1994-1 pioneering in expandable structures. the madrid i notebook by leonado da vinci

8/6/2019 1994-1 Pioneering in Expandable Structures. the Madrid I Notebook by Leonado Da Vinci

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BULLETIN OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES:IASS

PIONEERING IN EXPANDABLESTRUCTURES:THE "MADRID 1"

NOTEBOOK BY LEONARDO DA VINCI

J . PEREZ V ALCARCEL

University of La Corua (Spain)

F. ESCRIG

University of Sevilla (Spain)

MMARY

andab/e Structures based on scissors are o f g re at u ti /i ty and may seem to be a recentention . Neverthe/ess the discovery o f a reference to one o f such structures in a notebook bynardo da Vinci p roves that experiences were car ri ed out in 16th Century to use them as

ineering de v ices .

TRODUCTION

pandablebar

structuresare sets

of

culated parts that can be carried as smallkages, s ince their size is just that of the

ded bars. Using manual or mechanicalices, these structures can be unfoldederever their use is required. Among others, they can serve as architecturalelopes, military or civil ian construct ions,rgy collectors, reservoirs, containers andil ia ry e lement s such as stairs, antenna,

nel advert isements, bridges , e tc .

veral types of expandable structures arerently used . Among them, we will consider

structures , groupings of crossed barsnnected at their tips. Connect ions arequired to allow folding and unfold ing.

e s pa nis h architect Emilio Prez Piero ismmonly considered the pioneer of these

uctures . Such a recogn ition is based on hiss tan di ng c ont ri bu ti ons . H ow eve r olderorts can be found on the sub ject : even in

Antiquity one can f ind l i terary descr ip t ionsmechan isms that could be interpreted as

pandable bar s truc tu res. However some

uncertaintv exists in the interpretat ion ofthese ancient reports.

LEONARDO'S RESEARCH ON EXPANDABLESTRUCTURES

The discovery of an unambiguous reference toone of such structures in a notebook byLeonardo da Vinci p laces in th e Renaissancethe first design in this f ie ld . On page 24 in the

"Madrid-I" notebook , from the collection ofthe Spanish Nat ional Library, an X -dev ice isdepicted, with the follow ing legend: "Rendim iregione che forza debbe in m a llevare 4 librein n" ("Give me reason of th e power requiredin m to Iift 4 pounds in n" ) (Fig. 1). On page25 (right of th e former) an interest ing,expandable umbre ll a like structure can beobserved (Fig. 2) . This structure is a singleexample in the m an us cr ip t, t hu s making

diff icult an interpretation of it s significance.

Moreover on page 143 of the manuscr ipt th esame mechan ism is shown (Fig. 3), W it h twointeresting gear systems, based on an end lessscrew, which w ill be considered later.


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VOL. 35 (1994) n. 11 4

~, . j i ?.-

. ,

e Iifting mechanism descr ibed by Leonardonot exceedingly efficient, bu t it is th e first

scription of a flat unfoldable structure.erefore it has g reat h istorical interest.ditionally it ma y provide information aboutonardo's knowledge on Statics.

e should first inquire whether Leonardoew how to solve the problem or simplyended to establish it , in the hopes of aure solution . Th e second possib ility seems

kely. In other sections of the same manuspt Leonardo cons iders complex subjects

ch as beam deflect ion (tor which there wasm at he ma ti ca l s ol ut io n at his time);

yway , Leonardo described th e results of hisperiments. In th e abundant studies on

. r, ~ ,.

Fig. 1

equilibrium of wire -suspended weightsLeonardo always gives numerical solut ions. Itis interesting to point out that the "Madrid-I"notebook basically contains anything butresults, wh ile other notebooks c on ta in l on gexplanations and demonstrations (forinstance, the "Hammer" notebook) or detailedarithmetical operat ions (for instance, th e"Madrid- II " notebook, p robably written in adifferent year). Instead "M ad ri d-I " m ost lycontains results, with short and precise

comments . Therefore it seems Iikely that, ifLeonardo had known the way to solve theproblem, the N otebook w ou ld include th eresults or at least some guidel ines.


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Fig.2

Taking moment about crossing point (Fig. 4)

H L coso: + ~ l lseno: +!.-l. seno: - HJ. lJ' coso: =02 2 J

Since th e drawing mentioned does no t giveany scale

The mechanism described by Leonardoreceives th e charge through a ball or a roller.Since this action is made on t he t an ge nc ypoint , the equations will be:

1 = 0 ,70 0 m .14 = 0 ,425 m .

lB = 0,2625 m.la = 0 , 1625 m.

