The Secret to Trusses

Trusses are simply built out of triangles - and they are actually defined as such.  Some member will be in tension and some members will be in compression (about half).  The trick, and at times difficulty, is to determine which members are in tension and which members are in compression.   We can simplify things by recognizing there are only two types of triangles in trusses.  One that has two tension members, and one has only one tensions member (tension = rope, compression = wood in below image)...

If I extend this into a longer truss and simply replace the compression member with the tension member, we have...

Do you see the two types of triangle?  Also as you can see, the bottom chord is in tension and the top chord is in compression.   The trick to determining which diagonal or vertical member is in tension or compression, is ask if they follow the same curve as a hanging chain would between the two end points.  If they do, they are in tension, if not, they are in compression.

This is a stable truss for gravity loads...

What is really interesting about changing out the sticks for strings when they are in tension, is the fact that the entire truss can now be folded!   Check it out...

Think about how this can be extend to designing and building folding architecture, temporary construction, or tents etc.

Build a Lamella Dome

Lamella structures are created using many pieces of the exact same member shape and connection.   So 1000 pieces of something and stacked in a certain manner...

You can imagine it can get pretty big depending on the size of the individual beams and the quantity.   Here a pic with my RISD class a few years back that we built together from sticks I cut in my basement and eye hooks to temporarily hold the pieces together...

I wrote about the physics of this system as well... Published “Infinite Load Path?” October 2007 Structure Magazine if interested.

The Perfect Arch

For a arch to be prefect, it must be shaped such that it is in uniform compression.   Basically the inverse of a hanging chain.   So we can build what Robert Hooke describes in 1675 "as hangs the flexible line, so but inverted will stand the rigid arch".  So, we can hang strings from a sheet of plywood, trace and cut... As the thin shell innovator Heinz Isler would say "One does not actually create the form; one lets it become, as it has to according to its own law".  Also, this is an exceedingly complex mathematical shape called the hyperbolic cosine or catenary.

So we cut ...

Remove...

And build...

Build the Golden Ratio

Here is how you can build the most excellent rectangle ever conceived...

You start with a square, and you use one edge center point as a centroid of the quarter circle as shown above.   Then you can trace on top of this rectangle spiral growth rings...

Archimedes and Balance Beams

You can introduce concepts of moment and equilibrium by building a simple balance beam...

The weights times the distances on each side of the fulcrum have to equal.  I added nails at 1" spacings and use washers for the weights...works well.  And it is fun for the kids (Anders and Kinan in background)... force times distance has to be equal on each side.

Cable Y-Hangs and Pulleys

You can imagine a person hanging from a cable that forms a Y in shape between to cliffs.  What are the forces in the cables?   Well we can build this by adding a wight in the middle of a system, and measuring the cable tension by hanging other weights off pulleys like so...

The more weight you add to each end, the more shallow the cable gets.   In other words, the more horizontal the rope, the more tension it contains.  Can the rope be perfectly flat with a point load in the middle?   The answer is no, you would need infinite cable tension!   If cables are horizontal you have infinite tension, if cables are perfectly vertical, you have half the load carried.

What about uneven loading, or changes in cable angle like so?

The more vertical rope in this case takes more tension than the less steep rope, but why is that?   For discussion (sum forces in X and Y).

Platonic Solids from Perfect Triangles

Here are Plato's two perfect triangles from his seminal work "Timaeus" that can create the 5 platonic solids (which are according to him form the elements of earth, air, water, fire ...)

They are worth having as objects in the bookshelf to use and demonstrate how to build platonic solids, geodesic domes, tensegrity structures, etc.  Also they can be used to for statics class to demonstrate simple trigonometry (45/45/90 and 30/60/90 triangles), as well as x and y components of force vectors in statics.

Braced Frames vs Moment Frames

Comparing braced frames to moment frames, is like comparing triangles to squares... The square can easily deform and requires rigid joints if used as a lateral system to resist wind/seismic...

The triangle can only deform if the member itself shrinks or extends axially (along length) - which is difficult to do significantly (PL/AE).   The square with rigid joints does not need to deform axially, it simple needs to bend and it will translate.   So moment frames are about 20 times more flexible than braced frames (+/-10).   To make a moment frame into braced frame, simply add a diagonal...

Composite Beam Stiffness

You can make four strips of wood, say 1" wide by 1/8" thick.   Take a pair and tape the middle together, take another pair and only tape the ends.   Feel the difference in stiffness...

The one you one tape in the middle will act as two separate beams.  The one you taped at the ends will act as one beam with double the thickness.  How much stiffer will the beam that is taped at the ends be?   Since deflection is proportional to moment of inertia, and I is proportional to thickness cubed, it will be 4 times more stiff!  Two cubed is eight, and 1 cubed times 2 members is 2, and 8/2 = 4.  It will be double the strength and 4 times the stiffness - just by taping the ends so it resists horizontal shear and acts as a composite beam.  Go ahead and try this simple experiment.