Advanced Methods of Structural Analysis

22 2 General Theory of Influence Lines

As above, the expression for bending moment is transformed to the equation of

influence line for bending moment

IL.Mk/DaIL.RA/: (2.7)

For construction of this influence line, we need to construct the influence line for

reactionRA, then multiply all ordinates by parametera, and finally, to show the

operating range of influence line. Since loadPis located to therightof sectionk,

we obtainright-hand portionof influence line as the operating range. Corresponding

influence lineMkis presented in Fig.2.4.

Since units of ordinates IL.RA/and IL.RB/are dimensionless, then units of

ordinates IL.Mk/are units of length (for example, meter). Ordinate of influence line

at sectionkis.ab= l /. Sign of influence line for bending momentMkfor simply

supported beam is positive, which means that extended fibers at sectionkare located

below longitudinal axis for any position of the loadP.

Henceforward, all mathematical treatment concerning to construction of influ-

ence lines will be presented in tabulated form. The above-mentioned discussions

are presented in the following table.

LoadPD 1 left at sectionk LoadPD 1 right at sectionk

Mk!

P

MkrightD 0 , Mk!

P

MkleftD0;

MkDRBb!IL.Mk/DbIL.RB/ MkDRAa!IL.Mk/DaIL.RA/

To summarize, in order to construct the influence line for bending moment at section

kit is necessary to:

1.Plot ordinatesaandbon the left and right vertical lines passing through the
left-hand and right-hand support, respectively.
2.Join each of these points with base point at the other support; both lines intersect
at sectionk.
3.Show the operating ranges of influence line. The hatching (operating range) cor-
responds to thepositionof the load but not to thepartof the beam which is used
for computation of bending moment.

2.1.2.2 Influence LineQk

Since the expression for shear depends on whether the loadPis located to the

left or to the right from the sectionk, then two positions of the loadPneed to be

considered. The procedure of construction of influence line for shear is presented in

tabulated form. In the table we show thepositionof the load (left or right at section

k– this is the first line of the table) andpartof the beam, the equilibrium of which

is considered.