Warping functions

Warping Function and Torsion of Non-Circular Cylinders edit

The displacements are given by


where   is the twist per unit length, and   is the warping function.

The stresses are given by


where   is the shear modulus.

The projected shear traction is


Equilibrium is satisfied if


Traction-free lateral BCs are satisfied if




The twist per unit length is given by


where the torsion constant


Example 1: Circular Cylinder edit

Choose warping function


Equilibrium ( ) is trivially satisfied.

The traction free BC is




where   is a constant.

Hence, a circle satisfies traction-free BCs.

There is no warping of cross sections for circular cylinders

The torsion constant is


The twist per unit length is


The non-zero stresses are


The projected shear traction is


Compare results from Mechanics of Materials solution




Example 2: Elliptical Cylinder edit

Choose warping function


where   is a constant.

Equilibrium ( ) is satisfied.

The traction free BC is




where   is a constant.

This is the equation for an ellipse with major and minor axes   and  , where


The warping function is


The torsion constant is




If you compare   and   for the ellipse, you will find that  . This implies that the torsional rigidity is less than that predicted with the assumption that plane sections remain plane.

The twist per unit length is


The non-zero stresses are


The projected shear traction is

Shear stresses in the cross section of an elliptical cylinder under torsion

For any torsion problem where  S is convex, the maximum projected shear traction occurs at the point on  S that is nearest the centroid of S

The displacement   is

Displacements ( ) in the cross section of an elliptical cylinder under torsion

Example 3: Rectangular Cylinder edit

In this case, the form of   is not obvious and has to be derived from the traction-free BCs


Suppose that   and   are the two sides of the rectangle, and  . Also   is the side parallel to   and   is the side parallel to  . Then, the traction-free BCs are


A suitable   must satisfy these BCs and  .

We can simplify the problem by a change of variable


Then the equilibrium condition becomes


The traction-free BCs become


Let us assume that






Case 1: η > 0 or η = 0 edit

In both these cases, we get trivial values of  .

Case 2: η < 0 edit







Apply the BCs at   ~~ ( ), to get




The RHS of both equations are odd. Therefore,   is odd. Since,   is an even function, we must have  .



Hence,   is even. Since   is an odd function, we must have  .



Apply BCs at   ~~ ( ), to get


The only nontrivial solution is obtained when  , which means that


The BCs at   are satisfied by every terms of the series


Applying the BCs at   again, we get


Using the orthogonality of terms of the sine series,


we have








The warping function is


The torsion constant and the stresses can be calculated from  .

Prandtl Stress Function (φ) edit

The traction free BC is obviously difficult to satisfy if the cross-section is not a circle or an ellipse.

To simplify matters, we define the Prandtl stress function   using


You can easily check that this definition satisfies equilibrium.

It can easily be shown that the traction-free BCs are satisfied if


where   is a coordinate system that is tangent to the boundary.

If the cross section is simply connected, then the BCs are even simpler:


From the compatibility condition, we get a restriction on  


where   is a constant.

Using relations for stress in terms of the warping function  , we get


Therefore, the twist per unit length is


The applied torque is given by


For a simply connected cylinder,


The projected shear traction is given by


The projected shear traction at any point on the cross-section is tangent to the contour of constant   at that point.

The relation between the warping function   and the Prandtl stress function   is


Membrane Analogy edit

The equations


are similar to the equations that govern the displacement of a membrane that is stretched between the boundaries of the cross-sectional curve and loaded by an uniform normal pressure.

This analogy can be useful in estimating the location of the maximum shear stress and the torsional rigidity of a bar.

  • The stress function is proportional to the displacement of the membrane from the plane of the cross-section.
  • The stiffest cross-sections are those that allow the maximum volume to be developed between the deformed membrane and the plane of the cross-section for a given pressure.
  • The shear stress is proportional to the slope of the membrane.

Solution Strategy edit

The equation   is a Poisson equation. Since the equation is inhomogeneous, the solution can be written as


where   is a particular solution and   is the solution of the homogeneous equation.

Examples of particular solutions are, in rectangular coordinates,


and, in cylindrical co-ordinates,


The homogeneous equation is the Laplace equation  , which is satisfied by both the real and the imaginary parts of any { analytic} function ( ) of the complex variable




Suppose  .

Then, examples of   are


where  ,  ,  ,   are constants.

Each of the above can be expressed as polynomial expansions in the   and   coordinates.

Approximate solutions of the torsion problem for a particular cross-section can be obtained by combining the particular and homogeneous solutions and adjusting the constants so as to match the required shape.

Only a few shapes allow closed-form solutions. Examples are

  • Circular cross-section.
  • Elliptical cross-section.
  • Circle with semicircular groove.
  • Equilateral triangle.

There are a few other papers which propose closed-form or semi-closed-form solutions to the torsion problem for cross-sections with irregular shapes [1][2][3].

Example: Equilateral Triangle edit

Torsion of a cylinder with a triangular cross section

The equations of the three sides are


Let the Prandtl stress function be


Clearly,   at the boundary of the cross-section (which is what we need for solid cross sections).

Since, the traction-free boundary conditions are satisfied by  , all we have to do is satisfy the compatibility condition to get the value of  . If we can get a closed for solution for  , then the stresses derived from   will satisfy equilibrium.

Expanding   out,


Plugging into the compatibility condition




and the Prandtl stress function can be written as


The torque is given by


Therefore, the torsion constant is


The non-zero components of stress are


The projected shear stress


is plotted below

Stresses in a cylinder with a triangular cross section under torsion

The maximum value occurs at the middle of the sides. For example, at  ,


The out-of-plane displacements can be obtained by solving for the warping function  . For the equilateral triangle, after some algebra, we get


The displacement field is plotted below

Displacements   in a cylinder with a triangular cross section.

Thin-walled Open Sections edit

Examples are I-beams, channel sections and turbine blades.

We assume that the length   is much larger than the thickness  , and that   does not vary rapidly with change along the length axis  .

Using the membrane analogy, we can neglect the curvature of the membrane in the   direction, and the Poisson equation reduces to


which has the solution


where   is the coordinate along the thickness direction.

The stress field is


Thus, the maximum shear stress is


Thin-walled Closed Sections edit

The Prandtl stress function   can be approximated as a linear function between   and   on the two adjacent boundaries.

The local shear stress is, therefore,


where   is the parameterizing coordinate of the boundary curve of the cross-section and   is the local wall thickness.

The value of   can determined using


where   is the area enclosed by the mean line between the inner and outer boundary.

The torque is approximately


Related content edit

Introduction to Elasticity

  1. Approximate Torsional Analysis of Multi-layered Tubes with Non-circular Cross-Sections, Gholami Bazehhour, Benyamin, Rezaeepazhand, Jalil, Journal: Applied composite materials ISSN: 0929-189X Date: 12/2011 Volume: 18 Issue: 6 Page: 485-497 DOI: 10.1007/s10443-011-9213-z
  2. Simplified approach for torsional analysis of non-homogenous tubes with non-circular cross-sections, B Golami Bazehhour, J Rezaeepazhand, Date: 2012: Journal: International Journal of Engineering, Volume: 25, Issue: 3,
  3. Torsion of tubes with quasi-polygonal holes using complex variable method, Gholami Bazehhour, Benyamin, Rezaeepazhand, J. Journal: Mathematics and mechanics of solids ISSN: 1081-2865 Date: 05/2014 Volume: 19 Issue: 3 Page: 260-276 DOI: 10.1177/1081286512462836