If a beam is loaded as at W W W, Fig. 13, the weights produce reactions at the supports. These forces, or reactions, R1, and R2, oppose the action of the weights and their combined action must equal the total weight. The weights and reactions, constituting the external forces, tend to produce bending in the beam, and are resisted by the internal forces, consisting of the strength of the fibers composing the beam. In a simple beam, the effect of loading is to shorten the upper fibers, and to lengthen the lower ones. Somewhere between the top and bottom of the cross-section are located fibers which are neither shortened nor lengthened; this position is called the neutral axis (see page 75). In steel and like material of homogeneous nature, the neutral axis passes through the center of gravity of the section.

Theory Of Beams 240

At any point in the length of a beam, the tendency to produce bending is equal to the algebraic sum of the moments of the external forces at that point; this moment is called the bending moment. A beam resists bending at any point by the tance of its particles to extension or compression, the sum of the moments of which about the neutral axis of the cross-section is called the moment of resistance, or resisting moment. For a beam to be sufficiently strong to sustain the load, the moment of resistance must equal the bending moment If the moment of resistance is expressed in inch-pounds. the bending moment must likewise be reduced to inch-poonda.

Reactions

The reactions or supporting forces of any beam or structure must equal the loads upon it. If the load upon a simple beam is uniformly distributed, applied at the center of the span, or symmetrically placed and of equal amount upon each side of the center, the reactions R1 and R2 will each be equal to one-half the load. When the loads are not symmetrically placed, the reactions are found by the principle of moments in the following manner:

Fig. 14 represents a simple beam supporting loads W1 W2 , and W3; I is the span or distance between the reactions R1 and R2; a, b, and c are the distances from the reaction R1 to the loads W1, W2 W3.. ively. Then the right-hand reaction, R2 =

(W1x a)+( W2 x b)+(W3.x c) / l

This formula expressed in a general rule is: To find the reaction at either support, multiply each load by its distance from the other support, and divide the sum of these products by the distance between supports.

Theory Of Beams 241

Since the sum of the reactions must equal the sum of the loads, if one reaction is found, the other can he obtained by subtracting the known one from the sum of the loads.

Example

What are the reactions at R1 and R2, Fig. 15?

Solution

The lever arm of a uniformly distributed load is always the distance from the center of moments to the center of gravity of the load. The total uniform load a is 3,000 X 10 = 30,000 lb., and the distance of its center of gravity from R1 is 13 ft. The moments of the loads about R1 are as follows:

Theory Of Beams 242

4,000 lb. X 4 ft. =

16,000 ft.-lb.

30,000 lb. X 13 ft. =

390,000 ft.-lb.

9,000 lb. X 20 ft. =

180,000 ft.-lb.

Total =

586,000 ft.-lb.

The distance from R1 to R2 is 30 ft.; hence, 586,000 / 30 = 19,533 1/3 lb., the reaction at R2. As the sum of the loads is 43,000 lb., the reaction at R1 is 43,000 -19,533 1/3= 23,466 2/3 lb.