Torsional deflection and shear stress of a solid rectangular beam

[John2004]

Hi everyone,

Can anyone please give me the formula to calculate the torsional deflection and shear stress of a solid steel rectangular beam ?

I have a small beam .156" high X .216" wide X .491" long. One end is an integral part of a steel plate or wall, the other end is hanging free. At the free end, I have about 23 in-lb of torque applied to the beam (100 pounds .231" from beam center), torque twists beam along its lengthwise .491" axis.

Thanks for your help.
John

 

[rb1957]

from Bruhn, attributed to Roark,
twist per inch length = T/(KG)
K = ab^3*[16/3-3.36*(b/a)*(1-((b/a)^4)/12)]
a = long side, b = short side
maximum shear stress = T*(3/b-1.8/a)/(8*(ab))

 

[John2004]

Hi RB1957,

Thanks allot for your reply,

In the formula you gave does T = Torque "in-lb" ?

Is "G" the shear modulus / modulus of rigidity ? I am using 11500000 as the modulus of rigidity for most steels.

If I interpreted your formula and parentheses right, I calculate .000004 deflection per inch, but is this in degrees or radians ? It seems to small and I must have made an error.

Is the beam length part of the formula for torsional deflection of a rectangular beam, as it is with a round beam, i.e., torsional deflection angle for round beam = TL/JG in radians...

Where

T =Torque in-lb
L= Length in inches
J = Polar moment of inertia
G = Modulus of rigidity


I did not see length as part of the formula you gave, but it seems to me to be an important part of torsional deflection. It seems the torsional deflection for a rectangular beam is a bit more complex than for a round beam.

Thanks again for your help.
John

 

[prex]

The correct formula for maximum shear stress is
[τ]=T(3/b+1.8/a)/(8ab)
and for both formulae a and b are half side lengths.
The general formula for total angle of twist is indeed TL/KG, K being the torsional constant for a given section and G the shear modulus.
And yes, the formula for a rectangular section is more complex than for a round one: the above formulae are even approximations, as the actual values cannot be represented in a closed form. In the site below, under Beams -> Sections you'll see better approximation formulae and you can directly calculate the values of interest.
John2004, your beam may be hardly defined as such (length only two times the height), and you seem to have a limited knowledge of some basic engineering matters. If you elaborate more on your application, you could get more help, as you'll hardly calculate something meaningful for your 'beam' with the above formulae.

prex

http://www.xcalcs.com

Online tools for structural design

 

[rb1957]

john2004,
theta is in radians (check the units)
my equation (as stated) is twist per unit length, so total twist needs to be multiplied by L
prex is right about a and b being 1/2 side length (oops) ... ie the long side is 2a ... this should 1/2 the angle

 

[John2004]

Hi guys,

Thanks for the additional input and clarification.

>John, your beam may be hardly defined as such (length only two times the height), and you seem to have a limited knowledge of some basic engineering matters.<

Well, perhaps I am more of a draftsman than engineer, but I am always wanting & trying to learn more. In several cases my instincts & intuition have served me better than the PhD degrees of some of the people I have worked with in the past.

This is not a "beam" in a structure, but rather a small support-arm in a small mechanical device. I am treating the support arm as a "beam" for the purposes of trying to calculate torsional deflection and stress.

>If you elaborate more on your application, you could get more help, as you'll hardly calculate something meaningful for your 'beam' with the above formulae.<

Please let me know what additional information would be helpful? It seemed to me that the material, dimensions, load torque, and location of load torque, are basically what is needed. The material will be a basic free machining steel or tool steel, and the load torque will be applied to the very end of the beam or arm.

Are the formulas not valid for such a short beam with this aspect ratio ?

If the beam or arm fails, nobody can get hurt, it's not safety related, but the device will fail if the arm is overstressed or deflects too much.

Thanks again for your help.
John

 

[prex]

John2004,
of course I was not suggesting that you should not ask.
Concerning the details on the application, I can hardly believe that you critically rely on the torsional deformation of such a short thing, unless you are in a dimensional testing machine.
Moreover the elastic formulae for beams lead to big errors for lengths of the order of two times the depth, as the effect of boundary conditions is important over a length of the order of a beam depth.
Boundary conditions (the way the support end is fixed and the way the load is applied) are very critical for a beam in torsion. If the fixed end is integral with a steel wall then the beam model that assumes the sections can all distort in the same way is not valid.
Also the way the moment is applied is quite critical: there might be a non zero resultant that would give rise to a bending and shear deformation, with a possible contribution also from the support.
I guess that you will hardly obtain something meaningful concerning the deformation, even by using a FEM model.
As far as stress is concerned, I suggest you using the plastic torsional modulus (that you find in the site below): assuming torsion is really the only stressing load (and I can hardly believe this, see above), you'll get a more realistic estimate of the strength of your object.

prex

http://www.xcalcs.com

Online tools for structural design

 

[John2004]

Hi Prex,

Thanks for the additional feedback and comments. Your reply was helpful.

My instincts tell me this short arm *probably* won't be overstressed or deflect too much. It can be made and tested easy enough, but I figured I would try to calculate it to try to be more sure, & to satisfy my curiosity and have one less thing to worry about. I agree with you, I don't think the load force will be *pure* torsion.

As I started thinking about the possible torsion on the arm, I just wanted to learn how to accurately calculate it.

I suppose sometimes calculations and even FEA don't shed as much light on a problem as we would like. In the end I guess you always have to build and test, to be 100% sure.

Thanks again,
John

 

[rb1957]

i think prex is right too ... you'll have shear load on your section, which will produce it's own internal stresses ... you'd only have "pure" torsion if you loaded the beam with a couple.

don't take this personnally ... calculations and FEA are only as good as the user's knowledge. i think you're starting by asking questions. get a copy of Bruhn (i prefer him to Niu) or maybe some solid mechanics text books (depending on your conidence with structural analysis). don't get a copy of Roark (yet) untill you're confident about understanding how a structure responds to loads (the zen of stress analysis !)

good luck