28. Moment of a Force. By moment of a force with respect to a point is meant its tendency to produce rotation about that point. Evidently the tendency depends on the magnitude of the force and on the perpendicular distance of the line of action of the force from the point : the greater the force and the per-pendicular distance, the greater the tendency; hence the moment of a force with respect to a point equals the product of the force and the perpendicular distance from the force to the point.

A 300.000-lb. RIEHLE TESTING MACHINE The Chicago Physical Testing Laboratory of Robert W. Hunt & Co., Engineers, Chicago, New York, Pittsburg, and London.

The point with respect to which the moment of one or more forces is taken is called an origin or center of moments, and the perpendicular distance from an origin of moments to the line of action of a force is called the arm of the force with respect to that origin. Thus, if F1 and F2 (Fig. 7) are forces, their arms with respect, to O' are a1' and a2' respectively, and their moments are F1 a'1 and F2 a2"With respect to O" their arms are a1" and a2" respectively, and their moments are F1 a1' and F2 a 2".

If the force is expressed in pounds and its arm in feet, the moment is in foot-pounds; if the force is in pounds and the arm in inches, the moment is in inch-pounds.

20. A sign is given to the moment of a force for convenience; the rule used herein is as follows: The moment of a force about a point is positive or negative according as it tends to turn, the body about that point in the clockwise or counterclockwise direction *. Thus the moment (Fig. 7) of F1 about O' is negative, about O" positive; " F2 " O' " " , about O" negative.

30. Principle of Moments. In general, a single force of proper magnitude and line of action can balance any number of forces. That single force is called the eauilibrant of the forces, and the single force that would balance the equilibrant is called the resultant of the forces. Or, otherwise stated, the resultant of any number of forces is a force which produces the same effect. It can be proved that - The algebraic sum of the moments of any number of forces with respect to a point, equals the moment of their resultant about that point.

Fie. 7.

* By clockwise direction is meant that in which the hands of a clock rotate; and by counter-clockwise, the opposite direction.

This is a useful principle and is called "principle of moments." 31. All the forces acting upon a body which is at rest are said to be balanced or in equilibrium. No force is required to balance such forces and hence their equilibrant and resultant are zero.

Since their resultant is zero, the algebraic sum of the moments of any number of forces which are balanced or in equilibrium equals zero.

Fig. 8.

This is known as the principle of moments for forces in equilibrium; for brevity we shall call it also "the principle of moments."

The principle is easily verified in a simple case. Thus, let AB (Fig. 8) be a beam resting on supports at C and F. It is evident from the symmetry of the loading that each reaction equals one-half of the whole load, that is, ½ of 6,000=3,000 pounds. (We neglect the weight of the beam for simplicity.)

With respect to C, for example, the moments of the forces are, taking them in order from the left:

 - 1,000 X 4 = - 4,000 foot-pounds 3,000 X 0 = 0 " 2,000 X 2 = 4,000 " 2,000 X 14 = 28,000 " - 3,000 X 16 = - 48,000 " 1,000 X 20 = 20,000 "

The algebraic sum of these moments is seen to equal zero. Again, with respect to B the moments are:

 - 1,000 X 24 = - 24,000 foot-pounds 3,000 X 20 = 60,000 " - 2,000 X 18 = - 36,000 " - 2,000 X 6 = - 12,000 " 3,000 X 4 = 12,000 " 1,000 X 0 = 0 "

The sum of these moments also equals zero. In fact, no matter where the center of moments is taken, it will be found in this and any other balanced system of forces that the algebraic sum of their moments equals zero. The chief use that we shall make of this principle is in finding the supporting forces of loaded beams.

32. Kinds of Beams. A cantilever beam is one resting on one support or fixed at one end, as in a wall, the other end being free.

A simple beam is one resting on two supports.

A restrained beam is one fixed at both ends; a beam fixed at one end and resting on a support at the other is said to be restrained at the fixed end and simply supported at the other.

A continuous beam is one resting on more than two supports.

33. Determination of Reactions on Beams. The forces which the supports exert on a beam, that is, the "supporting forces," are called reactions. We shall deal chiefly with simple beams. The reaction on a cantilever beam supported at one point evidently equals the total load on the beam.

When the loads on a horizontal beam are all vertical (and this is the usual case), the supporting forces are also vertical and the sum of the reactions equals the sum of the loads. This principle is sometimes useful in determining reactions, but in the case of simple beams the principle of moments is sufficient. The general method of determining; reactions is as follows:

Fig. 9.

1. Write out two equations of moments for all the forces (loads and reactions) acting on the beam with origins of moments at the supports.

2. Solve the equations for the reactions.

3. As a check, try if the sum of the reactions equals the sum of the loads.

Examples. 1. Fig. 9 represents a beam supported at its ends and sustaining three loads. We wish to find the reactions due to these loads.

Let the reactions be denoted by R1 and R2 as shown; then the moment equations are: For origin at A,

1,000 X1 + 2,000 X 6 + 3,000 X 8 - R2 X10 = 0. For origin at E,