Three-dimensional strut-and-tie modelling of wind power plant foundations

Transcription

1 Three-dimensiona strut-and-tie modeing of wind power pant foundations Master of Science Thesis in the Master s Programme Structura engineering and buiding performance design NICKLAS LANDÉN JACOB LILLJEGREN Department of Civi and Environmenta Engineering Division of Structura Engineering Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 01 Master s Thesis 01:49

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3 MASTER S THESIS 01:49 Strut-and-tie modeing of wind power pant foundations Master of Science Thesis in the Master s Programme Structura engineering and buiding performance design NICKLAS LANDÉN JACOB LILLJEGREN Department of Civi and Environmenta Engineering Division of Structura Engineering Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 01

4 Strut-and-tie modeing of wind power pant foundations Master of Science Thesis in the Master s Programme Structura engineering and buiding performance design NICKLAS LANDÉN JACOB LILLJEGREN NICKLAS LANDÉN, JACOB LILLJEGREN, 01 Examensarbete / Institutionen för bygg- och mijöteknik, Chamers tekniska högskoa, 01:49 Department of Civi and Environmenta Engineering Division of Structura Engineering Concrete Structures Chamers University of Technoogy SE Göteborg Sweden Teephone: + 46 (0) Cover: Estabished 3D strut-and-tie mode for a wind power pant foundation. Chamers Reproservice Göteborg, Sweden 01

5 Master of Science Thesis in the Master s Programme Structura engineering and buiding performance design NICKLAS LANDÉN JACOB LILLJEGREN Department of Civi and Environmenta Engineering Division of Structura Engineering Concrete structures Chamers University of Technoogy ABSTRACT With an increasing demand for renewabe energy sources wordwide, a promising aternative is wind power. During the ast decades the number of wind power pants and their size has increased. Wind power pant foundations are subjected to a centric oad, resuting in a 3D stress distribution. Even though this is known, the common design practice today is to design the foundation on the basis of cassica beam-theory. There is aso an uncertainty of how to treat the fatigue oading in design. Since a wind power pant is highy subjected to arge variety of oad ampitudes the fatigue verication must be performed. The purpose with this master thesis project was to cary the uncertainties in the design of wind power pant foundations. The main objective was to study the possibiity and suitabiity for designing wind power pant foundations with 3D strutand-tie modeing. The purpose was aso to investigate the appropriateness of using sectiona design for wind power pant foundations. A reference case with fixed oads and geometry was designed according to Eurocode with the two dferent methods, i.e. beam-theory and strut-and-tie modeing. Fatigue assessment was performed with Pamgren-Miners aw of damage summation and the use of an equivaent oad. The shape of the foundation and reinforcement ayout was investigated to find appropriate recommendations. The centric oaded foundation resuts in D-regions and 3D stress fow which make the use of a strut-and-tie mode an appropriate design method. The 3D strut-and-tie method propery simuates the 3D stress fow and is appropriate for design of D- regions. Regarding the common design practice the stress variation in transverse direction is not considered. Hence the design procedure is incompete. If the ineareastic stress distribution is determined, regions without stress variation in transverse direction can be distinguished. Those regions can be designed with beam-theory whie the other regions are designed with a 3D strut-and-tie mode. Further, carications of fatigue assessment regarding the use of an equivaent oad for reinforced concrete need to be recognized. The method of using an equivaent oad in fatigue cacuations woud consideraby simpy the cacuations for both reinforcement and concrete. We found the use of 3D strut-and-tie method appropriate for designing wind power pant foundations. But the need for computationa aid or an equivaent oad are recommended in order to perform fatigue assessment. Key words: wind power pant foundation, gravity foundations, 3D, three-dimensiona strut-and-tie mode, fatigue, equivaent oad, concrete I

6 Dimensionering av vindkraftsfundament med tredimensionea fackverksmodeer Examensarbete inom Structura engineering and buiding performance design NICKLAS LANDÉN JACOB LILLJEGREN Institutionen för bygg- och mijöteknik Avdeningen för Konstruktionsteknik Betongbyggnad Chamers tekniska högskoa SAMMANFATTNING I takt med ökad efterfrågan på förnyesebara energikäor de senaste decennierna har både antaet vindkraftverk och dess storek vuxit. De större kraftverken har resuterat i större aster och därmed större fundament. På grund av en ständigt varierande vindast måste fundamenten dimensioneras för utmattning. Vidare är fundamenten centriskt beastade viket ger upphov ti ett 3D spänningsföde. Det verkar dock vanigt att dimensionera fundamenten genom att anta att spänningarna är jämt utspridda över hea fundamentet och använda bakteori. Ett sätt att ta större hänsyn ti det 3D spänningsfödet är att dimensionera fundamentet med en 3D fackverksmode. Det huvudsakiga syftet med examensarbetet var att undersöka möjigheten att dimensionera vindkraftsfundament med en 3D fackverksmode, men även undersöka om det är ämpigt att basera dimensioneringen på bakteori. Dessutom har oika armeringsutformningar studerats. För att utreda nämnda frågestäning utfördes en dimensionering av ett vindkraftsfundament med givna aster och dimensioner grundat på Eurocode. Fundamentet dimensionerades både med en 3D fackverksmode och genom att använda bakteori. Utmattningsberäkningarna utfördes med Pamgren-Miners deskadehypotes och med en ekvivaent spänningsvariation. Med hänsyn ti astförutsättningen, viket förutom att ge upphov ti ett 3D spänningsföde också resuterar i D-regioner. Därav finner vi det ämpigt att använda sig av 3D fackverksmodeer. Gäande dimensionering grundad på bakteori är denna ogitig då spänningsvariationen den transversea riktningen inte beaktas. Vi anser att det är ämpigt att använda sig av 3D fackverksmodeer, det krävs dock en automatiserad metod eer en ekvivaent ast för att kunna hantera hea astspektrumet. Gäande användandet av en ekvivaent ast krävs vidare studier på hur denna ska beräknas. Nyckeord: vindkraftsfundament, gravitationsfundament, 3D, tredimensione, fackverksmode, ekvivaent ast, betong II

7 Contents ABSTRACT SAMMANFATTNING CONTENTS PREFACE NOTATIONS I II III VII VIII 1 INTRODUCTION Background 1 1. Purpose and objective 1.3 Limitations 1.4 Method 3 WIND POWER PLANT FOUNDATIONS 4.1 Design aspects of wind power pant foundations 4. Function of gravity foundations 4.3 Connection between tower and foundation 5 3 DESIGN ASPECTS OF REINFORCED CONCRETE Shear capacity and bending moment capacity 6 3. Fatigue Fatigue in stee Fatigue in concrete Fatigue in reinforced concrete 10 4 STRUT-AND-TIE MODELLING Principe of strut-and-tie modeing 1 4. Design procedure Struts Ties Strut incinations Nodes Three-dimensiona strut-and-tie modes Nodes and there geometry 18 5 REFERENCE CASE AND DESIGN ASSUMPTIONS Design codes Genera conditions 0 CHALMERS Civi and Environmenta Engineering, Master s Thesis 01:49 III

8 5.3 Geometry and oading Tower foundation connection 5.5 Goba equiibrium 5 6 DESIGN OF THE REFERENCE CASE ACCORDING TO COMMON PRACTICE ON THE BASIS OF EUROCODE Bending moment and shear force distribution 9 6. Bending moment capacity Shear capacity Crack width imitation Fatigue Resuts Concusions on common design practice 4 7 DESIGN OF REFERENCE CASE WITH 3D STRUT-AND-TIE MODELS AND EUROCODE Methodoogy Two-dimensiona strut-and-tie mode Three-dimensiona strut-and-tie modes Reinforcement and node design Fatigue Resuts Concusions on the 3D strut-and-tie method 55 8 CONCLUSIONS AND RECOMMENDATIONS Reinforcement ayout and foundation shape Suggestions on further research 56 9 REFERENCES 58 APPENDICES A IN DATA REFERENCE CASE 60 A.1 Geometry 60 A. Loads 6 A.3 Materia properties 64 B GLOBAL EQUILIBRIUM 65 B.1 Eccentricity and width of soi pressure 65 IV CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

9 B. Shear force and bending moment distribution 68 B.3 Sign convention 74 C DESIGN IN ULTIMATE LIMIT STATE 75 C.1 Sections 75 C. Design of bending reinforcement 77 C.3 Star reinforcement inside embedded anchor ring 80 C.4 Minimum and maximum reinforcement amount 8 C.5 Shear capacity 8 C.6 Loca effects and shear reinforcement around stee ring 86 D CRACK WIDTHS SERVICE ABILITY LIMIT STATE 91 D.1 Loads 91 D. Crack contro 9 E FATIGUE CALCULATIONS WITH EQUIVALENT LOAD CYCLE METHOD 97 E.1 Loads and sectiona forces 97 E. Fatigue contro bending moment 10 E.3 Fatigue contro oca effects 109 G FATIGUE CONTROL WITH THE FULL LOAD SPECTRA 11 G.1 Loads and sectiona forces 11 G. Fatigue in bending reinforcement 11 G.3 Shear force distribution 13 G.4 Fatigue in U-bows 135 H UTILISATION DEGREE AND FINAL REINFORCEMENT LAYOUT 138 I FATIGUE LOADS 140 J SECTIONS OF STRUT AND TIE MODEL K REINFORCEMENT CALCULATIONS AND FORCES IN STRUTS AND TIES 149 CHALMERS Civi and Environmenta Engineering, Master s Thesis 01:49 V

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11 Preface This master s thesis project was carried out at Norconsuts office in Gothenburg in cooperation with the department of structura engineering at Chamers University of Technoogy. We woud ike to thank team Byggkonstruktion for making the stay so peasant. We especiay woud ike to thank our supervisor at Norconsut Anders Bohin for aways taking the time needed to answer questions and give usefu feedback. We are aso gratefu to our examiner Professor Björn Engström and supervisor Doctor Rasmus Remping for aiding us in this master s thesis project. CHALMERS Civi and Environmenta Engineering, Master s Thesis 01:49 VII

12 Notations Roman upper case etters Cross sectiona area of reinforcement in bottom Cross sectiona area of reinforcement in top Cross sectiona area of shear reinforcement Characteristic oad Soi pressure Compressive force component from moment Most eccentric tensie force component from moment Horizonta component of wind force in x direction Horizonta component of wind force in y direction Resuting horizonta component of wind force Tota sef-weight of foundation incuding fiing materia Bending moment Bending moment around x-axis Bending moment around y-axis Resuting bending moment Characteristic moment Equivaent number of aowed cyces Norma force Range of oad cyces Equivaent range of oad cyce Shear force Shear capacity for concrete without shear reinforcement Roman ower case etters b Width of soi pressure Concrete cover Effective depth Distance between force coupe from resisting moment Diameter of anchor ring eccentricity Eccentricity of soi pressure resutant Sef-weight of sab incuding fiing materia VIII CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

13 Concrete compressive strength Design vaue of concrete compressive strength Characteristic vaue of concrete compressive strength Design yied strength of stee Design yied strength of stee Exponent that defines the sope of the S-N curve Number of cyces Radius of anchor ring Length of interna ever arm Greek upper case etters Stress Design strength for a concrete strut or node Greek ower case etters Concrete strain Stee strain Load partia factor Fatigue oad partia factor Materia partia factor Reduction factor for the compressive strength for cracked strut (EC) CHALMERS Civi and Environmenta Engineering, Master s Thesis 01:49 IX

