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

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

Description Value UnitForce 300000 N

Hull/Track Length 8 mTrack Width 0.6 m

M1 Abrams Tank

Description Value UnitLength 8 mWidth 0.6 m

Thickness 0.1 mYoung's Modulus (E) 2.00E+11 Pa

Poisson's Ratio (n) 0.3

Density (r ) 7800 kg/m3

Beam

Analytical Formulation/Solution Moment of Inertia of a

Rectangular Cross Section of a Beam

Simply Supported Beam with Point Load at the Center

Simply Supported Beam with Uniformly Distributed Load

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

Simply Supported Beam with Uniformly Distributed Load

Mesh Nodes

Triangular Elements

Edge Elements

Vertex Elements

Max Element

Size

Min Element

Size

Degrees of

Freedom Solved xdmax (m) d

max (m) dmax (cm)

Extremely Fine 511 414 208 5 0.08 1.60E-04 2074 4.0 -0.29131 -29.131Finer 349 120 118 5 0.296 0.001 718 4.0 -0.29125 -29.125

Normal 349 120 118 5 0.536 0.0024 718 4.0 -0.29125 -29.125Coarser 349 120 118 5 1.04 0.048 718 4.0 -0.29124 -29.124

Extremely Coarse 313 103 105 5 2.64 0.4 624 4.0 -0.29123 -29.123

Mesh Nodes

Triangular Elements

Edge Elements

Vertex Elements

Max Element

Size

Min Element

Size

Degrees of

Freedom Solved xdmax (m) d

max (m) dmax (cm)

Extremely Fine 511 414 208 4 0.08 1.60E-04 2074 4.0 -0.18206 -18.206Finer 349 120 118 4 0.296 0.001 718 4.0 -0.18202 -18.202

Normal 349 120 118 4 0.536 0.0024 718 4.0 -0.18202 -18.202Coarser 349 120 118 4 1.04 0.048 718 4.0 -0.18203 -18.203

Extremely Coarse 313 103 105 4 2.64 0.4 624 4.0 -0.18202 -18.202

Results Comparison of COMSOL Modeling/Numerical and Analytical Method

Results

Comparison of ANSYS Modeling/Numerical and Analytical Method Results

Mesh Nodes Total Elements dmax (m)

Analytical

Result dmax (m) % ErrorCOMSOL

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%

Element Size Nodes Total Elements dmax (m)

Analytical Result

dmax (m) % Error

0.05 481 160 -0.20008 -0.20 0.038%0.075 322 107 -0.20006 -0.20 0.028%0.1 241 80 -0.20008 -0.20 0.038%0.33 76 25 -0.19969 -0.20 -0.154%1 25 8 -0.20008 -0.20 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