BYG DTU. Documentation for Calculations of Standard Fire Resistance of Slabs and Walls of Concrete with Expanded Clay Aggregate.

Transcription

1 BYG DTU Krisian Herz Documenaion for Calculaions of Sandard Fire Resisance of Slabs and Walls of Concree wih Expanded Clay Aggregae TECHNICAL UNIVERSITY OF DENMARK Repor BYG DTU R-48 December 22 ISSN ISBN (rev )

2 Krisian Herz Documenaion for Calculaions of Sandard Fire Resisance of Slabs and Walls of Concree wih Expanded Clay Aggregae Repor BYG DTU R-48 December 22 ISSN ISBN (rev )

3 1 Preface The Danish producers of expanded clay concree elemens and blocks have shown a remarkable iniiaive by foreseeing he need for reliable mehods for calculaing he load bearing capaciy of heir elemens exposed o fire. Since expanded clay aggregae concree mainly differ from he radiional heavy concree qualiies by he weigh and porosiy, bu no by oher subsanial mechanical differences, i is logical o presume, ha he basis for calculaion of expanded clay aggregae concree consrucions accord wih he basis for calculaing he fire resisance of heavy consrucions. This was previously esablished by he auhor and for example expressed in chaper 9 of he Danish Sandard DS 411 from 1999 [3] or he simplified calculaion mehod in chaper 4.3 of he CEN code prined as ENV from 1995 [4]. The basis for he calculaions of expanded clay aggregae concree consrucions was wrien in he repor "Calculaion mehod for fire safey design of consrucions of expanded clay aggregae concree" [1] from 1997 as a preliminary proposal for a ex for a Danish code of pracice for expanded clay aggregae concree. During he projec a number of full-scale ess has been made in order o provide a reasonable documenaion for he applicaion of he calculaion mehods, and he presen repor deals wih he second phase of full scale esing made in 22. The repor comprises also he resuls of he firs par of full-scale ess wih reference o he repor "Fire safey design of expanded clay aggregae concree - Calculaion of fire resisance ime" [2] from 21, and he resuls of he second phase of full-scale ess have been added. In addiion he resuls from a Norwegian full scale es on a Scan Brann Blokk wall has been released for he purpose of documening he calculaion mehods, and are adoped in his repor. Lyngby, May 22 In he revised ediion a small prining error of no significance for he conclusions has been correced in one of he spreadshees used for calculaing walls, and he numerical values of he wall calculaions are changed a few percen. Lyngby, November 22 Krisian Herz Acknowledgemens I would like o record my sincere hanks o he sociey of Danish producers of expanded clay aggregae concree elemens called Dansk Beonindusriforening (DBI), Elemen fracion (BIH) and Block fracion (BIB) and o Scan Brann Blokk for financing a number of full-scale ess, which can no only serve as a documenaion for he calculaion of expanded clay aggregae elemens bu conribue o he documenaion for calculaion of fire exposed concree consrucions in general. Krisian Herz

4 2 Conens Preface...1 Conens...2 Summary...3 Maerials...3 Temperaure calculaions...4 Anchorage calculaions for slab elemens...11 Slab elemens Wall elemens...21 Conclusions..35 References...36 Appendix 1 Tes daa recorded manually by he auhor...38 Appendix 2 Example of a slab calculaion...41

5 3 Summary A number of full-scale ess are made in order o documen calculaion mehods for fireexposed slabs and walls derived during a previous projec on fire exposed ligh-weigh aggregae concree consrucions. The calculaion mehods are derived, and hus have a logical connecion wih he calculaion mehods used for oher load cases. In addiion he mehods are shown o be valid for heavy concree consrucions by cooperaion wih ess for beams and columns, and a few slabs and walls. The wo es series phase 1 and 2 of his repor can herefore be seen as a necessary supplemen o show ha he mehods are applicable for slabs and walls of ligh weigh aggregae concree. I is shown ha he emperaures for sandard fire exposed cross secions can be calculaed, ha he ulimae momen capaciy can be calculaed for slabs, and ha he anchorage capaciy and he shear ension capaciy can be calculaed for slabs and ha a suppor of only 7 mm is sufficien for slabs wih deformed bars and he acual loads. I is also shown, ha he load bearing capaciy can be modeled for walls, if a deailed model for he hermal expansion is used, and if he calculaion is made in ime seps aking he ransien srains ino accoun. Maerials The concrees used for he ess of his repor is based on expanded clay aggregae and manufacured according o he requiremens of he Danish Sandards DS 414 and DS 42. The values for compressive srengh, ensile srengh and E-modulus are assessed as average values in a ho condiion in order make he calculaions comparable o he es resuls. This means ha differences beween calculaions and es resuls can only be ascribed o differences beween calculaion model and es mehod, and should no be influenced by a difference beween a characerisic and average values of he maerial properies. For he applicaion of he calculaion mehods in pracice, he characerisic values should be applied in sead, and herefore he load bearing capaciies mus be expeced o be somewha smaller. Fixed values of he hermal conduciviy used in simple emperaure calculaions are assessed as he values a 5 C. For a 6 kg/m 3 concree wih plasered surfaces (as used in phase 2 of he projec) and morar beween he blocks he final average densiy is 88 kg/m 3.

