Influence line

In engineering, an influence line graphs the variation of a function (such as the shear, moment etc.

[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.

The scaled maximum and minimum are the critical magnitudes that must be designed for in the beam or truss.

When adding the influence lines together, it is necessary to include the appropriate offsets due to the spacing of loads across the structure.

is applied and imposing a relative unit displacement that is kinematically admissible in the negative direction, represented as

When designing a beam or truss, it is necessary to design for the scenarios causing the maximum expected reactions, shears, and moments within the structure members to ensure that no member fails during the life of the structure.

Influence lines graph the response of a beam or truss as a unit load travels across it.

The influence line helps designers find where to place a live load in order to calculate the maximum resulting response for each of the following functions: reaction, shear, or moment.

[5] The second is to determine the influence-line equations that apply to the structure, thereby solving for all points along the influence line in terms of x, where x is the number of feet from the start of the structure to the point where the unit load is applied.

[1][2][5] This influence line will still provide the designer with an accurate idea of where the unit load will produce the largest response of a function at the point being studied, but it cannot be used directly to calculate what the magnitude that response will be, whereas the influence lines produced by the first two methods can.

Statics is used to calculate what the value of the function (reaction, shear, or moment) is at point A.

The slope of the inflection line can change at supports, mid-spans, and joints.

An influence line for a given function, such as a reaction, axial force, shear force, or bending moment, is a graph that shows the variation of that function at any given point on a structure due to the application of a unit load at any point on the structure.

An influence line for a function differs from a shear, axial, or bending moment diagram.

Influence lines can be generated by independently applying a unit load at several points on a structure and determining the value of the function due to this load, i.e. shear, axial, and moment at the desired location.

It is possible to create equations defining the influence line across the entire span of a structure.

This is done by solving for the reaction, shear, or moment at the point A caused by a unit load placed at x feet along the structure instead of a specific distance.

[5] According to www.public.iastate.edu, “The Müller-Breslau Principle can be utilized to draw qualitative influence lines, which are directly proportional to the actual influence line.”[2] Instead of moving a unit load along a beam, the Müller-Breslau Principle finds the deflected shape of the beam caused by first releasing the beam at the point being studied, and then applying the function (reaction, shear, or moment) being studied to that point.

Below are explanations of how to find the influence lines of a simply supported, rigid beam (such as the one displayed in Figure 1).

Instead, the engineer must use statics to solve for the functions value in that loading case.

For example, on a bridge the wheels of cars or trucks create point loads that act at relatively standard distances.

To calculate the response of a function to all these point loads using an influence line, the results found with the influence line can be scaled for each load, and then the scaled magnitudes can be summed to find the total response that the structure must withstand.

The integration of the influence line gives the effect that would be felt if the distributed load had a unit magnitude.

Therefore, after integrating, the designer must still scale the results to get the actual effect of the distributed load.

The methods above can still be used to determine the influence lines for the structure, but the work becomes much more complex as the properties of the beam itself must be taken into consideration.

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.