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Simple Beam with Variable Ix

Simple Beam with Variable Ix

Simple Beam with Variable Ix

(OP)
Problem:

A person installing my companies product wanted to do field modifications to what amounts to a piece of  3.56in. x  4.375in. x 0.24in-thick tubular steel with a 2.5in x2.5in x 0.25in-thick “strong-back” welded to the back of it.  They wanted to remove the strong back all together. I told them No because it would not meet code!  They then asked if they could remove 4 feet from each end the strong-back, the total length of the original member is 13 feet. I told them I would have to get back to them…of course they were annoyed.  

My shingleys Mechanical Eng. Design #7 does not go into detail of how to calculate a simply supported beam with a point load in the middle, given the beam has a Variable moment of inertia across the member. “The Google” (quote from G.W. Bush) has proved even less useful.

Could anyone point me to an internet resource that addresses this problem?
Or Kick my @$$ in the right direction…

Thank you!

RE: Simple Beam with Variable Ix

What is this member doing, is it a compression member or is it just in bending?

RE: Simple Beam with Variable Ix

(OP)
just in bending
a 1000lb load in the middle

finding max deflection

RE: Simple Beam with Variable Ix

Roark on Stress and Strain.  Lots of formulas and that is one.

Mike McCann
McCann Engineering

RE: Simple Beam with Variable Ix

(OP)
Downloading the trial right now, Looks pritty good

Thanks

RE: Simple Beam with Variable Ix

The actual shear and moment values are not dependent on the moment of inertia.  For a concentrated load at the midspan of a simply supported beam, the reaction is always P/2 and the moment is always P*L/4 (neglecting the self-weight).  The stresses are dependent on the member properties (e.g. bending stress = M*y/I).

RE: Simple Beam with Variable Ix

(OP)

Yes, mainly what I am looking for is y (assuming the beam is horizontal) which is dependent on I.

The typical equation being y=(W*l^3)/48E*I)
W= Weight
l=length of span
E= Modulus of Elasticity
I=Moment of Inertia

Although in my case I would be variable across the member.
The I=5 then I= 15 then back to I=5





RE: Simple Beam with Variable Ix

If you have access to a decent structural analysis text, look at Castigliano's method for deflection.  
This works very well for members with a varying EI along the length of the member.  
It will involve some integration, but will give you the answer you're looking for.
I have it in Hibbeler's text (I think 4th or 5th edition).
I'm sure it's in Timoshenko as well.
If you don't have access to one and would like the info, let me know and I'll scan the pages from the text and post them here.
I won't be able to do that until Wednesday, though.  I'm off until Wednesday and my texts are in the office.

RE: Simple Beam with Variable Ix

Hi Gymmeh

The strong back is a piece of angle I assume,whats its orientation to the tube underside ie;-

      ----|     or |
          |        |----


Regards

desertfox

RE: Simple Beam with Variable Ix

Hi Gymmeh

Using Strain energy method and a graphical method of intergration, I calculate that the deflection of the beam centre, with 4foot removed from the strong back at each end
will be 29.5mm or 1.161".
This assumes that the strong back is angle and one leg is welded to the underside of the tube ie the angle would be seen as an upside down L.
I'll try and post my working on Friday so you can see what I have done.

regards

desertfox

RE: Simple Beam with Variable Ix

(OP)
Structural, I have found a few structural engineering lecture notes on the Castigliano method and am working through them, its not to bad.

Desertfox.
Close, the strong back is 2.5"X 2.5" sq. tubular w/ wall thickness of 0.25"

using stain energy w/out a strong back, I get 1.06"
y =((W*L^3)/(48*I*E))

E=28.5E6 in^4
I=5 lbs/in^2
W=2000lbs
L=156in

My first calculation with the Castigliano method gave me 0.777in... now I just need to check, everything.
 



 

RE: Simple Beam with Variable Ix

Hi Gymmeh

Just check your figures I get "I"= 9.67 in^4 without strongback not 5 in^2 as you posted.
In addition when did the load change to 2000lbs your earlier post gave a 1000lbs.

regards

desertfox

RE: Simple Beam with Variable Ix

(OP)
Fox, Thanks for looking into this so much! I didnt think anyone would try to solve this so i didnt go into extreeme detail.

sorry, a little confusing, the original part was rated at 2000lbs,
so I am trying to prove after modification:
A) 2000lbs is will cause excessive deflection
B) De-Rating the part to 1000lbs will be sufficient.

Sorry again,I had the units backworks...
E=28.5E6 lbs/in^2
I=5 in^4

The geometry of the 3.56in. x  4.375in. x 0.24in-thick tubular steel is a little to complex to decribe in words,
but it reduces the value of I.

RE: Simple Beam with Variable Ix

Hi Gymmeh

Like to keep my number crunching and calculus skills in practice when I can.
Well what I worked out won't be much use I suppose, but I assumed the main part was a box section even though I couldn't find it as a standard size in any list and assumed
that the strongback was angle.
Placing the centroids of each section in line vertically I worked out the second moment of area of the combined section
as well as the second moment of area where there was no strong back.
I split the beam in half and like you based it on strain energy, but used a graphical method so it was only an estimate.
If you can give details of the main tube and in addition the position of the neutral axis I'll have another go and we can compare figures.
Can you derate to 1000lbs if service conditions are 2000lbs?

desertfox

RE: Simple Beam with Variable Ix

Hi Again

Is it possible to upload a sketch of the tube.

regards

desertfox

RE: Simple Beam with Variable Ix

(OP)
Main tubular section is proprietary and has walls that are at various angles and cuts for use in a particular application with objects running in the middle, I am not sure I have the right to give out to much info sad

The application of the system (luckily) allows for de-rating of the system, other wise I would not consider this modification.

Like you said, I am trying to keep my number crunching sharp and learn new techniques to solve these problems. If it was not Christmas and things weren’t a little slow, I would have told the workers to fly a kite! I normal have better things to do.

RE: Simple Beam with Variable Ix

Hi Gymmeh

Okay but seeing you have already given the Ixx as 5in^4
if I knew where that neutral axis was relative to say the vertical depth of tube which I assumed to be 4.375" thats all I would need to compare our calculations.
However I'll leave you to decide what you comfortable with.


best of luck

desertfox

RE: Simple Beam with Variable Ix

(OP)
I am going on vacation so I will not be able to get the information... I do have the neutral axis figured out, its in my office...

I will check with my boss when I get back, I am not sure how things work with regards to such information, I know I have a sheet of info to give customers but it does not go into much detail...so someone couldn't make their own part.

RE: Simple Beam with Variable Ix

If the higher I value is in the middle of the span and the stifened part extend for more than half the span, then I would conservatively use the average I value to calculate deflection.

To get an exact deflection, calculate M/EI across the section then apply this as a load to the beam. The resultant moment graph is actually the deflection graph. This is a simplified method for the direct integration method.

RE: Simple Beam with Variable Ix

Hi csd72

The stiffened part is less then half the span ie:- 13ft o/a
ten take 4ft off each end.

regards

desertfox

RE: Simple Beam with Variable Ix

(OP)
csd72,

After my two week vacation....I am just going to take the average method, with a design factor of two and call it good.... cuz their are now about 100 messages that I have to take care of smile

I did learn a little bit about the Castigliano methodd- intergration method... but will try to get back to it later....

IE: i will forget what I was doing and never do it until it comes up again smile

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