6 4 Temperaure calculaions The emperaure calculaions are made using a simplified mehod developed by he auhor and adoped in he Danish concree code DS 411 [3] represening an exac soluion o he Fourier equaion for hea conducion for a sinus variaion of he surface emperaure. This soluion is used approximaing he firs quarer of he sinus cycle wih he variaion of emperaure a he surface of a concree specimen exposed o a sandard fire. Hence his soluion is valid only for a sandard fire exposure and anoher calculaion mus be adoped in case he load bearing capaciy should be calculaed for an elemen exposed o a fully developed fire. The expression is given as Simple emperaure calculaion for a wall: T ( x, ) := 312 log( 8 + 1) exp( 1.9 k( ) x) sin π 2 k( ) x k( ) := π ρ c p 75 λ where is he ime in minues, ρ he densiy in kg/m 3, c p he specific enhalpy and λ he conduciviy of he concree. The empirical emperaure facor 312 is chosen o follow he surface emperaure of a sandard fire exposed concree aking he effec of evaporaion of waer from he concree ino accoun. In he firs phase of full scale ess repored in Herz and Hansen [2], a wall consruced of iles of qualiy 6, 12 and 18 kg/m 3 and hickness 1, 2 and 3 mm was esed (Andersen [7]) and emperaure profiles were recorded and compared o calculaed emperaure disribuions for he ime 6 minues giving a reasonable agreemen beween calculaion and es. T(x) Calculaed simple formula.. T1(x) Measured in 1 mm wall T2(x) Measured in 2 mm wall. T3(x) Measured in 3 mm wall = 6 min ρ= 18 kg/m 3 λ=.9 W/m C c p =1 J/kg C C T( x) T1( x) T2( x) T3( x) x m.3 T(x) Calculaed simple formula.. T1(x) Measured in 1 mm wall

7 5 = 6 min ρ= 12 kg/m 3 λ=.6 W/m C c p =1 J/kg C T(x) Calculaed simple formula.. T1(x) Measured in 1 mm wall T2(x) Measured in 2 mm wall. T3(x) Measured in 3 mm wall T( x) T1( x) 4 C T2( x) = 6 min ρ= 6 kg/m 3 λ=.3 W/m C c p =1 J/kg C T3( x) I is seen ha he curves calculaed by he simple formula gives oo low values a emperaures less han 1 C, because he formula is derived from an exac soluion for he harmonic oscillaion, which would proceed ino he negaive par, bu is cu off. The oher deviaions are comparable wih he uncerainy of he measuremens. The deviaions in he emperaure region less han 1 C have no effec on he load bearing capaciies, because no or small srengh reducions ake place here. From he following emperaure curved in fixed dephs is seen ha he measured emperaure is consan due o evaporaion a 1 C, and ha he simple formula is modified for his effec seen for heavy as well as for expanded clay aggregae concree x m.3

8 6 Temperaure developmen in he deph 25 mm for densiy 18 kg/m 3. Temperaure-ime curve from a 3 mm ile wih x = 25 mm ρ= 18 kg/m 3 λ=.9 W/m C c p =1 J/kg C C T( ) T3( ) m 12 Temperaure developmen in he deph 35 mm for densiy 18 kg/m 3. Temperaure-ime curve from a 3 mm ile wih x = 35 mm ρ= 18 kg/m 3 λ=.9 W/m C c p =1 J/kg C C T( ) T3( ) m Temperaure developmen in he deph 5 mm for densiy 18 kg/m 3. Temperaure-ime curve from a 3 mm ile wih x = 5 mm ρ= 18 kg/m 3 λ=.9 W/m C c p =1 J/kg C C T( ) T3( ) m 12

9 7 In he second phase of he full scale ess emperaures were measured a he reinforcing bars of deck elemens and a differen dephs of walls. Unforunaely he exac posiions of he poins of measuremen were no recorded, bu sill he resuls can serve as a furher documenaion by defining he deph obaining some agreemen beween calculaed and es recorded emperaure curve as a funcion of ime and hen comparing he emperaure profiles made by he recorded emperaures a he derived dephs agains he calculaed emperaure profiles for differen fixed imes. These resuls are shown for concree qualiy 18, 12 and 6 kg/m 3 from Andersen [11], [12] and [13] on he following pages, where he emperaures calculaed by he simple mehod are called T 2 and he emperaures from he ess are called T T and in addiion emperaures calculaed by a finie difference mehod are shown called T D.

10 8 Temperaure profiles for 18 wall a 3, 6 and 9 minues calculaed by proposed mehod T2 and by a Finie Difference mehod TD and measured a es TT. 3 minues T 2 ( x, w, 3) T D183 ( x) T T183 ( x) x T 2 ( x, w, 6) T D186 ( x) T T186 ( x) minues x T 2 ( x, w, 9) T D189 ( x) T T189 ( x) minues x

11 9 Temperaure profiles for 12 wall a 3, 6 and 9 minues calculaed by proposed mehod T2 and by a Finie Difference mehod TD and measured a es TT. 3 minues T 2 ( x, w, 3) T D123 ( x) T T123 ( x) x T 2 ( x, w, 6) T D126 ( x) T T126 ( x) minues x T 2 ( x, w, 9) T D129 ( x) T T129 ( x) minues x

12 1 Temperaure profiles for 6 wall a 3, 6 and 9 minues calculaed by proposed mehod T2 and by a Finie Difference mehod TD and measured a es TT. 3 minues T 2 ( x, w, 3) T D63 ( x) T T63 ( x) x T 2 ( x, w, 6) T D66 ( x) T T66 ( x) minues x T 2 ( x, w, 12) T D612 ( x) T T612 ( x) minues x

13 11 Anchorage calculaions for slab elemens By means of he emperaure calculaion program HEAT2 he emperaure condiions are calculaed in he corner, where a ligh weigh aggregae concree slab of hickness 2 mm ress on a suppor of a 2 mm hick expanded clay aggregae concree wall The anchorage srengh of a reinforcing bar wih cenre line 2 mm above he boom of he slab is calculaed. A ne of 1 mm masks is used exended 4 mm in he wall as well as in he slab. The following in-daa are used: Expanded clay aggregae concree wih λ=.6 W/mK, ρ = 1775 kg/m 3, c p = 1 J/kgK. Border condiions = 6 minues. No hea flux across surfaces excep he wo fire exposed inner surfaces, which are exposed by a surface emperaure developmen according o he one used for he simple emperaure calculaion from he previous chaper. The emperaure developmen in C is given by: sin(2π(-s)/144s), where 836 C is 312*log(8*6+1) C and 144s = 4*6*6s = 4 hours = he ime for a full period, where 1 hour is a quarer of a period equal o he heaing period. The resisance of hea ransfer a he surface is se o be raher low such as.1m 2 K/W, in order o make he surface emperaure vary as a sinus of max 846 C by he expression above, which is defined as a surface emperaure variaion. On he nex page he resul is presened as isoherms. Uilizing he faciliies of HEAT2 he emperaure profile in he deph 2 mm along he reinforcing bar is derived couned from he corner =. m and inwards. In addiion he emperaure profile in he disance 3 mm from he corner is derived, where he isoherms are parallel. The las profile is compared wih he measured profile from a 2 mm hick wall wih λ =.6 W/mK, ρ = 18 kg/m 3 and c p = 1 J/kgK, i.e. for approximaely he same concree. Temperaure profile along a reinforcing bar in he deph 2 mm above he boom of a slab calculaed by HEAT2 from he corner and inwards afer 6 min sandard fire a ligh aggregae concree slab 1775 kg/m 3. C 3 25 TC( x) x m.2

