MECH3300 Finite Element Methods Lecture 3

1
MECH3300 Finite Element Methods Lecture 3

• Practical aspects of implementing a direct
  stiffness solution for a structure made of beams

2
Exploiting the sparseness of K

• If degrees of freedom are numbered in the order
  of the nodes, then if nodes that are connected
  can be given similar numbers, off-diagonal terms
  of K will correspond to similar row and column
  numbers.
• If this is the case, then K is banded. The
  bandwidth is the maximum number of rows or
  columns over which non-zero terms occur.
• Bandedness can be exploited both when storing K
  and when solving the equations. Large reductions
  in disk space used and large increases in speed
  of solution are possible.

Non-zero terms
zeroes
Bandwidth
3
Exploiting the sparseness of K - 2

• Solvers in commercial codes typically either
• (a) store and process only terms of K within
  its bandwidth.
• (b) use a sparse matrix storage scheme in which
  only non-zero terms are stored, along with their
  row and column addresses.
• In the former case, several algorithms to
  renumber nodes are used to find the most compact
  form of the matrix. The matrix may be stored
  from the diagonal up to the last nonzero term in
  each column - this is called a skyline form of
  the matrix. STRAND7 will display the matrix in
  this form, before and after nodes are renumbered.

4
Assembly of element matrices in practice
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.
2
5
17
23
6
1
16
18
24
3
4
22
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
5
Differences between beam elements and a physical
beam

• One physical beam often needs to divided into
  several beam elements.
• (a) Elements only connect at nodes,
• as equations are only written at nodes.

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.
6
Refinements in modeling beams

• Intermediate or distributed loading can be
  represented by statically equivalent loading at
  the nodes. The loads to apply are minus the
  reactions that would occur if the nodes at each
  end of the element were fixed.

Moment reactions wL2/12
wL/2
wL/2
wL2/12
Load w per length
wL2/12
Loads applied to model - this causes the correct
nodal deflections
Force reactions wL/2
L
Applied load and fixed-end reactions - this
loading causes no nodal deflections.
Sum is the applied load only (what we wish to
model)
7
Refinements in modeling beams - 2

• If the centroidal axes of beams do not meet at a
  joint, one may need to be offset - that is, the
  node must be shifted some distance off the
  centroidal axis. This can be done in both
  principal axis directions.

Offset of node on the left element.
8
Refinements to modeling beams - 3

• The default connection is a rigid joint (all
  members displace and rotate the same at a joint).
  To create pin joints at particular nodes only,
  or to create a sliding joint, end-releases are
  used.
• An end-release creates 2 separate degrees of
  freedom, one on each beam - eg two independent
  rotations to give a pin joint.
• This is useful in modeling a mechanism.

9
Stresses in beams

• Stresses can only be found if the cross-sectional
  shape of a beam is specified in the data. Often,
  this is done by giving the positions of the
  extreme fibres - the corners furthest from the
  centroidal axis.
• Stresses consist of axial stress P/A, bending
  stresses in 2 principal planes, torsional and
  transverse shear stresses.
• A useful combined stress is total fibre stress
  - axial stress plus the stress due to bending in
  both transverse planes.

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.
10
Constraint equations

• Extra equations are often added to the set of
  equations solved, called constraint equations,
  that relate the motion of different nodes. The
  user is typically unaware of this, however
• The most common form of constraint equation is
  one prescribing rigid body behaviour. In STRAND7
  this is a rigid link. In NASTRAN it is a rigid
  element, or a multi-point constraint (MPC).
• Constraint equations also can be used to apply
  displacement boundary conditions. In STRAND7,
  the restraint menu allows a non-zero value to be
  specified. In NASTRAN, a node must first be
  fixed and then a load applied to it, with the
  load redefined as a displacement. ANSYS also
  regards imposed displacements as loads.

11
Local axes of a beam
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
Axis 2
End B
End A (1st node chosen when meshing)
Axis 3
Axis 1