Influence Lines

Influence line is the graphical representation of the response function of the structure as the downward unit load moves across the structure. The ordinate of the influence line show the magnitude and character of the function.
 

The most common response functions of our interest are support reaction, shear at a section, bending moment at a section, and force in truss member. Influence line can also be used to determine deflections.
 

With the aid of influence diagram, we can...

  1. determine the position of the load to cause maximum value to the function.
  2. calculate the maximum value of the function.

 

Properties of Influence Lines

  1. Value of the function due to single concentrated load
     
    influence-line-concentrated-load.jpg
     

    $\text{Function} = Ph$

     

  2. Value of the function due to multiple concentrated loads
  3.  

    influence-line-concentrated-loads.jpg
     

    $\text{Function} = P_1h_1 + P_2h_2 - P_3h_3$

     

  4. Value of the function due uniformly distributed load
     
    influence-line-uniform-load.jpg
     

    $\text{Function} = w \times \text{Area}$

     

  5. Value of the function for any type of load
     
    influence-line-any-load.jpg
     

    $\displaystyle \text{Function} = \int_a^b y_i (y \, dx)$