Beam Bending (U Beam-Rectangular)

[ACE123456]

Hello Everyone,
In the image file I have attached, I want to know the maximum force that can be applied to the lower part of the beam (shown by the red arrow). The cross section is rectangular(b*t).
If anyone could guide me to a link where the derivation of the formula is given, or just the formula would be good.
Thanks.

 

[drawoh]

ACE123456,

I found nothing in Roarks 7th edition. These things normally are solved using strain energy. Are you concerned about strength or rigidity?

--
JHG

 

[ACE123456]

drawoh,

I am concerned about the finding the relation between the force applied and the bending stress.
And to take it further, how much force to apply to displace it by 10mm. (But this is after the relation is developed)

 

[desertfox]

Hi

How big is this hair pin beam? Is it vertical or horizontally orientated like your sketch?
10mm deflection in which direction , I guess inline with the force but there will be a deflection horizontally as well if your sketch is anything to go on.

As the previous poster stated you could analyse it using Castiglano's method of strain energy.

If you can provide a bit more information we might help further.

 

[ACE123456]

Desertfox,

hair pin dia = d, rectangular cross section (b*t). Orientation is exactly like the sketch. right = +x axis, above = + y axis.
Yes you said it right, the displacement is in line with the force. I want to determine the force to displace 10mm.
The method you provided does not involve bending stresses. here the beam will bend during displacement.

 

[JStephen]

If you neglect deflection, you can easily find bending moment and stress in the straight parts from a free body diagram. If radius is not too tight, that would apply to the curved part as well.

For deflection, you could treat it as an assembly of two straight beams, one curved beams, and can find equations for deflection of each in Roark and other sources. It'd get tedious, but still, just a bunch of working through the details.

It'd get messier if you have to consider "large" deflections, large enough that the deflections appreciably change the moments.

 

[drawoh]

ACE123456,

This looks solvable with MathCAD, Octave, or a spreadsheet.

From the applied force to the start of the radius, you have something close to pure tension. As you go around the radius, you have an increasing bending moment, M=F(y), easily calculated. From the end of the radius to the fixed end, you have a beam loaded in compression and by a constant bending moment.

For stiffness purposes, you can ignore the tensile forces.

Since you can easily work out the bending moments, you can fairly easily work out the bending strain at each point along your structure. Then, you do numerical integration all along your structure to work out the deflection at the end. Each bending moment gives you a bending stress. If you really want to be accurate, you can add the tension and compression stresses to get the total stress.

All of this assumes that your structure is very much larger than 10mm, and that the mass of it is insignificant compared to your force. If the bending significantly affects your geometry, the problem becomes way more complicated.

--
JHG

 

[desertfox]

Hi

Go to chapter 9 of this link and I think you will find the method quoted does cover bending.

http://www.freestudy.co.uk/statics/complex/t5.pdf

I still can't see any lengths on your beam or what value diameter d is.

 

[rb1957]

i`d solve long-hand, using method of sections. bending moment is zero along the first flat segment (loaded in tension), constant along the upper straight segment, and changing around the radius.

another day in paradise, or is paradise one day closer ?

 

[ACE123456]

Hey Everyone
Thanks, I will try the methods you guys have proposed and let you know if I am able to get the desired result.

ACE