# What is the Limit State method, Working Stress method, and Ultimate Load Method

Written By Sourabh Negi

I cover structural design, construction technology and building materials.

Basically, there are three methods of structural design, working stress, limit state, and ultimate load method. Earlier, the working stress method was used to design the Reinforced Concrete structure as well as the Steel structure.

Keeping in view of the demerits of the Working Stress method, the Ultimate Load method was introduced. Nowadays most of the countries are using the Limit state method of design because it has advantages of both working stress and the ultimate load method of design.

## Methods of Structural Design

### Working Stress Method

This method of design was invented around 1900 and is also known as the elastic method. It assumes that both steel and concrete are perfectly elastic and obey hook’s law. It means that the stress is proportional to the strain.

It is assumed that the bond between steel and concrete is perfect and permissible stresses of material are obtained based on this assumption. Modular ratio (ratio of elasticity of steel and concrete) is used to determine the stresses between steel and concrete.

This method is based on the assumption that the permissible stresses will not increase anywhere in the structure even in the case of worst loading.

In this method, permissible stresses are used which is obtained after applying safety factors on material strength. The factor of safety of around 3 is used in respect of concrete and 1.8 for steel.

This method of design is the most traditional method of design for Reinforced Concrete structures, Steel structures, and Timber structures. The basic design philosophy of this method are  :

1. Assumption in this method is that the material behaves in a linear elastic manner. But we know that material does not strictly behave in a linear elastic manner. This assumption is justified only when allowable stress in the material is kept well below the strength of the material. In other words, allowable stress lies in the initial straight portion of the stress-strain curve.

2. The Second assumption is that adequate safety in the structure can be ensured by limiting the stresses in the structure induced due to the working loads.

#### Limitation of Working Stress Method of Design

• Major defects in this method are, since the method deals only with the elastic behavior of the member, it does not reflect real strength and the true factor of safety against the failure can’t be determined.
• This method assumes that concrete is elastic which is not true.
• This method of design gives a larger percentage of compression steel than the limit state method leading to uneconomical sections. The modular ratio is an imaginary quantity because of the creep and nonlinear stress-strain relationship of concrete, modulus of elasticity of concrete cannot be fixed as one true value.
• In this method of design, the concept of load factor is not present. It fails to distinguish the importance of different types of loads e.g. dead load, live load, earthquake load. It may be possible that some loads are relatively important to other loads. Due to this drawback, the method gives sometimes over-designed structures and sometimes under-designed structures.
• The working stress method of design gives a larger section compared to the limit state method of design and ultimate load method of design.
• The method does not take into account the effect due to shrinkage and creep which is time-dependent.

In this method of design ultimate load or load at collapse is taken as designed load. This collapse load is obtained by multiplying the working load with a suitable load factor. This method gives a fair idea of the exact margin of safety in terms of load.

The major disadvantage of the working stress method is that it does not give a true factor of safety against failure, to overcome it ultimate load method was invented. It is also called the load factor method or ultimate strength method, it is designed to overcome the shortcomings of the working load method.

Ultimate loads are obtained by magnifying the load at the working level by suitable load factors. In this method of design, stresses induced in the structure at the verge of failure (impending failure) are taken which means that the nonlinear part of the stress-strain curve is accounted for. Also to distinguish the importance of different loads, a term load factor is introduced.

The concept of load factor gives freedom to assign different load factors based on the relative importance of the load which was a major drawback in the working stress method of design.

The concept of modular ratio is avoided in this method. This method usually gives lighter sections(economical designs of beams and columns) that are economical as compared to the Working Stress method of design.

#### Limitations

1. We design structure for ultimate load using this method. One cannot guarantee that structure designed at the ultimate load will perform its function at service load also. In other words, large deflections, vibrations, cracks can occur at the working load.

2. Instead of linear portion of stress-strain curve, non-linear portion of the curve is used. But for analysis nonlinear stress-strain curve will be too cumbersome.

3. To determine the stress distribution at ultimate load, stress distribution at working load is calculated and then magnified by using load factor which is not a correct approach as significant stress redistribution can occur due to inelastic and non-linear behavior of the material.

• The ultimate load method takes into account the non-linear portion of the curve instead of the idealized straight curve in the case of the working stress method.
• This method gives a fair idea about the margin of safety which is not present in the working stress method.
• The sections designed using the ultimate load method are thinner thus ULM leads to economical design.
• Different loads can be assigned with different load factors based on the importance of load which is not possible in the working stress method.

### What is Limit State Method?

In the working stress method of design, the structure is designed at working loads only, and in the Ultimate load method of design, the structure is designed at ultimate load only. Limit State method of Design takes into account the safety at ultimate load and serviceability at service load.

In other words, this method of design is the combination of the working stress method of design and the ultimate load method of design. Different values of load factors are assigned for ultimate load and at working load. This method of design method is recommended as per Indian Standard Codes.

#### What is Limit State?

It is the state of failure beyond which either the structure collapses (impending failure) or becomes un-serviceable ( large defections, cracks will develop).

##### Limit State of Collapse

This limit state deals with the strength of the structure in terms of collapse, overturning, sliding, etc. Different limit states of collapse are Flexure, Compression, Shear, Torsion.

##### Limit State of Serviceability

It deals with the deformation of the structure beyond which, the structure will become unserviceable. Different limit states of serviceability are Deflection, Vibration, Cracking, etc.