The Secret to the Truss Strength!
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- เผยแพร่เมื่อ 25 พ.ย. 2022
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Truss structures are more common than you think. But why do we use them? Beams seem to work fine right, well yes but there is a catch! The trusses are mostly used in bridges, roofs of venues, some cars and many other places. But how do they work and what is their advantage over beams? In this video, we dive deeper on truss structures and the secret to their efficiency. The video is supplemented by a scaled experiment that practically shows the advantages of the truss.
This video was sponsored by Brilliant
References:
[1] M. Carver, "Tennessee’s Survey Report for Historic Highway Bridges," Ambrose Printing Company, Nashville, Tennessee , 2008.
[2] J. M. Gere and B. J. Goodno, Mechanics of Materials, Cengage Learning, 2013.
[3] R. C. Hibbeler, Structural Analysis, Upper Saddle River, New Jersey: Pearson Prentice Hall, 2015. - วิทยาศาสตร์และเทคโนโลยี
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I wonder when the moment in history will be, when every STEM-mind on the planet has brilliant 😆I love brilliant and for a while most of my family too now. Prime quality sponsor!
Pretty arrogant to say early engineers did not understand why trusses worked. Perhaps they just did not document that understanding enough to suit you.
@@dallasarnold8615 Hi Dallas, that was referenced to ref [1] which is where that information was taken from. I wasn't around back then so I cannot directly judge their understanding. However, someone (reference [1]) researched this and figured out that their understanding was not great.
You should delete your channel.
On your truss/beam testing table, drill a large hole in the table w/ a 6in holesaw, this would you allow you to load the test beams axially. I suspect the truss may have tested over 25% compared to the beam if the load was purely axial. The higher center of gravity and torsion induced by the rope twisted it significantly and in my opinion it failed prematurely. Great video, regards.
Hi Kurt, completely agree. In fact, I considered that exact idea. However my wife vetoed it ☹It would have been amazing! I just need to find an old table and then we have a proper loading table. Thanks for the suggestion!
Or just bridge between two tables.
@@tdkeyes1 or chairs, or saw horses, or just about anything. The pin concept was educational, but the engineering of the testing apparatus was horrendous.
@@TheEngineeringHuba table with removable leafs would work too and be non destructive. Thanks for the great video
I would expect the section to break between the pivots based on the description of how loads are transferred throughout the truss. Seems to me that the load needs to be placed at the pivots as that is where the load would be experienced in real scenarios.
I have been a pressed metal pated wood truss designer (and also engineered wood) for around 30 years now. This was a good video. I was worried your staple connections were too weak as the connections are where the engineering really matters, but they seemed to hold up well. In pressed metal plated wood trusses, when a truss is overloaded it always fails at a joint not in a member. A well overloaded joint actually causes the pressed metal plates to slowly roll out of the wood.
I remember two failures of metal gusset bridges. One, corrosion damaged a gusset until it failed, resulting in a failure of the bridge while it was in use. The other did use a more pin like design, misalignment and metallurgical faults resulted in its failure at the pin junction.
I-35W collapse and the Silver Bridge collapse, although the Silver Bridge was suspension, similar effects to gusset stress distribution resulted in the catastrophic collapse.
Bridges are all about stress distribution and redirection, but that's true of pretty much all structural engineering.
I also noticed that he didn't address two shear modes in a plain beam bridge, compressive and flex. But, that'd extend the scope to three times longer of a video...
Can you make a video of a well-overloaded truss losing its metal plate(s) over time? I'd love to see/hear/read more about that!
Do you know about the Tretten bridge failure in Norway? It was a wooden/metall bridge.
Well done. The most interesting part for me was the explanation of why it's so important to get all of the members connecting to the gusset to line up on the same point.
Hi William, thank you! The alignment is an important concept that doesn't get a lot of attention. I hope the explanation was not too brief.
Being involved with testing high voltage transmission towers, balancing the loads at joints and having the correct number of fasteners is important. I have seen a number of tower failures due to imbalanced joint fasteners or the need for just one more fastener at a joint to transfer load or torsion. Good video. Wished this was available in the 1960's when I was in engineering school.
How was this well done? His "experiment" wasn't even an experiment, it was just stupid.
@@maflones Well, I think we all know who else is.
@@carlhitchon1009 Bye troll.
