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.

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]

Demonstration from Betti's theorem

Influence lines are based on Betti's theorem. From there, consider two external force systems, ${\displaystyle F_{i}^{P}}$ and ${\displaystyle F_{i}^{Q}}$, each one associated with a displacement field whose displacements measured in the force's point of application are represented by ${\displaystyle d_{i}^{P}}$ and ${\displaystyle d_{i}^{Q}}$.

Consider that the ${\displaystyle F_{i}^{P}}$ system represents actual forces applied to the structure, which are in equilibrium. Consider that the ${\displaystyle F_{i}^{Q}}$ system is formed by a single force, ${\displaystyle F^{Q}}$. The displacement field ${\displaystyle d_{i}^{Q}}$ associated with this forced is defined by releasing the structural restraints acting on the point where ${\displaystyle F^{Q}}$ is applied and imposing a relative unit displacement that is kinematically admissible in the negative direction, represented as ${\displaystyle d_{1}^{Q}=-1}$. From Betti's theorem, we obtain the following result:

${\displaystyle -F_{1}^{P}+\sum _{i=2}^{n}F_{i}^{P}d_{i}^{Q}=F^{Q}\times 0\iff F_{1}^{P}=\sum _{i=2}^{n}F_{i}^{P}d_{i}^{Q}}$

Concept

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. When dealing with dead loads (loads that never move, such as the weight of the structure itself), this is relatively easy because the loads are easy to predict and plan for. For live loads (any load that moves during the life of the structure, such as furniture and people), it becomes much harder to predict where the loads will be or how concentrated or distributed they will be throughout 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. The designer can then scale the influence line by the greatest expected load to calculate the maximum response of each function for which the beam or truss must be designed. Influence lines can also be used to find the responses of other functions (such as deflection or axial force) to the applied unit load, but these uses of influence lines are less common.

Methods for constructing influence lines

There are three methods used for constructing the influence line. The first is to tabulate the influence values for multiple points along the structure, then use those points to create the influence line.^[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][3][4][5]} The third method is called the Müller-Breslau's principle. It creates a qualitative influence line.^[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.

Alternate loading cases

Indeterminate structures

While the influence lines of statically determinate structures (as mentioned above) are made up of straight line segments, the same is not true for indeterminate structures. Indeterminate structures are not considered rigid; therefore, the influence lines drawn for them will not be straight lines but rather curves. 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.

See also

• Beam
• Shear and Moment Diagram
• Dead and Live Loads
• Müller-Breslau's principle

References