GRAPHICAL STATICS. Graphical static deals with statics by purely graphical or draughting-room methods: constructions made with straight-edge, dividers, protractors, parallel rules, planimeters, etc. The analytic solution of a problem may give rise to a complicated form ula, an infinite series from which it is difficult to compute, or to many simultaneous equations laborious to solve. Graphical methods over come these disadvantages. They are very rapid in comparison with mathematical processes and quite precise when used carefully; they show at a glance the entire mechanism of solution. Al though they give numerical answers — not formulas — that are only approximately correct, these results need never in engineering prob lems be less precise than the only approximately correct data. With average dull and good in struments it is possible to keep the thickness of points and lines within 0.005 inch and to set off distances of this size and angles of 0.1 degree. No graphical construction need have an error of more than 0.5 per cent, which is about the pre cision attainable with a small slide rule or with four-place logarithmic tables. All work is done to scale. Land, (Die Maszstaebe bei der Zeich nerischen Loesung) (Zeitschr. fur Architektur u. Ingenieurtvesen, Heft 4, 1897) showed that the graphical units of length must satisfy the same relations as the functions themselves. Thus if z ---=f(x, y) then es = f (ex, el!) where is the scale value of x; i.e., if x is pounds and the scale of pounds is, say, five pounds to the inch, then ex--= 5 and is represented by a line inches long. Land gives a number of illustrations.
The entire subject is a direct development of the basic law of statics: the theorem of three forces. This is a form of the parallelogram of forces (see MECHANICS) and states that if three non-parallel, coplanar forces (all in one plane) are in equilibrium, the force vectors will be con current (through a common point) and will have such magnitudes and directions that they may be translated so as to form a closed tri angle with the arrows running head to tail. Consequently if any number of concurrent, coplanar forces are in equilibrium their force polygon must close with the arrows pointing head to tail. For, any two of them can be re placed by a single force, this partial resultant can be combined with one of the remaining forces, and so on until the set is reduced to three which must satisfy the fundamental con dition.
Four Forces.— Before dealing with non concurrent systems we shall discuss the special case of four coplanar forces on account of its importance in engineering. The method was first extensively employed by Culmann in 1866. Forces in space are treated at the end of this article. If four forces are in equilibrium the resultant of one pair balances that of the other pair, or since it equilibrates the two remaining forces it must be coplanar with, and concurrent with or parallel to them.
If in Fig. 1 P is given and only the lines of action of Q, R, S are known, we have the con struction shown, which fails, however, when (a) all four forces are concurrent or parallel, (b) three are concurrent or parallel.
Coplanar Forces in General.— If n forces P,Q,R,....Z are in equilibrium any one of them will balance the resultant of the other n-1. The construction is made as follows: Resolve P into any pair of components A and B, corn bine B with Q, this resultant with S, and so on until Z is left. Then A, Z, and the resultant of the others must satisfy the theorem of three forces. The reader should have this process clearly in mind while studying the following illustration.
Let P, Q, R, S in Fig. 2 be in equilibrium; their vector polygon in u will close. In ii re solve P into any pair of components A, B. Draw C (the resultant of B, Q) and D (the resultant of C, R, i.e., of B, Q, R) ; ii gives only magnitudes and directions, the actual positions being shown in .i. The three concurrent lines A, B, P in i correspond to the triangle ABP in ii ; similarly, in C must be concurrent with B and Q, D with C and R, and D with S and A. Before going further observe the peculiar cor respondence between the two diagrams: three forces forming a triangle in ii have three con current parallels in i; but there are exceptions to the converse. This is a simple case of recip rocal diagrams or figures, which were so called by Maxwell, On Reciprocal Figures and Dia grams of Forces' (Philosophical Magazine 1864) who was the first to discuss their proper ties with any degree of completeness. Cremona, 'Le figure reciproce nella statica grafica' (1872), developed them into a branch of geometry. ABCD in i was named the funicular polygon by Lame and Clapeyron in 1826 for a reason which will be evident later. In ii ABCD are the rays which meet at the pole. The funda mental property of the funicular polygon is that every vertex lies on a force, and conversely. Balanced forces form a closed vector polygon. This is necessary but not sufficient for equilib rium, and corresponds only to the conditions for concurrent forces. In order that the system may not have a resultant moment it is necessary that any force be at least collinear with the resultant of the rest of the set. Thus in i the sum of the moments of P, Q, R, S cannot van ish unless S lies on the line of action of the resultant of P, Q, R, i.e., unless S passes through A and D. This condition requires a complete or closed funicular polygon. Hence the graphical criteria of equilibrium are (1) For equilibrium of translation the force polygon must close.