During the course of my career I have recorded nearly a thousand bridges. Many of these were actual pin-connected bridges that did not use gussets. These were usually older railroad bridges from the late 19th and early 20th century. Some have been repurposed as local access bridges, but they are still around. That said, even through I have been immersed in bridge tech information, I learned quite a few new things from your video, always something new to learn. Thank you.
The first example I can think of, of a suspension bridge with a railway on it is the "Seto Ohashi Bridge" in Japan. It's actually a series of bridges, jumping between small islands to link the Japanese main island of Honshu with the large southern island of Shikoku over a total distance of 13km. There are six individual bridges: a truss bridge, two cable stayed bridges, and three suspension bridges, the longest with a 1100m centre span. All bridges are double decked with a highway on the upper deck and railway tracks on the lower deck. Currently it has two narrow gauge tracks, but the bridges were designed to carry four tracks to allow for an eventual Shinkansen line.
Stiffened with a truss, isn´t it?
@@patnor7354 The suspension bridges have trusses in their deck structures, but that's pretty common in suspension bridges, and if you look at pictures, it's pretty obvious that the load is being carried by the cables.
@@TonboIV I think he was looking for a suspension bridge without any trussed components.
@@MFcitrous That's a meaningless distinction. The deck of any suspension bridge has to be made of SOMETHING. If it's not trusses, then it's girders, which are are structurally equivalent to a truss without any holes in it anyway. If you're building a double-decker bridge, then a truss becomes pretty much the only practical way to build the deck.
A simple truss like in these bridges can't even support itself over such a span, let alone carry any load. These trusses only carry the load as far as the vertical cables, which is already a substantial job.
The trusses certainly do stiffen the deck structure, but any big suspension bridge has some degree of stiffening of the deck structure, and looking for a suspension bridge which carries anything heavy (including cars) and doesn't have any structure to stiffen the deck is looking for an engineering fiction.
Oresund sweden/denmark
Loved it - especially the work that went into the simulated catastrophe! The pin / convergence point explanation gave me exactly what I need for a roofing solution at home. Many thanks for the clarity you provided.
As a maker and not an engineer you helped me a lot with this description. Thanks
Make the truss bridge in a way that the load is hanging from a pin joint and not from the middle of a truss (place the joints in a way that one joint is placed exactly in the middle). Thus you can avoid breaking the truss by a bending effect.
7:33 apart from the support for your truss failing before the truss, the way you've loaded the truss with lines running near horizontal out to the table edges guarantees that you won't have a proper vertical load on your truss, and the way you wrapped the line around the bottom chord where it is unsupported also guarantees that the bottom chord is loaded in both bending and shear.
Wow thanks for another great video. Enjoyed the experiment--would be cool to see more of those!
Great demo; we can also so see that the bottom cord failed in tension… this demo is well balanced (picture/formula) linear tension-compression vs bending forces against deflection is a great introduction. There is great thinking and a lot of preparation for this very short fun Eng. lesson. 👍 thank you!
Great demo. You also inadvertently showed why good construction quality is important.
BURN!!!!
That was very insightful, would definitely be great to see more model experiments like this in the future :)
Glad you enjoyed it, more to come!
Very well explained, cleared a lot of concepts. Really good job!
Excellent video my dude
Even if the central axis of all members passes through the same point, the gussets still apply a moment to the members under load, particularly if the deflections are very high. That is why another condition of the pin connection assumption is fairly small allowed deflections.
Exactly, I was a bit disappointed he missed that assumption
Great video. The humble bicycle frame is also a perfect truss structure. The difficulty comes when other forces intervene such as the bending force on the head tube when you apply the front brake. Carbon frame engineers have learned a lot in recent years about laying the carbon mat is such a way that maximum stiffness is achieved with minimum weight and they have also learned how to vary the carbon layup so as to allow controlled flex to prevent an over-stiff ride.
Good practical, visual examples of what you were talking about. I might try your demonstration myself.
Doing engineering at Napier in the 70s we were given a sheet of balsa, a tube of glue and a craft knife (Aaargh! student risk assessment kills this idea in 2023!) and challenged to build a bridge across a gap between two benches in the lab. A truss just managed to outperform my box girder. It was a really great exercise. I have used the intuition about the distribution of structurally significant material that developed from that exercise many times since then!
Thanks for the video, I always watch for your new uploads.