14 12 Temperaure profile hrough 2 mm ligh aggregae concree slab 1775 kg/m 3 afer 6 min sandard fire TP calculaed by HEAT2 compared wih simple calculaion T and wih measured emperaures T2 from 2 mm wall 18 kg/m 3 C TP( x) T2( x) T( x) x m.2

15 13 Temperaure disribuion in a corner of a slab and a wall afer 6 minues sandard fire calculaed by HEAT2 for expanded clay aggregae concree of λ=.6 W/mK, ρ = 1775 kg/m 3, c p = 1 J/kgK. Along he reinforcing bar he following emperaures are calculaed in he cenre line a he disance.2m from he boom of he slab and in he mid poins of 3 lamellas of 5 mm hickness as a funcion of he deph from he corner and above he wall: Dybde T.2 T.125 T.75 T.25 m C C C C

16 14 The anchorage capaciy is assessed as he minimum of he bond capaciy for pulling ou he bar of he concree and he spliing capaciy for formaion of spliing cracks along he bar. The heory is presened in Herz [5]. The bond srengh is reduced along he bar by he reducion of he compressive srengh of he concree ξ c. Deph T.2 ξc m C Deph T.125 ξc T.75 ξc T.25 ξc ξc average m C C C From hese numbers i can be seen, ha he reducion of he bond capaciy for an anchorage lengh of 5 mm on average will be.979. The reducion of he concree conribuion o he spliing capaciy for an anchorage lengh of 5 mm on average will be.933.

17 15 The spliing capaciy of a 5 mm bar can hen be esimaed as 2π*.5m*.15m*.933*f c2 = 13.2 kn, if he ensile srengh of he concree a 2 C is assumed o be f c2 = 3. MPa. The bond srengh for a 5 mm Ø1 mm bar can be esimaed as.5m*.979*1.3*f cc2 *π*d/2 = 19.9 kn, if he concree compressive srengh is f cc2 = 19.9 MPa a 2 C. For a slab wih 8 reinforcing bars, he maximum shear capaciy will be 8*13.2 = 15.2 kn, where he load is 25 kn for a 5.5 m slab If he precondiions for his calculaion are valid, no shear failure should be seen in a massive slab of a expanded clay aggregae concree of qualiy 18 kg/m 3 wih a 5.5 m span and a wih of 1.2 m loaded wih kn in each of wo quarer poins even if he anchorage lengh is reduced o 5 mm in he second phase from 2 mm in he firs phase. One precondiion is ha he suppor is uniformly disribued a he 5 mm and ha he supporing consrucion is of he same hermal qualiy as he slab.

18 16 Slab elemens In phase 1 of full-scale ess (in he following marked 2) a massive slab and a sandwich slab were esed boh having a deph a suppor of 2 mm. The ulimae momen was shown o be decisive for he load bearing capaciy, and his varied as shown on he following figures according o he calculaions. Afer hese ess have been made he esing lab made some ess on presressed heavy concree slabs, which proved o fail very quickly afer he sar of he fire es. This gave reason o some debae, alhough here was a simple and predominan reason for he early failure: he slabs resed on bearing knos, and no concree was cased beween hem. Therefore he knos were able o expand side-wards and spliing cracks could develop along he reinforcing bars leading o a ension shear failure. Calculaions based on he principles shown in he previous chaper show ha an early failure should occur due o spliing even if he good bond properies of deformed bars were used. The debae caused by hese ess seem herefore o be quie irrelevan, alhough he bond srengh of he presressed wire sill have o be deermined if a precise calculaion of he anchorage capaciy should be made where spliing will no occur, i.e. where concree has been cased properly beween he bearing knos. However, caused by he debae, here was an undersandable wish o demonsrae ha a usual deph of 7 mm a he bearing is sufficien for he ligh weigh aggregae slabs. Therefore a calculaion of he anchorage capaciy of he deformed bars was made showing ha 7 mm should be more han sufficien o avoid a shear ension failure. The wo decks were placed a he oven wih he calculaed bearing deph of only 7 mm, and boh of hem clearly failed in bending almos a he prescribed ime proving ha he shear and anchorage capaciy can be calculaed for hese fire exposed decks. During he esing some verical ensile cracks were observed above he level of he reinforcemen a he end secions of he slabs. These cracks did also occur inbeween he reinforcing bars and herefore hey can no be iniiaed be hem. I was observed from he color of he concree how he moisure were conduced by he cracks. The obvious reason for he developmen of he cracks is hermal sresses, where his deph above he boom is subjeced o ension while he boom is compressed laerally during he es. Laer during he cooling phase afer he es he picure is reversed, and he cracks penerae down o he boom surface. The ess of phase 1 are repored in Andersen [1] and of phase 2 in Andersen [14] and [15].

