# Influence Lines for Beams

A downward concentrated load of magnitude 1 unit moves from *A* to *B* across the simply supported beam *AB* as shown below.

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 for the reaction at A**

$LR_A = 1.0(L - x)$

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

When *x* = 0, *R _{A}* = 1 and when

*x*=

*L*,

*R*= 0

_{A}

**Influence Line for the reaction at B**

$LR_B = 1.0x$

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

When *x* = 0, *R _{B}* = 0, and when

*x*=

*L*,

*R*= 1

_{B}

**Influence Line for Shear at C**

*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, *V _{C}* = 0, and when

*x*=

*a*(just before point

*C*),

*V*= -

_{C}*a*/

*L*

Unit load is beyond the segment *AC*

$V_C = R_A$

$V_C = 1 - \dfrac{x}{L}$

When *x* = *a* (just after point *C*), *V _{C}* = 1 -

*a*/

*L*= (

*L*-

*a*) /

*L*=

*b*/

*L*and when

*x*=

*L*,

*V*= 1 -

_{C}*L*/

*L*= 0

**Influence Line for Moment at C**

*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, *M _{C}* = 0, and when

*x*=

*a*(just just before point

*C*),

*M*= -

_{C}*a*

^{2}/

*L*+

*a*= (-

*a*

^{2}+

*aL*) /

*L*=

*a*(

*L*-

*a*) /

*L*=

*ab*/

*L*

Unit load is beyond the segment *AC*

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

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

When *x* = *a* (just after point *C*), *M _{C}* =

*a*-

*a*

^{2}/

*L*= (

*aL*-

*a*

^{2}) /

*L*=

*a*(

*L*-

*a*) /

*L*=

*ab*/

*L*and when

*x*=

*L*,

*M*=

_{C}*a*-

*aL*/

*L*=

*a*-

*a*= 0

**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.

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