# Beam Theory Explained - How Spars Work

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#### BBerson

##### Well-Known Member
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BB , vertical sheer and horizontal shear are equal ( for equilibrium) and as such not separated. In other words if you size web for one you are good for other.
Thanks.

#### Aviator168

##### Well-Known Member
Thanks for the post Billski. Now I am able to plot the shear and the moments diagram, and calculate the stress at different point of the beam. How to figure out number of layer of FG needed for the cap at any given point?

#### wsimpso1

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Post #8, where you stated the following:

More a case, of me over simplifying what you stated. The way I originally read it, I assumed simple addition of the stresses.

Tim
Close. Superposition in stresses is to calculate the stresses for each of the loadings and add the stresses at each location to get the total stresses at each location. Example:

A load in the up direction on a rectangular beam results in a positive axial stress along the bottom edge of the spar tapering linearly to an equal magnitude but negative axial stress along the top edge of the spar - max tensile stress is. This also makes a parabolically distributed shear stress with the maximum at mid hieght. You can do a numerical map of the cross section and find the maximum von Mises stress;

Another load in the aft direction on the same beam results in a positive axial stress along the forward edge of the spar tapering linearly to an equal magnitude but negative axial stress along the aft edge of the spar - max tensile stress is. This also makes a parabolically distributed shear stress with the maximum at mid-point between front and back of beam. You can do a numerical map of the cross section and find the maximum von Mises stress;

To get the stress picture when both of these act at the same time, you can do one of two things:

You can find the resultant shear and moment diagram for the combined load, find the shear distribution for a rectangular cross section rotated onto one of its corners, find I for the same rotated cross section, and search out the von Mises stresses in it, or;

Draw a little map of the cross section, label each of the corners, add the axial stresses at each corner, add the shear stresses at each corner, compute the von Mises stresses at each corner, and then if you really want to know the worst spot, run a search to map the von Mises stresses from calculated sums of shear, axial, then von Mises stresses around the cross section. You determine the total axial stress in each direction first, and the total shear stress, then compute von Mises. Like wise, the deflection of the beam can be calculated in each loading, and added, either at the ends or mapped over the whole beam. You gotta mind your directions, signs, and magnitudes when adding...

The beauty of superposition is that it allows you to analyze a complicated load situation using a pencil and paper and calculator or simple spreadsheet without doing all sorts of gymnastics or FEA to get there. And if you are doing FEA, and want to know if you can believe the numbers, run some cases that you can also do analytically and compare them... You would not believe the wild results I have seen from pro FEA folks. Took a while to convince them they had errors. You can actually run some pretty complicated cases using superposition. This was about two pages in my Mechanics of Materials text.

Billski

#### wsimpso1

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I gave up on engineering because I wanted to know why stuff worked not just learn what I was told too. I don't learn without a reason, so I was just treading water. GT is the family school, but I did not take the shot.
Altogether too common a story - the prof either does a lousy job of explaining what this stuff does or does not talk about it at all, then wonders why many of the students that stay are not terribly interested. Many classes are just teaching tools that have wide applicability, but no one bothers to talk that part up. Integral calculus, linear algebra, differential equations. Useful tools... Many Math profs in the US never tell you what you can do with the stuff, because once you are doing something with it, it becomes Engineering, and the Math profs view engineering as vulgar money grubbing. And I remember my class mates in k-8 that hated word problems - they were "hard". These weiners were complaining that their arithmetic problems actually were real... Take your pick, a lot of folks do not like things that require rigor. And rigor is required for learning to use tools that can be complicated.

Luckily, I learned my first bit of integral calculus in a physics class, where we integrated acceleration wrt time twice to get travel. By the time I got to statics, it got interesting with loads to move things like how much winch load is required to pull a truck out of a ditch.

Billski

#### wsimpso1

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I don’t think this really got answered so I’ll chip in.

Firstly, just to clarify, there is shear load, shear stress and shear strain:

Shear load – The load (force) acting perpendicular to the axis of the beam. This is the lift when considering the vertical forces acting on a wing spar.
Shear Stress – The internal shear forces acting within the material of the beam
Shear strain – The deformation of the beam material in response to the shear load

Now we’ve cleared that up, the ‘horizontal’ and ‘vertical’ shear BBerson is asking about is the shear stress...
This describes shear stress nicely. Shear stress IS the four equal force loads along the sides of a square element, or the two equal but opposite linear forces on the diagonal.

Look up Mohr's Circle. It relates the how shear and axial stresses can be related within a loading case.

Billski

#### wsimpso1

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Thanks David.
The books I have largely ignore vertical shear. (J.E. Gordon, Strojnic and others)
I am thinking that the web thickness is sized for horizontal shear, so perhaps it will be adequate for vertical shear, and so vertical shear can be ignored. At least in metal.
The web usually has vertical angles and ribs. These are in compression, I assume, and may or may not be related to vertical shear?

I will look for the book. (but don't want to complicate beam design more than needed)
In most beams shear is small and bending is big. Spending little effort on shear encourages folks to not fully consider what is happening and make the webs undersize. Happens all the time.

It sounds like the authors did a poor job on explaining shear. Shear IS all four loads along the edges of a square element. Shear is that square element being deformed into a diamond shape.

The magnitudes are the same on a square element, but in some materials, the shear strengths are different in different directions. Might that be what was going on?

Billski

#### wsimpso1

##### Super Moderator
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Thanks for the post Billski. Now I am able to plot the shear and the moments diagram, and calculate the stress at different point of the beam. How to figure out number of layer of FG needed for the cap at any given point?
Composites is coming.... Basic cap sizing is the same as in metal:

Get an estimate of max bending moment times FOS (2.0 in man rated composite structures);
Make an estimate of how far apart the centroids of the caps will be (educated guess);
Divide the moment by that dimension to get cap load;
Divide the cap load by the compressive strength or the tensile strength (which ever is smaller magnitude) to get cap area;
Refine with cap dimensions new distance between centroids, and the contribution to bending stiffness of web plies wrapping from web to caps.