19 17 Sandwich slab 2 phase m wide consising of 23 mm of 155 kg/m 3 and f cc2 = 17 MPa a op, 182 mm of 625 kg/m 3 and f cc2 = 2.8 MPa and f c2 = 1. MPa a middle and 35 mm of 15 kg/m 3 and f c2 = 3. MPa and λ=.6 W/mK a boom. The span was L=5.5 m and cover hickness d = 15 mm a 8 Y 8 mm bars of 55 MPa. The deck was loaded by 8.33 kn (2.5 kn/m 2 ) a a each 1/4 poin of he slab. 2 mm suppor kn m M u ( ) M( ) kn m The calculaed fire resisance ime was 63 minues. The esed fire resisance ime was more han 61 minues. Solid slab 2 phase m wide consising of 2 mm of 1775 kg/m 3 and f cc2 = 19.9 MPa and f c2 = 3. MPa and λ=.6 W/mK a boom. The span was L=5.5 m and cover hickness d = 15 mm a 8 Y 1 mm bars of 55 MPa. The deck was loaded by kn (4. kn/m 2 ) a a each 1/4 poin of he slab. 2 mm suppor kn m M u ( ) M( ) kn m The calculaed fire resisance ime was 57 minues. The esed fire resisance ime was 61 minues.

20 18 Sandwich slab Phase 2 a 7 minues. Deflecion and suppor. Sandwich slab 22 phase m wide consising of 23 mm of 155 kg/m 3 and f cc2 = MPa a op, 182 mm of 625 kg/m 3 and f cc2 = 2.8 MPa and f c2 =.3 MPa a middle and 35 mm of 15 kg/m 3 and f c2 = 2.7 MPa and λ=.6 W/mK a boom. The span was 5.63 m and cover hickness d = 15 mm a 8 Y 8 mm bars of 55 MPa. The slab was loaded by 8.11 kn (2.4 kn/m 2 ) a a each 1/4 poin of he slab. A he ime 6 minues he reinforcemen emperaure was measured o be 518 C and 486 C which gives an average of 52 C, where he calculaion has foreseen 53 C. The calculaion resuls are given below, where Q u means he ulimae shear force and Q he shear load. 7 mm suppor. Fire resisance ime a es was 73 minues. Q( ) = 15 kn M( ) = 21.2 kn m Q u ( 61) = 36 kn M u ( 61) = 21.4 kn m Calculaed fire resisance ime 61 minues = Q u ( ) = M u ( ) kn = (An increase of cover hickness from 15 o 25 mm kn m would give rise o an increase of fire resisance ime o 13 minues.) M u ( ) M( )

21 19 Solid slab Phase 2 a 8 min. Deflecion and suppor. Solid slab 22 phase m wide consising of 2 mm of 1775 kg/m 3 and f cc2 = 2 MPa and f c2 = 3.2 MPa and λ=.6 W/mK a boom. The span was 5.63 m and cover hickness d = 15 mm a 8 Y 1 mm bars of 55 MPa. The deck was loaded by 8.11 kn (2.4kN/m 2 ) a a each 1/4 poin of he slab. A he ime 6 minues he reinforcemen emperaure was measured o be 452 C, where he calculaion has foreseen 465 C. The calculaion resuls are given below, where Q u means he ulimae shear force and Q he shear load. 7 mm suppor. Fire resisance ime a es was 79 minues. Q( ) = 2.1 kn M( ) = 28.4 kn m := 5, Q u ( 69) = 112 kn M u ( 69) = 28.7 kn m Calculaed fire resisance ime 69 minues = Q u ( ) = M u ( ) kn = (An increase of cover hickness from 15 o 25 mm kn m would give rise o an increase of fire resisance ime o 114 minues.) M u ( ) M( )

22 As seen all ess show a good agreemen wih he ulimae load-bearing capaciies calculaed for shear, anchorage and bending, and i is possible o conclude ha he calculaion mehods seem o be well documened for bending as well as for shear and anchorage failure of hese ligh weigh aggregae slabs exposed o fire. 2

23 21 Wall elemens FIRE In order o calculae he load bearing capaciy of a fire exposed concree wall, some general problems have o be solved. Because he wall is exposed o fire a only one side, i will deflec ino he fire, giving rise o a considerable eccenriciy, which mus be aken ino accoun. A he same ime he concree secion is damaged a he fire exposed side giving an eccenriciy couneracing he eccenriciy of he hermal deflecion. Boh effecs will influence he disribuion of sresses in each ime sep of he fire, and he resuling sress disribuion again deermines he ransien hermal expansion. This is he free hermal expansion minus he srain, which can no ake place because he concree is loaded. This means ha he sress disribuion a a cerain ime is a funcion of he deflecion bu also influences he new hermal expansion and hereby he new deflecion and he new sress disribuion. I was herefore concluded in he firs phase of he projec, ha a calculaion has o be made in ime seps. And he walls esed in his par of he projec had deliberaely an iniial eccenriciy of he load, which gave compression owards he fire and herefore conribued he mos o he ransien srain. On he oher hand his eccenriciy also gave he mos sable condiions because he exernal load couneraced he hermal deflecion. In he second phase i was herefore decided o use load wih an eccenriciy away from he fire, which end o increase he hermal deflecion, and which mus be expeced o be he wors case and herefore he decisive load case for a wall. Doing his he es resuls would no only serve as a check for he calculaions, bu can also be used as a direc documenaion for he applicaion of he specific walls.

24 22 a c = damaged zone In he repor of he firs phase he suppor was modeled as a hinge a op and boom, bu i was learned ha in he es he walls had resed fla a he boom suppor. This gives rise o an eccenriciy, which was no calculaed correc. Since i seems o be he mos correc model of he suppor in pracice he supporing condiions were mainained in he es of he second phase, bu he calculaion model was modified in order o model i, and a quie new calculaion mehod was derived, and i was used calculaing all he ess. This means ha he ess of phase 1 were recalculaed wih he new border condiions, and ha he calculaions are no equal o hose in he firs repor [2]. The resuling eccenriciy a he boom is found as follows: Firs he widh of he supporing srip is calculaed as he load divided by he compressive srengh of he reduced cross secion. Then he eccenriciy from he middle of his srip o he cenre of he reduced cross secion is calculaed, where he srip sars in he deph of he damaged zone from he fire exposed side of he wall. This gives he momen load and he curvaure a he boom. The curvaure a op κ and a boom κ b is deermined, a he deflecion caused by his is calculaed as a funcion of he heigh z as ( ) := L z u z, κ, κ b, L ( ) z 6 L ( 2 L z) κ + ( L + z) κ b where L is he oal heigh of he wall.

