.
3. Uniform reliability: the probability of failure is nearly the same for all members, whereas WSM gives unknown, non-uniform safety margins.
4. Economy: because the variability of dead loads is smaller than of live loads, LSM gives lighter sections where uncertainty is less.
5. Serviceability check: LSM checks deflection, vibration, durability and fire explicitly, along with strength.
6. Probabilistic basis: characteristic values are statistical (e.g. 95% fractile of strength), allowing improvement as data increases.
7. Compatible with other codes: IS 456 (RCC), IS 1893 and international codes (Eurocode, AISC LRFD) all use limit states, so one design philosophy suits composite and mixed structures.
8. Tools for checking ultimate behaviour: design strength can be compared directly to the factored load, giving a clear margin to failure.

Thus LSM gives safer, more uniform and generally more economical design than WSM.

 WSM                         LSM
 stress                      stress
  fy |      /                 fy |  ______ (plastic)
 0.6fy|----/  perm.           fd |-/
     |   /                       |/ design
     |__/____ strain             |________ strain

In WSM the structure is designed to keep stress below $0.6f_y$ under working load; in LSM it is designed so that the factored load does not exceed the design strength $f_y/\gamma_{m0}$ and deflection stays within limits.

A structure must be safe against collapse and fit for use throughout its design life (IS 800:2007 cl. 5.1, 5.2).

Safety (strength) requirements

• Stability: the structure as a whole must be stable against overturning, sliding and uplift under factored loads.
• Strength: every member and connection must resist the factored forces without yielding, buckling or rupture: design strength $\ge$ factored load effect.
• Fracture / fatigue: brittle fracture, fatigue and progressive collapse (robustness) must be avoided.
• Ductility: failure should be gradual with warning, e.g. yield before rupture, ductile connections.
• Fire resistance for the required period and durability against corrosion.
• Partial factors: $\gamma_f$ on loads ($1.5$ DL+LL, $1.2$ DL+LL+WL), $\gamma_{m0}=1.10$, $\gamma_{m1}=1.25$.

Serviceability requirements

• Deflection under unfactored imposed loads within IS 800 Table 6, e.g. span/300 for floor and roof members not susceptible to cracking (span/360 if susceptible), span/150 for purlins and girts with elastic cladding, span/500 or more for crane gantry girders, and lateral drift of about height/325 for columns.
• Vibration: floors and bridges must not cause discomfort; natural frequency is checked.
• Durability: protection against corrosion by painting, galvanising or weathering steel.
• Fire: protection so that the structure remains serviceable for the required fire rating.
• Repair and maintenance accessibility, and no excessive cracking/damage to finishes.

Serviceability is checked with unfactored (characteristic) loads and strength with factored loads.

Example: for a beam under working load, WSM limits the stress to $0.66f_y$. In LSM the load is multiplied by 1.5 and the design strength is $f_y/1.10$.

• Characteristic strength: the value of material strength (e.g. yield stress $f_y$ or ultimate $f_u$) below which not more than 5% of test results are expected to fall. It is the specified minimum strength in IS 2062.
• Characteristic load: the value of load (dead, live, wind, seismic) which has an accepted probability (about 5%) of being exceeded during the life of the structure. It is taken from IS 875 / IS 1893 / NBC.
• Design strength: characteristic strength divided by the partial safety factor for material, $f_d=f_k/\gamma_m$ (e.g. $f_y/1.10$, $f_u/1.25$).
• Design load: the characteristic load multiplied by the partial safety factor for load, $F_d=\gamma_f F_k$ (e.g. $1.5\times$ DL+LL). Design loads are also called factored loads.

A limit state is a condition beyond which the structure no longer satisfies the design requirements. IS 800:2007 considers two groups.

1. Limit state of strength (ultimate limit state)

Concerns safety against collapse. It includes:

• Loss of equilibrium of the structure or part as a rigid body (overturning, sliding, uplift).
• Yielding, rupture and buckling (local, lateral-torsional, column) of members and connections.
• Transformation into a mechanism.
• Fracture due to fatigue or brittle behaviour.
• Excessive deformation causing instability or loss of strength.

Design check: factored load effect $\le$ design strength, using $\gamma_f$ and $\gamma_m$.

2. Limit state of serviceability

Concerns satisfactory performance under working loads:

• Deflection (including drift) beyond allowable limits (Table 6).
• Vibration causing discomfort or damage.
• Repairable damage or fatigue (cracking).
• Corrosion and durability.
• Fire resistance requirements.

Checked with unfactored (characteristic) loads.

Other

Accidental limit states (fire, explosion, impact) are treated as part of the strength group with accidental load factors.