Engineers, help me out?

[Seggybop]

I have a fairly simple project for one of my classes and I figured this would be a good place to ask about it, as the professor encouraged us to base our design on any available source.

I need to construct a structural beam that spans an 80cm gap and supports a load concentrated on the center of the top of the beam. The beam must be made entirely from a piece of softwood measuring 2" x 4" x 30cm (don't ask me why they used the mixed measurements) and yellow wood glue.

It's easy to use the google and see many of the various balsa/toothpick/popsicle stick bridge projects high school students commonly do. These are somewhat useful to examine, but it seems like they primarily aim to duplicate real bridges. On an actual bridge the load is spread across a roadway on the interior section of the bridge, whereas with my project the load is concentrated on a single area on the top of the beam. Based on that, it seems a design intended to receive the load at that single area would be more effective than one meant to support a road.

So, I am wondering if anyone has any recommendations for the design of this beam? It is probably a very simple problem as far as engineering goes, but I'm not an engineering major and it seems to me that the best plan is to consult people who are. Images, links, books, anything would be useful. Thanks for any assistance.

[Darth Wong]

It's easy to use the google and see many of the various balsa/toothpick/popsicle stick bridge projects high school students commonly do. These are somewhat useful to examine, but it seems like they primarily aim to duplicate real bridges.

There's a good reason for that. Real bridge designs are the product of centuries of refinement.

On an actual bridge the load is spread across a roadway on the interior section of the bridge, whereas with my project the load is concentrated on a single area on the top of the beam. Based on that, it seems a design intended to receive the load at that single area would be more effective than one meant to support a road.

The only particular danger of a point concentrated load is increased shear forces and bearing forces, and that is almost never the primary failure mode for a load-bearing beam. The primary failure mode is almost always going to be bending moment, which is distributed along the entire length of the span regardless of whether the load itself is concentrated at one point or distributed evenly along it. Putting the load at a single point merely increases the bending moment, but the bending moment is applied all along the beam (tensile on the bottom half of the beam, and compressive on the top half of the beam).

So, I am wondering if anyone has any recommendations for the design of this beam? It is probably a very simple problem as far as engineering goes, but I'm not an engineering major and it seems to me that the best plan is to consult people who are. Images, links, books, anything would be useful. Thanks for any assistance.

Just remember that bending moment tensile stresses are a function of the moment of inertia of the load-bearing beam in the z-axis. Therefore, the cross-sectional area should be designed for maximum moment of inertia. That's why the classic I-beam has the shape that it does.

In short, unless there is some perceived danger of the load actually shearing through the beam, you want to distribute the strength of the beam evenly across its length rather than doing anything silly to put more strength in the middle, because the bending moment is distributed all along the length of the beam.

[aerius]

Cheat. Use good strong softwoods like Redwood, Douglas Fir, or Sitka Spruce instead of the cheap pine found in lumber yards. Make sure your block has a tight clear grain and is free of knots or other imperfections. Same thing with glue, use the good high strength stuff instead of the $1 bottles at Home Depot. Take your time, try to maximize the area of the glued surfaces, and firmly clamp things together to ensure a firm joint whenever possible.

[Howedar]

To be clear, you need to machine a bridge out of a 2x4" cross-section piece of wood? Or you have to build a bridge that fits in a 2x4" envelope?

If it's the former, you could easily build a box beam that's almost arbitrarily strong, enough so that no classroom equipment is going to break it.

[Darth Wong]

Also take care to make sure that the joints are very strong on the bottom half. The joints will fail on the bottom, not the top. You might even want to glue reinforcing bits of wood on the bottom of each joint, because that's where they're going to come apart.

[Darth Wong]

To be clear, you need to machine a bridge out of a 2x4" cross-section piece of wood? Or you have to build a bridge that fits in a 2x4" envelope?

He has to take a 30cm long 2x4, cut it into pieces, and glue the pieces together so that they will cover a span 80cm long. Just remember, Seggybop: you want the cross-section to be tall, not wide.

[Howedar]

Within reason - he's dealing with a structure where the microstructure (maybe mini-structure, if we want to get cute ) is going to be relevant to the small-scale loading. It wouldn't be hard to build a 10cm tall 1cm wide built-up structure and have it twist itself to death. I don't think I'd try making it much narrower than h/2 or something, just as a guesstimate.

And the obvious thought - don't cheat yourself and saw off a piece of a 2x4, which is in fact 1.5"x3.5" in cross-section. If you're allowed two inches by four inches, make sure you're actually getting that much.

