hw4_s04_sln

# hw4_s04_sln - EMA 405 Spring 2004 Solutions for HW#4 1.1...

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Unformatted text preview: EMA 405, Spring 2004 Solutions for HW #4 1.1. Introduction This problem examines a basic axisymmetric configuration, a thick-walled pipe with internal pressure only, but we will model it in different ways. It gives us a chance to compare planar analysis with axisymmetric analysis in a problem that can be modeled accurately with either type of analysis. For planar analysis, our finite element mesh, i.e. the problem domain, covers a portion of the r- θ plane, and in the axisymmetric analysis, the finite-element mesh covers the r-z plane. We will also investigate the applicability of plane stress vs. plane strain in this configuration. 1.2. Preliminary Analysis This problem can be solved analytically (as described shortly just to help address whether the z- dimension has the behavior of a plane stress calculation or a plane strain calculation). Setting-up and solving the analytical problem are beyond what’s expected for this homework, however, so we can simply look up the solution in Roark, Table 13.5, case 1a on p. 683. For an external radius a , an internal radius b , and an internal pressure P , the hoop and radial stresses are given as ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 b a r r a b P b a r r a b P rr − − − = − + = σ σ θθ In particular, note that each of these stresses has a term that is proportional to r-2 and a term that is independent of r . The terms that are independent of r are the same in the two stresses, and the terms proportional to r-2 are the same except for a minus sign. With a =0.4 m, b =0.2 m, and P =100 MPa, the hoop stress at r = b is 166.7 MPa (tension), and the hoop stress at r = a is 66.7 MPa. The radial stress at r = b is –P or –100 MPa and increases with r , hitting 0 MPa at the outer surface r = a , where there is no applied pressure. Formulas for ∆ a and ∆ b are also provided by Roark: + − + = ∆ − = ∆ ν 2 2 2 2 2 2 2 2 b a b a E Pb b b a ab E P a As expected for an internal pressure, ∆ b is larger. Here, ∆ b evaluates to 0.19667 mm, and ∆ b evaluates to 0.1333 mm. 1/13 The above information from Roark is all that is needed for a PA, but let’s take a closer look at some of the analytics. The radial and hoop stresses are expressed in terms of the strains in Cook’s Eq. 6.1-4. Inserting the appropriate relations for strain (in particular, the somewhat unusual form of strain for hoop strain, r U r / = θθ ε , we have − + ∂ ∂ + ∂ ∂ − + = + ∂ ∂ + ∂ ∂ − − + = r U z U r U E r U z U r U E r z r r z r rr ) 1 ( ) 2 1 )( 1 ( ) 1 ( ) 2 1 )( 1 ( ν ν ν ν ν σ ν ν ν ν ν σ θθ For the force balance equation ( B = ⋅ ∇ σ ), there are no body forces ( B =0), and there is no net force in the azimuthal direction. For an axisymmetric problem, the radial and axial components of the force balance equation (in cylindrical coordinates) are then ( ) ( ) ( ) ( ) 1 1 = ∂ ∂ + ∂ ∂...
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hw4_s04_sln - EMA 405 Spring 2004 Solutions for HW#4 1.1...

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