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Engineering: Calculate Beam Loading (Intro)

As a simplistic introduction to beam loading, this article discusses the complications in calculating stresses and deflection.  While it’s pretty easy to calculate beam loading for simple theoretical cases, very few beams are actually simple.  A trailer frame, for instance, is a series of simple beams, but when we put them all together, it becomes much more complex.

Gantry Crane Failure Modes
Comparison Analysis: Steel vs. Aluminum For Trailer Frames & Crane Beams

Even the beams that seem easy, like a gantry crane top beam, are not so simple.  The image is from a previous article comparing Steel and Aluminum.  It shows stress rising near the main beam ends where simple theory says the stress goes down.  The extra stress comes because deflection of the top beam is trying to bend the leg.  While the connection is simple enough, it certainly makes an accurate full analysis more complex.

Don’t get discouraged.  Even with all the complications of reality, there are ways to get “close enough”, and mostly, that’s “good enough”.

Start With Beam Loading Concepts

Let’s emphasize, this is a simplistic introduction.  While reality is usually complex, the basic theory concepts still apply, so we’ll start with of basics to calculate beam loading.  We’ve touched on many of these concepts in other articles, so follow the links for more explanation.

Loading

Beam Loading IllustrationFirst, understand the beam loads.  What are the forces “trying” to do to the beam?  While the discussion is in “Choosing The Best Trailer Frame Material“, this image of bending, shear, and twist shows the concepts.  The beginning to calculate beam loading is knowing the forces, and their directions.  Do they “Bend” the beam?  Twist the beam?  Or, try to “Shear” it?  Often, it’s a combination.

Shape

Second, what does the beam shape do to resist the forces.  As in the discussion on “Beam Shapes To Build With“, the shape makes a big difference in how it handles forces.

I-Beam ProfileOne example.  The I-Beam is commonly recognized for it’s strength, and rightly so.  It’s a shape by engineering to handle big loads with less total weight.  YET, that statement is misleading.  The I-Beam does indeed handle bending loads very well (in the vertical direction looking at this image), but it doesn’t do so well with side loads (horizontal in the image), or twisting loads.  The direction and type of load matters with each shape.

Material

Material properties are the same for a given material no matter what the beam size or shape.  To calculate beam loading, we use these properties defined in mathematical terms by physics of the material.

One example is density.  How much does a given volume of the material weigh?  In metric terms, Steels are in the range of 7800 kg/m^3.  That means a block of steel that is 1 meter by 1 meter by 1 meter, is 7800 kg.  A block of aluminum the same size comes in at ~2700 kg.  Magnesium is more like 1700 kg.  Density is a property of the material, and it just is.

“Modulus of Elasticity” is a property we use in beam loading to calculate deflection (bending).  We often see it as “E” in equations.  To compare stiffness of materials we look at this number.  For instance, the “Modulus of Elasticity” of steel is roughly 30,000,000 psi.  Aluminum is roughly 10,000,000 psi.  Magnesium is 6,500,000 psi.  (English units.)  These articles comparing “Compare: Steel vs. Aluminum” and especially in “– Part 2” show how deflection is a big part of beam loading calculation.

Strength

We express Mechanical “Strength” in terms of “stress”.  Forces to a beam create a reaction that we calculate as stress.  If the stress exceeds the strength, then the material shape changes.  Sometimes that’s good, like if we’re making brackets and we want a new shape — yet, sometimes it’s bad like when a beam fails.  (See these crane failure issues.)

Material properties (above) don’t change with size or shape.  They also don’t change much with various alloys.  For example, raw stock steel has about the same density as high strength tool steel.  Same for aluminum alloys.  On the other hand, strength is very different for various alloys.  It also changes with temper — we’ve discussed weakening material near a weld.

To calculate beam loading we need to know the material strength, and in particular, the strength that is important to our loading situation.  The ultimate or tensile strength, for instance, is not so important when analyzing a trailer frame.  The frame members will bend long before they break, so knowing the “Yield” (bending) strength is the key.

One more, “Fatigue” strength, becomes important over time.  Unfortunately, fatigue numbers are not published because they depend on many things beyond the actual beam or material.  While we must be aware of it, calculating fatigue is beyond the scope of this article.

Size

This may seem a statement of the obvious.  Size is important to calculate beam loading stress, and it interplays with the Shape.

So, how do we quantify size for calculations?  We do it with 2 properties.

Graphic to Calculate Beam LoadingFirst, and easiest to understand is the “y”, or distance from the neutral axis.  For bending loads, typical of a trailer frame, we can use this image (from the article about welding spring brackets).  Looking at the force arrows (red and blue), we see the center of the beam has no bending load.  In this beam, because it is symmetric, the “neutral axis” is the center of the beam.  That’s not always the case, and angle iron is a good example.