1 = 0,9125 m .13 = 0,550 m .15 0 ,3 3 75 m.

17 = 0 , 2 00 m .

Leonardo cou ld no t solve th e problem, ity be of interest to solve it with ou r presentources, to determine it s difficulties and topothesize which ones could no t be overme in older t imes.

second question is whether Leonardo built

no t some experimental model of th e meni sm mentioned . Thi s is Iikewise unlikely.on trying lt , he would have realized themendous difference between the power

uired to keep th e mechanism folded andt of co mp le te u nfo ld in g. Th us he wouldve probably remarked this point. Howevers noteworthy that th e mechanism is drawni t a 45 unfolding , the only position that

be easi ly achieved .


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VOL. 35 (1994) n. 11 4

r,0 1 -- ~ I ' ' ' . . .-'.'---?

>-_. 6.1' ~ '-1. , "Te k " J.".::.. _:. .-' ' l ~ ' ' ' ' ' 1 : . ~ ~

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Fig.3

e material is no t specified, a lth ou gh t he

apeof th e articulations suggests

an iron barsquare section, f lat tened at t he t ips and th entre, with bolts. We will consider a 2x 2ction as th e basic designo

nder these conditions we have made equil ibum calculations, fo r diameter of 5, 10 and5 cm. and considering various bracket posi

ns. Since our s tudies on expandable struc-res have s ho wn t ha t major deflections can

importan t, the calculations have beenpeated considering non-linear effectsrough the matricial method although restrict

g ball's diameter to 10 cm . beca use some

analysis carried ou t show l it tl e inf luence con-cerning with th is variable at least in th e rangecons idered by Leonardo. Next table showshorizontal reactions on pinned ends beingfunction of t he d is tance between them. Loadis supposed 4 Lb. reactions are in these units.

The results a re as follows:


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I ILinear Analysis

1

Non Linear

1Analysis

Distance between 13 = 5 cm. 13 = 10 cm . 13= 1 5 c m . 13 = 10 cm .supports

d = 1,8 m. 76 ,9 4 77 ,05 77,16 78,1

d = 1,7 m. 33,03 33 ,16 33,28 33 ,26

d = 1,6 m . 23,56 23,70 23,84 23,71

d = 1, 5 m . 18,70 18,86 19,02 18 ,7 5

d = 1 ,4 m. 15,55 15,73 15,92 15,62

d = 1,3 m . 13,25 13,47 13,69 13,30

d = 1,2 m. 11,46 11,72 11,97 11,52

d = 1,1 m . 10,01 10,31 10,61 10,05

d = 1,0 m. 8,78 9,14 9,51 8,83

d = 0,75 m . 6,64 7,09 7,74 6,47

R= P2 sen o1jP/2

Fig .4


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VOL. 35 (1994) n. 11 4

,

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Fig .5

e first conclusion obtained from these

ults is th e poor performance of th e mechsm. In an a lm os t f ol de d p os it io n a bo ut 77

unds are required to lift th e 4 pounds prosed by Leonardo. Th is effort varies asnction of th e tangents as th e mechanismogressively unfolds. On the other hand ,re is ev idence that non -linear effects,

hich w er e co mp le te ly u nk no wn in Leonars time (although in th e same Notebook aerence is made to th e phenomenon ofckling), have Iittle importance for the d imenns and loads given. Thus an hypothet icalchanism constructed by Leonardo would

ve depar ted v er y li ttle fr om th e results

obtained through an equilibrium calculation,theoretically feasible.

Let's analyze br iefly the equilibrium calculationunder the light of Renaissance knowledge.The mechanism described is made of a num-ber of levers, which laws, established byArchimedes, were well known by medievalscient ists. In the notebook there are manyfigures and mechanisms that confirm Leonardo' s authority in th e matter. Particulary interesting is page 89 right (Fig. 5) where theproblem is well stablished , even in a way thatsuggest actual moment of a force. Thus,equilibrium equat ions can be replaced byLever Laws and used in his statements.


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BULLETIN OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES :IASS

Fig .6

me complication arises from considering an-punc tua l d iamete r in th e weight, albeit

results are little affected . The problem ofreactions of a ball wa s actually addressed

Leonardo in this same notebook. Theution is wrong. Specially interesting is page

right (Fig. 6) . Text is : this body hanging byfrom second, as is 8 Lb. weight, 4 are

ding on "f" and 4 on "r". An d this is shownmeans of central line "arn", that ends at

ntre " a" of th e line "rn" .. . Really it is a anfortunate texto No t o nl y t he result is wrongt th e demonstrat ion shows a lack of knowlge opposed to well d on e stud ie s on st r ingt ics contained in th e same Notebook. We

ppose that studies on no n punctual loadsre not advanced.