Thanks Johnathan, we appreciate you as a viewer 🙏
I’m looking forward to seeing your next program.
"Amazing work! As an aspiring engineer, you are a true inspiration. Your dedication and commitment to engineering excellence are truly remarkable!"
So flattering, comments like this make all the hard work and countless hours feel worth it. Thanks What2Read! 😊
Thanks for one more insightful video.
That was a great way to learn a little more about beams and trusses . Thanks
Thank you for your time and attention Michael, we are glad you found the video interesting!
Well explained...Appreciated
Since you have connected loading string twice to your bridge you created shear and twist. The shear failure is evident, 2 ropes cut you bridge. Put some metal clip first on the bridge then connect rope to the clip to get a better failure modes. To eliminate sideway force use pulleys at the edge of a table and make sure that the structure is centered
Hi Vlad! Yes, there are definitely lots of things to be improved. Thanks for the suggestions they make sense, cheers
Would love to hear more about how this applies to trusses loaded axially as in the case of lattice boom cranes!
This was very helpful thanks a lot
Will wait for more
Glad it was helpful, more to come!
Yes! MOAR! 😁
Exciting to make connections between my uni classes and these videos!
Nicely done!
I love the models and since you clearly state the drawbacks and simplifications, I am super fine with crude models (although YOUR woodworking skills still esceed mine BY FAR, obviously 🙂).
And remember, whe can't all be 'Practical Engineering' on the first tries 😉
Thanks for putting all the effort and work in these videos, much appreciated and future engineer generations will thank you 🥳
Building the truss turned out to be harder than initially anticipated but I think it illustrated the point alright. Practical Engineering is ana amazing channel, often inspirational to us as well.
Brilliant video-Thanx
Great video, thanks
Very interesting. Also very useful. Construction inadequacies are common in the real world, but the builders dont readily admit such things. Your construction issues aside, the demonstration at reduced scale is quite helpful, and fascinating.
Good investigation and research. Thanks.
Thank you Peter from Holland!
I really really like to see more experiments in the future video
Great explanation !!!!!!!!!!!!!!!!!!!!!!!!
Thanks Dilip, more experiments will come in the future for sure.
Love your videos. Please try to upload more frequently
Thanks Hossen, we will try our best!
Great video. That's amazing though how you were able to show the strength of a truss, I mean you whipped together a glued truss and it still supported nearly 100 lbs, awesome.
Great video!
I think the pin connection was glossed over in my structural mechanics class. It's not intuitive but the offset member causing a bending moment makes perfect sense.
The explanation here was weak IMO
Thanks that was a very nice video, additionally the stiffness of the gusset plate, and bolts tightening condition allows for deformation during loading and that adds up into the explanation of pinned connection.
One very important aspect of trusses which the common person is not aware off is weight reduction. There comes a point where the size of a solid beam required for a particular span makes it heavier than the load they have to support. Since trusses are more complex than solid beams, for short spans solid beams are preferred.
Yup, agreed! Intrinsically, this is related to the way the resist loads, bending vs axial stiffness.
Fabulous channel. This content is why I love TH-cam.
I agree that a central principle of a truss is that all intersecting beams must intersect at a common point in order to prevent inducing a moment.
However, even if this condition is met, when the beams are attached using gusset plates and lots of pins, simple geometry implies that there MUST BE some angular deflection of the beams in order for the aggregate deflection/bending of the bridge to occur. As a consequence, the pins furthest from the intersection point will experience the greatest shear loads.
There have been bridge failures caused by failure of pinned/gusseted joints.
Excellent content
Beam simplicity vs. truss is why I chose to build a beam to support an approximately 20ft (6m) span for a livestock shed roof. I also used the beam with a pair of hydraulic bottle jacks to raise the roof, correcting the near collapse when the previous support failed.
15% may have been 20-40%had the simulated abutment not yealded first putting the twisting moment into the truss causing it to fail.
Thanks for the memories.
8th grade in a class called Exploring Technology, we each built and tested a bridge to failure. 10' of 1/2"×1/8" pine, a 12x4" card stock as decking, and ample wood glue. 1st place got 150lbs I got 110lbs for 2nd place. Lowest was 10lbs for essentially a beam deck bridge ,( minimal effort, no super or sub structure) my failure was down to delamination of the wood grain where the super structure notched into the main horizontal member on top of the abutment . Had we been allowed more stick out, on top of the abutment , the delam would have been prevented.
