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Influence lines


Shear and moment diagram gives us a graphical representation of shear and moment for static concentrated or uniform loads. For moving loads such as vehicles, the variation of shear, and moment over the structural members is best described by influence lines. Influence lines can be used to determine the reaction, shear, moment, and deflections of a concentrated moving load over the member. By constructing an influence line for a member, we can determine the location at which a load can create the greatest influence on a member. Therefore, the influence line is important for the design of structures with moving loads such as bridges, crane rails, and others. 



We should distinguish between influence lines and shear or moment diagrams. Influence line representing the effects of a moving load at a specified point only. While the shear and moment diagram representing the effects of fixed loads at all points.

Procedure for Analysis:

The procedure for constructing an influence line is simple. There are two methods for constructing an influence line.


Tabulate values:

  • Place a unit load at different points over the member, and using equilibrium equations, determine the value of reaction, shear, and moment at a specified point.
  • For reaction at a specified point, consider the reaction to be positive if it acts upward. Shear and moment sign convention are similar to the one used for shear and moment diagram.
  • All statically determine beams will have a straight influence line. Influence lines will consist of a segment of straight lines. Practicing should minimize the computation and enable us to locate the point load at the endpoint of segments.
  • To avoid errors, we should construct a table. In this table, we will insert the x value, which represents the location of the point load from the specific point under investigation. Other columns will contain the reaction, shear, and moment at the specific point we investigate.


Influence-Line Equations:

  • ·The influence line can also be constructed by placing the unit load at a variable position x on the member and then computing the value of R, V, or M at the point as a function of x. In this manner, the equations of the various line segments composing the influence line can be determined and plotted.







Example: Construct the influence line for the vertical reaction at A of the beam shown in figure no:1.

 
Figure 1

Solution: here in this example, we will draw the influence line of a moving load for point A. we will place a unit load at a various location over the beam, then we will calculate the effects of the unit load at point A, then we can draw the influence line. We will construct a table that contains the x value, which represents the location of a unit load from the point A and the reaction at point A.

At x=0
Ay*10-1*10=0
Ay=1

At x=2.5

Ay*10-1*7.5=0
Ay=0.75





Influence line equation: When the unit load is placed a variable distance x from A,  the reaction as a function of x can be determined from


-Ay*10+1*(10-x)=0
Ay=1-x/10

This line is plotted in figure no:2

Figure 2

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