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15 1 Introduction There is a growing demand for renewabe energy sources in the word and wind power shows a arge growth both in Sweden and gobay. Both the number of wind power pants and their sizes have increased during the ast decades. 1.1 Background In the beginning of 1980 the first wind power pants were buit in Sweden. In 009 about 1400 wind power pants produced.8 TWh/year, which corresponds to % of the tota production in Sweden, Vattenfa (011). The Swedish government's energy goa for 00 is to increase the use of renewabe energy to 50 % of tota use. This means that the energy produced ony from wind power has to increase to 30 TWh/year. As wind has become a more popuar source of energy the deveopment of arger and more effective wind power pants has gone rapidy. The sizes of wind power pants have increased from a height of 30 m and a diameter of the rotor bade of 15 m in 1980 to a height of 10 m and a diameter of the rotor bade of 115 m in 005, se Figure 1.1. Figure 1.1 How the size of rotor bade and height have changed from 1980 to 005 adopted from Faber, T. Steck, M. (005). The increased sizes have ed to arger oads and consequenty arger foundations. In addition to the need for sufficient resting moment capacity the foundations are subjected to cycic oading due the variation in wind oads. The cycic oading requires that the foundations are designed with regard to fatigue. The tower is connected to the centre of the foundation where the rotationa moment is transferred to the foundation according to Figure 1.. The concentrated forces cause stress variations in three directions and aso resut in a Discontinuity region (Dregion) where beam-theory no onger is vaid. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 1

16 D-region Figure 1. The foundation is subjected to concentrated forces and centric oading causing need for oad transfer in two directions. In contrast to B-regions (Bernoui- or Beam-regions) the assumption that pane sections remain pane in bending is not vaid in D-regions. Figure 1.3 shows how a centric oading resuting in a stress variation in three directions, simiar to a fat sab. Figure 1.3 Left: boundary conditions. Midde: Loading appied aong the fu width, no stress variations aong the width. Right: Centric oading resuts in stress variation in three directions. Despite the centric concentrated oad it appears to be common practice to assume that the interna forces are spread over the fu width of the foundation and base the design on cassica beam-theory. D-regions can be designed with the so caed strut-and-tie mode, which is a ower bound approach for designing cracked reinforced concrete in the utimate imit state. The method is based on pastic anaysis and is vaid for both D-regions and B-regions. In addition the strut-and-tie mode can be estabished in three dimensions to capture the 3D stress fow. For this reason the strut-and-tie method seem to be a suitabe approach to design wind power pant foundations. 1. Purpose and objective The purpose with this master thesis project was to cary the uncertainties in the design of wind power pant foundations. The main objective was to study the possibiity and suitabiity for designing wind power pant foundations with 3D strutand-tie modeing. The purpose was aso to investigate the appropriateness of using sectiona design for wind power pant foundations. 1.3 Limitations In the project, focus wi be directed to the foundation, the behaviour of the surrounding soi and its properties shoud not be investigated in detai. The master thesis project ony considers on-shore gravity foundations. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

17 1.4 Method Initiay a itterateur study was performed to gain a better understanding of the dficuties in designing wind power pant foundations. Today s design practice was identied and the various design aspects were studied. Further information about the strut-and-tie method was acquired from iterature. For the purpose of studying the suitabiity for designing wind power pant foundations with the dferent approaches a reference case was used. The reference foundation was designed with both today s design practice, i.e. using sectiona design, and the use of a 3D strut-and-tie mode. The design of the reference foundation with fixed geometry and oading was performed according to Eurocode. The two dferent design approaches was compared in order to find advantages and disadvantages with the aternative methods. To hande the compex 3D strut-and-tie modes the commercia software Strusoft FEM-design 9.0 3D frame was used. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 3

18 Wind power pant foundations This chapter presents genera information about the function and oading of gravity foundations..1 Design aspects of wind power pant foundations The ocation of a wind power pant affects the design of the foundation in many dferent ways. One of the most important is obviousy the wind conditions. The design of the foundation changes depending whether the foundation is ocated on- or off-shore. On-shore foundation design is affected by the geotechnica properties of the soi. Three dferent types of on-shore foundations can be distinguished, gravity foundations, rock anchored foundations and pie foundations. In addition to the geotechnica conditions off-shore foundations must aso be designed for currents and upting forces. The most common type is gravity foundations, which is the ony type of foundations studied in this project. Gravity foundations can be constructed in many dferent shapes such as square, octagona and circuar. The upper part of the sab can be fat, but often has a sma sope of up to 1:5 from the centre towards the edges to reduce the amount of concrete and to divert water.. Function of gravity foundations The height of modern wind power pant can be over 100 m with amost the same diameter of the rotor bades. Consequenty the foundation is subjected to arge rotationa moments. As the name gravity foundations suggest, the foundation prevents the structure from titing by its sef-weight. In addition to restrain the rotationa moment the foundation must bear the sef-weight of turbine and tower. Depending on the height of the tower, size of the turbine and ocation of the wind power pant the foundation usuay varies between a thickness of m and a width of 15-0 m. Figure.1 shows how the structure resists the rotationa moment with a eve arm between the sef-weight and reaction force of the soi. Figure.1 The structure is prevented from titing by a eve arm (e) between the sef-weight (G) and the eccentric reaction force of the soi (F soi ). 4 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

19 Depending on oad magnitude and soi pressure distribution the eccentricity varies. To transfer the oad, the foundation must have sufficient fexura and shear force capacity, which must be provided for with reinforcement. Since the wind oads vary, the foundation is subjected to cycic oads which make a fatigue design mandatory to ensure sufficient fatigue e. Figure. shows a wind power pant where the oss of equiibrium has ed to faiure, even though the fexura capacity appears to be sufficient. Figure. A coapsed power pant due to oss of equiibrium SMAG (011)..3 Connection between tower and foundation There are dferent methods used to connect the tower to the foundation Faber, T. Steck, M. (005). Figure.3 shows three common connection types, so caed anchor rings or embedded stee rings. A consist of a top fange prepared for a bot connection to the tower. The anchor rings is ocated in the centre of the foundation surrounded by concrete. The first type (a) consists of an anchor ring in stee with an I- section. Aternative (b) ony has one fange casted in the concrete and is often used in smaer foundations. This soution requires suspension reinforcement to t up the compressive oad to utiise the concrete. The ast soution (c) consists of a pre-stressed bot connection between two fanges. Need for reinforcement Figure.3 Three common types of connections between the tower and foundation, adopted from Faber, T. Steck, M. (005). CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 5

20 3 Design aspects of reinforced concrete This chapter gives a genera description of design aspects regarding interna force transfer and fatigue in reinforced concrete. 3.1 Shear capacity and bending moment capacity For beams and sabs a inear strain distribution can be assumed since the reinforcement is assumed to fuy interact with the concrete. Hence sectiona design using Navier s formua can be used for design of reinforced concrete beams and sabs. The design must ensure that both the fexura and shear capacity is sufficient. In addition imitations on crack widths and deformations must be fufied to achieve an acceptabe behaviour in serviceabiity imit state. Three types of cracks can be distinguished in beams: Shear cracks, Figure 3.1 (1): deveop when principa tensie stresses reach the tensie strength of concrete in regions with high shear stresses. Fexura cracks, Figure 3.1 (3): deveop when fexura tensie stresses reach the tensie strength of concrete in regions with high bending stresses. Fexura-shear-cracks, Figure 3.1 (). A combination of shear and fexura cracks in regions with both shear and bending stresses Figure 3.1 Exampe of crack-types in a simpy supported beam. (1) Shear crack () fexura-shear-crack (3) fexura crack. To avoid faiure due to fexura cracks, bending reinforcement is paced in regions with high tensie stresses. The mode shown in Figure 3. can be used to cacuate bending moment capacity, assuming compressive faiure in concrete. In the mode tensie strength of concrete is negected and inear eastic strain distribution is assumed. 6 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

21 ε cu f cd M Rd d x β Rx z F c M Rd F s α r f cd b ε s y Figure 3. Cacuation mode for moment capacity in reinforced concrete assuming fu interaction between stee and concrete. This resuts in a inear strain distribution. The utimate bending moment capacity can be cacuated with the foowing equations: where: (3.1) ( ) (3.) Stress bock factor for the average stress Stress bock factor for the ocation of the stress resutant Shear forces in crack concrete with bending reinforcement are transferred by an interaction between shear transferring mechanisms shown in Figure 3.3. Figure 3.3 Shear transferring mechanisms in a beam with bending reinforcement. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 7

22 The shear capacity for beams without vertica reinforcement is hard to cacuate anayticay and many design codes are based on empirica cacuations. To increase the shear capacity vertica reinforcement (stirrups) can be used resuting in a truss action as shown in Figure 3.4. Figure 3.4 Truss action in a concrete beam with shear reinforcement. The mode in Figure 3.4 is used to cacuate the shear capacity for beams with vertica or incined reinforcement; in cacuations according to Eurocode effects from dowe force and aggregate interock are negected. The incination of the compressive stress fied ( ) depends on the amount of shear reinforcement; an increased amount increases the ange. In order to achieve equiibrium an extra norma force ( ) appears in the bending reinforcement. The reationship between the additiona tensie force of the shearing force and the ange of is that one increases, the other decreases and vice versa. To ensure sufficient shear capacity the faiure modes described in Figure 3.5 must be checked. Figure 3.5 Dferent shear faiure modes. Left: shear siding. Midde: Yieding of stirrups. Right: Crushing in concrete. A specia case of shear faiure is punching shear faiure which must be considered when a concentrated force acts on a structure that transfers shear force in two directions. When faiure occurs a cone is punched through with an ange reguary between 5 and 40 degrees, exempied in Figure CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

23 Figure 3.6 Punching shear faiure in a sab supported by a coumn. A cone is punched through the sab. 3. Fatigue Faiure in materias does not ony occur when it is subjected to a oad above the utimate capacity, but aso from cycic oads we beow the utimate capacity. This phenomenon is known as fatigue and is a resut of accumuated damage in the materia from cycic oading, fatigue is therefore a serviceabiity imit state probem. American Society for Testing and Materias (ASTM) defines fatigue as: Fatigue: The process of progressive ocaized permanent structura change occurring in a materia subjected to conditions that produce fuctuating stresses and strains at some point or points and that may cuminate in cracks or compete fracture after a sufficient number of fuctuations. The fatigue e is infuenced by a number of factors such as the number of oad cyces, oad ampitude, stress eve, defects and imperfections in the materia. Even though reinforced concrete is a composite materia, the combined effects are negected when cacuating fatigue e. Instead the fatigue cacuations are carried out for the materias separatey according to Eurocode. Concrete and stee behave very dferenty when subjected to fatigue oading. One important aspect of this is that the stee wi have a strain hardening whie the concrete wi have a strain softening with increasing number of oad cyces. Another is the effect of stress eves which affects the fatigue e of concrete more than stee. Cycic oaded structures such as bridges and machinery foundations are often subjected to compex oading with arge variation in both ampitude and number of cyces. A wind power pant foundation oaded with wind is obviousy such a case. Therefore, there are simpied methods for the compiation of force ampitude, one such exampe is the rain fow method. The basic concept of the rain fow method is to simpy compex oading by reducing the spectrum. The fatigue damage for the dferent oad-ampitudes can then be cacuated and added with the Pamgren-Minor rue Fatigue in stee Fatigue damage is a oca phenomenon; it starts with micro cracks increasing in an area with repeated oading which then grow together forming cracks. Fatigue oading accumuate permanent damage and can ead to faiure. Essentiay two basic fatigue CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 9

