Influence line

Last updated January 14, 2024

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, $F_{i}^{P}$ and $F_{i}^{Q}$ , each one associated with a displacement field whose displacements measured in the force's point of application are represented by $d_{i}^{P}$ and $d_{i}^{Q}$ .

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

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

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.

Tabulate values

To tabulate the influence values with respect to some point A on the structure, a unit load must be placed at various points along the structure. Statics is used to calculate what the value of the function (reaction, shear, or moment) is at point A. Typically an upwards reaction is seen as positive. Shear and moments are given positive or negative values according to the same conventions used for shear and moment diagrams.

R. C. Hibbeler states, in his book Structural Analysis, “All statically determinate beams will have influence lines that consist of straight line segments.”^[5] Therefore, it is possible to minimize the number of computations by recognizing the points that will cause a change in the slope of the influence line and only calculating the values at those points. 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. The calculated values for each function are then plotted where the load was applied and then connected together to generate the influence line for the function.

Once the influence values have been tabulated, the influence line for the function at point A can be drawn in terms of x. First, the tabulated values must be located. For the sections in between the tabulated points, interpolation is required. Therefore, straight lines may be drawn to connect the points. Once this is done, the influence line is complete.

Influence-line equations

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. This method is similar to the tabulated values method, but rather than obtaining a numeric solution, the outcome is an equation in terms of x.^[5]

It is important to understanding where the slope of the influence line changes for this method because the influence-line equation will change for each linear section of the influence line. Therefore, the complete equation is a piecewise linear function with a separate influence-line equation for each linear section of the influence line.^[5]

Müller-Breslau's Principle

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. The principle states that the influence line of a function will have a scaled shape that is the same as the deflected shape of the beam when the beam is acted upon by the function.

To understand how the beam deflects under the function, it is necessary to remove the beam's capacity to resist the function. Below are explanations of how to find the influence lines of a simply supported, rigid beam (such as the one displayed in Figure 1).

When determining the reaction caused at a support, the support is replaced with a roller, which cannot resist a vertical reaction.^[2]^[5] Then an upward (positive) reaction is applied to the point where the support was. Since the support has been removed, the beam will rotate upwards, and since the beam is rigid, it will create a triangle with the point at the second support. If the beam extends beyond the second support as a cantilever, a similar triangle will be formed below the cantilevers position. This means that the reaction’s influence line will be a straight, sloping line with a value of zero at the location of the second support.

When determining the shear caused at some point B along the beam, the beam must be cut and a roller-guide (which is able to resist moments but not shear) must be inserted at point B.^[2]^[5] Then, by applying a positive shear to that point, it can be seen that the left side will rotate down, but the right side will rotate up. This creates a discontinuous influence line that reaches zero at the supports and whose slope is equal on either side of the discontinuity. If point B is at a support, then the deflection between point B and any other supports will still create a triangle, but if the beam is cantilevered, then the entire cantilevered side will move up or down creating a rectangle.

When determining the moment caused by at some point B along the beam, a hinge will be placed at point B, releasing it to moments but resisting shear.^[2]^[5] Then when a positive moment is placed at point B, both sides of the beam will rotate up. This will create a continuous influence line, but the slopes will be equal and opposite on either side of the hinge at point B. Since the beam is simply supported, its end supports (pins) cannot resist moment; therefore, it can be observed that the supports will never experience moments in a static situation regardless of where the load is placed.

The Müller-Breslau Principle can only produce qualitative influence lines.^[2]^[5] This means that engineers can use it to determine where to place a load to incur the maximum of a function, but the magnitude of that maximum cannot be calculated from the influence line. Instead, the engineer must use statics to solve for the functions value in that loading case.

Alternate loading cases

Multiple loads

The simplest loading case is a single point load, but influence lines can also be used to determine responses due to multiple loads and distributed loads. Sometimes it is known that multiple loads will occur at some fixed distance apart. 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.^[5] The point loads can have different magnitudes themselves, but even if they apply the same force to the structure, it will be necessary to scale them separately because they act at different distances along the structure. For example, if a car's wheels are 10 feet apart, then when the first set is 13 feet onto the bridge, the second set will be only 3 feet onto the bridge. If the first set of wheels is 7 feet onto the bridge, the second set has not yet reached the bridge, and therefore only the first set is placing a load on the bridge.

Also, if, between two loads, one of the loads is heavier, the loads must be examined in both loading orders (the larger load on the right and the larger load on the left) to ensure that the maximum load is found. If there are three or more loads, then the number of cases to be examined increases.

Distributed loads

Many loads do not act as point loads, but instead act over an extended length or area as distributed loads. For example, a tractor with continuous tracks will apply a load distributed over the length of each track.

To find the effect of a distributed load, the designer can integrate an influence line, found using a point load, over the affected distance of the structure.^[5] For example, if a three-foot-long track acts between 5 feet and 8 feet along a beam, the influence line of that beam must be integrated between 5 and 8 feet. 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.

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.

Related Research Articles

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Structural analysis is a branch of solid mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and their components. In contrast to theory of elasticity, the models used in structure analysis are often differential equations in one spatial variable. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, aircraft and ships. Structural analysis uses ideas from applied mechanics, materials science and applied mathematics to compute a structure's deformations, internal forces, stresses, support reactions, velocity, accelerations, and stability. The results of the analysis are used to verify a structure's fitness for use, often precluding physical tests. Structural analysis is thus a key part of the engineering design of structures.

