Ballistic limit assessment for concrete slabs using the MHJC concrete model

Transcription

1 Ballisi limi assessmen or onree slabs using he MHJ onree model M. Polano-Loria INEF Maerials and hemisry, N-7465 rondheim, Norway O.. Hoersad &. Børvik ruural Ima Laboraory (IMLab, Dearmen o ruural Engineering, Norwegian Universiy o iene and ehnology (NNU, N-749 rondheim, Norway ABRA: A modiied version o he Holmquis-Johnson-ook (MHJ model was develoed by he auhors o handle ima and eneraion roblems in onree. In his modiied version a new oninuous ressure-shear union is adoed where he inluene o he hird deviaori sress invarian is onsidered; in addiion, a new srain-rae sensiiviy ormulaion is inluded and inally hree damage variables desribing he ensile raking, shear raking and ore omaion mehanisms are inrodued. Model arameers are obained or wo onree qualiies and eroraion o onree slabs by oined rojeiles is onsidered numerially and omared wih exerimenal resuls rom he lieraure. Ballisi limi assessmens wih deviaions under 8 % when omared o he exerimens are obained. INRODUION onree is a maerial o ariular ineres or he nulear indusry and oriiaion insallaions or deene uroses. Many imoran sudies o he behaviour o onree arges imaed by rojeiles have been o an emirial naure and mos o he exerimenal sudies have oused on he deerminaion o emirial ormulae, ennedy (976, Ben- Dor e al. (5. On he oher hand, he arallel develomen o numerial ools and new onsiuive models or onree has onribued o an inreased use o numerial simulaions or rediion o eneraion and ima roblems, Holmquis e al. (98, Bourlion (997, Riedel (998. Dieren onsiuive models or onree are available or he analysis o onree sruures; however, he develomen o reliable and robus maerial laws or onree subjeed o high-inensiy loading o shor duraion is subje o urren researh. omlex sae o sress is generaed in reinored onree sruures imaed by ree-lying rojeiles. Under a muli-axial sae o sress he damage mehanisms aivaed are highly deenden on he loading ah imosed. For onree maerials under sai loading ondiions, Mazars (984 showed ha deending on he load ah imosed, dieren damage mehanisms are aivaed: raking, shearing (e.g. mode II raking and omaion. In ariular, or ima and eneraion roblems hese hree mehanisms are always resen, as indiaed by Bourlion (997. his ruires he develomen o maerial models aable o desribe onree behaviour under low and high onining ressures. A lieraure review o exising exerimenal daa and onsiuive models or onree maerials under ima loading ondiions was underaken, Polano-Loria (, and i was onluded ha he HJ onree model, Holmquis e al. (98, reresened a good omromise beween simliiy and auray or large-sale omuaions. ome imrovemens o he original HJ model were reommended and evaluaed, Polano-Loria (, whih orms he basis o his modiied version (MHJ resened in his work. HE MODIFIED HJ ONREE MODEL. Pressure deendene In order o avoid he disoninuous desriion o he original model and irumven he ideniiaion o he ohesion arameer, he MHJ model ados a simle oninuous union deined by N σ = B P + D F ε R θ e ( ( (, max & ( and van- <. Nex, = is he normalized uivalen sress, P = P is he normalized ressure and & ε = & ε & ε is he normalized srain rae, where ε& is he uivalen deviaori srain rae. his uaion holds or P ( D ishes or P ( D σ σ

2 he normalizing arameers are he quasi-sai uniaxial omressive srengh and he reerene srain rae ε&. Furher, B is he ressure hardening oeiien, N is he ressure hardening exonen, is he srain rae sensiiviy oeiien, and max is he normalized maximum srengh ha an be develoed. Maerial degradaion is desribed by he damage variable D, resuling in reduion o he ohesive srengh. In he negaive ressure regime ( P < he normalized hydrosai ension = is inrodued wih as he maximum hydrosai ension he maerial an wihsand. he new unions F ( ε& and R ( θ, e are deined below. By assuming.8 B. and.6 N.8 he MHJ maerial model agrees wih exerimenal resuls or he omressive meridian, reored in he lieraure (hen 98, assuming an undamaged sae as illusraed in Figure. In a omlee damaged sae ( D = onree behaves as a granular maerial and he erm ( D in Equaion vanishes σ max B=. N=.8 B=.8, N=.6 P in whih is he deerminan o he deviaori sress ensor. he arameer e is a shae aor ha desribes he ou-o-roundness o he deviaori rae. By adoing he Willam-Warnke onree model, hen (98 