Funny m and its Tower Introduction Since the WTC towers at NY were destroyed on 9/11 2001 'live on (faked) TV' , there is an ongoing discussion whether steel structures just can one-way crush down or progressively collapse from top to bottom due to some local failures up top with the result that the complete structure becomes rubble - as shown right. It is also discussed at 1.18.50 in DVD2 - a film about the 911 incident. According some people, incl. religious fundamentalists and terrorists , it is a natural phenomenon that, however, cannot be modelled or explained by structural damage analysis, like, e.g. ship collisions . In order to clarify matters I have designed the Funny m structural assembly that you can use to build a Tower that you then can try to crush ... just for fun. Purpose is to establish what spring, if any, breaks first when a Funny m assembly, unrealistically, is dropped , free falls and then contacts another Funny m assembly in a Tower like structure or ground itself: Funny m Funny m is a simple structural 3-D assembly that consists of a horizontal element with a mass m (e.g. a floor) supported via solid connections by four vertical elements s (e.g. columns) that can compress like springs before breaking. Each s carries m/4. The height of this structural assembly is h. A picture of the Funny m assembly is seen right. Due to mass m the springs s compress elastically d = 0.03h. The structural Funny m assembly is really funny or at least the spring elements. They can compress 0.09h elastically and 0.1h plastically before they break (in this example). It means you must put on 3 m for the springs to start deforming plastically!

You can put Funny m assemblies on top of each other to get a bigger structure, e.g. a Tower like structure with n Funny ms as shown below. The springs then adjust themselves to the number of m carried! If you put a Funny m assembly on another Funny m assembly, the bottom springs become twice as strong, etc. Then you have to put on 6 m uniformly (3 m on each element) to start plastic deformation! In a Funny m tower structure with n assemblies any spring is just statically compressed 0.03h by n m. If you drop a number of Funny m assemblies, let's call it part C, on another number of Funny m assemblies, let's call it part A, of a Funny m tower structure, it doesn't slip off at contact, as you would expect, but it will 'fuse itself' to it, and both parts C and A will absorb the energy applied at the contact. If a spring breaks after compression/failure and m drops, parts C and A remain 'fused' after next contact. However, if no spring break at contact, there is no fusing, but part C bounces. Very funny! The Problem! Historically, when you drop a small weak structural part C (apply energy!) on a stronger, bigger structural part A, C breaks up, while there may be some local damages to A, unless C simply bounces on A. NIST suggests however in its report of WTC 1 destruction on 9/11 that, if you apply energy on the top (!) of a structure - e.g., a part C of the structure is dropped free fall on the remainder of the similar but stronger structure, part A - global collapse ensues, when the potential energy applied at contact plus potential energy released due to failures exceed the strain energy that can be absorbed by the structure. A professor Z P Bazant maintains the same things in numerous peer reviewed papers published in the USA. Bazant suggests that the smaller, weaker part C remains intact (!) and one-way crushes down the bigger, stronger part A. This article will explore, if this funny suggestion applies to a tower of n Funny m assemblies! Right we see a Funny m structure/tower on ground with n = 22 Funny m assemblies. The total mass of this structure is n m (or 22 m) The potential energy of each m is its distance above ground times g, where g is gravity acceleration. The total Potential Energy, PE, stored in the structure, relative ground, is the sum of the PE of each m or n * m * n * h * g /2 or PE = (n²mhg)/2 ....................................................................... (1) The spring elements adjust themselves to the number of m carried as explained above. Thus the spring elements below the top m can just carry one m. The bottom spring elements can carry n m, i.e. they are n times "stronger" than the top springs. The bottom springs can absorb n times more strain energy than the top ones! This means that all springs compress equal distance d in the funny tower under static load. Compressive stress is same throughout. It also means that a spring above will always break before a spring below, if you add extra m on top without adjusting the springs. The springs are really funny! They can compress 0.09h elastically and 0.1h plastically before they break (in this example) as already explained. In the intact tower right all springs compression is d = 0.03h, i.e. hardly visible. Thus n m will compress the tower n d or 0.03nh. The spring constants C for all springs are thus known. Each spring