Influence Lines for Beams

A downward concentrated load of magnitude 1 unit moves across the simply supported beam AB from A to B. We wish to determine the following functions:

  • reaction at A
  • reaction at B
  • shear at C and
  • moment at C

when the unit load is at a distance x from support A. Since the value of the above functions will vary according to the location of the unit load, the best way to represent these functions is by influence diagram.
 

Influence Line by Equilibrium Method

 

influence-line-simple-beam-given.jpg

 

Influence Line for the reaction at A

$\Sigma M_B = 0$

$LR_A = 1.0(L - x)$

$R_A = 1 - \dfrac{x}{L}$   ←   linear equation in x
 

When x = 0, RA = 1 and when x = L, RA = 0
 

influence-line-simple-beam-reaction-left.jpg

 

Influence Line for the reaction at B

$\Sigma M_A = 0$

$LR_B = 1.0x$

$R_B = \dfrac{x}{L}$   ←   linear equation in x
 

When x = 0, RB = 0, and when x = L, RB = 1
 

influence-line-simple-beam-reaction-right.jpg

 

Influence Line for Shear at C

 
influence-line-simple-beam-given-no-load.jpg

 

influence-line-for-shear-load-included.jpgUnit load is within the segment AC
$V_C = R_A - 1.0$

$V_C = \left( 1 - \dfrac{x}{L} \right) - 1.0$

$V_C = -\dfrac{x}{L}$
 

When x = 0, VC = 0, and when x = a (just before point C), VC = -a/L
 

Unit load is beyond the segment AC
$V_C = R_A$

$V_C = 1 - \dfrac{x}{L}$
 

influence-line-for-shear-load-beyond-ac.gif

 

When x = a (just after point C), VC = 1 - a/L = (L - a) / L = b/L and when x = L, VC = 1 - L/L = 0
 

influence-line-shear-at-c.jpg

 

Influence Line for Moment at C

 
influence-line-simple-beam-given-no-load.jpg

 

influence-line-for-moment-load-included.jpgUnit load is within the segment AC
$M_C = aR_A - 1.0(a - x)$

$M_C = a\left( 1 - \dfrac{x}{L} \right) - (a - x)$

$M_C = a - \dfrac{ax}{L} - a + x$

$M_C = -\dfrac{ax}{L} + x$
 

When x = 0, MC = 0, and when x = a (just just before point C), MC = -a2/L + a = (-a2 + aL) / L = a(L - a) / L = ab/L
 

Unit load is beyond the segment AC
$M_C = aR_A = a\left( 1 - \dfrac{x}{L} \right)$

$M_C = a - \dfrac{ax}{L}$
 

influence-line-for-moment-load-beyond-ac.jpg

 

When x = a (just after point C), MC = a - a2/L = (aL - a2) / L = a(L - a) / L = ab/L and when x = L, MC = a - aL/L = a - a = 0
 

influence-line-moment-at-c.jpg

 

Influence Line of Simply Supported Beam with Overhang
The ordinate of influence line at the overhang can be found by ratio and proportion of triangle.
 

influence-line-all.jpg