A space truss

In the Beams and frames pathway, we met truss members on a Pratt roof truss and checked its bars by the method of joints and the method of sections. Every bar of that truss lay in one plane, and so did every load. In this step, we'll use truss members in three dimensions. Our structure is a square pyramid of four bars meeting at an apex, carrying a vertical load in one case and a sideways load in another. We'll read the bar forces, check them by the method of joints at a 3D node, and see why a truss like this is only stable because it's three-dimensional. If you need a refresher on the method of joints, take a look at this EngineeringSkills tutorial, Truss Analysis using the Method of Joints and Method of Sections.

The model for this step (A square pyramid of four CHS 88.9 x 5.0 truss members, 3 m across at the feet and 2.5 m high, under 100 kN down and 20 kN sideways at the apex) isn't available yet.

1.0 The idea

A truss member carries axial force only. It's pinned at both ends, so no moment can be transmitted into it, and it can only carry tension or compression along its own length. In the 2D pathway, this meant that the equilibrium of a joint gave us two equations, one horizontal and one vertical, and we could solve for two unknown bar forces at a time. In 3D, a joint has three equations, one for each global direction, so we can solve for three unknown bar forces at a joint. That's the method of joints in three dimensions, and it's the only new piece of theory in this step.

The more important idea is about stability. A 2D truss is drawn in a plane, loaded in that plane, and analysed in that plane. Now, what holds it out of its plane? Nothing in the model. A real roof truss relies on purlins, bracing and the cladding to stop it falling sideways, and the 2D analysis simply assumes all of that is there. In 3D, Structure Bench makes no such assumption. A node is free to move in three directions, and if the bars meeting at it can't resist a movement in one of those directions, the structure is a mechanism and the solve stops with an error. In other words, a 3D truss has to be triangulated in every direction, not just on the elevation you drew.

Our pyramid is the simplest structure that shows this. It has four bars running from four feet to one apex. Whichever direction the apex moves in, at least one bar has to change length, so the apex is fully restrained. Take two of the bars away and the two that are left lie in a single plane with the apex, which is free to swing out of it.

	The solver	By hand
Each leg under Down at the apex	-32.8 kN	$P / 4\cos\alpha = 100 / (4 \times 0.7625) = 32.8$ kN
Each foot, vertical reaction	25.0 kN	$100 / 4 = 25$ kN
Each foot, horizontal reaction in X and in Y	15.0 kN	$32.8 \sin\alpha / \sqrt{2} = 15.0$ kN
Each leg under Sideways at the apex	10.9 kN	$20 / (4 \sin\alpha / \sqrt{2}) = 10.9$ kN
Legs on the +X side under Down + sideways	-43.7 kN	$32.8 + 10.9 = 43.7$ kN
Legs on the -X side under Down + sideways	-21.9 kN	$32.8 - 10.9 = 21.9$ kN
Apex drop under Down at the apex	0.51 mm	the shortening of a leg, divided by $\cos\alpha$

A space truss

1.0 The idea

2.0 The model

3.0 What you'll see

4.0 What a truss model assumes

5.0 What to take from this step