A good understanding of theories of failure are imperative in the design of civil structures or types of mechanical equipment. This lecture will give you a conceptual introduction on the theories of failure. So sit back and Enjoy

*Failure Theories*are given below.

## The Weight Lifter Analogy

Consider a weight lifter problem.

Fig.1 A weight lifter analogy |

## The Backbone of Failure Theories

In materials also we can apply the same concept of weight lifter failure theory.Here material will undergo a simple force test(simple tension test), so one can determine what's the maximum load capability the material has. Now, we will assume that in a complex loading condition also, the material has the same capability. This assumption forms the backbone of Failure theories.Concepts of Simple tension test and Principal stresses are the main 2 prerequisites to understand the Failure theories effectively.

## Simple Tension Test

In Simple tension test material is pulled from both the ends, the elongation of material(strain) with respect to the load is noted. From such an observation one can easily determine maximum strength of the material. For ductile material *upper yield point* is considered to be maximum strength of material, while for brittle material it is taken as *ultimate strength* of the material. From the maximum strength value of the material, values of various other parameters can easily be calculated.Simple tension graph and *upper yield point* value for a ductile material case is shown in the figure below.

Fig.2 Simple tension test |

## Principal Stress

Principal stress is the maximum normal stress occurring at a given point. In order to find out this value easy way is to do a Mohr circle analysis. Once you know Principal stress values you can go ahead with failure theories.Figure below shows principal stress values induced at point in a 3 dimensional complex loading case.

Fig.3 Principal stresses and planes |

## The Failure Theories

The interesting thing in the Failure theories is that, just by looking at the name of the theory you will be able to formulate condition of failure in an actual case. Just make sure that your concept of STT and Principal stresses are clear. The theories along with its usability is given below.

**Maximum principal stress theory**- Good for brittle materials***Maximum shear stress theory**- Good for ductile materials**Maximum normal strain theory**- Not recommended**Total strain energy theory**- Good for ductile material
According to this theory when the total strain energy in actual case exceeds the total strain energy in simple tension test at the time of failure, the material fails. The total strain energy in actual case is given by
The total strain energy in simple tension test at time of failure is given by
So failure condition can be simplified as
**Shear strain energy theory**- Highly recommended

According to this theory when the maximum principal stress induced in a material under complex load condition exceeds the maximum normal strength in a simple tension test the material fails. So the failure condition can be expressed as

According to this theory when the maximum shear strength in actual case exceeds maximum allowable shear stress in simple tension test the material case. Maximum shear stress in actual case in represented as

Maximum shear stress in simple tension case occurs at angle 45 with load, so maximum shear strength in a simple tension case can be represented as Comparing these 2 quantities one can write the failure condition asThis theory states that, when the maximum normal strain in actual case is more than maximum normal strain occurred in simple tension test case the material fails. The maximum normal strain in actual case is given by

Maximum strain in simple tension test case is given by So condition of failure according to this theory isWhere*E*is Youngs modulus of the material

According to this theory when the shear strain energy in the actual case exceeds shear strain energy in simple tension test at the time of failure the material fails. Shear strain energy in the actual case is given by

Shear strain energy in simple tension test at the time of failure is given by So the failure condition can be deduced as Where*G*is shear modulus of the material

*Since brittle materials does not have *yield point*, you can use *ultimate tensile stress* as failure criterion.

## Industrial Applications of Failure Theories

Nowadays FEA based solvers are well integrated to use failure theories. User can specify kind of failure criterion in his solution method. Shear strain energy theory is the most commonly used method. These softwares can produce *Von-mises stress* along material,which is based on *Shear strain energy theory*. So user can check whether maximum Von-mises stress induced in the body crosses maximum allowable stress value. It is a common practice to introduce Factor of Safety(F.S) while designing, in order to take care of worst loading scenario. So the engineer can say his design is safe if following condition satisfies.