Test: Working Stress Method - Civil Engineering (CE) MCQ

A balanced section is one in which the area of tension steel is such that the failure of both concrete and steel occurs simultaneously.
For WSM method:
The critical depth of the neutral axis is given by –

From stress distribution diagram,

Where, X_c = kd (k = Neutral axis depth factor for balanced section)

Put the value of modular ratio, 

Thus the neutral axis factor for the balanced section depends only on σ_st and is independent of σ_cbc.
Calculation:
σ_st = 140 N / mm²
Neutral axis depth factor for balanced section

Hence, the depth of neutral axis for a single reinforced rectangular balanced section is 0.40d.

The design methods of reinforced cement concrete structures are as follows:

• Working stress method
  • This method is based upon linear elastic theory or depends on the classical elastic theory.
  • This method ensured adequate safety by suitably restricting the stress in the materials induced by the expected working leads on the structures.
  • The basic assumption of linear elastic behavior is considered justifiable since the specified permissible stresses are kept well below the ultimate strength of the material.
  • The ratio of the yield stress of the steel reinforcement or the concrete cube strength to the corresponding permissible or working stress value is usually called a factor of safety.
• Ultimate load method:
  • This method is based on the ultimate strength of reinforced concrete at ultimate load is obtained by enhancing the service load by some factor called load factor for giving a desired margin of safety. Hence the method is also referred to as the load factor method or the ultimate strength method.
  • In the ULM, the stress condition at the state of in pending collapse of the structure is analyzed, thus using, the non-linear stress-strain curves of concrete and steel.
  • The safety measure in the design is obtained by the use of the proper load factor.
• Limit state method:
  • Limit states are the acceptable limits for the safety and serviceability requirements of the structure before failure occurs.
  • The design of structures by this method will thus ensure that they will not reach limit states and will not become unfit for the use for which they are intended.
  • It is worth mentioning that structures will not just fail or collapse by violating (exceeding) the limit states. Failure, therefore, implies that clearly defined limit states of structural usefulness have been exceeded.
  • Limit states are two types
    • Limit state of collapse
    • Limit state of serviceability

Kani’s method:
It is the method of structural analysis and its displacement method.

Working stress method:

• Working stress method is used for the design of Reinforced concrete, Steel, and Timber Structures. 
• The main assumption in the WSM is that the behavior of structural material is restricted within the linear-elastic region and the safety of it is ensured by restricting the stresses coming on the members by working loads.
• It assumes that both steel and concrete are elastic and obey hook's law.
• FOS for concrete is taken as 3 and for steel, it is 1.8

So, from the above table, the sequence of FOS will be
Short column = Tension member < Long column < Connection
Additional Information
Drawbacks of WSM:

• It assumes the concrete is elastic which is not true.
• It uses a factor of safety for stresses only and not for loads.
• It gives uneconomical sections.

The permissible stresses under bending and direct compression as per IS 456 for different grades of concrete are given below in the tabulated form.

A counterfort retaining wall is a cantilever wall with counterforts, or buttresses, attached to the inside face of the wall to further resist lateral thrust.

• In the counterfort retaining wall, the stem and the base and the base slab are tied together by counterforts, at suitable intervals
• Because of the provision of counterforts, the vertical stem, as well as the heel slab, acts as a continuous slab, in contrast to the cantilevers of cantilever retaining wall
• Counterforts are firmly attached to the face slab as well as the base slab; The earth pressure acting on the face slab is transferred to the counter forts which deflect as vertical cantilevers
• The back of the rear counter forts comes in tension and their front face is under compression
• Hence the inclined (back) face of rear counter forts should be provided with main reinforcement and in front counterforts, the tension develops at the bottom face so it is provided with main reinforcement
• Counterfort retaining walls are economical for height over about 6m

(i) As per CI 8.1.1 of IS 3370:2009, The minimum reinforcement in walls. floors and roofs in each of two directions at right angles. within each surface zone shall not be less than 0.35 percent of the surface zone.
(ii) For high strength deformed bars and not less than 0.64 percent for mild steel reinforcement bars. The minimum reinforcement can be further reduced to 0.24 percent for deformed bars and 0.40 percent for plain round bars for tanks having any dimension not more than 15 m.
(iii) In wall slabs less than 200 mm in thickness, the calculated amount of reinforcement may all be placed in one face. For ground slabs less than 300 mm thick  the calculated reinforcement should be placed in one face as near as possible to the upper surface consistent with the nominal cover. Bar spacing should generally not exceed 300 mm or the thickness of the section. whichever is less.
Additional Information

• Maximum cement content is 400 kg/m³ to take care of shrinkage effect.
• Minimum cement content is 320 kg/m³.
• Minimum grade of concrete is M30.
• Maximum w/c ratio is 0.45.
• Minimum nominal cover is 45 mm.
• Maximum allowed crack width is 0.2 mm in LSM design.

For the given condition:

Given
Grade of Concrete = M20 , Grade of Steel = Fe500
We know
Critical depth of Neutral axis is given by:
x_c = k × d

Where σ_st - Permissible stresses in Steel 
σ_cbc - Permissible stresses in concrete
For Grade M25 → σ_cbc = 8.5 N/mm²
For Fe500 → σ_st = 0.55 × f_y = 0.55 × 500 =  275 N/mm²
Equating in the formula

∴ x_c = 0.2528 × d ≈ 0.253 × d
Comparing it with x_c = k × d ⇒ k values come out to be 0.253
Important Points
For LSM Method,
As per IS 456:2000, cl.38.1

Steel beam theory is used to find the approximate value of the moment of resistance of a doubly reinforced beam specially when the area of compression steel is equal to or more than the area of the tensile steel.
In this theory, we assume concrete is weak in compression also then we get same amount of steel in both tension & compression, hence all the  moment will be resisted by steel only and concrete will be of no use.
In the steel beam theory :
(i) Concrete is completely neglected.
(ii) The moment of resistance is taken equal to the amount of the couple of compressive and tensile steel.
(iii) The permissible stress in compressive steel is taken as equal to the permissible stress in tensile steel.

In the elastic theory for the reinforced concrete structure, concrete and reinforcing steel are converted into one material. This is done by using the modular ratio ‘m’.
Modular ratio: It is the ratio of modulus of elasticity of steel and concrete.
m = E_s/E_c
However, concrete has varying moduli, as it is not a perfectly elastic material.
Therefore, its short-term modulus  is not considered and long-term modulus of elasticity is considered to take in account the effect of creep and shrinkage.
∴ The modular ratio is taken as  where σ_cbc is permissible compressive stress in concrete in bending.
Calculation:
Modular ratio ‘m’ (considering creep effect),

The permissible stresses under bending and direct compression as per IS 456 for different grades of concrete are given below in the tabulated form.