Advanced Methods of Structural Analysis

2.1 Analytical Method for Construction of Influence Lines 21

bending moment is

Mk!

X

MkrightD 0 W MkDRBb:

If position of the loadPis fixed, then reactionRBis anumberand the bending
moment is anumberas well. However, if loadPD 1 changes its position along the
left portion of the beam, then reactionRBbecomes afunctionof position of the load
Pand, thus, the bending moment is afunctiontoo. Thus, theexpressionfor bending
moment is transformed to theequationof influence line for bending moment

IL.Mk/DbIL.RB/: (2.6)

So for the construction of influence line for bending moment we need to construct
the influence line for reactionRB, after that multiply all ordinates by parameterb,
and, as the last step, show the operating range of influence line. Since loadPis
located to theleftof sectionk, then the operating range isleft-hand portionof in-
fluence line, i.e., the above equation of influence line is true when the loadPis
changing its position on theleftportion of the beam. Hatching the corresponding
part of the influence line reflects this fact.

Fig. 2.4 Simply supported
beam. Construction of
influence line for bending
moment at sectionk

x P= 1

RA l RB

k

ab

AB

Inf. line Mk (m)

a

b

ab

l

+

abl 1 ⋅b

Operating range

+

b·IL (RB)

Load P=1 left at section k

1 ⋅a

Load P=1 right at section k

Operating range

abl

+

a·IL(RA)

LoadPD 1 Is Located to the Right of Sectionk

In this case, it is convenient to calculate the bending momentMk using the
left forces. The only reactionRAis located to the left of pointk, so the bending
moment is
Mk!

X

MkleftD 0 W MkDRAa: