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7 Torsion of Prismatic Bars
Also, the function ψ(x, y), defining warping of cross-section must be determined
by the equations of equilibrium.
Therefore, we find that the function ψ must satisfy the equation
∂
2
ψ
∂ x 2 +
∂
2
ψ
∂ y 2 = 0
(7.3a)
Now, differentiating equation (d) with respect to y and the equation (e) with respect
to x, and subtracting we get an equation of compatibility.
Hence,
∂τ xz
∂ y
= −Gθ
∂τ yz
∂ x
= Gθ
∂τ xz
∂ y
−
∂τ yz
∂ x
= −Gθ − Gθ
= −2Gθ
(7.4)
Therefore, the stress in a bar of arbitrary section may be determined by solving
Eqs. (7.3) and (7.4) along with the given boundary conditions.
7.3 Boundary Conditions
Now, consider the boundary conditions given by
X = σ x l + τ xy m + τ xz n
Y = σ y m + τ yz n + τ xy l
Z = σ z n + τ xz l + τ yz m
For the lateral surface of the bar, which is free from external forces acting on the
boundary and the normal n to the surface is perpendicular to the z-axis. The first two
equations are identically satisfied and the third gives,
τ xz l + τ yz m = 0
(7.5)
which means that the resultant shearing stress at the boundary is directed along the
tangent to the boundary, as shown in Fig. 7.3.
Considering an infinitesimal element abc at the boundary and assuming that S is
increasing in the direction from c to a,