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Principal stress, Principal plane & Mohr's circle analysis | |Concepts of principal stress and plane form backbone of material stress analysis. Purpose of this video lecture is to give you a good introduction to concept of Principal stress, Principal plane and Mohr’s circle analysis.Summary of the video lecture is given below

Summary of Lecture

Engineers most often wants to determine maximum normal stress induced at a given point for their design purpose. But there can be infinite number of planes passing through a point, and normal stress on each plane will be different from one another.

There will be one plane on which normal stress value is maximum, this plane is known as Principal plane ( more precisely maximum principal plane) and normal stress on this plane is known as principal stress (more precisely maximum principal stress).

Similarly there will be one more plane on which normal stress value is minimum, this is also a principal plane (minimum principal plane) and normal stress on this plane is known as Principal stress (minimum principal stress).

2 Dimensional Stress Analysis – Stress acting on a 2D element is shown in figure below

Fig.1 Stress boundary conditions on a 2 dimensional element

Mohr’s circle method is the most easy and convenient way to do stress analysis The procedure to draw Mohr’s circle for above case is explained below

Step1 – Draw normal and shear axes with positive axes as shown

Fig.2 Normal and shear axes of a Mohr circleStep2 – Mark normal stress values with sign convention, tensile stress is positive and compression stress is negative

Fig.3 Marking normal stress values on normal axisStep3 - Draw shear stress values starting from already marked normal stress points.

Fig.4 Drawing shear stress valuesStep4 - Connect end of shear stress lines

Fig.4 Connecting end of shear stress linesStep5 - Draw Mohr’s circle assuming the connection line as diameter of the circle

Fig.5 Mohr circle constructionStep6 – Stress Analysis on Mohr circle - To get normal and shear stress values at any plane theta, take angle 2theta in Mohr circle starting from diagonal of the circle and locate a peripheral point as as shown. Shear stress value will be Y axis value and normal stress value will be X axis value.

Fig.6 Determination on normal and shear stress using Mohr cirlce

3 Dimensional Stress Analysis – Stress boundary condition of a 3 dimensional case is shown in left side of  Fig.7. There will be 3 normal stress values induced in a 3 dimensional case, this is shown in right size of the figure.

Fig.7 Stress boundary conditions in a 3 dimensional  body and normal stress values induced in it

There is no graphical method for 3 Dimensional stress analysis, instead we have to use analytical method for this. Values of Principal stress in a 3 dimensional systems are given by solution of following equation.

Where values of stress invariants I1,I2 and I3 are given by

Application of Principal Stresses

Values of principal stresses at a given point is vital design information. Material failure theories extensively use this data to predict whether the design will withstand given load at a specified location.

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What is Von Mises Stress ? | |

Von Mises stress is widely used by the designers to check whether their design will withstand a given load condition. In this lecture we will understand Von Mises stress in a logical way.

Detailed webpage version of the video lecture along with the industrial applications of Von Mises stress are listed below.

Use of Von Mises stress

Von mises stress is considered to be a safe haven for the design engineers.Using this information an engineer can say his design will fail, if the maximum value of Von Mises stress induced in the material is more than strength of the material. It works well for most of the cases, especially when the material is ductile in nature. In coming sections we will have a logical understanding of Von Mises stress and why it is used.

Distortion energy theory

Concept of Von mises stress arises from distortion energy failure theory. According to distortion energy theory failure occurs when the distortion energy in actual case is more than the distortion energy in a simple tension case at the time of failure.

Distortion energy

It is the energy required for shape deformation of a material. During pure distortion shape of the material changes, but volume does not change. This is illustrated in Fig.1.

Fig.1 Representation of a pure distortion case

Distortion energy required per unit volume, ud for a general 3 dimensional case is given interms of principal stress values as

Distortion energy for simple tension case at the time of failure is given as

Expression for Von Mises stress

Above 2 quantities can be connected using distortion energy, so the condition of failure will be as follows.

 Left hand side of above equation is denoted as Von Mises stress.

So as a failure criterion engineer can check, whether Von Mises stress induced in the material exceeds yield strength (for ductile) of the material.So the failure condition can be simplified as

Industrial Application of Von Mises Stress

Distortion energy theory is the most preferred failure theory used in industry. It is clear from above discussions that whenever an engineer resorts to distortion energy theory he can use Von Mises stress as failure criterion.Let's see one example.

Suppose an engineer has to design a cantilever beam using mild steel as material, with a load capacity of 10000 N. Materials properties of mild steel are also shown in figure. Yield stress value of mild steel is 2.5x108 Pa. He wants to check, whether his design will withstand the design load.

Fig.2 A design problem, the cantilever should be able to withstand design loadFollowing figure shows Von Mises stress distribution obtained by FEA analysis of the beam.

Fig.3 Distribution of Von Mises stress in the beam obtained from FEA analysisOne can note that Von Mises stress is maximum towards the fixed end of the beam, and the value is 1.32x108 Pa. This is less than yield point value of mild steel. So design is safe. In short an engineer's duty is to keep maximum value of Von Mises stress induced in the material less than its strength.

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Theories of Failure | | Civil, Mechanical

Good understanding of theories of failure are imperative in design of civil structures or mechanical equipments. This lecture will give you a conceptual introduction to theories of failure. So sit back and Enjoy

Summary of above lecture along with industrial application of Failure Theories are listed below.

Weight Lifter Analogy

Consider a weight lifter problem. In first case he is able to lift maximum up to 50 k.g in a relatively simple fashion. Now consider a second case, where he is lifting the same amount of weight in a different manner.Is it true to say here also his maximum lifting ability is 50 k.g?. Answer to this question could be Yes or No. But if you can well assume his lifting ability is same in second case also , then this can be considered as failure theory for a weight lifter.

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 material has got. Now we will assume that in a complex loading condition also material has same capability. This assumption forms backbone of Failure theories.Concepts of Simple tension test and Principal stresses are main 2 prerequisites to understand Failure theories effectively.

Simple Tension Test

In Simple tension test material is pulled from both the ends,elongation of material(strain) with respect to 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 maximum strength value of 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 figure below.

Fig.1 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 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 3 dimensional complex loading case.

Fig.2 Principal stresses and planes

Failure Theories

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

1. Maximum principal stress theory - Good for brittle materials*

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

2. Maximum shear stress theory - Good for ductile materials

According to this theory when 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 as

3. Maximum normal strain theory - Not recommended

This theory states that when maximum normal strain in actual case is more than maximum normal strain occurred in simple tension test case the material fails. 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 is

Where E is Youngs modulus of the material

4. Total strain energy theory - Good for ductile material

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

Total strain energy in simple tension test at time of failure is given by

So failure condition can be simplified as

5. Shear strain energy theory - Highly recommended

According to this theory when shear strain energy in actual case exceeds shear strain energy in simple tension test at the time of failure the material fails. Shear strain energy in 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 materialOut of 5 theories discussed Shear strain energy theory or Von-mises theory is the most valuable one.

*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.