Fatigue Assessment of Concrete Foundations for Wind Power Plants

Transcription

1 Fatgue Assessment of Concrete Foundatons for Wnd Power Plants Master of Scence Thess n the Master s Programme Structural Engneerng and Buldng Performance Desgn FRIDA GÖRANSSON, ANNA NORDENMARK Department of Cvl and Envronmental Engneerng Dvson of Structural Engneerng Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 011 Master s Thess 011:119

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3 MASTER S THESIS 011:119 Fatgue Assessment of Concrete Foundatons for Wnd Power Plants Master of Scence Thess n the Master s Programme Structural Engneerng and Buldng Performance Desgn FRIDA GÖRANSSON, ANNA NORDENMARK Department of Cvl and Envronmental Engneerng Dvson of Structural Engneerng Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 011

4 Fatgue Assessment of Concrete Foundatons for Wnd Power Plants Master of Scence Thess n the Master s Programme Structural Engneerng and Buldng Performance Desgn FRIDA GÖRANSSON, ANNA NORDENMARK FRIDA GÖRANSSON, ANNA NORDENMARK, 011 Examensarbete / Insttutonen för bygg- och mljöteknk, Chalmers teknska högskola 011:119 Department of Cvl and Envronmental Engneerng Dvson of Concrete Structures Chalmers Unversty of Technology SE Göteborg Sweden Telephone: + 46 (0) Chalmers Reproservce / Department of Cvl and Envronmental Engneerng Göteborg, Sweden 011

5 Fatgue Assessment of Concrete Foundatons for Wnd Power Plants Master of Scence Thess n the Master s Programme Structural Engneerng and Buldng Performance Desgn FRIDA GÖRANSSON, ANNA NORDENMARK Department of Cvl and Envronmental Engneerng Dvson of Concrete Structures Chalmers Unversty of Technology ABSTRACT The demands on cleaner and renewable energy have ncreased wth the depleton of natural resources and a way to meet these demands s to use wnd power. Wth an ncreasng amount of wnd power plants the desgn of foundaton slabs for these are of nterest. The foundaton slab for a wnd power plant s subjected to cyclc loadng from the wnd and there have been uncertantes concernng the effect of fatgue. From the begnnng of 011 all new structures n Sweden should be desgned accordng to Eurocode and as a result of ths a new approach to calculate the nfluence of fatgue s to be consdered. The am of ths project s to gve recommendatons for desgnng concrete foundaton slabs wth regard to fatgue. Ths s done by clarfyng the ssues of fatgue for a foundaton slab for a wnd power plant, comparng methods for fatgue assessment accordng to Eurocode and fb Model Code 010 as well as performng parametrc studes of the fatgue desgn of a foundaton slab and dentfyng parameters that have sgnfcant mpact on fatgue of a foundaton slab for a wnd power plant. When performng a fatgue assessment the choce of assessment method s mportant. In ths project two fatgue assessment methods regardng renforcement, accordng to Eurocode, have been ncluded. One of the methods takes the frequency of the load nto account and uses the Palmgren-Mner damage summaton to estmate the fatgue damage, whle the other method uses an equvalent value for the fatgue load and a reference number of cycles. For fatgue assessment of compressed concrete two methods accordng to Eurocode have been used, as well as a method accordng to fb Model Code 010. One of the Eurocode methods and the fb Model Code method are damage calculaton methods, whle the second Eurocode method just checks for an adequate fatgue lfe by usng the maxmum and mnmum compressve stress under the frequent load combnaton. The fatgue assessment and parametrc study performed shows that the dmensons of the slab have an mpact on fatgue of the foundaton slab. It s concluded that the damage calculaton methods are preferable when performng a fatgue assessment based on a spectrum load, lke the wnd actng on a wnd power plant. The fatgue assessment methods that do not take the frequency of the load nto account has been seen to not be vald for such a complex load used n ths project. Key words: Fatgue, fatgue assessment method, wnd power plant, concrete foundaton slab, Eurocode, fb Model Code 010 I

6 Utmattnngsbedömnng av Betongfundament för Vndkraftverk Examensarbete nom Structural Engneerng and Buldng Performance Desgn FRIDA GÖRANSSON, ANNA NORDENMARK Insttutonen för bygg- och mljöteknk Avdelnngen för Konstruktonsteknk Betongbyggnad Chalmers teknska högskola SAMMANFATTNING Kraven på renare och förnybar energ har ökat med snande naturresurser och ett sätt att möta dessa krav är att använda vndkraft. Med ett ökande antal vndkraftverk har desgnen av fundament för dessa blvt mer av ntresse. Ett vndkraftverks fundament är utsatt för cyklsk last från vnden och det har vart osäkerheter angående effekten av utmattnng. Från början av 011 ska alla nya byggnader Sverge desgnas enlgt Eurocode och och med detta ska nya metoder för beräknng av utmattnng användas. Syftet med det här projektet är att ge rekommendatoner för desgn av betongfundament med avseende på utmattnng. Detta görs genom att klargöra utmattnngsproblematken för ett fundament för ett vndkraftverk, jämföra metoder för utmattnngsverferng enlgt Eurocode och fb Model Code 010 samt att utföra parameterstuder avseende utmattnngsdesgn av ett fundament och att dentfera parametrar som har en betydande nverkan på utmattnng av ett fundament för ett vndkraftverk. Vd utmattnngsverferng är valet av metod vktgt. I det här projektet har två metoder enlgt Eurocode avseende armerngsutmattnng använts. En av metoderna tar hänsyn tll lastfrekvens och använder Palmgren-Mners delskadesummerng för att uppskatta utmattnngsskadan, medan den andra metoden använder ett ekvvalent värde för utmattnngslasten och ett referensvärde för antalet lastcykler. För utmattnngsverferng av tryckt betong har två metoder enlgt Eurocode använts, samt en metod enlgt fb Model Code 010. En av Eurocode-metoderna och metoden från fb Model Code använder delskadesummerngen, medan den andra Eurocodemetoden kontrollerar att lvslängden med avseende på utmattnng är tllräcklg genom att använda den maxmala och mnmala tryckspännngen under den frekventa lastkombnatonen. Utmattnngsverferngen och parameterstuden som har utförts vsar att fundamentets dmensoner har en nverkan på utmattnng av fundamentet. En slutsats som har dragts är att metoderna som använder delskadesummerng är att föredra när en utmattnngsverferng av en spektrumlast utförs. De metoder som nte tar hänsyn tll lastfrekvens har vsat sg vara otllfredsställande för den komplexa last som använts detta projekt. Nyckelord: Utmattnng, utmattnngsverferng, vndkraftverk, betongfundament, Eurocode, fb Model Code 010 II

7 Contents ABSTRACT SAMMANFATTNING CONTENTS PREFACE NOTATIONS I II III V VI 1 INTRODUCTION Background 1 1. Am and objectves 1.3 Method 1.4 Lmtatons 3 FOUNDATION SLABS FOR WIND POWER PLANTS 4.1 Ground condtons 4. Loadng condtons 4.3 Typcal foundaton slab desgns 5 3 FATIGUE OF REINFORCED CONCRETE General 7 3. Concrete Renforcng steel Fatgue of renforced concrete members 10 4 FATIGUE VERIFICATION ACCORDING TO EUROCODE AND FIB MODEL CODE Fatgue verfcaton of concrete accordng to Eurocode 1 4. Fatgue verfcaton of concrete accordng to fb Model Code Fatgue verfcaton of renforcng steel accordng to Eurocode 16 5 STATIC DESIGN PROCEDURE OF FOUNDATION SLABS FOR WIND POWER PLANTS Propertes and geometry Loads Desgn of the foundaton slab n the ultmate lmt state Stablty and szng 5.3. Force dstrbuton Strut and te model and desgn of compresson-compresson node 5 III

8 5.3.4 Flexural resstance Shear capacty Remarks and conclusons from the statc desgn 7 6 FATIGUE VERIFICATION OF FOUNDATION SLAB FOR WIND POWER PLANT Fatgue load calculatons 9 6. Strut and te models to fnd fatgue stresses Crtcal sectons for fatgue assessment Fatgue stresses n the crtcal sectons Fatgue verfcaton of foundaton slab Fatgue verfcaton regardng compresson of the concrete Fatgue verfcaton regardng tensle renforcement Fatgue verfcaton regardng shear 39 7 PARAMETRIC STUDY OF FOUNDATION SLAB SUBJECTED TO FATIGUE Parameters studed Varaton of wdth of slab Study of damage development n top and shear renforcement Varaton of heght of slab Study of damage development n top and shear renforcement Analyss of the results from the parametrc study Analyss of the results regardng renforcement Analyss of the results regardng concrete 70 8 CONCLUSIONS AND RECOMMENDATIONS Fatgue desgn of foundaton slabs 7 8. Suggested future research 73 9 REFERENCES 74 APPENDIX IV

9 Preface Ths study was ntated by Ramböll due to uncertantes regardng the effect of fatgue on a foundaton slab for a wnd power plant. The project was carred out between January and August 011 n cooperaton wth Ramböll and the Department of Structural Engneerng, Concrete Structures, Chalmers Unversty of Technology, Sweden. We would lke to thank Ramböll Byggteknk, Göteborg, for the opportunty to work wth ths thess and especally our supervsor, Per Lndberg, for hs support and for provdng us wth nformaton durng our work. Next, we would lke to show our grattude to our supervsor at Chalmers, Rasmus Remplng, for helpng us wth all our questons and thoughts, especally regardng fatgue. We would also lke to thank our examner, Björn Engström, for hs partcpaton, comments and nvaluable knowledge. Wthout the help of Karl Lundstedt at Skanska Teknk, Malmö, ths project would not have turned out as t dd and we deeply apprecate hs commtment and tme. We are also grateful to Pekko Sverge AB and Semens for provdng us wth materal durng the work wth ths project. Last but not least, we would lke to show are apprecatons to our opponents Maln Johansson and Terese Löfberg for always beng there for us wth comments and coffee breaks. Göteborg August 011 Frda Göransson & Anna Nordenmark V

10 Notatons Roman upper case letters A c0 A A A A s s s A' sv s max s the area the horzontal compressve force s actng on s the renforcement amount s the requred renforcement amount s the amount of bottom renforcement s the amount of shear renforcement s the amount of top renforcement C s the maxmum compressve concrete force n a cycle C mn s the mnmum compressve concrete force n a cycle D s the bendng dameter of a bent bar D s the fatgue damage D s the damage caused by the stress range E Ed cd max, equ Ecd max Ecd mn, equ F C, s the maxmum compressve stress level, s the maxmum compressve stress level, s the mnmum compressve stress level s the compressve force n the force couple F T s the tensle force n the force couple F s the horzontal force res F z s the normal force transferred from the tower to the slab G s the self weght of the foundaton slab M. s the peak to peak fatgue moment fat ampltude M res s the overturnng moment M z s the torson moment N s the number of load cycles untl fatgue falure N( σ ) s the ultmate number of cycles for stress range σ * N s a reference value for number of cycles untl fatgue falure dependng on renforcement type P s the horzontal sol pressure R s the stress rato eq R equ s the stress rato R res s the reacton force from the ground S s the maxmum compressve stress level c,max S c,mn s the mnmum compressve stress level T s the tensle force estmated n the strut and te model T s the maxmum tensle force n the renforcement n a cycle max T s the mnmum tensle force n the renforcement n a cycle mn VI

11 Roman lower case letters b s the dstance from the edge of the slab to the gravty centre of the reacton force f s the desgn compressve concrete strength n [MPa] cd f, s the desgn fatgue resstance for concrete cd fat f ck s the characterstc compressve strength n [MPa] f, s the reference fatgue compressve strength ck fat f ctk 0 s 10 MPa f s the desgn yeld strength of the renforcement k 4 yd s a parameter nfluencng the desgn strength for concrete struts k 1 s a coeffcent affectng the fatgue strength k 1 s the exponent that defnes the slope of the frst part of the S-N curve k s the exponent that defnes the slope of the second part of the S-N curve n R s the number of cycles causng falure at the same stress level and stress range s the number of actng stress cycles at a gven stress level and stress range n S n( σ ) s the number of cycles for stress range σ s s a coeffcent dependng on the type of cement t 0 s the concrete age n days when frst subjected to fatgue loadng Greek upper case letters M s half the ampltude of the fatgue moment s the stress range S c σ Rsk s the reference fatgue stress range after * ( N ) s the reference resstng fatgue stress range at σ Rsk σ S,max s the maxmum steel stress range * N cycles * N cycles Greek lower case letters β ( t 0 ) s a coeffcent for concrete compressve strength at frst load applcaton cc ( t ) β s a coeffcent that takes the effect of hgh mean stresses durng loadng φ γ C c, sus,t 0 f nto account s the dameter of the renforcement bar s a partal factor for concrete γ s a partal factor for loads, accordng to IEC γ F, fat s a partal factor for fatgue loadng γ G s a partal factor for permanent loads γ s a partal factor for materals, accordng to IEC m VII

12 γ n s a partal factor for consequence of falure, accordng to IEC γ s a partal factor for varable loads Q γ S S, fat s a partal factor for steel γ s a partal factor for fatgue that takes the materal uncertantes nto account ν s a parameter nfluencng the desgn strength for concrete struts σ s the maxmum compressve concrete stress n a cycle c0,max σ c0,mn s the mnmum compressve concrete stress n a cycle σ cd, max,equ s the maxmum equvalent compressve stress for 10 6 cycles σ cd, mn,equ s the mnmum equvalent compressve stress for 10 6 cycles σ c,max s the maxmum compressve stress n a cycle σ c,max s the maxmum concrete compressve stress under the frequent load combnaton σ s the mnmum compressve stress n a cycle c,mn σ c,mn s the mnmum concrete compressve stress n the secton where σ c, max found σ Rd.max s the largest stress that can be appled at the edge of compresson node n strut and te models σ s the maxmum tensle steel stress n a cycle s,max σ s,mn s the mnmum tensle steel stress n a cycle ξ s a reducton factor for the reference fatgue stress range for bent bars s VIII

13 1 Introducton 1.1 Background The demands on cleaner and renewable energy have ncreased wth the depleton of natural resources and a way to meet these demands s to use wnd power. Between 1999 and 010 the energy producton from wnd power plants n Sweden ncreased extensvely as a result of ths, Energmyndgheten (011). Year 010 the energy producton from wnd power plants was 3,0 TWh and the Swedsh government has set a goal of an ncrease of the energy producton from wnd power to 30,0 TWh year 00. Ths would result n a need for wnd power plants nstead of today s 1500, see Fgure Development of wnd power plants n Sweden Energy producton [MWh] Number of wnd power plants Year Fgure 1.1 The energy producton from wnd power and number of wnd power plants n Sweden and the plannng goal for 00. Based on tables from Energmyndgheten. The technology for wnd power plants s developng fast and plants become bgger and more effectve. Ths leads to that the lfetme of a wnd power plant s short, but the foundatons can be gven a longer lfe span. The foundaton slab for a wnd power plant s subjected to cyclc loadng from the wnd. In earler desgn there have been uncertantes concernng the effect of fatgue and ths has n many cases been assumed gnorable. Now wth desgn accordng to Eurocode and larger wnd power plants the nfluence of fatgue has become more of nterest. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 1

14 From the begnnng of 011 all new structures n Sweden should be desgned accordng to Eurocode. Ths European Standard s a desgn code wth purpose to smplfy the cooperaton between engneers from dfferent European countres and to functon as a legal document for the engneer to relate to. As a result of ths code a new approach to calculate the nfluence of fatgue s to be consdered. 1. Am and objectves The am of ths project s to gve recommendatons for desgnng concrete foundaton slabs wth regard to fatgue. Objectves of ths project are to: Clarfy the ssues of fatgue for a foundaton slab for a wnd power plant n relaton to the general desgn problems. Compare methods for fatgue assessment accordng to Eurocode and fb Model Code 010, as well as ther backgrounds. Perform parametrc studes of the fatgue desgn of a foundaton slab. Identfy parameters that have sgnfcant mpact on the fatgue of a foundaton slab for a wnd power plant. Formulate recommendatons for fatgue desgn of foundaton slabs. 1.3 Method To ncrease the knowledge of fatgue a lterature study was done. Ths study ncluded fatgue of renforced concrete as well as concrete and renforcement behavour under fatgue loadng. The approaches of fatgue accordng to Eurocode and fb Model Code 010 was also part of ths study. Further a lterature study concernng the desgn of foundaton slabs for wnd power plants has been carred out. A foundaton slab was desgned for ntal condtons dsregardng the fatgue loads. Ths slab was then nvestgated wth regard to fatgue usng both Eurocode and fb Model Code 010 to see how and f fatgue nfluences the desgn. A comparson between the approaches was done. When the desgn of the slab was made, the wdth and heght of the slab was vared wthn reasonable lmts. To do ths each parameter was altered to see the nfluence t has on the foundaton slab wth regard to fatgue. Durng the change of parameters the statc desgn was always verfed. After the study of the behavour of the foundaton slab wth respect to fatgue, an analyss of the results from the desgn of the slab and the fatgue assessment was done. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

15 1.4 Lmtatons Prmarly the foundaton slab for a wnd power plant was nvestgated but the results should be possble to apply on slabs n machne rooms. However the calculatons were only performed for foundaton slabs for wnd power plants. The project was lmted to nvestgatng onshore foundatons that are square and wth a constant heght. The most common type of foundatons for onshore wnd power plants are gravty foundatons, whle ths s the type studed. The transfer of loads between the tower and the slab s done by an nserted anchor rng n the concrete. The loads taken nto consderaton n the study were the self weght of the slab and the loads transferred from the tower to the slab. These loads consst of normal and horzontal force, and overturnng and torson moment. The fatgue assessment regardng concrete only ncluded a verfcaton of the concrete fatgue compressve strength snce ths enabled a comparson between the methods n fb Model Code 010 and Eurocode. For the fatgue verfcaton of renforcement the methods used was accordng to Eurocode and wll only verfy the fatgue tensle strength of the renforcement. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 3

16 Foundaton Slabs for Wnd Power Plants.1 Ground condtons Wnd power plants are used at locatons that fulfl requrements for wnd condtons. Every locaton s dfferent wth regard to the ground condtons and may requre dfferent types of foundatons. If a wnd power plant s bult on rock, a specal concrete rock adapter can be used for anchorage. For clay there s often a need to use pled foundatons due to the low bearng capacty of clay. The most common foundaton type for wnd power plants s gravty foundatons whch are sutable for sol types from sand and clay to hard rock. When decdng on the geometry of the slab the ground condtons are an nfluencng factor snce t s crucal that the ground can resst the pressure under the slab. The foundaton and sol must also have a suffcent rotatonal stffness whch s estmated as a combned stffness for foundaton and sol.. Loadng condtons A wnd power plant s subjected to a range of dfferent actons. Many of the loads are transferred from the wnd power plant tower to the foundaton slab. The loads can be dvded nto two categores, statc loads and fatgue loads. The loads consst for example of the self weght of the plant, the wnd load actng on the turbne at hubheght and a sectonal overturnng moment on the foundaton slab that the wnd load gves rse to. Fgure.1 shows the dfferent parts of a wnd power plant. Hub Rotor blades Tower Foundaton Fgure.1 The dfferent parts of a wnd power plant. 4 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

