Advanced Methods of Structural Analysis

26 2 General Theory of Influence Lines

Fig. 2.9 Construction of
influence lines for section 2
on the overhang

P= 1

2

x

c

The influence line for shear in section 1 for the simply supported beam should be
extended through support points to the end of overhangs.

Shear ForceQ 2 and Bending MomentM 2 (Fig.2.9)

Since expressions for bending moment and shear at section 2 depend on position of
the loadP(to the left or to the right of the section 2), then for deriving equations of
influence lines two positions of the load should be considered.

PD 1 left at section 2 PD 1 right at section 2

Q 2!

P

YleftD 0 !Q 2 DP Q 2!

P

YleftD 0 !Q 2 D 0

IL.Q 2 /D 1 IL.Q 2 /D 0

M 2!

P

M 2 leftD 0 !M 2 DPx M 2!

P

M 2 leftD 0 !M 2 D 0

IL.M 2 /Dx IL.M 2 /D 0

At xD 0 W IL.M 2 /D 0

At xDcW IL.M 2 /Dc

Note that for any position of the loadPD 1 (left or right at section 2) we use the
equilibrium equations

P

YleftD 0 and

P
M 2 leftD 0 , which take into account forces
that are locatedleftat the section. Similarly, for section 5 we will use the forces that
are locatedrightat the section 5.
Construction of influence lines for bending moments and shear at sections 3, 4,
and 5 are performed in the same manner.
Pay attention that influence lines of shear for sections 3 leftand 3 right,thatare
infinitesimally close to support point, (as well as for sections 4 leftand 4 right)are
different. These sections are shown as 30 and 300 , 40 and 400.
Influence lines for bending moments andshear forces at all pattern sections 1–5
are summarized in Fig.2.8. These influence lines are good reference source for prac-
tical analysis of one-span beams subjected toanytype of loads. Moreover, these
influence lines will be used for construction of influence lines for multispan hinged
statically determinate beams.