Advanced Methods of Structural Analysis

222 7 The Force Method

MN†M^0

P

EI

D

XZ 

MN 1 CMN 2 CCMNn



MP^0

ds

EI

D1PC2PCCnP:

The result of this multiplication equalsto the sum of all free terms of canonical
equations.

7.3 Analysis of Statically Indeterminate Structures.......................

This section contains application of the force method in canonical form to detailed
analysis of different types of structures. Among them are continuous beams, frames,
trusses, and arches.

7.3.1 Continuous Beams............................................

Let us consider two-span continuous beam on rigid supports (Fig.7.8a). Figure7.8b
shows one version of the primary system, which presents the set of statically de-
terminate beams. The primary unknownX 1 is bending moment at the intermediate
support.
Canonical equation of the force method isı 11 X 1 C1P D 0 ,whereı 11 is
a displacement in direction of first primary unknown due to unit primary unknown
X 1 D 1 ;1Pis displacement in the same direction due to applied load. This canon-
ical equation shows that for the adopted primary system themutualangle of rotation
at the support 1 caused by primary unknownX 1 and the given loadPis zero.
For calculation of displacementsı 11 and1P, it is necessary to construct the
bending moment diagrams in the primary system caused by unit primary unknown
X 1 D 1 and acting loads; they are shown in Fig.7.8c, d, respectively.
For calculation of unit displacement, we need to multiply the bending moment
diagramMN 1 by itself

ı 11 D

MN 1 MN 1

EI

D 2

1

EI

1

2

 1 l

2

3

 1 D

2l

3 EI



rad

kN m



:

For calculation of free term, we need to multiply the bending moment diagram
MP^0 byMN 1

1PD

MN 1 MP^0

EI

D

l

6 EI

Œ1.20:4/C 0 .1C0:4/0:24P l

D0:064

Pl

EI

.Ta b l e A:2;line5/