25 ε TE1 ( xx, w, ) ε h ( xx, w, ).5 ε TE ( xx, w, ) ε h411 ( xx, w, ) xx.5 In order o find a reliable mehod of calculaing he hermal curvaure, a wall was divided ino 1 lamellas, and he emperaure, he iniial hermal srain ε TE1 and he srengh reducion and siffness reducion was calculaed for each lamella. The resuling curvaure for a plane cross secion wih inernal hermal sresses was calculaed ε TE, and compared o a more simple expressions such as he one given as a guide line for columns in DS411 [3] ε h411 and he proposed expression ε h used in his repor. The resuls is shown above for a 1 mm wall of qualiy 18 kg/m 3 a he ime 6 minues. The expression used here and in he following calculaions for he increase of he hermal curvaure in a ime sep is ( ) 2 εacl pral εcl pr( w, L) a Tc ( w, L) h c ( w, L) 2 w is he widh of he cross secion, and L he ime. εacl is he increase of he free hermal srain in he deph of he inner edge of he damaged zone, εcl is he increase of he free hermal srain a he middle of he reduced cross secion, pral and pr are he reducions of hese due o ransiens, a Tc is he deph of he cenre of he reduced cross secion from he fire exposed surface, and h c is he wih of he reduced cross secion.

26 24 The formula is derived assuming, ha he ransien hermal srains (or he equivalen sresses if hindered) are disribued following a parabola from he cenre of he reduced cross secion o he edge of he damaged zone, and reduced from his level and ou o he surface, such ha he area under he curve in he las par (in he damaged zone) is equal o he area in he firs par. The momen of he hindered hermal sresses is found and divided by he siffness of he reduced cross secion. As seen from he graph, he expression gives a good approximaion o he more complicaed calculaed curvaure.

27 25 Since he hermal deflecion is oumos imporan for he calculaion, i has been necessary o adjus he simple formula (1.1*1-5 *T) previously used for mos concrees, and in sead use a parabolic expression, which gives a beer fi o he observed values. The hermal expansion has been measured a he beginning of he projec by differen producers, and he following approximae formulas are derived for he 3 qualiies. In he graphs he expressions are compared o measured expansions. For 18 kg/m3 ε T ( T) := ( T) 2 per 1 ε T ( T) Alfa18( T) HH18( T) For 12 kg/m3 ε T ( T) Alfa12( T) For 6 kg/m3 ε T ( T) Alfa6( T) T T T ε T ( T) := ( T) 2 per 1 ε T ( T) := ( T) 2 per 1

28 26 In he following he calculaions of he eccenric loaded walls are compared o he resuls from he full-scale ess. The ess called 21 are hose from phase 1 are all of hem have an iniial eccenriciy owards he fire. This gives he mos complicaed echnical problem, bu is on he safe side compared o an eccenriciy away from he fire, which mus be regarded as decisive, and herefore his was used he ess of phase 2. The ess of phase 1 are repored in Andersen [6]-[9], and of phase 2 in Andersen [11]-[13]. All es were made wih a fla foo bearing a he boom excep he 12 kg/m 3 wall of phase 2. I was he idea ha his wall should have a hinge a he boom in order o verify he calculaion mehod for his simpler border condiion. However, he hinge consruced by he esing lab proved no o be able o incline sufficien, and afer 45 minues he boom could be regarded as a fla foo suppor for his elemen also. This is aken ino accoun in he successive calculaion, where he momen a he boom is assessed o be fixed during he firs 45 minues, and following he fla foo principle hereafer. The 3 m high wall of qualiy 6 kg/m 3 of phase 1 failed afer only 36 minues. Calculaing he developmen of he load bearing capaciy of his wall, he resisance ime is found o be larger. The auhor has no been presen a he ess of phase 1, bu i can be seen from he phoos ha he wall is made of blocks wihou a plaser added o he surface, such as i has been done a phase 2. Therefore, i can be assumed, ha one reason for he difference beween calculaion and es is he missing filling of edges of he joins beween he blocks. A second calculaion is herefore made where he joins are presumed o miss 1 mm filling a he surface, reducing he cross secion used for he calculaion of he Navier load bearing capaciy. A phase 2 he 6 kg/m 3 wall was only 2.4 m high, loaded wih only 7.5 kn/m, bu wih an eccenriciy away from he fire. In his case he wall proved o have a fire resisance of more han 12 minues, and he calculaion is in agreemen wih he es. The graphs show he calculaed load bearing capaciy F ul as a funcion of ime, and he calculaed deflecion a he middle of he wall u m and he eccenriciy of he load relaed o he cenre of he reduced cross secion e una and he measured deflecions D. All calculaions are made in ime seps of 1 minues, and each page compare a calculaion wih a full-scale es. I seems ha here is a good agreemen beween calculaions and es resuls, and ha he ess can serve as documenaion for he calculaion mehods.

29 27 18 kg/m3, 22 Phase 2 Wall suppored by a hinge a op and a fla foo a he boom Load P = 4. kn/m f cc2 = 2. MPa λ =.9 W/mC Widh of wall w =.1 m E c2 = 18. GPa ρ = 18 kg/m3 Eccenriciy op e op =.2 m (Posiive owards he fire) f c2 = 3.2 MPa c p = 1 kj/kgc Heigh L = 3. m max = 12 min = 1 min Tes soped a 85 minues 168 mm deflecion 4 kn/m Thermal expansion e(t) = 1.2*T2 Deflecion in m calculaed um, inernal euna and measured D and Ful in kn/m as funcion of ime in min. = um = Ful = i i i ( ) u := u = Calculaed resisance ime 83 minues Phoos afer 8 minues and a he ime of brake afer 85 minues um D euna Ful

