triangular elements in finite element modeling · 5/22/2012 structures.aero page 2 structural...
TRANSCRIPT
5/22/2012
Page 1
IN THIS WEBINAR:
• Plate element calculation
• Examining the effects of triangle sin a model
• FEMAP tools and methods for meshing
Triangular Elements
in Finite Element Modeling
Andrew Nelson
Stress Engineer
Structural Design and Analysis
PRESENTED BY:
5/22/2012
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Structural Design and Analysis (Structures.Aero)
Structural Analysis
• Team of stress engineers that help our clients
design lightweight and load efficient structures.
• We service aerospace companies and other
industries that require high level analysis.
• Specialty in composites and lightweight
structures
• Tools used include hand analysis, HyperSizer,
Femap, NX Nastran, Fibersim, NX, Solid Edge,
Simcenter 3D, LS Dyna, and LMS.
Software Sales and Support
• Value added reseller providing software, training,
and support for products we use on a daily
basis.
• Support Femap, NX Nastran, Simcenter 3D,
Fibersim, Solid Edge, and HyperSizer.
5/22/2012
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• Worked at SDA since 2015
• Currently supporting Aurora Flight Sciences
• Virginia Tech graduate
Meet the Presenter
Andrew NelsonAerospace Stress Engineer
Structural Design and Analysis, Inc.
703-935-2816
5/22/2012
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Triangular Elements
• Why are they not as good?
• NASTRAN linear plates
• Constant strain elements
• Analytical Example
• When are they okay?
• FEMAP demonstration
• Meshing techniques
• FEMAP tools
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Plate Calculation
• The finite element equation for displacement in linear plate elements
• 4 node Quads
• 3 node Triangles
• Strain-displacement relations
𝜀𝑥𝑥 =𝜕𝑢
𝜕𝑥
𝜀𝑦𝑦 =𝜕𝑢
𝜕𝑦
𝜀𝑥𝑦 =1
2
𝜕𝑢
𝜕𝑥+𝜕𝑢
𝜕𝑦
𝑢 =ψ𝑖(𝑥, 𝑦)𝑢𝑖
(𝑥, 𝑦) (𝑥, 𝑦)
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Shape Functions
• Linear elements
• Quads result in linear strain over the element
• Triangles always result in constant strain over the element
• Nondimensional coordinates ξ and η are used along the element lines
1
ξ
ηψ1 =
1
4(1 − ξ)(1 − η)
ψ2 =1
4(1 + ξ)(1 − η)
ψ3 =1
4(1 − ξ)(1 + η)
ψ4 =1
4(1 + ξ)(1 + η)
2
3 4 ψ1 = ξψ2 = ηψ3 = 1 − ξ − η
2
η
3
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Analytical Example
• Simple linear plate model
• Examining the effects
of triangular elements
in our results
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Results Comparison
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When Triangles are okay
• 3 node Triangles are okay when:
• They’re being used to transfer load
• Strain changes gradually over an element
• Linear strain triangles have 6 nodes
• Quadratic element
• 2nd order equations
• Increases number of DOFs and subsequent run
times for analysis
• Can only be used with other polynomial elements
• Transition regions
ψ1 = ξ(2ξ − 1)ψ2 = η(2η − 1)ψ3 = 𝑅(2𝑅 − 1)
ψ4 = 4ξη
ψ5 = 4𝑅η
ψ6 = 4𝑅ξ
𝑅 = 1 − ξ − η
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Tools to Avoid Tris – Femap Demo
• Within the Meshing Toolbox:
• Meshing surfaces
• Free Meshing
• Mapped Meshing
• Editing geometry for mapping
• Pad/Washer
• Surface Split
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In Conclusion
• Triangular elements are not the best 2D elements for
a mesh since the solution methods are based on a
constant strain assumption and are artificially stiff.
• Triangles are okay in certain areas of a model where
they are unavoidable or where you know that your
strain/stress will be varying gradually.
• Maximizing the number of quadrilateral elements in
your model will provide a more reliable solution that
will capture a model’s behavior more effectively.
• FEMAP has some very useful tools and functionalities
to help either maximize the number of quadrilateral
elements in a finite element model or eliminate
triangular elements altogether.
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For questions on the material covered
today, please contact Andrew Nelson.
For questions about pricing, or to see a
demo, please contact Marty Sivic.
Questions?
Marty SivicDirector of Sales
724-382-5290
Andrew NelsonStress Engineer
703-935-2816