# Why do some struts used in aerospace vehicle structures taper at the ends?

This ISS truss segment S6 (in the Space Station Processing Facility) shows quite a few examples.

(Personal Photo)

I've also seen them in the shuttle Orbiter midbody and aft compartment.

It would seem to complicate fabrication of the struts.

I'm looking for a For-Dummies level explanation; structures is not my field.

• Note that the strut in the middle of the picture (which ends in a T-connector) does not taper on the side of the T-connector. – MSalters Sep 19 '19 at 6:59

A detailed discussion can be found in this report from NTRS.

To sum up, the limiting case driving the design of struts is the compressive load capability.

### To answer why it is tapered, we must first know why the struts are "fat" to begin with.

Notice all the tapered struts have pinned connections, which means they only transmit axial loads (tension and compression), and don't transmit any bending moments. A long, slender strut under compressive load is going to be limited not by its compressive strength directly, which is generally a function of the material and the cross-sectional area, but by its resistance to Euler buckling.

Buckling is a structural instability in which a column in compression deflects sideways, often suddenly. Sometimes this is used deliberately, as in the case of the ISS solar array mast flex battens, but in most structural cases, this is highly undesirable.

The buckling force of a column is determined by $$F = \frac{\pi^2EI}{(KL)^2}.$$

In particular, the term $$EI$$ refers to the bending stiffness, where $$E$$ is the Young's Modulus of the material and $$I$$ is the area moment of inertia. Sparing too much math, for a fixed cross-sectional area, $$I$$ is maximized by moving the material as far away from the center as possible.

Thus, the thicker the column or strut, the more load it can carry. Even more importantly, it's the material on the outside of the strut that provides the most resistance, so wide struts can be made hollow on the inside to minimize weight. Page 23 of the report linked above has two graphs that show how the weight of the strut that can support a given load changes with its inner diameter.

Thus: Struts are "fat" because that's the most weight-efficient way to resist buckling under compressive load.

### Now the taper.

Remembering that the strut is a hollow tube that terminates in a pinned connection at each end: compressive and tensile loads must flow from one pin to the other. Let's imagine a few alternative designs.

1. The tube ends in a short, solid cylindrical end cap that is attached to the end fitting.

• Stresses must flow from the center of the end cap to the periphery. This makes the limiting case the shear strength of the end cap.

• Further, the connection from the end cap to the strut tube is liable to experience stress concentrations under cyclic loading.

• Manufacturing this will either require the cap to be a separate piece, a welded piece, or bored out from a solid rod. All of these have disadvantages in higher part count, difficult control over material properties, and difficult machining.

2. A long cylindrical block connecting the tube to the end fitting

• Shear stress is less of an issue, due to the large area that can handle it.

• Large chunks of solid metal are HEAVY though, and it is also unfavorable from a vibration standpoint

• Very little of this material is actually participating in reacting the loads.

• Manufacturing this suffers the same issues as the short end cap.

3. Tapering

• Tapering from the tube to the end fitting allows the loads to "flow" smoothly, reducing stress concentrations.

• The length and wall thickness of the taper can be tuned to optimize the loading for minimal mass.

• It's not as hard as you might think to manufacture. Tapered metal tubes like this are made using a process called swaging, in which you force a tube through a series of dies that reduces its diameter. The process can be surprisingly fast.

• Using a swaging (i.e., forging) process rather than a cutting process ensures that the material grain flows along the taper, further improving strength.

Thus: Struts are tapered because that design is mass-efficient, manufacturable, and strong.

• Great answer, just what I was looking for. Thanks! – Organic Marble Sep 19 '19 at 14:54
• Could you direct me to some math-heavy but engineering-noobish references to buckling calculations? All I can find are either too scant on the math or assume I'm an engineering student, which I am not. – Anton Hengst Jun 22 '20 at 17:36
• @AntonHengst Even the mathematical derivation requires some engineering fundamentals -- a cursory understanding of Euler beam theory and how to interpret free body diagrams (roughly sophomore-level engineering classes in US schools). This wikipedia page is a good start: en.wikipedia.org/wiki/Euler%27s_critical_load – Tristan Jun 22 '20 at 22:37