17 The slab s also subjected to other varable actons lke snow and temperature. These actons are though small n comparson to the others. The self weght of the whole structure, the wnd power plant and the foundaton slab, s ressted by the resultng earth pressure actng on the foundaton slab. The foundaton slab s subjected to a fatgue load caused by the wnd actng on the tower and the movement of the turbne. Ths fatgue load s more complex and dffers from the fatgue load caused by traffc on a brdge for example. The wnd s actng on the tower wth varyng speed and drecton. When desgnng the foundaton slab the fatgue loads are often gven by the suppler of the wnd power plant and may dffer between dfferent supplers. The heght and sze of the tower are also nfluencng factors for the fatgue load actng on the slab. Fatgue loads specfed by the suppler are often gven n form of tables from whch fatgue damage can be calculated..3 Typcal foundaton slab desgns The foundaton slab for a wnd power plant s usually square, crcular or octagonal. The top of the slab s ether flat or wth a small slope of maxmum 1:5. Ths slope s a way of makng the slab more economcally benefcal due to the fact that less concrete s needed. It also ensures that water s led away from the slab. The maxmum slope s chosen so that no upper formwork has to be used when castng the concrete and the maxmum slab thckness s provded where the hghest shear force and moment are actng. The concrete strength class for foundaton slabs of wnd power plants normally ranges from C30/37 to C35/45. The sze of the foundaton slab s determned by the demand on foundaton stffness, set by the manufacturer of the wnd power plant. Ths s to avod self-oscllaton and lmt the rsk of settlements. Due to ths demand the foundaton slabs normally have a wdth of 15 to 0 meters and a thckness of 1.5 to.5 meters. The manufacturer also provdes nformaton about an anchor rng that s used to anchor the tower n the foundaton slab. For a prncpal drawng of an anchor rng see Fgure.. Fgure. Square foundaton slab of a wnd power plant wth anchor rng, Pekko Sverge AB. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 5

18 Dfferent foundaton slab shapes result n dfferences n the resultng earth pressure and dfferent renforcement layouts. For square and octagonal slabs the bottom renforcement s placed n two drectons, perpendcular, whle the top renforcement s radal through the anchor rng and smlar to the bottom renforcement outsde the rng, see Fgure.3. For crcular slabs both the top and bottom renforcement can be radal. a) b) c) Fgure.3 Renforcement layouts for a square foundaton slab of a wnd power plant: (a) Bottom renforcement (b) Radal top renforcement through the anchor rng (c) Top renforcement outsde the anchor rng. 6 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

19 3 Fatgue of Renforced Concrete 3.1 General When a structure s subjected to a cyclc load, falure can occur before the statc loadng strength of the materal s reached. Ths type of falure s known as a fatgue falure. Typcal cyclc loads are wnd, waves, traffc loads and vbratons from machnes. These affect structures lke brdges, tall buldngs, offshore structures and wnd power plants. The fatgue capacty of the materal s nfluenced by several dfferent parameters such as load frequency, maxmum load level, stress ampltude and materal composton. Cyclc load, or fatgue loadng, s dvded nto three dfferent categores: low-cycle, hgh-cycle and super-hgh-cycle fatgue. The number of load cycles determnes the type of fatgue. Few load cycles, up to 10 3 cycles, gve low-cycle fatgue. If the number of cycles s between t s referred to as hgh-cycle fatgue. The last category s super-hgh-cycle fatgue and t s wth more than 10 7 cycles. See Fgure 3.1 for examples of structures subjected to the dfferent types of fatgue loadng. LOW-CYCLE FATIGUE HIGH-CYCLE FATIGUE SUPER- HIGH-CYCLE FATIGUE Structures subjected to earthquakes Structures subjected to storm Brdges Wnd power plants Arport pavement Mass rapd transt structures Sea structures NUMBER OF CYCLES Fgure 3.1 Spectra of fatgue load categores and examples of structures subjected to the fatgue loads. When performng a fatgue lfe assessment of structural elements there are two basc approaches that can be used. One consders an analyss of crack propagaton at the pont under consderaton and s based on lnear elastc fracture mechancs. The second approach, whch s more commonly appled, uses a curve that shows the relaton between cyclc stress range and number of cycles to fatgue falure n logarthmc scales. Ths s known as a Wöhler or an S N curve where S s the stress range and N s the number of cycles. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 7

20 The S N curves are derved usng expermental data obtaned from fatgue tests. The curve s presented by the best ft lne wth constant slope. In practce the curve used s parallel to that obtaned from the fatgue tests, but wth a devaton to acheve a safety margn, see Fgure 3.. log S Curve obtaned from test data Curve used n practce Fgure 3. Prncpal S N curve. log N Fatgue falure s dvded nto three dfferent stages: crack ntaton, crack propagaton and falure. However concrete and steel behave dfferently wth regard to fatgue and renforced concrete s treated as two separate materals when desgnng for ths, Johansson (004). Investgatons have been made about how concrete behave and how renforcement react, but only a few studes have been made of renforced concrete members. Due to the fact that the self weght of concrete s normally a large part of the total load, ths counteracts the affect of fatgue. Wth the work towards more optmsed structures the self weght s reduced and the utlzaton of the concrete strength s ncreased. Ths could lead to that fatgue wll have a larger sgnfcance when desgnng n concrete. 3. Concrete Concrete s not a homogenous materal and durng hardenng of concrete pores and mcro cracks are formed. It s also common that macro cracks are formed before any load s appled, ths due to shrnkage and temperature dfferences. Because of the cracks and the nhomogenety concrete can be regarded as a stran-softenng materal.e. wth large strans the stffness decreases, see Fgure CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

21 Stress, σ Stran 7,81cm softenng f c Stran, ε Fgure 3.3 Stress-stran relaton for concrete showng the decreasng stffness due to stran softenng. When the concrete s subjected to a cyclc load there wll be no clear crack ntaton process, snce cracks already exst. Instead these cracks wll grow, slowly at frst and then qucker, untl falure n the remanng part of the concrete secton, Svensk Byggtjänst (1994). Unlke steel, fatgue cracks n concrete have no dentfable surface topography and due to ths fatgue falure n concrete structures s dffcult to dentfy. 3.3 Renforcng steel In contrast to concrete steel s a stran-hardenng materal, whch means that wth large strans above yeldng the strength ncreases, see Fgure 3.4. Ths gves steel a fatgue stress lmt and below ths lmt no fatgue falure wll occur, Thun (006). Stress, σ Stran hardenng 3,5cm f y Stran, ε Fgure 3.4 Stress-stran relaton for steel showng the ncreasng strength due to stran hardenng. For renforcng steel cracks are ntated by dscontnutes or stress concentratons n the bar; these cracks grow untl they cause a brttle falure n the remanng uncracked part of the bar. It has been seen that the strength of renforcement subjected to fatgue loadng s reduced to about 44 % of the yeld strength, CEB (1988). CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 9

22 The most mportant factor nfluencng the fatgue lfe of renforcement s the stress ampltude. Other nfluencng factors are geometry (whch affects stress concentratons n the bar) and envronment (.e. corroson). Wth ncreasng bar dameter the fatgue strength of the renforcement decreases. Ths s due to the ncreased rsk of flaws n the surface because of the larger dameter/area. Accordng to CEB (1988) bent bars, mechancal and welded connectons and corroson of the bars also result n a lowered fatgue performance of the renforcement. 3.4 Fatgue of renforced concrete members If a renforced concrete beam or slab s subjected to cyclc bendng loadng, cracks wll appear and ntal cracks wll grow. Wth the crack propagaton the tensle stresses n the concrete wll redstrbute to the renforcement. Deformaton of the member ncreases due to cyclc loadng, whch leads to that the mportance of deformatons, s generally greater than the actual fatgue lfe of the element. Fatgue falure n renforced concrete can occur n the concrete, renforcement or n the connecton between them. The falures are charactersed as bendng, whch results n compresson falure of the concrete or tensle falure of the renforcement, shear or punchng and bond falure. Bendng falure depends on how the concrete s renforced. For under-renforced sectons the fatgue resstance s manly determned by the fatgue propertes of the renforcement. Fatgue falure of the renforcement due to tensle forces, caused by bendng, occur wthout notceable stran and s therefore dffcult to predct. If the secton s over-renforced fatgue falure wll occur n the compresson zone n the concrete as a compresson falure, CEB (1988). Shear falure s dependng on whether the beam or slab has shear renforcement. For beams wthout shear renforcement a crtcal shear crack may develop after only a few loadng cycles. Ths shear crack appears when the tensle strength of the concrete s reached and the fatgue falure s determned by the propagaton of ths crack, CEB (1988). Two dfferent fatgue falure modes for beams wthout shear renforcement exst and are shown n Fgure 3.5. The frst mode s formaton of a dagonal crack and the second s compresson of the concrete at the top of the shear crack. Fgure 3.5 Possble shear fatgue falure modes n a beam wthout shear renforcement: (a) dagonal crackng (b) compresson of the concrete at the top of the shear crack 10 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

23 The fatgue behavour of beams wth shear renforcement s dependng on the fatgue propertes of the renforcement. In Fgure 3.6 four fatgue falure modes are shown. The frst two show when fatgue falure occur n the renforcement, ether n the strrups or n the longtudnal renforcement. The last two s when the concrete fals n compresson, ether at the top of the shear crack or n the mddle of the beam between cracks. Fgure 3.6 Possble shear fatgue falure modes n a beam wth shear renforcement: (a) fatgue of shear renforcement (b) fatgue of longtudnal renforcement crossng the shear crack (c) fatgue of the concrete n compresson at the top of the shear crack (d) fatgue of the concrete n compresson n the mddle of the beam. Bond falure s falure between the concrete and the renforcement. There are three dfferent types of bond falures; splttng of the surroundng concrete, concrete falng n shear along the permeter of the renforcement bar and break down of the shear strength of chemcal bonds between the renforcement bar and the concrete. These falures are affected by e.g. load level, frequency of load, strength of concrete and confnng effects, fb (000). Splttng of the surroundng concrete occur when the shear strength of the bond s suffcent and s caused by the radal pressure from the anchored renforcng bar. Ths pressure develops when the prncpal tensle stress cracks the concrete and causes the bond forces to be drected outward from the bar. The effect of cyclc load on ths type of falure s that under cyclc load the stress pattern changes due to stress redstrbuton. The repeated loadng opens up longtudnal cracks and cyclc creep of concrete causes the stress pattern at ultmate falure to emerge earler. When the concrete has suffcent splttng resstance the fatgue bond falure wll occur as shear falure of the concrete. Ths s the hghest bond resstance and s determned by the shear strength of confned concrete. Bond falure of chemcal type of bonds has rarely been studed wth regard to fatgue snce chemcal bonds are seldom used. However, when the concrete determnes the strength of the chemcal bond a reducton of the bond strength due to fatgue can be assumed, CEB (1988). CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 11

24 4 Fatgue Verfcaton Accordng to Eurocode and fb Model Code Fatgue verfcaton of concrete accordng to Eurocode The fatgue verfcaton methods for concrete n compresson, accordng to Eurocode, are presented n EN , SIS (005). In Eurocode there are two methods for fatgue verfcaton of compressed concrete that can be appled on foundaton slabs. These methods only take the compressve stresses n the concrete nto consderaton and not the number of load cycles untl fatgue falure occurs. Instead a reference value of 10 6 load cycles s appled. The methodology used s that a comparson s done for the stresses caused by the cyclc load and a reference concrete strength for statc load f,. Ths reference concrete strength s defned by equaton (4.5). cd fat Accordng to the frst fatgue verfcaton method the compressed concrete has a suffcent fatgue resstance f the followng condton s fulflled: Where: E 0,43 1 R 1 (4.1) cd, max, equ + equ R equ Ecd,mn, equ (4.) E cd,max, equ E E cd,mn, equ cd,max, equ σ cd,mn, equ (4.3) f cd, fat σ cd,max, equ (4.4) f cd, fat Where: R equ s the stress rato Ecd, mn, equ s the mnmum compressve stress level Ecd, max, equ s the maxmum compressve stress level f, s the desgn fatgue resstance for the concrete accordng to equaton cd fat (4.5) σ cd, max,equ s the maxmum equvalent compressve stress for 10 6 cycles σ cd, mn,equ s the mnmum equvalent compressve stress for 10 6 cycles 1 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

25 The desgn fatgue resstance of concrete accordng to Eurocode s determned by: Where: k 1 ( ) f ( ) ck fcd, fat k1 β cc t0 fcd 1 (4.5) 50 s a coeffcent dependng on the reference number of cycles tll falure. Accordng to EN :005-NA, ths coeffcent should be set to 1.0 β cc t 0 s a coeffcent for the concrete compressve strength at a certan age accordng to equaton (4.6) t 0 s the concrete age n days when frst subjected to the fatgue loadng f cd s the desgn compressve concrete strength n [MPa] f ck s the characterstc compressve concrete strength n [MPa] The coeffcent for estmaton of the concrete compressve strength for a certan age s calculated accordng to: Where: ( t ) s 1 8 t 0 β e (4.6) cc 0 s s a coeffcent dependng on the type of cement The second method n Eurocode checks the stresses n the concrete under the frequent load combnaton. It states that f the followng condton s fulflled the compressed concrete has adequate fatgue strength: σ σ c 0,5 + 0,45 f cd c,max,mn f cd, fat, fat 0,9 for f ck 50MPa 0,8 for f ck > 50MPa (4.7) CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 13

26 Where: σ c,max s the maxmum concrete compressve stress under the frequent load combnaton σ c,mn s the mnmum concrete compressve stress n the secton where σ c,max s found In ths project the frst Eurocode concrete method wll be referred to as EC:1 and the second method as EC:. 4. Fatgue verfcaton of concrete accordng to fb Model Code 010 Fatgue verfcaton of plan concrete accordng to fb Model Code 010 s found n Model Code 010, frst complete draft, fb (010). In ths method the number of load cycles untl fatgue falure s estmated for constant ampltude stress, n contrast to the approaches n Eurocode that only consders the stresses n the concrete. As well as n Eurocode a reference concrete strength for statc load,, s estmated and f ck, fat compared to the concrete stresses. The fatgue verfcaton n fb Model Code 010 can be used to check pure compresson, compresson-tenson and pure tenson falure of the concrete. Here the fatgue assessment wll only nclude a verfcaton of the concrete compressve strength snce ths can be compared to the methods n Eurocode. The maxmum and mnmum compressve stress levels caused by fatgue loadng are defned as: S c,max σ c,max (4.8) f ck, fat S c,mn σ c,mn (4.9) f ck, fat S S S (4.10) c c, max c,mn Where: S c,max s the maxmum compressve stress level S c,mn s the mnmum compressve stress level S c s the stress range σ c,max s the maxmum compressve stress n a cycle 14 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

27 σ c,mn s the mnmum compressve stress n a cycle For compresson of the concrete the condton below should be fulflled, and then the followng equatons apply: For S 0. 8 assume S 0. 8 c, mn > For S c, 0. 8 log N 0 mn c, mn (1 + 16S + 8S ) (1 S ) (4.11) 1 c,mn c,mn c, max log N 0. log N1 (log N1 1) (4.1) 1 ( ) log N 3 log N Sc, mn (4.13) Sc If log N 6 (4.14) 1 then log N log N1 If log 1 > 6 N and S ( S ) then log N log N (4.15) c 0 c,mn Where: If log 1 > 6 N and S < ( S ) then log N log N 3 (4.16) c 0 c,mn N s the number of load cycles untl fatgue falure The fatgue reference compressve concrete strength can be determned by: f ck, fat f β (4.17) ( ) ( ) ck t 0 β c, sus t, t0 f ck 1 5 f ctk cc 0 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 15

28 Where: f ck s the characterstc compressve strength n [MPa] f, s the reference fatgue compressve strength ck fat f ctk 0 s 10 MPa ( ) t 0 β cc s a coeffcent dependng on the age of the concrete at the frst applcaton of fatgue loadng ( t ) β s a coeffcent that takes the effect of hgh mean stresses durng c, sus,t 0 loadng nto account. May be assumed to 0.85 for fatgue loadng When the number of load cycles untl fatgue falure s known the Palmgren-Mner damage summaton may be appled to estmate the fatgue lfe. Where: ns D < 1 (4.18) n R D n S s the fatgue damage s the number of actng stress cycles at a gven stress level and stress range n R s the number of cycles causng falure at the same stress level and stress range 4.3 Fatgue verfcaton of renforcng steel accordng to Eurocode Fatgue verfcaton of renforcng steel, accordng to Eurocode, s presented n EN , SIS (005). Two dfferent approaches for fatgue assessment of the renforcement are gven. In the frst method the Palmgren-Mner damage summaton s used to estmate the damage. n( σ ) DEd < 1 N( σ ) (4.19) 16 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

29 Where: n( σ ) s the number of cycles for stress range σ N( σ ) s the ultmate number of cycles for stress range σ From the characterstc fatgue strength curve ( S N relaton) of renforcng and prestressng steel the ultmate number of cycles, N( σ ), can be estmated. As can be seen n Fgure 4.1 the S N relaton s a curve wth two dfferent slopes. The condtons stated n equatons (4.0) and (4.1) have been estmated from the S N relaton and gve the ultmate number of cycles for the slope vald for a certan stress range σ. log σ Rsk f yd k 1 1 k 1 N* log N Fgure 4.1 S N relatons for renforcng and prestressng steel. N( σ Rsk k1 * S, fat σ ) N f γ F, fat σ γ γ σ Rsk F, fat σ (4.0) γ S, fat N( σ Rsk k * S, fat σ ) N f γ F, fat σ γ γ σ Rsk F, fat σ < (4.1) γ S, fat Where: * N s a reference value for number of cycles untl fatgue falure dependng on renforcement type, Table 6.3N n EN :005 * σ Rsk s the reference fatgue stress range after N cycles, Table 6.3N n EN :005 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 17

30 γ S, fat s a partal factor for fatgue that takes the materal uncertantes nto account, Table.1N n EN : γ F, fat s a partal factor for fatgue loadng wth recommended value 1.0, EN : (1) k 1 k s the exponent that defnes the slope of the frst part of the S-N curve, Table 6.3N n EN :005 s the exponent that defnes the slope of the second part of the S-N curve, Table 6.3N n EN :005 The second method for fatgue verfcaton of the renforcement uses a damage equvalent stress range. Ths method can be appled on standard cases wth known loads, most commonly brdges. When usng t for buldngs the damage equvalent stress range can be approxmated wth the maxmum steel stress range under the exstng load combnaton. Accordng to ths method the renforcement has suffcent fatgue resstance f the followng condton s fulflled: γ σ ( N ) * Rsk F, fat σ S,max (4.) γ S, fat Where: σ S,max s the maxmum steel stress range * ( N ) s the reference resstng fatgue stress range at σ Rsk * N cycles The two methods for fatgue verfcaton of the renforcement wll n ths project be referred to as EC:1 and EC:. For shear renforcement the methods descrbed above can be used, however the value of σ Rsk s only vald for straght bars and therefore needs to be reduced for the bent bars n the shear renforcement. Ths s done accordng to the followng equaton: Where: D ξ (4.3) φ ξ D φ s a reducton factor for the reference fatgue stress range for bent bars s the bendng dameter of a bent bar s the dameter of the renforcement bar 18 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

