Advanced Methods of Structural Analysis

224 7 The Force Method

Q 0  1 D

0:096P l

l

D0:096P;

Q 1 nD

0:1824P l.0:096P l/

0:4l

D0:696P;

Qn 2 D

0:1824P l

0:6l

D0:304P:

Final shear force diagram is shown in Fig.7.8g.

Reactions of supportsHaving shear force diagram we can calculate the reaction of

all supports. They are following:

R 0 D0:096P; R 1 DQright 1 Qleft 1 D0:696P.0:096P /D0:792P;

R 2 D0:304P:

Static verificationEquilibrium condition for all structure in whole is

X

YD0:096PC0:792PC0:304PPD1:096PC1:096PD0:

Discussion:

1.Adopted primary system as a set of simply supported beams leads to the simple
(triangular) shape of bending moment diagrams ineach unit states. The bend-
ing moment diagram due to given load is located only within each loaded span.
Therefore, computation of coefficients and free terms of canonical equations is
elementary procedure.
2.Canonical equations of the force methodallow easy to take into account the
different bending stiffness for each span.

7.3.2 Analysis of Statically Indeterminate Frames

Design diagram of the simplest frame is presented in Fig.7.9a. The flexural stiffness

for all members isEI. It is necessary to construct the bending moment diagram and

calculate a horizontal displacement of the cross bar.

Primary system and primary unknown. The structure has four unknown reac-

tions, so the degree of redundancy isnD 4  3 D 1. Let us choose the primary

unknownX 1 be a vertical reaction at point 1. The primary system is obtained by

eliminating support 1 and replacing it byX 1 (Fig.7.9b).

Canonical equation of the force method isı 11 X 1 C1P D 0. This equation

shows that for the adopted primary system the vertical displacements of the left

rolled support caused by primary unknownXand the given loadqis zero.

Bending moment diagrams in the primary system caused by unit primary un-

knownX 1 D 1 and given load are shown in Fig.7.9c, d. These graphs also show the

displacements along eliminated constraint.