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Unformatted text preview: EMA 405, Fall 2003 Solutions for HW #4 1. The two cylinders slide over each other only when the outer cylinder has its temperature raised by 100 K. In the cooling process, the outer cylinder will shrink and squeeze the inner cylinder to a slightly smaller radius. Both will develop compressive radial stresses, but the contraction of the outer cylinder results in tensile hoop stresses, while the resistance of the inner cylinder leads to compressive hoop stresses. The critical aspect of the problem set-up is to provide an initial strain associated with the thermal contraction. When there are no shear stresses and strains, the stress-strain relation with an initial strain is: = z r z z r r E 1 1 1 1 in cylindrical geometry (see Cook, first parts of Ch. 3 & 6). Here, the outer cylinder undergoes an initial radial displacement of u = Tr . Thus, our initial radial and azimuthal strains are T r u T r u r = = = = assuming axisymmetry. The outer cylinder will also undergo an initial axial strain due to the temperature change, and the problem stated that the two cylinders do not slip past each other (to make the FEA set-up easier). In order to avoid a 2D computation in the preliminary analysis, we need to decide how to treat the axial strains. I chose to analyze the contraction as if the two cylinders were able to slide past each other, so that z 0, like a plane stress analysis for the cross section of the cylinder. An alternative is to use z- z0 0, like a plane strain analysis of the cross section. Since the radius and length are roughly the same, neither approximation is really suitable, but we can get an idea of the general behavior from either approximation. The FEA does treat the 2D nature of the actual problem, where r-z warping occurs from the two cylinders being stuck together. Combining the initial strains and stress-strain relations from above with the z 0 approximation, the radial and azimuthal stresses are given by = T T E r r 1 1 1 2 1/10 The radial and azimuthal strains are related to radial displacement, and well also ignore axial variations, so inserting the radial and azimuthal stresses into the radial force balance equation without body forces leads to an ODE for u in terms of radius. For completeness, the ODE is ( ) ( ) 1 = = r r r r r r Whats important for our preliminary analysis is that the radial displacement and stresses follow the same behavior as the cylinder with external pressure considered in HW3 #4, after taking the initial thermal strain into account....
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- Spring '04