Advanced Methods of Structural Analysis

7.1 Fundamental Idea of the Force Method 215

Fig. 7.5 Statically
indeterminate beam. Design
diagram and reactions of
supports

A B C

P

HA

RA RB RC

Ta b l e 7. 1 Analysis of statically indeterminate beam by superposition method
Ve r s i o n 1 Ve r s i o n 2
Primary
system A B C

P

HA

RA X=RB RC

A B C

P

HA

RA RB X=RC

Primary

unknown

XDRB XDRC

Displacement

caused by

given loads

A B C

P

yB(P)

A B C

P

yC(P)

Displacement

caused by

primary

unknownX

A

B

C

X=RB

yB(X)

A B

C

X=RC

yC(X)

Compatibility

equation

yBD 0

yBDyB.P/CyB.X/D 0

yCD 0

yCDyC.P/CyC.X/D 0

Xshould be determined from the followingcondition: behavior of the actual beam
and primary system must be identical. Since pointBhas no displacement in the
actual state, then the compatibility condition isyBD 0. The displacement of point
Bin the primary system is caused by given loadP, as well as by the primary
unknownRBDX. So the compatibility condition may be written in the following
form
yBDyB.P /CyB.X/D0; (7.1)

whereyB.P /andyB.X/are displacement of pointBin the primary system due
to given loadP, and primary unknownRB D X, respectively. The compati-
bility equation means that both structures – the given and primary ones – are
equivalent. These displacements may be calculated by any method, which are
described in Chap. 6. The solution of compatibility equation allows calculating
the primary unknownX. The obtained valueXDRBshould be considered as
active external load, which acts (together with given loadP) on the statically