31 5 Statc Desgn Procedure of Foundaton Slabs for Wnd Power Plants 5.1 Propertes and geometry When makng a prelmnary desgn of a foundaton slab for a wnd power plant, ntal condtons need to be determned. The condtons for the locaton of the wnd power plant are often gven by the suppler of the plant wth help from a geotechncal ste study. The type of sol, the depth to sold ground and the groundwater level are some of these condtons. There are also requrements on the foundaton slab and sol for mnmum rotatonal stffness around the horzontal axs and mnmum stffness for horzontal translaton. These demands are set by the suppler of the plant. For the desgn of the foundaton slab n ths project, the condtons were based on nformaton receved from Semens for a plant located near Tuggarp, Gränna n Sweden. Some of the condtons were based on what s common for buldng stes of wnd power plants. The condtons for the foundaton slab are; frcton sol, the groundwater level s below the foundaton slab, the mnmum demand for combned stffness of sol and foundaton s 1500MNm/deg wth regard to rotatonal stffness and the mnmum stffness wth regard to horzontal translaton s 500MN/m. The placement above the ground water level means that the pore pressure n the sol was dsregarded. The desgned slab s a gravty foundaton, snce ths s the most common type of foundaton for wnd power plants. The shape of the slab was assumed to be square and wth a constant heght,.e. no slope at the top of the slab. Concrete class C30/37 and renforcement type B500B was chosen. The anchor rng s placed n the centre of the slab, 1.7 meters down n the concrete. It has a heght of.45 meters, an outer dameter of 4. meters and an nner dameter of 3.6 meters. 5. Loads Statc loads actng on the slab are loads from the tower, snow and temperature actons, see Chapter, n partcular Secton.. The varable loads actng drectly on the foundaton are small n comparson and was dsregarded n ths analyss. The specfc wnd power plant has a hub-heght of 99.5 meters and the desgn wnd speed at the heght of the hub s 8.5 m/s. Ths has been used to fnd the loads actng on the foundaton slab. The desgn loads for the foundaton slab are calculated by Semens accordng to the standard for wnd power plants, IEC Ths standard descrbes several dfferent desgn load cases, for 8 desgn stuatons, that should be consdered n desgn of members of wnd power plants. From these load cases the most crtcal cases should be found, and the desgn of the member should be based on these. For each of the desgn load cases the approprate type of analyss s specfed, wth U for analyss of ultmate loads used to ensure structural stablty and materal strength, or wth F for load cases used n fatgue assessment of the member. The desgn load cases for CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 19

32 ultmate loads are dvded nto normal, abnormal and transport and erecton load cases. The normal load cases are expected to occur frequently and the turbne s then n a normal state or sufferng mnor faults. The abnormal load cases are less lkely to occur and often result n severe faults. Semens provde nformaton about fve desgn load cases and state that two of these should be used for the desgn of the foundaton slab. These two load cases are maxmum desgn loads under normal power producton and maxmum desgn loads under extreme condtons and they are ncludng partal safety factors from IEC In ths standard t s stated that partal safety factors from other desgn codes can be used together wth partal safety factors from IEC Ths s vald as long as the combned value of the partal safety factors for loads, materals and consequence of falure s not less than the combnaton of the partal safety factors that are stated n IEC In ths project the desgn loads provded by Semens were used ncludng the partal safety factors from IEC The partal safety factors for the materals, the self weght of the slab and the horzontal sol pressure were accordng to Eurocode. It was ensured that the combned value of the safety factors was at least what s stated n IEC Table 5.1 shows the partal safety factors accordng to IEC and Eurocode. Table 5.1 Partal safety factors accordng to IEC and partal safety factors n the ultmate lmt state accordng to Eurocode. IEC Eurocode Loads Normal γ 1, 0 f or 1, 35 Permanent γ G 1,0 or 1, 35 Abnormal γ 1, 1 Varable γ 1, 50 Transport γ 1, 5 Materal γ 1, 1 Concrete γ 1, 50 m f f Steel γ 1, 15 Q C S Consequence of falure Class 1 γ 0, 9 0,9 n Class γ 1, 0 1,0 n Class 3 γ 1, 3 1,1 n The foundaton desgn loads provded by Semens are gven as sectonal forces for the normal load case based on normal operaton and the abnormal load case whch s the load case wth the hghest overturnng moment. From the tower to the foundaton slab the followng sectonal forces due to loadng are transferred; normal force, horzontal force, overturnng and torson moment. The foundaton slab desgn n the ultmate lmt state was based on the desgn load case whch resulted n the hghest sectonal 0 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

33 forces and ths s the one wth the hghest overturnng moment, see Table 5.. For desgn of the slab n servceablty lmt state the desgn load case for normal operaton was used. Table 5. Foundaton desgn loads, ncludng partal safety factors accordng to IEC , for normal operaton and desgn load case wth hghest overturnng moment. Normal operaton Hghest overturnng moment Normal force, Horzontal force, Overturnng moment, Torson moment, F kn kn z F 800 kn kn res M knm knm res M knm knm z 5.3 Desgn of the foundaton slab n the ultmate lmt state For the statc desgn of the foundaton slab there are several aspects that need to be consdered. The stablty of the slab has to be verfed for resstance aganst overturnng moment and requred rotatonal stffness. The slab must also have a suffcent flexural resstance wth regard to postve and negatve bendng moments. Further the shear capacty of the slab must be adequate, and f necessary shear renforcement has to be desgned. For a foundaton slab for a wnd power plant the anchor rng transfers forces from the tower to the concrete slab by embedded anchor bolts. The lower part of the anchor rng, that s embedded n the concrete slab, provdes the anchorage of the tensle forces from the tower. The connecton wth the concrete s done by geometrcal lockng at the lower part of the anchor rng. The choce wth embedded bolts enables post-tensonng of the nterface between the tower and the foundaton slab and ths s preferable snce t s ncreasng the fatgue lfe of the structure. The compressve forces from the tower can be assumed to be transferred drectly from the upper part of the anchor rng, whch s not embedded n the concrete slab, to the top of the foundaton slab, see Fgure 5.1. Due to these compressve forces punchng shear needs to be checked under the loaded area of the rng. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 1

34 Secton where the compressve load from the tower s transferred to the slab Secton where the tensle load from the tower s anchored n the slab Fgure 5.1 Sectons where the compressve and tensle loads from the tower are transferred. Compresson of the concrete under the loaded area of the anchor rng also needs to be checked snce t can be consdered as a concentrated compresson node n a strut and te model. All of these verfcatons were performed accordng to Eurocode Stablty and szng The demands on stablty of the foundaton slab wll determne the sze of t. In order to check stablty, the slab sze was assumed and the reacton force from the ground was checked to be actng under the slab. If the reacton force s actng outsde the slab ths would mply that the stablty aganst overturnng moment s not suffcent. The locaton of the reacton force, R res, was found by moment equlbrum for the self weght of the wnd power plant and the slab, and the sectonal forces as well as the horzontal sol pressure, see Fgure 5.. Ths resulted n an estmaton of b, whch s the dstance from the edge of the foundaton slab to the gravty centre of the reacton force. The reacton force s unformly dstrbuted over the length b. The compressed area under the slab was then found to verfy that the ground could resst the sol pressure caused by the slab. F z F res M res G h b R res D/ P Fgure 5. Model of the loads actng on the slab and the reacton force from the ground. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

35 To check the combned rotatonal stffness of the slab and sol the slab was assumed to act as a cantlever and the rotaton caused by the sectonal moment was calculated. The reacton force from the ground s a unformly dstrbuted load over the length b and the self weght of the slab was assumed to counteract the rotaton Force dstrbuton The force dstrbuton n the slab was estmated by usng a load model where the overturnng moment, the horzontal force and the normal force were modelled as a force couple, see Fgure 5.3. Ths force couple conssts of a compressve force, F, and a tensle force, F T. These forces are transferred to the foundaton slab through the anchor rng and the reacton force from the ground, R res, s counteractng these forces and the self weght of the slab. C F z F C F T M res F res Fgure 5.3 The loads that are actng on the slab and how they are modelled as a force couple F C and F T. Compresson of the concrete s concentrated to a small part of the slab under the loaded area of the anchor rng. By usng the model shown n Fgure 5.4, assumng a smplfed stress block approach, the heght of the compressed part of the anchor rng was obtaned. When ths heght was known the dstance between the compressve and tensle force n the force couple could be found. Fgure 5.4 Model to fnd the heght of the compressed part of the anchor rng. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 3

36 In order to determne the heght of the compressve zone the part of the anchor rng that s n tenson had to be assumed. In ths case a quarter of the rng was assumed to be n tenson, whch means that the anchor bolts n ths quarter are the ones that are actve when t comes to transfer of tensle forces to the slab. The average depth to the anchor bolts n the quarter of the rng was estmated. Wth these assumptons the heght of the compressve zone could be calculated and then the dstance between the compressve and the tensle force could be found. A moment dagram was obtaned from the force dstrbuton, see Fgure 5.5. In the dagram t can be seen that n ths case the crtcal sectons for the bendng moment were found under the force couple. The maxmum bendng moment found under F C was used to estmate the amount of bottom renforcement needed. The bendng moment under the force F T was used to estmate the needed amount of top renforcement. Fgure 5.5 Assumed moment dstrbuton along the slab. The shear capacty of the foundaton slab was verfed n transverse sectons across the slab assumng the slab to behave lke a beam. See Fgure 5.6 for the shear force dstrbuton along the slab. The maxmum shear force was found between the force couple,.e. nsde the anchor rng. The slope at the left part of the dagram s due to the dstrbuted reacton force, R. res 4 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

37 Fgure 5.6 Shear force dstrbuton along the slab Strut and te model and desgn of compresson-compresson node A strut and te model was establshed to fnd the requred renforcement amounts and the horzontal compressve force actng on the compressed concrete area under the anchor rng. In the strut and te model the load model wth a force couple was used, see forces F C and F T n Fgure 5.7. The gravty centre of the reacton force from the ground s on a dstance b from the edge of the slab, as estmated prevously. Ths reacton force s a unformly dstrbuted load but n the strut and te model t was modelled as a concentrated force actng on the dstance b. The self weght of the slab was dvded nto two force components that are actng n nodes outsde the force couple. The forces n each strut and te were calculated and the stresses n the compresson-compresson node, node 5 n Fgure 5.7, were calculated to desgn ths node for compresson falure. G F C F T G 7 5 θ 1 3 θ 1 θ 1 θ 8 6 θ 4 R res Fgure 5.7 The strut and te model used n the prelmnary desgn of the slab. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 5

38 The capacty of the concrete n the compresson-compresson node under the anchor rng s ncreased due to embedded concrete: Where:. k ν (5.1) σ Rd max 4 f cd Where: f ck ν 1 (5.) 50 σ Rd.max s the largest stress that can be appled at the edge of the node, EN : (4) k 4 ν s a parameter nfluencng the desgn strength for concrete struts wth recommended value 3.0, EN : (4) s a parameter nfluencng the desgn strength for concrete struts, EN : () f cd s the desgn compressve concrete strength n [MPa] f ck s the characterstc compressve strength n [MPa] Flexural resstance From the strut and te model the tensle forces n the dfferent parts of the foundaton slab were known. There are tensle forces n a part of the top of the slab as well as along the entre bottom of t. Two vertcal tes are shown n the model, see Fgure 5.7 above, and these ndcate that there s a need for shear renforcement, both nsde and outsde the anchor rng. The amount of renforcement was estmated wth regard to resstance of the tensle forces n the slab, see equaton (5.3). Where: T A s (5.3) f yd A s T s the requred renforcement amount s the tensle force estmated n the strut and te model f yd s the desgn yeld strength of the renforcement Shear capacty The slab was checked for suffcent shear capacty by assumng the crtcal shear crack at the edge of the sol reacton. It was also necessary to verfy the shear capacty 6 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

39 n punchng shear, due to the transfer of forces through the anchor rng. By assumng that the tower acts lke a crcular column support, the calculaton was carred out accordng to Eurocode, EN : The anchor rng s assumed to transfer the compressve forces from the tower to the slab at the surface of the slab. The depth on whch punchng shear should be checked was therefore the whole depth of the slab. In the calculatons for punchng shear capacty dfferent crtcal permeters were assumed. The hghest nclnaton checked was 40 deg and the lowest 6 deg, see Fgure 5.8. If the shear capacty of the concrete slab s nsuffcent shear renforcement s needed and f ths s the case the shear renforcement s assumed to carry the whole shear force ,7 b Fgure 5.8 Crack nclnaton for the dfferent crtcal permeters used n calculaton of punchng shear. 5.4 Remarks and conclusons from the statc desgn The maxmum and the mnmum amount of renforcement were verfed n both ultmate- and servceablty lmt state. The crack wdth n servceablty lmt state was compared to the maxmum allowed crack wdth, whch s dependent on the lfe span of the slab and the exposure class. For ths slab the lfe span class was set to L50 and the exposure class to XC, ths gave a maxmum allowed crack wdth of 0.45 mm. To meet the demands for stablty of the slab the dmensons were chosen to 16x16 meters and the heght was set to meters. The renforcement arrangement was chosen to consst of two layers n the bottom of the slab, to ensure suffcent flexural resstance. The bars are 4 mm n dameter and the spacng s 150 mm. The top renforcement was also placed n two layers wth bars wth the same dameter as the bottom renforcement. The dmensons of the slab as well as sol pressure, rotatonal stffness and renforcement amounts are shown n Table 5.. Table 5. Prelmnary desgn of the foundaton slab, showng wdth, heght, sol pressure, rotatonal stffness, bottom, top and shear renforcement. Wdth [m] Heght [m] Sol pressure [kpa] Rotatonal stffness [MNm/deg] A s [mm ] A' s [mm ] A sv [mm ] 16 37, CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 7

40 In the strut and te model the force F T s appled at the top of the slab and transferred down to node at the level of the bottom renforcement, see Fgure 5.7 above. The force F T s however transferred nto the slab va the anchor rng and the bottom of the anchor rng s n realty not at the level of the bottom renforcement. In order to model the behavour of the slab accurately wth the strut and te model, a connecton between the bottom renforcement and the lower part of the anchor rng was assumed. The compresson node under the force F C was assumed to be a crtcal secton wth regard to crushng of the concrete. In the ntal statc calculatons the compressve capacty n ths node was not suffcent. No lmtatons on the heght of the node could be found, so to ncrease the capacty of the node the heght was ncreased and suffcent capacty obtaned. However, wth an ncrease of node heght the angles n the strut and te model change and a redstrbuton of forces wll occur. 8 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

41 6 Fatgue Verfcaton of Foundaton Slab for Wnd Power Plant 6.1 Fatgue load calculatons The fatgue loads are provded by Semens and are calculated for a 0 year operatng tme. They are shown as spectrum n three dfferent tables: one for horzontal force, one for overturnng moment and the last one for torson. Each of the fatgue loads obtaned n the table wll have a damagng mpact on the foundaton slab. The total fatgue damage of all the loads wll therefore be the damage caused over the whole perod of 0 years. In order to estmate ths fatgue damage, the moments affectng the fatgue calculaton has to be found. The values n the table are the fatgue loads that the wnd gves rse to. These fatgue loads come wth the number of cycles shown on the left axs, see Table 6.1. The numbers of cycles seen n the table s the sum of the cycles for all the loads n that row. Table 6.1 Fatgue load spectrum for overturnng moment. At the top of the table the mean values for the loads are shown. These mean values are gven by a range and the mean value can be chosen as any value wthn ths range. In the fatgue assessment performed n ths project the mean value was chosen as the mean value of the specfc range. There are 93 loads n the spectrum shown for overturnng moment. For each of these loads a mean value was estmated to smplfy the calculatons. If for example bn number 5 s used, the mean value should be between 1300 and 3000 knm, whch s the range for the bn. In ths project the mean value was chosen to the mean of the range for the bn,.e knm. The value n the table s then the peak to peak value,.e. the dfference between the mnmum and the maxmum value n a cycle, whch s the same as the ampltude, M., see Fgure 6.1. The maxmum fat ampltude CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 9

42 moment s the mean value plus half the ampltude, M, and the mnmum value s the opposte, the mean value mnus half the ampltude. Moment [knm] M M fat.ampltude Mean Value of cycles Tme Fgure 6.1 The fatgue load for one ampltude n the overturnng moment table. The moment that affects the fatgue calculatons s the dfference between the maxmum and the mnmum moment f both of these are postve. If the mnmum value s negatve the fatgue moment used n the calculatons s equal to the maxmum moment snce a negatve value mples that the slab s subjected to compresson nstead of tenson. If both the maxmum and the mnmum values are negatve the fatgue moment s set to zero. 6. Strut and te models to fnd fatgue stresses For the fatgue assessment the stresses due to fatgue loadng had to be checked. The strut and te model used n the statc desgn was assessed to see f ths could be used for servceablty lmt state, and descrbe the behavour of the slab when t s subjected to fatgue. Due to the large varatons n the fatgue moments the strut and te model was however found not to work properly for all the loads. The changes of the model for the dfferent loads are dependng on the poston of the reacton force from the ground. Ths reacton force s actng further towards the edge of the slab for ncreasng fatgue moments. In the statc desgn the reacton force was estmated to be actng on a dstance, b, from the edge of the slab, and was modelled as a concentrated force. For small fatgue moments the reacton force s actng nsde the anchor rng and ths result n a dfferent strut and te model. For the larger moments where the reacton force s actng closer to the dstance b, that was estmated n the statc desgn, a strut and te model smlar to the statc one can be used. When the reacton force s actng between the anchor rng and the dstance used n the statc desgn, the angles n the model wll determne the number of nodes and tes. The fatgue stresses were found by establshng the strut and te models for fatgue assessment. Because of the varyng fatgue loadng four dfferent strut and te models were establshed to descrbe the stress feld properly. In all four models the dstance between F T and F C was assumed to be the average dameter of the anchor rng nstead of the dstance 30 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

43 between the compressve and tensle part of the rng, whch was the dstance used n the statc desgn. The frst two strut and te models are both smulatng the stress feld n the slab when t s subjected to small fatgue moments, whch results n that the reacton force from the ground s actng nsde the anchor rng. The fatgue moments result n that compressve forces as well as small tensle forces are transferred to the slab. In order to get the models to functon as precsely as possble the reacton force from the ground was dvded nto two force components. As a smplfcaton when dvdng the reacton force n two components, they were assumed to be of equal sze on the same dstance from the gravty centre of the reacton force. The frst strut and te model s shown n Fgure 6.3. In ths model the component of the reacton force actng close to the tensle force F T was locked under the force n node, whle the reacton force and the component on the other sde changes for the dfferent fatgue moments. Ths was to smplfy the calculatons regardng the reacton force from the ground. Ths strut and te model s vald untl the dstance between the locked component of the reacton force, and the actual reacton force s too large. In order to ensure a proper result ths model was used untl the component on the left sde of the reacton force and the node t s actng n, whch s node 6, was under node 7. G F C F T G 7 θ 67 5 θ 3 θ θ θ 56 4 R res R res Fgure 6.3 The strut and te model for when the reacton force s actng nsde the anchor rng and one of the components s locked under the tensle force. The second strut and te model, see Fgure 6.4, was also used when the reacton force s actng nsde the anchor rng. In ths model the component of the reacton force actng close to the tensle force, F T, was locked n the mddle of the slab n node 4. Ths model s vald untl the reacton force s actng under node 5,.e. untl the reacton force s no longer actng nsde the rng. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 31

