Dynamics of Collapse of a High Building

Jou nal o Mechanics Enginee ing and Au oma ion 2 (2012) 349-354 Dynamics o Collapse o a High Building I an Němec1, Ma ina Ju áňo á2, I an Še čík1, Pe F an ík1 and Zbyněk Vlk1 1. Facul y o Ci il Enginee ing, B no Uni e si y o Technology, B no 602 00, Czech Republic 2. Ins i u e o Fo ensic Enginee ing, B no Uni e si y o Technology, B no 602 00, Czech Republic Recei ed: Ap il 28, 2012 / Accep ed: May 23, 2012 / Published: June 25, 2012. Abs ac : This pape is a con ibu ion o he discussion abou he all o he New Yo k Wo ld T ade Cen e (WTC) owe s in 2001. The di e en ial equa ion o he collapse o a high building is de i ed by aking in o accoun many in luences. A compu e simula ion o he collapse o he WTC building is p esen ed using wo independen p og ams wi h pa ame e a ia ions. The esul s o bo h, di e en ial equa ion and compu e simula ion, a e compa ed esul ing co espondences. The au ho s conside ce ain p obable pa ame e s which would ha e lessened bo h he obse able speed o he collapse and i s ex en , howe e unce ain y ega ding he magni udes o he pa ame e s emains. Key wo ds: Dynamics, collapse, WTC (Wo ld T ade Cen e ). 1. In oduc ion This pape deals wi h he all o a high building. I s aim is no o in es iga e he cause o he all as Bazan did in Re . [1], bu o examine he alling p ocess i sel . The pape is based on a inal hesis pu o wa d by Ju ano a [2] and in oduces heo y o he all o a high building. The p ocess o he all is in es iga ed om he poin o iew o he basic laws o mechanics and conside s only he dynamics o he collapse in e ms o speed and he ex en o he all. A di e en ial equa ion o a high building collapse is de i ed and all majo in luences o he alling p ocess a e included. Resul s o bo h, he nume ical solu ion o he di e en ial equa ion and he compu e simula ion a e p esen ed o he a ious pa ame e s along wi h he co espondence ound be ween he app oaches. The p esen ed solu ion was also compa ed wi h a s udy by Ku le [3] based on a disc e e app oach, which achie ed a e y posi i e Ma ina Ju áňo á, Ing., esea ch ield: o ensic enginee ing. I an Še čík, Ing. Ph.D., esea ch ield: enginee ing mechanics. Pe F an ík, Ing. Ph.D., esea ch ield: dynamics. Zbyněk Vlk, Ing. Ph.D., esea ch ield: dynamics. Co esponding au ho : I an Němec, associa e p o esso , Ing. Ph.D., esea ch ield: nonlinea mechanics. E-mail: [email p o ec ed]. conco dance. The pape is o ganized as ollows: Sec ion 2 p esen s he de i a ion o he di e en ial equa ion o he all o a high building; sec ion 3 in oduces he solu ion o he di e en ial equa ion de i ed in sec ion 2; sec ion 4 discusses he magni udes o he pa ame e s used in he di e en ial equa ion p esen ed abo e; sec ion 5 pu s o wa d a compu e simula ion o he all o a high building and i s compa ison wi h he nume ical solu ion o he di e en ial equa ion; sec ion 6 p esen s he s udy o K. Ku le and i s compa ison wi h he solu ion o he abo e p esen ed di e en ial equa ion; in sec ion 7, conclusions a e s a ed. 2. De i a ion o a Di e en ial Equa ion o he Fall o a High Building A simple scheme o a alling building is illus a ed in Fig. 1. The au ho s assume ha columns in he loca ion be ween he coo dina es ´ x and 0 x lose s abili y and a op pa o he building abo e ´ x s a s o all and hi s he s ill undamaged lowe pa o he building unde he loca ion 0 x wi h eloci y 0 . The equa ion o dynamical equilib ium o he loca ion x is DAVID PUBLISHING D Dynamics o Collapse o a High Building 350 Fig. 1 The scheme o alling building. 