If th e load is considered point -l ike (a logicalassumption, given th e scale of th e d ra ft) , th eproblem can be solved simply by decomposingth e load in th e directions of th e bars of th efirst X; t he n th e lever laws can be sequentiallyapplied.

The only doubt in the former rationale iswhether Leonardo knew or no t th e parallelo -gram rule that a ll ow s t he decomposition of aforce in two directions. Histor ical evidencea bo ut th is subject is contrad ictory. It is wellknown that Pierre Varignon gave in 1725 anexpl icit formulation of th e parallelogram rule.With respect to Leonardo's studies, Timo

shenko rel ies on t he f igure 7 on t he Notebookto state that Leonardo d id know that la w


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VOL. 35 (1994) n. 11 4

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Fig.8

hout any doubt . However our research onMadrid-I notebook, which is probably the

st complete on th e subject, advises someti on about .

. 8 shows page 77 right of the Madr id Itebook along with their transcription in Fig.By means of string equilibrium rules it is

own that laws proposed by Leonardoested and general rule" are a lways t rue. Itsurprising because he does no t demostrates

it is of difficult practice, and to show it,ual knowledge was no t enough.

xt pages 77 left and 78 right, shown ings. 10 and 11, show different constructionsere that general rule is used.

Nevertheless, once achieved this notablegeneral rule, th e Notebook uses it only fo rparticular cases which can be solved withinteger or s imple fractional numbers.

It is well known that Leonardo usuallyattempted to salve problems with integernumbers or at least with simple fractionary

Fig.7


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m f bTranscr pton of page 77 right ,

"Tested and general rule"

- - - - - - - - - ~ - - -- - - - -IIIII

etwe en "tS" and "Sv " it will be the same prop ortion than betw een we ight "b " and "r ". And prop ortion be tweeneight ''f '' and "b" and "r" togheth er w ill be the same than betw een "x v" di stance and "Sx" line o

etw een wei ght "e" and w eight hangin g at "a " will be the same proportion than bet ween distances "ab " and "be ",the same wa y proportion between weights "m" and "ac " togeth er will be th e same than betw een lines "bm " a nd

me" ,

o not for get that tw o strin gs that han g the weight mu st be al ways e qual. l f on e was infinitety l onger than the other,ou w ill tak e an equidistant mea sure from h anging point of the weight and so you will carry on the anal ysis as d oneefore to a chieve a general rul e.

9

bers. Hence we may pay attention to th estruction of Figure 12 (page 156, right). Itesents weight hanging from a thread,ilibrated by two addit ional threads eachng through a pulley. The geometr ic con-

ction allows th e design of angles whosene is a fractionary number. In th e casen in th e figure, cos a = 3/4 . The tensionth e thread and hence the necessarynterweight would be 12 .09, that, is aboutvalue 12 given by Leonardo (incidentally,ten aboye ano ther unident if ied result,ly a wrong one).

wever th e most surprising feature of the

is that there are very fe w rational valuesh e cosine allowing integer or almost inte -

values. Within a reasonable range ofues only 3/ 5 and 4/5 wh ich correspond to

egyptian t ri angle, wel l known in Leonar -time, allow integer values of the tension.

The mentioned value 3/4 allows an almostinteger value and probably at tracted Leonar-do's curiosity , since he pays more attention toits descr ip tion than to any other one. Withrespect to th e angles of simple geometric

construction, integer values are only a llowedby 30 (and almost integer by 15). These areth e cases found in th e pages of the Notebook.

The conclusion is that Leonardo surely carriedou t many exper imen ts with threads. Theseexperiments enabled him to find even a hardlyintuit ive case like th e one cited. He likelysolved all th e cases in which an integer sol-ution wa s possible.

He also found a general rule. With it he couldsolve correctly s tr ing equ il ib rium problems.Bu t he could no t obtains reactions of sus-pended weight.


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BULLETIN DF THE INTERNATIDNAL ASSDCIATIDN FDR SHELL AND SPATIAL STRUCTURES :IASS

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KNOWLEDGMENT :

research has been achieved with financialport provided by CICYT (Comisin Interisterial de Ciencia y Tecnologa) .

Fig. 77


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VOL. 35 (1994) n. 11 4

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Fig. 12

ANDELA, r .. PREZ PIERO, E.;CALATRA-VA, S.; ESCRIG, r. & PREZ VALCARCEL,J.