24 design concepts are used for stee, cacuation of inear eastic fracture mechanics and use of S-N curves. In genera fatigue faiure is divided in three dferent stages, crack initiation, crack propagation and faiure. Cacuations of the fatigue e with fracture mechanics is divided into crack initiation e and crack propagation e. These phases behave dferenty and are therefore governed by dferent parameters. The other method is Whöer diagram or S-N curves which are ogarithmic graphs of stress (S) and number of cyces to faiure (N), see Figure 3.7. These graphs are obtained from testing and are unique for every detai, Stephens R (1980). Figure 3.7 S-N curves for dferent stee detais. Note that the cut-off imit shows stress ampitudes which do not resut in fatigue damage. 3.. Fatigue in concrete Concrete is a much more inhomogeneous materia than stee, Svensk Byggtjänst (1994). Because of temperature dferences, shrinkage, etc. during curing micro cracks deveop even before oading. These cracks wi continue growing under cycic oading and other cracks wi deveop simutaneousy in the oaded parts of the concrete. The cracks grow and increase in numbers unti faiure. It shoud be noted that is very hard to determine where cracking wi start and how they wi spread Fatigue in reinforced concrete As stated before the fatigue capacity of reinforced concrete is determined by checking concrete and stee separatey. When a reinforced concrete structure is subjected to cycic oad the cracks wi propagate and increase, resuting in stress redistribution of tensie forces to the reinforcement Svensk Byggtjänst (1994). Fatigue can occur in the interface between the reinforcement bar and concrete which can ead to a bond faiure. There are dferent types of bond faiure such as spitting and shear faiure aong the perimeter of the reinforcement bar. Regarding concrete without shear reinforcement the capacity is determined by the friction between the cracked surfaces. The uneven surfaces in the cracks are degraded by the cycic oad which can resut in faiure. When shear reinforcement is present, it is the fatigue properties of the shear reinforcement that wi determine the fatigue e. 10 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

25 Fatigue faiure in reinforcement can be considered more dangerous than in concrete, since there might not be any visua deformation prior to faiure. For concrete on the other hand there is often crack propagation and an increased amount of cracks aong with growing deformations, which form under a reativey ong time. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 11

26 4 Strut-and-tie modeing In this chapter the basic principes of strut-and-tie modeing wi be described. Design of the dferent parts of strut-and-tie modes wi be expained, such as ties, struts and nodes. 4.1 Principe of strut-and-tie modeing The strut-and-tie mode simuates the stress fied in reinforced cracked concrete in the utimate imit state. The method provides a rationa way to design discontinuity regions with simpied strut-an-tie modes consisting of compressed struts, tensioned ties and nodes in-between and where externa concentrated forces act. A strut-and-tie mode is we suited for Bernoui regions (B-regions) as we as in shear critica- and other disturbed (discontinuity) regions (D-regions). A D-region is where the Euer-Bernoui assumption that pane sections remain pane in bending is not vaid. Consequenty, the strain distribution is non-inear and Navier s formua is not vaid. The stress fied is indeterminate and an infinite number of dferent stress distributions are possibe with regard to equiibrium conditions. A D-region extends up to a distance of the sectiona depth of the member. The strut-and-tie mode is a ower bound soution in theory of pasticity, which means that the pastic resistance is at east sufficient to withstand the design oad. For this to be true the foowing criteria must be fufied: The stress fied satisfies equiibrium with the externa oad Ideay pastic materia response The structure behaves ductie, i.e. pastic redistribution can take pace The strut-and-tie method is beneficia to use when designing D-regions since it takes a oad effects into consideration simutaneousy i.e., and. Another advantage is that the method describes the rea behaviour of the structure. Hence, it gives the designer an understanding of cracked reinforced concrete in utimate imit state in contrary to many of the empirica formuas found in design codes. 4. Design procedure When designing structures with the strut-and-tie method, it is important to keep in mind that it is a ower bound approach based on theory of pasticity. This means that many soutions to a probem may exist and be acceptabe, even for exampe the reinforcement amount or ayout become dferent. The reason for this is that in the utimate imit state a the necessary pastic redistribution has taken pace and the reinforcement provided by the designer is utiised. However, it is sti important that the structure is designed so that the need of pastic redistribution is imited. This can be achieved by designing the structure on the basis of a stress fied cose to the inear eastic stress distribution, which wi give an acceptabe performance in serviceabiity imit state. There are no unique strut-and-tie modes for most design situations, but there are a number of techniques and rues which hep the designer to deveop a suitabe mode. To find a reasonabe stress fow there are dferent methods that can be used such as the oad path method purposed by Schaich, J. et.a (1987), stress fied approach 1 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

27 according to Muttoni, A. et.a. (1997) or by inear finite eement anaysis. These methods can aid the designer in choosing an appropriate stress feid. In order to show how a strut-and-tie mode can be estabished the methodoogy wi be used on a simpe D probem. The first step is to determine the B- and D-region. The second step is to choose a mode to simuate the stress fied. To find the stress fied the oad path method wi be used in the exampe beow. When using the oad path method there are certain rues that must be fufied: The oad path represents a ine through which the oad is transferred in the structure, i.e. from oaded area to support(s) Load paths do not cross each other The oad path deviates with a sharp bent curve near concentrated oads and supports The oad path shoud deviate with a soft bent curve further away from supports and concentrated oads At the boundary of the D-region the oad path starts in the same direction as the oad or support reaction The oad must be divided in an adequate amount to avoid an oversimpistic representation When a oad paths that fufi a these requirements have been estabished, areas where transverse forces are needed to change the direction of the oad paths are ocated. These are areas where there is a need for either a compressive or tensie force in transversa direction. It is aso important to note the change in transverse direction shoud deveop abrupty or graduay, since this wi decide the corresponding nodes wi be concentrated or distributed, which is further expained in Section 4.6 about nodes. Figure 4.1 iustrates the creation of a strut-and-tie mode by means of the oad path method. However before the strut-and-tie mode can be accepted ange imitations and contro of concentrated nodes described beow must be fufied. Figure 4.1 Exampe of how a strut-and-tie mode can be estabished by means of the oad path method. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 13

28 4.3 Struts The struts represent the compressed concrete stress fied in the strut-and-tie mode, often represented by dashed ines in the mode. Struts are generay divided in three types, prismatic-, botte- and fan-shaped struts, see Figure 4.. The prismatic-shaped strut has a constant width. The botte-shaped strut contracts or expands aong the ength and in the fan-shaped strut a group of struts with dferent incinations meet or disperse from a node. The capacity of a strut is in Eurocode reated to the concrete compressive strength under uniaxia compression. The capacity of the strut must be reduced, the strut is subjected to unfavourabe muti-axia effects. On the other hand, the strut is confined in concrete (i.e. muti-axia compression exists), the capacity of the strut becomes greater. If the compressive capacity of a strut is insufficient, it can be increased by using compressive reinforcement. Figure 4. The dferent strut shapes with exampes in a beam, Chanteot, G. and Aexandre, M. (010). 4.4 Ties Ties are the tensie members in a strut-and-tie mode, which represent reinforcement bars and stirrups. Athough concrete has a tensie capacity, its contribution to the tie is normay negected. There are two common types of ties, concentrated and distributed. Concentrated ties connected concentrated nodes and are usuay reinforced with cosey spaced bars. Distributed ties are in areas with distributed tensie stress fieds between distributed nodes and here the reinforcement is spread out over a arger area. A critica aspect when detaiing especiay concentrated ties is to provide sufficient anchorage. It can be beneficia to use stirrups, since the bends provide anchorage. 14 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

29 4.5 Strut incinations When a strut-and-tie mode is estabished, it needs to fufi rues concerning the ange between the struts and ties. The reason for this imitation is that too sma or arge anges infuence the need for pastic redistribution and the service state behaviour. The recommended anges vary between design codes, but aso depending on how the strut(s) and tie(s) intersect. When designing on the basis of more compex strut-and-tie modes, a situation may arise where a ange requirements cannot be satisfied. Then the heaviy oaded struts shoud be prioritised and the requirements for ess critica struts may be disregarded, Engström (011). Recommended anges according to Schäfer, K. (1999) Distribution of forces sha take pace directy, with approximatey 30 but not more than 45 Recommended minimum anges between struts and ties, Schäfer, K. (1999) Between strut and tie, approximatey 60 but not ess than 45 Figure 4.3 (a) and (b) In case of a strut between two perpendicuar ties, preferred 45 but not smaer than 30, see Figure 4.3 (c) and (d) Figure 4.3 Ange imitations adopted from Schäfer (1999). 4.6 Nodes Nodes represent the connections between struts and ties or the positions where the stresses are redirected within the strut-and-tie mode. Nodes are generay divided in two categories, concentrated and distributed. Distributed nodes are not critica in design and therefore not further expained. The concentrated nodes are divided into three major node types, CCC-, CCT- and CTT-nodes iustrated in Figure 4.4, Martin, B. and Sanders, D (007). The etter combinations expain which kind of forces that acts on the node, C for compression and T for tension. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 15

30 Figure 4.4 The dferent nodes, from eft to right CCC-node, CCT-node and CTTnode. When nodes are designed they are infuenced by support condition, oading pate, geometrica imitations etc. The node geometry for two common nodes is shown in Figure 4.5. Figure 4.5 Left: node region of a CCC-node. Right: node region for a CCT-node, Schäfer, K. (1999). An exampe of ideaised node geometries for a CCC-node and a CCT-node is shown in Figure 4.5. The noda geometry can be defined by determining the ocation of the node and the width of the bearing pate. It is important that the detaiing of concentrated nodes are designed in an appropriate way, especiay nodes subjected to both compression and tension forces. For exampe it is important to provide sufficient anchorage for reinforcement and confining the anchored reinforcement with for instance stirrups. Concentrated nodes shoud be designed with regard to the foowing stress imitations according to Eurocode. The compressive strength may be increased with 10 % at east one of the conditions in Eurocode is fufied, EN : For exampe, the reinforcement is paced in severa ayers the compressive strength can be increased with 10 %. Note that nodes with three-axia compression may have a compressive strength up to three times arger than for a CCC-node. CCC-nodes without anchored ties in the node where: (4.1) 16 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

31 CCT-nodes with anchored ties in one direction where: (4.) CTT-nodes with anchored ties in more than one direction where: (4.3) 4.7 Three-dimensiona strut-and-tie modes Structures subjected to oad that resut in a 3D stress variation are not adequate to mode in D. Exampes of structures with a 3D stress variation are pie caps, wind power pant foundations and deep beams. There are two dferent approaches for construction a 3D strut-and-tie mode, by mode in 3D or by combining D modes. A 3D strut-and-tie mode for a centric oaded pie cap is shown in Figure 4.6. Figure 4.6 Exampe of a 3D strut-and-tie mode and corresponding reinforcement arrangement for a pie pinth, Engström, B. (011). Figure 4.7 iustrates how two D strut-and-tie modes can be used, one in pane of the fanges and one in pane of the web. For such a mode each strut-and-tie mode transfers the oad in its own pane. The two modes are joined with common nodes. The resut is a combination of D modes which is appicabe on structures with a 3D behaviour. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 17

32 Figure 4.7 A combination of D strut-and-tie modes, Engström, B. (011) Nodes and there geometry A 3D strut-and-tie mode can resuts in nodes with muti-axia stress for which there are no accepted design rues or recommendations. This is not the case for ange imitations in 3D which often can be adopted from the D recommendations. A soution for designing 3D node regions is proposed in a master thesis Strut-and-tie modeing of reinforced concrete pie caps, Chanteot, G. and Aexandre, M. (010). The basic concept was to simuate 3D noda regions with rectanguar paraeepiped and struts with a hexagona cross-section shown in Figure 4.8. Figure 4.8 Geometry of the 3D noda zone above the pies, Chanteot, G. and Aexandre, M. (010) 18 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

