Finite Element Method Approach to an Elasticity Problem
By A. H. Haji
The task is to compute the deformation along z (vertical to the plate) for the case of a circular section plate (Fig.1) under a distributed load using finite element method.
Fig.1: A circular section plate
The governing equation for this problem is:
where 
is the z – deformation,
, 
, 
, 
, 
are the plate material stiffnesses and
is the distributed load. The FEM model of the above equation
is as follows:
Mesh Generation and relation between local and global numbers of the nodes
Assemblage processes requires a proper system of local and global numbering for the nodes. A typical method is shown in Fig. 2:
Fig.2: The numbering system
The tangential and radial division of the section are
and 
respectively. Fig.3 shows the local and global numbers
relation for a sample element (the element number 1).
Fig.3: The local and global numbers relation for element number 1
Let 
be the global number of the node number
(in local numbering) of the element number
. This numbering system uses the following relations:
Load and Stiffness Relations
The geometry shape functions considering the trapezoidal shape of the elements which are transformable to a square master element by a linear transformation (Fig.4) are as follows:
Fig.4: Transformation of a sample element to the master element
The variable shape functions considering the governing equation are of the Hermite type:
Now regarding the equation for 
:
we need the second derivatives of the variable shape functions. The following relation is an answer:
The above expression states the relation between
and 
. Now for :
The distributed load is computed using the relation
where in the transformed coordinates takes this form:
Considering Fig.5 we can obtain
as follows:
and finally for 
:
Fig.5: Vertical load distribution on the circular section
Assemblage of Stiffness and Load Matrices
The assemblage process requires the knowledge about the effect of the stiffness
and load matrices of each element on the entire stiffness and load matrices.
First assume we have assigned the following global node numbers to the nodes of
element number:
The effect of the elements of this element stiffness matrix
on the entire stiffness matrix
is as follows:
Actually the first 4 rows of are divided to four matrices (4 by 4) and added to proper
locations of . This procedure extends to the other parts of
:
Also for the load matrix, assemblage is achieved in a similar way:
Applying Boundary Conditions
The boundary conditions on the boundaries
and 
are:
Fig.6: The plate boundaries
The conditions on can be stated in 
coordinate:
As we see for each node on there are four unknowns:
(where 
and 
are the external moment and shear force on the element) and
four equations. Noting that the unknowns on node
are related to those on node
(Fig.7) the stiffness (and load) matrix can be rearranged as
follows:
by 
and add to the column number
. 
by 
and the column number
by 
and add them to the column number
.
Fig.7: The external moment and shear force (directed out of the plate) on an element
Now the load matrix unknowns should be clustered. First note that the external
load contribution 
is computed as:
where changes from 5 to 8 for node number 2 (in local system) and
from 9 to 12 for node number 3.
The clustering can be accomplished through these steps :
and number 
by proper coefficients and add to the elements
to 
in order to cancel out the factors including
and in these elements.
Following this procedure the loads corresponded to the unknown displacements
i.e. 
and 
turn to be fully known. So
and can be calculated.
Calculating the Integral Due to The External Loads
The calculation of the integral of () regarding the following points can be simplified:
from 5 to 13
also we have:
and for 
:
where 
is the size of element side on the boundary ()
Finally:
References:
mfiles may be sent under request.