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Influence line

A simply supported beam and four different influence lines.
Figure 1: (a) This simple supported beam is shown with a unit load placed a distance x from the left end. Its influence lines for four different functions: (b) the reaction at the left support (denoted A), (c) the reaction at the right support (denoted C), (d) one for shear at a point B along the beam, and (e) one for moment also at point B.
A statically determinate beam BMD and influence line for BM at B.
Figure 2: The change in Bending Moment in a statically determinate Beam as a unit force moves from one end to the other. The bending moment diagram and the influence line for bending moment at the centre of the left-hand span, B, are shown.

In engineering, an influence line graphs the variation of a function (such as the shear, moment etc. felt in a structural member) at a specific point on a beam or truss caused by a unit load placed at any point along the structure.[1][2][3][4][5] Common functions studied with influence lines include reactions (forces that the structure's supports must apply for the structure to remain static), shear, moment, and deflection (Deformation).[6] Influence lines are important in designing beams and trusses used in bridges, crane rails, conveyor belts, floor girders, and other structures where loads will move along their span.[5] The influence lines show where a load will create the maximum effect for any of the functions studied.

Influence lines are both scalar and additive.[5] This means that they can be used even when the load that will be applied is not a unit load or if there are multiple loads applied. To find the effect of any non-unit load on a structure, the ordinate results obtained by the influence line are multiplied by the magnitude of the actual load to be applied. The entire influence line can be scaled, or just the maximum and minimum effects experienced along the line. The scaled maximum and minimum are the critical magnitudes that must be designed for in the beam or truss.

In cases where multiple loads may be in effect, influence lines for the individual loads may be added together to obtain the total effect felt the structure bears at a given point. When adding the influence lines together, it is necessary to include the appropriate offsets due to the spacing of loads across the structure. For example, a truck load is applied to the structure. Rear axle, B, is three feet behind front axle, A, then the effect of A at x feet along the structure must be added to the effect of B at (x – 3) feet along the structure—not the effect of B at x feet along the structure.

Many loads are distributed rather than concentrated. Influence lines can be used with either concentrated or distributed loadings. For a concentrated (or point) load, a unit point load is moved along the structure. For a distributed load of a given width, a unit-distributed load of the same width is moved along the structure, noting that as the load nears the ends and moves off the structure only part of the total load is carried by the structure. The effect of the distributed unit load can also be obtained by integrating the point load's influence line over the corresponding length of the structures.

The Influence lines of determinate structures becomes a mechanism whereas the Influence lines of indeterminate structures become just determinate.[7]

  1. ^ Kharagpur. "Structural Analysis.pdf, Version 2 CE IIT" Archived 2010-08-19 at the Wayback Machine. 7 August 2008. Accessed on 26 November 2010.
  2. ^ Dr. Fanous, Fouad. "Introductory Problems in Structural Analysis: Influence Lines". 20 April 2000. Accessed on 26 November 2010.
  3. ^ "Influence Line Method of Analysis". The Constructor. 10 February 2010. Accessed on 26 November 2010.
  4. ^ "Structural Analysis: Influence Lines". The Foundation Coalition. 2 December 2010. Accessed on 26 November 2010.
  5. ^ a b c Hibbeler, R.C. (2009). Structural Analysis (Seventh Edition). Pearson Prentice Hall, New Jersey. ISBN 0-13-602060-7.
  6. ^ Zeinali, Yasha (December 2017). "Framework for Flexural Rigidity Estimation in Euler-Bernoulli Beams Using Deformation Influence Lines". Infrastructures. 2 (4): 23. doi:10.3390/infrastructures2040023. S2CID 125406249.
  7. ^ "Influence Lines | Structural Analysis Review". www.mathalino.com. Retrieved 2019-12-25.

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