33 5 Reference case and design assumptions This chapter contains a description of the reference case, used design codes and assumptions made in design. The fixed parameters in design such as oads and the geometry are presented aong with specications on concrete strength cass and minimum shear reinforcement prescribed by the turbine manufacturer is aso presented. The design of the foundation was performed with Eurocode and IEC These codes were used for dferent design aspects. The design was mainy performed with Eurocode, but the partia safety factors for the oads are cacuated according to IEC standard. 5.1 Design codes Eurocode is a reativey new common standard in the European Union and repaced in Sweden the od Swedish design code BKR in May 011. The standard is divided in 10 dferent main parts, EC0-EC9, each with nationa annexes. EC0 and EC1 describe genera design rues and rues for oads respectivey. The other eight codes are specic for various structura materias or structura types and EC Design of concrete structures together with EC0 and EC1 are reevant for this project. In order to ensure safe design Eurocode uses the so caed partia coefficient method. The partia coefficients increase the cacuated oad effect and decrease the cacuated resistance, in order to account for uncertainties in design. This is done to ensure that the probabiity of faiure is sufficienty ow, shown in Figure 5.1. Figure 5.1 Method of partia safety factors. S is the oad effect and R the resistance. The d index indicates the design vaue. IEC is an internationa standard for designing wind turbines; the standard is deveoped by the Internationa Eectrotechnica Commission, IEC (005). The IEC standard is based on the same principes as Eurocode concerning partia factors on both materias and oads. The oads given by the turbine manufacturer foow the IECstandard and the standard was therefore used for oad cacuations. The standard aows the designer to impement partia factors based on Eurocode. The partia safety factors for oads are in IEC cassied with regard to the type of design situation and the oad is favourabe or unfavourabe. Instead of cassying the oading in serviceabiity imit state and utimate imit state, IEC uses norma and abnorma oad situations. The used partia factors for oads are presented in Tabe 5.1. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 19

34 Tabe 5.1 Partia safety factors on oads according to IEC Genera conditions Abnorma (ULS) Norma (SLS) Fatigue unfavourabe favourabe - The considered wind power pant foundation ocated in Skåne in the south of Sweden. The soi consists of sand and grave. The project has been imited to ony study the foundation and the ground conditions are assumed good and are not further investigated. 5.3 Geometry and oading The foundation is square shaped with 15.5 m ong sides and a height that varies with a sope of approximatey 4.5 %. The tower is 68.5 m high and both the tower and turbine are suppied by the turbine manufacturer. The wind power pant is designed for a e time of 50 years. The foundation consists of concrete strength cass C45/55 and is designed for the exposure cass XC3. Figure 5. shows the section and pan of the foundation with fixed geometry from the reference case. After construction the foundation is to be covered with fiing materia, which in the design was incuded in a constant surface oad ( ). Figure 5. Section and pan of the foundation. 0 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

35 The sectiona forces at the connection between tower and foundation are specied by the turbine suppier with safety factors according to the standard IEC The foowing oads must be resisted; rotationa moment from wind forces and the unintended incination of the tower, a twisting moment from wind forces (which are excuded in this project), a transverse oad from wind forces and a norma force from sef-weight of the tower (incuding turbine and bades). Besides the oads acting on the anchor ring, described in Chapter the foundation, is subjected to sef-weight of reinforcement, concrete and potentia fiing materias. Figure 5.3 shows the definition of the oad from the tower and the characteristic vaues are presented in Tabe 5.. The design oads are cacuated in Appendix A. Figure 5.3 Tabe 5. Definition of sectiona forces from the tower at the connection between tower and foundation, adopted from ASCE/AWEA (011). Characteristic vaues of sectiona forces acting on top in the centre of the anchor ring and sef-weight of foundation. The oad effects are based on design oad case 6. extreme wind speed mode with a recurrence period of 50-years. Load type Size Remark + Incuding moment from misaignment of 8mm/m and dynamic ampication Excuded Incuding fiing materia CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 1

36 In serviceabiity imit state the characteristic crack width shoud be imited to 0.4 mm specied in the nationa annex of Eurocode. The crack width imitation given in Eurocode depends on exposure cass (XC3) and e time (50 years). Since the wind power pant is subjected to arge wind oads of variabe magnitude, the foundation s fatigue capacity is of great importance. The fatigue oad ampitudes are suppied by the turbine manufacturer, consisting of 80 unique oads (presented in Appendix I). The fatigue oad ampitudes are presented in a tabe with number of cyces. It is however uncear for how ong time the presented oad ampitudes are vaid. The mean vaues are aso presented aong with the used safety factor see Tabe 5.3. Tabe 5.3 Mean vaues of sectiona forces for fatigue design of reinforced concrete structures [kn] [kn] [kn] [kn] [kn] 5.4 Tower foundation connection The reference case is designed with an anchor ring of type (b) described in Section.3. This type of anchor ring has ony one fange in the bottom, which means that both the compressive and tensie force is appied at the same eve in the foundation. The anchor ring used in the reference case is shown in Figure 5.4. Figure 5.4 The anchor ring in the reference case during reinforcement instaation. In the cacuations the resuting moment ( ) was repaced by a force coupe consisting of a compressive and tensie resutant. In order to cacuate the eve arm between the force coupe a inear eastic stress distribution was assumed at the interface between concrete and the stee fange. Navier s formua was used to cacuate the maximum stresses in concrete subjected to compression by the fange of the anchor ring: CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

37 ( ) (5.1) ( ) The second moment of inertia (I) for an annuar ring with dimension of the bottom fange of the anchor ring is cacuated as: where: ( ) (5.) To very the assumption of inear eastic behaviour the cacuated stresses were compared with the stress-strain reationship for concrete shown in Figure 5.5. According to Figure 5.5 the stress strain reation is amost inear to about 50% of. The maximum stress was cacuated to approximatey 56% of and a inear eastic stress distribution in the compressed concrete coud be assumed. Figure 5.5 Stress-strain reation for concrete in compression according to EC As a simpication the inear stress distribution was assumed to correspond to a unorm stress distribution in two quarters of the anchor ring according to Figure 5.6. The eve arm was then cacuated as the distance between the arcs centres of gravities according to equation 5.3. ( ) =3.6m (5.3) where: =m CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 3

38 Figure 5.6 Resisting moment acting on the anchor ring with resuting force coupe and simpied stress distribution, =3.6m The sef-weight of the tower and turbine was assumed to be equay spread over the anchor ring and the resutant,, was divided in 4 equa parts. Two of the components coincide with the force coupe from the moment. The mode shown in Figure 5.7 was used in cacuations. Figure 5.7 Ideaised mode of the forces acting on the anchor ring, where is the diameter of the anchor ring (4m) and is the distance between the resuting force coupe from the rotationa moment (3.6m). As described in Section.1 anchor type (b) requires reinforcement in order to t up the compressive force and to pu down the tensie force. The compressive force is ted in order to utiise the fu height of the section. The two other types of anchor rings that are presented in Section.1 take the compressive force directy in the top of the sab, i.e. does not need to be ted by reinforcement to utiise the fu height of the section. The distance between the vertica bars of the suspension reinforcement or U- bow reinforcement was prescribed by the turbine manufacture to be minimum 500 mm. How the compressive and tensie forces from the anchor ring are assumed to be transferred is shown in Figure 5.8. Cacuations are found in Appendix B. 4 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

39 Figure 5.8 Force coupe from the rotationa moment acting in the bottom of the anchor ring. The compressive force ( ) is ted by the U-bow and the tensie force ( ) pued down by the U-bow. 5.5 Goba equiibrium As briefy described in Section.1 the foundation must prevent the tower from titing by a resisting moment created by an eccentric reaction force ( ). To ensure stabiity in arbitrary wind directions the stabiity was checked with two wind directions, perpendicuar and diagona (wind direction 45 degrees), see Figure 5.9. By fufiing equiibrium demands these two oad cases, stabiity for a intermediate oad directions were assumed to be satisfied. b 45 Figure 5.9 Left: Wind direction perpendicuar to foundation Right: Wind direction 45 degrees direction to foundation. In order to be abe to determine the soi pressure ( ) and its eccentricity ( ), the stress distribution of the soi pressure needed to be assumed. The exact distribution of the soi pressure is hard to determine, because of the compex oading situation, with concentrated oad at the centre of the foundation. As a simpication the soi pressure was assumed to be equay spread in the transverse direction (over the fu width of CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 5

40 the foundation). In the ongitudina direction two dferent assumptions are considered; unorm soi pressure and trianguar soi pressure, see Figure σ soi σ soi Figure 5.10 Dferent distributions of soi pressure within the ength (b). Left: Unorm soi pressure distribution. Right: Trianguar soi pressure distribution. The resutant of the soi pressure ( ) and its eccentricity ( ) can be determined from goba equiibrium with the foowing equations: (5.4) (5.5) With trianguar distribution the size of the soi pressure varies over the ength. The soi pressure is distributed over the ength b, which is determined by the eccentricity. The maximum soi pressures per unit width for a perpendicuar wind direction can be cacuated as: (5.6) (5.7) With a wind direction of 45 degrees and an assumed unormed stress distribution the soi pressure can be cacuated in a simiar manner as for the trianguar soi distribution in case of perpendicuar wind direction. The unormed soi pressures resutant is then trianguar because of the shape of the oaded area. (5.8) With known eccentricity and assumed soi distribution the bending moment and shear force distributions in the foundation sab can be cacuated. To identy the most critica wind direction the dferent bending moment and shear force distributions are compared in Figure 5.11 and Figure 5.1. These distributions was ony used for compression and the width of the sab is not considered. 6 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

41 Figure 5.11 Bending moment distributions for dferent oad cases. The index uni correspond to unorm soi distribution and index 45 is with a wind direction of 45 degrees. Figure 5.1 Shear force distributions for dferent oad cases. The index uni correspond to unorm soi distribution and index 45 is with a wind direction of 45 degrees. The concusions that can be drawn from the diagrams are that the dferences are sma and it was assumed sufficient to design the foundation for a perpendicuar wind direction. To simpy cacuations the argest need for bending and shear reinforcement is provided a the way to the edges of the foundation. By providing reinforcement to the edges, more than sufficient capacity is assumed in the corners, see Figure CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 7

42 Extend reinforcement to the corners Figure 5.13 To achieve sufficient capacity a reinforcement shoud be extended to the corners. Regarding the soi pressure distribution it is common to assume a unorm distribution when designing in the utimate imit state. In the serviceabiity imit state and for fatigue cacuations, a trianguar soi pressure distribution is more appropriate. The distributions with unorm soi pressure and trianguar soi pressure was compared, see Figure Figure 5.14 Shear and bending moment distribution for unorm and trianguar soi pressure distribution. The trianguar soi pressure distribution resuted in sighter higher bending moment and shear force. The dferences are however sma and in addition the rea soi pressure distribution is rather a combination of the two distributions, see Figure σ soi Figure 5.15 A combination of unormed and trianguar soi pressure distribution. Therefore the design in the utimate imit state was performed assuming unorm soi pressure distribution, whie the trianguar distribution was used in the serviceabiity imit state and for fatigue assessment. The fu cacuations are found in Appendix B. 8 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

43 6 Design of the reference case according to common practice on the basis of Eurocode In this chapter it is described how the reference case is designed according to Eurocode considering the conditions and assumptions presented in Chapter 5. Obtained resuts are presented. Detaied cacuations are shown in Appendix A-H. The sectiona design was performed in various sections which are presented in Figure 6.1. F c F t Figure 6.1 The sectiona design was performed in four dferent sections on each side of the foundation. The design was performed according to the foowing design steps: Design of top and bottom reinforcement in the utimate imit state using sectiona design (Appendix C). Design of shear reinforcement and the zone around the anchor ring in the utimate imit state (Appendix C) Design with regard to serviceabiity imit state (Appendix D) Design with regard to fatigue of reinforcement and compressed concrete (Appendix E using equivaent stress range and G using fu oad spectra) 6.1 Bending moment and shear force distribution The foundation was regarded simiar to a fat sab where the oad is transferred to the support using crossed reinforcement in two perpendicuar directions, see Figure 6.. y x Figure 6. Reinforcement in principa direction transfers the oad in two directions separatey. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 9

