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Physics/Small swinging bridge construction

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QUESTION: I want to build a walking bridge across a creek on my property. I plan to use galvanized cable for the supports and pressure treated 4x4s for the walking surface. The bridge will be 4 feet wide. I will drill holes in the 4x4s and pass a cable through each end of the 4x4s. I will put a nut on the cable in between each 4x4 to space the 4x4x out inch. This will reduce the number of 4x4x needed and allow for better water run off and prevent moisture from being trapped between the 4x4s.  4x4s are a little over kill, but will not require fasteners to attach to the cable and will have better longevity compared to a 2x6.  The walking surface of the bridge will be 35 feet but the total span of the cable will be 55 feet as I do not want the anchor point to be too close to the edge of the bank.

I have calculated my bridge to weigh 2,400 pounds.
The breaking strength of 3/8 inch galvanized cable is 14,400 pounds, but the safe working load is 20 percent. I have also been told to use 10 percent when personnel are suspended. This brings me down to only 1,440 pounds per cable or 2,880 for the 2 cables I plan to use.  I am not far off with my weight but there seems to be quite a bit of safety built in with my working load being only 10 percent of the breaking strength.  I also plan to use the 3/8 inch cable as railings and attach them to the bridge as well for further support.

I am pretty comfortable with all of this except I believe the numbers I am using are for cable being used to support objects vertically rather than a load on the top of a horizontal cable. I think there is a physics equation that I need to use to figure the force on the cable instead of just using the weight of my bridge.  I planned to stretch the cable tightly to minimize sag, but then it occurred to me that this will increase the force on the cable and decreases the reserve the cable has to support the bridge. I am interested in an equation or any other advice

Thanks

Dave

ANSWER: This is a question more for an engineer, as I'm (disclaimer) in no way an expert on building bridges.  I can offer you personal advice as someone who grew up with many such bridges and a boy scout, but if you want a professional opinion then you want an engineer.  Keep in mind that this would've been far easier with a picture, I wish experts could attach them in responses.

First, to the major dimensions, 55 ft long and 4 ft wide.  From an engineering standpoint, I rarely see things this small and static designed for more than a safety factor of 5 when I taught engineers.  It's not a nuclear reactor (which are designed for safety factors of 100 in some cases and still get hit with earthquakes), so I wouldn't worry too much about it if it can hold 1,440 lbs...each.  The cables at the ends will have a distributed load. You mention an odd geometry, stretching across the bridge?  It's pretty long for a suspension bridge...where are you anchoring this cable?  

You need torque equations to find the tension in the cable.  I'm not sure what you mean with 4x4's because you provide no details about structure of the plank itself aside from dimensions and spacing...are there rails?  Is it just a bunch of 4x4's nailed/fastened together with spacers?  What's the deal?  

Yeah, reading over it again, I need a sketch or some extensive details before I can answer this.

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bridge plan
bridge plan  
QUESTION: I am primarily concerned with choosing the correct size cable because I do not know how to calculate the load on the cable other than adding up the weight of the walkway and people walking on the bridge. I have provided a sketch of the plan. I am not much of an artist, but I think the sketch will answer your questions. If it is a torque equation I need I hope maybe you can provide the equation.  

Thanks

Dave

ANSWER: You need some give in the cable (it won't ever stay flat like that) in order for it to have vertical components to the tension.  In other words, you need to make the ends way taller and let that top support cable dip in the middle.  The way you have it drawn, the whole bridge will dip like that naturally.  Also, you need to figure out what's going to connect the 4x4" sections of wood on the sides (just cable might work, but you'll need spacers at least).  Getting tension in that cable will be the really interesting trick, as the cable ages, it will need to be tightened.  I think your cable idea might work fine, but if not then you can always double-up any particular cable.

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QUESTION: If I use the 4x4s I will run the cable through the hole, space them out with the nut, pull them tight with a come along, and then put a spring and rope clip on the end to hold them tight together.  The spring will allow for expansion and contraction of the 4x4s according to moisture.

I will probably use some turn buckles so that the cables can be tightened later if needed. Getting the bridge across the creek after the 4x4s are installed on the cable will be interesting, but I have tractors and come alongs so it should be doable.  It seems like the price to strength ratio becomes less efficient as you move higher than 3/8s cable so I think doubling up the cable will be more economical.

I think I will use 2 top support cables with sag as well as a hand rail cable. I will use the top cable like a suspension bridge to lift the bottom cable up close to parallel. I do not want the bridge to sag much or it will be in danger of being in contact with the water during the rare flood where it may be close to the top of the bank. Do you have any advice for the top cable height at the post on each side of the creek? It seems to me that a higher post is supposed to put somewhat less stress on the cable but will require a better anchoring system.

Answer
I'd say that if you got the connectors to go up to make a 20 degree angle with the bridge, then the sin(30)=1/3 (close, not exact).  So for vertical tension you only need 1/3 of the full tension, but that's just for the 20 degree parts.  The rest of the bridge with higher angles near the anchor points will have a more favorable tension applied to them, vertically.

Your plan seems good, aside from the come-alongs (might not be strong enough, you might need a real nautical crank or something).  But you have reached the end of what basic physics can help you with.  Send a picture when you build it, bridge building is a classic problem in physics/engineering!

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Dr. Stephen O. Nelson

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I can answer most basic physics questions, physics questions about science fiction and everyday observations of physics, etc. I'm also usually good for science fair advice (I'm the regional science fair director). I do not answer homework problems. I will occasionally point out where a homework solution went wrong, though. I'm usually good at explaining odd observations that seem counterintuitive, energy science, nuclear physics, nuclear astrophysics, and alternative theories of physics are my specialties.

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I was a physics professor at the University of Texas of the Permian Basin, research in nuclear technology and nuclear astrophysics. My travelling science show saw over 20,000 students of all ages. I taught physics, nuclear chemistry, radiation safety, vacuum technology, and answer tons of questions as I tour schools encouraging students to consider careers in science. I moved on to a non-academic job with more research just recently.

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Ph. D. from Duke University in physics, research in nuclear astrophysics reactions, gamma-ray astronomy technology, and advanced nuclear reactors.

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