ound ha he shae aor hanges rom.684 o.75 when he ressure goes rom. o.. For moderae ressure levels < P <, Launay & Gahon (97 reored a shae aor around e =.7. Based on hese observaions, a linear deendeny beween he ressure and he shae aor was assumed. he reduion aor R( θ, e is inrodued in a muliliaive way in Equaion as roosed by Riedel (998.. Rae deendene In he original HJ onree model he srain rae inluene F ( ε& is deined as a linear union on a logarihmi sale o he srain rae and haraerized by he sloe. However, in order o avoid negaive values o F ( ε& or relaive srain rae values & ε <, while keeing he wo-arameer ormulaion, he ollowing exression roosed by amaho & Oriz (997 and largely used by Børvik e al. (999 or meals is adoed, viz. ( F & ε = + & ε (4 where he arameer desribes he non-linear haraer o he rae ee. Praially he relaive inrease in omression srengh, ound in he lieraure, (EB 988, Bisho & Perry 99, is always relaed o he sai ase whih involves srain rae 5 values o abou s. For his reason we assumed suh a value as he reerene srain rae ε&. Figure Undamaged omressive meridian srengh resonse o he MHJ model.. Inluene o he hird shear sress invarian ri-axial exerimens on onree learly demonsrae he subsanial dierene o shear srengh beween he omressive and ensile meridian, hen (98. A reduion o he shear srengh on he omressive meridian an be onsidered by inroduing a union R deending on he deviaori olar angle θ and he normalized ou-o-roundness arameer e, as roosed by Willam & Warnke (975, viz. R ( θ, e = ( θ ( ( 4 ( e osθ + ( e e + e e + e e os 4 os θ 5 4 where θ is deined as 7 θ = os σ ( (.4 Pressure-volume resonse he ressure-omaion resonse is illusraed in Figure. he ressure P is deined as a union o he volumeri srain, as in he original aer o Holmquis e al. (98 μ = (5 where and are he iniial and aual densiies, reseively. In omression, he behaviour is divided ino hree regions. he irs region is linear elasi and limied by ( μ rush, Prush. A his sae he seond region sars, whih involves rushing o he onree and roduion o lasi volumeri lok lok. he air voids are hen assumed o be ully omressed ou o he onree (omaion damage. In he hird region, he onree is ully dense, i.e. all air voids are removed rom he maerial. he irs and seond regions are modelled by a lassial inremenal elaso-lasi-damaging ormulaion wih a linear srain hardening and salar damage srains, and i oninues unil ( μ, P

3 (omaion assumions, while he hird region is modelled by assuming ha onree is omleely elasi (rushed maerial wih no ensile aaiy. P P lok P rush P lok P rush D= μ rush (-D av Η μ lok μ lok D= μ [ D D ] ( + P = μ + μ + μ μ = Figure. Pressure-volume resonse o he MHJ model In ariular, he oal volumeri srain inremen is searaed ino is elasi and lasi onribuions, aording o e Δ μ =Δ μ +Δ μ (6 he volumeri srain hardening modulus H, see Figure, is deined by Plok Prush H = (7 μ lok he volumeri lasi srain inremen Δμ is ound wih a lassial elasi redior-lasi orreor roedure. For his a rial ressure P rial is alulaed assuming an elasi behaviour as P = μ (8 rial av where he averaged elasi bulk modulus av is deined aording o he omaion damage value ( D, o be deined laer as = ( D + D (9 av By omaring he ressure in he revious se and he rial alulaion one an dedue he inremenal volumeri lasi srains aording o i Prial Pi Δ μ = Prial P ( i i Prial > Pi ( av + H Finally, an udaing o he lasi volumeri srain and ressure is ruired. he ressure-volume behaviour in he ully omaed region ( P> Plok ollows a non-linear elasi behaviour. For his a modiied volumeri srain is inrodued μ μ lok grain μ = = ( + μ lok where grain is he grain densiy and i shows ha μ reresens he volumeri srain o he grain maerial. his maerial ollows a non-linear elasi behaviour o ye P= μ + μ + μ ( his uaion is used as long as D = (ully omaed maerial. he modiied volumeri srain μ is used in he onsiuive relaion so ha he onsans, and are uivalen o hose or a maerial wihou voids. Beause no volumeri lasi srains an be generaed or ensile (negaive ressures, he elasi redior sheme works also in his regime by udaing he ressure aording o e P = max( μ, ( D ( i av i I is ineresing o observe ha in he ully omaed zone D= D =, he ensile hreshold vanishes..5 Damage behaviour he main idea in his work is o rea he hree basi damage mehanisms searaely, and or his urose hree inernal damage variables D, D and