Advanced Methods of Structural Analysis

10.1 Construction of Influence Lines by the Force Method 343

In case of arbitrary external load the internal force in any member can be

calculated using the influence line for primary unknownX 1. For example, the truss

is loaded by concentrated forceP 1 D 60 kN at joint 3 andP 2 D 20 kN at joint 6 as

shown in Fig.10.10i. For this loading the primary unknownX 1 which is the reaction

at supportCbecomesRCDX 1 D

P

PiyiDP 1 0:803CP 2 0:443D57:04kN.

Now we can calculate the reaction at supportA:

RA!

X

MBD 0 WRA6dCP 1 4dRC3dCP 2 dD 0 !RAD34:81kN:

After that we can calculate any internal force. For example, for forceU 1  3 (sec-

tionaa)weget:

U 1  3!

X

M 2 leftD 0 WRAdCU 1  3 hD 0 !U 1  3 D

34:81 4

3

D46:41kN:

Summary

1.The reciprocal displacementstheorem allows to considerıP1asmodelof vertical
displacements of joints of the truss (within a constant multiplier which equals
1=ı 11 ).
2.For the calculation ofıP1the elastic loads method have been applied. This
method is more effective in comparison of Maxwell–Mohr formula by the fol-
lowing reason. When a set of four loads is applied, then the internal forces arise
only in the members, which form the two adjacent panels of the truss. There-
fore, procedure of summation is related only to elements, whichbelongs to these
two panelsof the truss, while using the Maxwell–Mohr integral, a summation is
related toall elementsof the truss.
3.The elastic loads method also allows calculating unit displacementı 11 .Itis
obvious thatı 11 could be calculated using Maxwell–Mohr formula

ı 11 D

XNNl

EA

:

However, since in the unit state practically in all members of the truss arise inter-
nal forces (Fig.10.10d), then this procedure becomes cumbersome. In fact, the
elastic load method allows calculatingıP1for all joints andı 11 in one step, as
shown in Fig.10.10g.
4.Influence line of internal force,which is induced at any memberkof the truss,
may be constructed by formula (10.9).
5.Influence line for primary unknownX 1 should be treated as key influence line.
Indeed, using this influence line we can calculateX 1 in case of arbitrary fixed
load. After that, the truss should be considered as statically determinate one,
which is subjected to the given loads and the forceX 1.