Introduction to Elasticity/Sample midterm 4

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Sample Homework Problem 4[edit | edit source]

Part (a)[edit | edit source]

A solid is subjected to stresses as shown by the arrows in the figure below. Indicate the indices for each of the stress components, and whether the stresses should be positive or negative.

Solution[edit | edit source]

Stress components

Part (b)[edit | edit source]

Suppose that the stress tensor field in a body is given by

Find the body force distribution required to maintain equilibrium. (, , and are in meters). Show units.

Solution[edit | edit source]

The equation of equilibrium is

Therefore,

or,

The required body forces are (in MN/m)

Part (c)[edit | edit source]

Find the surface tractions at the internal point in the body on an internal surface with a surface normal .

Solution[edit | edit source]

The surface traction is given by

The stress at point is

Therefore,

The traction vector is (after converting into a unit normal)

Part (d)[edit | edit source]

Find the hydrostatic and deviatoric stress at the point .

Solution[edit | edit source]

The hydrostatic stress is given by

In this case,

Therefore,

The deviatoric stress is given by

Therefore,

Part (e)[edit | edit source]

Find the principal stresses at this point.

Solution[edit | edit source]

The principal stresses can be found using the equation

where is a principal stress. In expanded form,

Substituting the values of stress into the above equation,

Expanding out,

or,

or,

or,

Thus, the first possible value of MPa. Also,

Therefore,

The principal stresses are (in MPa)

Part (f)[edit | edit source]

Find the principal direction corresponding to the intermediate principal stress.

Solution[edit | edit source]

The directions of the principal stresses can be found using the equation

For the principal direction (corresponding to the principal stress , we have,

Hence,

gives us and

Now, . Therefore,

Taking squares of both sides,

So we get,

Therefore,

The direction corresponding to the intermediate principal stress is

Part (g)[edit | edit source]

The symmetry of the stress tensor can be derived from a certain balance principle. Name the principle and write it down in index notation.

Solution[edit | edit source]

The balance principle is

In index notation