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Elasticity/Constitutive example 1

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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