44 The design of bending reinforcement was based on the assumption that the bending moment in a section is unormy distributed over the fu width of the sab. The assumption requires a redistribution of sectiona forces since the inear eastic stress distribution has a stress variation in the transverse direction. The assumption is used when designing fat sabs according to the strip method. Hierborg suggests that the reinforcement shoud be concentrated over interior supports in fat sabs in order to achieve a better fexura behaviour in the serviceabiity imit state, shown in Figure 6.3. Figure 6.3 Bending moment capacity in a corner supported sab with reinforcement concentrated over the coumn In design practise it appears to be common to assume that the sectiona shear force is unormy distributed over the fu width of the sab, i.e. the same assumption as for bending moment. However, this assumption is not true near the reaction of the anchor ring. Figure 6.4 iustrates the oaded sab with two dferent sections, 1 and. Figure 6.4 Equiibrium conditions in a sab Section 1 is far away from the anchor ring and it is therefore reasonabe to assume that the sectiona shear force is unormy distributed over the fu width of the sab: 30 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

45 where: ( ) fu width of the sab (6.1) Aong section this assumption is not reasonabe because of the concentrated oad, i.e. the shear force varies in y-direction aong section : ( ) ( ( )) (6.) The equiibrium condition is staticay undetermined and it is hard to assume a distribution without determining the inear eastic stress distribution. It is doubtfu a redistribution of the sectiona shear forces is possibe in the same manner as for bending moment. Since the common practice is to assume that the interna forces are spread over the fu width the assumption was used despite the ack of a transition from a unormed distribution to a more concentrated near the anchor ring. The bending moment and shear force distribution are shown Figure 6.5 and Figure 6.6. As previousy stated a unorm soi distribution was assumed for design in the utimate imit state. Figure 6.5 Bending moment distribution used for sectiona design. The moment was assumed to be unormy distributed in the transverse direction and a unorm soi pressure is assumed. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 31

46 Figure 6.6 Shear force distribution used for sectiona design. The shear force was assumed to be unormy distributed in the transverse direction and a unorm soi pressure is assumed. 6. Bending moment capacity The reinforcement was designed according to Eurocode assuming an idea eastopastic materia mode of the stee. For concrete the stress-strain reation presented in Figure 5.5 was used. Since the height of the foundation varies both over the ength and across the foundation, the mean height over the width was used in each section, see Figure 6.7. Figure 6.7 Variation of mean height aong the ength. The variation is equa on both sides of the foundation. To simpy both cacuations and reinforcement arrangement required reinforcement amounts was cacuated ony in section 0 shown in Figure 6.1. Specia consideration of the top reinforcement near the anchor ring was required, since it is not possibe to continue the bars through the anchor ring. The effect of the incination of top reinforcement with approximatey 4.5 % was negected. The design of top reinforcement near the anchor ring was performed using so caed star reinforcement. Figure 6.8 shows the anchor ring and the ayout of the star reinforcement. 3 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

47 Figure 6.8 Left: Exampe of an anchor ring with hoes where star reinforcement is paced. ESB Internationa (010). Right: Principe arrangement of star reinforcement. The star reinforcement was paced within 56 hoes spread equay around the upper part of the anchor ring. The capacity of the star reinforcement was determined by cacuating an equivaent reinforcement area of the star reinforcement. The equivaent reinforcement area was then mutipied with the number of bars in the anchor ring. The product corresponds to the equivaent amount of reinforcement bars, which can be compared to the required amount of straight bars. If the equivaent star reinforcement is greater than the required amount of straight bars, sufficient capacity of star reinforcement was assumed. The foowing cacuations were performed: (6.3) where: ( ) (6.4) Area of top reinforcement bar Diameter of anchor of ring Spacing of top reinforcement Design moment in critica section Moment capacity in controed section (with bars ony in x- direction) Area of star reinforcement bar Ange of each bar, see Figure 6.9 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 33

48 bar i φ i Y Figure 6.9 X 6.3 Shear capacity The equivaent amount of reinforcement is cacuated as the equivaent number of bars in x-direction within a 90 degree circe sector. The region near the anchor ring must be designed with regard to concentrated anchor and compressive forces and with regard to punching shear. In other regions the design with regard to shear capacity was based on the assumption that the shear force was unormy distributed in the transverse direction The shear capacity without shear reinforcement was cacuated according to EN : equation 6..a: [ ( ) ] (6.5), d in mm (6.6) (6.7) (6.8) where: Characteristic concrete compression, in MPa Constant found in nationa annex Area of horizonta bars Width of section Effective depth According to the cacuations the capacity without shear reinforcement was sufficient except in the area cosest to the anchor ring. Even though no shear reinforcement was required in outer parts of the foundation, the turbine manufacturer specied a minimum shear reinforcement amount depending on the concrete cass. This is the 34 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

49 reason for the chosen minimum shear reinforcement of mm with spacing 500 mm. In the anaysis of the region near the anchor ring the maximum stress ( ) was cacuated according Section 5.4 with Navier s formua and the second moment of inertia for an annuar ring. A bar diameter of mm was used and the required spacing was cacuated according to the mode in Figure The maximum compressive stress ( ) was aso compared with the compressive strength of concrete. Figure 6.10 Mode for cacuating required spacing of the U-bows. Regarding punching shear it is not obvious how the capacity shoud be veried. The arge bending moment coud resut in a punching faiure where haf the anchor ring is punched down whie the other haf is punched up. Eurocode provides methods for verication of punching shear at coumns subjected to bending moment, but the actua situation dfers from the one described in Eurocode since the bending moment dominates. Instead of treating the oaded area as a coumn that is punched, a cone aong the perimeter of the anchor ring was assumed to be punched according to Figure Figure 6.11 A cone under the anchor ring was assumed to be punched out. Note that a simiar cone must aso punch through the upper part of the foundation sab for punching shear to occur. The critica sections were chosen according to EC and are shown in Figure 6.1. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 35

50 r r mean =m Figure 6.1 Left: Contro perimeter for punching shear according to EC. Left: The used mode. The described assumptions were used together with equation 6. for determining the punching shear capacity for concrete without shear reinforcement, see equation 6.. Instead of using the sectiona area the perimeter area in Figure 6.1 was used. The area of the contro perimeter section, d from the appied oad, marked A in Figure 6.1 was cacuated as: (6.9) For this specia type of punching shear the two contro perimeters sections have dferent radius and the tota area was cacuated using the mean radius. Observe that it is ony the perimeter of a haf circe according to Figure 6.11 that shoud be considered. ( ) (6.10) To have sufficient punching shear capacity the resut must be greater than the resutant of the compressive force( ). The edge areas of the cone shown in Figure 6.11 may contribute to the capacity. Since haf of the ring is punched up and haf is punched down, parts of the edge area wi coincide, see Figure Edge area Figure 6.13 One cone is punched up whie the other is punched down. 36 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

51 It is therefore uncertain how the contribution from the edges shoud be handed. If this edge area was incuded the capacity was sufficient and no extra shear reinforcement was needed. However without the contribution from this area the capacity was insufficient and extra reinforcement was needed. The minimum reinforcement, with spacing 500 mm, is enough to ensure that the cracks cross at east two reinforcement bars which is enough to provide sufficient capacity. 6.4 Crack width imitation The design with regard to permissibe crack width was performed in the serviceabiity imit state assuming a trianguar soi pressure. The crack width cacuations were performed by first determining the maximum stee stresses in state II, i.e. assuming that the tensie part of the concrete section is fuy cracked. The characteristic crack width was then cacuated according to EN : where: ( ) (6.11) Maximum crack spacing Concrete cover thickness (6.1) Coefficient considering the bond properties between concrete and reinforcement Coefficient considering the strain distribution Vaue from nationa annex Vaue from nationa annex Reinforcement bar diameter Strain dference between the mean vaues for stee and concrete Reinforcement ratio in effective concrete area The reinforcement amount needed for fexura resistance was not sufficient to fufi the crack width imitations. As expected a arger reinforcement amount was needed both in the top and bottom. The most critica part of the foundation with regard to crack widths was the bottom side of the sab cose to the anchor ring where the argest bending moment was ocated. In addition to the need of bending reinforcement the foundation needed reinforcement near the edges to imit the crack widths. 6.5 Fatigue When designing a wind power pant foundation the fatigue anaysis cannot be omitted. In this project the fatigue anaysis have been performed separatey for concrete and reinforcement. The fatigue e was veried for bending reinforcement, U-bows and the compressed concrete under the fange of the anchor ring. The need for shear reinforcement was sma, except for the region near the anchor ring. Fatigue verication is therefore ony performed on the U-bows with a oca anaysis. The CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 37

52 shear capacity outside the oca area around the anchor ring was assumed to be sufficient. The fatigue anaysis for stee was performed with two approaches, Pamgren-Miner cumuative damage aw and the use of an equivaent oad. Both mentioned approaches exist in Eurocode, but no description for estabishing the equivaent oad exists. However, the fatigue e of concrete can ony be veried with an equivaent oad since there are no S-N curves for concrete, which are necessary in order to use Pamgren-Miner cumuative damage aw. In order to cacuate an equivaent oad a method described in Fatigue equivaent oad cyce method by H.B Hendriks and B.H. Buder was used, Hendriks and Buder (007). They purpose a method to cacuate one equivaent oad ampitude ( ) which is based on the fu oad spectra. This equivaent oad can be used to cacuate equivaent stress variations which then can be used to very the capacity according to Eurocode. With an equivaent stress range both fatigue verication of reinforcement and compressed concrete are possibe. Equation 6.13 shows the equation used for determine, and the equations used for verication is shown in equation ( ) (6.13) Equivaent range of oad cyce Equivaent number of aowed cyces Exponent that defines the sope of the S-N curve Range of oad cyces Number of cyces The method is deveoped to compare dferent fatigue oad spectrum on a quantitative basis, Hendriks and Buder (007). From our understanding the equivaent fatigue oad in Equation 6.1 is not intended for fatigue cacuation of reinforcement, but instead for other components of the wind power pant such as the rotor bades, Stiesda, H (199). Equation 6.13 can ony be used with the sope of one S-N curve. In Eurocode two dferent sopes are presented depending on the oad magnitude. The two dferent sopes presented in Eurocode for reinforcement are and ( in Eurocode EN ). The vaue was assumed to be the mean vaue of the given sopes, i.e.. The equivaent stress range was cacuated for oad cyces, which was used together with the mean vaue given by turbine manufacturer to cacuate a minimum and a maximum of fatigue oads. The compete cacuation together with the oad spectra can be found in Appendix I. The variation of moment oad was cacuated as: where: = kNm (6.14) 38 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

53 Cacuation of was performed in the same manner. The determined maximum and minimum oads are used to cacuate dferent eccentricities for the dferent oads as described in Section 5.5 but with a trianguar distribution of the soi pressure. The smaer oads resuts in smaer eccentricities, hence the soi pressure is distributed over the fu ength, shown in Figure σ 1 σ Figure 6.14 Soi pressure distribution used in fatigue cacuations. The size of and can be determined by estabishing the expression for the distance to the gravity centre and horizonta equiibrium. The equivaent moment and shear force distribution is presented in Figure Figure 6.15 Variation in bending moment and shear force for the two used equivaent fatigue oads. The stress-ampitudes for reinforcement and concrete can be determined from the moment and shear force distribution. The stress-ampitudes in reinforcement can be used in Equation 6.14 (EN : ) to determine the fatigue damage for the reinforcement. where: ( ) ( ) ( ) Stress range of oad cyces ( ) Damage equivaent stress range for cyces Partia safety factor for fatigue oading Partia safety factor for materia uncertainties (6.14) CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 39

