Text-book of mechanics . E PRINCIPLE OF WORK 193 Section XXI DEFLECTIONS DUE TO SHEAR The formulas giving the deflections of beams undervarious loadings are usually deduced from the equation This equation is deduced under the assumption that^the beams are under the action of purebending stresses only. Thus the deflections due toshearing stresses are not taken into account. The prin-ciple of work furnishes a simple means for the determi-nation of these (up to this point disregarded) Due to Shear. — In Fig. 99 is shown adifferential element subject to a shearing stress, q.
Text-book of mechanics . E PRINCIPLE OF WORK 193 Section XXI DEFLECTIONS DUE TO SHEAR The formulas giving the deflections of beams undervarious loadings are usually deduced from the equation This equation is deduced under the assumption that^the beams are under the action of purebending stresses only. Thus the deflections due toshearing stresses are not taken into account. The prin-ciple of work furnishes a simple means for the determi-nation of these (up to this point disregarded) Due to Shear. — In Fig. 99 is shown adifferential element subject to a shearing stress, q. Theforce q dydz = qdA acting uponone face of this element suffersa displacement dx, if repre-sents the shearing strain due toq. Thus the work done on, orthe resilience of, this elementdue to shear is \{qdA) (ct>dx),provided we assume the loading and thus the shearingstress to increase gradually from zero to their greatestvalues. 2 As G = *-, the resilience also equals -~^dx dA, or, as (j> 2 Lr. it is often stated, the resilience per unit volume is 2G Exercise 228. What is the resilience per unit volume fornormal stress? 194 MECHANICS OF MATERIALS Exercise 229. Compute the resilience per unit volumefor the element shown in Fig. 82. Deflection Due to Shear. —• As a special case considerthe deflection at the free end of a cantilever (Fig. 100)due to shear only, the cantilever to be loaded at its free -h Fig. 100 W Is end and to be regarded as weightless and of constantrectangular section. If Fig. 99 represents an element of this beam, theinternal work is readily seen to be where the first integration must cover the section of thebeam normal to its axis, and the second integration mustbe taken over the entire length of the beam. WSThe external work is — > where 8 is the required deflec-tion. Thus, *-f*m If we assume that the shearing force is uniformlydistributed over the cross section, q is constant andequation (1) becomes ?-&/*/?• THE PRINCIPLE OF
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