Strength of materials/Lesson 3

Lesson 3: Stress transformation and Mohr's circle

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Last time we talked about Hooke's law and plane stress. We also discussed how the normal and shear components of stress change depending on the orientation of the plane that they act on. In this lecture we will talk about stress transformations for plane stress.

For the rest of this lesson we assume that we are dealing only with plane stress, i.e., there are only three nonzero stress components  ,  ,  . We also assume that these three components are known.

We want to find the planes on which the stresses are most severe and the magnitudes of these stresses.

Stress transformation rules

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Let us consider an arbitrary plane inside an infinitesimal element. Let this plane be inclined at an angle   to the vertical face of the element. A free body diagram of the region to the left of this plane is shown in the figure below.

 
Stress transformation

A balance of forces on the free body in the  -direction gives us

 

or,

 

Using the trigonometric identities

 

we get

 

or,

 

Similarly, a balance of forces in the  -direction leads to

 

or

 

or,

 

Now let us look at a section that is perpendicular to the one we have looked at. This situation is shown in the figure below.

 
Stress transformation

In this case, a balance of forces on the free body in the  -direction gives us

 

or,

 

or,

 

or,

 

A balance of forces in the  -direction gives

 

or,

 

or,

 

From equations (2) and (4) we see that the shear stresses are equal. However the normal stresses on the two planes are different as you can see from equations (1) and (3).

You can think of the two cuts as just the faces of a new infinitesimal element which is at an angle   to the original element as can be seen form the following figure.

 
Stress transformation

If we label the new normal stresses as   and   and the shear stresses as  , then we can write

 

Maximum normal stresses

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What is the orientation of the infinitesimal element that produces the largest normal stress and the largest shear stress? This information can be useful in predicting where failure will occur.

To find angle at which we get the maximum/minimum normal stress we can take the derivatives of   and   with respect to   and set them to zero. So we have

 

or,

 

The angle at which   is a maximum or a minimum is called a principal angle or  .

Now, from the identities (or we can think in terms of a right angled triangle with a rise of   and a run of  )

 

we have

 

Taking another derivative with respect to   we have

 

Plugging in the expressions for   and   we get

 

Clearly   is a maximum while   is a minimum value.

Principal stresses

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The normal stresses corresponding to the principal angle   are called the principal stresses.

We have

 

Plugging in the expressions for   and   we get

 

These principal stresses are often written as   and   or   and   where  .

The value of the shear stress   for an angle of   is

 

Plugging in the expressions for   and   we get

 

Hence there are no shear stresses in the orientations where the stresses are maximum or minimum.

Maximum shear stresses

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Similarly, we can find the value of   which makes the shear stress a maximum or minimum. Thus

 

or

 

In that case

 

The value of the shear stress   for an angle of   is

 

Plugging in the expressions for   and   we get

 

We can show that this is the maximum value of  .

Note that, at the value of   where   is maximum, the normal stresses are not zero.

Mohr's circle

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Mohr's idea was to express these algebraic relations in geometric form so that a physical interpretation of the idea became easier. The idea was based on the recognition that for an orientation equal to the principal angle, the stresses could be represented as the sides of a right-angled triangle.

Recall that

 

We can represent this in graphical form as shown in the figure below. In general, the locus of all points representing stresses at various orientations lie on a circle which is called Mohr's circle.

 
Mohr's circle

Notice that we can directly find the largest normal stress and the small normal stress as well as the maximum shear stress directly from the circle. In three-dimensions there are two more Mohr's circles.

Negative shear stress

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Also note that there is a region where the shear stress   is negative. The convention that we follow is that if the shear stress rotates the element clockwise then it is a positive shear stress. If the element is rotated counter-clockwise then the shear stress is negative.

In the next lecture we will get into some more detail about actually plotting Mohr's circles.


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Professor Brannon's notes on Mohr's circle