An influence line is constructed by plotting the position of the unit load as the abscissa and as the ordinate at that position, to some scale, the value of the effect being studied. For example, Fig. 3.83a shows the influence line for reaction A in simple-beam AB. The sloping line indicates that when the unit load is at A, the reaction at A is 1.0. When the load is at B, the reaction at A is zero. When the unit load is at midspan, the reaction at A is 0.5. In general, when the load moves from B toward A, the reaction at A increases linearly: RA = (L - x) /L, where x is the distance from A to the position of the unit load.
Figure 3.83b shows the influence line for shear at the quarter point C. The sloping lines indicate that when the unit load is at support A or B, the shear at C is zero. When the unit load is a small distance to the left of C, the shear at C is -0.25; when the unit load is a small distance to the right of C, the shear at C is 0.75. The influence line for shear is linear on each side of C.
Figure 3.83c and d show the influence lines for bending moment at midspan and quarter point, respectively. Figures 3.84 and 3.85 give influence lines for a cantilever and a simple beam with an overhang.
Influence lines can be used to calculate reactions, shears, bending moments, and other effects due to fixed and moving loads. For example, Fig. 3.86a shows a simply supported beam of 60-ft span subjected to a dead load w = 1.0 kip per ft and a live load consisting of three concentrated loads. The reaction at A due to the dead load equals the product of the area under the influence line for the reaction at A (Fig. 3.86b) and the uniform load w.
The maximum reaction at A due to the live loads may be obtained by placing the concentrated loads as shown in Fig. 3.86b and equals the sum of the products of each concentrated load and the ordinate of the influence line at the location of the load. The sum of the dead-load reaction and the maximum live-load reaction therefore is
R = 1â„2 x 1.0 x 60 x 1.0 + 16 x 1.0 + 16 x 0.767 + 4 x 0.533 = 60.4 kips A
Figure 3.86c is the influence diagram for midspan bending moment with a maximum ordinate L/4 = 60â„4 = 15. Figure 3.86c also shows the influence diagram with the live loads positioned for maximum moment at midspan. The dead load moment at midspan is the product of w and the area under the influence line. The midspan live-load moment equals the sum of the products of each live load and the ordinate at the location of each load. The sum of the dead-load moment and the maximum live-load moment equals
M = 1â„2 x 15 x 60 x 1.0 + 16 x 15 + 16 x 8 + 4 x 8 = 850 ft-kips
An important consequence of the reciprocal theorem presented in Art. 3.25 is the Mueller- Breslau principle: The influence line of a certain effect is to some scale the deflected shape of the structure when that effect acts.
The effect, for example, may be a reaction, shear, moment, or deflection at a point. This principle is used extensively in obtaining influence lines for statically indeterminate structures (see Art. 3.28).
Figure 3.87a shows the influence line for reaction at support B for a two-span continuous beam. To obtain this influence line, the support at B is replaced by a unit upward-concentrated load. The deflected shape of the beam is the influence line of the reaction at point B to some