44 G F C F T G 7 θ 67 5 θ 3 θ θ θ 56 R res 4 R res Fgure 6.4 The strut and te model for when the reacton force s actng nsde the anchor rng and one of the components s locked n the mddle of the slab. The thrd strut and te model, see Fgure 6.5, was used to smulate the stress feld n the slab when t s subjected to greater fatgue moments. Ths wll result n that the reacton force from the ground s actng outsde the anchor rng. The reacton force was modelled as a concentrated force actng on the dstance b from the edge of the slab, snce ths gves a proper strut and te model for ths case. Ths model s closer to the strut and te model used n the statc desgn. However, the dstance b that the reacton force s actng on s larger, and closer to the anchor rng, than the one estmated n the statc desgn. The model smulates the stress feld n the slab properly untl the reacton force s actng under node 7. G F C F T G 7 θ 67 5 θ 3 θ θ θ 56 4 R res Fgure 6.5 The strut and te model for when the reacton force from the ground s actng outsde the anchor rng, but not further away than node 7. The fourth strut and te model smulates the stress feld n the slab when the reacton force from the ground s actng outsde the anchor rng, on a dstance further away than node 7, see Fgure 6.6. The model s vald when the fatgue moments are great and result n that large tensle forces as well as large compressve forces are transferred to the slab. Ths model s the one closest to the strut and te model used n the statc desgn and s used for the smallest values of b. The reacton force was n ths model, as well as n the statc one, modelled as a concentrated force actng under the slab. As can be seen n the fgure below the dfferences from the thrd strut and te model s the addtonal vertcal te and node 8 was added. 3 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

45 G F C F T G 7 5 θ 3 θ θ 1 1 θ 78 θ R res Fgure 6.6 The strut and te model for when the reacton force from the ground s actng outsde the anchor rng and further away than node Crtcal sectons for fatgue assessment For fatgue verfcaton of the foundaton slab a number of sectons n the slab can be assessed. For the verfcaton n ths project these sectons needed to be lmted to a reasonable amount. Ths was done by fndng the crtcal sectons and assessng these. The statc analyss of the slab showed what sectons were crtcal n the ultmate lmt state, and these are then probably also crtcal wth regard to fatgue. In order to check fatgue of the concrete as well as the renforcement the crtcal sectons for each of these needed to be found. Ths resulted n that some sectons were assessed only wth regard to fatgue of the renforcement and some only wth regard to fatgue of the concrete, snce the crtcal sectons for these do not concde. As can be seen n the statc analyss, for a foundaton slab of a wnd power plant compresson of the concrete under the tower s a crtcal pont. Ths secton was therefore assessed wth regard to fatgue. The bottom renforcement close to the anchor rng s subjected to fatgue stresses and needed to be verfed. The secton of nterest for the top renforcement was chosen to the te between the force couple. Ths s because ths s the only secton of the top renforcement where there are any fatgue stresses. In the te between outsde and nsde the anchor rng the only force actng s the self weght of the slab and no fatgue stresses were obtaned. For the load model where the moment s modelled nto a force couple, there are hgh compressve stresses that need transfer through the nsde of the anchor rng. Ths stress affects the shear renforcement and ths needed to be checked for fatgue effects. The shear renforcement outsde the anchor rng was found to not be affected by the fatgue load, only the self weght. Strut and te models 1 and were used to assess fatgue of the top renforcement, the concrete n the compresson node under the compressve force and the shear renforcement nsde the anchor rng. These sectons are shown n Fgure 6.7. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 33

46 Concrete n compresson F C Top renforcement n tenson F T Shear renforcement Fgure 6.7 The crtcal sectons n strut and te models 1 and that were assessed wth regard to fatgue. The sectons assessed wth strut and te model 3 were the bottom and top renforcement, the shear renforcement nsde the anchor rng and the concrete n the compresson node under the compressve force, see Fgure 6.8. Concrete n compresson F C Top renforcement n tenson F T Bottom renforcement n tenson Shear renforcement Fgure 6.8 The crtcal sectons n strut and te model 3 that were assessed wth regard to fatgue. Strut and te model 4 was used to assess the bottom renforcement, the shear renforcement nsde the anchor rng and the concrete n the compresson node. In Fgure 6.9 the sectons assessed wth ths strut and te model are shown. Concrete n compresson F C F T Bottom renforcement n tenson Shear renforcement Fgure 6.9 The crtcal sectons n strut and te model 4 that were assessed wth regard to fatgue. 34 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

47 6.3.1 Fatgue stresses n the crtcal sectons In order to perform a fatgue assessment of the desgned concrete foundaton slab the sectonal stresses needed to be determned. The stresses n the slab are dependng on f the secton s cracked or not. For a cracked renforced concrete member the renforcement s assumed to carry all the tensle forces. If the member would be uncracked ths would mply that no tensle forces are carred by the renforcement, whle there would be no fatgue of ths. In all fatgue calculatons however, the crosssecton should be assumed to be cracked and the tensle strength of the concrete should be dsregarded n the assessment. The stresses n the crtcal sectons, whch were chosen for the fatgue assessment, were found wth the use of the results from the strut and te models. The forces n all struts and tes were calculated for the mnmum and maxmum value n every cycle of fatgue loadng. The stresses were calculated for both the mnmum and maxmum values and the tensle stresses n the dfferent types of renforcement were estmated by equatons (6.1) and (6.). T max s, max As σ (6.1) Where: T mn s, mn As σ (6.) σ s,max s the maxmum tensle steel stress n a cycle σ s,mn s the mnmum tensle steel stress n a cycle T max s the maxmum tensle force n the renforcement n a cycle T mn s the mnmum tensle force n the renforcement n a cycle A s s the renforcement amount The fatgue stresses n the concrete were only checked n the compressoncompresson node under the anchor rng, snce ths s the most crtcal secton for the concrete. These stresses were estmated n the same way as n the statc desgn, by fndng the area that the horzontal compressve force s actng on. Ths stress s the hghest stress n the node and t was therefore the one checked for the fatgue assessment of the concrete. C max σ c0,max (6.3) Ac 0 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 35

48 C mn σ c0,mn (6.4) Ac 0 Where: σ c0,max s the maxmum compressve concrete stress n a cycle σ c0,mn s the mnmum compressve concrete stress n a cycle C max s the maxmum compressve concrete force n a cycle C mn s the mnmum compressve concrete force n a cycle A c0 s the area the horzontal compressve force s actng on When performng the fatgue assessment the concrete strength was ncreased n the same way as n the statc desgn. Ths s due to the fact that the compressoncompresson node that was checked s embedded n the concrete. 6.4 Fatgue verfcaton of foundaton slab The desgned foundaton slab was assessed to ensure a fatgue lfe of at least 0 years, snce ths s the estmated lfe span for the wnd power plant. A frst check of the stresses affectng the fatgue lfe of the slab, showed that the horzontal force and torson moment are small n comparson to the overturnng moment and have no or very small mpact on the fatgue lfe. The fatgue loads due to horzontal force and torson were therefore dsregarded n the fatgue verfcaton of the foundaton slab. The prelmnarly desgned slab has a concrete strength class of C30/37 and renforcement amounts accordng to Table 6.1. Table 6.1 Renforcement amounts estmated n the prelmnary desgn. Bottom renforcement [mm ] Top renforcement [mm ] Shear renforcement [mm ] In the fatgue assessment presented below the followng abbrevatons wll be used; Eurocode wll be referred to as EC and fb Model Code 010 as MC CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

49 6.4.1 Fatgue verfcaton regardng compresson of the concrete Fatgue verfcaton of the concrete n the compresson-compresson node under the anchor rng was performed accordng to methods n EC and MC010. The methods n EC uses, as mentoned prevously, a reference value of number of cycles and do not take the frequency of the fatgue load nto account. For a spectrum load, lke the wnd actng on a wnd power plant, these methods were found to not be vable. In order to use the frst of these methods an equvalent stress at the reference number of cycles has to be found. For a proper use of the second method a mean value of the maxmum and mnmum stresses has to be estmated, wth the assumpton that ths s the stress under the frequent load combnaton. Ths mples that smplfcatons of the spectrum load are needed, such as fndng a mean value for the fatgue stresses for fatgue loads wth number of cycles close to the reference value. Ths was though not the choce for the fatgue assessment n ths project. Instead the expresson for the frst method for fatgue verfcaton of concrete was re-wrtten accordng to the method n EN 199-:005 that s commonly used for brdges, see equaton (6.5). When the number of cycles untl fatgue falure was known, the Palmgren-Mner rule was used to estmate the damage. N 1 14 Ecd,max 1 Req 10 (6.5) Where: N s the number of load cycles untl fatgue falure Ecd, max s the maxmum compressve stress level R eq s the stress rato The method from MC010 also estmates the number of cycles untl fatgue falure, for each of the gven fatgue moments. Ths approach was found to work well for the spectrum load snce t takes the varyng frequency of the dfferent loads nto account, by the use of the Palmgren-Mner summaton. The fatgue assessment regardng compresson of the concrete, showed that there s no substantal damage of the concrete under the anchor rng, see Table 6.. Ths s vald for all the methods used n the fatgue verfcaton of the concrete. The MC010 method resulted n no damage at all, whle the EC:1 method showed a damage of Ths damage s however nsgnfcant. The second EC method also ndcated that the compressed concrete had suffcent fatgue lfe. Table 6. Damage of the concrete n compresson accordng to EC:1 and MC010, and the result for EC:. MC010 Damage [ - ] EC:1 Damage [ - ] EC: Concrete 0 3, OK CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 37

50 6.4. Fatgue verfcaton regardng tensle renforcement The fatgue assessment of the renforcement was performed accordng to two methods from Eurocode. One of these methods estmates the number of cycles untl fatgue falure and the other uses a reference value for the number of cycles. The frst method was found to work well for the spectrum load for the same reasons as the concrete method accordng to MC010; t takes the frequency of the fatgue load nto account. The second method s based on the same approach as the concrete methods n EC and s therefore found to not work properly for the spectrum load, as can be seen n the results. The fatgue assessment regardng tensle renforcement showed that the damage of the top renforcement was hgh, whle ths amount had to be ncreased due to fatgue, see Table 6.3. Accordng to the results from EC: both top and bottom renforcement amounts would need to be ncreased. However, the damage of the bottom renforcement was low and ths was left wthout further consderaton. Table 6.3 Damage of top and bottom renforcement accordng to EC:1 and results for EC:. EC:1 Damage [ - ] EC: Top renforcement 6,155 NOT OK Bottom renforcement 0,0054 NOT OK To ensure that the slab had a 0 year fatgue lfe the top renforcement amount was ncreased. When ncreasng the amount of renforcement the damage s reduced, see Table 6.4. The amount was ncreased untl the damage was below one,.e. untl the fatgue lfe of the top renforcement was suffcent. Ths resulted n a fnal fatgue desgn, accordng to EC:1, wth an amount of top renforcement of mm. For a proper fatgue desgn accordng to EC: the amount of top renforcement needed to be further ncreased to mm. Table 6.4 Increase of top renforcement amount for fatgue desgn, accordng to methods n EC. Renforcement amount [mm ] Statc desgn Fatgue desgn: EC:1 Fatgue desgn: EC: EC:1 Damage [ - ] 6,155 0,997 0,086 EC: NOT OK NOT OK OK 38 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

51 6.4.3 Fatgue verfcaton regardng shear To check the fatgue resstance of the shear renforcement, the same methods as for the top and bottom renforcement were used, wth a reducton of the fatgue stress range because of bent bars, see Chapter 4 n partcular Secton 4.3 and equaton (4.1). Table 6.5 shows the result from the fatgue verfcaton of the shear renforcement. As can be seen the fatgue assessment of the shear renforcement ndcated damage above one, whch means that the fatgue lfe s nsuffcent. The result for the second method also showed that the amount of shear renforcement would need to be ncreased due to fatgue. Table 6.5 Damage of shear renforcement accordng to EC:1 and result for EC:. EC:1 Damage [ - ] EC: Shear renforcement 8,73 NOT OK The shear renforcement amount was ncreased due to the nsuffcent fatgue lfe, see Table 6.6 Accordng to the frst method the requred amount of shear renforcement was mm. For ths amount the second method stll ndcated that the fatgue lfe was nadequate. Wth a further ncrease to mm the shear renforcement had a proper fatgue desgn accordng to both EC methods. Table 6.6 Increase of shear renforcement amount for fatgue desgn, accordng to methods n EC. Renforcement amount [mm ] Statc desgn Fatgue desgn: EC:1 Fatgue desgn: EC: Damage [ - ] 8,73 0,99 0,003 EC: NOT OK NOT OK OK CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 39

52 7 Parametrc Study of Foundaton Slab Subjected to Fatgue 7.1 Parameters studed The fatgue assessment of the prelmnarly desgned foundaton slab showed more damage than acceptable for the top and shear renforcement, whle the other crtcal components of the slab had suffcent fatgue lfe. The parametrc study was performed to gve a deeper understandng of how fatgue affects a foundaton slab subjected to cyclc loadng. The study also nvolved a more thorough comparson between the dfferent fatgue assessment methods. The parametrc study was carred out by assessng the desgned foundaton slab, f nothng else s stated, and changng the parameters affectng the fatgue lfe. The prelmnarly desgned slab has a wdth of 16 meters, a heght of meters and a concrete strength class of C30/37. Parameters that nfluence the fatgue lfe of the slab, and that were altered n ths study, are the wdth and the heght of the slab. In order to choose ranges for the parameters studed the statc desgn was verfed for all the values. For changes of the wdth and heght the foundaton stffness n ultmate lmt state s what determnes the range for the varaton of the parameters. The sol pressure s dependent on the self weght of the slab and wth a larger wdth or heght the volume and self weght of the slab ncreases. The resultng reacton force from the ground s then actng on a larger area of the slab and the sol pressure decreases for an ncreased heght or wdth. The upper lmt for the sol pressure caused by the slab s set to 400 kpa, to ensure suffcent sol stablty. No lower lmt s set for the sol pressure. Fgure 7.1 shows the sol pressure versus wdth of the slab. The sol pressure s decreasng wth ncreasng wdth. A wdth of the slab of 15 meters results n a sol pressure of 437 kpa; ths s slghtly above the lmt for the sol pressure. 40 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

53 Fgure 7.1 Sol pressure for a slab heght of meters and varyng wdths. In Fgure 7. the decrease of sol pressure wth ncreasng heght s shown. For the heght 1.5 meters the sol pressure s 470 kpa and the lmt of 400 kpa s exceeded. Fgure 7. Sol pressure for a slab wdth of 16 meters and varyng heghts. The slab also has to be verfed for adequate rotatonal stffness snce ths, as well as the sol pressure, s dependent on the wdth and heght of the slab. The demand for rotatonal stffness s, as mentoned prevously, 1500 MNm/degree. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 41

54 The change of slab wdth never results n a rotatonal stffness below the lmt, see Fgure 7.3. Fgure 7.3 Rotatonal stffness for a slab heght of meters and varyng wdths. The rotatonal stffness ncreases when the heght s ncreasng, see Fgure 7.4. The heghts ncluded n ths study all have suffcent rotatonal stffness, whle t wll not be possble to choose the range for the heght wth ths check. Fgure 7.4 Rotatonal stffness for a slab wdth of 16 meters and varyng heghts. 4 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

55 The range for the varaton of the wdth of the slab was set to 15 to 0 meters. The wdth of 15 meter was chosen as the lower lmt for the parametrc study as ths wdth results n a sol pressure above the demanded. The upper lmt of 0 meters was chosen snce ths s wthn the normal range of wdths for a foundaton slab of a wnd power plant. The heghts ncluded n the parametrc study are 1.5 to.5 meters. The smallest heght, 1.5 meters, results n sol pressure just above the acceptable level. Snce no upper lmt for the heght s determned by the stffness of the slab, the lmt was chosen as a reasonable maxmum heght for a foundaton slab of a wnd power plant. In the parametrc study presented below the followng abbrevatons wll be used; Eurocode wll be referred to as EC and fb Model Code 010 as MC Varaton of wdth of slab The study regardng varaton of the wdth of the slab was executed for the desgned slab, whch has a heght of meters and concrete strength class C30/37. The frst part of the assessment was performed for the varyng wdths and wth renforcement amounts that were changed n order for the statc desgn to have suffcent flexural and shear capacty n the ultmate lmt state, see Table 7.1. Table 7.1 Varyng wdths and correspondng sol pressure, rotatonal stffness and renforcement amounts that ensure adequate flexural and shear capacty n the ultmate lmt state. Wdth [m] Sol pressure [kpa] Rotatonal stffness [MNm/deg] A s [mm ] A' s [mm ] A sv [mm ] , , , , , , , , , , , CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 43

56 The fatgue assessment for the varyng slab wdths was performed for both the concrete and the dfferent types of renforcement. Table 7. shows the damage of the concrete for the methods from EC and MC010 as well as the results from the second EC method. As can be seen there s no or nsgnfcant damage of the concrete accordng to the two damage calculaton methods. The second EC method also shows that the concrete has an adequate fatgue lfe. Snce there s nsgnfcant fatgue damage of the concrete, and the results are cohesve, compresson of the concrete was left wthout further consderaton n ths part of the study. Table 7. Damage of concrete accordng to EC:1 and MC010 and results for EC:, for the varyng wdths. Wdth [m] MC010 Damage [ - ] EC:1 Damage [ - ] EC: , OK , OK , OK , OK 17 0, OK , OK , OK , OK 19 0, OK , OK 0 0 3, OK Table 7.3 below shows the damage of the renforcement accordng to EC and the results from the second EC method for fatgue assessment of the renforcement. The results from these two methods are consstent, except for the bottom renforcement. The second EC method ndcates that the bottom renforcement would have an exceeded fatgue lfe for wdths of 15 to 18 meters. However, the damage calculaton method shows a damage of just 0.0% to 0.5% for these wdths. The amounts of top and shear renforcement need to be ncreased for all the wdths ncluded n the study, accordng to both EC methods. 44 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

57 Table 7.3 Damage of the renforcement accordng to EC:1 and results for EC:, for the varyng wdths. Wdth [m] A s EC:1 Damage [ - ] A' s A sv A s EC: A' s A sv 15 0, ,010 3,773 NOT OK NOT OK NOT OK , ,415 7,57 NOT OK NOT OK NOT OK 16 0, ,155 8,73 NOT OK NOT OK NOT OK , ,36 11,46 NOT OK NOT OK NOT OK 17 0, ,335 14,583 NOT OK NOT OK NOT OK , ,647 17,089 NOT OK NOT OK NOT OK 18 0, ,767 19,16 NOT OK NOT OK NOT OK , ,64 1,631 OK NOT OK NOT OK 19 0, ,59 4,497 OK NOT OK NOT OK ,179 7,690 OK NOT OK NOT OK ,603 31,336 OK NOT OK NOT OK The fatgue assessment of the bottom renforcement shows that the damage s constantly low. The hghest damage s found for the wdth 15 meters, and that s estmated to 0.5 %. Fgure 7.5 shows the damage of the bottom renforcement for the varyng wdths. Wth an ncreasng wdth the fatgue damage decreases. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 45

58 Fgure 7.5 Damage of the bottom renforcement, for varyng wdths, accordng to EC:1. The fatgue lfe of the top renforcement s exceeded for all of the renforcement amounts estmated for the dfferent wdths. The damage becomes sgnfcantly hgher for the decreased renforcement amounts used for the larger wdths. The slab wth wdth 0 meters has a damage of the top renforcement of 18.6, whle the 15 meter slab has a damage of the top renforcement of See Fgure 7.6 for the relaton between damage of the top renforcement and wdth of slab. Fgure 7.6 Damage of the top renforcement, for varyng wdths, accordng to EC:1. 46 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

