Analisis Estructural

23

Matrix methods of structural analvsis

23.1

Introduction

This chapter describes and applies the matrix displacement method to various problems in structural analysis. The matrix displacement method first appeared in the aircraft industry in the 1940s7,where it was used to improve the strength-to-weight ratio of aircraft structures. In today's terms, the structures that were analysedthen were relatively simple, but despite this, teams of operators of mechanical, and later electromechanical, calculators were required to implement it. Even in the 1950s,the inversion of a matrix of modest size, often took a few weeks to determine. Nevertheless, engineers realised the importance of the method, and it led to the invention of the finite element method in 1956', whlch is based onthe matrix displacement method. Today, of course, with the progress made in digital computers, the matrix displacement method, together with the finite element method, is one of the most important forms of analysis in engineering science. The method is based on the elastic theory, where it can be assumed that most structures behave like complex elastic springs, the load-displacement relationship ofwhich is linear. Obviously, the analysis of such complex springs is extremely difficult, but if the complex spring is subdivided into a number of simpler springs, whch can readily be analysed, then by considering equilibrium and compatibility at the boundaries, or nodes, of these simpler elastic springs, the entire structure can be represented by a large number of simultaneous equations. Solutionof the simultaneous equations results in the displacements at these nodes, whence the stresses in each individual spring element can be determined through Hookean elasticity. In this chapter, the method will first be applied to pin-jointed trusses, and then to continuous beams and rigid-jointed plane frames.

23.2

Elemental stiffness matrix for a rod

A pin-jointed truss can be assumed tobe a structure composed of line elements, called rods, which possess only axial stiffness. The joints connecting the rods together are assumed to be in the form of smooth, fnctionless hinges. Thus these rod elements in fact behave llke simple elastic springs, as described in Chapter 1. Consider now the rod element of Figure 23.1, which is described by two nodes at its ends, namely, node 1 and node2.

'Levy, S., Computation of Influence Coefficients for Aircraft Structures with Discontinuities and Sweepback,
J. Aero. Sei., 14,547-560, October 1947.

Martin, H.C. and Topp, L.J., Stiffness and Deflection Analysis of Complex Structures, 'Turner, M.J., Clough, R.W., J. Aero. Sei., 23,805-823, 1956.

566

Matrix methods of structural analysis

Figure 23.1 Simple rod element.

LetX,
X2
u,

=
=

axial force at node 1 axial force at node 2 axial deflection at node 1 axial deflection at node 2 cross-sectional area of the rod element elemental length Young's modulus of elasticity

=
=

u2

A
1

= =
=

E

Applying Hooke's law to node 1, -(I = E
&

but
( I

=

X,IA

and
E

=

(uI

- u*y1

so that

X,

=

AE

(u,

-

ldzyl(23.1)

From equilibrium considerations

X,

=

-XI

=

AE

( ,.

141y/

(23.2)

System stiffness matrix [K]

567

Rewriting equations (23.1) and (23.2), into matrix form, the following relationship is obtained:

};{
(PI}
where,
=

=

5E[1 -1
lkl

(23.3)
- 1 ] { 5u* 1 ]

or in short form, equation (23.3) can be written

{ UI}

(23.4)

(PI}

=(uI}

=

6) [::}
=
=

a vector of loads

a vector of nodal displacements

Now, as Force

=

stiffhess x displacement

[k]

=

g
I

[

1 -1
1

-1

(23.5)

=

the stifmess matrix for a rod element

23.3

System stiffness matrix [K]

A structure such as pin-jointed truss consists of several rod elements; so to demonstrate how to form the system or structural...