[aerius]

Personally I'd lean towards something like this, but with some key changes. Get rid of the piece which encloses the bottom and make the sides a lot deeper, use the full 4" height of the board and make the beam 4" high and around 2"-3" wide. Unfortunately you'll turn half your block of wood into sawdust while cutting out the pieces unless you use a bandsaw with a really thin blade.

You'll probably want to make a test version first out of a cheap block of pine to make sure you can actually cut out all the pieces in the final version.

[Howedar]

That's a pretty terrible idea. The moment of inertia would be negligible compared to a nice wide-flange beam.

[aerius]

Problem is he's working with natural wood & normal glue. While there are wide flange beams made from wood, they generally require the use of strand oriented lumber and other engineered wood products along with high-strength epoxy adhesives.

[Howedar]

Yes, I'm quite aware of that. That doesn't change the simple fact that you've turned your beam the wrong direction.

[aerius]

Yup. I know the wide flange is better in theory, I did take a property of materials class back in my 2nd year of university. Problem is building a 4"x4" version is going to be next to impossible without running out of wood, 2"x2" is definitely doable though.

I've also forgotten 99% of what I learned in that class, so I'll be damned if I can come up with any numbers to support anything.

[Howedar]

It doesn't get much more basic that moments of inertia. It's not like I-beams have some inherit magic to them that makes them better - it's a function of the amount of material as far from the centroid as possible. One could easily build a box beam with a similar moment of inertia, and the fact that your retort was "it's hard to build an I-beam" rather than "yeah, my bad, we should just turn that box beam 90deg" demonstrates that you indeed haven't retained a whole lot.

[Ariphaos]

Is there a limit to the amount of glue you can use?

[Darth Wong]

Is there a limit to the amount of glue you can use?

I would be quite surprised if there is no weight limit for the finished beam.

[aerius]

I defer to your superior wisdom. I have a hard time remembering stuff I haven't touched in nearly 8 years and what I do remember often gets confused, to be honest I vaguely remember moment of inertia from my 1st year physics class and not my materials class. Electronics is the only area where I've retained a reasonable amount of knowledge.

[Seggybop]

There's a good reason for that. Real bridge designs are the product of centuries of refinement.

Trusses appear to be the most commonly used bridge design for these types of models. If I were to use a truss, would it be better to have a simple trapezoidal outline or a more complicated shape like this? Are vertical members important?

Or is there another basic design such as an I-beam more appropriate for this than a truss, despite its popularity?

If it's the former, you could easily build a box beam that's almost arbitrarily strong, enough so that no classroom equipment is going to break it.

Box beam as in a wooden box girder? Or a variation of an I-beam?

The strongest project from last year's class apparently survived up to 1800lbs and was some type of truss, but I don't know anything about its design beyond that.

Personally I'd lean towards something like this, but with some key changes. Get rid of the piece which encloses the bottom and make the sides a lot deeper, use the full 4" height of the board and make the beam 4" high and around 2"-3" wide. Unfortunately you'll turn half your block of wood into sawdust while cutting out the pieces unless you use a bandsaw with a really thin blade.

The constraints allow for extra wood to be used to make up for what's lost to the kerf, so luckily that's not a problem. I did get enough douglas fir to make around 3 of these beams, so I should be ok.

Is there a limit to the amount of glue you can use?

The finished beam's weight isn't supposed to exceed the weight of the uncut block of wood + 20ml glue.

[aerius]

The constraints allow for extra wood to be used to make up for what's lost to the kerf, so luckily that's not a problem. I did get enough douglas fir to make around 3 of these beams, so I should be ok.

That's going to open up a lot of possibilities. 4" beam designs of all sorts are now possible without worrying about glue failure.

[Darth Wong]

1800 lbs, eh? That's pretty impressive. When we had to do a wooden load-bearing project in first-year, we only hit about 400 lbs or so. Mind you, we were told to use balsa wood (which is shit) and the weight limit was really low, so we didn't have much to work with.

If the weights are approaching a ton, then the question of precisely how the weight is affixed to the beam becomes important. If it's concentrated on a small point, it could easily tear through, so you have to be careful what the middle of the beam looks like.

Assuming that it's affixed in some way that limits stress concentrations, you don't want to be too close to a real bridge. Remember that a real bridge always has a flat wide part in the middle for traffic to pass through, and you don't need that. Moreover, you just don't have a lot of wood to work with, so you can't afford to get too fancy.

As for the engineering, my old textbook says that Douglas Fir has a UTS of ~50 MPa (my first reaction was "whoa, that's way too low", but I'm used to working with metal; it's a reasonable figure), and a shear strength of 7.5 MPa.

