The following will be discussed in this chapter: Strain energy, strain energy density, elastic strain energy for normal stresses, strain energy for shearing stresses, strain energy for a general state of stress, design for impact loads, work and energy under a single load,. | Third Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Energy Methods Lecture Notes: J. Walt Oler Texas Tech University © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Energy Methods Strain Energy Strain Energy Density Elastic Strain Energy for Normal Stresses Strain Energy For Shearing Stresses Sample Problem Strain Energy for a General State of Stress Impact Loading Example Example Design for Impact Loads Work and Energy Under a Single Load Deflection Under a Single Load © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Sample Problem Work and Energy Under Several Loads Castigliano’s Theorem Deflections by Castigliano’s Theorem Sample Problem 11 - 2 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Strain Energy • A uniform rod is subjected to a slowly increasing load • The elementary work done by the load P as the rod elongates by a small dx is dU = P dx = elementary work which is equal to the area of width dx under the loaddeformation diagram. • The total work done by the load for a deformation x1, x1 U = ∫ P dx = total work = strain energy 0 which results in an increase of strain energy in the rod. • In the case of a linear elastic deformation, x1 2 U = ∫ kx dx = 1 kx1 = 1 P x1 2 1 2 0 © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 11 - 3 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Strain Energy Density • To eliminate the effects of size, evaluate the strainenergy per unit volume, U = V x1 P dx ∫A L 0 ε1 u = ∫ σ x dε = strain energy density 0 • The total strain energy density resulting from the deformation is equal to the area under the curve to ε1. • 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. • Remainder of