54 The Pamgren-Miner cumuative damage aw approach was used with the fu oad spectrum suppied from the turbine manufacturer to cacuate accumuated damage with both sopes of the S-N curves for reinforcement. To do this the compete oad spectra are exported to Mathcad, where the bending moment and shear force distribution for each dferent oad is cacuated in order to determine the stress variations for each unique oad. The size of the oad is then checked to see which sope of the S-N curve that shoud be used. The two dferent sops given in Eurocode are presented beow. The tota damage For For ( ) ( ) can then be cacuated as: (6.15) Where ( ) is the tota number of cyces unti faiure for the stress range ( ) cacuated as: ( ) ( ) (6.16) For the fatigue verication of the compressed concrete, two approaches exist in Eurocode. The used method is based on the equivaent oad, where a reference number of oad cyces,, is used instead of the fu oad spectra. There is an aternative method of cacuating equivaent oad described in the bridge part of Eurocode EN :005 that takes account for the frequency of the oad. However, there was no time to evauate this method within the imited time for this project. The used equations for fatigue verication of concrete are, EN : : (6.17) (6.18) (6.19) (6.0) where: Stress ratio Lowest compressive eve Highest compressive eve Concretes design strength Lowest compressive at stress change for Highest compressive at stress change for cyces cyces 40 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

55 6.6 Resuts In the static design of the reference case both the bottom and top reinforcement amounts cacuated in utimate imit state had to be increased in order to fufi the crack width imitations. Shear reinforcement was ony required to avoid punching shear faiure. The provided U-bows and minimum shear reinforcement prescribed by the turbine manufacturer was however sufficient to avoid punching shear faiure and no extra reinforcement was needed. The highest degrees of utiisation are presented in Tabe 6.1. Wind power pants are subjected to a arge number of oad cyces and the fatigue anaysis becomes of great importance. Two dferent fatigue verication methods were performed; Fatigue equivaent oad cyce method and Pamgren-Miner cumuative damage aw. The Pamgren-Miner cumuative damage aw can ony be used together with fu oad spectra and requires appicabe S-N curves. Hence, this method cannot be used to check compressed concrete, since no S-N curves for concrete exist. Further, the Fatigue equivaent oad cyce method is more straightforward and requires ess cacuations. Though it is uncear this method is suitabe for fatigue anaysis of reinforced concrete structures. Both fatigue cacuation methods resuted in ess damage than expected, in a checked regions and components apart from the U-bows. However, there are uncertainties regarding which time period the oad spectra provided by the turbine manufacturer represent which make the resuts hard to evauate. The fatigue cacuations performed with the equivaent oad gave higher damage than the damage summation method in a checks, except for the anaysis of the U-bows. In anaysis of the U-bows the equivaent oad method gave a damage of 80 % and the Pamgren-Miners damage summation aw resuted in fatigue faiure ( ). Since the cacuation was performed ony on the outermost U-bow, which is subjected to the argest stress variations, the resuts were accepted even the damage was above 1. Since the U-bows are eveny distributed around the perimeter of the anchor ring and stress redistribution is possibe in case of faiure. The dference in resut between the two cacuation methods indicates that the Fatigue equivaent oad cyce method may be improper for reinforced concrete structures. At east the method must be investigated regarding which assumptions the method is based on. The concrete fatigue e was ony cacuated with the equivaent oad, the fu oad spectra coud not be used since S-N curves for concrete do not exist. The cacuated fatigue damage for concrete was ow. The reason for this coud be the high required concrete strength cass C45/55 specied by the turbine manufacturer. Tabe 6.1 presents some utiisation ratios from the design. A resuts are presented in Appendix H. The utiisation ratios are cacuated by dividing required capacity divided by provided capacity. Tabe 6.1 present utiisation ratios from the design, a resuts are presented in Appendix H. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 41

56 Tabe 6.1 Highest utiisation ratios Part ULS Fatigue Remark Bending reinforcement bottom Bending reinforcement top Star reinforcement U-bow reinforcement Concrete compression Shear reinforcement Section 0, Equivaent oad Section 0, Equivaent oad cacuated with required area compared to the used Loca anaysis, Pamgren-Miner Loca anaysis under anchor ring, Equivaent oad - Section 0 Crack width - Section 0, at the bottom Tabe 6.1 ceary shows that the critica design aspects of the reference foundation were the crack width imitation and the U-bows subjected to fatigue oading. The utiisation ratio for shear reinforcement was cacuated with shear reinforcement spacing 500 mm, which was specied by the turbine manufacturer. Shear reinforcement was however ony needed with regard to punching shear faiure. Besides the resut for the star reinforcement, the utimate imit state utiisation ratio and the fatigue e is rather simiar. The ow utiisation ratios in the utimate imit for bending reinforcement are an effect of the crack imitations in the serviceabiity imit state, may expain the rather sma fatigue damage. The resut for star reinforcement was cacuated dferenty and coud not be compared with the other resuts for bending reinforcement. The U-bow reinforcement is not designed with regard to crack width imitations, which expains the arge utiisation, both in the utimate imit state and in case of fatigue. 6.7 Concusions on common design practice Design according to common practice is based on the idea of distributing the sectiona forces unormy across the fu width of the foundation and using sectiona design. However, this assumption is unreasonabe near the anchor ring because of the concentrated reaction from the anchor ring. By concentrate the reinforcement to the 4 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

57 centre of the sab the effects of stress variation in transverse direction is accounted for. The bending capacity can be regarded as sufficient as ong as the tota bending reinforcement is enough and pastic redistribution is possibe. Regarding the shear design it is necessary to construct a truss mode in order to ensure sufficient shear resistant. Therefore a 3D truss mode is recommended in order to consider the 3D behaviour of the sab. In common design practice the stress variation in transverse direction is disregarded and the design procedure is incompete. If the inear eastic stress fied is known, regions where 3D-aspects need to be considered can be identied. Hence, regions where beam-theory is vaid can be recognised and designed with sectiona design. Sectiona design is straight forward and it is easy to determine how sectiona forces change depending on oad magnitude. This makes fatigue cacuations based on the fu oad spectra and Pamgren-Miners damage summation aw rather simpe. The 3D aspects must aso be considered in the fatigue assessment. Because of the reative sma fatigue oads it is unreasonabe to assume that the interna forces wi redistribute. Therefore it is recommended to assume that both the shear force and bending moment are concentrated to the centre of the foundation. There are uncertainties regarding which time period the oad spectra used for fatigue assessment represent, which make the resuts from these cacuations hard to evauate. It is aso uncertain an equivaent oad is reasonabe for design of reinforced concrete structures. The resuts from the cacuations with Pamgren-Miners damage summation aw dfer from the one performed with an equivaent oad. The equivaent oad was used, because the fatigue verication of concrete in Eurocode requires one equivaent stress range. Because of the arge bending moment in the anchor ring the verication of capacity against punching shear faiure is conducted with a modied version of the one proposed in Eurocode. The used method for verication of capacity against punching shear faiure must be studied further before it can be accepted in design. The square shape of the foundation is we suited for a reinforcement ayout with bars paced ony perpendicuar and parae with the edges. In case of circuar foundations a design where the reinforcement is paced radia may be more suitabe. With a circuar foundation the ength of radiay paced bars can be constant, whie they need to be shortened in a square foundation. With the same reasoning a circuar foundation is ess suited for reinforcement with crossed bars, se Figure Figure 6.16 Dferent reinforcement ayouts in square and circuar foundations Unike crossed bars the use of bars paced radiay resuts in probem with the spacing in the centre of the foundation. If the bars are paced radiay the need of reinforcement is reduced due to the fact that the oads do not need to be transferred in two directions separatey. In Figure 6.17 this is exempied with a corner supported sab. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 43

58 Figure 6.17 Left: Reinforcement in x- and y-direction. Left: Radiay paced reinforcement. With radiay paced reinforcement the need for reinforcement decreases. 44 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

59 7 Design of reference case with 3D strut-and-tie modes and Eurocode With regard to the boundary conditions and the concentrated centric oad a 3D-mode was used to capture the behaviour of the foundation. This chapter describes the design methodoogy that was used to estabish 3D strut-and-tie modes for the reference object described in Chapter Methodoogy The previousy described methodoogy in Chapter 4 to describe the stress fow in D- regions known as the oad-path method can be used in 3D. There is however a very compex oading situation and without great experience or advanced computer anaysis a reasonabe stress fied is hard to assume. The chosen procedure was to simpy the oading and start to construct a suitabe D mode that then was deveoped into a 3D mode. The sef-weight and soi pressure needed to be divided in an adequate amount of nodes to avoid an oversimpistic mode. With a chosen division of oads the modes were estabished based on the oad path method. The modes were constructed in the commercia software Strusoft FEM-design 9.0 3D frame. Strut-and-tie modes are ony based on equiibrium conditions, i.e. no deformations shoud be assumed in the struts or ties. Therefore the eements were represented by truss members with properties chosen to according to Figure 7.1. Figure 7.1 Used eements in anaysis. The first modes in FEM-design were constructed with fictitious bars, but because of probems with setting the fexura rigidy to zero ordinary truss members were used instead. These eements can ony transfer norma forces and a connections are hinged. In order to avoid infuence from deformations or bucking the oads were scaed to 1/100 and arge stee sections of high strength were used. To very the resuts from FEM-design the freeware Fachwerk was used, deveoped by Vontobe, A (010). Fachwerk is designed for anaysing strut-and-tie modes and uses ony equiibrium conditions, i.e. does not consider any materia behaviour. 7. Two-dimensiona strut-and-tie mode To simpy the oading situation the sef-weight was represented by two resutants acting on top of the structure. The soi pressure was modeed as unormy distributed and represented by one resutant in the strut-and-tie mode. The position of the U- bows is fixed and the distance between vertica bars is 500 mm. The first mode was estabished with ony the criterion of equiibrium and did not consider ange imitations or node stresses. In order to keep baance so caed u-turns were needed above the resutants and in order to take care of the bending moment. Ony vertica and horizonta ties were accepted with regard to practica reinforcement arrangement. The deveoped D mode aong with used oading conditions is shown CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 45

60 in Figure 7.. This D mode was used as the base for deveopment to 3D modes. u-turns Figure 7. Estabished D strut-and-tie mode for the wind power pant foundation. 7.3 Three-dimensiona strut-and-tie modes A wind power pant foundation is subjected to many dferent oad ampitudes and a unique strut-and-tie mode coud be estabished for each oad case both in D and 3D. The 3D strut-and-tie modes were estabished for the utimate imit state. The dference in the serviceabiity imit state is the ocation of since the soi pressure and eccentricity varies with the oad magnitude. When deveoping the D mode to 3D, the reactions acting on the foundation must be represented by an adequate amount of nodes over the width of the foundation. The soi pressure was assumed to be eveny distributed over the width of the foundation. Choices made regarding the distribution of nodes were the foowing: The sef-weight incuding the fiing materia was divided into six parts of the same size The soi pressure was divided into three equa parts over the width How the oads were divided is shown in Figure 7.3. In the strut-and-tie modes a node were paced in the centre of each oaded area. 46 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

61 Y X Figure 7.3 Load dividing ines for the nodes. With the chosen oad distribution two dferent oad paths were used, one with oad transferred in one pane at a time (mode 1) and another with oad transfer radia (mode ). Mode 1 was based on the idea to ony use reinforcement parae or perpendicuar to the edges, i.e. in x- and y-directions. Mode transfers the oad in diagona paths to and from the anchor ring. The dferent modes are iustrated in Figure 7.4. Y X Figure 7.4 The dferent oad path modes. Left: mode 1 oad paths in x- and y- directions. Right: mode with diagona oad paths. Dotted ine: division of, dot-dashed ine division of. As stated earier the D strut-and-tie mode was used as a base for the 3D mode. The diagona egs and the parae egs are simiar to the D mode. These egs were connected with a strut-and-tie mode for the anchor ring. Figure 7.5 shows the principe ideas for the estabishment of the strut-and-tie modes and the so caed egs. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 47

