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Introduction
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 astructure 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 momentdiagram. 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 linefor the function.For example, the influence line for the support reaction at A of the structure shown inFigure 1, is found by applying a unit load at several points (ee Figure ! on the structureand determining what the resulting reaction will be at A. This can be done by solving thesupport reaction #
A
as a function of the position of a downward acting unit load. $ne suche%uation can be found by summing moments at upport &.Figure 1 ' &eam structure for influence line exampleFigure ! ' &eam structure showing application of unit load
&
) *
(Assume counter-clockwise positive moment)
'#
A
(+ 1(+'x ) *#
A
) (+'x -+ ) 1 ' (x-+
1
The graph of this e%uation is the influence line for the support reaction at A (ee Figure . The graph illustrates that if the unit load was applied at A, the reaction at A would bee%ual to unity. imilarly, if the unit load was applied at &, the reaction at A would be e%ual to*, and if the unit load was applied at /, the reaction at A would be e%ual to 'e-+.Figure ' Influence line for the support reaction at A$nce an understanding is gained on how these e%uations and the influence lines they produce are developed, some general properties of influence lines for
statically determinatestructures
can be stated.1.
For a statically determinate structure
the influence line will consist of onlystraight line segments between critical ordinate values.!.The influence line for a shear force at a given location will contain a translationaldiscontinuity at this location. The summation of the positive and negative shear forces at this location is e%ual to unity..0xcept at an internal hinge location, the slope to the shear force influence line will be the same on each side of the critical section since the bending moment iscontinuous at the critical section..The influence line for a bending moment will contain a unit rotational discontinuityat the point where the bending moment is being evaluated.2.To determine the location for positioning a single concentrated load to producemaximum magnitude for a particular function (reaction, shear, axial, or bendingmoment place the load at the location of the maximum ordinate to the influenceline. The value for the particular function will be e%ual to the magnitude of theconcentrated load, multiplied by the ordinate value of the influence line at that point.3.To determine the location for positioning a uniform load of constant intensity to produce the maximum magnitude for a particular function, place the load along
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those portions of the structure for which the ordinates to the influence line have thesame algebraic sign. The value for the particular function will be e%ual to themagnitude of the uniform load, multiplied by the area under the influence diagram between the beginning and ending points of the uniform load.There are two methods that can be used to plot an influence line for any function. Inthe first, the approach described above, is to write an e%uation for the function beingdetermined, e.g., the e%uation for the shear, moment, or axial force induced at a point due tothe application of a unit load at any other location on the structure. The second approach,which uses the
Müller Breslau Principle
, can be utili4ed to draw %ualitative influence lines,which are directly proportional to the actual influence line.The following examples demonstrate how to determine the influence lines for reactions, shear, and bending moments of beams and frames using both methods describedabove.0xamplesInfluence lines for a simple beam by developing the e%uations5ualitative influence lines using the 6ller &reslau 7rinciple5ualitative influence lines for a statically determinate continuous beam/alculation of maximum and minimum shear force and moments on a statically determinate continuous beam5ualitative influence lines and loading patterns for a multi'span indeterminate beam5ualitative influence lines and loading patterns for an indeterminate frame
3
ource 8 http8--www.public.iastate.edu-9fanous-ce!-influence-homepage.html:ate searching 8 !; eptember !*1
Influence line
Figure 18 (a This simple supported beam is shown with a unit load placed a distance
x
fromthe left end. Its influence lines for four different functions8 (b the reaction at the left support(denoted A , (c the reaction at the right support (denoted / , (d one for shear at a point &along the beam, and (e one for moment also at point &.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. ome of the common functions studied with influence linesinclude reactions (the forces that the structure<s supports must apply in order for the structureto remain static , shear , moment, and deflection. 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. The influence lines show where a load willcreate 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. Tofind the effect of any non'unit load on a structure, the ordinate results obtained by theinfluence line are multiplied by the magnitude of the actual load to be applied. The entire
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