#### 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