Concrete Structures and Design Methods

The working stress method (WSM) designs a member so that stresses under working (service) loads stay within permissible elastic limits. The limit state method (LSM) designs a member so that it will not reach any limit state (collapse or serviceability) when factored loads act, using partial safety factors for loads and materials.

Stress and strain diagrams at section level

   WSM (service load)        LSM (collapse)
   strain    stress          strain    stress
   |\        |\              0.0035    0.446fck
   | \       | \ f_c<=σcbc    |\        |========|
 x |  \    x |  \           xu |  \    xu |        |
 --+---  ----+---        ----+---    ----+--------+
   |  /      |   T=σst.Ast    |  /       |   T=0.87fy.Ast
   | /       |                | /        |
   linear   linear      linear   parabola+rectangle

In WSM, a triangular concrete stress block with $f_c \le \sigma_{cbc}$ and steel stress $\le \sigma_{st}$ is used. In LSM, the strain at the extreme concrete fibre is $0.0035$ and the compressive block is parabola plus rectangle (average stress $0.36 f_{ck}$, centroid at $0.416x_u$ from the top), with steel at $0.87 f_y$.

Characteristic strength

The characteristic strength of a material is the value of strength below which not more than 5 percent of the test results are expected to fall (95 percent reliability). It is the basis of design in limit state method (IS 456:2000, cl. 36.1).

• Concrete: characteristic compressive strength $f_{ck}$ is the 28-day strength of 150 mm cubes. For M20, $f_{ck} = 20$ N/mm$^2$.
• Steel: characteristic strength $f_y$ is the yield stress (or 0.2% proof stress). For Fe415, $f_y = 415$ N/mm$^2$.
• Statistically, $f_{ck} = f_m - 1.65\,s$, where $f_m$ is the mean strength and $s$ the standard deviation.

Characteristic load

The characteristic load is the value of a load which has a 95 percent probability of not being exceeded during the life of the structure (IS 456:2000, cl. 36.2). Since statistical data are rarely available, the loads given in the codes are taken as characteristic loads: dead load from IS 875 (Part 1), live load from IS 875 (Part 2), wind load from IS 875 (Part 3), and earthquake load from IS 1893.

Use in design

Design values are obtained with partial safety factors:

• Design strength $= \dfrac{f_k}{\gamma_m}$, with $\gamma_m = 1.5$ for concrete and $1.15$ for steel.
• Design load $= \gamma_f \times$ characteristic load, with $\gamma_f = 1.5$ for DL+LL, for example.

Reinforced concrete members can be designed by the following methods.

1. Working stress method (WSM)

• Based on elastic theory; concrete and steel are assumed to behave elastically, with strain proportional to distance from the neutral axis.
• Concrete tension is ignored; steel is transformed into an equivalent concrete area using modular ratio $m = 280/(3\sigma_{cbc})$.
• Safety is obtained by limiting stresses under service load to permissible values ($\sigma_{cbc}$, $\sigma_{st}$) which are fractions of the strength.
• Drawbacks: ignores non-linear behaviour, creep and shrinkage effects, gives no idea of true load factor.

2. Ultimate load method (load factor method)

• Design is based on the ultimate strength of the section at failure, using non-linear stress-strain of concrete.
• Working loads are multiplied by a load factor (e.g. 1.5 for DL, 2.2 for LL in the older ACI/CP114 practice) to get ultimate loads, and section strength is compared with it.
• Gives a realistic factor of safety against collapse but does not control deflection and cracking at working loads.

3. Limit state method (LSM)

• Adopted as the main method of IS 456:2000. The structure is designed so that it does not reach a limit state, i.e. a state beyond which it becomes unfit for use.
• Limit state of collapse (flexure, shear, torsion, compression) is checked with factored loads and design material strengths ($f_{ck}/1.5$, $f_y/1.15$).
• Limit state of serviceability (deflection, cracking, durability, vibration) is checked at service loads.
• Uses partial safety factors for loads and materials, so safety is more rational and economical.

4. Probabilistic method

• Loads and strengths are treated as random variables; design is done for a target probability of failure (reliability). It is the basis of the partial factors in LSM but is not used directly in routine design.

Summary: IS 456:2000 uses LSM as the primary method, with WSM kept as an alternative (Annex B) mainly for liquid-retaining and special cases.

Working stress method (WSM)

Stresses under working load are limited to permissible values, using linear elastic theory and modular ratio $m$.

Advantages

• Simple, quick calculations; long experience in use.
• Stresses at service load are known, so deflection and cracking are reasonably controlled.
• Suitable for liquid-retaining structures (cracking governs).

Disadvantages

• Real stress-strain behaviour is non-linear, so the method does not predict true strength.
• Single factor of safety; the margin against collapse is unknown and varies with the load combination.
• Ignores creep, shrinkage and different variability of dead and live loads.
• Uneconomical sections.

Limit state method (LSM)

The structure is designed for the limit states of collapse and serviceability using characteristic loads, partial safety factors and the non-linear stress-strain curves.

Advantages

• Realistic, uses ultimate strength so a known margin against collapse.
• Different partial factors for loads and materials reflect their different uncertainty.
• Serviceability (deflection, cracking) is checked explicitly.
• More economical sections.

Disadvantages

• More complex; separate checks for several limit states.
• Serviceability checks may need trial and revision.

Comparison of balanced, under-reinforced and over-reinforced sections

Under-reinforced sections are always preferred because they give warning (large deflection and cracks) before failure.

In WSM safety is given by one factor of safety applied to the material strength to get a permissible stress. In LSM safety is split into partial safety factors, one for material strength ($\gamma_m$) and one for loads ($\gamma_f$).

Working stress method

• Concrete: $\sigma_{cbc} = \dfrac{f_{ck}}{\text{F.O.S.}}$. For M20, $\sigma_{cbc} = 7$ N/mm$^2$, so the factor of safety is about $20/7 \approx 2.86$ (nearly 3 on cube strength).
• Steel: $\sigma_{st} = \dfrac{f_y}{\text{F.O.S.}}$. For Fe415, $\sigma_{st} = 230$ N/mm$^2$, giving $415/230 \approx 1.8$; for mild steel, $\sigma_{st} = 140$, so $250/140 \approx 1.8$.
• Loads are taken at working value with no factor.

Limit state method (IS 456:2000, cl. 36.4)

Design strengths: concrete $0.67f_{ck}/1.5 = 0.446f_{ck}$ in the stress block; steel $0.87f_y$ (as $f_y/1.15$).

Comparison

A lower material factor is acceptable in LSM because uncertainty in loads is accounted separately by $\gamma_f$.