30 28 18 kg/m3, 21 Phase 1 Wall suppored by a hinge a op and a fla foo a he boom Load P = 7. kn/m f cc2 = 2. MPa λ =.9 W/mC Widh of wall w =.1 m E c2 = 18. GPa ρ = 18 kg/m3 Eccenriciy op e op =.2 m (Posiive owards he fire) f c2 = 3.2 MPa c p = 1 kj/kgc Heigh L = 3. m max = 12 min = 1 min Tes soped a 78 minues 137 mm deflecion 7 kn/m Thermal expansion e(t) = 1.2*T2 Deflecion in m calculaed um, inernal euna and measured D and Ful in kn/m as funcion of ime in min. = i um = i Ful = i u := 6 + ( 7. 7) u = 6 Calculaed resisance ime 6 minues.13.1 um.7 D18.4 euna Ful

31 29 12 kg/m3, 22 Phase 2 Wall suppored by a hinge a op and a fla foo a he boom Load P = 25. kn/m f cc2 = 1.5 MPa λ =.45 W/mC Widh of wall w =.1 m E c2 = 8. GPa ρ = 12 kg/m3 Eccenriciy op e op =.2 m (Posiive owards he fire) f c2 = 2.2 MPa c p = 1 kj/kgc Heigh L = 3. m max = 18 min = 1 min Tes soped a 96 minues 173 mm deflecion 25 kn/m Thermal expansion e(t) = 1.1*T2 Deflecion in m calculaed um, inernal euna and measured D and Ful in kn/m as funcion of ime in min. = um = Ful = D12 = i i i r ( 26 25) u := u = Calculaed resisance ime 13 minues Phoos of he wall afer 95 minues from ouside and afer he es from inside he furnace um.1 D12.75 euna Ful

32 3 12 kg/m3, 21 Phase 1 Wall suppored by a hinge a op and a fla foo a he boom Load P = 35. kn/m f cc2 = 1.5 MPa λ =.45 W/mC Widh of wall w =.1 m E c2 = 8. GPa ρ = 12 kg/m3 Eccenriciy op e op =.2 m (Posiive owards he fire) f c2 = 2.2 MPa c p = 1 kj/kgc Heigh L = 3. m max = 18 min = 1 min Tes soped a 154 minues 13 mm deflecion 35 kn/m Thermal expansion e(t) = 1.1*T2 Deflecion in m calculaed um, inernal euna and measured D and Ful in kn/m as funcion of ime in min. = i um = i Ful = i D12 = r u := 9 + ( 36 35) u = 91.8 Calculaed resisance ime 91 minues um.1 D12.75 euna Ful

33 31 6 kg/m3, 22 Phase 2 Wall suppored by a hinge a op and a fla foo a he boom Load P = 7.5 kn/m f cc2 = 3.75 MPa λ =.3 W/mC Widh of wall w =.15 m E c2 = 2.5 GPa ρ = 6 kg/m3 Eccenriciy op e op =.15 m (Posiive owards he fire) f c2 =.3 MPa c p = 1 kj/kgc Heigh L = 2.4 m max = 2 min = 1 min Tes soped a 12 minues 7 mm deflecion loaded up o 17.5 kn/m Phoo wall 6 afer 124 min Thermal expansion e(t) =.9*T2 Deflecion in m calculaed um, inernal euna and measured D and Ful in kn/m as funcion of ime in min. = um = Ful = D6 = i i i r A 12 minues he wall is calculaed o have a load bearing capaciy of 14.2 kn/m, and i is calculaed o fail afer 188 minues.1.8 um.6 D6 euna Ful

34 32 6 kg/m3, 21 Phase 1 Wall suppored by a hinge a op and a fla foo a he boom Load P = 1. kn/m f cc2 = 3.75 MPa λ =.3 W/mC Widh of wall w =.1 m E c2 = 2.5 GPa ρ = 6 kg/m3 Eccenriciy op e op =.2 m (Posiive owards he fire) f c2 =.3 MPa c p = 1 kj/kgc Heigh L = 3. m max = 12 min = 1 min Tes soped a 36 minues? mm deflecion loaded up o 1. kn/m Thermal expansion e(t) =.9*T2 Deflecion in m calculaed um, inernal euna and measured D and Ful in kn/m as funcion of ime in min. = i um = i Ful = i um.4 euna Calculaed resisance ime 79 minues Ful

35 33 6 kg/m3, 21 Phase 1 correced for joins by reduced hickness Wall suppored by a hinge a op and a fla foo a he boom Load P = 1. kn/m f cc2 = 3.75 MPa λ =.3 W/mC Widh of wall w =.9 m E c2 = 2.5 GPa ρ = 6 kg/m3 Eccenriciy op e op =.2 m (Posiive owards he fire) f c2 =.3 MPa c p = 1 kj/kgc Heigh L = 3. m max = 12 min = 1 min Tes soped a 36 minues? mm deflecion loaded up o 1. kn/m Thermal expansion e(t) =.9*T2 Deflecion in m calculaed um, inernal euna and measured D and Ful in kn/m as funcion of ime in min. = i um = i Ful = i um.4 euna Calculaed resisance ime 5 minues Ful

36 34 9 kg/m3, 1999 Wall of blocks suppored by a hinge a op and fla foo a boom Load P = 18. kn/m f cc2 = 3. MPa λ =.4 W/mC Widh of wall w =.15 m E c2 = 3. GPa ρ = 9 kg/m3 Eccenriciy op e op =.1 m (Posiive owards he fire) f c2 = 1. MPa c p = 1 Heigh L = 2.5 m max = 24 min = 1 min Tes soped a 198 minues -5 mm deflecion 18 kn/m Thermal expansion e(t) = 1.*T2 Deflecion in m calculaed um, inernal euna and Ful in kn/m as funcion of ime in min. FNav is Navier ension and FNavc is Navier compression crierion and FR is he rankine capaciy. kj/kgc = i um = i Ful = i FNav =FNavc =FR = i i i um.6 euna Ful ( ) u := u = Calculaed resisance ime 24 minues Noice ha he inernal eccenriciy e una of his wall firs grows, hen declines and finally grows again, and ha he ulimae capaciy is firs caused by compression failure (FNav), hen Rankine insabiliy (FR) and finally ension Navier failure (FNav).

