Text-book of mechanics . 197 Let Lq be the shearing stress due to the load L, xq be the shearing stress due to the load Wi, etc.;then the resilience per differential element, provided L isthe only load upon the beam, is i(v^4 When the load Wi has been gradually but whollyapplied, the additional resilience per element due to theadditional sinking of the load L only, will be where (? the additional displacement of the shear-ing force, Lq dA, this force being due to the load L resilience due to the load L displaced by 5 underaction of all loads upon the beam will then be I-(LqdA)\
Text-book of mechanics . 197 Let Lq be the shearing stress due to the load L, xq be the shearing stress due to the load Wi, etc.;then the resilience per differential element, provided L isthe only load upon the beam, is i(v^4 When the load Wi has been gradually but whollyapplied, the additional resilience per element due to theadditional sinking of the load L only, will be where (? the additional displacement of the shear-ing force, Lq dA, this force being due to the load L resilience due to the load L displaced by 5 underaction of all loads upon the beam will then be I-(LqdA)\ Lq+Xmq I y~—dxj per element. In general the vertical component of the shearingstress at any point within a beam may be expressed inthe form ^ = hT \ ydA (see page 66). Substituting this value of q, the resilience per elementdue to L becomes m= n 2 rs^1 fydA iG _ L^ bl J y L& m= 1 dA dx. 198 MECHANICS OF MATERIALS Now LQX being the shearing force on any section dueto the load L only, we may put LQX = L4> (*),. CydA ~\2 IdA. It may happen that no load L exists at the point atwhich the deflection is desired. Under these conditionsplacing L = o in the above equation simply removes theterm LQX. m=n Moreover, as LQX + £. mQx always represents the total 771 = 1 shearing force at any section due to all existing loads, wemay replace it by the symbol Qx previously formula for the deflection can then be written Here the expression within the parenthesis must firstbe evaluated as on page 68, and expressed in terms of , when dA is expressed in terms of y, the integrationindicated may be extended over the beam section. Theresult of this integration is either a constant or it mustbe expressed in terms of x when the beam section allows the final integration to be extended over thewhole length of the beam. THE PRINCIPLE OF WORK 199 The (x) used in the above formula is defined byLQX = Lcj> (x), or when L is unity that is, (x) is the shearing force at any se
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