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.

**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.

ANSAH NANA BENYI-GHANA

I WOULD BE HAPPY TO OBTAIN SOLVED EXAMPLES ON INFLUENCE LINES FOR FRAMES AND BEAMS INCLUDING CONTINUOUS BEAMS. THANK U>

Osama Omran

good approach

Abdur Rakib

very helpful

Devendra Dhakal

VERY VERY HELPFUL….

Gabriel Gathecha Kamau

will get to you but was looking to strain energy method.

Md Ziaul

I want continues beam.

Md Naheyan

UITS (university of information technology and science).

I Like this site……………

Ravi Shankar

very useful for student.

Ravi Shankar

please more solveed problem.

Joseph Njuguna

what about un.

proped cantilever

Engr Noushad Ali

very usefull

Yousof Khmd

good

Hafiz Muhammad Awais Attari

helpful for me

Abdul Basir Rahmati

liked it

Subin Gajurel

ILD for fix end moments of a fixed beam???

Rakesh-Cool Sparsha

needs improvement.

James Kisawuzi

Good