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Chapter: Civil : Principles of Solid Mechanics : Torsion

Prandtl’s Stress Function

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


Prandtl’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 ϕ(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 ϕ 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  ϕ 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 ϕ) can be plot-ted and:

 

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

 

b. τmax occurs where the stress-hill contours become closest togeth-er, which will be on the boundary of the cross-section (generally where r = Rt(x2+y2) is a minimum.

 

From Figure 9.6 it is easy to show that the twisting moment Mt is propor-tional to the volume of this stress-function hill. The shear force dS on a differ ential element E will be


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 ϕedge = 0. The volume of the stress-function hill (a paraboloid) is 0.5 π abh and therefore:



 

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


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