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15 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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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]


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.

zeroes

Non
-
zero
terms

Bandwidth

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.

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.

Numbering of
rows/columns in [K
e
]

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 [K
e
] = row 16 of [K]

column 5 of [K
e
] = column 23 of [K]

1

2

3

5

4

6

16

17

18

23

22

24

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.

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.

Load
w

per
length


Moment
reactions

wL
2
/12

Loads applied to model
-

this
causes the correct nodal
deflections

wL/2

wL/2

wL
2
/12

wL
2
/12

L

Force
reactions

wL/2

Applied load and fixed
-
end
reactions
-

this loading
causes no nodal
deflections.

Sum is the applied
load only (what
we wish to model)

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.

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.

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.


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.

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

Axis 1

Axis 3

End A

(1st node chosen
when meshing)

End B