Structural Analysis Methods I: Beam Elements - Mechanics of Structures: Variational and Computational Methods - page 179

4

Structural Analysis Methods I:
Beam Eleme nts

The sources for many contemporary computational methods of solid mechanics can be
traced to structural analysis techniques. In this and the following chapter, we will out-
line structural analysis methodology. We begin with the study of structural members, with
primary emphasis given to the plane beam element. In the next chapter, the mixed, displace-
ment, and force methods of joining the elements into structural systems will be outlined.

4.1 Sign Convention

The sign convention of Chapter 1, which is frequently employed for structural members,
where the distribution of the internal bending moment and shear force are of concern, is
illustrated in Fig. 4.1a and labeled Sign Convention 1. It is convenient to introduce another
sign convention which is better suited for use in the stiffness methods of analysis of network
structures where values of the bending moment and shear force at the ends of the elements
are to be calculated. This new sign convention, which will be referred to as Sign Convention
2, is shown in Fig. 4.1b. For this second convention, on both ends of the beam, the forces
and moments along the positive coordinate directions are considered to be positive.
Designate the forces on the ends of the beam element by s for Sign Convention 1 and by
p for Sign Convention 2. More specifically, define for Sign Convention 1

V a
M a

V b
M b

s a
s b

s a = s b = s = (4.1)

and for Sign Convention 2

V a
M a

V b
M b

p a
p b

p a = p b = p = (4.2)

The sign conventions are related by V a | Sign Convention 1 =− V a | Sign Convention 2 and M a | Sign Conven -

=− M a | Sign Convention 2 . In matrix notation,

tion 1

10 .

− 00

p a . s a

0 − 1 00

··· = ··· ··· ··· ··· ··· (4.3a)

00 . 10

p b s b

00 .

01

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