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: hxx=−+4060032.. 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/in3 Loading: Tip Load: P=10,000 lbs Figure 4.1 Geometry, material, and loading specifications for a tapered beam. P 2h L xApplication 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 The procedure for creating the finite element model and obtaining the finite element solution for each type of model is presented at the end of this chapter. The 1D beam analysis should be Figure 4.2 Idealized geometry for a tapered beam. P L xApplication of the Finite Element Method Using MARC and Mentat 4-3 performed three times, each with a different mesh. Meshes consisting of 2, 4 and 6 elements should be developed. The 2D analysis should be performed only one time, using the mesh described within the procedure. 4.5 Model Validation Simple hand calculations can be performed to estimate the stresses and deflections in this beam structure. The results of these calculations should be used to assess the validity of the finite element results (i.e., to make sure that the finite element results are reasonable and do not contain any large error due to a simple mistake in the model). The vertical displacement at the end of the bar can be approximated by assuming the bar is of uniform cross-sectional shape. The cross-sectional shape used in this calculation may be, for example, the cross-sectional shape at the mid-point of the bar (at x = 5”). Then the vertical displacement can be estimated using the well-known relation: EIPL33=δ where δ is the tip displacement of the bar, I is the (uniform) second moment of the cross-sectional area about the bending axis, and the other parameters are defined in Figure 4.1. Note that the above relation depends strongly on the value of I, which varies along the length of the actual beam. Thus, the approximation above cannot be expected to be accurate in the current situation, but it should provide a reasonable first-order estimate. The axial stress at any cross-section in the beam can be estimated by neglecting all other stress components and assuming that the axial stress is linearly distributed over the cross-section according to beam theory. From equilibrium, it is found that the resultant moment M at any cross-section is P(L-x), so the axial stress can be estimated using the relation: ()IyxLPIMy−==σ where I is the actual second moment of the area at the cross-section under consideration and y is the vertical coordinate with its origin at the centroid of the beam. 3.6 Post Processing A total of four finite element models were developed – three using 1D two-noded thin beam elements, and one using 2D four-noded bilinear plane stress elements. Based on the results of these analyses, perform and submit the following postprocessing steps.Application of the Finite Element Method Using MARC and Mentat 4-4 (1) Complete the following table: Model ID Tip Displacement Maximum Stress at x = 5” 1D – two elements 1D – four elements 1D – six elements 2D – plane stress elements Validation hand calculation (2) Create a plot of the distribution of vertical displacement along the x-axis as predicted by the four models. Put all of the results on a single plot so that comparisons among the solutions can be made. (3) Create a plot of the distribution of maximum axial stress along the x-axis as predicted by the four models. Put all of the results on a single plot so that comparisons among the solutions can be made. For the beam element models, use hand calculations to calculate the stress based on the predicted bending moment at