The “y” (some literature will use a different variable) is the distance from the neutral axis to the point of analysis.  Since the highest forces are at the farthest point out (looking at a section of the beam) we usually analyze there.  We call it “y-max”.

Second, is the “Area Moment of Inertia”  or sometimes called “second moment of area”.  Often in equations we see it as “I” (capital i).  We won’t get into detail (you can find the math details here) because it gets complicated, but it has to do with the amount of material and it’s distance from the neutral axis.

I-Beam ProfileFor an I-Beam, the bulk of material is at the extremes (near y-max), which gives the shape a larger “I” for it’s total weight and size.

Remember, it’s a combination of shape, and size.  Width matters as well, and so does thickness.  Also, if you stack beams, the “Area Moment of Inertia” changes to take in the full stack.

Simple Engineering Equations

Now we look to calculate beam loading.  We have a little knowledge about the loading, material properties, strength, size and shape.

Simple Beam Loading GraphsSince simple is a good place to start, we’ll use a solid beam on simple end supports.  Then, a single center load.  The top portion of the graphic shows the beam and loads.

The graphic includes some equations to calculate beam loading.  The Shear graph (Blue) shows the way we represent what’s happening with forces along the beam.

The next graph is a representation of moments, or bending forces along the beam.

Finally, there is a deflection graph.  (There are a few steps between the Moments calculations and the deflections, but they are beyond the scope here.)

This is a very simple case.  Actually too simple for most real life situations, but it gives the idea.  So, what does it mean?  If this beam is Titanium, 1″ x 1″ solid (25.4 mm x 25.4 mm), 6 ft long (~2 m), having a Yield Strength of 35,000 psi (241 MPa), then it would support a load of 324 lbs (147 kg) before before bending permanently.  The center would deflect just over 1.7″ (43 mm) prior to yield.  This is theoretical, of course, because it is based on published numbers.  In reality, it might be a little more or a little less because things are never perfect, but it’s close.

Of course, simple basics do not take in motion, bouncing, or anomalies which happen in real life.

Calculate Beam Loading For A Trailer Frame

Now for a question we are asked fairly frequently.  How do you calculate beam loading for a trailer?

For simplicity, let’s use the example from the article “Where Does The Axle Go” since it already has the equations for calculating forces and position.  Here’s a review.

Forces to Calculate Trailer Beam Loading

Loading Summary:

Tongue Length = 42″
Bed Length = 96″
Frame Weight = 450 Lbs
Distributed Load = 2250 Lbs
Toolbox Weight = 300 Lbs

Tongue Load = 360 Lbs (12%)
Axle Load = 2640 Lbs @ 94.3″ from ball.

Calculate Trailer Axle Position

Details for these numbers are in the the article “Where Does The Axle Go“.

So, that’s the load condition.  Now, here is a graph (below) showing how the above loading applies to the trailer.  This is like the loading graphs above, but here the 3 graphs are together with scaling so it’s easy to compare from line to line.

Graphs of how to Calculate Beam Loading

For brevity, we’ll skip the equations for these graphs.  They are like the equations above, but with more complexity having multiple loads, various load types, and not just at the ends.

In this second graph, we don’t show deflection, but we do show “Stress” for the trailer frame.  The Stress line highlights areas of transition like where the deck meets the tongue.  Since stress is a function of beam size and shape, this graph assumes beams already.  However, that’s getting a step ahead.

Actually, we use the forces and moments to choose the beams.  By applying beam and material properties, we can choose beams that will handle the loads.  The “Stress” line is for the beams in this example, but the choosing process is not so simple, and is therefore beyond the scope of this article.

Understanding the Engineering

The above calculations are the beginning.  We start with a loading graph showing the forces.  The forces UP, must be equal to the forces DOWN.  (See the shear diagram.)  Forces also must sum ZERO when calculating the force at the appropriate distance.  (See calculations in the axle position article.)  We call it summing the moments.  If these two conditions are true, then we’ve probably done it right and the other calculations will work out.

Next is the shear forces.  It’s kind of a sanity check that we’ve done the load calculations correct.  (Blue Line in graph.)  Using a graph is a simple way to see the relative magnitude of the forces, and it makes a good guide in thinking about trailer loading.

Then, the moments graph.  (Green Line)  You can think of it as bending, but that’s not totally true.  For beams like those on a trailer, bending is the primary condition, so we must really pay attention.  While it’s tempting to just look at the high points, the areas where loading transitions from one set of beams to another is super important.  For instance, the point where the bed ends, and the tongue continues.  From our experience, that is the weakest point for most trailers.

