STRUCTURE MECHANICSINFLUENCE LINE DIAGRAMDefinitionUsesPractical application

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2012-BT-CIVIL-11

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2012-BT-CIVIL-28

Introduction of influence line diagramAn influence line for a given function, such as reaction, axial force, shear force or bending moment, is a graph that is shows the variation of that function at any given point on a structure due to application of unit load at any point on the structure.An influence for a function differs from a shear, axial or bending moment diagram. Influence line can be generated byIndependently apply a unit load at several points on a structure and determining the value of function due to this load, i.e. shear, axial and moment at the desired location. The calculated values of 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 section 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 lineAn influence line represents the variation of the reaction, shear, moment or deflection at a specific point in a member as a concentrated force moves over the member.

OrAn influence line is a graph of a response function of a structure as a function of the position of a downward unit load moving across the structure

NOTE: Influence lines for statically determinate structures are always piecewise linear. Once an influence line is constructed:

1. Determine where to place live load on a structure to maximize the drawn response function; and2. Evaluate the maximum magnitude of the response function based on the loading

Use of Influence LinesPoint Response Due to a Single Moving Concentrated LoadEach ordinate of an influence line gives the value of the response function due to a single concentrated load of unit magnitude placed on the structure at the location of that ordinate.

1. The value of a response function due to any single concentrated load can be obtained by multiplying the magnitude of the load by the ordinate of the response function influence line at the position of the load.

2. Maximum positive value of the response function is obtained by multiplying the point load by the maximum positive ordinate. Similarly, the maximum negative value is obtained by multiplying the point load by the maximum negative ordinate.

APPLICATION OF INFLUENCE LINESResponse at a particular location due to a single moving concentrated load

i. The value of a response function due to any single concentrated load can be obtained by multiplying the magnitude of the load by the ordinate of the response function influence line at the position of the load

ii. To determine the maximum positive value of a response function due to a single moving concentrated load, the load must be placed at the location of the maximum positive ordinate of the response function influence line, whereas to determine the maximum negative value of the response function, the load must be placed at the location of the maximum negative ordinate of the influence line.

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iii. Suppose that we wish to determine the bending moment at B when the load P is located at a distance x. MB=PY

Maximum Positive bending moment at B

*Place the load P at point B

* MB=PyB

Maximum Negative bending moment at B* Place the load P at point D* MB=-

Response at a particular location due to a uniformly distributed live loadConsider, for example, a beam subjected to a uniformly distributed live load wl. Suppose that we want to determine the bending moment at B when the load is placed on the beam, from x=a to x=b.The bending moment at B due to the load dP as

The total bending moment at B due to distributed load from x=a to x=b:

This equation also indicates that the bending moment at B will be maximum positive if the uniformly distributed load is placed over all those portions of the beam where the influence-line ordinates are positive and vice versa and vice versa.

To determine the maximum positive (or negative) value of a response function due to a uniformly distributed live load, the load must be placed over those portions of the structure where the ordinates of the response function influence line are positive (or negative).

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Response at a particular location due to a series of moving concentrated loads

Suppose we wish to determine the shear at B of the beam due to the wheel loads of a truck when the truck is located as in figure

Influence lines can also be used for determining the maximum values of response functions at particular locations of structures due to a series of concentrated loads.Suppose that our objective is to determine the maximum positive shear at B due to the series of four concentrated loads

During the movement of the series of loads across the entire length of the beam, the (absolute)maximum shear at B occurs when one of the loads of the series is at the location of the maximum positive ordinate of the influence line for SB.

We use a trial-and-error procedure.

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