2. Finite Element Analysis
Randall Bock, Professor of Continuing Education
The Pennsylvania State University

3. rgb@psu.edu
• Research Engineer
• Professor of Continuing Education
• Happy Valley SWUG, Leader
• CSWP

5. Mission Statement
Students should be proficient in
communicating and analyzing
designs as 3-D models.

7. www.hvswug.org

8. Pedagogy
• Instructive theory
• Trainee teachers learn their subject and also the
pedagogy appropriate for teaching that subject.

9. Andragogy
• The art and science of teaching adults.

10. Here’s the Plan
• SolidWorks Simulation
The Design Cycle
Finite Element Method
SolidWorks Simulation
Defining a Simulation Study
Controlling the Mesh
Accessing the Results
Indentifying Singularities
Instructional Examples
Tips and Tricks

11. A Traditional Design Cycle
• Build a 3D model. SolidWorks
• Manufacture prototype.
$ Prototype
• Test the prototype.
• Analyze results
Test
modify the model
build a new prototype
No
test it again Satisfied?
repeat
Yes
Mass Production

12. FEA Integrated Design Cycle
SolidWorks FEA
No
Satisfied?
Yes
$ Prototype
No Test
Mass Production

13. Tools > SimulationXpress
SimulationXpress
• Limitations
1. PARTs (one solid body)
2. Static analysis only (stress)
3. Optimize one variable
4. Isotropic materials
Sidebar
5. Uniform loads
6. Fixed restraints

14. Tools > Add-Ins
SolidWorks Simulation
• Advantages
Parts & Assemblies
Non-linear, thermal, buckling, frequency, drop test,
optimization, fatigue
Isotropic & orthotropic materials
Uniform & non-uniform loads
Multiple restraints
More…

15. SolidWorks Simulation 2010
Simulation Premium Nonlinear
Flow
Simulation
Simulation Professional
Static* Frequency Buckling Thermal Drop Test
Linear
Dynamics
Sustain-
Pressure Event-based
Event-
ability
Motion* Fatigue Optimization Motion
Vessel
Composites
*Included with SolidWorks Premium

16. FEA Fundamentals
• Define and discretize the domain
• Specify approximating function and B.C.
• Create and converge system of equations
• Resolve for quantities of interest

17. FEA Fundamentals
• Define the domain

18. FEA Fundamentals
• Discretize the domain
MESH

19. FEA Fundamentals
• Discretize using tetrahedrons: 1st order (linear)
1 element
First Order Structural Tetrahedron Element:
4 nodes 4 nodes
12 dof
∴ 12 x 12 matrix

20. FEA Fundamentals
• Discretize using tetrahedrons: 2nd order (quadratic)
1 element
Second Order Structural Tetrahedron Element:
10 nodes 10 nodes
30 dof
∴ 30 x 30 matrix

21. FEA Fundamentals
• Discretize using tetrahedrons: 2nd order
1 element
Second Order Structural Tetrahedron Element:
10 nodes 10 nodes
30 dof
∴ 30 x 30 matrix
Curved

22. FEA Fundamentals
• Neighboring elements share nodes
4 element
7 nodes
DOF ?

23. FEA Fundamentals
• Cube
12 element
9 nodes
DOF ?

24. FEA Fundamentals
• Specify approximating function

26. And also an…

28. FEA Fundamentals
• Specify approximating function

29. FEA Fundamentals
• Specify approximating function (2D Triangle, linear)
u ( e ) ( x, y ) = α o + α 1 x + α 2 y
EACH ELEMENT

30. FEA Fundamentals
• Specify approximating function (new node values)
u ( e ) ( x, y ) = α o + α 1 x + α 2 y

31. FEA Fundamentals
• Specify approximating function (unknown coefficients)
u ( e ) ( x, y ) = α o + α 1 x + α 2 y

32. FEA Fundamentals
• Specify approximating function (current node values)
u ( e ) ( x, y ) = α o + α 1 x + α 2 y

33. FEA Fundamentals
• Specify approximating function (2D Triangle, quadratic)
u( x, y ) = αo + α1 x + α 2 y + α 3 x 2 + α 4 xy + α5 y 2
ν ( x, y ) = β o + β1 x + β 2 y + β 3 x 2 + β 4 xy + β 5 y 2
Sidebar

34. FEA Fundamentals
• Specify approximating function
u( x, y ) = αo + α1 x + α 2 y
ν ( x, y ) = β o + β1 x + β 2 y

35. FEA Fundamentals
• Specify approximating function
u ( x, y ) = α o + α 1 x + α 2 y
ν ( x, y ) = β o + β1 x + β 2 y
• Equilibrium equations (Hook’s Law 2D)
E ∂ ⎛ ∂u ∂v ⎞ ∂ ⎛ ∂u ∂v ⎞
+ ⎜ + ⎟+G ⎜ + ⎟ = 0
1 − υ 2 ∂x ⎜ ∂x ∂y ⎟
⎝ ⎠ ∂y ⎜ ∂y ∂x ⎟
⎝ ⎠
∂ ⎛ ∂u ∂v ⎞ E ⎛ ∂u ∂v ⎞
⎜ ∂y ∂x ⎟ 1 − υ 2 ⎜υ ∂x + ∂y ⎟ = 0
+G ⎜ + ⎟+
∂x ⎝ ⎜ ⎟
⎠ ⎝ ⎠

