The elasticity and resistance of the materials of engineering . all the preceding formulae, a and t are to be taken ininches ; w in pounds per square inch, and P in pounds. The investigations can only be considered provisional. Al-though they give, as a whole, tolerably satisfactory results, therange of the experiments is far too small for the establishmentof thoroughly reliable formulae. Experiments on which aproper exponent of a can be based, are yet wholly lacking; andas the only resort, that found in the rough analysis has beenretained. Art. 82.—Special Cases of Flexure. There are a few ca
The elasticity and resistance of the materials of engineering . all the preceding formulae, a and t are to be taken ininches ; w in pounds per square inch, and P in pounds. The investigations can only be considered provisional. Al-though they give, as a whole, tolerably satisfactory results, therange of the experiments is far too small for the establishmentof thoroughly reliable formulae. Experiments on which aproper exponent of a can be based, are yet wholly lacking; andas the only resort, that found in the rough analysis has beenretained. Art. 82.—Special Cases of Flexure. There are a few cases of flexure which, while not frequentlyfound in engineering experience, are of some practical impor-tance, and are occasionally required. The two or three whichfollow involve the integration of some linear differential equa-tions that are treated in all the advanced works on the integralcalculus; consequently the operations of integration will notbe given here, but the general integrals will be assumed. Flexure by Oblique ForcesIn Fig. I let OX represent. beam acted upon by theoblique forces P, whichmake angles a with theaxis of X. The origin 0is supposed to be takenanywhere on the axis ofthe beam. If right-hand Art. 82.] SPECIAL CASES OF FLEXURE. 675 moments are positive and left-hand negative, the componentP sin a will have the negative moment — P sin ax about O,The lever arm of P cos a^ if the deflection w is positive, is-f- Wj and its moment P cos a ,w is positive. Hence the result-ant moment of any force, P, in reference to the origin O is : EI —.— = — P sin a , X -\- P cos a , w . (i)dx^ If a is greater than 90°, cos a is negative, so that if . P cos a . rr P sin a . x A — ± —^^-y— and F = ^^7 , EI EI the two cases may be expressed by the equation : ^^J^Aw=V .^ . (2) dx ^ ^ ^ If a = -{- V — A, and b—— V — Ay the general integralof Eq. (2) is : ^ix w = Ce + Ce^ H ^^ I Ve- ^^ dx ^^—-j Ve- *^ dx ; (3) a — 0 ] a — b) in which C and C a
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