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Steel Designers' Manual - 6th Edition (2003)
316 Introduction to manual and computer analysis
The same principle can be employed to determine the displacement due to other
causes, viz. axial load or shear or torsion.
9.6 Finite element method
The advent of high-speed electronic digital computers has given tremendous
impetus to numerical methods for solving engineering problems. Finite element
methods form one of the most versatile classes of such methods which rely strongly
on the matrix formulation of structural analysis. The application of finite elements
dates back to the mid-1950s with the pioneering work by Argyris, 4 Clough and
others.
The finite element method was first applied to the solution of plane stress
problems and subsequently extended to the analysis of axisymmetric solids,
plate bending problems and shell problems. A useful listing of elements developed
in the past is documented in text books on finite element analysis. 5
Stiffness matrices of finite elements are generally obtained from an assumed displacement
pattern. Alternative formulations are equilibrium elements and hybrid
elements. A more recent development is the so-called strain based elements. The
formulation is based on the selection of simple independent functions for the linear
strains or change of curvature; the strain–displacement equations are integrated to
obtain expressions for the displacements.
The basic assumption in the finite element method of analysis is that the response
of a continuous body to a given set of applied forces is equivalent to that of a system
of discrete elements into which the body may be imagined to be subdivided. From
the energy point of view, the equivalence between the body and its finite element
model is therefore exact if the strain energy of the deformed body is equal to that
of its discrete model.
The energy due to straining of the element, U, written in two-dimensional form
is
or in matrix form
1 T
U = Ú Ú { e} { s}
d( vol)
2
T
in which {} e = { e , e , g }
and
1
2
Ú Ú
( es es g t ) d( vol)
U = x x + y y + xy xy
x y xy
{ s} = { s , s , t }
x y xy
c
{} e = [ f ]{ d }
d c { } = { u1, u2,..., un} T
T
(9.22)
(9.23)
(9.24)