Page 1 Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem Page 2 Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem o A uniform rod is subjected to a slowly increasing load o The elementary work done by the load P as the rod elongates by a small dx is which is equal to the area of width dx under the load-deformation diagram. dU Pdx elementary work == o The total work done by the load for deformation x 1 , which results in an increase of strain energy in the rod. 1 0 x U P dx total work strain energy = == ò 1 2 11 1 11 22 0 x U kx dx kx Px = == ò o In the case of a linear elastic deformation, Strain Energy Strain Energy Page 3 Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem o A uniform rod is subjected to a slowly increasing load o The elementary work done by the load P as the rod elongates by a small dx is which is equal to the area of width dx under the load-deformation diagram. dU Pdx elementary work == o The total work done by the load for deformation x 1 , which results in an increase of strain energy in the rod. 1 0 x U P dx total work strain energy = == ò 1 2 11 1 11 22 0 x U kx dx kx Px = == ò o In the case of a linear elastic deformation, Strain Energy Strain Energy Strain Energy Density Strain Energy Density o To eliminate the effects of size, evaluate the strain energy per unit volume, 1 1 0 0 x xx U P dx V AL u d strain energy density e se = == ò ò o As the material is unloaded, the stress returns to zero but there is a permanent deformation. Only the strain energy represented by the triangular area is recovered. o Remainder of the energy spent in deforming the material is dissipated as heat. o The total strain energy density is equal to the area under the curve to e 1 . Page 4 Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem o A uniform rod is subjected to a slowly increasing load o The elementary work done by the load P as the rod elongates by a small dx is which is equal to the area of width dx under the load-deformation diagram. dU Pdx elementary work == o The total work done by the load for deformation x 1 , which results in an increase of strain energy in the rod. 1 0 x U P dx total work strain energy = == ò 1 2 11 1 11 22 0 x U kx dx kx Px = == ò o In the case of a linear elastic deformation, Strain Energy Strain Energy Strain Energy Density Strain Energy Density o To eliminate the effects of size, evaluate the strain energy per unit volume, 1 1 0 0 x xx U P dx V AL u d strain energy density e se = == ò ò o As the material is unloaded, the stress returns to zero but there is a permanent deformation. Only the strain energy represented by the triangular area is recovered. o Remainder of the energy spent in deforming the material is dissipated as heat. o The total strain energy density is equal to the area under the curve to e 1 . Strain Strain- -Energy Density Energy Density o The strain energy density resulting from setting e 1 = e R is the modulus of toughness. o If the stress remains within the proportional limit, 1 22 11 0 22 xx E u Ed E e es ee = == ò o The strain energy density resulting from setting s 1 = s Y is the modulus of resilience. 2 2 Y Y u modulus of resilience E s == Page 5 Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem o A uniform rod is subjected to a slowly increasing load o The elementary work done by the load P as the rod elongates by a small dx is which is equal to the area of width dx under the load-deformation diagram. dU Pdx elementary work == o The total work done by the load for deformation x 1 , which results in an increase of strain energy in the rod. 1 0 x U P dx total work strain energy = == ò 1 2 11 1 11 22 0 x U kx dx kx Px = == ò o In the case of a linear elastic deformation, Strain Energy Strain Energy Strain Energy Density Strain Energy Density o To eliminate the effects of size, evaluate the strain energy per unit volume, 1 1 0 0 x xx U P dx V AL u d strain energy density e se = == ò ò o As the material is unloaded, the stress returns to zero but there is a permanent deformation. Only the strain energy represented by the triangular area is recovered. o Remainder of the energy spent in deforming the material is dissipated as heat. o The total strain energy density is equal to the area under the curve to e 1 . Strain Strain- -Energy Density Energy Density o The strain energy density resulting from setting e 1 = e R is the modulus of toughness. o If the stress remains within the proportional limit, 1 22 11 0 22 xx E u Ed E e es ee = == ò o The strain energy density resulting from setting s 1 = s Y is the modulus of resilience. 2 2 Y Y u modulus of resilience E s == Elastic Strain Energy for Normal Stresses Elastic Strain Energy for Normal Stresses o In an element with a nonuniform stress distribution, 0 lim total strain energy V U dU u U u dV V dV D® D = = == D ò o For values of u < u Y , i.e., below the proportional limit, 2 2 x U dV elastic strain energy E s == ò o Under axial loading, x P A dV A dx s== 2 0 2 L P U dx AE = ò 2 2 PL U AE = o For a rod of uniform cross section,Read More

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