C-02 (Slender Cantilever) |
  
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C-02 (Slender Cantilever) |
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Objective
To verify membrane formulation of the shell element using regular and irregular element shapes
Problem Description
The slender cantilever beam shown below is modeled with a). regular shape elements; b). trapezoidal shape elements; c). parallelogram shape elements. Trapezoidal and parallelogram shapes take 450 angle. All elements have equal volume.
Material properties: E = 1.0e7 psi, ν = 0.3
Section properties: Length = 60 in, height = 0.2 in, thickness t = 0.1 in
Loads: a). unit axial force; b). unit in-plane shear
Finite Element Model
6 shell elements
Model type: 2D Plane Stress
Results
The tip displacements are given by [Ref 1]. Theoretical stresses at the root are also given here for comparison.
Units: displacement – in; stress - psi
Element |
Load type |
ENERCALC 3D |
[Ref 1] |
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Displacements @ tip |
Stresses @ root |
Displacements @ tip |
Stresses @ root |
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Compatible Regular |
Axial force |
3.0e-5 |
-50 |
3.0e-5 |
-50 |
In-plane shear |
-0.01009 |
-846.2 |
0.1081 |
-9000 |
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Incompatible Regular |
Axial force |
3.0e-5 |
-50 |
3.0e-5 |
-50 |
In-plane shear |
-0.1073 |
-8250.0 |
0.1081 |
-9000 |
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Incompatible Trapezoidal |
Axial force |
3.0e-5 |
-50 |
3.0e-5 |
-50 |
In-plane shear |
-0.02385 |
-7071.6 |
0.1081 |
-9000 |
|
Incompatible Parallelogram |
Axial force |
3.0e-5 |
-50 |
3.0e-5 |
-50 |
In-plane shear |
-0.08608 |
-6510.1 |
0.1081 |
-9000 |
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Comments
The results given by ENERCALC 3D are mixed in comparison with the referenced values.
All meshes behave correctly in the axial force loading. For in-plane shear, the regular mesh using incompatible membrane formulation behaves the best. The behavior of the regular mesh using compatible formulation and the irregular mesh using compatible or incompatible formulation can be improved by using more elements. In practice, a rectangular element shape with small aspect ratio should be used whenever possible.
Reference
[1]. MacNeal & Harder, “A Proposed Standard Set of Problems to Test Finite Element Accuracy”, Finite Elements in Analysis and Design, 1 (1985) 3-20
