An alternative solution procedure that leads to a much simpler boundary condition, but a somewhat more difficult field equation.

**Prandtlâ€™s Stress
Function **

An alternative solution procedure that
leads to a much simpler boundary condition, but a somewhat more difficult field
equation, involves the intro-duction of a stress function Ï•(*x*,*y*)
defined as:

By this definition, the equilibrium
Equation (9.10) is automatically solved and the boundary condition Equation
(9.15) becomes:

As noted in previews, this is a Poisson
equation for which a solution can always be found and the boundary condition

is particularly nice.

However, the power of the Prandtl stress
function approach is in visualiz-ing torsional shear fields. It is easy to show
that the stress function Ï• will transform in any
direction as an invariant scalar function, and therefore in a new coordinate
orientation *x*', *y*'

Thus at any point the
shear stress in one direction is equal in magnitude to the slope of the Ï• surface in the perpendicular direction, and the maximum shear is the maximum
slope of the stress function curve and acts tangent (along a contour) to that
curve. If we visualize the stress function plotted in the *z* direction
above the cross-section, then contours (of equal Ï•)
can be plot-ted and:

a. The
shear Ï„ acts along these contours and is
proportional to the shortest distance (slope normal) to the next contour;

b. Ï„_{max}
occurs where the stress-hill contours become closest togeth-er, which will be
on the boundary of the cross-section (generally *where r = Rt(x*^{2}*+y*^{2}*) is a minimum.*

From Figure 9.6 it is
easy to show that the twisting moment *M _{t}* is propor-tional to
the volume of this stress-function hill. The shear force

Therefore, between two
contour lines the resisting moment

which is twice the
volume of the stress-function hill.

For example, consider
the elliptic cross-section discussed previously. Assume a stress function

which must satisfy
Equation (9.22), and the boundary condition Ï•_{edge}
=
0. The volume of the stress-function hill (a paraboloid) is 0.5 Ï€ abh and therefore:

which corresponds to
the result obtained by the direct St. Venant approach* using the warping
function.

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Civil : Principles of Solid Mechanics : Torsion : Prandtlâ€™s Stress Function |

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