0−−−−= NmCa GF F F F (1) whe e: Gis weigh o a pa o he building abo e he loca ion x, o which equilib ium equa ion is o mula ed; N Fis esis ance pu up by he columns agains he collapse; m F is esis ance o igina ed by impac o he alling pa o he building in o a mo ionless mass; C F is a iscous damping; a F is an ine ial o ce o a alling mass. Following is a discussion o he indi idual pa s o he equilib ium equa ion. y The weigh o he building abo e he loca ion x: β =Gmg (2) whe e m is he mass o he building abo e he loca ion x , g is accele a ion o g a i y, β is po ion o he o al mass abo e he loca ion x which pushes o a lowe pa o he building. Mass which alls ou side o he building is sub ac ed. y The columns esis ance: κ = N Fmgs (3) whe e s is a a e o he ul ima e o ce o columns o he cu en o ce in columns in he momen o he collapse, κ is he ac o o he ul ima e o ce o columns which ep esen s a e age column esis ance du ing i s de o ma ion ela ed o an ul ima e o ce. Compu a ion o column p essing was done using he me hod o con olled de o ma ion in o de o in es iga e his ac o and ecei e i s ope a ional cha (See Fig. 2). Fac o κ is hen he a e o he ul ima e o ce o i s median alue. y The esis ance o a mo ionless mass: I is ine ial o ce o a s ill mass dm accele a ed in ime d o he speed (/=ad d ). The e m / = a d can be used o accele a ion in he equa ion due o accele a ion s a ing om ze o up o he speed . The o ce m F can be hen exp essed in his way: m Fdmadm d = ⋅= ⋅ (4) When conside ing /, dxd = Eq. (4) can be ew i en as ollows: 2 2 μ =⋅= m Fdm dx (5) whe e / μ = dm dx is a line densi y o he building. y The iscous damping: α =⋅= c FC m (6) whe e C is a ac o o he iscous damping. The au ho s a e conside ing Rayleigh damping he e which depends on mass quan i y α =Cm only. y The ine ial o ce o a alling mass: βββ =⋅= = a d d Fmam m d dx (7) Again, only he ine ial o ce o a mass which does no all ou side o he building is conside ed he e. 3. Di e en ial Equa ion o he Building Collapse The ollowing equa ion can be go en by subs i u ing ela ions de i ed abo e in o Eq. (1): 20 βκμαβ − −− − = d mg mgs m m dx (8) The au ho s di ide he equa ion wi h speed and mass m and adjus 0 0 β α − −− = + b d xx dx (9) whe e ( ) β κ =−bg s . The ela ion () 0 μ += x xm is used when adjus ing he equa ion. Analy ical solu ion o he di e en ial equa ion was ound only when he in luence o damping was Dynamics o Collapse o a High Building 351 omi ed: () () () 2 0 01 2 2 β β − + =++⋅ + bx x x x x C (10) To speci y he cons an 1 C, he magni ude o he speed 0 is needed. This is he speed o he mass 0 m abo e 0 x alling in o he undamaged pa o he building. The au ho s s a om he same di e en ial equa ion whe e ac o bis modi ied in o () 00 β κ =−bsg, whe eas 01 < s: () ( ) ( ) 00 ´ β −− = + x d x b x x x dx (11) The solu ion will hus ha e a simila o m. Bounda y condi ions ( ) 00= will be used o inding he magni ude o he in eg a ion cons an . Then we a e looking o ( ) ( ) 0 ´´ x x=. A e ha he au ho s can e u n o a sea ch o he in eg a ion cons an 1 C: () () () 2 00 001 2 2 β β − + =++⋅ + bxx x xx C (12) ( ) ( ) () () () 2 000 12 0 22 2 β β β − +− + = ++ x bxx C xx (13) The au ho will es ablish 1 C in (10) and ob ain he solu ion o Eq. (9) wi hou he in luence o damping: () () () ()( ) () () () 0 2 2000 02 0 2 2 22 2 β β β β β − − ++ + =+− + ++ ⋅ ++ bx x x x bxx xx xx (14) Then i can be ound ou ha when he collapse o he building will s op by assuming he speed o be ze o ( ) 0 = x . A solu ion o his equa ion was no ound in he closed o m so he au ho s a e going o e u n o ea Eq. (9) wi h a nume ical me hod. The Eule implici