=113. Method of Least Work.=—The so-called theorem or principle of “Least Work” is closely related to the subject of elastic deflections just considered in its availability for furnishing equations of condition in addition to those of a purely statical character in cases where indetermination would result without them. This principle of least work is expressed in the simple statement that when any structure supports external loading the work performed in producing elastic deformation of all the members will be the least possible. Although this principle may not be susceptible of a complete and general demonstration, it may be shown to hold true in many cases if not all. The hypothesis is most reasonable and furnishes elegant solutions in many useful problems. The application of this principle requires the determination of expressions for the work performed in the elastic lengthening and shortening of pieces subjected either to tension or compression, and for the work performed in the elastic bending of beams carrying loads at right angles to their axes. Both of these expressions can be very simply found. Let it be supposed that a piece of material whose length is _L_ and the area of whose cross-section is _A_ is either stretched or compressed by the weight or load _S_ applied so as to increase gradually from zero to its full value. The elastic change of length will be _SL/AE_, _E_ being the coefficient of elasticity. The average force acting will be ½_S_, hence the work performed in producing the strain will be 1 _S²L_ --- -----. (48) 2 _AE_ It will generally be best, although not necessary, to take _L_ in inches. The expression (48) applies either to tension or compression precisely as it stands. To obtain the expression for the work performed by the stresses in a beam bent by loads acting at right angles to its axis, a differential length (_dL_) of the beam is considered at any normal section in which the bending moment is _M_, the total length being _L_. Let _I_ be the moment of inertia of the normal section, _A_, about an axis passing through the centre of gravity of the latter, and let _k_ be the intensity of stress (usually the stress per square inch) at any point distant _d_ from the axis about which _I_ is taken. The elastic change produced in the indefinitely short length _dL_ when the intensity _k_ exists is (_k/E_)_dL_. If _dA_ is an indefinitely small portion of the normal section, the average force or stress, either of tension or compression, acting through the small elastic change of length just given, can be written by the aid of equation (5) as _Md_ ½_k.dA_ = ----- ._dA._ (49) 2_I_ Hence the work performed in any normal section of the member, for which _M_ remains unchanged, will be, since ∫_k.dA.d_ = _M_, ⌠ _M M²_ ⌡ ------ _kd.dA.dL_ = ----- _dL._ (50) 2_IE_ 2_IE_