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:

INFLUENCE LINE METHOD OF ANALYSIS

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.

clip_image003

As’ x’ increases, clip_image004 increases and clip_image005 decreases.

At x=0, clip_image004[1]=0, clip_image005[1]=1

At x=l, clip_image004[2]= 1, clip_image005[2]= 0

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

Uses of an Influence Line Diagram:

  1. To determine the value of the quantity for a given system of loads on the span of the structure.
  2. 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 METHOD OF ANALYSIS

INFLUENCE LINE DIAGRAMS:

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

INFLUENCE LINE METHOD OF ANALYSIS

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,

clip_image009

When the unit load is between D and B

clip_image010

(2) For the bending moment at a given section

When the unit load is between A and D.

clip_image011

When clip_image012, clip_image013

When clip_image014, clip_image015

When the unit load is between D and B,

clip_image016

When clip_image014[1],clip_image017

When clip_image018, clip_image013[1]

INFLUENCE LINE METHOD OF ANALYSIS

(3) Simply supported beam with overhanging

INFLUENCE LINE METHOD OF ANALYSIS

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 clip_image005[3]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 clip_image005[4]. Thus after removing the support at A, apply an external forces clip_image005[5] at A so as to maintain the beam in its position.

INFLUENCE LINE METHOD OF ANALYSIS

Now from Belti’s theorem,

clip_image025

This means that the ordinate of the influence line for clip_image005[6] at the point n is obtained by dividing the deflection at n by the deflection at point a.

Thus, influence line diagram for clip_image005[7] is given by

INFLUENCE LINE METHOD OF ANALYSIS

(5) Propped Cantilever

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

Let clip_image005[8] and clip_image004[3] be the reacting forces at A and B respectively.

To draw the influence line for clip_image004[4]

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,

clip_image028

As per Belti’s Law;

clip_image029

Thus, we can write

clip_image030

INFLUENCE LINE METHOD OF ANALYSIS

Thus the ordinate of the influence line for clip_image004[5] is obtained by dividing the deflection at any section X by the deflection at the point B due to unit load B.

clip_image033

INFLUENCE LINE METHOD OF ANALYSIS