36. FEA Fundamentals
• Specify approximating function
u( x, y ) = αo + α1 x + α 2 y
ν ( x, y ) = β o + β1 x + β 2 y
• Equilibrium equations (Hook’s Law 2D)
E ∂ ⎛ ∂u ∂v ⎞ ∂ ⎛ ∂u ∂v ⎞
+ ⎜ + ⎟+G ⎜ + ⎟ = 0
1 − υ 2 ∂x ⎜ ∂x ∂y ⎟
⎝ ⎠ ∂y ⎜ ∂y ∂x ⎟
⎝ ⎠
∂ ⎛ ∂u ∂v ⎞ E ⎛ ∂u ∂v ⎞
⎜ ∂y ∂x ⎟ 1 − υ 2 ⎜υ ∂x + ∂y ⎟ = 0
+G ⎜ + ⎟+
∂x ⎝ ⎜ ⎟
⎠ ⎝ ⎠

37. FEA Fundamentals
• Specify approximating function
u( x, y ) = αo + α1 x + α 2 y
ν ( x, y ) = β o + β1 x + β 2 y
• Equilibrium equations (Hook’s Law 2D)
E ∂ ⎛ ∂u ∂v ⎞ ∂ ⎛ ∂u ∂v ⎞
+ ⎜ + ⎟+G ⎜ + ⎟ = 0
1 − υ 2 ∂x ⎜ ∂x ∂y ⎟
⎝ ⎠ ∂y ⎜ ∂y ∂x ⎟
⎝ ⎠
∂ ⎛ ∂u ∂v ⎞ E ⎛ ∂u ∂v ⎞
⎜ ∂y ∂x ⎟ 1 − υ 2 ⎜υ ∂x + ∂y ⎟ = 0
+G ⎜ + ⎟+
∂x ⎝ ⎜ ⎟
⎠ ⎝ ⎠

38. Moduli conversion
Sidebar

39. E ∂ ⎛ ∂u ∂v ⎞ ∂ ⎛ ∂u ∂v ⎞
+ ⎜ + ⎟+G ⎜ + ⎟ = 0
⎜ ∂x ∂y ⎟
1 − υ ∂x ⎝
2
⎠ ∂y ⎜ ∂y ∂x ⎟
⎝ ⎠ u( x, y ) = αo + α1 x + α 2 y
+G
∂ ⎛ ∂u ∂v ⎞
⎜ + ⎟+
E ⎛ ∂u ∂v ⎞
⎜ ∂y ∂x ⎟ 1 − υ 2 ⎜υ ∂x + ∂y ⎟ = 0
⎜ ⎟ ν ( x, y ) = β o + β1 x + β 2 y
∂x ⎝ ⎠ ⎝ ⎠

40. FEA Fundamentals
• Specify approximating function
u( x, y ) = αo + α1 x + α 2 y
ν ( x, y ) = β o + β1 x + β 2 y
• Equilibrium equations (Hook’s Law 2D)
E ∂ ⎛ ∂u ∂v ⎞ ∂ ⎛ ∂u ∂v ⎞
+ ⎜ + ⎟+G ⎜ + ⎟ = 0
1 − υ 2 ∂x ⎜ ∂x ∂y ⎟
⎝ ⎠ ∂y ⎜ ∂y ∂x ⎟
⎝ ⎠
∂ ⎛ ∂u ∂v ⎞ E ⎛ ∂u ∂v ⎞
⎜ ∂y ∂x ⎟ 1 − υ 2 ⎜υ ∂x + ∂y ⎟ = 0
+G ⎜ + ⎟+
∂x ⎝ ⎜ ⎟
⎠ ⎝ ⎠

41. FEA Fundamentals
• Specify approximating function
u( x, y ) = αo + α1 x + α 2 y
ν ( x, y ) = β o + β1 x + β 2 y
• Equilibrium equations (Hook’s Law 2D)
E ∂ ⎛ ∂u ∂v ⎞ ∂ ⎛ ∂u ∂v ⎞
+ ⎜ + ⎟+G ⎜ + ⎟ = 0
1 − υ 2 ∂x ⎜ ∂x ∂y ⎟
⎝ ⎠ ∂y ⎜ ∂y ∂x ⎟
⎝ ⎠
∂ ⎛ ∂u ∂v ⎞ E ⎛ ∂u ∂v ⎞
⎜ ∂y ∂x ⎟ 1 − υ 2 ⎜υ ∂x + ∂y ⎟ = 0
+G ⎜ + ⎟+
∂x ⎝ ⎜ ⎟
⎠ ⎝ ⎠

42. ∂v
Slope Field: =v−x
∂x
v
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x x
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x
∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x

