Fig. 32.
The numerical elements of the expression for the work done in the members of the triangular frame are:
| Member. | Stress. | Length. | Area of Section. | ||
| BC | ½W₂ tan α | 360 inches = l | 140 | square | inches |
| DC | ½W₂ sec α | 204.5 ” | 4.14 | ” | ” |
| DG | W₂ | 96 ” | 80 | ” | ” |
| I = | 10 × 14³ | = | 27440 | = | 2286.7. |
| 12 | 12 |
The substitution of those quantities in the first term of the second member of equation (52) will give
| 1 | ∑ | S²L | = | 1 | ( | W₂² tan² α . 360 | + | W₂² . 96 | ) |
| 2E | A | 2,000,000 | 4 × 140 | 80 |
| + | 2 | W₂² sec² α .204.5 | = | .000,003,73 W₂². | |
| 56,000,000 | 4 × 4.14 |
The substitution of numerical quantities in equation (54) gives
| 1 | W₁²l³ | = | .000,213W₁². | |
| EI | 96 |
Or, since W - W₂ = W₁,
e = .000,003,73W₂² + .000,213(W - W₂)². (55)
Hence
| de | = | .000,007,46W₂ - .000,426(W - W₂) = 0. (56) |
| dW₂ |
The solution of this equation gives
W₂ = .893W = 19,660 pounds.
W₁ = 340 ”
It is interesting to observe that the first term of the second member of equation (56) is the deflection of the point of application of W₂ as a point in the frame, while the second term is the deflection of the point of application of W₁ considered as a point of the beam. In other words, the condition resulting from the application of the principle of least work is equivalent to making the elastic deflections by W₁ and W₂ equal. Indeed equation (53) expresses the equivalence of deflections whenever the features of the problem are such as to involve concurrent deflections of two different parts of the structure.
116. Removal of Indetermination by Methods of Least Work and Deflection.—The indetermination existing in connection with the computations for such trusses as those shown in Fig. 22 and Fig. 23 can be removed by finding equations of condition by the aid of the method of least work or of deflections. It is evident that the component systems of bracing of which such trusses are composed must all deflect equally. Hence expressions may be found for the deflections of those component trusses, each under its own load. Since these deflections must be equal, equations of condition at once result. A sufficient number of such equations, taken with those required by statical equilibrium, can be found to solve completely the problem. Such methods, however, are laborious, and the ordinary assumption of each system carrying wholly the loads resting at its panel-points is sufficiently near for all ordinary purposes.
The method of least work can be very conveniently used for the solution of a great number of simple problems, like that which requires the determination of the four reactions under the four legs of a table, carrying a single weight or a number of weights, and many others of the same character.