Influence Lines

STRUCTURAL ANALYSISBFC 21403

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Introductionwww.uthm.edu.my With Wisdom, We Explore

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Influence lines describe the variation of an analysis variable(reaction, shear force, bending moment, twisting moment,deflection, etc.) at a specific point (any distance fromsupport), say at C:-

Why do we need the influence lines? For instance, whenlarge loads pass over a structure such as a bridge, one needsto know when and where the maximum values ofreaction/shear/bending-moment will occur at a point, sothat the section can be designed regarding its loadingbehaviour.

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Moving Loads: Loads applied to a structure with points ofapplication (including their magnitude) can vary as afunction of positions on the structure. Examples of movingloads include live load on buildings, traffic loads on bridges,loads induced by wind. In the analysis, the moving loads canbe modelled as varying distributed loads, a series ofconcentrated loads, or the combination of distributed loadsand concentrated loads.

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A moving unit load : a concentrated load of unit magnitudewith its point of application varies as a function of positionon the structure.

Responses due to moving loads : quantities of interest thatindicate the effect of the moving loads on a structure(internal forces, support reactions, displacements androtations, deformations)

Responses due to a moving unit load :quantities of interestat a particular point within a given structure (internal forces,support reactions, deformations, displacements androtations) due to an applied moving unit load. Thequantities are given in terms of functions of a position of amoving unit load on the structure; these response functionsare termed as the influence functions and their graphicalrepresentations are known as the influence lines.

Influence Line for Beamwww.uthm.edu.my With Wisdom, We Explore

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Procedure:

1) Allow a unit load (either 1kN, 1N, 1kip, or 1 tonne) to move over beam from left to right.

2) Find the values of shear force or bending moment, at the point under consideration, as the unit load moves over the beam from left to right.

3) Plot the values of the shear force or bending moment, over the length of the beam, computed for the point under consideration.

1kN

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Notations: (Sign convention)

1) Normal Forces: +ve forces cause +ve displacements in+ve directions.

2) Shear Forces: +ve shear forces cause clockwise rotation& -ve shear force causes anti-clockwise rotation.

3) Bending Moments: +ve bending moments cause cuprotations.

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When a point load move from left to right, the threefollowing variation will be produced:

Variation of Reactions RA and RB as functions loadposition

Variation of Shear Force at C as a function of loadposition

Variation of Bending Moment at C as a function of loadposition

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Variation of reaction forces

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Moving Concentrated Load www.uthm.edu.my With Wisdom, We Explore

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Moving Concentrated Load www.uthm.edu.my With Wisdom, We Explore

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Variation of Shear force and Moment

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Moving Concentrated Load www.uthm.edu.my With Wisdom, We Explore

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Moving Concentrated Load www.uthm.edu.my With Wisdom, We Explore

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Moving Concentrated Load www.uthm.edu.my With Wisdom, We Explore

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Moving Concentrated Load www.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Principle of Muller-Breslau The principle gives only a procedure to determine of the

influence line of a parameter for a determinate/indeterminate structure

But using the basic understanding of the influence lines,the magnitudes of the influence lines also can be computed

In order to draw the shape of the influence lines properly,the capacity of the beam to resist the parameterinvestigated (reaction, bending moment, shear force, etc.),at that point, must be removed

The principle states : The influence line for a function (e.g.reaction, shear, moment) at a point, is to the same scale asthe deflected shape of the beam when the beam is actedupon by that function. The capacity of the beam to resist that parameter, at

that point, must be removed. Then allow the beam to deflect under that parameter Positive directions of the forces are the same as before

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Influence Line for a Determinate Beam by Muller-Breslau’sMethod (Simply Supported Beam-Pin Join)

Influence line for Reaction at A

Deflection, y

Remove the ability to resist movement in the vertical direction at A by using a guided roller

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Influence Lines for a Determinate Beam by Muller-Breslau’s Method (Cantilever beam-fixed support)

Influence line for Reaction at A

•Use a double roller to transmit moments and axial load at the fixed support. Vertical reaction will not transmit horizontal reaction.

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Influence Lines for a Determinate Beam by Muller-Breslau’s Method (Indeterminate Beam-Shear)

Influence Line for Reaction at indeterminate beam

•At roller support, consider the vertical reaction at this point.

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Influence Lines for a Determinate Beam by Muller-Breslau’s Method (Shear at the mid-span)

Influence Line for Shear at C

•Remove the ability to resist shear at point C with roller connection. Applying a positive shear force to the beam at C and allows the deflection.

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Influence Lines for a Determinate Beam by Muller-Breslau’s Method (Shear at the cantilever beam)

Influence Line for Shear at C

•A roller guide must be placed at C, the positive shear is applied. The left segment of the beam does not deflect due to fixed support.

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Influence Lines for a Determinate Beam by Muller-Breslau’s Method (Moment at the mid-span)

Influence Line for Bending Moment at C

•Remove the ability to resist moment at C by using hinge. Moments will occur at both segments.

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Example 5.3

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Example 5.4

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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Example 5.4

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Qualitative Influence Linewww.uthm.edu.my With Wisdom, We Explore

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General Shape Influence Line (Shear)www.uthm.edu.my With Wisdom, We Explore

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General Shape Influence Line (Shear)www.uthm.edu.my With Wisdom, We Explore

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General Shape Influence Line (Moment)www.uthm.edu.my With Wisdom, We Explore

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General Shape Influence Line (Moment)www.uthm.edu.my With Wisdom, We Explore

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