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

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Application of the Finite Element Method Using MARC and Mentat 4-1 Chapter 4: Tapered Beam Keywords: elastic beam, 2D elasticity, plane stress, convergence, deformed geometry Modeling Procedures: ruled surface, convert 4.1 Problem Statement and Objectives A tapered beam subjected to a tip bending load will be analyzed in order to predict the distributions of stress and displacement in the beam. The geometrical, material, and loading specifications for the beam are given in Figure 4.1. The geometry of the beam is the same as the structure in Chapter 3. The thickness of the beam is 2h inches, where h is described by the equation: h x x = - + 4 0 6 0 03 2 . . 4.2 Analysis Assumptions Because the beam is thin in the width (out-of-plane) direction, a state of plane stress can be assumed. The length-to-thickness ratio of the beam is difficult to assess due to the severe taper. By almost any measure, however, the length-to-thickness ratio of the beam is less than eight. 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: Tip Load: P=10,000 lbs Figure 4.1 Geometry, material, and loading specifications for a tapered beam. P 2h L x
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Application of the Finite Element Method Using MARC and Mentat 4-2 Hence, it is unclear whether thin beam theory will accurately predict the response of the beam. Therefore, both a 2D plane stress elasticity analysis and a thin elastic beam analysis will be performed. 4.3 Mathematical Idealization Based on the assumptions above, two different models will be developed and compared. The first model is a beam analysis. In this model, the main axis of the bar is discretized using straight two- noded 1D thin beam finite elements having a uniform cross-sectional shape within each element. Thus, the geometry is idealized as having a piecewise constant cross-section, as shown in Figure 4.2. The uniform thickness within each element is taken to be equal to the actual thickness of the tapered beam 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 4.1. The 2D finite element model of this structure will be developed using 2D plane stress bilinear four-noded quadrilateral finite elements. In the present analysis, the geometry and material properties are symmetric about the mid-plane of the beam. However, the loading is not symmetric about this plane, so the response of the structure (i.e., displacements, strains, and stresses) will not be symmetric about this plane. Hence, it is necessary to model the entire domain of the beam, as shown in Figure 4.1. 4.4 Finite Element Model
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Marc.TaperedBeam - Application of the Finite Element Method...

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