21-05-2012, 12:18 PM
On the reduction of domain integrals to theboundary
for the BEM formulation of plates onelastic foundation
On the reduction of domain.pdf (Size: 290.31 KB / Downloads: 55)
INTRODUCTION
Application of the BEM for the analysis of plate bending
problems generally requires the evaluation of load domain
integrals. These integrals in th~ case of dimensionality are
computationally inefficient. Domain integrals can in certain
cases introduce singularities, such as when they are used to
calculate the internal moments. Hence it is highly desirable
that they are transformed into boundary integrals when
possible.
This paper describes in detail a procedure which can be
applied to formulate equivalent boundary integrals. This
procedure has been used by the authors in previous papers, 1'2
and it is exact when the Laplacian of the load gives a
constant.
Numerical results are presented for various geometries,
computing the presented approach with results obtained
integrating over the domain. Some results are also compared
against analytical solutions for a particular plate problem.
The paper presents the Fortran program used to compute
the equivalent boundary integrals on the assumption that
this may be of interest to other workers, hoping that this
will expedite the application of the procedure.
NUMERICAL EXAMPLES
Two problems were solved in order to validate the formulation
and demonstrate the accuracy of using equivalent
boundary integrals for the solution of plate problems.
Example 1
A number of plates have been solved using the classical
domain integration scheme and the equivalent boundary
integrals are computationally more efficient, as they require
the different cases is shown in Fig. 1, as well as the discretisations
used. The data for this example was as follows:
a (radius, or side length of plate geometry) = 10 in;
flexural rigidity D = 106 in lb;
modulus of the foundation K = 2 x 104 lb/in3;
uniform pressure load q = I000 psi.
The results obtained are provided in Table 1. These were
obtained by computing the integrals using Gaussian quadrature
formulae. Four and seven points were used for the
boundary and domain integrals, respectively. The boundary
integral results agree very closely with those obtained by
computing the domain integrals. In addition the boundary
integrals are computationally more efficient as they require
a small number of boundary elements and Gaussian integration
points.
EQUIVALENT BOUNDARY INTEGRALS FOR THE
DOMAIN LOAD INTEGRALS
Given two functions ~b and ~, which are continuous in
~2 + P and differentiable to the second order in ~2, Green's
Second Identity 6 can be expressed as:
* = f(* %=iiann ] (5)
I2 P
where h is the outer direction normal to the boundary P.
It is assumed that the load distribution q satisfies V2q = 0.
Using this property, together with relation (5), the domain
integrals in the equation (1) can be written.
CONCLUSIONS
The following advantages can be pointed out when using
the equivalent boundary integrals for the representation of
domain integrals of plates on elastic foundations:
(i) The technique gives accurate results using a small
number of boundary elements, resulting in a highly
efficient computational procedure.
(ii) The calculation of internal moments which otherwise
requires using a semi-analytical approach, is greatly
simplified.
(iii) The method preserves the reduced dimensionality of
the problem, characteristic of the boundary element
method.