Influence line:

An influence line is a curve, an ordinate of that at any point is equivalent to the value of some structural quantity, when a unit load is placed at that point.

The structural quantity could be external support reactions, (e.g. vertical or horizontal reactive forces or bending moments), internal stress resultants (e.g. axial force, SF or BM) or deformations (e.g. slope and deflections). The above definition can be explained by the following simple examples.

Figure shows a beam which is simply supported at A and B and has a span of 10 m. If we want to find the influence line for the reaction R_A at the support A, we have to place a unit load (e.g. 1 kN) at several points on the beam and to search the corresponding values of R_A. Those values are plotted at the points whereas the load is placed. For instance we know that if the unit load is placed at A, the reaction R_A = 1, and if it is placed at B, R_A = 0. Now if the moveable unit load is placed at any point P on the beam, that is among A and B, at a distance of x from A, then taking moments of all forces about B, we have here

R_A  10-1×(10-x)=0 ⇒ giving  R_A =(10-x)/10

Eq. (1.1), as can be seen, is the equation of a straight line shown in Figure (b). Thus, Figure is the influence line for the reaction RA of the beam.

Similarly, we may draw influence lines for bending moment or shearing forces at a point on the beam.