Chapter6 - AE 321 Practice Problems Chapter 6: Extension,...

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AE 321 – Practice Problems Chapter 6: Extension, Bending and Torsion 1. A rectangular bar of a linearly elastic homogeneous and isotropic material, and unit thickness, is loaded in extension with a load P per unit length linearly distributed over its width b as shown. If the bar is very long (i.e. a>>b), find (a) the traction (three dimensional) on the cylindrical surface shown in the figure as a function of the applied load and bar dimensions. (b) the displacement field in the bar (solve only for the in-plane two dimensional displacement field). Sketch the deformed shape of the cylindrical surface as predicted by your solution. a/2 b P cylindrical surface
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2. (a) Show that the stresses ! xx = ky + mz yy = zz = xy = xz = yz = 0 represent the solution to some problem in elasticity for a linear elastic, isotropic and homogeneous body with no body forces. (b) Determine the loading which must be applied to the bar below so that the resulting stress components are those given in (a). 3. (a) Show that the stress function for the torsion of a long cylinder of solid triangular cross section (shown below) subjected to a given torque T is given by: ! x , y ( ) = C x " 3 y " 2 3 h # $ % & ( x + 3 y " 2 3 h # $ % & ( x + 1 3 h # $ % & ( (b) Find the stress, strain and displacement state for this bar assuming a homogeneous, isotropic linear elastic solid ( E and ν ) and no body forces. L x y a 2c y 2 h 3 h 3 x = 3 y + 2 h 3 h 3 h x y
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4. A solid right circular cylinder of radius R has one end at z =0 fixed and the one at z = L rotated about the cylinder axis ( z axis). The rotation angle is α ( <<1) and the torque needed is T . The cylinder is linearly elastic, isotropic, homogeneous and has no body forces. ( r, θ , z ) is a cylindrical coordinate system with z aligned with the cylinder axis.
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Chapter6 - AE 321 Practice Problems Chapter 6: Extension,...

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