## 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 is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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