62 Parae egs Diagona egs Y X Figure 7.5 Principe of the 3D strut-and-tie modes. Left: The oads are transferred in x- and y-direction separatey. Right: Load transferred diagonay. The parae and diagona egs are marked red. In order to achieve equiibrium the nodes representing the soi pressure must be connected with the reaction force of the anchor ring. In the mode that transfer oads in x- and y- directions the egs is connected with the anchor ring in the midde and on the edges to utiise the fu width of the anchor ring. In the diagona mode the position of egs were chosen to go between the positions of the nodes representing the sef-weight. In the D mode the bending moment was represented by a force coupe. The same method was used in the 3D mode, but instead 3 force coupes represented the bending moment. To determine the magnitude of each force a simiar approach was used as in the design based on common practice, i.e. assume that pane sections remain pane in the interface between the anchor ring and the concrete. In this case, six components must be determined and their resutants must act in the node position corresponding to the connection between the egs and the anchor ring. The cacuation of the forces was carried out with a FEM-anaysis. The FEM mode consisted of a thick anchor ring to avoid deformations in the anchor ring. It was supported with point supports paced at the chosen node positions and oaded with the bending moment. The mode is shown in Figure 7.6, where the stress resutants of the supports were paced at the corresponding nodes in the strut-and-tie modes. The argest resutants were ocated in the most eccentric part of the anchor ring. The magnitudes of the dferent forces are presented in Tabe 7.1 and their ocation in Figure CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

63 F c F c F c F t Figure 7.6 Used mode to determine and in the 3D strut-and-tie mode. Tabe 7.1 F t F t Utimate oads cacuated with the FEM-anaysis incuding With chosen oad distribution on the foundation, positions and size of the forces corresponding to the rotationa moment two strut and tie modes were estabished. These 3D strut-and-tie modes are presented in Figure 7.7 and Figure 7.8. Figure 7.7 shows mode 1 that was estabished from the concept of using ties in x- and y- direction for simpied reinforcement ayout. Figure 7.7 Mode 1, were the detaiing for the centre of the strut-and-tie mode is shown separatey. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 49

64 Figure 7.8 shows mode estabished with the idea of transferring the oad radia. Figure 7.8 Mode, were the detaiing for the centre of the strut-and-tie mode is shown separatey. Note that the modes have dferent centre, the reason for this was to achieve equiibrium by ony using straight bars for Mode 1. When the 3D strut-and-tie modes are estabished, the anges and node capacities shoud be checked. The ange recommendations used in D can be adapted to 3D, by checking the ange in each pane separatey. The foundation must be abe to resist arbitrary wind directions, but the strut-and-tie modes can ony be estabished for one oad case at a time. As described in Section 6.1, performing the design of the foundation for a parae wind directions is regarded as sufficient since the reinforcement is crossed. If mode 1 is rotated to restrain a perpendicuar wind directions the mode is assumed to resist a wind directions. Mode 1 becomes doube symmetric when rotated, which is not the case for mode. In Figure 7.9 both modes are rotated. For mode it is not sufficient to ony check parae wind directions, since the oad is not transferred in two directions the diagona wind direction can resut in arger need for reinforcement and therefore it must be veried separatey. Figure 7.9 Rotated strut-and-tie modes: Left: mode 1 Right: Mode. 50 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

65 Since the foundation of the reference case is square Mode 1 is preferabe due to the probematic connection in the centre of the foundation. 7.4 Reinforcement and node design Designing the reference foundation with radia paced reinforcement was regarded as inappropriate because of the square shape. Therefore the reinforcement cacuations were ony performed for Mode 1. Mode 1 was divided into dferent sections, which were designed separatey. The definition of sections is shown in Figure 7.10 and the corresponding forces and sections can be found in Appendix J. 3.6 m 3.6 m Figure 7.10 Definitions of sections for Mode 1, each section is presented in Appendix J. The design of shear, top and bottom reinforcement in each section was performed according to the foowing steps: 1. Determine the argest tensie force for vertica, top and bottom tie separatey.. Cacuate the amount of shear, top and bottom reinforcement required for the corresponding ties. 3. Spread the needed reinforcement over the width of the approximated tensie stress fied, which the corresponding tie represents. For exampe Section 1-1 s argest tensie force in the bottom ayer is spread over a width of 3.6 m, see Figure This resuted in a spacing of 00 mm of bars. 3.6 m is the distance between Sections 1-1 and -, which is the width where the corresponding tensie stress fied of the tie is assumed to occur. The reinforcement needed to transfer the soi pressure and sef-weight are spread over the same widths as CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 51

66 used for the oad paths, shown in Figure 7.4. To resist oad from arbitrary wind directions the reinforcement cacuated must aso be provided in the transverse direction. The suspension reinforcement was designed under the same assumptions and with the same design procedure as described in Section 6.3. Accordingy U-bows with a bar diameter of mm and a spacing of 100 mm were chosen. An exampe of detaiing around the anchor ring is iustrated in Figure Figure 7.11 Exampe of U-bows that are paced very dense around the anchor ring. In 3D compex node geometries can arise which cannot be designed by directy adapting the design rues from D design. There are no accepted design rues for how to design these nodes. However, a soution for designing compex 3D node regions is purposed by Chanteot, G. and Aexandre, M. (010) and is briefy described in Section The wind power pant foundation is subjected to distributed forces from the soi pressure and sef-weight. The sectiona forces are distributed over the circumference of the anchor ring fange at the interface to concrete. The sectiona forces at the anchor ring interface connection are distributed over the circumference of the anchor ring fange. Hence, the corresponding nodes are distributed and do not need to be checked. When confirming the strut-and-tie mode it is not enough to very the concentrated nodes. The compressive force in the struts does aso need to be imited. This can be done by cacuating the concrete area required to take the compressive forces in the struts and compare it to the avaiabe. The struts are assumed to be spread over the same width as the corresponding tie. To very the capacity of the struts the required concrete area for each strut is cacuated in Appendix K. There are however, struts that are critica within the anchor ring but the estabished mode of the detaiing around the anchor ring needs to be refined. This can be achieved by subdividing the force coupe in more than six nodes. In addition the design shoud be improved with minimum reinforcement. 5 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

67 The tensie forces in the vertica ties in the strut-and-tie mode outside the anchor ring are assumed to be spread over the same ength as the bottom and top reinforcement. This gave the required spacing of the shear reinforcement which were arger than the required, bars spaced 500 mm from the turbine manufacturer. But since the design ony have been performed for the utimate imit state the design must be suppemented with service abiity cacuations. 7.5 Fatigue No fatigue verication has been performed on the strut-and-tie modes, since every dferent oad case woud resut in a unique strut-and-tie mode. Without an automatic routine it is unreasonabe to estabish a 3D strut-and-tie mode for every oad case. Two strut-and-tie modes coud be estabished for the two equivaent oads to find the stress ampitude in these cases, but with regard to uncertainty of the accuracy of these oads this has not been performed. With either an automatic routine or a reduced number of oad cases the strut-and-tie method is we suited for fatigue cacuations, since the 3D behaviour of the foundation is taken into consideration. However, the strut-and-tie mode is used for fatigue cacuations the mode must be cose to the inear eastic stress fied, i.e. have a sma need for pastic redistribution. Further, the reinforcement ayout cannot change between the modes, i.e. one reinforcement soution must fit a oad cases and corresponding modes. The master thesis Fatigue Assessment of Concrete Foundations for Wind Power Pants Göransson, F. Nordenmark, A. (011) describes how fatigue verication of D strut-and-tie modes can be performed. Instead of using one equivaent oad as in our project a reduced oad spectrum was used, which was provided by the turbine manufacturer. To simuate the stress fied four unique D strut-and-tie modes were estabished in the fatigue anaysis. The strut-and-tie modes were dferent, but a modes had the same reinforcement ayout. 7.6 Resuts The design of the foundation with a 3D strut-and-tie mode resuted in a reinforcement ayout shown in Figure 7.1 and Figure CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 53

68 Figure 7.1 Bottom reinforcement ayout, a measurements are in mm. Figure 7.13 Top reinforcement ayout, a measurements are in mm. The reinforcement ayout shows that the horizonta reinforcement is paced denser in the centre of the foundation. The need for bottom reinforcement is considerabe arger than the need for top reinforcement. The shear reinforcement is paced with a spacing of 500 mm, Figure 7.14 iustrates the type of shear reinforcement that was used. 54 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

69 Figure 7.14 Shear reinforcement 7.7 Concusions on the 3D strut-and-tie method By designing the wind power pant foundation based on a 3D strut-and-tie mode the 3D stress distribution is taken into consideration. By conducting a inear eastic FEM-anaysis of the foundation the inear eastic stress fied coud be cacuated and a more refined mode can be estabished. A more refined strut-and-tie mode better simuates the eastic stress fied and reduce the need of pastic redistribution. A reduced need for pastic redistribution wi improve the behaviour of the foundation in the serviceabiity imit state. The estabished mode resuts in two dferent reinforcement ayouts: one with radiay paced reinforcement bars and one with reinforcement bars ony in parae and perpendicuar directions to the edges. Due to the square shaped foundation reinforcement paced ony in parae and perpendicuar directions to the edges was preferabe. Without an automatic routine for estabishment of strut-and-tie modes or a reduced oad spectra it is very time consuming to perform fatigue cacuations on a strut-andtie mode. The reason for this is that a unique mode must be estabished for each fatigue oad case. Except for these requirements the strut-and-tie mode is we suited for fatigue cacuations since the stress variations is easy to evauate. It shoud be kept in mind that the strut-and-tie mode is designed for the utimate imit state and the fatigue oads are we beow the utimate oads. It is therefore of great importance that the mode is based on a stress fied cose to the inear eastic. If the strut-and-tie mode is based on a stress fied far away from the eastic, the mode wi not simuate the stress fied for the reative sma fatigue oads and pastic redistributions are sma or non-existing. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 55

70 8 Concusions and recommendations The centricay oaded foundation resuts in D-regions and 3D stress fow which make the use of a 3D strut-and-tie mode an appropriate design method. The 3D strut-andtie mode propery simuates the 3D stress fow of reinforced concrete and is appropriate for design of both B-and D-regions. The design according to common practice does not capture the 3D behaviour and is therefore unsatisfactory. Shear design with a sectiona mode is not possibe, i.e. a truss mode is required. And in order to capture the 3D behaviour a 3D truss mode is necessary. By conducting a inear eastic FEM-anaysis of the foundation the inear eastic stress fied can be cacuated and D-regions can be distinguished. With this known a more refined strut-and-tie mode can be estabished that foow the inear eastic stress fow more accuratey. It aso possibe to distinguishes where sectiona design can be used, i.e. where the stress variation in transverse direction do not need to be considered. We found it rather compex to estabish the 3D strut-and-tie modes, it was particuary hard to mode the region around the anchor ring in an appropriate way. This might be due to ack of experience of modeing in 3D. Suitabe software might simpy the estabishment of 3D strut-and-tie modes. Another dficuty with strutand-tie modeing for the design of the wind power pant foundation is the fatigue verication. Fatigue verication with the fu oad spectra are not reasonabe to perform with strut-and-tie modes without an automated routine since a unique mode must be estabished for each fatigue oad. Without an automated routine the use of an equivaent oad becomes necessary. The uncertainties regarding the equivaent oad resuts in a need for a separatey research before it can be accepted in design. 8.1 Reinforcement ayout and foundation shape A square foundation seems more suitabe for the use of reinforcement in the two main directions than radiay paced reinforcement. It is an easier ayout that avoids probematic connection in the centre of the foundation and the need of reinforcement bars in many dferent engths. One disadvantage is that it requires more reinforcement since the oad must be transferred in two directions separatey. If a circuar foundation instead is used, radiay arrangement of the reinforcement bars appears to be more appropriate. The reinforcement ayout from the design according to common practice was suggested to be concentrated towards the centre of the foundation for both top and bottom reinforcement. This choice is motivated by the simiarities with a fat sab, where the soution is used to improve the behaviour in service state. The resuts from the strut-and-tie mode aso impy that this is a good reinforcement arrangement, with regard to the concentration of interna forces near the anchor ring. 8. Suggestions on further research In this thesis ony one type of connection between the tower and foundation has been studied. It woud be interesting to study aternative connection types and how they infuence the design. Aso how to perform reevant verication of punching shear faiure of the anchor ring need to be further studied. Further a design for serviceabiity imit state is desirabe. 56 CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

