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.
 

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

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

 

Value of the function for any type of load
 

if-beam-any-type-of-load.gif

 

$\displaystyle \text{Function} = \int_{x_1}^{x_2} y_i (y \, dx)$
 

Problem 854 | Continuous Beams with Fixed Ends

Problem 854
Solve for the moment over the supports in the beam loaded as shown in Fig. P-854.
 

854-i-span.gif

 

Answers:
$M_1 = \dfrac{w_o L^2}{12} \cdot \dfrac{1}{\alpha + 2} = M_4$

$M_2 = -\dfrac{w_o L^2}{12} \cdot \dfrac{2}{\alpha + 2} = M_3$
 

Problem 853 | Continuous Beams with Fixed Ends

Problem 853
For the continuous beam shown in Fig. P-853, determine the moment over the supports. Also draw the shear diagram and compute the maximum positive bending moment. (Hint: Take advantage of symmetry.)
 

853-shear-diagram.gif

 

Problem 852 | Continuous Beams with Fixed Ends

Problem 852
Find the moments over the supports for the continuous beam in Figure P-852. Use the results of Problems 850 and 851.
 

852-fixed-ended-continuous-beam.gif

 

Answers
$M_1 = -146.43 ~ \text{N}\cdot\text{m}$

$M_2 = -307.14 ~ \text{N}\cdot\text{m}$

$M_3 = -521.43 ~ \text{N}\cdot\text{m}$