Now I'll try to go from memory, so hopefully I haven't gotten too rusty over the years: the bending moment for a beam supported at both ends would be M=(P/2)*(L/2)=PL/4, where P is the load and l is the length of the beam. Therefore, at any given point along the z-axis, the normal stress will be PLz/4I, where z is the distance from the neutral axis and I is the moment of inertia. For a simple rectangle, I=bh^3/12, where b is the base width and h is height.

If we had a straight 80cm long 2x4 (I know you can't do that, but just for the sake of an example), this means that the bending moment for an 1800lb load would be roughly 1600 Nm. I for a 2x4 would be .05*.1^3/12, or 4.2E-6, and z would be 2 inches max (or .05m). Therefore, the stress at the bottom surface of an 80cm long 2x4, supported at both ends, holding up an 1800 lb load, should be roughly 20 MPa, or less than half of the UTS of Douglas Fir.

Of course, you don't get to use an 80 cm long 2x4, and an I-beam shape is not too practical with wood. Let's just say that you slice it up into three skinny 2/3" x 4" pieces (let's just ignore the possibility of torsioning the beam). This means your moment of inertia would be 1.4E-6 instead of 4.2E-6, so you're about 3 times weaker, which means you'd be looking at 60 MPa, compared to the 50MPa UTS figure in my old textbook for Douglas Fir. Of course, it's always possible that someone got some unusually strong wood last year.

Personally, I don't see how you could fabricate anything fancier than that if you're stuck working with a saw and a little 30cm long 2x4, but if you think you can do something fancier, go right ahead. Mind you, I'm curious how the beam is supported at each end, and how the weight is affixed to the beam.

[Howedar]

I'm a little less rusty on that class, and my book is probably closer at hand, and I didn't see any goofs in there.

[Beowulf]

You could shave it into thin strips and laminate it into a curved structure. I don't mean veneer thickness, but rather a bit thicker. It'd be longer than a straight beam, but possibly allow you to get a better strength for the bridge. This would be because the curve helps resist the bending of the bridge. I don't know the exact best shape to do it though. Of course, this assumes that the testing jig is able to resist horizontal forces.

I did the similar problem in HS Physics. Popsicle sticks and white glue. Weighed two pounds, and could support the teacher. Couldn't get enough weight to break it. Of course, there was no weight limit. Score was based on strength to weight ratio though. They changed the rules the next year to encourage more spindly bridges.

Also, it's measured by weight of the wood plus a dollop of glue. There's not much that really prevents you from using a longer piece than can be cut out of a 2"x4", unless they specifically say that all pieces must be cut out of a 2"x4"x30cm.

[Seggybop]

Personally, I don't see how you could fabricate anything fancier than that if you're stuck working with a saw and a little 30cm long 2x4, but if you think you can do something fancier, go right ahead. Mind you, I'm curious how the beam is supported at each end, and how the weight is affixed to the beam.

The test jig won't provide any support, so the beam must stand on its own. The force is being applied to a small area in the center of the top of the beam by a press. Unfortunately I wasn't given the exact dimensions, but it's a circle several cm in diameter.

Weak attempt at diagram:

Code: Select all

                 [] <-load, centered
   =============== <-beam
[][]                          [][] <-stand

You could shave it into thin strips and laminate it into a curved structure. I don't mean veneer thickness, but rather a bit thicker. It'd be longer than a straight beam, but possibly allow you to get a better strength for the bridge. This would be because the curve helps resist the bending of the bridge. I don't know the exact best shape to do it though. Of course, this assumes that the testing jig is able to resist horizontal forces.

Yeah, I thought about making a laminated arch, but since it's not going to be supported horizontally it would probably end up flattening out like a leaf spring.

Also, it's measured by weight of the wood plus a dollop of glue. There's not much that really prevents you from using a longer piece than can be cut out of a 2"x4", unless they specifically say that all pieces must be cut out of a 2"x4"x30cm.

Unfortunately, they do, so any long pieces are going to need to be made from laminated strips.

[Seggybop]

I spoke to some classmates and the diameter of the pressure-applying implement is actually even smaller than I thought. Closer to 1" diameter. This is much less area than was used last year when they had bridges supporting almost a ton, and it seems like with the pressure so concentrated the chance of it poking straight down is way higher.

[Darth Wong]

I spoke to some classmates and the diameter of the pressure-applying implement is actually even smaller than I thought. Closer to 1" diameter. This is much less area than was used last year when they had bridges supporting almost a ton, and it seems like with the pressure so concentrated the chance of it poking straight down is way higher.

Try doing the math on bearing stress and shear stress if you're worried, using the 50 MPa UTS I gave you earlier as a benchmark (we'll not worry about safety factor since I assume these things are being tested to failure anyway).

[CJvR]

How about an arc? It would distribute the load better that a simple beam design. IIRC you can bend heated wood fairly well.