Shear stress is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

<span class="mw-page-title-main">Beam (structure)</span> Structural element capable of withstanding loads by resisting bending

A beam is a structural element that primarily resists loads applied laterally across the beam's axis. Its mode of deflection is primarily by bending, as loads produce reaction forces at the beam's support points and internal bending moments, shear, stresses, strains, and deflections. Beams are characterized by their manner of support, profile, equilibrium conditions, length, and material.

<span class="mw-page-title-main">Buckling</span> Sudden change in shape of a structural component under load

In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column.

In applied mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

<span class="mw-page-title-main">Euler–Bernoulli beam theory</span> Method for load calculation in construction

Euler–Bernoulli beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko–Ehrenfest beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.

In fluid dynamics, shear flow is the flow induced by a force in a fluid. In solid mechanics, shear flow is the shear stress over a distance in a thin-walled structure.

In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending moments is the beam. The diagram shows a beam which is simply supported at both ends; the ends can only react to the shear loads. Other beams can have both ends fixed ; therefore each end support has both bending moments and shear reaction loads. Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever, which is fixed at one end and is free at the other end. In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely.

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<span class="mw-page-title-main">Neutral axis</span>

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Shear force and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear forces and bending moments at a given point of a structural element such as a beam. These diagrams can be used to easily determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure. Another application of shear and moment diagrams is that the deflection of a beam can be easily determined using either the moment area method or the conjugate beam method.

<span class="mw-page-title-main">Deflection (engineering)</span> Degree to which part of a structural element is displaced under a given load

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<span class="mw-page-title-main">Structural engineering theory</span>

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The Müller-Breslau principle is a method to determine influence lines. The principle states that the influence lines of an action assumes the scaled form of the deflection displacement. OR, This principle states that "ordinate of ILD for a reactive force is given by ordinate of elastic curve if a unit deflection is applied in the direction of reactive force."

The conjugate-beam methods is an engineering method to derive the slope and displacement of a beam. A conjugate beam is defined as an imaginary beam with the same dimensions (length) as that of the original beam but load at any point on the conjugate beam is equal to the bending moment at that point divided by EI.

<span class="mw-page-title-main">Size effect on structural strength</span> Deviation with the scale in the theories of elastic or plastic structures

According to the classical theories of elastic or plastic structures made from a material with non-random strength (f_t), the nominal strength (σ_N) of a structure is independent of the structure size (D) when geometrically similar structures are considered. Any deviation from this property is called the size effect. For example, conventional strength of materials predicts that a large beam and a tiny beam will fail at the same stress if they are made of the same material. In the real world, because of size effects, a larger beam will fail at a lower stress than a smaller beam.

This glossary of engineering terms is a list of definitions about the major concepts of engineering. Please see the bottom of the page for glossaries of specific fields of engineering.

This glossary of structural engineering terms pertains specifically to structural engineering and its sub-disciplines. Please see glossary of engineering for a broad overview of the major concepts of engineering.

References

1 2 3 Kharagpur. "Structural Analysis.pdf, Version 2 CE IIT" Archived 2010-08-19 at the Wayback Machine . 7 August 2008. Accessed on 26 November 2010.
1 2 3 4 5 6 7 8 Dr. Fanous, Fouad. "Introductory Problems in Structural Analysis: Influence Lines". 20 April 2000. Accessed on 26 November 2010.
1 2 "Influence Line Method of Analysis". The Constructor. 10 February 2010. Accessed on 26 November 2010.
1 2 "Structural Analysis: Influence Lines". The Foundation Coalition. 2 December 2010. Accessed on 26 November 2010.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Hibbeler, R.C. (2009). Structural Analysis (Seventh Edition). Pearson Prentice Hall, New Jersey. ISBN 0-13-602060-7.
↑ Zeinali, Yasha (December 2017). "Framework for Flexural Rigidity Estimation in Euler-Bernoulli Beams Using Deformation Influence Lines" (PDF). Infrastructures. 2 (4): 23. doi: 10.3390/infrastructures2040023 .
↑ "Influence Lines | Structural Analysis Review". www.mathalino.com. Retrieved 2019-12-25.

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[onlinefreeebooks-1] 1 2 3 Kharagpur. "Structural Analysis.pdf, Version 2 CE IIT" Archived 2010-08-19 at the Wayback Machine . 7 August 2008. Accessed on 26 November 2010.

[iastate-2] 1 2 3 4 5 6 7 8 Dr. Fanous, Fouad. "Introductory Problems in Structural Analysis: Influence Lines". 20 April 2000. Accessed on 26 November 2010.

[theconstructor-3] 1 2 "Influence Line Method of Analysis". The Constructor. 10 February 2010. Accessed on 26 November 2010.

[foundationcoalition-4] 1 2 "Structural Analysis: Influence Lines". The Foundation Coalition. 2 December 2010. Accessed on 26 November 2010.

[Hibbeler-5] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 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" (PDF). Infrastructures. 2 (4): 23. doi: 10.3390/infrastructures2040023 .

[7] "Influence Lines | Structural Analysis Review". www.mathalino.com. Retrieved 2019-12-25.

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