Warping functions

Warping Function and Torsion of Non-Circular Cylinders

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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

 

or,

 

The twist per unit length is given by

 

where the torsion constant

 


Example 1: Circular Cylinder

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Choose warping function

 

Equilibrium ( ) is trivially satisfied.


The traction free BC is

 

Integrating,

 

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

 

and

 


Example 2: Elliptical Cylinder

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Choose warping function

 

where   is a constant.


Equilibrium ( ) is satisfied.


The traction free BC is

 

Integrating,

 

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

 

where

 

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

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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

 

Then,

 

or,

 

Case 1: η > 0 or η = 0

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In both these cases, we get trivial values of  .


Case 2: η < 0

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Let

 

Then,

 

Therefore,

 

Apply the BCs at   ~~ ( ), to get

 

or,

 

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


Also,

 

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


Therefore,

 

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

 

or,

 

Now,

 

Therefore,

 

The warping function is

 

The torsion constant and the stresses can be calculated from  .

Prandtl Stress Function (φ)

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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

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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

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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

 

Thus,

 

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

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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

 

Therefore,

 

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

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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

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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

 
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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