Calculation the deflection of a box section steel beam supported at each end

[GreensandsParticipant @greensands]

I wish to be able to calculate the deflection at the centre of a box-section steel beam supported at each end subject to a load distributed across a central section of the beam. Can somebody pleased direct me to a suitable source of information showing how it is done with examples of a typical calculation. Thanks in advance.

[duncan webster 1Participant @duncanwebster1]

Send me a pm with your email and I’ll scan the relevant page of Roark. Not being able to attach screen shots to this makes it too complicated to do it direct

[DC31kParticipant @dc31k]

If you type ‘beam deflection calculator’ into a nearby search engine, there are many useful leads.

Whatever calculator you find, you will need the Young’s modulus of the material (E value). This is standard for steel.

You will also need the second moment of area of the section (I value). This is purely a function of the geometry of the section, so you need width and height and wall thickness. Manual calculation is easier if you simplify the section to give it square corners. A half-decent CAD program would calculate a more refined I value which takes account of the radiussed corners of a real world box section.

“Structures” by JE Gordon is a superb text on this sort of thing.

Edit:

This one has the ability to model a ‘partial uniform load’, which is what is asked for above. Many of them only cater for a full-length uniform load:

https://calcresource.com/statics-simple-beam.html

[duncan webster 1]

On duncan webster 1 Said:

…I’ll scan the relevant page of Roark

Or cite the page numbers in the pdf here:

https://jackson.engr.tamu.edu/wp-content/uploads/sites/229/2023/03/Roarks-formulas-for-stress-and-strain.pdf

[Nigel Graham 2Participant @nigelgraham2]

The type of support affects it too, i.e. whether “simply supported” (not clamped) or clamped rigidly at its ends.

[Alan DonovanParticipant @alandonovan54394]

Hi Greensands.

Machinery’s Handbook is my ‘go to’ source for that information.  It will also offer  formulae to take into account of the end supports / configuration as well (as mentioned by Nigel Graham 2).

Hope that helps.

Alan.

[Howard LewisParticipant @howardlewis46836]

How much of the load is “Distributed across ” the central section?

Is this a uniformly distributed load, across much of the beam, or effectively a point load applied in the centre of the beam?

The answer will affect the deflection calculated, since the bending moment diagram will be either parabolic, or linear.

Howard

[Howard Lewis]

On Howard Lewis Said:

How much of the load is “Distributed across ” the central section?

One assumes “all of it”. The initial post is clear – the entire length of the beam is not loaded, only some of its length.

Details concerning support conditions are perhaps irrelevant at this stage.

It would be useful, instead of referring to 800 pages plus of Machinery’s Handbook, to actually go and look at a copy (online ones are available) and cite a specific page where the actual question can be answered.

—

If a person wants a conceptual understanding, the easiest place to start is a point load midspan of a simply-supported solid rectangular section. For that, the midspan deflection formula is:

delta = PL^3/48EI

where P is the load, L is the length, E Young’s modulus, I second moment of area. Side note that for a rectangualr section I = bd^3/12 where b is width and d is height.

This tells us a lot, all of which is intuitive on reflection: the deflection is directly proprtional to the load P. The deflection varies with the cube of the span length (so twice the span gives eight times the deflection). The deflection is inversely proportional to the Young’s modulus – use a material twice as stiff for half the deflection. The deflection is inversely proportional to the I-value, so double the I  value for half the deflection.

The I value, second moment of area, reflects how the material of the section is distributed relative to the bending (neutral) axis of the section. With the rectangular section, double the width, double the I value. Double the depth, the I value increases eightfold. That is why an I-beam is the shape it is – it puts as much of the material in the flanges, as far away from mid-depth as possible.

Other loading cases and support conditions are just more complex cases of this most basic example. The important take away is how the four factors (load, span, material stiffness, material distribution) affect the deflection.

[Nicholas FarrParticipant @nicholasfarr14254]

Hi, Greensands has asked for the deflection of the central section. In my old maintenance job, the company used Lloyds British, and the deflection on overhead hoists and lifting beams was always done in the central position. With overhead moveable hoists on a cross beam, the testing was done with the cross beam midway between the upright supports.

Regards Nick.

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