## 122. Explanation Of The Principles Of Graphic Statics

Except for a few forms of trusses, it is much easier to determine the stresses by means of a diagram drawn accurately to a scale than by mathematical calculation, while the representation to the eye of the forces which exist in the several parts of a frame or truss gives one a better understanding of the actual conditions which exist than when the stresses are worked out by means of formulas.

In determining stresses by the graphic method, also, any mistakes or errors are much more likely to be discovered, than in mathematical calculations, while with ordinary care the stresses may be determined as accurately as the several parts of the frame can be proportioned.

Statics is that branch of mechanics which treats of forces in equilibrium, i. e., as balanced.

Graphic Statics is the representation of the different forces graphically, or by means of lines.

The loads and supporting forces of a truss, which are commonly designated as the external forces, produce stresses in the members of the truss, which are also represented as forces.

A Stress Diagram is a drawing made to a scale, usually of pounds to the inch, which represents the stresses and also the external forces which act in and on a frame or structure.

123. Stress diagrams, for trusses and framed structures, are based upon the following propositions, which, although quite simple in themselves, must be well understood before one can draw a stress diagram intelligently.

(1) Any force may be represented by a straight line, drawn to a scale. The length of the line, measured by the scale, gives its magnitude; the position of the line shows the line of action of the force, and an arrow head, the direction in which the force acts.

(2) If two forces applied at a point and acting in the same plane be represented in direction and magnitude by two intersecting straight lines, the resultant force will act in the direction of, and be equal to, the diagonal of the parallelogram formed on these lines.

Thus, if the lines A B and A C (Fig. 267) represent two forces acting on one point A, and in the same plane, then, to obtain the force which would have the same effect as the two forces, we complete the parallelogram A, B. C. D, and draw the diagonal A. D. This line will then represent the resultant of the two forces. It also gives the direction and magnitude of the single force that would balance the two given forces, only the balancing force must act in the opposite direction.

Also to obtain the diagonal, it is not necessary to draw the full parallelogram, for as we know the magnitude and direction of the two forces, we can draw one, as A C (Fig. 268), and from the end of the line, draw the other, as C D, and the line connecting the points A and D will be the required diagonal. It is also evident that it makes no difference which force we draw first, thus, if we draw A B and then B D we obtain the same diagonal as in the first operation. The principle elucidated in this proposition is the same as explained in § 4, page 16.

Fig. 267.

Fig. 268.

Fig. 269.

Fig. 270.

124. (3) - If any number of forces acting at a point can be represented in magnitude and direction by the sides of a polygon taken9 in order, they will be in equilibrium. Thus, let A, B, C and D (Fig. 269) represent four forces applied at one point and acting in the direction of the arrow heads; now, if the four forces balance each other, if we draw a line a, equal and parallel with A, and from the end of a, a line b, equal and parallel to B, and so on with c and d, the line d will just close the polygon. If it does not close the polygon the forces are not balanced and either one or more of the forces must be changed, or another force added. Thus, if the forces were of such magnitude that when drawn in order to a scale, they formed an open polygon, as in Fig. 270, either the magnitude of two of the forces must be changed or an additional force added, equal and parallel to e, to produce equilibrium, also if all but two of the forces are known, we can obtain the magnitude of those two forces, provided their direction is known. Thus, in Fig. 269, if we know the magnitude of the forces A and B, and the direction of C and D we can find the magnitude of each of these forces required to produce equilibrium by means of the polygon, for we can draw the lines a and b (Fig. 269) to a scale, and parallel to the line of action of the forces, and from the end of b draw a line parallel to C and from the end of a, a line parallel to D. These two lines will intersect and the intersection will give the length of c and d, and by measuring the length of these lines we can obtain the required magnitude of the forces C and D. It is upon this principle particularly that the stress diagrams are drawn.

125. - How to tell the character of a force. But two kinds of forces are involved in the stress diagram of a truss - compressive forces, and tensile or pulling forces, the former indicating compression and the latter indicating tension. In § 4, it was shown that if a piece of material is subject to compression, each end of the piece pushes against the opposing force, or the force acts outward at each end, and if it is in tension, there is a pull at each end, or the force acts inward at each end. Consequently, when we have a force, with the arrow head pointing towards the point of application, it must be a compressive force, and if the arrow head points from the point of application, it must be a tensile force.

In a truss the forces are considered as applied at the joints and as the forces in the equilibrium polygon must follow each other in rotation, i. e., the arrow heads must point as in Figs. 269 and 270, around the polygon, we can readily tell in which direction each force acts in relation to any given joint, and if the arrow head points toward the joint it indicates compression. If it points from the joint it indicates tension. In most trusses one can tell by general principles which pieces are in tension and which are in compression, but it sometimes happens that the only way in which one can tell is by the direction of the arrow head, so that the direction of the arrow heads should always be noted in the mind if not drawn on the stress diagram. The foregoing principles in conjunction with those given in §117 and 118 are the only ones ordinarily involved in the construction of a stress diagram, and if these are well understood there should be no difficulty in understanding the diagrams.