Marc.TaperedBar - Application of the Finite Element Method...

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Application of the Finite Element Method Using MARC and Mentat 3-1 Chapter 3: Tapered Bar Keywords: 1D elasticity, 2D elasticity, plane stress, model symmetry, convergence Modeling Procedures: ruled surface, convert 3.1 Problem Statement and Objectives A tapered bar subjected to an axial load will be analyzed in order to predict the distributions of stress and displacement in the bar. The geometrical, material, and loading specifications for the bar are given in Figure 3.1. The thickness of the bar is 2h inches, where h is described by the equation: h x x = - + 4 0 6 0 03 2 . . 3.2 Analysis Assumptions Because the bar is thin in the width (out-of-plane) direction, a state of plane stress can be assumed. Even though the load is exclusively axial, the taper in the bar may cause the state of stress to be two-dimensional in nature. The effect of taper on the stress state depends upon the degree of the taper, and is difficult to assess a-priori . Therefore, both a 2D plane stress elasticity analysis and a 1D elasticity analysis will be performed. Geometry: Material: Steel Length: L=10” Yield Strength: 36 ksi Width: b=1” (uniform) Modulus of Elasticity: 29 Msi Thickness: 2h (a function of x) Poisson’s Ratio: 0.3 Density = 0.0088 slugs/in 3 Loading: Axial Load: P=10,000 lbs Figure 3.1 Geometry, material, and loading specifications for a tapered bar. P 2h L x
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Application of the Finite Element Method Using MARC and Mentat 3-2 3.3 Mathematical Idealization Based on the assumptions above, two different models will be developed and compared. The first model is a 1D elasticity analysis. In this model, the main axis of the bar is discretized using linear two-noded 1D bar/truss finite elements having a uniform thickness within each element. Thus, the geometry is idealized as having a piecewise constant cross-section, as shown in Figure 3.2. The uniform thickness within each element is taken to be equal to the actual thickness of the tapered bar at the x-coordinate corresponding to the centroid of that element. The second model is a 2D plane stress model of the geometry as shown in Figure 3.1. The 2D finite element model of this structure will be developed using 2D plane stress bilinear four-noded quadrilateral finite elements. The present analysis can be greatly simplified by taking advantage of the horizontal plane of symmetry in the bar. The geometry, material properties, and loading conditions are all symmetric about this plane. Therefore, the response of the structure (i.e., displacements, strains, and stresses) will also be symmetric about this plane. Hence, it is necessary to model only a one-half of the bar, as shown in Figure 3.3. The boundary conditions on the symmetry plane are those that occur naturally on this plane, as can be verified by obtaining a solution using the entire bar domain. In particular, the vertical displacement and the shear traction are zero along the symmetry plane. Taking advantage of symmetry reduces the modeling effort,
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This note was uploaded on 07/25/2008 for the course ME 424 taught by Professor Averill during the Spring '05 term at Michigan State University.

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Marc.TaperedBar - Application of the Finite Element Method...

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