CHAPTER 6: 3D SOLIDS - UFL MAE · 1 CHAPTER 6: 3D SOLIDS A 3D solid can have any shape, size,...
Transcript of CHAPTER 6: 3D SOLIDS - UFL MAE · 1 CHAPTER 6: 3D SOLIDS A 3D solid can have any shape, size,...
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CHAPTER 6: 3D SOLIDS ■ A 3D solid can have any shape, size, boundary conditions, etc.
■ All six stress components and all three displacement components must be considered.
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AXISYMMETRIC SOLID
■ A solid of revolution generated by revolving a plane shape about an axis in the plane.
■ Nothing varies with the circumferential coordinate θ if loads, supports and elastic properties are also axisymmetric.
■ No circumferential displacement and τrθ=τθz=0.
■ The problem is mathematically two-dimensional and is usually modeled with plane triangles and quadrilaterals.
■ Hence we model a longitudinal cross-section for Rin < r < Rout.
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Axisymmetric Solid Example
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Constitutive Equation in 3D
where σ0 is the initial stress produced by the temperature change and
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Constitutive Equation for Axial Symmetry
■
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Strain-Displacement Relations ■ Strain-Displacement Relations in Cartesian Coordinates:
■ Strain-Displacement Relations for a solid of revolution in cylindrical coordinates:
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Stiffness Matrix■ Displacements are interpolated from nodal dof as usual, u=Nd with N being the shape function matrix.
■ When u is substituted in strain-displacement equation, the strain-displacement matrix B is obtained, which is used to compute the element stiffness matrix:
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Constant Strain Tetrahedron
■ Has four nodes with three translational dofat each one. (total of 12 dof)
■ Like CST, accurate only when the true strains are almost constant over its domain.■ CTETRA in Genesis
■ The displacement field is
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Linear Strain Tetrahedron
■ Has ten nodes with three translational dof at each one. (total of 30 dof)
■ Like the plane LST, has a linear strain field. Therefore it can model pure bending.
■ Edges can be curved.
■Not available in Genesis
■ The displacement field is quadratic, for example:2 2
1 2 3 4 5 6
27 8 9 10
u x y z x yz xy xz yzβ β β β β β
β β β β
= + + + + +
+ + + +
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Trilinear Hexahedron (8-node brick)
■ Has eight nodes with three translational dof at each one. (total of 24 dof)
■ Of arbitrary shape; formulated as an isoparametric element.■CHEXA in Genesis
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Trilinear Hexahedron: Displacement Field
which is a product of three linear functions in x, y, and z.
Genesis also has a CPENTA element for connecting a CTETRA to
a CHEXA
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Quadratic Hexahedron
• Has 20 nodes and 60 dof. CHEX20 in Genesis
• Extension of quadratic quadrilateral (plane element)
• Edges of an undeformed element can be straight or curved.