Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 311

or

b 1 fb 1 X b 1 fb 0

=−

(12)

V

FX =

This is the same result given by Eqs. (5.130) and (5.131).
This completes the direct derivation of the fundamental equations for the force method.
We can now return to the example problem to complete the calculations for the response
of the beam of Fig. 5.31. The essential ingredients of the force method formulation are
the matrices b 0 , b 1 , and f Unfortunately, the computation of b 0 and b 1 , which contain

equilibrium information, is difficult to accomplish systematically. This is a shortcoming
that retards the use of the force method in general purpose structural analysis computer
programs. This contrasts with the displacement method which is readily systematized due
to the ease in formulating the equivalent of the kinematic matrix a (using assembly by
superposition of element matrices).
For our beam, the calculations for P x or X involve F and V

.

. We find

48

2

/ 942

2

/ 9

12 EI

b 1 fb 1 =

F = (13)

42

2

/ 948

2

/ 9

24

/ 9 − 30 / 9

=−
12 EI

V b 1 fb 0

=− (14)

30 / 9 − 24 / 9

With Eq. (5.129) ( FP x = V ) , the redundants are evaluated from

48 R b

R c

24 M b

M c

42 − 30

=− (15)

42 48 30 − 24

F P x P

Alternatively, if X is to be utilized, it can be obtained from [Eq. (5.130)]

48 X 24

42 − 30

=− (16)

42 48 30 − 24

V F

This gives

0

. 20 . 8

=

X (17)

− 0 . 8 − 0 . 2

The displacements and forces of interest can now be com p uted. If the redundants are
determined using (15), the forces can be found from p = b 0 P + b 1 P x . If X of (1 7 ) is to be

utilized, first obtain b = b 0 + b 1 X and then use this to find the forces p = bP . Finally,

displacements are available using

b T v b 0 v b 0 fbP v = fp = fbP and V = = = (18)