| TRUE ECONOMY IN DESIGN. | 31 |
| that are very nearly true but it holds good only for parallel chords. It is this: |
| Let A = weight of the chords, B = weight of the web, C = weight of the truss, and D = depth of truss. |
| Then C = A + B |
| But the weight of the chords varies inversely as the depth, or |  |
| and the weight of the web varies directly as the depth, or B = bD, |
| where a and b are constants; and therefore |
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| If C is to be made a minimum, we shall have, by differentiation, |
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| or |
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| or A = B |
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As the second differential coefficient, after substitution according to the usual method for maxima and minima, comes out positive, the result obtained corresponds to a minimum. From this it is evident that, for trusses with parallel chords, the greatest economy of material will prevail when the weight of the chords is equal to the weight of the web. The author has verified this conclusion by checking the weights of chords and webs in a number of finished designs, finding it to be absolutely reliable. However, it is not of much practical value, because the economic depths of trusses with parallel chords are pretty well known; and, again, when spans are in excess of 175 or 200 feet, the chords of through-bridges are seldom made parallel. Moreover, the best depth to use is not
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