43. ∂v
Solutions: =v−x
∂x
v
x

44. ∂v
Solutions: =v−x
∂x
v
x

45. FEA Fundamentals
• Specify the boundary conditions
Load
Restraint

46. FEA Fundamentals (basic strategy)
• Create and converge system of equations
1) Plug initial mesh geometry into
u( x, y ) = αo + α1 x + α 2 y
ν ( x, y ) = β o + β1 x + β 2 y
2) Plug new node values from 1) into +
E ∂ ⎛ ∂u ∂v ⎞ ∂ ⎛ ∂u ∂v ⎞
+ ⎜ + ⎟+G ⎜ + ⎟ = 0
⎜ ∂x ∂y ⎟
1 − υ ∂x ⎝
2
⎠ ∂y ⎜ ∂y ∂x ⎟
⎝ ⎠
∂ ⎛ ∂u ∂v ⎞ E ⎛ ∂u ∂v ⎞
+G ⎜ ∂y ∂x ⎟ 1 − υ 2 ⎜υ ∂x + ∂y ⎟ = 0
⎜ + ⎟+
∂x ⎝ ⎜ ⎟
⎠ ⎝ ⎠
3) Subtract 1) from 2) =
4) Difference 3) drives next iteration

47. FEA Fundamentals
• Resolve for other quantities of interest
Strain (є)
Stress (σ)
von Mises Stress: σ eq = 0.5[(σ 1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 3 − σ 1 )2 ]

48. Linear elastic stress analysis
• Small displacements with constant b.c.
• Material properties constant (ductile)
Ultimate or Tensile
Yield

49. Defining a Study (fea fundamentals)
1. Define the type of analysis. (specify approx. function)
2. Create a study.
3. Define material for each component.
4. Define Connections. (b.c.: component-component)
5. Define Fixtures. (b.c.: reduce model DOF)
6. Define External Loads. (b.c.: force, pressure, torque)
7. Define the Mesh. (discretize the domain)
8. Run the analysis. (solve linear system)
9. View Results.
10. Interpret results.

51. Analytical vs. FEA (node)
FEA Solution
σmax = Kn ×σn
Stress concentration factor, (flat plate, circular hole, D/W>0.65):
3
⎛ D⎞
K n = 2 + ⎜1 − ⎟
⎝ W⎠
Normal stress, at hole cross section:
P
σn =
(W − D) × T
Plate geometry:
D = hole diameter = 70 mm
W = plate width = 100 mm
T = plate thickness = 10 mm
Load:
P = tensile load = 50,000 N
∴ σ max = 338 MPa

52. Flat Plate: vonMises (MPa) vs DOF
Mesh vonMises* DOF
• Coarse 327 7,944
• Default 341 46,728
• Fine 349 281,142
• Default w/ mesh ctrl. 348 52,368
* σ eq = [
0.5 (σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 )
2 2 2
]

53. It’s all about the Mesh

54. vonMises (MPa): Default Nodes

55. vonMises (MPa): Default Elements

56. vonMises (MPa): Fine Nodes

57. vonMises (MPa): Fine Elements

58. Consider Symmetry

59. Consider Symmetry

60. Consider Symmetry

61. Simplification: ¼ Symmetry

62. Simplification: ¼ Symmetry & 3D

63. Simplification: ¼ Symmetry & 2D

64. vonMises vs. Degree of Freedom
Fine
Mesh
Control Shell
Default
Coarse

65. Flat Plate: vonMises (MPa) vs DOF
Mesh vonMises DOF
• Coarse 326 7,902
• Default 340 46,581
• Fine 349 280,218
• Default w/ mesh ctrl. 352 52,197
• ¼ Symmetry Shell 347 1,531

66. Singularity: Reentrant Corner
Force
Ou ch!
Force

67. Consider Singularity: L-Bracket (MPa)
Study Sharp Corner Radius Corner
• Mesh Control 1 76 101
• Mesh Control 2 92 101
• Mesh Control 3 194 101
• Adaptive Mesh - 115

68. McMaster-Carr

69. Work Load Limit Given

70. Simplify Geometry

71. Suppress nut & thread

72. Insert > Curve > Split Line
Split Line

73. Assembly Analysis

74. SolidWorks Tutorials

75. SolidWorks Web Help
• http://help.solidworks.com/2010/English/SolidWorks/s
ldworks/Overview/StartPage.htm

76. National Agency for Finite Element
Methods and Standards
• NAFEMS Benchmarks...
Help > SolidWorks
Simulation > Validation >
NAFEMS Benchmarks.

77. FEA Reference

78. Make Math Fun w/ Direction Field Plotter
• http://www.math.psu.edu/cao/DFD/Dir.html

79. Formula Plotter with Integral Curves

80. ZoomIt

81. MWSnap

82. This was the Plan
• SolidWorks Simulation
The Design Cycle
Finite Element Method
SolidWorks Simulation
Defining a Simulation Study
Controlling the Mesh
Accessing the Results
Indentifying Singularities
Instructional Examples
Tips and Tricks

83. Questions