me hod o sol ing a di e en ial equa ion was ound o be he mos sui able me hod. I s p inciple is ( ) () 1, +=+⋅ ii ii h x x (15) The ollowing equa ion can be go en: ()() () () 1 1 110 α β + + ++ ⎛⎞ =+ − − ⎜⎟ ⎜⎟ + ⎝⎠ i ii ii x hb x x x x x (16) A e deduc ion o he speed ( ) 1+i x , he ollowing ela ion will be go en: () () () () ()()() () () 1 10 1100 2 10 1 0 1 1 0 0 1 2 4 i i iii i ii iii i x hx x hx x x hx x x h x x bhx bhx hx x x hx x x ββ αβ αβ ββ α β αβ + + ++ ++ ++ =++ ⎧ ⎫ −+ −+ + ⎪ ⎪ ⋅⎨ ⎬ +− + + − − + − + − ⎪ ⎪ ⎩ ⎭ (17) The speed () 0 will be compu ed in a simila way. I is also impo an o say ha his equa ion will s and only i he ollowing condi ions a e p esen : y When he mass is e enly sp ead ou o e he heigh o he building (app op ia e o high- ise buildings); y When he c i ical s eng h o he columns (including hei esis ance) is p opo ional o he weigh o he building abo e he espec i e columns. 4. Discussion o he Magni ude o Damping, he Sa e y Fac o s and he Ul ima e Fo ce Ra io κ Be o e he au ho s began o seek a solu ion o he equa ion in a nume ical way, hey a e going o cla i y he magni ude o he damping α : 22 ω ξ α ωξ == = n n m C mm (18) Fo he a io o damping, he au ho s will conside he alue 10-30% and he limi ing alue 0 ha yields he damping a ios 0.147, 3.18, 0 α αα = ==. Value 0 will be conside ed in o de o sol e he collapse wi hou he e ec o damping. The e is less ce ain y ega ding he alue o he pa ame e s . I is assumed o be a ound 2.5-3. The coe icien o he ul ima e s eng h in columns κ was compu ed om simula ion o p essing o he columns by he me hod o con olled de o ma ion. The pe inen nume ical analyses we e pe o med u ilizing he RFEM p og am [4]. A de o med shape and pe inen esponse diag am is shown in Fig. 2. Dynamics o Collapse o a High Building 352 0 50 100 150 200 250 300 350 400 450 500 o ce [kN] Fig. 2 Response diag am and he de o med shape o a box column. I is clea om he g aph ha he alue o he κ coe icien will be a ound 0.25. 5. Compu e Simula ion Fo he compu e simula ion o he all o a high building, he RFEM and FyDiK p og ams we e used. Bo h p og ams use he explici me hod. The RFEM is a ini e elemen p og am whe eas he FyDiK uses an in e se app oach, mass poin s connec ed by elas oplas ic sp ings. The di e ences be ween he esul s o bo h p og ams we e small. Figs. 3-4 show he esul ing de o ma ion o he building a e he all has s opped. Fo he chosen pa ame e s, which he au ho s belie e could be p obable, he all o he building would s op a e alling cca 80 m. The main nume ical esul s o he compu e simula ion and i s compa ison wi h he solu ion o he di e en ial equa ion is p esen ed in Table 1. 6. A S udy by P o esso Kenne h Ku le and I s Compa ison wi h he Abo e P esen ed Solu ion o he Di e en ial Equa ion In 2006, p o esso Ku le o Bi gham Young Uni e si y also ca ied ou a s udy o he WTC collapse [3]. In his s udy, he conside s only he impac o he alling mass on o he mo ionless mass as i he loo s we e loa ing in he ai unsuppo ed by columns. The collapse i sel is slowed only by he c ash o alling loo s on o he mo ionless mass (hi ing he lowe loo s). Fig. 3 De o med building om RFEM (α = 0.5; s = 3). Fig. 4 De o med building om FyDiK (α = 0.5; s = 3). Dynamics o Collapse o a High Building 353 Table 1 Times and ex ends o he all o a ious pa ame e s. s (-) α The heo y wi h alling away o a mass x (m) RFEM x (m) FyDiK x (m) The heo y wi h alling away o a mass (s) RFEM (s) FyDiK (s) 2.0 0.147 330 331.0 278.7 25.8 15.7 19.0 2.0 3.18 330 18.2 63.2 357.1 7.4 31.5 3.0 0 330 259.8 287.1 36.1 17.9 20.4 3.0 0.147 330 