Problem 3 Consider the beam bending problem shown below. The beam is of length L, stiffness E, and density p. It has a r

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Problem 3 Consider The Beam Bending Problem Shown Below The Beam Is Of Length L Stiffness E And Density P It Has A R 1 (136.61 KiB) Viewed 14 times

Problem 3 Consider the beam bending problem shown below. The beam is of length L, stiffness E, and density p. It has a rectangular cross-section of width 2a and thickness a (with the thickness defined as the dimension in the y-direction), and is subjected to the effect of gravity (with acceleration g in the negative y-direction), and a concentrated transverse load P applied at x=2L/3 as indicated in the figure. K P fim k * The boundary conditions are described by a torsional spring of stiffness K at x=0 and by a linear spring of stiffness k at x=L. (a) Provide the full boundary value problem (i.e., governing differential equations and boundary conditions) describing the equilibrium of the beam. Give the expression of the quantities (e.g., moment of inertia) entering these equations. (b) Derive the solution for the deflection of the beam in the absence of gravity and in the absence of the linear spring (i.e., k=0). Put your solution in a non-dimensional form. Check that the non-dimensional parameter describing the torsional spring is indeed non- dimensional. (c) Explain in details how you would compute the transverse deflection at x=2L/3 for the 'original problem' (i.e., with the linear spring and the effect of gravity) using the Castigliano method. Important: You do not need to compute all the integrals and derivatives. Just explain in details (and with sentences) the steps you need to get the solution (including the expression of the internal bending moments, ...) so that the only thing left to do would be use a symbolic math software to perform the calculations. (d) What do you think happens in the limiting case when both k and K tend to infinity? Sketch the type of beam bending problem you would then get.