14-06-2012, 06:05 PM
MECH3300 Finite Element Methods
Finite Element Methods.ppt (Size: 529.5 KB / Downloads: 66)
Concepts
Node - a generalised joint
- connection point at which equations are written
- there are at most 6 unknowns (degrees of freedom) at a node (3 displacements, 3 rotations)
Element - connection between a group of nodes representing stiffness or other properties approximately
- interpolation of displacement between the nodal values is used - this becomes more accurate as element size is reduced
Reference node - an extra point used to orient the cross-section of a beam.
Beam elements
Beam elements were developed first, as a stiffness matrix [Ke] of a beam can be found exactly for elastic behaviour and small deflections if there are boundary conditions and loading at each end only.
The matrix [Ke] links forces and moments (a vector F) to displacements and rotations (a vector u) at each end (each node).
In 3D 3 force components and 3 moment components act on each end - hence the element matrix is 12 by 12.
Individual terms are listed on the fifth slide. They depend on the length L, the area of cross-section A, the second moments of area, an effective area deforming in shear etc.
Terms in a Beam Element Matrix
With Euler-Bernoulli beam theory, the following types of terms arise in a beam element matrix, for bending a beam of length L on a principal axis of the cross-section, with 2nd moment of area I.
Transverse force/deflection relations give ±12EI/L3
Transverse force/rotation or bending moment/displacement relations give ±6EI/L2
Bending moment/rotation relations give 4EI/L or 2EI/L
Twisting a beam gives GIP/L
Axial deformation gives EA/L (A = cross-sectional area0
Transverse shear deformation gives GAS/L
where AS = effective cross-sectional area for shear.
Rotation of coordinates
A beam will not in general be aligned with the “global” xyz axes. To rewrite the stiffness terms in terms of forces/displacements in the global axis directions, a rotation matrix R is found, the individual terms of which are direction cosines.
The rotation R applies to both forces and displacements. The inverse of R is just is transpose. This corresponds to rotating back the other way.
Hence if in local axes aligned with a beam F = K u, then in global coordinates RFG = K RuG or FG = RTKR uG
Even for a single spring, this transformation leads to a 6 by 6 matrix, containing products of direction cosines, as each end of the spring can move in 3 directions, giving 6 equations.
A beam has rotations at each end as
well, giving 12 equations.