My bridge had a minimal failure 1 cracked member. 1st place was a catastrophic failure where it shattered into 100's of pices my trophy was my bridge, his was a pile of toothpicks.
Another way to look at how a truss works is that it disperses the load across a wider two dimensional plane whereas a beam is basically one dimensional so the load becomes more focused and less force is needed to compromise it. The longer the beam the more one dimensional it becomes.
You can achieve the same strength as a truss with a solid plane attached to a boxed frame but trusses accomplish this with less material and save weight.
The preview pic is from a truss bridge accident in Norway. There was a suspected construction error. So all the bridges had to be checked and some were closed until checked. It is a very common bridge in Norway since it is both cost efficient and somewhat aesthetically pleasing.
As the engineer said, “Truss me, I know what I’m doing.”
I’ll see myself out.
🤣🤣 the last line got me laughing
pretty interesting-never though about trusses like that!
Great instruction and loved the description of avoiding couples by proper alignment the members. Your drawings made sense on that point. I'm curious as to a comparison of deflection to failure of beams to trusses. It seems to me that the beam would flex more prior to failure than a truss.....
That is true. The beam deflected more because its material was engaged in bending. The truss was engaged axially, which is much stiffer and therefore deflects less.
Interesting video boss.👍👍👍
if you want to do a test like that in the future with a setup like that, mount the test member on it's side and use a pulley, that will make the stress force vector down with respect to the test member, and you wont have to worry about getting extra tables, or cutting a hole in a table.
Hi Alexander, pulleys are definitely considered for the next loading!! Thanks for the suggestion!
The Patullo Bridge in Vancouver BC has a commuter train - 'The Sky Train' - running on it. In 1990, when it opened, it was the world’s longest transit span and the world’s only cable-stayed bridge designed solely for carrying rapid transit. It held its distinction until 2019 when it was surpassed by the Egongyan Rail Transit Bridge in Chongqing, China.
Tilikum Crossing is another transit only cable-stayed bridge in Portland, Oregon, that carries light rail.
Lisbon, Portugal, 25 April Suspended Bridge, was after-built adapted to accommodate a railway underneath. It work fine, and heavily, the train is of suburban type.
Suspension bridges that I know of that carry subway trains - Manhattan Bridge and Williamsburg Bridge.
Nice detailed video. Defining the terms was helpful, but there was one left out. The term "axial" is confusing, and I'm not sure what it represents in the illustrations.
Axial means a force applied to the member in the direction of its length. For example, if you have a vertical column the axial load would be a force pressing (or pulling) vertically down on it. If the force is applied horizontally (e.g. wind) to the column then it would cause the column to bend hence producing flexural forces in the column. In a truss, all members are loaded axially which means they do not bend but only compress and stretch since the load is aligned to their length.
@@TheEngineeringHub Thanks for the explanation!
Great explaination. Biggest issue with the test was the load was not at a joint in the middle.that was probably why it prematurely failed
Agreed Brett and the lateral instability of the support, which cause bi-axial bending
Brilliant video
Great video. Would like to see a discussion of the I-35W bridge collapse in 2007, which was blamed at least partially on undersized gusset plates. At least one gusset plate had exhibited bending before the collapse.
1/2 thick....How could the contractor and engineer not have spotted this when erecting it..???
Insane
7:55 bro did can carry me 😂
Hi, really liked your video. At a few placed you showed what I thought was a reference to the source of something you said (e.g. a "[1]" in the lower left at 00:27). But I don't see anything in the description. What do these references mean?
Hi John, sorry about that, the references will be uploaded in the description in the next couple of hours. Cheers!
@johnreedch6909 the references are uploaded now!
The Tyne Bridge at the beginning of the video has bridge beside it with a truss system with arches intergrated into the truss. This bridge has rail transport on top and motor vehicles on the lower deck.
Is a truss in a roof possible if the ridge line of the roof is not straight, buts runs diagonal????
Great video
Glad you enjoyed it
4:43 - I built a model of the Florida concrete truss pedestrian bridge that fell down, replacing the tension elements with chains to demonstrate which elements are tension (chains) or compression (sticks). I think there is no way to add a photo of it here?
