An influence line for any given point or section of structure is a curve whose ordinates represent to scale the variation of a function such as shear force, bending moment, deflection etc at a point or section, as the unit load moves across the structure.

Table of Contents

- 1 Influence line for determinate structures:
- 2 (1) Simply supported beam:
- 3 I.L.D. for reactions at the supports:
- 4 Uses of an Influence Line Diagram:
- 5 INFLUENCE LINE DIAGRAMS:
- 6 (1) For shear force at a given section of a simply supported girder
- 7 (2) For the bending moment at a given section
- 8 (3) Simply supported beam with overhanging
- 9 MULLER BRESLAU PRINCIPLE:
- 10 (4) Continuous Beams
- 11 (5) Propped Cantilever

**Influence line for determinate structures:**

**(1) Simply supported beam:**

**I.L.D. for reactions at the supports:**

Let a unit load waves from the left end A to the right end B of the beam.

As’ x’ increases, increases and decreases.

At x=0, =0, =1

At x=l, = 1, = 0

Thus, I.L.D. is shown in the figure below.

**Uses of an Influence Line Diagram:**

- To determine the value of the quantity for a given system of loads on the span of the structure.
- To determine the position of live load for the quantity to have the maximum value and hence to compute the maximum value of the quantity.

**INFLUENCE LINE DIAGRAMS:**

**(1) For shear force at a given section of a simply supported girder**

Let a unit load move along the span of a simply supported girder AB of span l.

Let D be a given section.

When the unit load is between A and D,

When the unit load is between D and B

**(2) For the bending moment at a given section**

When the unit load is between A and D.

When ,

When ,

When the unit load is between D and B,

When ,

When ,

**(3) Simply supported beam with overhanging**

**MULLER BRESLAU PRINCIPLE:**

This principle states that if a reaction (or internal force) acts through an imposed displacement, the corresponding displaced shape of the structure is to some scale the influence line for force quantities only.

**(4) Continuous Beams**

Consider a continuous beam ABC shown in figure below. Let it be required to obtain the influence line for the vertical reaction at A. Assume that the support A is removed (as in fig b). Let the given beam carry unit load at a. Let the corresponding reaction at A be . Thus after removing the support at A, apply an external forces at A so as to maintain the beam in its position.

Now from Belti’s theorem,

This means that the ordinate of the influence line for at the point n is obtained by dividing the deflection at n by the deflection at point a.

Thus, influence line diagram for is given by

**(5) Propped Cantilever**

Figure (a) below shows a propped cantilever AB fixed at A and simply supported at B.

Let and be the reacting forces at A and B respectively.

**To draw the influence line for **

Remove the support B and apply unit load at any point distance ‘x’ from left support (fig b). Again apply a unit load at right end and displacements were measured at those two points.

As deflection at B is zero, therefore,

As per Belti’s Law;

Thus, we can write

Thus the ordinate of the influence line for is obtained by dividing the deflection at any section X by the deflection at the point B due to unit load B.