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2D MODELING OF THE DEFLECTION OF A SIMPLY SUPPORTED BEAM UNDER POINT OR DISTRIBUTED LOADS YIN-YU CHEN MANE4240 – INTRODUCTION TO FINITE ELEMENT ANALYSIS APRIL 28, 2014 Introduction/Background Maximum deflection of a simply supported elastic beam subject to point or distributed loads M1 Abrams tank (67.6 short tons) equally supported by two simply supported steel beams M1 Abrams Tank Description Value Force 300000 Hull/Track Length 8 Track Width 0.6 Beam Description Value Length 8 Width 0.6 Thickness 0.1 Young's Modulus (E) 2.00E+11 Poisson's Ratio (n) 0.3 Density (r ) 7800 Unit N m m Unit m m m Pa kg/m3 Land mine flush with the ground under the center of each beam Determine the required height of the beam from the ground in order to avoid setting off the land mine Moment-Curvature Equation 2 2 = Analytical Formulation/Solution Moment of Inertia of a Rectangular Cross Section of a Beam = ℎ3 12 Simply Supported Beam with Point Load at the Center = − 3 48 Simply Supported Beam with Uniformly Distributed Load = 54 − 384 Modeling COMSOL Multiphysics 2D Structural Mechanics, Solid Mechanics and Stationary presets Rectangular geometry with prescribed displacements of 0m at bottom corners (x & y for one, y only for the other) to represent a simply supported beam Point load case: -300000N at center (x=4m) Distributed load case: -37500N/m Mesh Extension Validation • Extremely Fine • Finer • Normal • Coarser • Extremely Coarse Results Simply Supported Beam with Point Load at the Center Degrees Max Min of Triangular Edge Vertex Element Element Freedom Size Size Solved x dmax (m) d max (m) Mesh Nodes Elements Elements Elements Extremely Fine 511 414 208 5 0.08 1.60E-04 2074 4.0 -0.29131 Finer 349 120 118 5 0.296 0.001 718 4.0 -0.29125 Normal 349 120 118 5 0.536 0.0024 718 4.0 -0.29125 Coarser 349 120 118 5 1.04 0.048 718 4.0 -0.29124 Extremely Coarse 313 103 105 5 2.64 0.4 624 4.0 -0.29123 d max (cm) -29.131 -29.125 -29.125 -29.124 -29.123 Simply Supported Beam with Uniformly Distributed Load Degrees Max Min of Triangular Edge Vertex Element Element Freedom Size Size Solved x dmax (m) Mesh Nodes Elements Elements Elements Extremely Fine 511 414 208 4 0.08 1.60E-04 2074 4.0 Finer 349 120 118 4 0.296 0.001 718 4.0 Normal 349 120 118 4 0.536 0.0024 718 4.0 Coarser 349 120 118 4 1.04 0.048 718 4.0 Extremely Coarse 313 103 105 4 2.64 0.4 624 4.0 d max (m) d max (cm) -0.18206 -18.206 -0.18202 -18.202 -0.18202 -18.202 -0.18203 -18.203 -0.18202 -18.202 Results Comparison of COMSOL Modeling/Numerical and Analytical Method Analytical Results Mesh Nodes Total Elements d max (m) Result d max (m) % Error COMSOL Normal Mesh Point Load 349 243 -0.29125 -0.32 -8.985% COMSOL Normal Mesh Uniformly Distributed Load 349 242 -0.18202 -0.20 -8.988% Comparison of ANSYS Modeling/Numerical and Analytical Method Results Element Size Nodes 0.05 481 0.075 322 0.1 241 0.33 76 1 25 Total Elements 160 107 80 25 8 d max (m) -0.20008 -0.20006 -0.20008 -0.19969 -0.20008 Analytical Result d max (m) -0.20 -0.20 -0.20 -0.20 -0.20 % Error 0.038% 0.028% 0.038% -0.154% 0.038% Conclusions Maximum deflection of a simply supported elastic beam subject to point or distributed loads may be achieved using either the modeling/numerical or analytical methods Appears that the shape of the cells for the mesh is a major factor in the accuracy of the maximum beam deflection results • Quadrilateral cell mesh may offer the most accurate solution The steel beam requires a minimum height of 0.2m from the ground for the tank to avoid setting off the land mine This study highlights necessity for verifying the reliability of the approximate solution by comparing the results to: A theoretical/exact solution A different modeling approach A mesh extension validation • If results from the COMSOL analysis of the uniformly distributed load across the beam were used without a factor of safety > 1.1 for the height of the beam from the ground, the maximum deflection due to the tank would set off the land mine