99 3. " Ar qu itectura Transformable" . Escuelade Arquitectura. Sevilla.

IMOSMENKO, S. 19530 "History of thetrength of Materials" oMe Graw Hill.NoY.


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BULLETIN OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIA L STRUCTURES:IASS

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VOL. 35 ( 1 9 9 ~n. 11 4

TORROJA MEDAL AWARDED TO PROFESSOR

A. SCORDELIS

emarks at Presentation dur ing the Opening Cereony of the lA SS-A SCE International Syrnposium,pril 24, 1994, A ti en te, Georgia, USA, by Prossor John F. Abel, Vice President of the IASS

h e Eduardo Torroja Medal of th e IASS is awardednce every three years in recognition of outstand-g and distinguished contributions to the developent of th e field of shell and spat ial structures orr exceptional service to th e Association.

Eduardo Torroja was the founding presient of th e IASS. I think it is app ropr iat e to beginis a wa rd c er em on y with

a fe w remarksabout

orro ja and his work in order to help set the conxt both of t he a wa rd and of the IASS.

Tor re ja , a Spaniard, was an outstandingructural e ngineer. He wa s one of t hos e unusualeople wh o excel in many facets of his professionalfe -- he wa s a designer of structures; a humanist;

professor and teacher ; a productive researcher;nd the d irec tor and manager of such enterprisess th e Laboratorio Cent ra l and the Techn ical Ins tite of Construction and Cemento It was his visionhich led to th e creation of the IASS in 1959 as a

means fo r bringing together -- from all over the

orld -- architects, engineers, and builders with anterest in shell and spatial structures . We are

ortunate to have as his legacies the Association,e bui lt s truc tures he designed, the continuingaboratorio , and his students and proteges (manyf whom are active in th e IASS). We also have aonderful book , "The Structures of Eduardo Torro," in which he himself describes th e rat ional and

maginative process by which he created hisesigns. This book, like his structures, is elegant,parse, and r ati ona l. In one sentence which I woldke to read from his preface, he distills his philosphy: "My final aim has always been for theunctional , structural, and aesthetic aspects of aroject to present an integrated whole, both inssence and appearance." No more needs to beaid to define th e ar t and science of structuralngineering. Bu t I do wish to mention and showwo of Torroja 's most famous and enduring strucures.

The Zarzuela Racecourse in Madrid (1935)an e legan t example of the integrated project as

well as of hyperbolic paraboloid concrete shells.he Market Dome at Al ge ci ras from 1933 isno th er w el l known example of Torroja's work.

hisspher

icaldome

issupported

on only 8 colmns. T he c yl in dr ic al shell canopies stiffen thedges between th e columns and help red irec t theoof load to the columns while still maintainingg ht ne ss . T he columns themselves are no t sub

ected to hor izon ta l forces at their tops becauseh ff t b l l t i ti hi h

ciras dome "probably the first example of a prestressed concrete thin-shell roof."

Now, it is my great privilege to announcethe recipient of this year's Torroja medal -- Prof.Alex C . Scordelis. Professor Scordelis c1earlyfulfills all th e cr iteria of t he a wa rd , b ot h in contribu-tions to our field and in long ded icated serv ice toth e IASS. He is well known as a distingu ishedteacher at the Univers ity of Califor nia at Berkeleysince 1948. Moreover, his research on concreteshell s and box-g irde r bridges has had an enormousimpact on the art and science of structural engineering. His many f ormer students continue towiden this impact o He has also been consultant onthe design of a number of major projects -- fo rexample, the hypar roof of the stadium in Ponce,Puerto Rico. As a professor emeri tus he continuesto serve the profession on such commissions asthe California Governer's Commission on R eco mmendations for Se ismic Design and th e AdvisoryCommission on th e Retrofit of th e Golden GateBridge. He is a member of the U. S. Na t ionalAcademy of Engineering and an Honorary Memberof both ASCE and IASS.

He joined IASS in 1960 (one year after itsfounding) and has been active in many ways ,i nc luding in th e organizat ion of this Symposium .For example, he wa s the general secretary of th eI AS S Wor ld Congress on Shell Structures held inSan Francisco in 1962 ; he has been a member ofthe IASS Execut ive Council since 1976 ; and he hasserved faithfully and effectively in var ious irnport-an t capacities such as Chair of the NominatingCommittee from 1983 until th e presento

It is with great pleasure that I call uponPresident Stefan Medwadowski to present th emedal to vou , Professor Scordeli s.

1994-1 pioneering in expandable structures. the madrid i notebook by leonado da vinci

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