3D 6. Solids and Solids of Revolution
MEEM4405 Introduction to Finite Element Analysis
6.1 Introduction
Problems of beam bending, plane stress, plates, etc. may be considered as special cases of 3D solids. So why not use 3D solids all the time?
Harder to prepare Harder to check for errors Much greater demand on computer resources
MEEM4405 Introduction to Finite Element Analysis
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Introduction
Stress can vary in all three directions Nodes are located in 3 space Nodes have displacements in 3 directions
MEEM4405 Introduction to Finite Element Analysis
Stress-strain relations
Stress-strain relations are now described by a 6x6 matrix
MEEM4405 Introduction to Finite Element Analysis
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Strain-Displacement Relations
If strains are small:
These are the most general equations. In deriving previous elements we made assumptions about some of these values (e.g. in plane strain we assumed z=0)
MEEM4405 Introduction to Finite Element Analysis
Displacement Interpolation
Displacements within an element are interpolated from nodal displacements using u=Nd, as before, however there are now interpolations in three directions.
MEEM4405 Introduction to Finite Element Analysis
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General Formula for k
The general, energy-based formula for k is the same as for previous elements, except that there are more terms in the integration due to the bigger matrices.
MEEM4405 Introduction to Finite Element Analysis
6.2 3D Solid Elements
The 3D solid elements are analogous to planar counterparts:
Constant Strain Tetrahedron Const. Strain Triangle Linear Strain Tetrahedron Linear Strain Triangle Trilinear Hexahedron Bilinear Quadrilateral Quadratic Hexahedron Parabolic Quadrilateral
MEEM4405 Introduction to Finite Element Analysis
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Constant Strain Tetrahedron
As with the Constant Strain Triangle, it is only accurate when strains are almost constant over the element span.
MEEM4405 Introduction to Finite Element Hexahedron
Analysis
Trilinear Is also called the 8 node brick element.
Solving for shape functions:
Isoparametrically:
MEEM4405 Introduction to Finite Element Analysis
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Distributed Loading
The work equivalent nodal loads for a constant distributed pressure p are as shown.
MEEM4405 Introduction to Finite Element Analysis
6.3 Axisymmetric Solid Elements
In axisymmetric problems, the geometry is axisymmetric and the loads and support are usually also axisymmetric, but they do not need to be, if certain tricks are employed. Meshing is similar to plane problems, however each node represents a circle and each element represents the cross section of an annulus.
MEEM4405 Introduction to Finite Element Analysis
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Axisymmetric Solid Elements
Stress-strain relations
Strain-displacement relations
MEEM4405 Introduction to Finite Element Analysis
Axisymmetric Solid Elements
Displacement interpolation
General formula for k (axisymmetric stress field)
MEEM4405 Introduction to Finite Element Analysis
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Axisymmetric Elements
Except for circumferential strain of =u/r, these elements are similar to plane elements. Three node triangl...