Torsion of Non-Circular Prismatic Bars: Finite Element Modeling and AnalysisPertinent reading: Section 11.5 in textbook

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Idealized Problem

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Torsion of a Slot Cross-SectionSee Figure 11.14 on p. 432 in textbook.

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Torsion of a Hollow Slot Cross-SectionG = 1.154e+07 psiQ = 4.433e-04 rad/in

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Torsion of a Hollow Slot Cross-SectionSolution Domain & its BoundaryWGGGG

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Torsion of a Hollow Slot Cross-SectionNodal SpecificationsWF = 0F = 0F = 0F = 0F = 0

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Torsion of a Hollow Slot Cross-SectionCoarsest MeshWill yield a trivial solution!F = 0F = 0F = 0F = 0

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Torsion of a Hollow Slot Cross-SectionFiner MeshThe key is to also provide nodes not along the boundary!

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Taking Advantage of Symmetry in Finite Element AnalysesIf a specific problem has one or more lines (axes) of symmetry, the resulting FE analysis can be simplified.This results in fewer elements being used in the solution.Consequently, the computational effort is reduced.

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Taking Advantage of Symmetry in Finite Element AnalysesTo qualify for conditions of symmetry, the following must be shown to be symmetric with respect to a given axis:

Geometry of the solution domain Material constitution and distribution Applied loading Nodal specifications (boundary conditions)

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Torsion of a Hollow Slot Cross-SectionSymmetry ConsiderationsWGWe thus only need to analyze one-quarter of W!

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Torsion of a Hollow Slot Cross-SectionSolution Domain & its BoundaryWGGGSymmetry boundarySymmetry boundaryAlong symmetry boundariesxy

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First mesh(36 elements)

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Second mesh(144 elements)

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Third mesh(576 elements)

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Torsion of a Filled Slot Cross-SectionG1 = 12.0e+06 psiQ = 4.0e-04 rad/inG2 = 6.0e+06 psi

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Torsion of a Filled Slot Cross-SectionSolution Domain & its BoundaryWGGG

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First mesh(20 elements)

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Second mesh(80 elements)

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Torsion of a Semi-Circular Cross-Sectionxyr = 2.0 cm

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Torsion of a Semi-Circular Cross-SectionSolution Domain & its BoundaryxyWGG

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First mesh(4 elements/side)

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Second mesh(8 elements/side)

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Third mesh(16 elements/side)

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