Advanced Methods of Structural Analysis

10.2 Construction of Influence Lines by the Displacement Method 345

In case of fixed load, the shear at any sectionkis a number

QkDQkZ 1 CQ^0 k:

Similarly in case of traveling load, the shear at any sectionkbecomes a function,

therefore

IL.Qk/DQkIL.Z 1 /CIL



Q^0 k



: (10.19)

For then-times kinematically indeterminate structure, the canonical equations of

the Displacement method in case of a fixed loadPD 1 are

r 11 Z 1 Cr 12 Z 2 CCr1nZnCr1PD 0

 (10.20)

rn1Z 1 Crn2Z 2 CCrnnZnCrnPD 0

Unit reactionrikare caused by unit primary unknowns. They can be calculated using

the typical procedures of the displacement method; these reactions are presented as

the specificnumbers.

ReactionriPcan be calculated using Tables A.3–A.6. Fundamental feature of

system (10.20) is that the free termsriPare somefunctionsof position of unit load

PD 1. Therefore, a solution of the system (10.20) leads to the primary unknowns

Zias thefunctionsof the load position, in fact, to IL.Zi/,iD1;:::;n.

Bending moment at any sectionkin case of fixed load may be calculated by

formula

MkDM 1 Z 1 CM 2 Z 2 C:::CMP^0 :

In case of moving load, the bending moment at the any specified sectionkbecomes

function, so influence line should be constructed by formula

IL.Mk/DMk1IL.Z 1 /CMk2IL.Z 2 /C:::CIL



Mk^0



: (10.21)

Influence lines for shear and axial force at any section can be constructed similarly.

Similarly to the force method, the construction of influence lines for internal

forces in statically indeterminate structures by the displacement method starts from

construction of influence lines for primary unknowns IL.Z 1 /;IL.Z 2 /;:::.

The following procedure may be recommended for construction of influence

lines for statically indeterminate structures by the displacement method:

1.Determine the degree of kinematical indeterminacy, construct the primary system
and formulate the canonical equations of the displacement method.
2.Compute the coefficientsrikof canonical equations; they are unit reaction at
introduced constrainticaused by unitdisplacementsof introduced constraintk.
These unit reactions present the some specificnumbers.
3.Construct the expressions for free termsriPof canonical equations. These load
terms presents reaction at introduced constraintidue to load, depends of location
of unit load and therefore present somefunctions.