Influence line
diagram
(ILD) can be developed by the following ways;
(a)
by independently applying a
unit load at several positions on the structure and determining the
structural response (reaction, shear, moment etc.) by applying the
principles of statics. Tabulate these values and then plot the graph of
load position(along xaxis) vs the structural response (along
yaxis). You can also use
online calculator
to get the ordinates of ILD.
(b)
write the equation for
structural response due to a unit load at a distance 'x' by applying
the principles of statics. This will give the equation of
influence line in terms of 'x' for different parts of the span.
(c) MullerBreslau Principle
which gives qualitative influence lines. This principle states
that the influence line for a function is to the same scale as the
deflected shape of the beam when the beam is acted upon by that
function. By using these techniques we can quickly draw the shape of
an influence line.
To explain the procedure of
developing influence line we consider a beam of span L and
apply a unit load at a distance x from the support A as
shown in figure 1.
Influence line for Reaction
at a support
Calculating the reaction at
supports we get;
R_{B} = x/L
(straight line equation)
R_{A} = 1 x/L
(straight line equation)
The influence line diagrams for
R_{B}
is plotted in figure 1(a) which shows that the value of
R_{B} is x/L when the
unit load is at a distance x from A and it would be equal to 1 when
the unit load will be at B.
Similarly the ILD for
R_{A } plotted
in figure 1 (b) shows that the value of
R_{A }
will be 1 when unit load is at A and it would be equal to 1x/L when
the unit load is at a distance x from support A.
Influence line for Shear Force at a section
F_{C} = R_{B}
=  x/L ;when load is
between A and C
F_{C} = R_{A}
= 1 x/L ;when the
load is between C and B
The ILD for
F_{C}
plotted in figure 1(c) clearly indicates the above values. Maximum
positive or negative shear force will occur at C when the load is
placed at C.

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