The work performed throughout the entire piece will then be ⌠ _M²_ ⌡ ------ _dL._ (51) 2_IE_ Each of the expressions (48) and (51) belongs to a single piece or member of the structure. The total work performed in all the pieces subjected either to direct stress or to bending, and which, according to the principle of least work, must be a minimum, is found by taking the summation of the two preceding expressions: 1 ⎲ _S²L_ 1 ⎲ ⌠ _M²_ _e_ = ----⎳ ------ + ----- ⎳ ⌡ ----- _dL_ = minimum. (52) 2_E_ _A_ 2_E_ _I_ In making an application of equation (52) it is to be remembered that _S_ is the direct stress of tension or compression in any member, and that _M_ is the general value of the bending moment in any bent member expressed in terms of the length _L_. =114. Application of Method of Least Work to General Problem.=—The problem which generally presents itself in the use of equation (52) is the finding of an equation which expresses the condition that the work expended in producing elastic deformation shall be a minimum, some particular stress in the structure or some external load or force being the variable. If _t_ represent that variable, then the desired equation of condition will be found simply by placing the first differential coefficient of _e_ in equation (52) equal to zero: _de_ 1 (⎲ _S dS_ ⎲ ⌠ _M dM_ ) ----- = ----(⎳ --- --- _dL_ + ⎳ ⌡ ---- --- _dL_) = 0. (53) _dt_ _E_( _A dt_ _I dt_ ) The solution of equation (53) will give a value of _t_ which will make the work performed as expressed in equation (52) a minimum. This method is not a difficult one to employ in such cases as those of drawbridges and stiffened suspension bridges. In the latter case particularly it is of great practical value. =115. Application of Method of Least Work to Trussed Beam.=—The method of least work may be illustrated by the application of the preceding equations to the simple truss shown in Fig. 32. The pieces _BC_ and _GD_ are supposed to be of yellow-pine timber, the former 10 inches by 14 inches (vertical) in section and the latter 8 inches by 10 inches, while each of the pieces _BD_ and _DC_ are two 1⅝-inch round steel bars. The coefficient of elasticity _E_ will be taken at 1,000,000 pounds for the timber and 28,000,000 for the steel. The length of _BC_ is 360 inches; _GD_ 96 inches; _BD_ = 96 × 2.13 = 204.5 inches. tan α = 1.875 and sec α = 2.13.