D, reresening he ensile, shear and omaion damage, reseively, are inrodued..5. ensile damage (brile raking Prediion o ensile raking has been largely sudied and basially hree yes o models are ommonly used: disree, smeared and damage models. In hese models he inroduion o a rak ormaion rierion is ruired. By simliiy, i was deided here o use he hydrosai ensile srain as he main indiaor or rak ormaion. For his, he minimum value o he volumeri srain μ (in he ensile regime aained during he loading hisory is assumed as he uivalen srain or rak ormaion and i reads ε = min( ε, min(, μ (4 where he suersris and - indiae he aual and revious inremen, reseively. he ensile damage rierion is simly deined by or ε > ε D = (5 or ε ε where ε = is he volumeri ensile srain hreshold or rak ormaion..5. hear damage he umulaive damage develomen roosed in he HJ original model, Holmquis e al. (98, is adoed here. However, damage rom shear and volumeri sraining is searaed. he evoluion o he shear damage variable D is deined by

4 Δε Δ D = (6 ε he lasi srain o raure ε is here adoed as in he original model in he orm ε = α P β + ( ε MIN (7 where α and β are onsans. he hird damage onsan ( ε MIN is inrodued o allow or a inie amoun o lasi srain o raure he maerial..5. omaion damage Damage omaion due o lasi volumeri srain is deined by he ressure-volume law. For his mehanism he ohesive srengh o he onree is los during air voids ollase (i.e. he ore omaion onribues o damage and he bulk siness inreases (i.e. he bulk modulus aroahes o ha o a omaed maerial. hus, an inernal omaion damage variable D an be deined as Δμ Δ D = (8 μ lok Here, Δμ is he inremenal lasi volumeri srain (see Equaion and μ lok is he lasi volumeri srain o he ully omaed granular maerial (an inu arameer..5.4 oal damage he ombinaion o hree dieren damage mehanisms, deined by he damage variables D, D and D, ino one reresenaive salar value D is quesionable. Under he ramework o he salar damage heory we deide o exlude D in he averaging roosal based on he argumen ha ensile raking ee is redued by he resene o he seel reinoremen (i resen and he omeiion beween rak losure and oening aused by he omressive and ensile waves roagaion. hus, he damage variable D is used only as a ensile damage indiaor (is used only or os-roessing uroses. We onsider D and D in he averaging roedure where he oal damage ee is alulaed aording ( D = ( D ( D (9 A similar averaging roedure was roosed by Riedel (998. A omlee desriion o he MHJ onree model an be ound in Polano-Loria e al. (6. PARAMEER IDENIFIAION A irs, a omlee alibraion o he arameers involved in MHJ seems omlex and exensive beause he ideniiaion o 9 arameers ruired. However, his ask an be simliied i one onsiders a redued number o ess omlemened wih inverse modelling and assumions relaed o some values. For insane, he maerial srengh haraerizaion ruires he uniaxial omressive srengh and he iniial densiy, while he uniaxial ensile srengh an be relaed o he omressive srengh, i.e. =.54, aording o he FIB/EB (99. Also he Young s modulus E an be relaed o he omressive srengh, i.e.. E = 7( (ellevold e al A Poisson s raio o ν =. an be assumed. he shear-ressure behaviour an be reresened by assuming values o.8 B. and.6 N.8, see seion.. I should be oined ou ha seial onrees an deviae largely rom he B and N values reviously roosed. As a maer o a, in he exerimenal rogram on eroraion o onree slabs, ondued by Hanhak e al. (99, wo onree qualiies _48 and _4 wih seial glassy gravel (alled eilaoon Glaial were used. he hardness o his ye o gravel is slighly lower han a quarz mineral ye. he shear-ressure resonse o a series o riaxial ess reored by Hanhak e al. (99 is illusraed in Figure. By a iing roedure one inds values o B =.4, N =.65 and B =., N =.45 or he onree qualiies o _48 and _4, reseively. o sum u, wo riaxial omression ess (in addiion o he uniaxial omression would be reerable or aurae ideniiaion o B and N σ _48 MHJ: B=.4, N=.65 _4 MHJ: B=., N= Figure. hear-ressure resonse. Ideniiaion o B and N Nex, some o he arameers involved in he ressure-volume law an easily be ideniied. For insane, rom elasiiy Prush = and μrush = ( ν E, while μ lok an be assessed rom he densiy o he gravel maerial ( grain and he iniial densiy o onree ( using μlok = grain. Unorunaely, he ideniiaion o P lok and (,, ruires hydrosai ess a very high ressure levels. Based on ommon resuls rom he lieraure Plok 6 MPa = 85 MPa = 7 MPa and = 8 MPa are reresenaive values or onree maerials (Holmquis e al. 98, Johnson e al In order o avoid he ideniiaion o duiliy diagrams (raure srain P