While the above is a nice way to look at forces, in reality things are much more complex.  Even in loading, because the example is static (not moving).  When things move, dynamics really change the game.

More To Calculate Beam Loading

All the above is a first step to selecting trailer frame beams.  We said the next step is choosing appropriate beams, but that’s not quite true.  The next step is repeating this process for all potential load conditions.  We did a simple, static evenly distributed load — like filling the trailer with water where the load is even all over the deck.  That’s interesting, but not so realistic.  Even when you haul sand, there is usually a hump in the middle which makes the load not quite evenly distributed.

So, what about the case of hauling a RZR?  A small tractor?  Or an ATV?  Since the tires contact the trailer deck at only 4 points, it gives very different forces.  We must calculate the beam loading for this case too.  If it drives it onto the trailer, all the weight is at the back, and that’s another a load case.  We need to do yet another beam loading calculation.

The loading list goes on.  How about when a refrigerator is at the front of the trailer, with a washer and dryer right by it?  (Nothing at the rear.)  Yup, another set of calculations.

When the calculations are done, the next step is thinking about how realistic they are, and what modifiers apply.  For instance, hauling sand you will likely hit a bump while driving down the road.  We need to compensate with dynamic calculations.  We probably won’t hit a bump while loading an ATV.  However, there is probably some bouncing, so our dynamic calculations for that are different.

One more thing, the above is a 2D analysis, but most trailers are 3D . . . Hopefully you’re getting the idea.

Choosing Beams

As much as we’d like it to be simple, it’s not.  Unfortunately, that’s why many trailer manufactures don’t do engineering.  Yes, that’s the truth of it.  Most just slap the beams together based on experience, copying, and what “feels” good.  It works for most, until it doesn’t.

As an Engineer doing this a long time, I probably see the worst of it.  People contact me, so I see it, but there’s not much I can do.  Once a beam bends or a weld breaks, the time for engineering is long past.

Anyway, we’re getting off topic.  Beam choices include a melding of all the load cases with appropriate dynamics and situational mods.  Yes, safety factors enter as well.  All of it establishes the needs, then we can then select beams.  Basically, we calculate the stress of any given beam (shape, size, material), then compare the stress against the material strength.  If the stress is lower than the strength, then it will probably work.  If the predicted stress is higher, the beam may fail.

That’s the essence of how to calculate beam loading.  While it is reasonably easy for simple beams and single use cases, it complications come fast when dynamics and loading cases overlap.

Optimize For Light Weight

One word of caution.  A frequent goal with trailer design is that of minimizing weight.  It’s a good goal, but it’s deceiving.  If the use cases are well defined, then greater optimization is appropriate.  However, optimizing for one use case, will leave other areas vulnerable.  I’ve seen a lot of “too light” trailers that didn’t make it.  If you’re looking for light weight, make sure you don’t sacrifice functionality for the perception of “light”.  There are ways to achieve the goals without taking short cuts.

Also, think about the real benefits.  It’s pretty easy to get sucked into the numbers, like weight, in a design.  Ask yourself about the value, like what is the value of 50 lbs?  If that is the difference between a frame you are confident in, and a “light weight” frame that might be sketchy, then is 50 lbs worth it?  What about 100 lbs?  For a trailer that’s designed to carry 3500 lbs, a slight increase in frame weight is trivial compared to the cost of a frame that fails.

I suggest a design to handle the difficult tasks that will inevitably come.  There is always a practical balance, so keep that in mind if you want light weight.

Let The Computer Calculate Beam Loading

The examples here are simple, and 2D.  In reality, analysis of a trailer frame is much more complex, which is why we use special CAD tools to assist.

Finite Element Analysis on the Tiny House FoundationSometimes you’ll see colored images like this on our website.  They come from FEA, Finite Element Analysis, which does all the complicated beam calculation for us.  Unfortunately, it is loading case at a time, so a full analysis is time consuming — especially though iterations for optimization.

The big advantage is it will calculate beam loading in ways the above methods will not.  It makes complex easy.  While this is a great tool, it has one major flaw — garbage in equals garbage out.  It will give pretty pictures easily, but the inputs must be right, or the outputs are trash.  Just food for thought.

Using tools and engineering knowledge we produce the best trailer plans on the market.  We do it for the trailer plans on this site, because it’s the right way to design.  We do it for all the custom trailers we design as well.  That’s one of the values you get at Mechanical Elements.

As a side note, one of our competitors claim they have the “only” engineered trailer plans — Now you know the truth.  They sell on several websites, but it’s all the same plans.  I won’t knock the engineering of others, but let’s just say those plans don’t meet my standards.

Build with confidence, that’s what we say.  Good luck in your design endeavors.

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