37 35 Conclusions The simplified emperaure calculaion mehod seems o be verified by comparison wih measured emperaure profiles and developmens in a special block wall wih varying hickness and concree qualiy and by comparison wih emperaure profiles and developmens measured in he esed wall elemens and a he reinforcing bars of he esed slabs. In addiion he simplified mehod prove a good agreemen wih he resuls of finie difference calculaions. The simplified emperaure calculaion seems herefore o be well documened. For slabs he calculaions of ulimae bending capaciy seems o be well documened, and i seems o be verified ha even very small bearing dephs can be foreseen o give a sufficien shear and anchorage resisance for deformed bars. On he precondiion ha he calculaions are made in reasonable ime seps, and a deailed assessmen is used for he ransien hermal srain i can be concluded ha here is a good agreemen beween calculaions and es resuls for walls, and ha he ess can serve as documenaion for he calculaion mehods. Furher he es resuls can serve as a direc documenaion for he acual slabs and walls wih he prescribed loads.

38 36 References [1] Herz, K: Beregningsmeode il brandeknisk dimensionering af konsrukioner i leklinkerbeon. (Calculaion mehod for fire safey design of consrucions of expanded clay aggregae concree). Repor R-17. Deparmen of Buildings and Energy, Technical Universiy of Denmark p. [2] Herz, K. de Place Hansen, E.: Brandeknisk dimensionering af leklinkerbeon - Beregning af brandmodsandsid. (Fire safey design of expanded clay aggregae concree - Calculaion of fire resisance ime). BYG.DTU SR-1-1. Augus p. [3] Danish Sandards: DS 411Norm for beonkonsrukioner. (Code of pracice for he srucural use of concree) p. [4] CEN/TC25/SC2: ENV Design of concree srucures, General rules - Srucural fire design p. [5] Herz, K.: The anchorage capaciy of reinforcing bars a normal and high emperaures. Magazine of concree research, Vol. 34, Number 121, December pp [6] Niels Andersen: Prøvningsrappor, Flisevæg. Dansk Brandeknisk Insiu. Sag nr. PG1642 A [7] Niels Andersen: Prøvningsrappor, Blokvæg. Dansk Brandeknisk Insiu. Sag nr. PG1642 B [8] Niels Andersen: Prøvningsrappor, Helvæg 12 kg/m3. Dansk Brandeknisk Insiu. Sag nr. PG1642 D

39 37 [9] Niels Andersen: Prøvningsrappor, Helvæg 18 kg/m3. Dansk Brandeknisk Insiu. Sag nr. PG1642 C [1] Niels Andersen: Prøvningsrappor, Massivdæk, Sandwichdæk. Dansk Brandeknisk Insiu. Sag nr. PG1742 E [11] Niels Andersen: Prøvningsrappor, Blokvæg 6 kg/m3. Dansk Brandeknisk Insiu. Sag nr. PG [12] Niels Andersen: Prøvningsrappor, Helvæg 12 kg/m3. Dansk Brandeknisk Insiu. Sag nr. PG [13] Niels Andersen: Prøvningsrappor, Helvæg 18 kg/m3. Dansk Brandeknisk Insiu. Sag nr. PG [14] Niels Andersen: Prøvningsrappor, Dæk 1, Sandwich dæk. Dansk Brandeknisk Insiu. Sag nr. PG [15] Niels Andersen: Prøvningsrappor, Dæk 2, Massiv dæk. Dansk Brandeknisk Insiu. Sag nr. PG

40 38 Appendix 1 Tes daa recorded manually by he auhor Tes daa sandwich deck manually recorded a es a DIFT Time Deflecion Reinforcemen emperaure Commens min mm C One pison replaced (average of 518 and 486) One pison replaced (average of 577 and 545) Acceleraing deflecion wihou load defines he brake Tes daa massive deck manually recorded a es a DIFT Time Deflecion Temperaures Commens min mm C , , , , , , Acceleraing deflecion wihou load defines he brake Afer es verical cracks were observed from he boom a he suppors also beween he reinforcing bars. Two days laer he cracks were closed. (Thermo cracks) During es moisure seems o spread from hese cracks.

41 39 Tes daa for block wall 6 kg/m3 manually recorded a es a DIFT Time Deflecion Temperaures Commens min mm C Hinge a op. Fla foo a boom Verical crack in he middle Horizonal crack 15 cm up Horizonal crack 3 cm up Horizonal crack 45 cm up Afer 2 hours he wall was loaded up from 7.5 kn/m (15bar hydraulic pressure) and i broke ino he oven a 17.5 kn/m (35 bar hydraulic pressure). Tes daa for wall 18 kg/m3 manually recorded a es a DIFT Time Deflecion Temperaures Commens min mm C Hinge a op. Fla foo a boom Verical cracks The wall broke ino he oven. The line of fracure was.7 imes he oal heigh from he boom.

42 4 Tes daa for wall 12 kg/m3 manually recorded a es a DIFT Time Deflecion Temperaures Commens min mm C Hinge a op and boom a firs Verical cracks The deflecions acceleraed and he es was sopped. I was observed ha he boom hinge was no capable of aking he maximum inclinaion A calculaion show ha he limi of movemen has been reached a a deflecion of 64 mm, i.e. afer 45 minues, where he suppor mus be considered o be an inclined fla foo wih eccenriciy +5 mm in sead of -2 mm.

