MECH3300 Finite Element Methods Lecture 3 . Practical aspects of implementing a direct stiffness solution for a structure made of beams. Exploiting the sparseness of [K].
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Practical aspects of implementing a direct stiffness solution for a structure made of beams
Conceptually, element matrices are expanded with rows and columns of zeroes and added. In practice, this amounts to finding the right row and column addresses in the full matrix into which each stiffness term should be placed.
To do this a list of the degrees of freedom at each node of an element is first stored for each element, called an element destination vector.
Numbering of rows/columns in [Ke]
Numbering of rows/columns of [K] for the same pair of nodes.
Element destination vector: [16 17 18 22 23 24]
ie row 1 of [Ke] = row 16 of [K]
column 5 of [Ke] = column 23 of [K]
If this is one element, it is NOT connected to the vertical one
- it needs subdividing.
(b) To apply an intermediate load, a beam must be subdivided in order to place a node where the load is applied.
Load w per length
Loads applied to model - this causes the correct nodal deflections
Applied load and fixed-end reactions - this loading causes no nodal deflections.
Sum is the applied load only (what we wish to model)
Offset of node on the left element.
In the absence of intermediate or distributed loading, the worst stresses are at the nodes.
To see stresses, typically a visualisation of the cross-section of a beam is first turned on in a package.
The usual convention for local beam axes is as follows.
Axis 1 (or local x) is the major principal axis.
Axis 2 (or local y) is the minor principal axis and points toward the reference node.
Axis 3 (or local z) is along the beam.
Note that this means that forces in local axes may have inconsistent signs for different elements, where there is a change in reference node.
Reference node in plane of axes 2 and 3
(1st node chosen when meshing)