324 10 Influence Lines Method
Influence line for bending moment In case offixedload, a bending moment at any
sectionkequals
MkDMkX 1 CMk^0 (10.3)
whereX 1 is the primary unknown of the force method;Mkis the bending moment
at sectionkin a primary system due to unit primary unknownXD 1 ;andMk^0 is
the bending moment at sectionkin a primary system due to given load.
Now we need to transform (10.3) for the case ofmovingload. The bending mo-
mentMkat any sectionkpresents thenumber, because this moment is caused
by the unit primary unknownX 1 ; the second componentX 1 according to (10.2)
presents a function. The last component, the bending momentMk^0 , is caused by
the given load, which is considered as moving load now; therefore, the bending
momentMk^0 also becomes afunctionof the position of this load. As a result, the
bending moment at any sectionkbecomes afunction:
IL.Mk/DMkIL.X 1 /CIL
Mk^0
: (10.4)Influence line for shear force In case of fixed load, the shear at any sectionkis a
number
QkDQkX 1 CQ^0 k: (10.5)Similarly in case of traveling load, the shear at any sectionkbecomes a function,
therefore
IL.Qk/DQkIL.X 1 /CIL
Q^0 k
: (10.6)For then-times statically indeterminate structure, the canonical equations of the
force method in case of a fixed loadPD 1 are
ı 11 X 1 Cı 12 X 2 CCı1nXnCı1PD 0
(10.7)
ın1X 1 Cın2X 2 CCınnXnCınPD 0Unit displacementsıik which are caused by unit primary unknowns present
numbers.
DisplacementsıiPare caused by unit moving load. Fundamental feature of sys-
tem (10.7) is that the free termsıiPare somefunctionsof position of unit load
PD 1. Therefore, a solution of the system (10.7) leads to the primary unknowns
Xias thefunctionsof the load position, in fact, toIL .Xi/,iD1;:::;n. In this case
the bending moment at the any specified sectionkbecomes function, so influence
line should be constructed by formula
IL.Mk/DMk1IL.X 1 /CMk2IL.X 2 /CCIL
Mk^0
: (10.8)Influence lines expressions for shear and axial force at any section may be con-
structed similarly.