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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.