5 versus riaxialiy, values o α =.4 and β =. an be assumed o reresen he onree damage behaviour wih reasonable auray (Holmquis e al. 98, Johnson e al he raure arameer ( ε MIN wih values o. (low srengh qualiy and.5 (high srengh qualiy an be inrodued o reresen dieren embrilemen haraerisis. Finally, or eneraion roblems he rae sensiiviy arameer an be ideniied by inverse modelling in he high ima veloiy region (i.e. rom 5 o m/s. Wih his roedure values o =.4 (_48 and =.5 (_4 were ound. hese indings urher orroborae he exerimenal observaions ha high srengh onrees are less sensiive o variaion in srain raes (EB 988, Bisho & Perry NUMERIAL IMULAION OF PERFORAION OF ONREE LAB 4. Exerimenal sudy o Hanhak e al. he ballisi limi omuaions are based on he es erormed by Hanhak e al. (99, where square reinored onree laes o mm were esed. hree layers o square-aern reinoremen seel rods wih a diameer o 5.6 mm were used. wo onree qualiies _48 and 4 wih uniaxial omressive srengh o = 48 and 4 MPa and uniaxial ensile srengh o = 4 and 5 MPa were reored. In addiion o he ressureomaion urves, riaxial ess were erormed under various onining ressure levels suh ha shear srengh versus ressure urves ould be esablished, see Figure. A -mm, smooh-bore owder gun was used o launh.5 kg ogival-nose seel rojeiles wih a lengh o 4.7 mm and a diameer o 5.4 mm. In he ess, iniial and residual rojeile veloiies were measured. hese values were used o onsru iniial versus residual veloiy urves or he wo onree qualiies and rom hese diagrams ballisi limis were dedued. Hanhak e al. s main onlusion was ha even hough he unonined omressive srengh was inreased by a aor o hree, he ballisi limi veloiy only inreased by %. 4. onree slabs wih = 48 MPa he MHJ onree model was imlemened in L- DYNA (999 and inie elemen analyses wih D axisymmeri elemens were used in he simulaions. A redued inegraion sheme wih hourglass onrol was adoed. For he onree slab a oal o elemens were used hrough he hikness and 5 elemens along he radius. he seel reinoremen was no inluded in he simulaions sine is ee on he eroraion resisane was ound negligible (Holmquis e al. 98. he se o inu arameers assumed are dedued rom he reommendaions o seion. he seel rojeile was modelled using a von Mises maerial model (Ma_ in L-DYNA wih linear isoroi hardening. he main daa used or he rojeile inluded: E = GPa, ν =., σ Y =.7 GPa and E = 5 GPa (angen modulus. No srain rae ee was onsidered. he original densiy o he rojeile (8 g/m was slighly modiied o 8 g/m o obain he oal launh akage mass o.5 g. In he resen alulaions, we adoed he elemen erosion oion o L- DYNA wih a rierion based on he maximal rinial srain wih a ailure srain value o ( ε MAX =. (based on similar values used in Holmquis e al. 98. he D_auomai _single _surae ona oion o L-DYNA was used o deine he ona behaviour beween seel and onree wihou riion. he rae sensiiviy arameer value o =.4 was used based on he residual veloiy rediions o he revious seion. he numerial rediions o he residual veloiy are omared wih he exerimenal indings (Hanhak e al. 99 or he _48 onree in Figure 4. he MHJ omares very well wih he exerimenal values, in ariular or ima veloiies higher han 4 m/s. he redied ballisi limi (5 m/s deviaes by less han 5% rom he exerimenal value (4 m/s. he ensile damage (raking evoluion during he eroraion roess is illusraed in Figure onree slabs wih = 4 MPa he uniaxial omressive and ensile srengh o his onree was = 4 MPa and = 5MPa, reseively. A new se o inu arameers based on he reommendaions o seion is used. We used he maerial daa o he revious examle or he rojeile. he elemen erosion oion wih a ailure srain value o ( ε MAX =. and he D_auomai_single _surae ona oion wihou riion o L- DYNA were one again adoed. A rae sensiiviy arameer value o =.5 was used or his ase (see seion. In addiion, he lasi raure srain limi is redued o ( ε MIN =.4 in order o aoun or he brile naure o high srengh onree. he numerial rediions o residual veloiy are omared wih he exerimenal indings as shown in Figure 4. Also in his ase he MHJ omares well wih he exerimens, and he redied ballisi limi (7 m/s deviaes by less han 8% rom he exerimenal one (4 m/s.