59 The fatgue assessment of the shear renforcement shows that the slab wll have damage above one for all the dfferent wdths, whch would mean that the fatgue lfe of the renforcement s exceeded. In Fgure 7.7 the damage development n the shear renforcement s shown. The damage of the shear renforcement s ncreasng wth ncreasng wdth of slab. Fgure 7.7 Damage of the shear renforcement, for varyng wdths, accordng to EC: Study of damage development n top and shear renforcement When the frst part of the parametrc study regardng the slab wdth was performed a deeper analyss of the response of the top and shear renforcement followed. Both the top and shear renforcement showed an nsuffcent fatgue lfe for all the wdths ncluded n the study. Due to ths the damage development n ths renforcement s of nterest and t was therefore further studed. For every full meter of the wdth the top renforcement amount was ncreased to see the damage development wth the ncreasng renforcement amount, see Fgure 7.8. The damage was found to decrease rapdly for an ncrease of the renforcement amount up to mm. As the renforcement amount was further ncreased the rate of the reducng damage was decreased. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 47

60 Fgure 7.8 Damage of the top renforcement for ncreasng renforcement amount and varyng wdths of the slab. In order for the slab to have a suffcent fatgue lfe the damage should be below one. The amount of top renforcement was further ncreased to fnd the requred amount for every wdth of the slab, see Fgure 7.9. Fgure 7.9 Damage of the top renforcement and varyng wdths of the slab, for renforcement amounts ensurng damage below one. 48 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

61 The fnal amounts of top renforcement needed for the dfferent wdths, wth regard to fatgue, are shown n Table 7.4. The dfference between the amount needed for a wdth of 15 meters and the amount for a wdth of 0 meters s mm. The dfference n fatgue damage for the renforcement amounts estmated n the statc desgn was very large, but as seen n these results the damage reduces rapdly wth ncreasng amount of renforcement. Table 7.4 Varyng wdths and top renforcement amounts that result n a suffcent fatgue lfe accordng to EC:1 and results for EC:. Wdth [m] Top renforcement amount [mm ] EC:1 Damage [ - ] EC: Statc Fatgue ,989 NOT OK ,997 NOT OK ,988 NOT OK ,998 NOT OK ,991 NOT OK ,991 NOT OK Table 7.5 shows the top renforcement amounts needed for suffcent fatgue capacty accordng to the second EC method. As seen n the table these renforcement amounts are consderably larger than the ones estmated by the damage calculaton method. It s also notceable that EC:1 ndcates that the needed amount of top renforcement ncreases wth mm when changng the wdth from 15 to 0 meters. EC: however, shows that the same change of wdth would result n an ncrease of top renforcement of mm. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 49

62 Table 7.5 Varyng wdths and top renforcement amounts that result n a suffcent fatgue lfe accordng to EC: and damage accordng to EC:1. Wdth [m] Top renforcement amount [mm ] EC:1 Damage [ - ] EC: Statc Fatgue ,116 OK ,086 OK ,019 OK ,015 OK ,011 OK ,011 OK In order to try to solate the parameters affectng the fatgue damage the amount of top renforcement was set to mm, whch s the amount estmated n the fatgue assessment (EC:1) of the prelmnarly desgned foundaton slab. The slab heght s also set as a constant of meters. Fgure 7.10 shows the damage development n the top renforcement for constant renforcement amount and heght of slab. Fgure 7.10 Damage of the top renforcement accordng to EC:1, for varyng wdths and wth constant renforcement amount and heght of slab. 50 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

63 When the analyss of the top renforcement was done the same analyss was performed for the shear renforcement n order to see f the damage development s smlar. Fgure 7.11 shows the development of damage n the shear renforcement, for ncreasng renforcement amount and varyng wdths. The damage reduces wth the ncreasng renforcement amount, but wth a slower pace than the damage of the top renforcement. Fgure 7.11 Damage of the shear renforcement for ncreasng renforcement amount and varyng wdths of the slab. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 51

64 The renforcement amount was ncreased untl the damage was below one, and the fatgue lfe suffcent, for all the wdths. The result s shown n Fgure 7.1. Fgure 7.1 Damage of the shear renforcement and varyng wdths of the slab, for renforcement amounts ensurng damage below one. The amounts of shear renforcement needed accordng to the frst EC method are shown n Table 7.6, together wth the result from the second method. The requred renforcement amount s decreasng wth ncreasng wdth of slab. As seen n the table the second method ndcates an nsuffcent fatgue lfe, whle the damage accordng to the frst method s below one. 5 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

65 Table 7.6 Varyng wdths and shear renforcement amounts that result n a suffcent fatgue lfe accordng to EC:1 and results for EC:. Wdth [m] Shear renforcement amount [mm ] EC:1 Damage [ - ] EC: Statc Fatgue ,993 NOT OK ,99 NOT OK ,996 NOT OK ,998 NOT OK ,994 NOT OK ,994 NOT OK To further compare the two methods the needed amount of shear renforcement accordng to the second method was estmated, see Table 7.7. These amounts are consderably larger than the amounts estmated by the damage calculaton method. It can also be seen that EC: ndcate that the requred amount of shear renforcement decreases wth ncreasng wdth, whch s the opposte to the result of EC:1. Table 7.7 Varyng wdths and shear renforcement amounts that result n a suffcent fatgue lfe accordng to EC: and damage accordng to EC:1. Wdth [m] Shear renforcement amount [mm ] EC:1 Damage [ - ] EC: Statc Fatgue ,001 OK ,003 OK ,006 OK ,018 OK ,011 OK ,011 OK CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 53

66 The amount of shear renforcement was set to mm, whch was the requred amount accordng to the fatgue verfcaton of the prelmnarly desgned slab. The heght of the slab was kept constant at meters and then the damage for the dfferent wdths was calculated, see Fgure The damage ncreases wth ncreasng wdth up to 17.5 meters, then the rate of the ncrease of damage decreases. Fgure 7.13 Damage of the shear renforcement accordng to EC:1, for varyng wdths and wth constant renforcement amount and heght of slab. 54 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

67 7.3 Varaton of heght of slab The parametrc study regardng varaton of the heght of the slab was performed for a slab wth wdth and concrete strength class as for the prelmnarly desgned slab, whch means a wdth of 16 meters and strength class C30/37. The frst part of the fatgue assessment of the varyng heght was performed for the dfferent heghts wth renforcement amounts changed for suffcent statc capacty of the slab, see Table 7.8. Table 7.8 Varyng heghts and correspondng sol pressure, rotatonal stffness and renforcement amounts that ensure adequate flexural and shear capacty n the ultmate lmt state. Heght [m] Sol pressure [kpa] Rotatonal stffness [MNm/deg] A s [mm ] A' s [mm ] A sv [mm ] 1,5 471, ,6 366, ,7 310, ,8 75, ,9 53, , ,1 6, , 17, ,3 11, ,4 06, ,5 03, CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 55

68 Wth the renforcement amounts as shown n the table above the fatgue assessment of the dfferent slab heghts was executed. Table 7.9 shows the results for both the EC and MC010 methods regardng fatgue of the concrete. As can be seen the damage s nsgnfcant for both damage calculaton methods and the second EC method also ndcates a suffcent fatgue lfe of the concrete. Table 7.9 Damage of concrete accordng to EC:1 and MC010 and results for EC:, for the varyng heghts. Heght [m] MC010 Damage [ - ] EC:1 Damage [ - ] EC: 1,5 1, , OK 1,6 5, , OK 1,7 7, , OK 1,8 0, OK 1,9 0 3, OK 0 3, OK,1 0 3, OK, 0 3, OK,3 0 3, OK,4 0, OK,5 0, OK The MC010 method shows a small damage for heghts 1.5 to 1.7 meters, whle the EC method shows a slghtly larger damage for all the heghts ncluded n the study. Fgure 7.14 shows the damage of the compressed concrete accordng to the methods from EC and MC CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

69 Fgure 7.14 Damage of the concrete, for varyng heghts, accordng to EC:1 and MC010. The dfferent types of renforcement were assessed wth the two Eurocode methods and these results are shown n Table The bottom renforcement was found to have nsgnfcant damage for all the heghts, but the second method stll ndcated an nsuffcent fatgue lfe. The fatgue assessment regardng top and shear renforcement showed that for all the dfferent heghts the fatgue lfe of ths renforcement was exceeded. The result of the second EC method agreed wth the result of the frst one, and also ndcated that an ncrease of the top and shear renforcement s needed to ensure a 0 year fatgue lfe. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 57

70 Table 7.10 Damage of the renforcement accordng to EC:1 and results for EC:, for the varyng heghts. Heght [m] A s EC:1 Damage [ - ] A' s A sv A s EC: A' s A sv 1,5 0,0067 6,813,591 NOT OK NOT OK NOT OK 1,6 0, ,119 3,146 NOT OK NOT OK NOT OK 1,7 0, ,911 4,548 NOT OK NOT OK NOT OK 1,8 0, ,947 6,360 NOT OK NOT OK NOT OK 1,9 0,003 55,78 7,431 NOT OK NOT OK NOT OK 0,005 6,155 8,73 NOT OK NOT OK NOT OK,1 0,000 55,516 10,947 NOT OK NOT OK NOT OK, 0, ,468 11,655 NOT OK NOT OK NOT OK,3 0,001 66,335 13,366 NOT OK NOT OK NOT OK,4 0, ,495 16,65 NOT OK NOT OK NOT OK,5 0, ,30 18,036 NOT OK NOT OK NOT OK In Fgure 7.15 the result of the fatgue verfcaton regardng the bottom renforcement s shown. The damage of the bottom renforcement s constantly low; t vares between 0.1 and 0.7 %. Ths shows that the fatgue lfe s suffcent for all the heghts, even though the amount of bottom renforcement for the.5 meter deep slab s mm less than the amount for the 1.5 meter deep slab. 58 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

71 Fgure 7.15 Damage of the bottom renforcement, for varyng heghts, accordng to EC:1. The damage of the top renforcement versus heght s shown n Fgure 7.16 and shows an nsuffcent fatgue lfe for all the heghts. Ths was expected snce the top renforcement amount was ncreased n the fatgue assessment of the 16 meter wde slab; see Chapter 6 n partcular Secton The maxmum amount of top renforcement used n ths study was stll lower than the amount needed accordng to the prevous assessment. Fgure 7.16 Damage of the top renforcement, for varyng heghts, accordng to EC:1. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 59

72 When assessng the shear renforcement wth regard to fatgue and varyng heght, the results showed that for all the heghts the fatgue lfe of the shear renforcement was nadequate, see Fgure Wth ncreasng slab heght the damage of the shear renforcement ncreased. Fgure 7.17 Damage of the shear renforcement, for varyng heghts, accordng to EC: Study of damage development n top and shear renforcement The second part of the parametrc study regardng the heght of the slab conssted of a more thorough analyss of the damage development n the top and shear renforcement. For sx of the dfferent heghts the amount of top and shear renforcement was ncreased to see how the damage changes and the requred amounts of renforcement were found accordng to both the fatgue assessment methods. Fgure 7.18 shows the damage development wth ncreasng amount of top renforcement. The damage decreases consderably at frst, then the rate decreases. 60 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

73 Fgure 7.18 Damage of the top renforcement for ncreasng renforcement amount and varyng heghts of the slab. To further study the damage development n the top renforcement wth ncreasng heght the renforcement amount was ncreased untl the damage became below one, and a suffcent fatgue lfe was reached, see Fgure Fgure 7.19 Damage of the top renforcement and varyng heghts of the slab, for renforcement amounts ensurng damage below one. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 61

74 In order to show the dfferences between the two fatgue assessment methods for the renforcement, the needed amount of top renforcement was estmated accordng to both the methods, see Table 7.11 and 7.1. Table 7.11 Varyng heghts and top renforcement amounts that result n a suffcent fatgue lfe accordng to EC:1 and results for EC:. Heght [m] Top renforcement amount [mm ] EC:1 Damage [ - ] EC: Statc Fatgue 1, ,993 NOT OK 1, ,993 NOT OK 1, ,98 NOT OK, ,986 NOT OK, ,98 NOT OK, ,996 NOT OK Table 7.1 Varyng heghts and top renforcement amounts that result n a suffcent fatgue lfe accordng to EC: and damage accordng to EC:1. Heght [m] Top renforcement amount [mm ] EC:1 Damage [ - ] EC: Statc Fatgue 1, ,16 OK 1, ,168 OK 1, ,086 OK, ,046 OK, ,034 OK, ,06 OK 6 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

75 As can be seen n the tables the amount of top renforcement needed accordng to the second method s sgnfcantly larger than the amount needed accordng to the frst method. The dfference between the needed amounts for a 1.5 and a.5 meter slab s however about mm accordng to both the methods. The damage of the top renforcement was also studed by settng the amount of renforcement to mm, snce ths s the estmated amount needed for a suffcent fatgue lfe of a slab wth a heght of meters. Fgure 7.0 shows the damage development of the top renforcement for ths renforcement amount. The damage s below one for slab heghts of to.5 meters. The damage s gradually decreasng, rapdly for the smaller slab heghts and slower for the larger heghts. Fgure 7.0 Damage of the top renforcement accordng to EC:1, for varyng heghts and wth constant renforcement amount and wdth of slab. The same type of study as for the top renforcement was performed for the shear renforcement. Fgure 7.1 shows the damage development n the shear renforcement for ncreasng renforcement amount. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 63

76 Fgure 7.1 Damage of the shear renforcement for ncreasng renforcement amount and varyng heghts of the slab. Fgure 7. shows the damage development n the shear renforcement for renforcement amounts ensurng a suffcent fatgue lfe and damage below one. Here t can be seen that the damage decreases slowly when t s close to one. Fgure 7. Damage of the shear renforcement and varyng heghts of the slab, for renforcement amounts ensurng damage below one. 64 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

77 The two EC methods for fatgue assessment of the renforcement were compared by estmatng the requred amount of shear renforcement accordng to each of the methods. The results are shown n Tables 7.13 and As well as for the top renforcement the second EC method ndcates that a sgnfcantly larger amount of renforcement s needed. Table 7.13 Varyng heghts and shear renforcement amounts that result n a suffcent fatgue lfe accordng to EC:1 and results for EC:. Heght [m] Shear renforcement amount [mm ] EC:1 Damage [ - ] EC: Statc Fatgue 1, ,991 NOT OK 1, ,998 NOT OK 1, ,996 NOT OK, ,996 NOT OK, ,991 NOT OK, ,996 NOT OK Table 7.14 Varyng heghts and shear renforcement amounts that result n a suffcent fatgue lfe accordng to EC: and damage accordng to EC:1. Heght [m] Shear renforcement amount [mm ] EC:1 Damage [ - ] EC: Statc Fatgue 1, ,0003 OK 1, ,001 OK 1, ,00 OK, ,004 OK, ,007 OK, ,017 OK CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 65

78 An nterestng result from ths comparson s that for the frst EC method the needed amount of shear renforcement s ncreasng wth ncreasng heght, and the second method shows the opposte; a decrease of shear renforcement for ncreasng heght. Ths dfference between the two methods was also seen n the parametrc study regardng the wdth of the slab. The last part of the study of the damage development n the shear renforcement was done by settng the renforcement amount to mm, whch was the amount needed for the 16 meter wde and meter hgh slab. The damage for ths renforcement amount s estmated for all the heghts ncluded n the study. The result s shown n Fgure 7.3. Fgure 7.3 Damage of the shear renforcement accordng to EC:1, for varyng heghts and wth constant renforcement amount and wdth of slab. 7.4 Analyss of the results from the parametrc study To determne whch structural model to use for the fatgue calculatons the dstance between the edge of the slab and the reacton force from the ground s used. Dfferent models, see Secton 6. and Fgures , can be used for the maxmum and mnmum values of a force. For some values of the overturnng moment ths s necessary, especally f there s a large dfference between the maxmum and the mnmum values. There are however occasons n the parametrc study when the dfference between the maxmum and mnmum values of the overturnng moment s small, but dfferent models are stll used snce the reacton force s actng close to the lmt between two models. The effect of ths can be observed n the dagrams from the parametrc study and ths s the reason why the curve does not always follow a trend. 66 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

79 Fgure 7.4 shows the S-N curve for steel. In the fgure the stress ranges n the top renforcement are plotted aganst the number of cycles for the fatgue loads. As can be seen there are values above the S-N curve. Ths mples that for each of these values the estmated damage s above one. The amount of top renforcement was ncreased untl the stress ranges were decreased enough for suffcent fatgue resstance. In the fgure t s seen that the values for the ncreased renforcement amount are all below the S-N curve. Fgure 7.4 S-N curve for steel and stress ranges n the top renforcement versus number of cycles for the fatgue load Analyss of the results regardng renforcement The damage of the bottom renforcement was found to decrease wth ncreasng slab wdth and heght. When analysng these results t s seen that the reason for ths behavour s that wth the smaller wdths and heghts strut and te models 3 and 4 are used to a larger extent and these results n hgher stresses n the bottom renforcement. For the larger wdths and heghts strut and te models 1 and are used more. The choce of strut and te model s, as mentoned earler, dependent on where the reacton force from the ground s actng. For larger wdths and heghts the self weght of the slab s ncreasng and ths s causng the reacton force to act closer to the mddle of the slab and nsde the anchor rng. When the reacton force s actng nsde the anchor rng, there are no or small tensle forces n the bottom of the slab. Hence, the decrease of damage of the bottom renforcement wth ncreasng wdth or heght of the slab s due to the ncreased self weght of the slab beng favourable. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 67

80 The fatgue damage of both shear and top renforcement was found to be hgh and to ncrease wth ncreasng slab wdth and heght. For the top renforcement the damage rapdly decreases wth ncreasng renforcement amount and the needed amount of renforcement s wthn reasonable lmts. For the shear renforcement the decrease rate of the fatgue damage slows down before the damage s below one and the needed amount of renforcement s therefore great. For one of the fatgue assessment methods the amount of shear renforcement needed s even mpossble to ft nto the slab. For the damage calculaton method n EC the amount of shear renforcement needed for suffcent fatgue desgn s however reasonable and therefore t can be assumed that even f the ntally calculated fatgue damage s hgh t s also reasonable. Wth ncreasng slab wdth and heght the self weght of the slab s ncreased and ths s unfavourable for fatgue of the top and shear renforcement. When lookng at the strut and te models used t s seen that wth ncreasng self weght models 1 and are used more, as mentoned above. In these models the fatgue load s taken up by the top and shear renforcement and there s not much tenson n the bottom of the slab. Wth ncreasng dmensons the self weght causes hgher stress n the slab, whle the fatgue load s unchanged. The entre fatgue load s concentrated to the secton under the tower and when changng the wdth ths secton stays the same. When ncreasng the heght however, t changes. A consequence of ths change s that wth ncreased heght more fatgue load s carred vertcally and less horzontally. Ths s seen n the fgures that show the damage of the top and shear renforcement for constant renforcement amount and heght or wdth of slab. For the varyng wdth and constant renforcement amount t s seen that the damage of the top and shear renforcement s ncreasng wth ncreasng wdth. For the varyng heghts and constant renforcement amount the damage of the top renforcement s decreasng wth ncreasng heght, whle the damage of the shear renforcement s ncreasng wth ncreasng heght. The second fatgue assessment method for renforcement, EC:, dd not work well for the spectrum load used n ths project. The reason for ths s that the method checks f the fatgue lfe s suffcent, by comparng the capacty of the renforcement wth the hghest stress caused by the fatgue load. It does not take the number of cycles nto account, and ths s crucal for ths type of load snce the maxmum stress only comes wth a number of cycles of 1000 n 0 years. Ths s also the reason why the renforcement amounts estmated by the use of ths method are sgnfcantly larger than the ones estmated by the damage calculaton method. Fgures shows the renforcement amounts estmated n the statc analyss, as well as the requred amounts for fatgue desgn accordng to EC:1 and EC:. 68 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

