Moment At Midspan

FIGURE 3.85 Influence diagrams for a beam with overhang.

FIGURE 3.86 Determination for moving loads on a simple beam (a) of maximum end reaction (b) and maximum midspan moment (c) from influence diagrams.

Ra = 12 X 1.0 X 60 X 1.0 + 16 X 1.0 + 16 X 0.767 + 4 X 0.533 = 60.4 kips

Figure 3.86c is the influence diagram for midspan bending moment with a maximum ordinate L/4 = 6% = 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 = 1/2 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

INFLUENCE LINE FOR SHEAR AT P FIGURE 3.87 Influence lines for a two-span continuous beam.

scale. To show this, let SBP be the deflection at B due to a unit load at any point P when the support at B is removed, and let SBB be the deflection at B due to a unit load at B. Since, actually, reaction RB prevents deflection at B, RBSBB - SBP = 0. Thus RB = SBP/SBB. By Eq. (3.124), however, SBP = SPB. Hence sBP sPB

"BB JBB

Since SBB is constant, RB is proportional to SPB, which depends on the position of the unit load. Hence the influence line for a reaction can be obtained from the deflection curve resulting from replacement of the support by a unit load. The magnitude of the reaction may be obtained by dividing each ordinate of the deflection curve by the displacement of the support due to a unit load applied there.

Similarly, influence lines may be obtained for reaction at A and moment and shear at P by the Mueller-Breslau principle, as shown in Figs. 3.87b, c, and d, respectively.

(C. H. Norris et al., Elementary Structural Analysis; and F. Arbabi, Structural Analysis and Behavior, McGraw-Hill, Inc., New York.)

Renewable Energy 101

Renewable Energy 101

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