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Influence line represents the variation of either the reaction, shear, moment, or deflection at a specific point in a member, as a concentrated force moves over the member. Influence lines are important for the design of a structure that resists large live loads moving along the spans such as bridges and cranes.
Tabulated method and Influence line equations technique can be employed to construct influence lines of a member over which a concentrated live load moves along its span. When the influence line is drawn, the location of the live load that leads to the greatest influence on the structure can be found easily and quickly. Moreover, the magnitude of the associated reaction, shear, moment, or deflection at the point can then be calculated from the ordinates of the influence-line diagram.
Furthermore, the procedure for constructing an influence line is simple. However, it is important to remember the difference between constructing an influence line and constructing a shear or moment diagram. The former represents the effect of a moving load only at a specified point on a member whereas the latter represents the effect of fixed loads at all points along the member.
Tabulated Method for Constructing Influence Lines
- Place a unit load at various locations, x, along the span member.
- At each location use statics to compute the value of the function (reaction, shear, or moment) at the specified point.
- If the influence line for a vertical force reaction at a point on a beam is to be constructed, consider the reaction to be positive at the point when it acts upward on the beam.
- If a shear or moment influence line is to be drawn for a point, take the shear or moment at the point as positive, Fig. 1.
- All statically determinate beams have influence lines that consist of straight-line segments. (Note: After some practice, one should be able to minimize computations and locate the unit load only at points representing the endpoints of each line segment).
- To avoid errors, it is advised that one should first construct a table, listing “unit load at x” versus the corresponding value of the function calculated at the specific point; that is, “reaction R,” “shear V,” or “moment M.”
- Once the load has been placed at various points along the span of the member, the tabulated values can be plotted and the influence-line segments constructed.
Example 1
Construct the Influence line for the vertical reaction at A of the beam in Fig. 2.
Solution:
Firstly, a unit load is placed on the beam at each selected point x. Secondly, the value of Ay is calculated by summing moments about B. Thirdly, place unit loads at specified points on along the span of the beam; the value of Ay is computed for the unit load placed at 2.5ft and 5 ft as shown in fig. 3.
The results for are entered in Table 1. A plot of these values yields the influence line for the reaction at A, Fig. 4.
Table 1 - Placement of Unit load at specified points along the span of the beam versus the value of reaction Ay
x | Ay |
0 | 1 |
2.5 | 0.75 |
5 | 0.5 |
7.5 | 0.25 |
10 | 0 |
Also Read: Statics of Bending: Shear and Bending Moment Diagrams