81 Fgure 7.5 Top renforcement amounts for varyng wdths for statc desgn and fatgue desgn accordng to EC:1 and EC:. Fgure 7.6 Shear renforcement amounts for varyng wdths for statc desgn and fatgue desgn accordng to EC:1 and EC:. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 69

82 Fgure 7.7 Top renforcement amounts for varyng heghts for statc desgn and fatgue desgn accordng to EC:1 and EC:. Fgure 7.8 Shear renforcement amounts for varyng heghts for statc desgn and fatgue desgn accordng to EC:1 and EC: Analyss of the results regardng concrete The fatgue assessment of the concrete n compresson showed very lttle or no damage for both damage calculaton methods used n ths project. The EC method gves a larger damage than the MC010 method though. Even though the results n ths parametrc study showed very small dfferences n the results from the two 70 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

83 methods t could stll be worth pontng out that there stll are dfferences. It should also be noted that the frst fatgue assessment method for compressed concrete, accordng to EC, was rewrtten accordng to EN 199-:005. The orgnal method that should be used for buldngs dd not work well for the spectrum load, for the same reason as one of the renforcement methods. The second fatgue assessment method for concrete, accordng to EC, does not work well for the spectrum load ether snce t does not take the frequency of the load nto account. In ths project however the stresses n the concrete are small enough, so ths method gves a result showng a suffcent fatgue lfe of the concrete. If the damage calculaton methods would show damage closer to the lmt however, ths method would mply an exceeded fatgue lfe. In order to ensure a suffcent statc desgn the node heght, n the compresson node under the tower, was ncreased. Ths s where fatgue of the concrete s checked snce ths s the most crtcal secton for compresson of the concrete. In the fatgue assessment the ncreased node heght was used n the calculatons, snce a suffcent statc desgn should be verfed. Ths has naturally resulted n a lower damage of the concrete than what would have been f the ntal node heght had been used. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 71

84 8 Conclusons and Recommendatons 8.1 Fatgue desgn of foundaton slabs The am of ths project has been to gve recommendatons for desgn of foundaton slabs wth regard to fatgue. The statc desgn of the foundaton slab has however been proved to be more complex than what was frst thought. Ths s due to the anchor rng that s used to anchor the tower to the slab. Ths rng wll transfer the forces to the slab and wll result n large sectonal forces actng on a small area. Both the concrete and the renforcement n the slab are subjected to fatgue loadng. In the statc desgn of the slab some crtcal sectons are dentfed. These sectons are lkely to be crtcal also for the fatgue assessment. In ths project t s seen that the part of the slab that s affected by the fatgue load s the secton nsde the anchor rng. For the concrete the crtcal secton s the compresson node, at the top of the slab, under the tower. A fatgue verfcaton of the slab and a parametrc study has been performed to nvestgate the behavour of the dfferent members of the slab, when t s subjected to fatgue loadng. Three fatgue assessment methods regardng concrete have been used n the fatgue verfcaton and the parametrc study. Two of these methods are accordng to Eurocode and one s accordng to fb Model Code 010. Fatgue of the renforcement was assessed wth two fatgue assessment methods accordng to Eurocode. The parametrc study has been performed both to fnd the parameters that have an mpact on the fatgue lfe of the foundaton slab and to compare the dfferent fatgue assessment methods. The parametrc study showed that the dmensons of the slab have an mpact on the fatgue lfe. Wth a larger slab the self weght ncreases, gvng a lower damage to the bottom renforcement. The work towards more optmzed and mnmzed structures mght result n that the fatgue load s governng for the bottom renforcement. An ncreasng heght of slab resulted n more fatgue load beng carred vertcally and less horzontally, whch gave hgher fatgue of the shear renforcement and less fatgue of the top renforcement. An ncreasng wdth of slab resulted n decreasng fatgue of the bottom renforcement and ncreasng fatgue of the top and shear renforcement. The parametrc study showed that fatgue s governng for the top and shear renforcement for all the slab dmensons ncluded n ths study. For fatgue desgn of a foundaton slab for a wnd power plant the method used for the fatgue assessment should be chosen so the whole spectrum of fatgue loads can be assessed. In ths project t has been seen that the damage calculaton methods are advantageous compared to the other methods. The methods for fatgue verfcaton that do not take the frequency of the load nto account are found to not be vald for such a complex fatgue load lke the wnd. In order to make these methods vable an equvalent value for the spectrum needs to be obtaned. It s seen when comparng the two fatgue assessment methods for the renforcement that the needed amount of shear renforcement accordng to the damage calculaton method s ncreasng wth ncreasng self weght of the slab. The results from EC: 7 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

85 show the opposte, a decreasng amount of shear renforcement s needed for ncreasng self weght. Ths s due to that EC: only checks the maxmum stress range n the renforcement and the maxmum stress range n the shear renforcement s decreasng wth an ncreasng self weght. The total fatgue load taken up by the shear renforcement s however ncreasng wth an ncreased self weght, and ths s seen n the results from the damage calculaton method. When calculatng the damage of the concrete accordng to MC010 and EC:1 no sgnfcant damage was seen for ether method. In order to compare the two methods very small damages were looked at and from these t could be concluded that the method accordng to MC010 resulted n a lower damage than EC:1. Fatgue assessment regardng shear s n ths project performed accordng to Eurocode. There are however uncertantes n the desgn code concernng how to perform the fatgue verfcaton. The calculatons regardng fatgue of the shear renforcement are n ths project performed accordng to the same methods as for the bottom and top renforcement. The shear renforcement s assumed to carry all of the shear force caused by the fatgue loadng. The methods used could however be more accurate f the shear capacty of the concrete could contrbute to the shear fatgue capacty. 8. Suggested future research To lmt the project only one shape of slab has been nvestgated and the connectons between the tower and the slab have not been checked for fatgue effects. One concluson drawn was that the self weght of the slab has an mpact on the fatgue damage of the members n the slab, whle the shape of slab would be an nterestng area to contnue studyng. Further, only gravty foundatons have been studed and there are other types of foundatons that may also be of nterest wth regard to fatgue, e.g. pled foundatons. In ths project there have been uncertantes regardng the fatgue verfcaton methods and whch models to use. In the parametrc study a connecton between the fatgue damage and the choce of model could be seen, especally for fatgue loads close to lmts between two dfferent strut and te models. To further nvestgate the force dstrbuton n the slab, when subjected to fatgue loadng, a FEM analyss could be done. The structural models used n ths project are two dmensonal; ths s however a smplfcaton and t would be more accurate to develop three dmensonal models of the foundaton slab. The uncertantes regardng the fatgue verfcaton methods are concernng fatgue assessment of the shear renforcement. Further gudelnes are needed to clarfy the methods as well as mprovements to make t possble to take the concrete capacty nto consderaton. The speedy development of wnd power plants wth regard to effcency s resultng n a short lfe span of the exstng plants. In order to encourage a sustanable development n the constructon ndustry a foundaton slab could be used for several wnd power plants. The lfe span of a wnd power plant s today set to 0 years but a foundaton slab, desgned wth regard to fatgue, can have a much longer lfe span. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 73

86 9 References CEB (1988): Fatgue of concrete structures, CEB (Comté Euro-nternatonal du Béton), No 188, Lausanne, Swtzerland. fb (000): Bulletn 10: Bond of renforcement n concrete, fb (fédératon nternatonal du béton), Lausanne, Swtzerland. fb (010): Bulletn 56: Model Code 010, frst complete draft, Volume, fb (fédératon nternatonal du béton), Lausanne, Swtzerland. IEC (005): Wnd turbnes-part 1: Desgn requrements, IEC (Internatonal Electrotechncal Commsson) IEC :005, Geneva, Swtzerland. Johansson, U. (004): Fatgue Tests and Analyss of Renforced Concrete Brdge Deck Models. Lc. thess. Department of Archtectural and Structural Engneerng, Royal Insttute of Technology. Stockholm, Sweden, 004. SIS (005): Eurokod : Dmensonerng av betongkonstruktoner Del 1-1: Allmänna regler och regler för byggnader (Eurocode : Desgn of concrete structures Part 1-1: General rules and rules for buldngs. In Swedsh), SIS Swedsh standards nsttute, Svensk standard SS-EN :005, Stockholm, Sweden. SIS (005): Eurokod : Dmensonerng av betongkonstruktoner Del : Broar (Eurocode : Desgn of concrete structures Part : Concrete brdges. In Swedsh), SIS Swedsh standards nsttute, Svensk standard SS-EN 199-:005, Stockholm, Sweden. Svensk Byggtjänst (1990): Betonghandbok Konstrukton (Handbook on Concrete Structures, Desgn. In Swedsh), Svensk Byggtjänst, Stockholm, Sweden. Svensk Byggtjänst (1994): Betonghandbok Materal (Handbook on Concrete Structures, Materal. In Swedsh), Svensk Byggtjänst, Stockholm, Sweden. Swedsh Energy Agency (011). Vndkraftsstatstk 010. [retreved ] Accessble at: Thun, H (006). Assessment of Fatgue Resstance and Strength n Exstng Concrete Structures. Ph.D thess. Department of Cvl and Envronmental Engneerng, Luleå Unversty of Technology. Luleå, Sweden, CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

87 Appendx A. Intal condtons for the slab 1 A.1 Loads 1 A.1 Materal propertes 1 A. Geometry of slab A.3 Self weght of the slab 3 A.4 Sol pressure 4 B. Structural analyss 5 B.1 Force dstrbuton 5 B.1 Ultmate lmt state 9 B. Servceablty lmt state 19 C. Fatgue assessment background calculatons 3 C.1 Fatgue load nput 3 C. Maxmum stresses n struts and tes 4 C.3 Mnmum stresses n struts and tes 35 D. Fatgue Assessment 46 D.1 Fatgue assessment of renforcng steel 46 D. Fatgue assessment of concrete n compresson 49 D.3 Fatgue assessment of shear renforcement 5

88

89 A. Intal condtons for the slab A.1 Loads The loads were calculated by Semens accordng to the desgn code IEC Ed.3. The foundaton loads are based on: Annual average wnd speed at hub heght...vave 8.5 m/s 10 mn. extreme wnd speed wth return perod of 50 years at hub heght...vref 4.5 m/s 3 sec. gust wnd speed wth return perod of 50 years at hub heght...v50e 59.5 m/s Average ar densty kg/m3 Desgn sectonal forces, ultmate lmt state Abnormal load case accordng to IEC Ed.3 Ultmate lmt state F z F res M res M z : 3600 kn Normal Force : 1080 kn Shear Force : knm Overturnng Moment : 3800kNm Torsonal Moment Desgn sectonal forces, servceablty lmt state Normal load case accordng to IEC Ed.3 Servceablty lmt state F z.sls F res.sls M res.sls M z.sls : 3600 kn Normal Force : 800 kn Shear Force : 7500 knm Overturnng Moment : 7900 knm Torsonal Moment A.1 Materal propertes Concrete Assume a concrete strength class of C30/37. Materal propertes accordng to [EN : table 3.1] f ck f cm : 30MPa Characterstc compressve strength : 38MPa Mean compressve strength at 8 days f ctm :.9MPa Mean tensle strength E cm : 33GPa Mean Young modulus for concrete ε cu : Ultmate stran CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 1

90 Desgn compressve strength f cd : f ck γ c 0 MPa Renforcement Assume renforcement of type B500B f yk E s : 500MPa Characterstc yeld strength : 00GPa Young modulus for steel γ s : 1.15 Partal coeffcent accordng to [EN : table.1n] Desgn yeld strength f yd : f yk γ s MPa α : E s E cm Post-tensoned anchor bolts f p0.1k : 900MPa γ s : 1.15 Partal coeffcent for prestressng steel [EN : table.1n] f pd : f p0.1k γ s MPa A. Geometry of slab D h t : 16000mm Length of slab : 000mm Heght of slab : 750mm The heght over the slab whch the horzontal force s actng on c con : 50mm Concrete cover φ : 5mm Dameter of the tensle renforcement φ s : 1mm Dameter of the shear renforcement Depth to bottom renforcement, placed n two drectons and layers. d 1 : h c con φ s 0.5φ 1.95 m Depth to the frst renforcement layer, unfavourable drecton CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

91 d : h c con φ s 1.5 φ 1.9m Depth to the second renforcement layer, unfavourable drecton d 1 + d d m : m Depth to top renforcement, placed n two drectons and layers d' 1 : c con + φ s + 0.5φ m d' : c con + φ s + 1.5φ 0.1m d' 1 + d' d' m : m Propertes anchor rng (Semens) 400 r o : mm.1m Outer radus 3560 r : mm 1.78m Inner radus r o + r r average : 1.94m Average radus A.3 Self weght of the slab Volume and unt weght V con : D h 51 m 3 Volume of concrete slab γ con : 5 kn m 3 Unt weght of renforced concrete γ sol: 18 kn m 3 Unt weght of the sol Total self weght of the concrete Average depth to the bottom renforcement Depth to the frst renforcement layer, unfavourable drecton Depth to the second renforcement layer, unfavourable drecton Average depth to the top renforcement G tot : V con γ con 1.8 MN Partal safety factors should be added to the self weght of the slab for ULS, SLS and EQU ULS Favourable γ G : 1.0 G ULS.fav : γ G G tot 1.8 MN Unfavourable γ G : 1.35 G ULS : γ G G tot 17.8 MN CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 3

92 SLS G SLS : G tot 1.8 MN EQU Favourable γ G.EQU : 0.9 G EQU.fav : γ G.EQU G tot 11.5 MN Unfavourable γ G.EQU : 1.1 G EQU : γ G.EQU G tot MN A.4 Sol pressure Desgn sectonal forces n ULS F z 3600 kn F res 1080 kn M res knm For the check of stablty n ULS the self weght of the slab wth partal factor for EQU s used [EN 1990/A1:005 ] R res : F z + G EQU MN Resultng vertcal force The horzontal sol pressure at the depth of the slab, wth the assumpton that the ground water level s below the slab k : 0.35 Coeffcent dependng on the type of sol ( ) p : k γ sol h kpa Multply wth 1.5 due to stff constructon 1 P h p h : 5. knm Horzontal sol pressure 3 m The horzontal sol pressure s favourable: γ G.EQU : 0.9 P EQU : γ G.EQU P knm.68 m Moment around the lower left corner gves the eccentrcty of the resultng force from the sol b : ( F z + G EQU ) D F res ( h + t) M res + P EQU D.37m R res Compressed area A com : b D m Vertcal sol pressure, ULS R res σ ULS : A com kpa 4 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

93 B. Structural analyss B.1 Force dstrbuton A part of the anchor rng wll be under compresson due to the transfer of forces. To check how large part of the anchor rng that s under compresson a smplfed stress block approach s used. Assumng a rng wth outer dameter r o and nner dameter r. [EN : fg 3.5] M Ed : M res + F res t knm η : 1.0 λ : 0.8 f ck 50MPa η f cd ( ) b λ x c d β x c F z d r o M Ed 0 where β x c s the dstance from the compressed edge to the gravty centre of the smplfed stress block and s calculated by: β x c k 1 λ x c A 1 z 1 A z A 1 A ( r o λ x c ) k λ x c r o r s 1 s r o asn.r asn ( ) k 1 r o k r A 1 s the area of the whole crcular segment, consstng of the anchor rng and the concrete area nsde the rng. A s the area of the "nner" crcular segment whch s not ncluded n the actual rng area. r λ x c r o r ( ) Assume that a quarter of the anchor rng s n tenson and that the anchor bolts are placed at the dstance r average from the centre of the anchor rng. The average depth from the top of the rng to the actve anchor bolts then become: r average sn( 45deg) d : r o m 45deg Areas of crcular segments 1 and. ( ) r o s 1 k 1 A 1 ( ) r s k A + k 1 λ x c + k λ x c r o r ( ) CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 5

94 Dstances to gravty centres for segment 1 and. 3 k 1 z 1 r o 1 A 1 3 k z r o 1 A Gven x c :.056m η f cd x c ( ) b λ x c d β x c F z d : Fnd( x c ) x c.056 m Heght of stress block: h c : λ x c m Area of the entre anchor rng A rng : π ( ) r r o r 4 o ( ) m k 1 : λ x c ( r o λ x c) s 1 : r o asn k 1 r o k : λ x c r o r ( ) 5.68m M Ed 0 r λ x c r o r ( ) s :.r asn k r 4.67 m Areas of crcular segments 1 and. ( ) r o s 1 k 1 A 1 : ( ) + k 1 h c m r s k + k h c r o r A : ( ) m 6 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

95 Dstances to gravty centres for areas 1 and. 3 k 1 z 1 : r o 1 A m 3 k z : r o 1 A m Area of effectve stress block A c0 : A 1 A m A c % The part of the anchor rng that s compressed A rng Dstance to gravty centre of the compressed part of the anchor rng z : A 1 z 1 A z A 1 A m Moment and shear force dagram The horzontal sol pressure has a small nfluence on the structural analyss, whle ths wll be dsregarded n the analyss. In order to fnd the force dstrbuton n the slab the sectonal forces n ULS are replaced by a force couple. The dstance between the compressve and tensle force actng on the slab s found from the calculatons above: l : d z m Force couple F C and F T Compressve force transferred from the anchor rng to the slab ( ) F z F res t + M res F C : MN l Tensle force transferred from the anchor rng to the slab ( ) F z F res t + M res F T : MN l The self weght of the slab s favourable and wll therefore be used wth the partal safety factor for favourable loads n ULS. R res : F z + G ULS.fav 16.4 MN CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 7

96 b : ( F z + G ULS.fav ) D F res t M res 1.993m R res Lengths of sectons between forces x : 0 m, 0.01 m.. D b m D b : r o + z m D b 3 : r o + d m From the edge to the second force of the force couple Dstrbuted forces G ULS.fav g : 0.8 MN Self weght D m R res q s : MN Reacton force b m Shear force dstrbuton V( x) : g x + q s x f x b g x + q s b f b < x b g x + q s b F C f b < x b 3 g x + q s b F C + F T f x > b 3 From the edge to the mddle of the dstrbuted reacton force From the edge to the frst force of the force couple V( b) MN ( ) V b 3 0 MN Maxmum shear force Shear force Dagram 10 Shear force [MN] V ( x) MN F C MN F T MN Length [m] 8 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 x

97 Moment dstrbuton M( x) : g x q s x f x b g x q s b ( x b) + f b < x b g ( D x) g ( D x) F T b 3 x f b < x b 3 f ( ) x > b 3 0 Moment Dagram M top : MNm Moment [MNm] 0 M( x) 0 MNm 40 M 60 bottom : MNm B.1 Ultmate lmt state x Length [m] Bottom and top renforcement Amounts of bottom and top renforcement are estmated n order to check the demand for mnmum rotatonal stffness. L crtcal : b 6.583m Moment that needs to be ressted by the bottom renforcement The crtcal secton s obtaned from the moment dagram and s counted from the edge of the slab. M bottom : R res ( L crtcal b ) G ULS.fav L D crtcal knm CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 9

98 Moment that needs to be ressted by the top renforcement G ULS.fav L D crtcal M top : Bottom renforcement amount knm A s : M bottom f yd d d' ( ) mm Top renforcement amount A' s : M top f yd d d' ( ) mm Rotatonal stffness Requred combned mnmum rotatonal stffness around horzontal axs of sol and foundaton: 1500 MNm/deg Rotatonal stffness s checked n ULS by assumng a cantlever. The reacton force from the ground s a unformly dstrbuted load over the length b. The self weght of the slab s counteractng the rotaton. Gven x II : 0.95m x II D + ( α 1) A' s ( x II d' ) α A s ( d x II) 0 x II : Fnd( x II ) x II 0.95m 3 D x II I II : + ( α 1) A' 3 s ( x II d' ) + α A s ( d x II) 1.97 m 4 h M tot : F res + t E cm D a : θ 33 GPa b m q s ( b) 3 q s ( b) a : E cm I II 4 E cm I II + M res MNm q s ( b) D b E cm I II a g D 3 6 E cm I II deg 10 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