can in fact carry 3 times more m than it is certified for, before it starts to deform plastically. Thus, the total elastic and plastic Strain Energy, SE, stored in the intact tower structure (the springs) is known. It is actually the total mass of the tower, n m, times g times 0.03h! It is also known that the total structure can absorb 3 times the elastic strain energy before any spring is overloaded, but then - of course - the extra loads or masses must be applied uniformly inside the whole structure! The maximum SE that the Funny m Tower then can absorb uniformly is SE max = 0.09nmgh ............................................................... (2) or with n = 22, SE max = 1.98mgh . It is very little compared to PE - but we are just considering the springs. Now we just suddenly apply energy at specific locations! Top, bottom and in the middle! Experiment 1 - Top loading! We first remove, suddenly, the springs below element n-1, #21 in this case, the tower structure decompresses accordingly, and then we drop the top (n/11) m (part C = 1/10 of part A) or 2 m on the tower below (part A) or 20 m distance h! It is assumed that part C really contacts part A below and compresses part A and that part A compresses part C and the ground. The energy applied by part C on part A at the collision is (n/11)mhg or 2mgh or 200/(11n) % of the total Potential Energy, PE, of the tower. It is not much! It seems to be about SE max (2) !!! (The parameters have been adjusted to get this result)



Evidently all springs in both parts C and A compress elastically at contact by the forces that develop at the interface C/A, before one spring breaks, if any. Which spring breaks first, if any? These questions are best answered by removing other springs! Experiment 2 - Bottom loading! Let's remove the springs below the bottom #1 m and drop the upper part on ground h below! Then part C consists of n=22m, it becomes completely decompressed under free fall and the energy applied by part C on ground (part A!) is 22mgh. It is 11 times more than in Experiment 1! Evidently the ground applies energy and force on part C! Which springs in part C break first, if any? Experiment 3 - Middle part loading! We can also suddenly remove, say springs between levels 6 and 12, so that part C - 11 m - drops 5 h 'free fall'. Then part C, again completely decompressed, applies 55mgh energy on part A - 6 m. The 5 m between parts C and A also drop and add 15mgh energy on part A. Total energy applied at 6 contacts - which take some time - is 70mgh. It is 35 times more than in Experiment 1! Which springs break then first? The top ones in part A or the bottom ones in part C? Answers In Experiment 2 it is evidently a spring in part C that breaks first, unless all springs in big part C just compress, apply a big force on ground, the ground applies a big force on part C that bounces. Please note that bottom #1springs are 22 times stronger than top #22 springs. One question remains unanswered! How much energy is absorbed by ground and how much of it is absorbed by part C? In Japan they put shock absorbers between towers and ground to avoid earthquakes to shake towers into pieces from below. Evidently the ground can absorb plenty of energy. In Experiment 1 it is evidently the top #22 springs in part C that break first, if any (compare Experiment 2), when both parts C and A first compress, apply forces on one another, as the C springs are the weakest. The energy available, 2mgh, may be split 50/50 to parts C and A and big part A can handle 1mgh. Top #20 springs in part A are three times (!) stronger than the #22 C springs, etc. Top #22 m of part C may then drop on part A, as #21 m did a little time before and the same thing will happen again: part A, decompressed due to top #22 springs having failed, part A compresses again (!) when #22 m contacts, applies a force on the #22 top m of part C that bounces, as there are no more springs above to break. Part C cannot crush part A! You could say that part A crushed part C! This is what normally happens! In Experiment 3 it is again the weaker springs in part C that breaks first but as the energy applied is big also springs in part A may break at the same time or a little later. This effect is what we call Controlled Demolition, CD. So a Funny m tower may be demolished by CD! Conclusions Suggestions by NIST and professor Z P Bazant that you can one-way crush down and destroy a steel framed structure part A by dropping a little top part C on it due to lack of strain energy that can be absorbed is utter nonsense. Towers are built so they are stronger at bottom than at top and energy applied will also be transmitted to ground and to horizontal elements! Also, when something drops on a structure, it is not the total mass of the dropped part C that matters! It is the energy applied and the associated forces and what damage they do that count. It seems NIST has watched too much TV! The myth about the Twin Towers had its genesis in the immediate aftermath.

