(1) Derive the equations of beam theory based upon the Euler-Bernoulli assumption by expressing the elastic...

Transcribed Image Text:

(1) Derive the equations of beam theory based upon the Euler-Bernoulli assumption by expressing the elastic strain energy density in terms of the applied moment, M, and the curvature of the beam, , rather than the more fundamental quantities of stress and strain. The strain energy density per unit length, V, is then defined as dy = Mdk. The linear-elastic constitutive relationship between M and is M = EIK, where E is Young's modulus and I is the moment of inertia of the beam. The cross-sectional height of the beam is h and the cross-sectional width is b, both of which are constant. (a) Write an expression for the potential energy, 7, of the system in terms of and q(), which is the distributed force per unit length applied along the length of the beam. Refer to the sketch in Figure 1 for the sign conventions. The -axis coincides with the neutral axis of the beam in its reference state so that the vertical displacement of the beam is given as w(x). (b) Approximate the curvature as dw/dar and invoke the Principle of Minimum Potential Energy to derive the equation that governs the deformation of the beam as well as the appropriate boundary conditions. Discuss the physical interpretation of the boundary conditions. (c) Using these equations, find the displacement of a cantilever beam of unit cross- sectional width and a uniform distributed force of g(x) =q. The end of the beam at x =0 is built into a rigid wall to form the cantilever. (d) Find an approximate solution for the displacement of the same beam assuming a displacement (which is not exact) of the form w(x) = cx + cx and substituting it directly into the Principle of Minimum Potential Energy. Demonstrate that this assumed displacement satisfies the boundary conditions at x = 0. Why is this important? Integrate the expression for to derive the approximate form for the potential energy of the system the result will be the function 7 = T(C, C2). Since this is no longer a functional, but rather a function, find the stationary value of by setting On/Oc = 0 and an/a = 0, and solving the resulting expressions for and c. Hint: 7(C, C) = EI [4cL+12cL + 12cL] q [L +L]. (e) Compare the deformed shape according to both the exact and approximate solutions by plotting them. Also compare the expressions for displacement at = L. Are the boundary conditions satisfied at 2 = = L? q(x) X FIGURE 1. Schematic diagram of beam with a distributed load. (1) Derive the equations of beam theory based upon the Euler-Bernoulli assumption by expressing the elastic strain energy density in terms of the applied moment, M, and the curvature of the beam, , rather than the more fundamental quantities of stress and strain. The strain energy density per unit length, V, is then defined as dy = Mdk. The linear-elastic constitutive relationship between M and is M = EIK, where E is Young's modulus and I is the moment of inertia of the beam. The cross-sectional height of the beam is h and the cross-sectional width is b, both of which are constant. (a) Write an expression for the potential energy, 7, of the system in terms of and q(), which is the distributed force per unit length applied along the length of the beam. Refer to the sketch in Figure 1 for the sign conventions. The -axis coincides with the neutral axis of the beam in its reference state so that the vertical displacement of the beam is given as w(x). (b) Approximate the curvature as dw/dar and invoke the Principle of Minimum Potential Energy to derive the equation that governs the deformation of the beam as well as the appropriate boundary conditions. Discuss the physical interpretation of the boundary conditions. (c) Using these equations, find the displacement of a cantilever beam of unit cross- sectional width and a uniform distributed force of g(x) =q. The end of the beam at x =0 is built into a rigid wall to form the cantilever. (d) Find an approximate solution for the displacement of the same beam assuming a displacement (which is not exact) of the form w(x) = cx + cx and substituting it directly into the Principle of Minimum Potential Energy. Demonstrate that this assumed displacement satisfies the boundary conditions at x = 0. Why is this important? Integrate the expression for to derive the approximate form for the potential energy of the system the result will be the function 7 = T(C, C2). Since this is no longer a functional, but rather a function, find the stationary value of by setting On/Oc = 0 and an/a = 0, and solving the resulting expressions for and c. Hint: 7(C, C) = EI [4cL+12cL + 12cL] q [L +L]. (e) Compare the deformed shape according to both the exact and approximate solutions by plotting them. Also compare the expressions for displacement at = L. Are the boundary conditions satisfied at 2 = = L? q(x) X FIGURE 1. Schematic diagram of beam with a distributed load.