66 CHAPTER 3 Torsion of Solid Sections
Fig.3.1
Torsion of a bar of uniform, arbitrary cross section.
which identically satisfies the third of the equilibrium equations (3.1) whatever formφmay take.
Therefore,wehavetofindthepossibleformsofφwhichsatisfythecompatibilityequationsandthe
boundary conditions, thelatter being, in fact, therequirement that distinguishes onetorsion problem
fromanother.
Fromtheassumedstateofstressinthebar,wededucethat
εx=εy=εz=γxy=0 (seeEqs.(1.42)and(1.46))Further, sinceτxzandτyzand henceγxzandγyzarefunctions ofxandyonly, then the compatibility
equations(1.21)through(1.23)areidenticallysatisfiedasisEq.(1.26).Theremainingcompatibility
equations,(1.24)and(1.25),arethenreducedto
∂
∂x(
−
∂γyz
∂x+
∂γxz
∂y)
= 0
∂
∂y(
∂γyz
∂x−
∂γxz
∂y)
= 0
Substituting initially forγyzandγxzfrom Eqs. (1.46) and then forτzy(=τyz)andτzx(=τxz)from
Eqs.(3.2)gives
∂
∂x(
∂^2 φ
∂x^2+
∂^2 φ
∂y^2)
= 0
−
∂
∂y(
∂^2 φ
∂x^2+
∂^2 φ
∂y^2)
= 0
or
∂
∂x∇^2 φ= 0 −∂
∂y∇^2 φ=0, (3.3)