Once you get the caps and shear web basic size, you will usually find out that the web fails early, and that when you start iterating the design, you will find that you get to strength at lowest weight when both the caps and the web have been enlarged. Much heavier to just beef up the web alone. So caps end up lower stressed while the web ends up with part of it being closer to failure than anything else.

Billski

#### BJC

##### Well-Known Member
HBA Supporter
Altogether too common a story - the prof either does a lousy job of explaining what this stuff does or does not talk about it at all, then wonders why many of the students that stay are not terribly interested.

Billski
My conclusion, born out by my later experience in hiring some engineering professors with PhD’s (who, to their credit, realized that they would become better professors if they got some experience) for summer jobs, is that they didn’t really understand practicable applications. The profs learned from our working engineers, and the engineers learned from the Profs. It turned out to be a real win-win.

BJC

#### BBerson

##### Well-Known Member
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I make little spars out of balsa or cardboard and test bend them to see the force distortions.
Gluing threads on the web at 45° really helps.

#### Aviator168

##### Well-Known Member
Gluing threads on the web at 45° really helps.
The threads make the part that's in tension to fight the part in compression.

#### Aviator168

##### Well-Known Member
Composites is coming.... Basic cap sizing is the same as in metal:

Get an estimate of max bending moment times FOS (2.0 in man rated composite structures);
Make an estimate of how far apart the centroids of the caps will be (educated guess);
Divide the moment by that dimension to get cap load;
Divide the cap load by the compressive strength or the tensile strength (which ever is smaller magnitude) to get cap area.

Billski
Thanks again for the clear explanation. I will wait for your composite post to figure out the number of layers of FG given the cap height.

BTW. I learnt most of my mechanical and aero stuff from here and lectures on youtube.

#### Aviator168

##### Well-Known Member
Useful video

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#### tspear

##### Well-Known Member
Many Math profs in the US never tell you what you can do with the stuff....

Billski
That's funny for me. My math professors spent all sorts of time showing how the math is used in practical applications, mostly engineering or physics.
On the flip side, my engineering professors didn't explain anything and did not show any practical applications; and were super boring to boot.

Tim

#### wsimpso1

##### Super Moderator
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My conclusion, born out by my later experience in hiring some engineering professors with PhD’s (who, to their credit, realized that they would become better professors if they got some experience) for summer jobs, is that they didn’t really understand practicable applications. The profs learned from our working engineers, and the engineers learned from the Profs. It turned out to be a real win-win.
BJC
I have twice been a guest lecturer at University of Michigan College of Engineering, telling graduate students in a Mechanics of Composites about designing the composite structures for my homebuilt airplane. I am preparing to be a guest lecturer for a senior/graduate level class in airplane configuration and design at Western Michigan University later on this winter. A dose of reality is what I have been told is the purpose. Both profs have told me this and I do my best to help with that mission.

Billski

#### wsimpso1

##### Super Moderator
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Log Member
That's funny for me. My math professors spent all sorts of time showing how the math is used in practical applications, mostly engineering or physics.
On the flip side, my engineering professors didn't explain anything and did not show any practical applications; and were super boring to boot.

Tim
What university? Math profs that are not purists have become rare...

#### BJC

##### Well-Known Member
HBA Supporter
I have twice been a guest lecturer at University of Michigan College of Engineering, telling graduate students in a Mechanics of Composites about designing the composite structures for my homebuilt airplane. I am preparing to be a guest lecturer for a senior/graduate level class in airplane configuration and design at Western Michigan University later on this winter. A dose of reality is what I have been told is the purpose. Both profs have told me this and I do my best to help with that mission.

Billski
Bill, you have twice the experience as a guest lecturer that I have. I enjoyed discussing process control engineering with engineering students at the University of Florida. It would be good if there were more interaction between academia and industry in the USA. Apparently, such interaction is common in Europe.

BJC

#### wsimpso1

##### Super Moderator
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Bill, you have twice the experience as a guest lecturer that I have. I enjoyed discussing process control engineering with engineering students at the University of Florida. It would be good if there were more interaction between academia and industry in the USA. Apparently, such interaction is common in Europe.
BJC
The professors are people I know through flying. Maybe that makes a difference... but I still think it is important to do this sort of thing when you can.

Billski

#### tspear

##### Well-Known Member
What university? Math profs that are not purists have become rare...
Prince George's Community College in Maryland, then Towson State University outside Baltimore.
Almost all my math was at PGCC, since I finished Calc 1&2 as a junior in highschool (DeMatha in MD)

Tim

#### choppergirl

##### Banned
Which is better on an airplane fuselage, an "X" where the center is joined (like with a through bolt), or an "X" where it's not joined at the center.

The former, it shortens the span of the cross supports in half and keeps them from flexing and makes it more rigid

The later it allows it to flex, and flexing is a good thing against breaking, like a reed flexes in the wind but does not break (if it were more ridgid).

On bridges, I see both used. I guess it depends on what you need or expect to happen most or want to prevent against most in that area.

Which is stronger, an X shape, or a V shape? I'm thinking V shape, because triangles are super strong, but an X shape would spread out the loads further and not put all the load a single bottom point. But it would flex more (if unconnected in the center), where a V would not.

And then there are I's, H's, U's, C's, M's, L's, N's, T's, Y's, Z's, A's, W's, and O's...

My ABC's of geometrical structural engineering... the math don't interest me much, but I do like the geometry.

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