Swing set loading.
May 16, 2013 12:00 AM Subscribe
Actual question: what is the formula for the deflection of an intermediately loaded simply supported beam from multiple point loads? Reason for the question so you can question my assumptions (and know that I'm not doing homework):
I am building a swing-set, from pretty much standard swing set parts, but want to deviate from the standard dimensions a bit - in particular, I want to make the top beam in a single bay setup 1.75' longer than the usual 10'. I am doing this so I can hang a longer bench swing that would hold a couple of people, so this is more length supporting more load, and I want to calculate some things. More inside!
First, my assumptions - I feel like if I calculate the deflection from static loading in both the standard 10', two swing configuration, and the deflection from static loading on my 13.75', 1 swing, 1 bench configuration and compare them, I'll get an idea if I'm being insane or just not within normal tolerances for a public swingset. I'm okay with the latter. This is ultimately for use at little outdoor parties of the sort where I can set up a swing-set, and burning man - but out in the suburbs, not a super highly trafficed location. If it ultimately bends it's not going to be unattended long enough to lead to disaster, and I can handle losing a piece of pipe... but I want that to be less than 100% likely within the first 10 minutes.
I also know that load from swinging is going to be much different from static load, but I am hoping that it varies linearly! So part one of my question is - is that assumption close to correct?
It's easy enough to find that the maximum deflection from a single load here at wikipedia, and I've found the constants I believe I need (moment of inertia and modulus of elasticity) for the 2" schedule 40 steel pipe that I plan on using.
But it doesn't seem like I could just add the results of the 4 loads I have from two swings, and I can't find an expansion of that after reading a few chapters of the textbooks I can find online (or a path to deriving it. I am not that clever.)
So as I said, the real question is: what is the formula for the deflection of an intermediately loaded simply supported beam from multiple point loads? And what percentage crazy am I being. I'm sure it's a little, I just don't want it to be a lot.
First, my assumptions - I feel like if I calculate the deflection from static loading in both the standard 10', two swing configuration, and the deflection from static loading on my 13.75', 1 swing, 1 bench configuration and compare them, I'll get an idea if I'm being insane or just not within normal tolerances for a public swingset. I'm okay with the latter. This is ultimately for use at little outdoor parties of the sort where I can set up a swing-set, and burning man - but out in the suburbs, not a super highly trafficed location. If it ultimately bends it's not going to be unattended long enough to lead to disaster, and I can handle losing a piece of pipe... but I want that to be less than 100% likely within the first 10 minutes.
I also know that load from swinging is going to be much different from static load, but I am hoping that it varies linearly! So part one of my question is - is that assumption close to correct?
It's easy enough to find that the maximum deflection from a single load here at wikipedia, and I've found the constants I believe I need (moment of inertia and modulus of elasticity) for the 2" schedule 40 steel pipe that I plan on using.
But it doesn't seem like I could just add the results of the 4 loads I have from two swings, and I can't find an expansion of that after reading a few chapters of the textbooks I can find online (or a path to deriving it. I am not that clever.)
So as I said, the real question is: what is the formula for the deflection of an intermediately loaded simply supported beam from multiple point loads? And what percentage crazy am I being. I'm sure it's a little, I just don't want it to be a lot.
You may also wish to consider the strength of the pipe at the joints. If you are using threaded rather then welded construction, be aware that the threading will cut quite deeply into the pipe. Note that a failure here will be a catastrophic failure, not just a bent pipe.
posted by ryanrs at 1:06 AM on May 16, 2013 [1 favorite]
posted by ryanrs at 1:06 AM on May 16, 2013 [1 favorite]
Regardless of what you calculate, you may also want to proof load test it once it's built. easy to do and easy to measure the deflection and compare it to your expectations. the real world has ways of intruding on your work.
posted by FauxScot at 1:15 AM on May 16, 2013
posted by FauxScot at 1:15 AM on May 16, 2013
I would be more worried about catastrophic collapse than just bending.
Secondly, I built a similar structure one time, and the side to side forces were what I didn't calculate correctly. Wobbly as shit.
posted by gjc at 2:28 AM on May 16, 2013
Secondly, I built a similar structure one time, and the side to side forces were what I didn't calculate correctly. Wobbly as shit.
posted by gjc at 2:28 AM on May 16, 2013
This is a classic strength of materials question - go to the library and find a textbook! In short, bending questions do follow superposition principles, so problems like this are generally solved by starting at one end of the beam and adding up the loads as you move across.
posted by backseatpilot at 5:04 AM on May 16, 2013
posted by backseatpilot at 5:04 AM on May 16, 2013
Also, are you sure this is simply-supported? If your cross support is being threaded into pipe fittings, that is not a simply-supported boundary condition.
posted by backseatpilot at 5:07 AM on May 16, 2013
posted by backseatpilot at 5:07 AM on May 16, 2013
Best answer: what is the formula for the deflection of an intermediately loaded simply supported beam from multiple point loads?
