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Energy_02_Elastic_Strain_Energy

Energy_02_Elastic_Strain_Energy - Section 5.2 5.2 Elastic...

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Section 5.2 Solid Mechanics Part I Kelly 180 5.2 Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for stored energy will then be used to solve some elasticity problems using the energy methods mentioned in the previous section. 5.2.1 Strain energy in deformed Components Bar under axial load Consider a bar of elastic material fixed at one end and subjected to a steadily increasing force P , Fig. 5.2.1. The force is applied slowly so that kinetic energies are negligible. The initial length of the bar is L . The work dW done in extending the bar a small amount Δ d is 1 Δ = Pd dW (5.2.1) Figure 5.2.1: a bar loaded by a constant stress It was shown in §4.3.2 that the force and extension Δ are linearly related through EA PL / = Δ , Eqn. 4.3.5, where E is the Young’s modulus and A is the cross sectional area. This linear relationship is plotted in Fig. 5.2.2. The work expressed by Eqn. 5.2.1 is the white region under the force-extension curve (line). The total work done during the complete extension up to a final force P and final extension Δ is the total area beneath the curve. The work done is stored as elastic strain energy U and so EA L P P U 2 2 1 2 = Δ = (5.2.2) If the axial force (and/or the cross-sectional area and Young’s modulus) varies along the bar, then the above calculation can be done for a small element of length dx . The energy stored in this element would be EA dx P 2 / 2 and the total strain energy stored in the bar would be 1 the small change in force during this small extension may be neglected P L Δ d Δ
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