## 27 Influence Lines

Bridges, industrial buildings with traveling cranes, and frames supporting conveyer belts are often subjected to moving loads. Each member of these structures must be designed for the most severe conditions that can possibly be developed in that member. Live loads should be placed at the positions where they will produce these severe conditions. The critical positions for placing live loads will not be the same for every member. On some occasions it is possible by inspection to determine where to place the loads to give the most critical forces, but on many other occasions it is necessary to resort to certain criteria to find the locations. The most useful of these methods is the influence line.

An influence line for a particular response, such as reaction, shear force, bending moment, and axial force, is defined as a diagram, the ordinate to which at any point equals the value of that response attributable to a unit load acting at that point on the structure. Influence lines provide a systematic procedure for determining how the force in a given part of a structure varies as the applied load moves about on the structure. Influence lines of responses of statically determinate structures consist only of straight lines whereas they are curves for statically indeterminate structures. They are primarily used to determine where to place live loads to cause maximum force and to compute the magnitude of those forces. Knowledge of influence lines helps one to study the structural response under different moving load conditions.

### 2.7.1 Influence Lines for Shear in Simple Beams

Figure 2.50 shows influence lines for shear at two sections of a simply supported beam. It is assumed that positive shear occurs when the sum of the transverse forces to the left of a section is in the upward direction or when the sum of the forces to the right of the section is in the downward direction. A unit force is placed at various locations and the shear force at sections 1-1 and 2-2 are obtained for each position of the unit load. These values give the ordinate of influence line with which the influence line diagrams for shear force at sections 1-1 and 2-2 can be constructed. Note that the slope of the influence line for shear on the left of the section is equal to the slope of the influence line on the right of the section. This information is useful in drawing the shear force influence line in other cases.

2.7.2 Influence Lines for Bending Moment in Simple Beams

Influence lines for bending moment at the same sections 1-1 and 2-2 of the simple beam considered in Figure 2.50 are plotted as shown in Figure 2.51. For a section, when the sum of the moments of all the forces to the left is clockwise or when the sum to the right is counterclockwise, the moment is taken as positive. The values of bending moment at sections 1-1 and 2-2 are obtained for various positions of unit load and plotted as shown in the figure.

It should be understood that a shear or bending moment diagram shows the variation of shear or moment across an entire structure for loads fixed in one position. On the other hand, an influence line for shear or moment shows the variation of that response at one particular section in the structure caused by the movement of a unit load from one end of the structure to the other.

Influence lines can be used to obtain the value of a particular response for which it is drawn when the beam is subjected to any particular type of loading. If, for example, a uniform load of intensity q0 per unit length is acting over the entire length of the simple beam shown in Figure 2.50, the shear force at section 1-1 is given by the product of the load intensity, q0, and the net area under the influence line diagram. The net area is equal to 0.3 and the shear force at section 1-1 is therefore equal to 0.3q0. In the same way, the bending moment at the section can be found as the area of the corresponding influence line diagram times the intensity of loading, q0. The bending moment at the section is equal to 0.08q0l2.

### 2.7.3 Influence Lines for Trusses

Influence lines for support reactions and member forces may be constructed in the same manner as those for various beam functions. They are useful to determine the maximum load that can be applied to the truss. The unit load moves across the truss, and the ordinates for the responses under consideration may be computed for the load at each panel point. Member force, in most cases, need not be calculated for

every panel point, because certain portions of influence lines can readily be seen to consist of straight lines for several panels. One method used for calculating the forces in a chord member of a truss is by the method of sections discussed earlier.

The truss shown in Figure 2.52 is considered for illustrating the construction of influence lines for trusses.

The member forces in UjU2, L1L2, and UjL2 are determined by passing a section 1-1 and considering the equilibrium of the free-body diagram of one of the truss segments. Unit load is placed at L1 first and the force in U1U2 is obtained by taking the moment about L2 of all the forces acting on the right-hand segment of the truss and dividing the resulting moment by the lever arm (the perpendicular distance of the force in U1U2 from L2). The value thus obtained gives the ordinate of the influence diagram at L1 in the truss. The ordinate at L2 obtained similarly represents the force in U1U2 for unit load placed at L2. The influence line can be completed with two other points, one at each of the supports. The force in the member L1L2 due to unit load placed at L1 and L2 can be obtained in the same manner and the corresponding influence line diagram can be completed. By considering the horizontal component of force in the diagonal of the panel the influence line for force in U1L2 can be constructed. Figure 2.52 shows the respective influence diagram for member forces in U1U2, L1L2, and U1L2. Influence line ordinates for the force in a chord member of a ''curved-chord'' truss may be determined by passing a vertical section through the panel and taking moments at the intersection of the diagonal and the other chord.

### 2.7.4 Qualitative Influence Lines

One of the most effective methods of obtaining influence lines is the use of Muller-Breslau's principle, which states that the ordinates of the influence line for any response in a structure are equal to those of the deflection curve obtaining by releasing the restraint corresponding to this response and introducing a corresponding unit displacement in the remaining structure. In this way, the shape of the influence lines for both statically determinate and indeterminate structures can be easily obtained, especially for beams.

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