warping and bending functions of common cross sections in St Venant problems in elasticity

[robinho]

Where can one find warping and bending function formulae of common cross sections of a beam in St Venant problems in elasticity?

[studiot]

Just now, robinho said:

Where can one find warping and bending function formulae of common cross sections of a beam in St Venant problems in elasticity?

I see you are a new member so welcome.

😀

 

Can you elaborate on what you want ?

 

The standard works for these are

Roark's Formulae for Stress and Strain

and

Kleinlogel Rigid Frame Formulae.

 

How many handbooks  and textbooks contain shorter tables eg

Fiona Cobb

The Structural Engineers Pocketbook

 

Edited January 26 by studiot
correct book title

[robinho]

thank you so much! I checked the books you suggested. For example, I couldn't find warping function formulae of common cross sections of a beam.  Imagine having a straight bar of known cross section, then twist it from both ends, and compute the longitudinal displacement, then we need a warping function satisfying a poisson equation. The solution depends only on the cross section, not anything else. Is there a table listing such warping functions?

[studiot]

Just now, robinho said:

thank you so much! I checked the books you suggested. For example, I couldn't find warping function formulae of common cross sections of a beam.  Imagine having a straight bar of known cross section, then twist it from both ends, and compute the longitudinal displacement, then we need a warping function satisfying a poisson equation. The solution depends only on the cross section, not anything else. Is there a table listing such warping functions?

Glad you came back, because I am not sure what you mean by warping functions. I think I must call them something different. I'm sure there must be an old name for what you want.

So

Are you referring to torsion ?

Are you doing this for analystical solutions of finite element analysis ?

Are you aware that any analysis is only valid for a range of width to length or depth to length ratios.
There is things called wide beam theory,  slenderness ratio and so on.

or are you referring to products of inertia as opposed to moments of inertia ?

 

Please elaborate a bit.

Can you give an example?

My original references came off the top of my head, near my midnight as I was about to shut down.

Just some I could grab quickly.

 

Edited January 29 by studiot

[studiot]

Just now, robinho said:

thank you so much! I checked the books you suggested. For example, I couldn't find warping function formulae of common cross sections of a beam.  Imagine having a straight bar of known cross section, then twist it from both ends, and compute the longitudinal displacement, then we need a warping function satisfying a poisson equation. The solution depends only on the cross section, not anything else. Is there a table listing such warping functions?

 

This large more modern book deals with both analytical and FE methods for warping, and has tables along with some discussion and some useful references.

Pilkey  chapter 14 page 705.

 

Please note the special notation used in this chapter as there are cross references to the rest of the book, including more twisting and torsion.
This chapter includes a tables of equivalent symbols.

 

This large more modern book deals with both analytical and FE methods for warping, and has tables along with some discussion and some useful references.

Pilkey  chapter 14 page 705.

 

Please note the special notation used in this chapter as there are cross references to the rest of the book, including more twisting and torsion.
This chapter includes a tables of equivalent symbols.

 

 

 

 

Edited January 29 by studiot

[robinho]

 Thank you very much for your reply😛. I look for analytic solutions. I'm aware that analysis is only valid for a range of width to length or depth to length ratios. But analytic solutions for torsion isn't aspect ratio sensitive. Yes, I refer to torsion. When you twist a bar and if the bar is not of round cross section, there is longitudinal (z) displacement and this is where the warping function psi(x,y) comes from. When the cross section is round, psi=0.

I don't have the book Formulas for Stress, Strain, and Structural Matrices. Don't know if I can find it online.

Edited January 29 by robinho

[studiot]

Just now, robinho said:

 Thank you very much for your reply😛. I look for analytic solutions. I'm aware that analysis is only valid for a range of width to length or depth to length ratios. But analytic solutions for torsion isn't aspect ratio sensitive. Yes, I refer to torsion. When you twist a bar and if the bar is not of round cross section, there is longitudinal (z) displacement and this is where the warping function psi(x,y) comes from. When the cross section is round, psi=0.

I don't have the book Formulas for Stress, Strain, and Structural Matrices. Don't know if I can find it online.

Pages 700+ are right in the middle of a 1450 page book so it doesn't scan very well, but here is a table from chap14 if it is of any use.

[studiot]

Most approach this by extending Timoshenko theory. Here are some links.

 

https://www.sciencedirect.com/science/article/pii/S0020768316300725

https://ntrs.nasa.gov/api/citations/19910000994/downloads/19910000994.pdf

https://jresm.org/archive/resm2015.19me0827.pdf

file:///C:/Users/somerset_2/Downloads/Mit_2001-06.pdf

https://www.ias.ac.in/article/fulltext/reso/007/10/0054-0058

https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/gss/default.htm?startat=ch06s02.html

[robinho]

 Thank you very much, though I haven't found any warping or bending stress functions for bars with such simple cross sections as circles or rectangles yet. I'll keep scanning. Maybe I should've called them stress functions from the beginning, stress functions satisfying harmonic PDEs or biharmonic PDEs.

On 1/30/2025 at 12:15 AM, studiot said:

😛

 

Edited February 1 by robinho

[robinho]

 By the way what is the best branch on this website to post elasticity questions?