If you're really interested in deriving this from first principles, you probably want Euler-Bernoulli beam theory. Basically, the equation for the deflection would be
E I w''''(x) = Σ (mi g δ(x - xi) )
where E is the elastic modulus of the beam material, I is the second moment of area, mi are the masses of the loads, xi are their positions, and δ(x) is the Dirac delta-function. This is a fourth-order differential equation, but at least it's linear.
posted by Johnny Assay at 5:18 AM on May 16, 2013 [1 favorite]
If you're really interested in deriving this from first principles, you probably want Euler-Bernoulli beam theory. Basically, the equation for the deflection would be
E I w''''(x) = Σ (mi g δ(x - xi) )
where E is the elastic modulus of the beam material, I is the second moment of area, mi are the masses of the loads, xi are their positions, and δ(x) is the Dirac delta-function. This is a fourth-order differential equation, but at least it's linear.
posted by Johnny Assay at 5:18 AM on May 16, 2013 [1 favorite]
Another thing to keep in mind is area moment of inertia for a pipe is something like OD^4 - ID^4. So if 2 inch pipe isn't quite stiff enough, going to 3 inch will solve the problem. Quartic functions are powerful.
posted by ryanrs at 6:03 AM on May 16, 2013
posted by ryanrs at 6:03 AM on May 16, 2013
Best answer: The equation for deflection of a simple beam with a concentrated load at any point:
deflection = ( P * b * x ) / (6 * E * I * L) * ( L^2 - b^2 - x^2 )
P = load
b = distance from right support to load
x = distance at which you want to know the deflection
E = elastic modulus
I = moment of inertia of the section
L = distance between supports
Make sure all your units match - don't use feet for L and inches for everything else.
Not shown in the equation is a, the distance from the left support to the load.
For this equation, x must be less than a (i.e. location at which you want the deflection must be left of the load). So, depending on your load locations, turn the equation around.
To get your total deflection, calculate this equation multiple times and add up the deflections (superposition works, as long as the deflection is elastic. Not sure why backseatpilot thinks they don't). Also, simply supported would be worst case of deflection. Partial fixity at the ends would help for deflection (but may cause other practical problems)
(from the AISC Manual of Steel Construction)
posted by dforemsky at 7:08 AM on May 16, 2013
deflection = ( P * b * x ) / (6 * E * I * L) * ( L^2 - b^2 - x^2 )
P = load
b = distance from right support to load
x = distance at which you want to know the deflection
E = elastic modulus
I = moment of inertia of the section
L = distance between supports
Make sure all your units match - don't use feet for L and inches for everything else.
Not shown in the equation is a, the distance from the left support to the load.
For this equation, x must be less than a (i.e. location at which you want the deflection must be left of the load). So, depending on your load locations, turn the equation around.
To get your total deflection, calculate this equation multiple times and add up the deflections (superposition works, as long as the deflection is elastic. Not sure why backseatpilot thinks they don't). Also, simply supported would be worst case of deflection. Partial fixity at the ends would help for deflection (but may cause other practical problems)
(from the AISC Manual of Steel Construction)
posted by dforemsky at 7:08 AM on May 16, 2013
superposition works, as long as the deflection is elastic. Not sure why backseatpilot thinks they don't
Because if the pipe is threaded into a fitting at each end, the interaction with the fitting acts as a moment connection rather than a pin. Assuming that the support can be made absolutely rigid, the material of the pipe and fitting will be absorbing some of the torque loads in addition to bending and shear.
posted by LionIndex at 7:14 AM on May 16, 2013
Because if the pipe is threaded into a fitting at each end, the interaction with the fitting acts as a moment connection rather than a pin. Assuming that the support can be made absolutely rigid, the material of the pipe and fitting will be absorbing some of the torque loads in addition to bending and shear.
posted by LionIndex at 7:14 AM on May 16, 2013
Eh? I did say that superposition works. I'm just questioning the assumption of simply-supported ends, since if they crosspiece is being put into pipe fittings that's not simply-supported, it's a rigid/fixed boundary condition. Superposition still works there.
posted by backseatpilot at 7:18 AM on May 16, 2013
posted by backseatpilot at 7:18 AM on May 16, 2013
Response by poster: The ends are not threaded, and I'm using commercially available swing set joints - which is also what determined the pipe material. 2" schedule 40 actually has a 2 3/8" OD.