99 M tot rotatonal stffness : MNm OK θ deg deg Strut and te model and desgn of compresson - compresson node A strut and te model s establshed to fnd the requred renforcement amounts. Under the compressve force from the tower there s a rsk of crushng of the concrete. Therefore a check of the compressve stresses n node 5 needs to be done. Node heght: ( ) u node : h d m m The node heght s ncreased to ensure that the concrete n node 5 has suffcent compressve strength. b : θ θ 1 ( F z + G ULS.fav ) D F res ( h + t) M res 1.86m R res : atan : atan ( d d' ) D l b ( d d' ) l deg deg F T : F C : F z F res ( h + t) + M res MN l F z F res ( h + t) + M res MN l G ULS.fav G : 6.4 MN CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 11

100 Node 1 C 1. : T 1.3 : Node G sn( θ ) cos( θ ) C 1. F T MN F T C.3 : sn θ 1 T.4 : cos θ 1 Node MN ( ) ( ) C MN sn( θ ) C 1. ( ) sn θ 1 cos( θ ) C MN 1.65 MN ( ) C.3 C 3.5 : cos θ 1 T MN T 3.4 : sn θ 1 Node 4 ( ) C MN T 3.4 C 4.5 : MN sn( θ 1) T 4.6 : T.4 + cos θ MN Node 5 F C MN ( ) C 4.5 F C sn θ 1 C 5.6 : sn( θ ) ( ) C MN ( ) C 4.5 C 5.7 : C cos θ 1 cos( θ ) C MN Node 6 T 6.7 : sn( θ ) C MN T 6.8 : T 4.6 cos( θ ) C 5.6 Node MN C 5.7 C 7.8 : cos( θ ) T 6.7 sn( θ ) C MN G 1 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

101 Node 8 T 6.8 R res cos( θ ) C sn( θ ) C 7.8 Compressve stress under the compressve force F C ( ) C 4.5 C c0.5 : C cos θ MN k 1 + k k average : A c0.5 : m u node k average 0.656m σ c0.5.statc : C c0.5 A c MPa The maxmum allowable stress n the node s: [EN : (6)] k 4 : 3 Natonal parameter f ck ν : MPa Natonal parameter f cd.c : k 4 ν f cd 5.8 MPa σ Rd.max: f cd.c 5.8 MPa σ c0.5.statc σ Rd.max 1 Amount of bottom renforcement ( ) MN T 1 : max T.4, T 4.6, T 6.8 T 1 A s : mm f yd Maxmum force n bottom renforcement φ 5 mm assumed dameter of the renforcement bar. A s 4 n : π φ n : 157 Number of bars requred ( ) D n φ c con cc : mm Maxmum spacng between bars ( n 1) cc : 150mm CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 13

102 Amount of top renforcement T : T MN There s only one te n the top of the slab T A' s : mm f yd Amount of shear renforcement outsde nsert rng T MN T 6.7 A ss.out : 3000 mm f yd Amount of shear renforcement nsde nsert rng T MN T 3.4 A ss.n : mm f yd The dfferences between the renforcement amounts estmated earler and the ones from the strut and te model depends for example on where the moment equlbrum s calculated. In the strut and te model a total equlbrum s assumed and the moment equlbrum equaton s therefore calculated for the horzontal force actng on the heght of the slab plus the heght of the anchor rng. For the renforcement amounts estmated earler the moment equlbrum s estmated n the nterface between the tower and the slab. The tensle force F T and the compressve force F C are actng at a dstance l from each other. In the strut and te model the nodes these forces are actng n are placed at a dstance l/ from the mddle of the slab. In realty the dstance from the mddle of the slab to the compressve force s smaller than the dstance from the mddle of the slab to the tensle force. In the strut and te model the self weght of the slab s modelled as concentrated loads, whle n the force dstrbuton above t s modelled as a unformly dstrbuted load. These are all reasons for the dfferences between the calculated renforcement amounts. Shear force [EN : ] The worst case for checkng the shear capacty s usng the effectve depth for the second layer of renforcement, d V Ed : R res b g MN Crtcal shear force V Rd.c C Rd.c k 1 ( 100ρ f 3 ck) D d v mn D d v mn k f ck 14 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

103 0.18 C Rd.c : γ c mm k : k.0 1 d A s1 : A s mm ρ : A s1 D d 0.53 % V Rd.c : f ck C Rd.c MPa k 100 ρ MPa 1 3 D d 9.49 MN 1 3 f ck V Rd.c MPak D d MPa 1 V Ed V Rd.c 0 Shear renforcement requred! Amount of shear renforcement. The shear renforcement ressts all of the crtcal shear force. A ss : V Ed mm f yd Check of how far out n the slab the shear renforcement has to be placed. Shear renforcement s needed n every secton where the shear force s hgher than the shear capacty, V Rd.c. From the shear force dstrbuton t s seen that ths secton wll be wthn the dstance b from the edge of the slab. Gven x :.865m g x + q s x V Rd.c x : Fnd( x) x.865 m Choose to place the shear renforcement out to a dstance of.8 meters from the edge of the slab. In ths secton the shear force s checked to be lower than the shear capacty. x :.8m V x : g x + q s x V x V Rd.c 1 OK! CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 15

104 Punchng Shear [EN : ] The anchor rng transfers forces from the tower to the concrete slab. The lower part of the anchor rng, that s embedded n the concrete slab, s the anchorage of the tensle forces from the tower. The compressve forces from the tower can be assumed to be transferred drectly from the upper part of the anchor rng, whch s not embedded n the concrete slab, to the top of the foundaton slab. Due to the compressve forces punchng shear needs to be checked. Ths check s done by assumng punchng shear under a crcular column. : φ d m 1.913m d ef : d m 1.9m Effectve depth of the slab v Rd.c θ 40 1 C Rd.c k ( 100ρ l f 3 ck) v mn 35 : deg Dfferent angles that are checked wth regard to punchng to fnd the 30 crtcal secton. 6 d ef.714 d :... d m Dstance to the control secton tan θ 3.9 ( ) r cont : r o + d m Radus of the control secton u : π r cont m The length of the control secton s the permeter of the crcle wth the radus r cont k 1.34 C Rd.c 0.1 Same as those calculated for shear capacty V Rd,c ρ A s1 A : s1 D d ρ l : ρ : D d ρ l ρ CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

105 Bearng capacty for punchng shear v Rd.c : C Rd.c MPa k 100 ρ f ck MPa 1 3 f ck v Rd.c MPak 1 MPa Eccentrcty MPa M Ed e ecc : 7.34m M V Ed : M res + F res t MNm V Ed : F z 3.6 MN Ed e ecc β : π... For crcular nner columns r o + 4 d σ j : F z bd kpa Load nsde the control permeter, reduced wth the self weght of the slab. D V Ed : 0 f r cont < 4 3 b r ( cont ) D D b r cont b σ j otherwse Shear force at the control permeter, reduced wth the compressve force from the ground.... MN V Ed.red : V Ed V Ed... MN Maxmum shear stress v Ed : β V Ed.red u d ef 1... MPa 1 v Ed v Rd.c OK! 1 1 v Ed MPa CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 17

106 Mnmum and maxmum renforcement amount Ultmate lmt state [EN : ] b t : D Average wdth of the tensle zone of the secton. Mnmum renforcement area ULS f ctm A s.mn.uls : 0.6 b f t d m mm [EN : (1)] yk A s.mn.uls > b t d m 1 OK! Check that the mnmum renforcement amount s smaller than the estmated renforcement area. A s > A s.mn.uls 1 OK! Maxmum renforcement area ULS Area of the concrete cross secton A c : D h 3 m Maxmum renforcement area ULS A s.max : 0.04 A c 1.8 m Check that the maxmum renforcement amount s larger than the estmated renforcement area. A s < A s.max 1 OK! Accordng to EN : the maxmum dstance between renforcement bars should not be larger than: s max.slabs : 3 h f 3 h 400mm for man renforcement where h s the total heght of the slab 400 mm otherwse cc s max.slabs 1 OK! Post-tensoned anchor bolts The propertes for the anchor bolts are provded by the suppler of the wnd power plant and are n ths project M4 bolts wth a pretstressng force of 400 kn. The number of bolts s 100 n both the nner and outer bolt crcle. 18 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

107 φ b : 4mm Area of one anchor bolt A p : π φ b mm Desgn capacty of one anchor bolt F : f pd A p kn As assumed before one quarter of the anchor rng s n tenson. Ths gves that the number of actve bolts s: n p : Wth 50 actve anchor bolts the capacty s: F tot : F n p MN F tot n b : > F T 1 OK! F T F B. Servceablty lmt state Crack wdth [EN : ] Steel stress n SLS state II The needed amount of actve anchor bolts to anchor the tensle force. In servceablty lmt state the self weght of the slab s used wth a partal factor for SLS. The horzontal sol pressure s dsregarded. R res.sls : F z.sls + G SLS 16.4 MN ( ) D F z.sls + G SLS F res.sls ( h + t) M res.sls b SLS : 3.445m R res.sls M bottom.sls : Gven x II : 0.301m R res.sls ( L crtcal b SLS) x II D + ( α 1) A' s ( x II d' ) α A s ( d x II) 0 G SLS D L crtcal knm CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 19

108 x II : Fnd( x II ) x II 0.301m 3 D x II I II : + ( α 1) A' 3 s ( x II d' ) + α A s ( d x II) 1.34 m 4 σ s : α M bottom.sls d I x II II ( ) w k s r.max ε sm ε cm ε sm ε cm σ s ( ) MPa f ct.eff k t ( 1 + α ρ e ρ p.eff ) p.eff σ s 0.6 E s E s Maxmum allowed crack wdth s dependent on lfe tme and exposure class etc. (L50 and XC) w k.max : 0.45mm Maxmum allowed crack wdth accordng to [EN :005 NA] k t : 0.4 Long term duraton of load f ct.eff : f ctm.9 MPa ( ) h x II h A c.eff : mn.5 ( h d m),, 3 D 3.48 m Effectve area A s ρ p.eff : 0.0 A c.eff E s 00 GPa α e σ s : E s E cm MPa Steel stress ε ε sm ε cm ε : σ s f ct.eff k t ( 1 + α ρ e ρ p.eff ) p.eff E s σ s σ s ε : 0.6 f ε 0.6 E s E s ε otherwse 0 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

109 s r.max k 3 c con + k 1 k k 4 ρ p.eff φ φ 5 mm Dameter of the renforcement c con 50 mm Concrete cover k 1 : 0.8 Good nteracton between concrete and renforcement k : 1.0 No prestressed renforcement k 3 : 7 φ 3.5 Accordng to natonal annex c con k 4 : 0.45 Accordng to natonal annex s r.max : k 3 c con + k 1 k k m ρ p.eff φ w k : s r.max ε 0.37 mm Crack wdth w k w k.max 1 OK! Check f estmated crack wdth s smaller than maxmum allowed crack wdth accordng to natonal annex n EN :005 Mnmum renforcement amount for lmtaton of crack wdth Servceablty lmt state [EN : ] f yk 500 MPa k c : 1.0 Pure tenson k : 0.65 h > 800 mm f ct.eff.9 MPa Average concrete tensle strength at the tme when the frst crack s expected. Assume ths wll occur after 8 days. α : E s E cm A I : A c + ( α 1) A' s + ( α 1) A s m x I : h A c + ( α 1) A' s d' m + ( α 1) A s d m 1.008m A I CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 1

110 ( ) A ct : D h x I m Area of the tensle zone before the frst crack Mnmum renforcement area SLS A s.mn : k c k f ct.eff A ct 0.06 m f yk Check that the mnmum renforcement amount s smaller than the estmated renforcement area. A s A s.mn 1 OK! A s m CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

111 C. Fatgue assessment background calculatons C.1 Fatgue load nput Fatgue moment and horzontal force mported from excel. M Maxover : Maxmum overturnng moment n a cycle. M Mnover : Mnmum overturnng moment n a cycle n over : Number of cycles overturnng moment F Fatmax : Maxmum horzontal force n a cycle F Fatmn : Mnmum horzontal force n a cycle n F : Number of cycles horzontal force M fat.max : M Maxover knm F fat.max : F Fatmax kn M fat.mn : M Mnover knm ( ( ) 1) : 0.. length M fat.max F fat.mn : F Fatmn kn A frst check of the fatgue stresses shows that the horzontal force s small n comparson to the overturnng moment and has no mpact on the fatgue lfe of the slab. The horzontal force s therefore dsregarded n the fatgue assessment. D b : r average 6.06 m D b 3 : + r average 9.94 m CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 3

112 To fnd the stresses that are affectng the fatgue lfe of the slab 4 dfferent strut and te models are establshed. Four models are needed n order to descrbe the behavour of the slab properly snce the fatgue moments are of varyng sze and results n dfferences n the locaton of the reacton force from the ground. To smplfy the calculaton procedure the force couple F C and F T are assumed to be actng on the anchor rng wth a dstance between them of r average. C. Maxmum stresses n struts and tes F M z fat.max F C.max : + r... average F T.max : F M z fat.max... r average R res.max : F T.max + F C.max + G SLS... G SLS G : 6.4 MN ( ) D F z + G SLS M fat.max b :... R res.max u node 0.174m Strut and te model 1, when the resultant of the reacton force s actng nsde the anchor rng. θ 1 : atan ( d d' ) 3 D 4 b 3... deg 4 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

113 θ : atan θ 56 : atan θ 67 : atan ( d d' ) r average... deg ( d d' ) D r average ( d d' ) ( b b 3 ) D 4 ( b b 3)... deg... deg Node 1 G C 1. : sn θ 1 T 1.3 : Node C.3 : ( ) cos θ 1 ( ) C MN... MN R res.max F T.max ( ) sn( θ ) C 1. sn( θ 1 ) ( ) C 1. C.4 : cos θ C.3 + cos θ 1... MN... MN Node 3 T 3.5 : ( ) cos θ C.3 + T MN T 3.4 : sn( θ ) C.3... MN Node 4 T 3.4 C 4.5 : sn θ ( )... MN C 4.6 : C.4 cos θ ( ) C MN Node 5 F C.max sn θ C 5.6 : sn( θ 56 ) T 5.7 : T 3.5 cos θ ( ) C 4.5 ( ) C MN ( ) C cos θ MN CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 5

114 Node 6 C 6.7 : C 4.6 Node 7 R res.max C 5.6 sn θ 56 cos θ 67 ( ) sn θ 67 ( ) C 6.7 T 5.7 C 6.7 cos( θ 67 ) sn θ 67 ( ) C 6.7 G ( ) cos θ ( ) C 5.6 Compressve stress under the compressve force F C (node 5) C c0.5 : T 3.5 ( ) + cos θ ( ) C 4.5 A c0.5 : σ c0.5 : u node k average 1.31 m C c0.5 A c0.5 Maxmum stresses from strut and te model 1 σ cc.1.max : 0 f σ c0.5 < 0 σ c0.5 f σ c0.5 0 σ st.1.max : 0 f C ( C 4.6 ) f C A 4.6 < 0 s T 3.5 σ st'.1.max : f T A' s 0 f T 3.5 < 0 V 1.max : T CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

115 Strut and te model, when the resultant of the reacton force s actng nsde the anchor rng. θ 56 : atan Node 1 G C 1. : sn θ 1 T 1.3 : cos θ 1 Node C.3 : ( d d' ) θ 1 : atan 3 D b 4 3 ( d d' ) θ : atan r average... deg ( d d' ) D r average ( ) ( ) C MN... MN b ( ) ( F T.max ) C 1. sn θ 1 ( ) ( ) sn θ D ( ) C MN... deg C.4 : cos θ C.3 + cos θ 1... MN Node 3 θ 67 : atan ( ) T 3.5 : cos θ C.3 + T MN ( d d' ) D b... deg D 4... deg T 3.4 : sn( θ ) C.3... MN CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 7

116 Node 4 R res.max T C 4.5 : sn θ ( )... MN C 4.6 : C.4 cos θ ( ) C MN Node 5 F C.max sn θ C 5.6 : sn( θ 56 ) T 5.7 : T 3.5 cos θ Node 6 C 6.7 : C 4.6 Node 7 ( ) C 4.5 ( ) C 4.5 R res.max C 5.6 sn θ 56 cos θ 67 ( ) sn θ 67 ( ) C 6.7 T 5.7 C 6.7 cos( θ 67 ) sn θ 67 ( ) C 6.7 G... MN ( ) C cos θ MN ( ) cos θ ( ) C 5.6 Compressve stress under the compressve force F C (node 5) C c0.5 : T 3.5 ( ) + cos θ ( ) C 4.5 A c0.5 : σ c0.5 : u node k average 1.31 m C c0.5 A c0.5 Maxmum stresses from strut and te model σ cc..max : 0 f σ c0.5 < 0 σ c0.5 f σ c CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

117 σ st..max : 0 f C ( C 4.6 ) f C A 4.6 < 0 s T 3.5 σ st'..max : f T A' s 0 f T 3.5 < 0 V.max : T 3.4 f T f T 3.4 < 0 Strut and te model 3, when the reacton force from the ground s actng outsde the anchor rng and not further away from t than node 7. θ 1 : atan θ 56 : atan θ : atan θ 67 : atan ( d d' ) 3 D 4 b 3 ( d d' ) D r average ( d d' ) ( r average ) ( d d' ) D b 4 b CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 9

118 Node 1 G C 1. : sn θ 1 T 1.3 : Node C.3 : ( ) cos θ 1 ( ) C 1. ( F T.max ) sn θ 1 ( ) sn θ ( ) C 1. T.4 : Node 3 C 3.5 : T 3.4 : ( ) C.3 cos θ cos θ ( ) C.3 sn( θ ) C.3 ( ) C 1. cos θ 1 T 1.3 Node 4 T 3.4 C 4.5 : sn θ ( ) T 4.6 : T.4 + cos θ ( ) C 4.5 Node 5 F C.max sn θ C 5.6 : sn( θ 56 ) T 5.7 : cos θ 56 ( ) C 5.6 ( ) C 4.5 Node 6 R res.max sn θ 56 C 6.7 : sn( θ 67 ) T 4.6 Node 7 G ( ) C 5.6 cos θ 56 ( ) C 6.7 sn θ 67 ( ) C 4.5 cos θ C 3.5 ( ) C 5.6 ( ) C 6.7 cos θ 67 T 5.7 cos θ 67 ( ) C CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

119 Compressve stress under the compressve force FC (node 5) C c0.5 : C cos θ ( ) C 4.5 A c0.5 : u node k average 1.31 m C c0.5 σ c0.5 : A c0.5 Maxmum stresses from strut and te model 3 σ cc.3.max : 0 f σ c0.5 < 0 σ c0.5 f σ c0.5 0 σ st.3.max : 0 f T 4.6 < 0 T 4.6 f T A s ( ) C 3.5 σ st'.3.max : f C A' s 0 f C 3.5 > 0 V 3.max : T 3.4 f T f T 3.4 < 0 Strut and te model 4, when the reacton force from the ground s actng outsde the anchor rng and further away from t than node 7. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 31

