An Overview of Truss Designs

Introduction

I want to close the recent examples of trusses by providing a sampler of truss designs. This is far from encyclopedic. In fact, this post is limited to planar trusses.

First, however, let me give you a link to an online calculator. I checked it out on the Howe truss with a snow load.

As you can see, I scaled the loads by a factor of 10. I had to, for the program – not a big deal.

There are plenty of websites with information. You can search for yourself… you could start with the usual wiki article – the merit of which is that it has a lot of external links.

Personally, I have kept a link to bridge trusses in western PA and a link to roof trusses by an Australian contractor.

Okay, what do we have?

Trusses – Snow Load on Howe, Fink, and Pratt trusses

Intoduction

I want to work another snow load problem… using three different trusses. I found a pair of these, for a Fink truss and a Howe truss, on a professor’s university website – his name is Zig Herzog and his main statics page is this. Individual links for the two problems will follow.

Each of these trusses is 12 meters across the bottom, and 4 meters high. Each has a total snow load of 2400 Newtons. As ever, I have used Mathematica® for the computations and graphics.

One of the things I liked was that Herzog asked us to find the maximum values of stress (both compression and tension), and the total length of the beams used in each truss. In addition to using Fink and Howe trusses, I will do the Pratt truss again, with the parameters of this problem.

The purpose of this post was to see how different the trusses are, under the same load. In one respect they are the same: the maximum values of both compression and tension are the same for all three trusses. And that’s the summary!

Here are screenshots of the author’s assigned problem… for which he does provide solutions, so I’m not giving away any secrets.

For comparison, here’s the Pratt truss I used in the previous post.

Trusses – Example 4, Pratt roof truss with snow

This problem, like the Howe truss, comes from J.L. Meriam, “Statics”, John Wiley & Sons, 1966.

This diagram actually shows concentrated forces at the joints – which is where they must be for what we’re doing. Nevertheless, the problem as posed says that we have a total weight of 4000 pounds distributed uniformly across the top of the truss. We take the 1000 pounds on each of these four beams, and assign 500 pounds to each of its endpoints. Points A and E have 500 pounds, but the other three joints connect two loaded beams and get 1000 pounds.

Trusses – Example 3, a Howe truss

Introduction

This is a sample problem from “Statics”, by J.L. Meriam, John Wiley & Sons, 1966; #105, p. 88.

It is called a Howe truss – or, sometimes, a double Howe truss. Here’s a link that will give you the names of some trusses so you can look for more information. There’s a lot to be had, but it doesn’t seem all that standardized.

And here’s a link to some class notes, in case you want more explanation than I have been providing. And, since this is the third post about trusses, you could go look at my first truss post and my second truss post.

Anyway, we see that it is fixed at points A and G… and it has three external loads, each of 2 kips (a kip is a kilogram-force), at points B, C, I.

Trusses – Example 2

Let’s try a slightly more complicated truss. You may want to read the previous post, if you have not already.

Trusses – Example 1

Given the following picture of three beams… with a given force applied at the apex… let’s see if we can work out the internal forces in the beams, and the reaction forces at the two bottom points.

This is an example of a “statically determinate” problem. We will be able to solve this assuming that the three beams do not bend or compress or stretch.

Calculus – deriving the equations for simple projectile motion

We have worked several examples of simple projectile motion – meaning that the acceleration of gravity was constant and vertical, and there was no acceleration in the horizontal direction. (In particular, there is no air resistance.)

I simply handed us four equations and used them in a first… second… and third post. I said I would show how to derive them.

It was junior year in high school that I learned the equations for position and speed as a function of a constant acceleration. I didn’t take calculus until I was a college freshman… and at some point I decided that I knew enough calculus to derive the equations I had been told to memorize two years before.

This is about as elementary as it gets in calculus, but when it was all new to me, it was a thrill to see what it could do for me in physics. I will actually derive them in two slightly different ways.

Here we go.

Mechanics: Simple Projectile Motion – 2 (Fort and Ship)

Here is a projectile problem that fascinated me, and I’ve been meaning to show it to you. It comes from Neville de Mestre, “The Mathematics of Projectiles in Sport”, 1990. This is the second post about simple projectile motion, so you might want to look at the first one.

Here we go.

A fort is on top of a cliff h meters directly above the ocean. Approaching the fort is a ship whose guns have the same muzzle velocity vo as the guns at the fort….

Find over what range the ship can be fired on, from the fort, without being able to effectively return the fire.

If gh is small compared with vo^2 show that this distance is approximately double the height of the cliff.

So, we need to find two distances: max from fort to ship, and max from ship to fort.

Orbits: Vector derivations for 2-body orbits

Introduction

First, let’s take it for granted that the differential equation for the two-body problem (e.g. the earth orbiting the sun in an otherwise empty universe) is

where is the vector from the primary (e.g. the sun with mass M) to the secondary (e.g. the earth, with mass m) and ; and a simple is the magnitude of the vector .

I want to derive

• conservation of energy
• conservation of angular momentum
• a neat equation for r and v
• the scalar equation for the orbit.

Furthermore, I want to use vector operations wherever possible. (I like vectors!)

rotating coordinate systems: example 1

conventions and setup

As far as possible, I am going to stay with my notation. r and are the old and new (fixed and rotating) components of the position vector; v and are derivatives wrt time of r and respectively; a and are derivatives wrt time of v and respectively. (But R is a convenient scalar value, and will no longer denote the position vector whose components are r and .)

The rotating frame is the same in all these problems, so get its matrices early (hence not often). The z-axis is our axis of rotation.

The attitude matrix for a CCW rotation of the axes (about the z-axis) is…

The transition matrix is…