Transcript Project
2D M
ODELING OF THE
D
EFLECTION OF A
S
IMPLY
P
OINT OR
S
UPPORTED
D B
ISTRIBUTED EAM
L U
OADS NDER
Y IN -Y U C HEN MANE4240 – I NTRODUCTION TO F INITE E LEMENT A NALYSIS A PRIL 28, 2014
Introduction/Background
M1 Abrams Tank Description Value
Force 300000
Hull/Track Length
8
Track Width
0.6
Beam Description Length Value
8
Width Thickness
0.6
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/m 3
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
𝐸𝐼 𝑑 2 𝑦 𝑑𝑥 2 = 𝑀
Analytical Formulation/Solution
Moment of Inertia of a Rectangular Cross Section of a Beam
𝐼 = 𝑏ℎ 3 12
Simply Supported Beam with Point Load at the Center
𝛿 𝑚𝑎𝑥 𝑃𝐿 3 = − 48𝐸𝐼
Simply Supported Beam with Uniformly Distributed Load
𝛿 𝑚𝑎𝑥 5𝑤𝐿 4 = − 384𝐸𝐼
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 Mesh Extremely Fine Finer Normal Nodes
511 349 349
Coarser Extremely Coarse
349 313
Triangular Elements Edge Elements Vertex Elements
414 120 120 120 103 208 118 118 118 105 5 5 5 5 5
Max Min Degrees of Element Size
0.08
Element Freedom Size Solved x
d
max (m)
d
max (m)
1.60E-04 2074 4.0
-0.29131
0.296
0.536
0.001
0.0024
718 718 4.0
4.0
-0.29125
-0.29125
1.04
2.64
0.048
0.4
718 624 4.0
4.0
-0.29124
-0.29123
d
max (cm)
-29.131
-29.125
-29.125
-29.124
-29.123
Simply Supported Beam with Uniformly Distributed Load Mesh Extremely Fine Finer Normal Nodes
511 349 349
Coarser Extremely Coarse
349 313
Triangular Elements Edge Elements Vertex Elements
414 120 120 120 103 208 118 118 118 105 4 4 4 4 4
Max Size
0.08
Min Size
1.60E-04
Degrees of Element Element Freedom Solved x
2074 d
max
4.0
(m)
0.296
0.536
1.04
2.64
0.001
0.0024
0.048
0.4
718 718 718 624 4.0
4.0
4.0
4.0
d
max (m)
-0.18206
-0.18202
-0.18202
-0.18203
-0.18202
d
max (cm)
-18.206
-18.202
-18.202
-18.203
-18.202
Results
Comparison of COMSOL Modeling/Numerical and Analytical Method Results
d
max (m) Analytical Result
d
max (m) % Error Mesh COMSOL Normal Mesh - Point Load Nodes Total Elements
349 243
COMSOL Normal Mesh - Uniformly Distributed Load
349 242 -0.29125
-0.18202
-0.32
-0.20
-8.985% -8.988%
Comparison of ANSYS Modeling/Numerical and Analytical Method Results Element Size
0.05
0.075
0.1
0.33
1
Nodes
481 322 241 76 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
% Error
0.038% -0.20
-0.20
-0.20
-0.20
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