6 8 6 4 Residual Veloiy (m/s Ex_48 MHJ Ex_4 MHJ Ima veloiy (m/s Figure 4. Ballisi limi rediions using he MHJ Figure 5. ensile damage develomen during onree eroraion aer. ms 5 ONLUION A modiied version o HJ model or onree maerials subjeed o ima loading has been invesigaed. In his modiied version a new oninuous ressure-shear union is adoed where he inluene o he hird deviaori sress invarian is onsidered; in addiion, a new srain-rae sensiiviy ormulaion is inluded and inally hree damage variables desribing he ensile raking, shear raking and ore omaion mehanisms are inrodued. A roosal or arameer ideniiaion is rovided. Ballisi limi assessmens wih deviaions under 8%, when omared o exerimenal resuls rom he lieraure, were ound, indiaing ha he MHJ model reresens a good omromise beween simliiy and auray or large saleomuaions o onree laes imaed by rojeiles. REFERENE Ben-Dor, G., Dubinsky, A. & Elerin,. 5. Ballisi Ima: Reen advanes in analyial modeling o lae eneraion dynamis-a review. Alied Mehanis Reviews, 58: Bisho, P.H. & Perry,.H. 99. omressive behaviour o onree a high srain raes. Maerials and ruures, RILEM, 4: Burlion, N omaion des beons: elemens de modelisaion e araerisaion exerimenale. Ph.D. Disseraion, LM, EN de ahan, Frane. Børvik,., Langseh, M., Hoersad, O.. & Malo,.A Ballisi eneraion o seel laes. Inernaional Journal o Ima Engineering, 8 (: amaho G.. & Oriz M Adaive Lagrangian modelling o ballisi eneraion o mealli arges. omuer Mehods in Alied Mehanis and Engineering, 4: 69-. EB 988. onree sruures under ima and imulsive loading. EB bullein d'inormaion No. 87, Laussanne, wizerland. hen, W.F. 98. Plasiiy in reinored onree. M Graw Hill, New York. FIB/EB 99. ae o he ar reor: High srengh onree wizerlad. Hanhak,.J., Forresal, M.J., Young, E.R. and Ehrgo, J.Q. 99. Peroraion o onree slabs wih 48 MPa and 4 MPa unonined omressive srenghs. Inernaional Journal o Ima Engineering, (: -7. Holmquis,.J., Johnson, G.R. and ook, W.H. 99. A omuaional onsiuive model or onree subjeed o large srains, high srain raes and high ressures. Proeedings o 4h Inernaional ymosium on Ballisis, anada. Johnson, G.R. Beissel,.R., Holmquis,.J. & Frew, D.J omuer radial sresses in a onree arge eneraed by a seel rojeile. Proeedings o 5 h Inernaionla onerene on ruures under hok and Ima V. omuaional Mehanis Publiaions: ennedy, R.P A review o roedures or he analysis and design o onree sruures o resis missile ima ees. Nulear Engineering and Design, 7: 8-. Launey, P. & Gahon H. 97. A sudy o he ailure o onree under ombined omressive sress. Universiy o Illinois, Engineering Exerimenal aion, Bullein No. 85. UA. L-DYNA eyword User's Manual. Ver. 95, Livermore oware ehnology ororaion, L. Mazars, J Aliaion de la meanique de l'endommagemen au omoremen nonlineaire e la ruure du beon de sruure. hese de Doorae d'ea, LM, Universie de Paris, Frane. Polano-Loria, M.. onsiuive models or onree maerials under ima loading ondiions. INEF reor F4 F5, rondheim, Norway. Polano-Loria, M.. Imrovemens o he HJ onree model in L-DYNA. INEF reor F4 F86, rondheim, Norway. Polano-Loria, M., Hoersad, O.., Børvik,. & Bersad,. 6. Numerial rediions o ballisi limis or onree slabs using a modiied version o he HJ onree model. ubmied or ubliaion o he Inernaional Journal o Ima Engineering. Riedel, W Ein makroskoishes, modulares Beonmodell ür Hydroodes mi Veresigung, ädigung, Enesigung, Drei-Ivarianenabhägigkei und ae", Berih 7/98, Fraunhoer Insiu ür urzzeidynamik-erns-mah- Insiu, Freiburg i. Br. ellevold, E., Jusnes, H, melass,. & Hansen, E.A eleed roeries o high erormane onree. Advanes in emen and onree, Ed. Gruzek and arkar, New England: Willam,.J. & Warnke, E.P onsiuive model or he riaxial behaviour o onree. eminar on onree sruures subjeed o riaxial sresses, IABE Pro. 9, Ialy.