Just after the destructions of the Towers, FOX News cut to a "man in the street", an eyewitness (?) who explained what would later become the official story born at Ground Zero. FOX News interviewed the "passerby", who somehow explained, " . . . I witnessed both Towers collapse, one first and then the second, mostly due to structural failure because the fire was just too intense." In a state of near shock, using the jargon of structural engineering, this man speculated on the cause of the catastrophe. In doing so, he foreshadowed what later became the official view and myth. However, the clown on the street was wrong! A tower or any steel structure of elements joined together cannot collapse due to structural failures up top and by gravity! This does not prevent other clowns and true believers of fundamentalism at Internet forums to continue to maintain that steel tower structures globally collapse due to gravity, when a little part drops free fall (!) due to local structural failures up top. They even try to build structures of different sizes from 1 to 10 meters height to prove it - with various built in weaknesses at the bottom plus a heavy top that can be dropped free fall. This is very good! As they will always fail - the structure will collapse by itself during assembly - they might finally come to the conclusion that a global collapse of a normal steel tower structure after a drop of a small part C on the remainder big part A due to gravity alone is impossible. In reality real steel structures do not behave as outlined above - only springs compressing vertically! In reality top parts do not suddenly drop free fall , and, if it happens, the springs s will punch holes in the m elements or slip off and the m elements will get entangeled into one another, friction develops and arrest follows, etc. I have described it here ! In reality a steel structure also contains much more strain energy, strength, than the one in the columns. There are horizontal elements and numerous connections, which all deforms in tension and compression and absorb energy. Explanation: To destroy a steel structure you really have to manually cut all supporting elements, the columns/springs of a big section using energy other than the one supplied by gravity and stored inside the building as per Experiment 3. That WTC 1 and WTC 7 were destroyed using energy other than provided by gravity on 9/11 should be clear to anybody that does not believe in fairy tales. But we are living in a funny world full of religious fundamentalists of all types and that's not funny. They need their myths to go about their evil ... or holy, in thier eyes ... work. Recommended homework Many persons agreeing to above still maintain that a one-way crush down is still possible (!!) because it is not the springs s that fail but the connections between the springs (columns) and m (floor)! They are the same throughout the structure! And at WTC 1 on 9/11 it was not 2 m but 14 m that dropped on the top #97 m (and 96 other m below) and overloaded it, etc, etc. This is the so called pancake theory. For that to be valid all connections c between 14 masses (floors) and springs (columns) in upper part C must suddenly fail and that is not possible. There were 1000's of connections c in WTC 1 upper part C. Even the fundamentalists do not believe that! Recommended homework is however to make a Funny m tower, where eight connections c fail, i.e. the springs (above and below) get detached from m due to gravity loads at contact part C with part A - see picture right - while the springs s remain intact. The homework includes explaining connections c, incl. details, safety factors, intact load transfers (m to s), failure mode(s) and why c would fail in the first place and why various m should drop and the time table when the two ms contact lower structure.

When homework is ready, please send a copy of the updated pancake model to Anders Björkman Heiwa Co and it will be published here! For the Advanced Student The Funny m assembly is very simple. Only one m and four identical springs s. You can also join 9 Funny ms to a Super Funny m assembly shown below: Then you have 9 m supported by 16 springs s. Four of the springs just carry 1/4 m each - the outer corner ones . Eight springs carry 1/2 m - the intermediate, external ones . Four springs carry 1 m each - the core springs . The springs compress exactly the same as described above and when you put Super Funny m assemblies on top of each other, the springs adjust themselves for the load carried above. Imagine a Tower with 22 Super Funny m assemblies (see left) and that you redo the Experiment 1 above with it; thus all 16 of the assembly #21 springs are suddenly removed and top part C - now 18 m - drops down free fall (quite a feat to remove 16 springs of different sizes in one go!) and that part C collides after free fall with part A below and 32 springs fuses perfectly. Which one of the 16 #22 springs fails first? The weaker outer corner ones or the stronger core ones? Or the intermediate, external ones? Answer: The weakest springs always fail first, i.e. the outer corner ones, and the four corner #22 ms will drop before any other level #22 m. Then, other #22 springs will fail, e.g. the intermediate, external ones and other #22 ms will drop, i.e. the four external, intermediate ones, and finally the core springs may fail and the core m will drop. Now the lower part A has even less problem to resist as the 9 ms of level #22 of part C do not impact simultaneously. Recap! In a collision between two similar structures but of different size, e.g. a small part C dropping on a big part A (or two ships in a horizontal collision), it is always the weakest sub-elements of the composite parts that fail first. Stronger elements need more time to fail! The various masses or elements supported by the weak, failed elements will then displace first. Other masses may displace later but as they do not act in uniform any intact structure, e.g. part A below, has little problem to deflect them, entangle them or simply arrest them with assistance of friction. Only religious fundamentalists and terrorists believe that a weak element can destroy a strong element, or a weak part C can destroy a strong part A and fair enough! You may believe what you want. But try to prove it! It is not easy! Much easier to prove the opposite. Other articles by Anders Björkman - WTC 1 - The Case for Collapse Arrest , Why a tower cannot destroy itself from top Read also Anders Björkman about WTC 7 and JME ! Heiwa Co home page