# CHAPTER 6: 3D SOLIDS - UFL · PDF file1 CHAPTER 6: 3D SOLIDS A 3D solid can have any shape,...

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### Transcript of CHAPTER 6: 3D SOLIDS - UFL · PDF file1 CHAPTER 6: 3D SOLIDS A 3D solid can have any shape,...

1

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 if loads, supports and elastic properties are also axisymmetric.

No circumferential displacement and r=z=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 0 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 yz

= + + + + +

+ + + +

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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.

CHAPTER 6: 3D SOLIDSAXISYMMETRIC SOLIDAxisymmetric Solid ExampleConstitutive Equation in 3DConstitutive Equation for Axial SymmetryStrain-Displacement RelationsStiffness MatrixConstant Strain TetrahedronLinear Strain TetrahedronTrilinear Hexahedron (8-node brick)Trilinear Hexahedron: Displacement FieldQuadratic Hexahedron