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Purposes for a Zero Force Member In a Truss

A few months ago I commented on a post of a very impressive wooden bridge contest (LinkedIn post below). I made the comment: “Excellent use of the vertical ‘zero force members’ in the center…” Well if there is no force on the member, what purpose does it have?

A Zero Force Member can be used to stabilize other members in the structure in compressive loads from buckling under a column load. By supporting a member at the midpoint, you can increase the critical buckling load by a factor of 4.

What Is a Zero Force Member

A Zero Force Member in a truss structure is a member that attaches to a node in which any force would cause static instability. The most common example of this is two members in line and a third member at the node coming in at any angle. The third member has Zero Force.

A truss structure is made up of nodes linked together by members. The node it self is considered a pinned joint (this might not always be the case); therefore, it is able to pivot and cannot support a bending load. As a result, the forces summed at the pin must cancel out to zero.

Using the force table, we can demonstrate how applied tensile forces must balance to keep the center ring off the center pin.

We can treat the above force table as a pinned joint in the truss with three members. F1 acts at 35°, F2 at 180° and F3 and 270°. In order for the blue ring to stay off the aluminum center pin, the forces must balance out in both the X (left / right) and Y (top / bottom) directions. That is accomplished by using the two following equations.

Σ Fx = 0 lb = F1 cos(35°) – F2

Σ Fy = 0 lb = F1 sin(35°) – F3

If we know F1, F2, or F3, we can find the other two values.

Locating a Zero Force Member in a Truss

Zero force members are easy to look for when you know what you are looking for. Be on the lookout for two members 180° apart and a third one that comes in at a different angle. See below.

Since two forces come in at 0° and 180°, the force at 270° must be zero otherwise the ring will touch the center pin.

The main two forces come in at 0° and 180°. For this to balance, in the X direction as shown below.

Σ Fx = 0 lb = F1 – F2

When the force at 270° is applied, it will pull the ring down under any amount of load. This happens at any load and can mathematically be expressed as this.

Σ Fy = 0 lb = F3 sin(270)

The Y equation above mandates that F3 = 0 lb at any angle (except 0° and 180°). F3 is by definition a Zero Force Member and cannot take any load.

If there is another member present or an external force applied, F3 may not be a zero force member. So be careful.

Typical Truss Structures. Notice how the Baltimore and K bridges have a lot of (potentially) zero force members, but the center span is supported.

How Does a Zero Force Member Help In Buckling

The buckling formula is Pcr=n2*E*I/L2. The variable n represents the harmonic plus 1. For a free span column, n is 1. If we support the column at the midspan, we now force the column to deform at the first harmonic and n is now 2, making it four times as strong.

Buckling of columns can be a hidden danger. In the bridge truss above, the bottom member is in tension and the top member is in compression. This means that the top member is subject to a hidden failure mode.

If you are wondering how I determined so easily I was able to determine this without math, I removed these members in my mind and asked which way the structure moves. When I remove the top member, the structure will bend down requiring a force under each load pushing out (compression). When I remove the bottom member, structure still bends down, but a force pulling in (tension) at each support is needed.

Once we determine which members are in compression and subject to buckling, we might want to support them. In the bridge truss above, the node in the top member at the center has a zero force member pointing down. There is no way that the member can take load.

But this vertical member does serve a critical purpose. It prevents the horizontal member from buckling by changing the harmonic at which it can buckle. The standard Euler buckling equation is as follows.

Pcr=n2*E*I/L2

Where, Pcr is the critical buckling load, n is the harmonic + 1, E is the modulus of elasticity, I is the area moment of inertia and L is the length. When we keep E, I and L the same, that leaves us only n to vary if we want more buckling strength.

Failure in Buckling

If we look at how members in buckling fail, we notice that like this ruler below, the center will bow out at the middle (red line). This is a zero harmonic failure mode, where n=1. When we grab a hold of the center, we have now changed the failure mode so that it is likely the ruler will failure at 1/4 and 3/4 of the length. This is illustrated by the blue line below and is the first harmonic.

Red – Natural frequency, Blue – 1st harmonic, Green – 2nd Harmonic

Notice how the amplitude of the orange curve is so much less than that of the black curve. This is caused because more force is required to make this deflection happen. The increase in buckling strength is 4x that of the natural frequency.

Natural Buckling
1st Harmonic – 4x harder to buckle

When we increase to the second harmonic, we see a displacement of zero at 1/3 and 2/3 the length and the amplitude decreases again. The buckling strength is now 9x that of the natural frequency.

Summary

Zero force members do serve a critical role in truss structures even though they don’t take any force. By preventing buckling on long compressive members, they allow engineers to use the materials better leading to lighter and stronger structures.

Clean Up Messy Wiring Diagrams and Schematics with Simple Ladder Logic

I’m often asked the question, “It is so difficult to troubleshoot my wiring. Corey, is there any way to clean it up?” Yes there is!

Traditional wiring schematics have all of the inputs and outputs on the component. Wires will need to cross over each other to make the needed connections. Ladder Logic separates the inputs and outputs and organizes them into rungs. This dramatically simplifies the schematic and provides easy troubleshooting.

Take a look at this seemingly simple electrical schematic for a car horn, headlights and a dome light. Too complicated.

Simple 3 Function Schematic

Compare that to this same representation using ladder logic.

Simplified version in ladder logic

Let’s learn more! Click below.

Easy Launch Drum Design Overview- Tilley’s Terror Backyard Coaster

I Need Your Help!

Projects like this are expensive and take a while. I need help. I’ve created a YouTube Membership channel to help others design and build your own backyard coaster. Take a minute to check out the channel. Perks of joining are:

  • Invitation to Ride Once Complete
  • Shout out in up coming videos
  • Your name engraved on a plaque to be mounted on the coaster

Click the link here and join today!

Is Pipe Scream a Coaster Credit? Panel Finally Decides

As I was nearing 200 coaster credits (that is riding 200 unique coasters), I wanted to make sure that my count was accurate before I left on my summer coaster trip. Provided everything is open, I would hit 204!

My kids were looking over my list and noticed one coaster that didn’t count. The debate ensued. After a lot of discussion and research, those kiddos were right; it wasn’t a coaster.

Now we are here to clarify what is and is not a coaster credit.

What Is a Roller Coaster Inversion? Heated Panel Discussion

Join the great debate and we determine what is and is not a coaster inversion. Most of us inherently know what an inversion is, but can we define it? There are several edge cases where the answer is not so clear. What about an overbanked turn or a beyond 90° drop? Inversions? Maybe…

Join this channel to get access to perks:
https://www.youtube.com/channel/UC8G89qtU0HMpKCuXUr1hvWQ/join

Cover image from Wikimedia Commons

Best Way to Charge High Capacity Capacitors

As we get started building our roller coaster, we need a sufficient power source. I would like this to be “portable” so plugging it into the wall is an attractive idea. Well, you can’t plug a 3 hp motor into the wall. It’ll trip the breaker every time.

The solution is to store energy in capacitors! But how do we safely use capacitors charged to 96V? Since we only get 15-20 amps out of a socket, how do we limit the in rush current?

Traditional charging methods waste power and take too long to charge. Let’s to this smarter.

But I Need Your Help!

Projects like this are expensive and take a while. I need help. I’ve created a YouTube Membership channel to help others design and build your own backyard coaster. Take a minute to check out the channel. Perks of joining are:

  • Invitation to Ride Once Complete
  • Shout out in up coming videos
  • Your name engraved on a plaque to be mounted on the coaster

Click the link here and join today!

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