43 41 Appendix 2 Example of a slab calculaion Calculaions of fire exposed ligh concree srucures. K. Herz A his firs page, he unis are defined and he basic formulas are given for laer use. N := 1 newon kn := 1 3 N MN:= 1 6 N kpa:= 1 3 Pa MPa:= 1 6 Pa GPa:= 1 9 Pa λ :=.6W/mC ρ := 1775 kg/m3 c p := 1 J/kgC T ( x, ) := 312 log( 8 + 1) exp( 1.9 k( ) x) sin π 2 ξ si ( T) := if( T 1, 1, ) ξ sj ( T) := if 1 < T 4, 1.3 T 1, 3 ξ s ( T) := ξ si ( T) + ξ sj ( T) + if 4 < T 65,.7.6 T 4, + if 65 < T 12,.1.1 T 65, The reducion of he compressive srengh ξc(t) of a Danish ligh aggregae concree: ξ c ( T) := if( T 2, 1, ) + if 2 < T 8, 1.4 T 2, + if 8 < T 1,.6.6 T 8, 6 2 The reducion in each of 5 zones of he half of a wo sided exposed wall w. w ξ c1 ( w, ) ξ c T 2 1, w, 3 w := ξ c2 ( w, ) ξ c T 2 1, w, w := ξ c3 ( w, ) := ξ c T 2 2, w, 7 w ξ c4 ( w, ) ξ c T 2 1, w, 9 w := ξ c5 ( w, ) := ξ c T 2 1, w, The average of he concree srengh reducions in 5 zones of he half of a wo sided exposed wall w: ξ cave ( w, ) := 5 ( ξ c1 ( w, ) + ξ c2 ( w, ) + ξ c3 ( w, ) + ξ c4 ( w, ) + ξ c5 ( w, ) ) The emperaure TM(w,) and he srengh reducion xcm(w,) in he cenre line of his wall: ( ) T M ( w, ) := T 2 ( w, w, ) ξ cm ( w, ) := ξ c T 2 ( w, w, ) The reducion ab(w,) and ac(w,) of a cross secion of widh w a he ime of a sandard fire exposure ís hen: a b ( w, ) := w 1 a c ( w, ) w 1 ξ cave ( w, ) ξ c T 2 ( w, w, ) ( ) for a beam 1.3 ξ cave ( w, ) := for a column η w ξ c T 2 ( w, w, ) ( ) k( ) x T1(x,) is he emperaure in he deph x of a semi infinie specimen a he ime calculaed by he simple formula: T2(x,w,) is he emperaure in he deph x of a wo sided exposed wall or web of hickness 2w a he ime :. The reducion of he.2% srengh ξs(t) of he slack reinforcemen: ( ) T 2 ( x, w, ) := T 1 ( x, ) + T 1 ( 2 w x, ) k( ) := π ρ c p 75 λ T 1 ( x, ) := if x π <, T 2 k( ) ( x, ), ( ) T 1 ( x, ) := if T 1 ( x, ) > 2, T 1 ( x, ), 2 T 1, T 1 (, ) ( ) + T 1 ( 2 w, ) ( ) T 2 ( x, w, ) := if T 2 ( x, w, ) > 2, T 2 ( x, w, ), 2 In some exbooks he sress disribuion facor h is used in sead of a, and his is: ξ cave ( w, ) (, ) := ξ c T 2 ( w, w, ) ( )

44 42 SOLID SLAB Free span L := Concree srengh 5.63 m f cc2 := Widh 2. MPa Time of sandard fire b := 1.2 m f c2 := Heigh 3.2 MPa := 6 min h :=.2 m w := 2 h Seel.2% srengh f s2 := 55 MPa Reinforcemen D :=.1 m D Cover hickness c y :=.15 m c := c y + c =.2 m Anchorage 2 D cs := c + cs =.25 m 2 Exernal load q b := 2.4 kn Q b := q b 1.2 m L Q b = 16.2 kn m 2 Q b Exernal load in each 1/4 poin for each slab elemen F := 2 Dead load per slab Shear force G := b h Q( ) := G L 2 + F kn m m 3 1 Momen load M( ) := 8 G L2 + F m D 2 Toal reinforcemen area A s := n s π m 2 4 A s = m 2 n s := 8 G = 4.26 kn m Q( ) = 2 kn l a := M( ) = 28 kn m.7 m F = 8.11 kn Temperaure of reinforcemen Yield srengh of reinforcemen T s ( ) := T 1 ( c, ) ( ) F s ( ) := A s ξ s T s ( ) f s2 T s ( ) = 465 C F s ( ) = 188 kn ξ s ( T s ( ) ) =.544 Temperaure a op Deph of compression zone Momen capaciy T ( ) := T 2 ( h, w, ) y( ) M u ( ) := F s ( ) h m Temperaure in he weakes shear layer T ( ) = 2 F s ( ) := b ξ c ( T ( ) ) f cc2 c m y( ) 2 C T cs ( ) := T 1 ( cs, ) ξ c ( T ( ) ) = 1. y( ) =.784 m M u ( ) = 33.1 kn m T cs ( ) = 396 C Shear capaciy F shear ( ) := ( ) f c2 ξ c T cs ( ) b 2 h m c m y( ) 2 F shear ( ) = kn Esimaed ensile srengh of 15 mm concree cover on a reinforcing bar F cs ( ) :=.5 ξ c T 1 (.25, ) F cs ( ) = kn Ulimae bond srengh Ulimae anchorage capaciy Ulimae shear capaciy ( ( ) + ξ c ( T 1 (.75, ) )) Ulimae spliing srengh ( ).5 F ab ( ) := ξ c T 1 ( c, ) 1.3 f cc2 π D m l a ( ) F a ( ) := n s min F ab ( ), F as ( ) ( ) Q u ( ) := min F shear ( ), F a ( ) ( ) +.5 ξ c T 1 (.125, ) F as ( ) := F cs ( ) 2 π m F ab ( ) = 23.5 kn F a ( ) = 115 ( kn) f c2 l a F as ( ) = 14.3 kn

45 43 Q( ) = 2.1kN M( ) = 28.4kN m := 5, Q u ( 69) = 112 kn M u ( 69) 28.7kN m = Calculaed fire resisance ime 69minues = Q u ( ) = M u ( ) kn = (An increase of cover hickness from 15 o 25 mm kn m would give rise o an increase of fire resisance ime o 114 minues.) M u ( ) M ( )