I assumed that my assumptions were incorrect for determining the ACTUAL deflection, but I'm going to look at the comparison between the cases to make my determination. Thank you all!
posted by flaterik at 12:33 PM on May 16, 2013
I assumed that my assumptions were incorrect for determining the ACTUAL deflection, but I'm going to look at the comparison between the cases to make my determination. Thank you all!
posted by flaterik at 12:33 PM on May 16, 2013
1: I have a copy of The Machinery's Handbook right here at my desk and can easily reference it for deflection data on very specific loading scenarios if you want to MeFi mail me. It confirms dforemsky's equation for that basic scenario.
2: I think modeling as simply-supported is fine. The deflection (or potential deflection) of the cross beam would have to be quite high for the rigidity/counterforce of the endpoint fixture to influence the dynamics appreciably. Plus, if you are modeling for strength, the act of tossing out the moments at the end is giving you a little safety margin implicitly.
3: If I am you, I model the current loading arrangement and solve for the highest deflection and the highest stress. Then when I design the new scenario, I set those levels as design criteria and find out what is necessary from a crossbar/loading arrangement.
Good luck man!
posted by milqman at 12:41 PM on May 16, 2013
2: I think modeling as simply-supported is fine. The deflection (or potential deflection) of the cross beam would have to be quite high for the rigidity/counterforce of the endpoint fixture to influence the dynamics appreciably. Plus, if you are modeling for strength, the act of tossing out the moments at the end is giving you a little safety margin implicitly.
3: If I am you, I model the current loading arrangement and solve for the highest deflection and the highest stress. Then when I design the new scenario, I set those levels as design criteria and find out what is necessary from a crossbar/loading arrangement.
Good luck man!
posted by milqman at 12:41 PM on May 16, 2013
I've got PATRAN/NASTRAN open and ready to go if you want an FEA solution :-)
Though as per the other's I'd use superposition. End constraints are probably not purely simply supported, but you could bound them by trying the fixed / pinned solutions.
posted by trialex at 4:55 PM on May 16, 2013
Though as per the other's I'd use superposition. End constraints are probably not purely simply supported, but you could bound them by trying the fixed / pinned solutions.
posted by trialex at 4:55 PM on May 16, 2013
Note that the swing set kits you linked require substantial concrete anchors at each leg. Don't expect to get away with skipping this part. The concrete is necessary to keep the frame rigid (see gjc's comment).
posted by ryanrs at 12:13 AM on May 17, 2013
posted by ryanrs at 12:13 AM on May 17, 2013
Response by poster: Thanks ryanrs! I have thought of that, and was hoping to get away with heavy duty 18" forged steel stakes and ratchet straps, one per leg. I should probably be drawing force diagrams for that.
This whole adventure is a bit safety 3rd, I know. I'm trying to avoid it jumping up to safety 4th or 5th, though.
I did my maths and the deflection on a 10' pole with two 200 pound people is 0.55"; a 12'9" pole with 3 200 pound people is 1.56". Kind of an alarming ratio, but still a worst case (IE if it's at the center, which it wouldn't be) deflection angle of 1.2 degrees. I... still have no idea how okay that is. But it's enough that I am looking for more info, not just jumping into it.
posted by flaterik at 12:32 AM on May 17, 2013
This whole adventure is a bit safety 3rd, I know. I'm trying to avoid it jumping up to safety 4th or 5th, though.
I did my maths and the deflection on a 10' pole with two 200 pound people is 0.55"; a 12'9" pole with 3 200 pound people is 1.56". Kind of an alarming ratio, but still a worst case (IE if it's at the center, which it wouldn't be) deflection angle of 1.2 degrees. I... still have no idea how okay that is. But it's enough that I am looking for more info, not just jumping into it.
posted by flaterik at 12:32 AM on May 17, 2013
Response by poster: It does occur to me that the straps would only prevent tipping, not keep the structure rigid.
posted by flaterik at 12:40 AM on May 17, 2013
posted by flaterik at 12:40 AM on May 17, 2013
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As to the dynamic load from centripetal acceleration, if your swingers traverse a 180 degree arc, reversing direction at the 3 o'clock and 9 o'clock positions, then the centripetal force will be 2*m*g. Add the static force of m*g from their normal weight gives a total of 3*m*g, or 3 times their normal weight. If they make a full 360, then maximum force at bottom dead center will be 5 times their normal weight.
Of course actual people won't be swinging in perfectly smooth arcs, so there will be jerks that increase peak forces above these numbers. On the other hand, the more flexible the pipe, the lower the force will be during these short term excursions.
posted by ryanrs at 12:58 AM on May 16, 2013 [1 favorite]