Elasticity/Constitutive example 1

Example 1

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Take a uniform bar of length "56", cross-section area "6", density "4", Young's modulus "8", and hang it from a rigid, hypothetical ceiling. Calculate how much the bar increases in length due to its own weight. Assume that the acceleration due to gravity is "6" and the bar is fixed rigidly to the ceiling.

Solution

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The tensile stress at a point P, at a distance   from the ceiling is

 

From Hooke's law

 

Now,

 

Integrating,

 

Applying the boundary conditions,   at  , we get  . Therefore, the increase in length is