Working Stress Method (WSM) and Limit State Method (LSM)

Working Stress Method (WSM)                                           

The working stress method was developed during the 20th century and it follows the elastic theory of reinforced concrete sections. In simple words, it will follow the Hooke’s Law. The WSM will follow the assumptions that the structural materials will behave linearly elastic till a point. 

Features of Working Stress Method

• In this method, the stresses will be limited to this point value so that the structure will remain safe. Or in other words, the stresses are limited under this safe stress value. This stress or load value is called as the service or working loads.
• The permissible stress for steel and concrete can be obtained by dividing the characteristics strength of the material by the factor of safety. 
• This is done to restrict the working stress acting on the material under the action of service loads to be within the linear elastics phase of the material.

What is factor of safety (FOS) and Modular Ratio in Working Stress Method?

The factor of Safety is the ratio of the yield stress of the steel reinforcement or the concrete cube strength to the corresponding permissible or working stress.

• The factor of safety used with respect to cube strength of the concrete is 3 and with respect to the yield strength of steel, the value is 1.8. 
• Reinforced concrete is a composite material of concrete and steel. The WSM method takes into consideration the strain compatibility. This means the strain in steel is assumed to be equal to that of concrete.This will make the stress in steel to have a constant relationship with the concrete adjoining by means of a constant factor. This factor is called as Modular Ratio.

Modular Ratio is the ratio of modulus of elasticity of steel (Es) to that of concrete (Ec).This makes the WSM method to be known as Modular Ratio Method.

m = Es/Ec

As mentioned above, the modular ratio is the ratio of modulus of elasticity of steel to that of concrete. (i.e. Es/Ec) But this value will vary for all the grades of concrete. Hence the below formula for modular ratio is taken for the calculation in reinforced concrete designs. Here, 𝛔cbc is the  permissible compressive stress in concrete in bending.

 m = 280/3𝛔cbc 

Table.1. Modular Ratios for Different Grades of Concrete

   

Concrete Grade	σcbc (MPa)	Modular Ratio (m)
M 15	5.0	18.67
M 20	7.0	13.33
M25	8.5	10.98
M30	10.0	9.33
M35	11.5	8.11
M40	13	7.18
M45	14.5	6.44
M50	16.0	5.83

Assumptions in Working Stress Method

The basic assumptions in WSM are:

• Plane Section before bending will remain plane after bending
• The stress-strain relationship will follow Hooke's law
• The tensile stress is taken by steel only
• The modular ratio is given by m = 280/3𝛔cbc

Merits of Working Stress Method

The merits of working stress method are:

• Simplicity in concept - This helps in ease of application
• Large sections are obtained through WSM - This will provide better serviceability performance, like fewer cracks and less deflection. This must be within the working loads
• Only method available to investigate the R.C. section serviceability and service stresses

Knowledge of WSM is essential as it is a part of presently followed Limit state design (LSD) for serviceability Condition.

Demerits of Working Stress Method

The demerits of WSM are:

• The real strength of the structure is not completely utilized by the structure. This won't give the true factor of safety of the structure when it is under failure.
• The modular ratio design will end up giving a structure with a large percentage of steel and uneconomical design.
• Concrete won't possess a definite value of modulus of elasticity because of creep and nonlinear stress strain relationships
• The discrimination of loads that are acting simultaneously and are different in nature is not done by WSM.

Also Read: Why is Working Stress Method Preferred for the Design of Liquid Retaining Structures?

Limit State Method (LSM)                                                         

Before moving into limit state method we also have a method called the ultimate load method. The method will take into consideration the ultimate strength of the reinforced concrete at an ultimate load. 

The WSM method gives good serviceability but it is not realistic under an ultimate state of collapse. But the ULM method will provide realistic safety but does not provide serviceability conditions.

Hence,

An Ideal Method is the one which takes into account not only the ultimate strength of the structure but also the serviceability and durability requirements.

Limit State Method is the new method that is developed by considering all these mentioned requirements. The LSM method is designed for safety against collapse and checked for the serviceability condition at the working loads. Thus the structure is fit for intended use. The LSM will consider the structure both at working and ultimate load levels so that they satisfy the requirements of safety and serviceability.

There are two main limit states:

1. The Limit State of Collapse
2. The Limit State of Serviceability

The Limit State of Collapse

The LSM for collapse will deal the strength and stability of the structure. This will is done by applying the maximum load from all possible combination of loads. This limit state will ensure that any structural part or element will undergo collapse or become unstable under any combination of the expected overloads.

The Limit State of Serviceability

This limit state will deal with the deflection and the cracking of the structures under all possible service loads, the durability under the working environment. This is by considering all anticipated exposures like fire resistance, the stability of structures as a  whole etc.

In order to ensure an adequate degree of safety and serviceability, the relevant limit states have to be considered in the design. The structure must be designed for a most critical combination of load and checked for other limit state combinations.

Limit State Load Combinations and Partial Safety Factors

We have different loads acting on the structure.

1. Deal Load - DL
2. Live Load - LL
3. Wind Load - WL
4. Earthquake Load - EL

The assumption that is followed is in 95% cases the load won't exceed the characteristic loads (Fck). But in certain situations, the structure may be subjected to over loads. This will demand for designing the loads by multiplying it with a suitable factor of safety that is dependent on the nature of loads or their combinations and the limit state that is considered. 

These factors that are multiplied are called as partial safety factor  (γ_f ). Hence,

Design Load =Characteristic Load * Partial Safety Factor for the Load

==>  Fd = F * γ_f

Also Read: Analysis of Composite Sections by Elastic Theory ( Working Stress Method)