The last paragraph brings us to the composition and resolution of forces, of which some knowledge will be required to understand the explanations which follow. For the benefit of those who have not studied mechanics, a brief explanation will be given of the laws relating to the action of forces when applied at a single point.
The "resultant" of two or more forces is that single force which would exactly replace them and have the same effect on the body acted upon as the given forces.
The "components" of a force are the two or more simple forces by which it may be replaced: hence any force may be considered as the resultant of two or more other forces by which it may be replaced.
For example: if a force represented in direction and magnitude by the line F, Fig. 4, is applied to a ball resting on a plane horizontal surface it will cause the ball to move forward in the direction of the dotted line. The ball may be made to move in the same direction, however, by means of two forces, f and f1, provided they are properly proportioned. The forces f and f1 may therefore be considered as the components of the force F, as they fulfill the conditions of the above definition. The component forces may be at any angle with the given force less than 900, but they must be applied on opposite sides of the given force. In trusses oblique forces are generally, although not always, resolved into horizontal and vertical components.
Again, if two forces, f and f1, Fig. 5, act on a ball, the ball will move forward in the direction of the diagonal of a parallelogram, of which the forces form two sides, and the resultant of any two forces is indicated in magnitude and direction by the diagonal of the parallelogram formed upon them. Conversely, the magnitude of two components of any force is represented by the sides of the parallelogram having the given force as its diagonal.
Another application of this law is that any given force may be "balanced" by two other forces acting in the opposite direction to its components, or two forces may be balanced by a single force acting in the opposite direction to the resultant of the given forces. Thus, the force F in Fig. 6 (either diagram) acting on the ball may be exactly balanced by the two forces f and f1 if equal in magnitude respectively to the sides a and b of the parallelogram, having the given force F for its diagonal. Conversely, the forces f and f1 will be balanced by the force F.
A given force may also have any number of components, but as in trusses it is seldom that any given force is resisted by more than two other forces, it is not deemed necessary to consider more than two components.
Another principle that requires notice at this time is that whenever a strut or tie is subjected to compression or tension, that it shall be kept in position, the resistance at each end must equal the stress in the piece, and must act in opposite directions. Thus, if two boys are pulling on a rope so that one balances the other, and one boy is pulling with a force of 50 pounds, the other boy will be exerting exactly the same force, but the strain in the rope will be but 50 pounds. This is a truth not always comprehended. It may perhaps, best be seen by considering a person pulling on a rope the other end of which is attached to the hook of a spring balance. In this case it will readily be seen that the gauge of the balance will indicate the force with which the person is pulling and also the stress on the rope, but it should also be remembered that the spring is pulling back with exactly the same force that the person is exerting.
These are very simple illustrations, but the author has found that the truth which they illustrate is not always comprehended, or, at least, is not considered.
To return to our truss, Fig. 3. It should now be readily understood that if the horizontal component of the thrust in the struts is resisted by the tie T only the vertical component will need to be resisted by the walls.
Again, the strain in the tie T will only be equal to the horizontal component of the stress in one strut, and the horizontal components of the strut stresses must equal each other. If the weight W is nearer one support than the other, so that the struts are not equally inclined, the thrust in the struts will not be alike, and the vertical component of the steeper strut will be greater than the vertical component of the other; but the horizontal components must be equal, otherwise the tie would be pulled along endways.