One factor, given a uniform linear load on the truss (vehicle traffic but mostly just the weight of the bridge itself), is that the vertical elements on the ends must support nearly the entire weight, while those halfway to the middle support only the weight of the middle half, so the end elements must be stronger and heavier. This is not true of the horizontal elements, where (in the video drawing) the middle tension element supports the full tension of the end horizontal elements, PLUS part of the tension of the vertical tension elements, PLUS part of the compression of the middle two vertical elements, so it is the middle horizontal elements that must be stronger than those on the ends! A beam could be designed for similar factors, where the thickness/strength of the flanges and web are varied along the beam length for maximum strength to weight ratio.
I use a similar principle when propping form work to hold back tons of concrete
I form up do all the internal steel work that takes care of the tension an then I concrete it an that takes care of the compression...
Good show,,,,, I enjoyed that
Thanks maggot, we are glad to hear you liked the video 🙏
Very good, and actually, seeing the truss fail because of the lateral movement I think was very interesting to watch.
7,47 the word you're looking for is "Torsion."
Old trusses actually did use very large pins for connections instead of gusset plates, though likely it was more to do with the staged method of construction resulting in rotations at the nodes.
Excellent point! Yes, the bolted gusset plate provides rotational fixity during construction that makes it easier to install rather than a pinned member that will need to be held in place until another element is attached.
Interesting!
I think if you are testing bridge strength, and your moment of weight is coming from an angle, it might make sense to have the bridge sit at the same angle? I guess it depends on if the sideways force is desirable. Otherwise, maybe balancing between two chairs might be better, if you don't have two equal height tables.
This was fun to watch either way.
3:30 Brooklyn Bridge has Subway lines running on it.
5:58 FIU pedestrian bridge collapse.
Would the western span of the SF/Oakland bridge fit your request for a suspension bridge running rail and pavement?
It was originally designed to handle light rail on the lower deck and auto traffic on the upper deck.
I do really want to know about what we need to know to calculate curved reinforced concrete shells. Like, a rectangular pilar and rectangular beam is already a bit hard to give dimentions and to chose the steel. A lot of rules and things like that. What we nee to know to advance to curved or nor linear shapes?
I remember a documentary saying that the Golden Gate Bridge was designed to carry commuter rail on the lower level. I THINK I heard that the BART system has considered making that happen.
25 de Abril bridge in Lisbon, Portugal is very similar to the Golden Gate in San Francisco and it carries a double track railway in the under deck.
Non truss rail bridge - East Link: Sound Transit's expansion project which travels from Seattle to Redmond over a floating bridge.
When building with lumber, to support across a horizontal span. It's much better to put the 2×4's on their side. The wood is much better at supporting downward 👇🏻 pressure when it's in that position rather than flat because it has much more ability to bend due to downward 👇🏻 pressure across horizontal spans.
Hi Ben, that's correct, and the reason why is the property called moment of inertia. This same property is maximized for wide flange beams (or I-beams) which is why they take that shape. Check out our video on the secret of the I beam if you are interested to hear more about it.
A boy at school explained to me how some materials stand up to compressive forces and others to the kind that tends to pull things apart.
It must have been interesting enough for me to remember that so long
Thank you for the video which gave me an idea about truss bridges. Please correct me, if I mention that the important supports are the piers (vertical columns) between the truss. Should they not take the major load of the movement of traffic moving? They should be able to prevent the bending of the truss.
Hi Varadarajan ! All of the loading has to go through the piers so you are right they are very important. But the loads travel through the structure rather than being individually taken by the element. So the load from the traffic would go into the truss, travel through it, and then go into the piers. In essence, the size of the piers dose not affect the strength of the truss or its deflection. The spacing between them does! Very much so! Spacing the piers close together would reduce the deflection with (roughly) the cube power. But closely spaced piers are not always possible and they are expensive to build.
@@TheEngineeringHub thank you for the explaining of the principal. I also agree that having too many piers is costly with reduced spaces between them . The bending of the truss is compensated by the piers' reaction.
The Smithfield Street Bridge is 140 years old.
The Bridge is in Pittsburgh. The interesting aspect of the Bridge is the small piece of steel as part of the tresses.
Also if you look at the parts of steel with "Carnegie Steel".
Another interesting fact is that it a suspension Bridge without cables
It has an oval shaped suspension.