120 θ 1 : atan ( d d' ) 3 D 4 b 3 d d' θ 56 : atan D r 4 average d d' θ 78 : atan D b 4 ( d d' ) θ : atan r ( average ) Node 1 G C 1. : sn( θ 1 ) T 1.3 : cos θ 1 Node C.3 : T.4 : cos θ Node 3 ( ) ( ) C 1. ( F T.max ) sn θ 1 ( ) C.3 ( ) C.3 ( ) sn θ ( ) C 1. ( ) C 1. cos θ 1 C 3.5 : cos θ T 1.3 T 3.4 : sn( θ ) C.3 Node 4 T 3.4 C 4.5 : sn( θ ) T 4.6 : T.4 + cos θ ( ) C 4.5 Node 5 ( ) C 4.5 F C.max sn θ C 5.6 : sn( θ 56 ) C 5.7 : C cos( θ ) C 4.5 cos θ 56 ( ) C CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

121 Node 6 T 6.8 : T 4.6 cos θ 56 T 6.7 : Node 7 sn θ 56 ( ) C 5.6 ( ) ( ) C 7.8 C 5.7 C 7.8 : cos θ 78 T 6.7 Node 8 T 6.8 R res.max sn θ 78 cos θ 78 ( ) C 5.6 G ( ) C 7.8 sn θ 78 ( ) C 7.8 Compressve stress under the compressve force F C (node 5) C c0.5 : C cos θ ( ) C 4.5 A c0.5 : u node k average 1.31 m C c0.5 σ c0.5 : A c0.5 Maxmum stresses from strut and te model 4 σ cc.4.max : 0 f σ c0.5 < 0 σ c0.5 f σ c0.5 0 σ st.4.max : 0 f T 4.6 < 0 T 4.6 f T A s ( ) C 3.5 σ st'.4.max : f C A' s 0 f C 3.5 > 0 V 4.max : T 3.4 f T f T 3.4 < 0 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 33

122 Maxmum stresses n crtcal sectons b 3 σ cc.max : σ cc.1.max f b + D 8 σ cc..max f D r average b < b 3 + D 8 σ cc.3.max f D r average b > D 4 σ cc.4.max otherwse b 3 σ st.max : σ st.1.max 0 f b + D 8 σ st..max 0 σ st.3.max σ st.4.max otherwse D f r average b < D f r average b > D 4 b 3 + D 8 b 3 σ st'.max : σ st'.1.max f b + D 8 σ st'..max f D r average b < b 3 + D 8 σ st'.3.max f D r average b > D 4 σ st'.4.max otherwse b 3 V max : V 1.max f b + D 8 V.max f D r average b < b 3 + D 8 V 3.max f D r average b > D 4 V 4.max otherwse 34 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

123 C.3 Mnmum stresses n struts and tes F C.mn : F T.mn : F M z fat.mn + r... average F M z fat.mn... r average R res.mn : F T.mn + F C.mn + G SLS... ( ) D F z + G SLS M fat.mn b :... R res.mn Strut and te model 1, when the resultant of the reacton force s actng nsde the anchor rng. θ 1 : atan θ : atan ( d d' ) 3 D 4 b 3 ( d d' ) r average... deg... deg θ 56 : ( 90 deg) atan D f D r average ( d d' ) r average ( b b 3) ( b b 3 ) 0 otherwse... deg CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 35

124 θ 67 : atan ( d d' ) ( b b 3 ) D 4... deg Node 1 G C 1. : sn θ 1 T 1.3 : Node C.3 : ( ) cos θ 1 ( ) C MN R res.mn F T.mn ( ) sn( θ )... MN C 1. sn( θ 1 ) ( ) C 1. C.4 : cos θ C.3 + cos θ 1... MN Node 3 ( ) T 3.5 : cos θ C.3 + T MN T 3.4 : Node 4 sn( θ ) C.3 T 3.4 C 4.5 : sn θ ( )... MN... MN... MN C 4.6 : C.4 cos θ ( ) C MN Node 5 F C.mn sn θ C 5.6 : sn( θ 56 ) T 5.7 : T 3.5 cos θ Node 6 C 6.7 : C 4.6 ( ) C 4.5 ( ) C 4.5 R res.mn C 5.6 sn θ 56 sn( θ 67 ) ( ) C 6.7 cos θ MN ( ) C cos θ MN ( ) cos θ ( ) C CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

125 Node 7 T 5.7 C 6.7 cos( θ 67 ) sn θ 67 ( ) C 6.7 G Compressve stress under the compressve force FC (node 5) C c0.5 : T 3.5 ( ) + cos θ ( ) C 4.5 A c0.5 : u node k average 1.31 m C c0.5 σ c0.5 : A c0.5 Maxmum stresses from strut and te model 1 σ cc.1.mn : 0 f σ c0.5 < 0 σ c0.5 f σ c0.5 0 σ st.1.mn : 0 f C ( C 4.6 ) f C A 4.6 < 0 s T 3.5 σ st'.1.mn : f T A' s V 1.mn : T f T 3.5 < 0 Strut and te model, when the resultant of the reacton force s actng nsde the anchor rng. CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 37

126 θ 1 : atan θ : atan θ 56 : atan ( d d' ) 3 D 4 b 3 ( d d' ) r average... deg... deg ( d d' ) D r average ( d d' ) θ 67 : atan D b Node 1 G C 1. : sn( θ 1 ) T 1.3 : cos θ 1 ( ) C 1. D 4... MN... MN b... deg D... deg Node C.3 : ( ) ( F T.mn ) C 1. sn θ 1 ( ) ( ) sn θ ( ) C MN C.4 : cos θ C.3 + cos θ 1... MN Node 3 ( ) T 3.5 : cos θ C.3 + T MN T 3.4 : sn( θ ) C.3... MN Node 4 R res.mn T C 4.5 : sn θ ( )... MN C 4.6 : C.4 cos θ ( ) C MN 38 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

127 Node 5 F C.mn sn θ C 5.6 : sn( θ 56 ) T 5.7 : T 3.5 cos θ Node 6 C 6.7 : C 4.6 Node 7 ( ) C 4.5 ( ) C 4.5 R res.mn C 5.6 sn θ 56 sn( θ 67 ) ( ) C 6.7 cos θ 67 T 5.7 C 6.7 cos( θ 67 ) sn θ 67 ( ) C 6.7 C c0.5 : T 3.5 G ( ) + cos θ... MN ( ) C cos θ MN ( ) cos θ 56 ( ) C ( ) C 5.6 Compressve stress under the compressve force F C (node 5) A c0.5 : u node k average 1.31 m C c0.5 σ c0.5 : A c0.5 Maxmum stresses from strut and te model σ cc..mn : 0 f σ c0.5 < 0 σ c0.5 f σ c0.5 0 σ st..mn : 0 f C ( C 4.6 ) f C A 4.6 < 0 s T 3.5 σ st'..mn : f T A' s 0 f T 3.5 < 0 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 39

128 V.mn : T 3.4 f T f T 3.4 < 0 Strut and te model 3, when the reacton force from the ground s outsde the anchor rng and not further away from t than node 7. θ 1 : atan θ 56 : atan ( d d' ) 3 D 4 b 3 ( d d' ) D r average ( d d' ) θ : atan ( r average ) ( d d' ) θ 67 : atan D b 4 Node 1 G C 1. : sn θ 1 T 1.3 : cos θ 1 Node C.3 : T.4 : cos θ ( ) ( ) C 1. ( F T.mn ) sn θ 1 sn( θ ) ( ) C.3 b ( ) C 1. ( ) C 1. cos θ 1 40 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

129 Node 3 C 3.5 : T 3.4 : cos θ ( ) C.3 sn( θ ) C.3 T 1.3 Node 4 T 3.4 C 4.5 : sn θ ( ) T 4.6 : T.4 + cos θ ( ) C 4.5 Node 5 F C.mn sn θ C 5.6 : sn( θ 56 ) T 5.7 : cos θ 56 Node 6 ( ) C 5.6 ( ) C 4.5 R res.mn sn θ 56 C 6.7 : sn( θ 67 ) T 4.6 Node 7 G T 5.7 ( ) C 5.6 cos θ 56 ( ) C 6.7 cos θ 67 ( ) C 6.7 sn θ 67 ( ) C 4.5 cos θ C 3.5 ( ) C 5.6 ( ) C 6.7 cos θ 67 Compressve stress under the compressve force F C (node 5) C c0.5 : C cos θ ( ) C 4.5 A c0.5 : u node k average 1.31 m σ c0.5 : C c0.5 A c0.5 Maxmum stresses from strut and te model 3 σ cc.3.mn : 0 f σ c0.5 < 0 σ c0.5 f σ c0.5 0 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 41

130 σ st.3.mn : 0 f T 4.6 < 0 T 4.6 f T A s ( ) C 3.5 σ st'.3.mn : f C A' s 0 f C 3.5 > 0 V 3.mn : T 3.4 f T f T 3.4 < 0 Strut and te model 4, when the reacton force from the ground s actng outsde the anchor rng and further away from t than node 7. θ 1 : atan θ 56 : atan θ 78 : atan θ : atan ( d d' ) 3 D 4 b 3 ( d d' ) D r 4 average d d' D b 4 ( d d' ) ( r average ) 4 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

131 Node 1 G C 1. : sn θ 1 T 1.3 : Node C.3 : T.4 : cos θ Node 3 ( ) cos θ 1 ( ) C 1. ( F T.mn ) sn θ 1 ( ) C.3 ( ) C.3 ( ) sn θ ( ) C 1. ( ) C 1. cos θ 1 C 3.5 : cos θ T 1.3 T 3.4 : sn( θ ) C.3 Node 4 T 3.4 C 4.5 : sn θ ( ) T 4.6 : T.4 + cos θ ( ) C 4.5 Node 5 F C.mn sn θ C 5.6 : sn( θ 56 ) ( ) C 4.5 ( ) C 4.5 C 5.7 : C cos θ cos θ 56 Node 6 T 6.8 : T 4.6 cos θ 56 T 6.7 : Node 7 sn θ 56 ( ) C 5.6 ( ) ( ) C 7.8 C 5.7 C 7.8 : cos θ 78 T 6.7 sn θ 78 ( ) C 5.6 G ( ) C 5.6 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 43

132 Node 8 T 6.8 cos θ 78 ( ) C 7.8 R res.mn sn θ 78 ( ) C 7.8 Compressve stress under the compressve force F C (node 5) C c0.5 : C cos θ ( ) C 4.5 A c0.5 : u node k average 1.31 m C c0.5 σ c0.5 : A c0.5 Maxmum stresses from strut and te model 4 σ cc.4.mn : 0 f σ c0.5 < 0 σ c0.5 f σ c0.5 0 σ st.4.mn : 0 f T 4.6 < 0 T 4.6 f T A s ( ) C 3.5 σ st'.4.mn : f C A' s 0 f C 3.5 > 0 V 4.mn : T 3.4 f T f T 3.4 < 0 Maxmum stresses n crtcal sectons b 3 σ cc.mn : σ cc.1.mn f b + D 8 σ cc..mn f D r average b < b 3 + D 8 σ cc.3.mn f D r average b > D 4 σ cc.4.mn otherwse 44 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

133 σ cc.mn.ec : σ cc.mn f σ cc.max 0 0 otherwse b 3 σ st.mn : σ st.1.mn 0 f b + D 8 σ st..mn 0 σ st.3.mn σ st.4.mn otherwse D f r average b < D f r average b > D 4 b 3 + D 8 b 3 σ st'.mn : σ st'.1.mn f b + D 8 σ st'..mn f D r average b < b 3 + D 8 σ st'.3.mn f D r average b > D 4 σ st'.4.mn otherwse b 3 V mn : V 1.mn f b + D 8 V.mn f D r average b < b 3 + D 8 V 3.mn f D r average b > D 4 V 4.mn otherwse CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 45

134 D. Fatgue Assessment D.1 Fatgue assessment of renforcng steel Steel stress range n bottom renforcement σ st : σ st.max σ st.mn σ st : σ st Steel stress range n top renforcement σ st' : σ st'.mn σ st'.max σ st' : σ st' f σ st' > 0 σ st'.1.mn σ st'.1.max otherwse Fatgue assessment of renforcng steel, accordng to Eurocode [EN : ] γ Sfat : 1.15 Partal factor that takes uncertantes n the materal nto account, [EN : table.1n] γ Ffat : 1.0 σ Rsk : 16.5MPa Partal factor for fatgue loadng, [EN : (1)] The fatgue stress range after N * cycles [EN :005 table 6.3N] k 1 s the exponent that defnes the slope of the frst part of the S-N curve [EN :005 table 6.3N]. k 1 : 5 k s the exponent that defnes the slope of the second part of the S-N curve. k : 9 Fatgue assessment of the bottom renforcement Reference value for number of cycles untl fatgue falure, dependng on renforcement type [EN :005 table 6.3N] N ref : 10 6 Number of cycles untl fatgue falure, overturnng moment. ( ( ) 1) : 0.. length σ st 46 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

135 N st : N ref N ref σ Rsk γ Sfat γ Ffat σ st 0 otherwse σ Rsk γ Sfat γ Ffat σ st k 1 k f f γ Ffat σ st γ Ffat σ st σ Rsk γ σ st 0 Sfat σ Rsk < γ σ st 0 Sfat Total fatgue damage of the bottom renforcement, caused by overturnng moment, accordng to the Palmgren-Mner rule. j : ( length( N st ) 1) j D st : 0 n over D st 1 1 D st f N st 0 N st otherwse Fatgue assessment of the top renforcement Reference value for number of cycles untl fatgue falure, dependng on renforcement type [EN :005 table 6.3N] N ref : 10 6 Number of cycles untl fatgue falure, overturnng moment. ( ( ) 1) : 0.. length σ st' N st' : N ref N ref σ Rsk γ Sfat γ Ffat σ st' 0 otherwse σ Rsk γ Sfat γ Ffat σ st' k 1 k f f γ Ffat σ st' γ Ffat σ st' σ Rsk γ σ st' 0 Sfat σ Rsk < γ σ st' 0 Sfat CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 47

136 Total fatgue damage of the top renforcement, caused by overturnng moment, accordng to the Palmgren-Mner rule. j : ( length( N st' ) 1) j D st' : 0 n over D st' 1 0 D st' f N st' 0 N st' otherwse Fatgue assessment of renforcng steel, accordng to Eurocode [EN : ] The followng condton should be fulflled for requred fatgue resstance γ Ffat σ s.max Bottom renforcement σ Rsk γ Sfat Maxmum steel stress range n the bottom renforcement σ st.max: max σ st Check of condton γ Ffat σ st.max Top renforcement ( ) MPa σ Rsk 0 γ Sfat Maxmum steel stress range n the top renforcement σ st'.max: max σ st' Check of condton γ Ffat σ st'.max ( ) MPa σ Rsk 0 γ Sfat 48 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

137 D. Fatgue assessment of concrete n compresson Fatgue assessment of concrete n compresson, accordng to Eurocode [EN : ] The followng condton should be fulflled for suffcent fatgue resstance E cd.max R eq 1 γ Cfat : 1.5 Coeffcent takng the materal uncertantes nto account f cdfat : f ck γ Cfat 0 MPa k 4 : 3 natonal parameter f cd.c : k 4 ν f cdfat k 1c : 1.0 Coeffcent dependng on the reference number of cycles t 0 : 8 Concrete age at frst fatgue load applcaton s c : 0.5 Coeffcent dependng on the type of cement Coeffcent for estmated concrete compressve strength at frst fatgue load applcaton 8 s c 1 t 0 β cc : e 1 Desgn fatgue resstance of concrete f cd.fat : k 1c β cc f cd.c... MPa Mnmum compressve stress level n a cycle ( ( ) 1) : 0.. length σ cc.mn E cd.mn : σ cc.mn.ec... f cd.fat Maxmum compressve stress level n a cycle E cd.max : σ cc.max... f cd.fat CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 49

138 Stress rato R eq : 0 0 f f E cd.mn E cd.max logn cc : 14 E cd.mn < 0 E cd.max E cd.mn 1 E cd.max otherwse 1 E cd.max 1 R eq 307 otherwse f 1 E cd.max R eq logn cc N cc : 10 Number of cycles untl fatgue falure [EN 199-:005] j D cc.ec : 0 n over logn cc 10 D cc.ec 0 E cd.max R eq check : E cd.max R eq 1... check 85 check length( M fat.max ) 1 50 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

139 Fatgue assessment of concrete n compresson, accordng to Eurocode [EN : ] σ cc.max σ cc.mn check : f cd.fat f cd.fat check 85 check length( M fat.max ) 1 Fatgue assessment of concrete n compresson, accordng to fb Model Code 010 β csus : 0.85 Coeffcent that takes the effect of hgh mean stresses durng loadng nto account. Can be assumed to 0.85 for fatgue loadng. f ctk0 : 10 MPa Fatgue reference compressve strength f ck.c : k 4 ν f ck... MPa f ckfat : β cc β csus f ck.c... MPa The maxmum compressve stress level caused by overturnng moment. S cmax : σ cc.max f ckfat The mnmum compressve stress level S cmn : σ cc.mn f ckfat Check f S cmn s between 0 and 0.8, otherwse S cmn s put equal to S cmn 0.8 S cmn : 0.8 f S cmn > 0.8 S cmn f 0 S cmn 0.8 S c : S cmax S cmn... CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 51

140 ( ) logn 1 : S cmn + 8 S cmn ( ) ( ) logn : 0. logn 1 logn 1 1 logn 3 : 307 f S c 0 1 S cmax ( ) logn S cmn ( ) logn fat : logn 1 f logn S c otherwse logn f S c S cmn logn 1 > 6 logn 3 otherwse logn fat : logn fat f logn fat otherwse Check the damage wth Palmgren-Mner summaton j : ( length( logn fat ) 1) j D cc : 0 D cc 0 n over logn fat 10 D.3 Fatgue assessment of shear renforcement Fatgue assessment of shear, accordng to Eurocode. Assume the slab s cracked and no contrbuton from the concrete. V max V mn σ ss.max : A σ ss.mn : ss.n A ss.n Fatgue assessment of the shear renforcement Steel stress range n shear renforcement σ ss : σ ss.max σ ss.mn If σ ss s negatve ths means that the maxmum and mnmum values have been taken from dfferent strut and te models. For all these values t s true that one of them comes from model 1 and the other from model -4. Snce model 1 s the most used model the stress range s chosen from ths model n these cases. 5 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119

141 σ ss : σ ss f σ ss > 0 V 1.max A ss.n V 1.mn A ss.n otherwse Reference value for number of cycles untl fatgue falure, dependng on renforcement type [EN :005 table 6.3N] Number of cycles untl fatgue falure, overturnng moment. ( ( ) 1) : 0.. length σ ss φ bendng: 64mm N ref : 10 6 φ σ Rsk.strrups σ Rsk bendng : + σ Rsk.strrups N ss : N ref N ref 0 otherwse MPa σ Rsk.strrups γ Sfat γ Ffat σ ss σ Rsk.strrups γ Sfat γ Ffat σ ss k 1 k f f φ s γ Ffat σ ss γ Ffat σ ss σ Rsk.strrups γ σ ss 0 Sfat σ Rsk.strrups < γ σ ss 0 Sfat Total fatgue damage of the shear renforcement, caused by overturnng moment, accordng to the Palmgren-Mner rule. j : ( length( N ss ) 1) j D ss : 0 D ss 1 0 D ss f N ss 0 n over N ss otherwse CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119 53

142 Maxmum steel stress range n the shear renforcement σ ss.max: max σ ss Check of condton γ Ffat σ ss.max ( ) MPa σ Rsk.strrups 0 γ Sfat 54 CHALMERS, Cvl and Envronmental Engneerng, Master s Thess 011:119