324.5 304.2 103.0 28.6 33.1 3.0 0.5 330 79.6 72.3 330.5 12.1 28.1 3.0 1.06 330 64.6 66.5 707.3 15.9 37.4 3.2 0.147 36.5 69.0 6.0 29.0 12.5 76.0 His solu ion is based on he law o he conse a ion o ine ia and is disc e e: () 2 312 22 2 211 211 1 12 2 222 2 11 1 1 12 1 2 21 celk n kkk jj hh gh gh gh h k j gh j gh kk k =−− == =+ ⎛⎞⎛ ⎞ ⎛⎞ ⎛⎞ ⎜⎟⎜ ⎟ ++ + ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠ +⎛⎞ ⎛⎞ ⎛⎞ − ⎜⎟ ⎜⎟+++ ⎜⎟ ⎜⎟ ⎜⎟ − ⎝⎠ ⎝⎠ ⎝⎠ ∑ ∑∑ (19) As an example, he au ho s p esen a demons a ion o Ku le ’s solu ion in compa ison wi h his app oach. A building o 411 m high wi h 110 s o ies is aken. h will be 411/110 = 3.74 m, g = 9.81 m/s2 and n = 110. F om Eq. (19), i will be calcula ed ha such a building would all acco ding o his app oach o 14.96 s. In his app oach, he au ho s ha e u ilized he same assump ions as p o esso Ku le , he di e en ial equa ion aking he o m: () ( ) 0 0−−= + x gd x x x dx (20) I is possible o a i e a a solu ion in his case in closed o m. The au ho s will s ill ake a quan i y xA, which will be he place, whe e he all will s op. Ma ginal condi ions a e clea , he speed is ze o a he poin o all cessa ion. The esul is () ( ) 33 6 3 − =A g xx x x (21) I is possible o de e mine om his esul he all ime by an in eg a ion such as: () 2 33 00 66 3 6 == == ∫∫ AA xx AA A A xx dx x dx x x g xg xg (22) P o ided conside ing he same inpu da a as abo e, all ime is 15.85s. In ac he all o he buildings ook abou 11 s ( he ime di e s in a ious epo s, i is he accu acy o he ime span can be e i ied om ideo eco dings). The ma e ial p esen ed in his sec ion is o cou se g ea ly o e simpli ied. In ac uali y he si ua ion is mo e complex. O he mo e complica ing in luences may be p esen , o ins ance ela ing o column esis ance, which migh b ing in o being a subs an ial inc ease in all ime. Fu he mo e conside a ion should be gi en o he possibili y ha he all could e en cease be o e he whole building des uc s. 7. Conclusions I was possible o sol e a di e en ial equa ion o he building collapse in closed o m only when he damping was omi ed. Real damping mus be a necessa y elemen when he massi e des uc ion o all he componen s o he building s uc u e. The e o e his solu ion ep esen s limi o he speed o he all and he ex en o he collapse. The gene al o m o he di e en ial equa ion was sol ed only in a nume ical way. Two independen compu e p og ams we e used o he simula ion, named RFEM [4] and FyDik (by P. F an ík). Despi e he di e ence in he app oaches bo h compu e p og ams ga e compa a i ely simila solu ions. The di e ence be ween he solu ion o he di e en ial equa ion and ha o he compu e simula ion is g ea e . The di e ences ha e wo sou ces. The i s cause is he ac , ha in he di e en ial Dynamics o Collapse o a High Building 354 equa ion i is assumed ha he a e age esis ance o he columns is he mean alue om he diag am in Fig. 2. On he o he hand he compu e simula ion wo ks wi h a mo e gene al app oxima ion o he column esponse diag am. In his case he compu e simula ion is be e . The compu e simula ion howe e did no ake in o accoun he a io β assuming a uni alue whe eas he di e en ial equa ion can assume an a bi a y alue o his pa ame e . Re e ences [1] Z.P. Bažan , Y. Zhou, Why did he Wo ld T ade Cen e collapse?—Simple analysis, Jou nal o Enginee ing Mechanics ASCE 128 (1) (2002) 2-6. [2] M. Ju áňo á, Dynamická analýza kolapsu ýško é budo y, Final Thesis, VUT, B no, 2011. [3] K.L. Ku le , A sho compu a ion, Jou nal o 9/11 S udies 1 (2006) 1-3. [4] I. Němec, e al., Fini e Elemen s Analysis o S uc u es, Shake Ve lag, Aachen, 2010.