This section is from the book "Building Construction And Superintendence", by F. E. Kidder. Also available from Amazon: Building Construction And Superintendence.

Before the stress diagram of a truss can be drawn, it is neces-sary to know not only the loads which the truss has to support; but also the supporting forces or reactions.

These are calculated on the principle that when a beam or truss supports any number of loads acting vertically, the supports of the beam or truss, taken together, must offer an upward resistance equal to the sum of the loads.

Thus if we have a beam loaded as in Fig. 257, and supported by two posts, the load on the two posts will be equal to the sum of the weights, and the posts are assumed to push against the under side of the beam just as much as the loads bear down. If we assume that the balls in Fig. 257 have the weights indicated by the numbers, then the total load on the beam is 40 lbs., and the two posts, together, must push upwards, as it were, with a force of 40 lbs. If the weights on the beam were symmetrically disposed in respect to the supports, then the supports would each offer the same resistance; but when the weights are unsymmetrical, either in amount or position, the posts will not receive the same load, but one will be more heavily loaded than the other.

Fig. 257.

The supports of beams and trusses are usually passive, so that they cannot really push upwards, but in considering the forces which act on a truss, they are considered as an active force, as much so as the loads, and must be determined both in direction and amount with accuracy. The supports of a beam or truss are considered as reacting against the under side, and the weight or pressure which comes on the supports is called the reaction. The algebraic sum of the reactions must always be equal to the sum of the loads, but in the case of cantilever beams or trusses, it may sometimes happen that one of the supporting forces must pull down on the beam in order to balance it. In that case the reaction is considered as a minus or negative quantity.

Before describing the method of determining the amount of each supporting force, it is necessary to take up the subject of moments.

A moment in mechanics is the tendency to cause rotation, and when we speak of a moment we must have in mind some fixed point about which the moment is taken. Moments can only be produced by forces, hence we speak of the moment of a force, or have some particular force in mind.

The moment of a force about any given point may be defined as the product of the force into the perpendicular distance from the point to the line of action of the force, or, in other words, as the product of the force by the arm with which it acts. The arm must always be measured square or at right angles to the direction of the force.

Fig. 258.

Fig. 259.

As an example of moment, we will assume that the body, shown by the irregular figure, Fig. 258, is pivoted at the point P, and a weight of 5 lbs. is suspended from the point a by a string. The weight is a force, or, more strictly, represents the force of gravity, and as gravity always acts vertically, the force is a vertical one. Consequently its arm must be measured horizontally. Now, it is evident that a body in the condition shown in Fig. 258 will rotate about the point P in the direction of the arrow until the point a is directly under P, assuming that the body itself has no weight. The tendency to rotate is the moment of the weight, and the amount of this moment is the product of the weight into the arm shown by the dotted line, or 5 x 6 = 30 inch pounds. A body may have several moments acting upon it at the same time, and every force exerts a moment, except when the force is applied at the pivot point.

In Fig. 259 we have three forces, F1, F2 and F3, acting on the body, and all tending to turn it in the same direction about the point P. Assuming that the forces have the values represented by the numbers, and that the lengths of arms in inches are as indicated, then the sum of the moments tending to rotate the body in the direction of the arrow is 48. Now it is evident that the body can be kept from rotating by a force applied in the direction R, and to just balance the body, so that it will not rotate in either direction the moment of R about P must equal the sum of the other three moments, or 48. The perpendicular distance between P and the direction of the force R is 3, therefore if the moment of R must be 48, and the arm is 3, R must equal 48/3, or 16. This leads us to a second proposition, viz.:

If any number of forces act on a body, then for the body to be in equilibrium, the sum of the moments tending to turn the body in one direction must equal the sum of the moments tending to turn the body in the opposite direction about any given point.

Also, if a force and its moment are known, the arm may be found by dividing the moment by the force, or if the moment and arm are known, the force must be the quotient obtained by dividing the moment by the arm. By means of the two propositions given in this section, the supporting forces for any form of truss or variations in loads may be computed.

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