Consider the deformation of a uniform, solid cylinder under its own weight. Give
ID: 1412055 • Letter: C
Question
Consider the deformation of a uniform, solid cylinder under its own weight. Given: the cylinder has a uniform mass density , modulus of elasticity E, cross-sectional area A, and a total height L. Treat these quantities (, E, A, and L) as constants. They may appear in any of the final results requested below. a) Using statics (i.e. draw a free body diagram then write and solve an equation of equilibrium based on the free body diagram), determine the axial force resultant N in the cylinder as a function of x (the variable defining the height of the cut above the base) in terms of the given constants , A, and L and the gravitational constant g. Use the coordinate system with x = 0 at the base, and x = L at the top of the cylinder. [Hint: Consider the free body extending from the cut to the top of the cylinder. The height of this free body will be (Lx). Check your result: the internal resultant should be biggest at x = 0, and should equal zero at x = L.] b) Express the stress and strain at cross-section x in terms of the given constants , E, A, L and g. c) By direct integration, determine the total deflection of point B resulting from the weight of the cylinder in terms of , E, A, L and g. [Hint: see Equation 2.9 of the text or use the more general expression: = B A x x dx AB .]
Explanation / Answer
a) the axial force resultant N in the cylinder = * A * L * g * x
b) the stress = E * * A * L * g
=> strain at cross-section x = E * * A * L * g/E
= * A * L * g
c) the total deflection of point B = * A * g * x2/L2