Influence line
Encyclopedia
In engineering, an influence line graphs the variation of a function (such as the shear felt in a structure member) at a specific point on a beam
or truss
caused by a unit load placed at any point along the structure. Some of the common functions studied with influence lines include reactions (the forces that the structure’s supports must apply in order for the structure to remain static), shear, moment
, and deflection
. Influence lines are important in the designing beams and trusses used in bridges
, crane rails, conveyor belts
, floor girders, and other structures where loads will move along their span. 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
. 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, the influence lines for the individual loads may be added together in order to obtain the total effect felt by the structure 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, if it is known that load A will be three feet in front of load B, 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.
and 
, each one associated with a displacement field whose displacements measured in the force's point of application are represented by 
and 
.
Consider that the
system represents actual forces applied to the structure, which are in equilibrium. Consider that the 
system is formed by a single force, 
. The displacement field 
associated with this forced is defined by releasing the structural restraints acting on the point where 
is applied and imposing a relative unit displacementwhich is cinematically admissible in the negative direction, represented as 
. From Betti's theorem, we obtain the following result:
Influence lines graph the response of a beam or truss as a unit load travels across it. The influence line allows the designers to discover quickly 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 is less common.
. It creates a qualitative influence line. 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.
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.” 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.
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.
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 will be a piecewise linear function which has a separate influence-line equation for each linear section of the influence line.
In order to understand how the beam will deflect 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).
The Müller-Breslau Principle can only produce qualitative influence lines. 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.
In order to calculate the response of a function to all of 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 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 will increase.
In order 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. 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.
Beam (structure)
A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.- Overview...
or truss
Truss
In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes. External forces and reactions to those forces are considered to act only at the nodes and result in...
caused by a unit load placed at any point along the structure. Some of the common functions studied with influence lines include reactions (the forces that the structure’s supports must apply in order for the structure to remain static), shear, moment
Bending Moment
A bending moment exists in a structural element when a moment is applied to the element so that the element bends. Moments and torques are measured as a force multiplied by a distance so they have as unit newton-metres , or pound-foot or foot-pound...
, and deflection
Deflection (engineering)
In engineering, deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load and can be calculated by...
. Influence lines are important in the designing beams and trusses used in bridges
Bridges
-People:* Alexander Bridges, English footballer* Angelica Bridges , American actress, model and singer* Barry Bridges , English footballer* Beau Bridges , American actor* Bles Bridges , South African singer...
, crane rails, conveyor belts
Conveyor belt
A conveyor belt consists of two or more pulleys, with a continuous loop of material - the conveyor belt - that rotates about them. One or both of the pulleys are powered, moving the belt and the material on the belt forward. The powered pulley is called the drive pulley while the unpowered pulley...
, floor girders, and other structures where loads will move along their span. The influence lines show where a load will create the maximum effect for any of the functions studied.
Influence lines are both scalar
Scalar
Scalar may refer to:*Scalar , a quantity used to multiply vectors in the context of vector spaces*Scalar , a quantity which is independent of specific classes of coordinate systems...
and additive
Additive function
In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...
. 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, the influence lines for the individual loads may be added together in order to obtain the total effect felt by the structure 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, if it is known that load A will be three feet in front of load B, 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.
Demonstration from Betti's theorem
Influence lines are based on Betti's theorem. From there, consider two external force systems, and , each one associated with a displacement field whose displacements measured in the force's point of application are represented by and .Consider that the
system represents actual forces applied to the structure, which are in equilibrium. Consider that the system is formed by a single force, . The displacement field associated with this forced is defined by releasing the structural restraints acting on the point where is applied and imposing a relative unit displacementwhich is cinematically admissible in the negative direction, represented as . From Betti's theorem, we obtain the following result: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 in order to ensure that no member will fail 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 will be moved 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 allows the designers to discover quickly 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 is 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. 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. The third method is called the Müller-Breslau principleMüller-Breslau principle
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 that the structure displays after removing the restraint on the point where the action is evaluated and applying a point that...
. It creates a qualitative influence line. 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
In order 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. StaticsStatics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...
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
Shear and moment diagram
Shear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of an element. Using these diagrams the type and size of a member of a given material can...
.
R. C. Hibbeler states, in his book Structural Analysis, “All statically determinate beams will have influence lines that consist of straight line segments.” 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.
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.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 will be a piecewise linear function which has a separate influence-line equation for each linear section of the influence line.
Müller-Breslau 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.” 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.In order to understand how the beam will deflect 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. 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. 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 which 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. 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. 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.
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.In order to calculate the response of a function to all of 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 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 will increase.
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.In order 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. 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 a 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
- BeamBeam (structure)A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.- Overview...
- Shear and Moment DiagramShear and moment diagramShear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of an element. Using these diagrams the type and size of a member of a given material can...
- Dead and Live Loads
- Müller-Breslau principleMüller-Breslau principleThe 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 that the structure displays after removing the restraint on the point where the action is evaluated and applying a point that...