## INFLUENCE LINE METHOD OF ANALYSIS

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

Jul 15, 2011@ 14:10 pmI WOULD BE HAPPY TO OBTAIN SOLVED EXAMPLES ON INFLUENCE LINES FOR FRAMES AND BEAMS INCLUDING CONTINUOUS BEAMS. THANK U>

Osama Omran

Oct 12, 2011@ 08:47 amgood approach

Abdur Rakib

Dec 02, 2011@ 15:46 pmvery helpful

Devendra Dhakal

Apr 15, 2012@ 13:06 pmVERY VERY HELPFUL….

Gabriel Gathecha Kamau

Apr 16, 2012@ 04:46 amwill get to you but was looking to strain energy method.

Md Ziaul

Jul 07, 2012@ 17:53 pmI want continues beam.

Md Naheyan

Aug 12, 2012@ 08:01 amUITS (university of information technology and science).

I Like this site……………

Ravi Shankar

Sep 09, 2012@ 14:14 pmvery useful for student.

Ravi Shankar

Sep 09, 2012@ 14:15 pmplease more solveed problem.

Joseph Njuguna

Nov 29, 2012@ 13:09 pmwhat about un.

proped cantilever

Engr Noushad Ali

Oct 08, 2013@ 14:51 pmvery usefull

Yousof Khmd

Oct 14, 2013@ 09:57 amgood

Hafiz Muhammad Awais Attari

Nov 22, 2013@ 10:12 amhelpful for me

Abdul Basir Rahmati

Jan 25, 2014@ 13:53 pmliked it

Subin Gajurel

Feb 21, 2014@ 15:33 pmILD for fix end moments of a fixed beam???

Rakesh-Cool Sparsha

Feb 28, 2014@ 17:20 pmneeds improvement.

James Kisawuzi

Jun 06, 2014@ 08:31 amGood

Emaleu Michael

Jul 24, 2014@ 16:47 pmhelpful

Lawrence Martins

Jul 25, 2014@ 08:46 amGreat! this gives basic principles of INFLUENCE LINE METHOD OF ANALYSIS.

Mbah Humphrey Teke

Jul 25, 2014@ 22:47 pmMichael, thanks for your comments on the platform on the exam.

Emaleu Michael

Jul 26, 2014@ 08:25 amyou are well come Mrs Mbah. Yours to was very nice and important

Ricardo Balbuena

Aug 05, 2014@ 10:27 amme like it

Pisoluposh Sospeter

Aug 21, 2014@ 12:32 pmNimekuona uko juu bwana poa