The Smithfield Street Bridge replaced a Bridge built by Roebling.
Roebling's Bridge was tore down.
The truss is amazing in its visual simplicity and elegance, which hides its technical complexity.
Many years ago I built a small truss bridge from Lego Technic. It spanned about 5 feet, and had almost zero deflection..
So I put a question to myself: how far COULD it span..
What is the physical limitations of Lego in a truss system..
Long story short, I am the designer and builder of the world's second longest and fully self-supporting Golden Gate Bridge..
It is a suspension bridge, of course, but the deck sections are trusses, the mid span of the bridge is an impressive 8 metres (26 feet) with a total length of over 13 metres (46 feet).. as long as a semi trailer truck.
It's a 1:160 scale replica and has been exhibited and a number of Lego expos here in Australia over the years..
It stuns all who see it, children and adults alike.
I'd love to be able to take it to California and set it up on the hill in Marin County overlooking the real bridge..
wow that's amazing!! Would love to see pictures. Send us a link if available!
brother which software you use , because i have need of this software for college presentation
This video would be even more useful for the uninitiated if you took a moment to define terms like "axially loaded", or explain how the resistance to sheer in a beam transfers force to the supports.
Axial load means a load that is aligned with the long axis of a structural member. In other words, it is either pure tension, or pure compression, and no bending, twisting or shearing.
Check out the Ben Franklin bridge in Philadelphia. It might be an appropriate study, tho I haven’t seen or been on it in over 30 years.
"Use whatever unit you prefer he says. Mm, inch, whatever you prefer...". I'm taken aback. He doesn’t include the light year, a critical unit!
can you provide a truss design for me with Tbar or 50X50 ANGLE for my shed 22m long x 10.5m wide x 7m high in the middle, legs are 4m high
I found this video very informative, I have passion in engineering but never took up the subject. I live near an interesting bridge and would enjoy your take on the skills used in its construction. It is the Ironbridge in Shropshire UK, it is said to be the first bridge in the world to used iron as it main material. Before this bridge most were built using stone or timber. If you are interested there are many site that show it constitution methods. Have great Christmas or should I say holidays sorry don't what the PC term is. kind regards.
the oakland to san francisco. was built to handle freight and passenger rail cars pulled by stem locos. this was done on the lower deck, and the sf part is and still is a suspension.
sad the steel oakland side was taken down and replaced with a weaker concert structure.
no trains have been on this bridge in a very long time.
Can you tell me what is the difference between stiffness and rigidity when applied to bridges? Thanks.
Think of stiffness as resistance to a change in size, while rigidity is a resistance to change in shape.
More formally, the stiffness of a material is its Young's Modulus, while the rigidity of a material is its shear modulus. I'll leave it to you to Google these terms if you want more details.
The idea is that when you apply only axial loads, an element that starts as a rectangle will still be a rectangle after it deforms. The deformation causes a change in size and aspect ratio, but not shape. When you load it in shear by contrast, a stress element might start as a rectangle, but turn into a parallelogram, as the shear warps it to a different shape.
You opened the video with the collapsed Tretten bridge. In the investigation they talked a lot about bolts possibly failing , the wood breaking etc. But I said from day one : that is NOT even a truss bridge!! All the diagonal truss beams in that bridge was LEANING THE SAME WAY. now Im not an engineer, but surely that will PROPAGATE the stress to more parts of the bridge , an not, as it should distribute the load
Where does aisc show tolerance for the gusset plate?
1:05 Padma Bridge
Just searched for it: Cool bridge! Thanks for the share!
Thanks
I think Matt needs to see this.
At about o.40 you say that one part of the beam is being stretched and the other compressed. As both top and bottom flange are growing in length are they both not being stretched ?
The displacement is exaggerated on the animations which may give the impression of elongation of the entire beam. But the top flange is not growing in length; its length is actually reducing which is what generates the compressive stresses.
The demo would be greatly improved by pulling down or pushing down on the top beam at its center and drilling a hole in the table and pulling straight down, or even better using a press
Dale
On the truss failure, the right end support failed before the truss.
The results of experiment is not occurate because some of the load are transferred to both sides of the table through ropes. It's better to put holes on the center of the table so that the rope is free. Another thing is, it should be two parallel trusses and the apply the load on top of the trusses so that it will equally distributed the load to each member of the truss.