71 The uncertainties regarding how to hande the fatigue oads, i.e. an equivaent oad can be used for design of reinforced concrete needs to be caried. If the use of an equivaent oad coud be veried, this woud make the fatigue cacuations consideraby simper. Further the interaction between the soi and the foundation infuence the design and studies about the actua soi pressure distribution is of great interest. CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 57

72 9 References ASCE/AWEA (011): Recommended Practice for Compiance of Large Onshore Wind Turbine Support Structures (draft), American society of civi engineering and American wind energy association, USA. Martin, B. and Sanders, D. (007): Verication and Impementation of Strut-and-Tie Mode in LRFD Bridge Design Specications, 1-14pp Boverket (004): Boverkets handbok om betongkonstruktioner BBK 04 (Boverket s handbook on Concrete Structures BBK 04, Vo. 3 Design. In Swedish), Boverket, Byggavdeningen, Karskrona, Sweden, 181 pp. Chanteot, G. and Aexandre, M. (010): Strut-and-tie modeing of reinforced concrete pie caps. Master s Thesis. Department of Structura Engineering, Chamers University of Technoogy, Pubication no. 010:51, Göteborg, Sweden, pp. Engström, B. (011): Design and anaysis of deep beams, pates and other discontinuity regions. Department of Structura Engineering, Chamers University of Technoogy, Göteborg, Sweden, 011. ESB Internationa (010): Wind Turbine Foundations Risk Mitigation of Foundation Probems in the Industry (Eectronic), ESB Internationa. Accessibe at< undations010.pdf?uid= > ( ). Faber, T. Steck., M. (005): Windenenergieaagen zu. Wasser und zu Lande Entwickung und Bautechnik der Windenergie (Construction deveopment of offshore and onshore wind energy. In German), Germanischer Loyd WindEnergie GmbH, Hamburg, 005 fib (1999): Buetin 3 Structura Concrete, fib (fédération internationa du béton) Vo. 3, Stuttgart, Germany, 1999, 141 pp. Göransson, F. Nordenmark, A. (011): Fatigue Assessment of Concrete Foundations for Wind Power Pants. Master s Thesis. Department of Structura Engineering, Chamers University of Technoogy, Pubication no. 011:119, Göteborg, Sweden. Hendriks, H.B., Buder, B.H. (1995): Fatigue Equivaent Load Cyce. ECN, ECN-C , pp. 3. IEC (005): IEC (Internationa Eectrotechnica Commission). third edition, Geneva, Switzerand, 005. Muttoni, A., Schwartz, J., and Thürimann, B. (1997): Design of Concrete Structures with Stress Fieds. Birkhäuser Verag, Base, Boston and Berin, Switzerand, 1997, 145 pp. Rogowsky, M. MacGregor, J. (1983): Shear strength of deep reinforced concrete continuous beams. Structura engineering report no 110 Department of civi engineering, University of Aberta. Edmonton, Aberta Candada. Russo, G., Venir, R., Pauetta, M. (005): ACI Structura journa, Vo. 10, No. 3, May-June 005, pp Sanad, A., Saka, M. (001): Journa of structura engineering. Vo. 17, No. 7, Juy 001, pp CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49

73 Schaich, J., Schäfer, K., and Jennewein, M. (1987): Toward a Consistent Design of Structura Concrete. Journa of Prestressed Concrete Institute, V. 3, No. 3, May- June 1987, pp Schäfer, K. (1999): Nodes. Section in Structura Concrete, Vo., fédération internationae du béton (fib), Buetin, Lausanne, Suisse, pp SMAG (011): Wind Turbines Are Safe. (Eectronic), Saddeworth Moors Action Group Accessibe at: < ( ) Stephens R, et a. (1980): Meta fatigue in engineering. John Wiey & sons, inc, New York. Stiesda, H. (199): Journa of Wind Engineering & Industria Aerodynamics. Vo. 39, No. 1, pp Svensk Byggtjänst (1994): Betonghandbok Materia (Concrete Handbook Materia. In Swedish), Svensk Byggtjänst, Stockhom. Vattenfa (011): Vind i framtiden (Wind in the future. In Swedish) (Eectronic), Vattenfa. Accessibe at: < ( ) Vontobe, A. (010): Fachwerk (Eectronic), Accessibe at: < ( ) CHALMERS, Civi and Environmenta Engineering, Master s Thesis 01:49 59

74 A In data reference case A.1 Geometry h mm Section h 1700mm h 3 00mm h 4 900mm pane 15500mm 45 d sr 4m Length of foundation Diagona of foundation Outer diameter of stee ring c 50mm c soi 100mm Concrete cover tempate Concrete cover to soi (beow) x 00.01m 15.5m Variation of foundation section height 60

75 h h 1 hx ( ) h 1 3.3m x 00.01m h h 1 h 3.3m h otherwise 15.5 m x x x 3.3m 3.3m 3.3m x Variation of foundation mean height (height varies in two directions): h m ( x) hx ( ) x x ( hx ( ) 1.5m) x 3.3m 1.7m6.6m ( 6.6m) 1.7m 1.5m otherwise 1.65 Variation of mean height 1.6 [m] h m ( x) x [m] 61

76 A. Loads Coordinate system Characteristic oads Loads from tower F z 11kN M z M xy 5863kNm 51115kNm F xy 800kN Loads of foundation Incuded moment from misaignment Dead weight of concrete foundation incuding fiing materia and reinforcement G kN g kn m 6

77 Partia factors for oads according to IEC :005 edition 3 According to tabe p. 35 IEC :005 DLC 6. "Extreme wind speed mode 50-year recurrence period" Utimate anaysis, Abnorma IEC use another standard where: Abnorma corresponds to ULS Norma corresponds to SLS Live oads: ULS Dead oads: γ Q 1.1 Unfavourabe γ G 1.0 γ Qf 0.9 Favourabe γ Gf 0.9 Unfavorabe Favorabe SLS γ Qss 1.0 γ Gss 1.0 Fatigue γ f Partia factors for consequences of faiure to IEC :005 edition 3 Component cass 1: γ n 1.0 Design oads ULS M xyd F xyd F zd M zd G d g d SLS γ Q M xy γ Q F xy γ Gf F z γ Q M z γ Gf G γ Gf g M xysls F xysls F zsls M zsls 56.7MNm 0.88MN 1.909MN 6.449MNm MN kn m γ Qss M xy γ Qss F xy γ Gss F z γ Qss M z MNm 0.8MN.11MN 5.863MNm 63

78 G dsls g dsls γ Qss G γ Qss g 1.575MN kn m A.3 Materia Properties Materia properties and partia factors according to Eurocode Partia safety factors [EN : tabe.1n] γ mc 1.5 Materia partia factor for concrete γ ms 1.15 Materia partia factor for stee Concrete strength cass C45/55 [EN : tabe 3.1] f ck 45MPa f cm 53MPa f ctm 3.8MPa E cm 36GPa ε cu Characteristic compressive strength Mean compressive strength Mean tensie strength Mean Young's moduus Utimate strain Reinforcement KS600S f yk 600MPa E s 00GPa Characteristic yied strength Young's moduus for stee Design vaues f cd f ck γ mc f cd 30MPa f yd f yk γ ms f yd MPa Design compressive strength of concrete Design yied strength of stee E s α α E cm Note that the fatigue oads are presented in respective chapter 64

79 B Goba equiibrium B.1 Eccentricity and width of soi pressure Find minimum eccentricity of soi pressure resutant with extreme oads. e M xyd F xyd h 4 F zd G d m Minimum eccentricity for soi pressure Soi pressure (shaded area) in case of dferent wind direction Left: Wind direction 90 degree. Right: Wind direction 45 degree. A intermediate direction is assumed to be fufied when those two are checked. Width of soi pressure with unorm soi pressure and wind direction 90 degree (the soi resutant at b ) b uni e 6.61 m Width of soi pressure with trianguar soi pressure and wind direction 90 degree (the soi resutant at b 3 ) b 3 e m Width of soi pressure with unorm soi pressure and wind direction 45 degree. 3 b 45.uni e m Width of soi pressure with trianguar soi pressure and wind direction 45 degree (resut in a rectanguar soi pressure ( b )) b 45 e m b

80 Ideaisation of oading case. Moment repaced by a force coupe. F c and F t incuding Left: rectanguar soi pressure Right: Trianguar soi pressure Cacuation of soi pressure F zd 4. f soi F zd G d b F zd G d f soi.uni b uni.667 MN m MN m Resuting soi pressure with trianguar soi pressure and wind direction 90 deg Resuting soi pressure with unorm soi pressure and wind direction 90 deg Resuting soi pressure with trianguar soi pressure and wind direction 45 deg. F zd G d f 45.soi.uni b 45.uni.706 MN m Resuting soi pressure with unorm soi pressure and wind direction 45 deg g d G d kn m Resuting soi pressure per meter 66

81 Since the cacuation is made in D it is important to cacuate where the resutants on the anchor ring acting. Assume that the stresses is concentrated in two quarters of the anchor ring with each resutant in its gravity center. d sr r o r o m Outer radius of stee ring d s πr o π 4 π 4 r o cos( φ) r o dφ Cacuation of distance between compressive and tensie forces with gravity center under the assumption of a fourth part of the stee ring being active for the compressive and tensie side d s d s m r s The norma force F z is equay spread on the anchor ring and resuting in: d s m d sr 4m Distance between tensie F t and compressive force F c Diameter of anchor ring 67

82 Transformation of moment to force coupe M xyd F xyd h 4 F zd F c ds 4 M xyd F xyd h 4 F zd F t ds MN MN Compressive force from moment and vertica force Tensie force from moment and vertica force F c F t g d b f soi F zd 0MN Check of goba equiibrium F c F t g d f soi.uni b uni F zd 0MN Check of goba equiibrium F c F t g d b 45.uni f 45.soi.uni F zd 0MN Check of goba equiibrium B. Shear force and bending moment distribution Assume that the bending moment and shear force are equay spread over the fu width of the foundation Shear force and moment distribution for wind direction 90 degree x 00.01m 15.5m f soi x V( x) f soi x b f soi x f soi x b g d x g d x d s x F c d s x f soi x f soi x F zd g b d x F c x b b F zd d s f soi g d x F c b x b F zd d s f soi g d x F c F t x V uni ( x) f soi.uni x g d x x b uni f soi.uni b uni g d x d s b uni x f soi.uni b uni g d x F c d s x f soi.uni b uni g d x F c F zd d s x f soi.uni b uni g d x F c F zd d s F t x 68

83 x f soi x 3 F soi ( x) f soi b 6 x Mx ( ) F soi ( x) g d d s x x d s F soi ( x) g d F c x x d s F soi ( x) g d F c x b f soi b x x d s 3 g d F c x d s x F zd x b f soi b x x d s 3 g d F c x F zd x d s F t x x b F zd x b x d s x d s x x M uni ( x) f soi.uni g d x b uni b uni x d s f soi.uni b uni x g d b uni x x x x b uni x d s f soi.uni b uni g d F c x b uni x d s f soi.uni b uni g d F c x F zd x b uni x d s f soi.uni b uni g d F c x F zd d s x F t x d s x d s x d s x 69

84 Shear force and moment distribution for wind direction of 45 degree The forces is assumed to be spread aong the fu width ( eff x 45 ) which vary with x, se figure beow. x m 45 x 45 eff x x x 45 x 45 eff x x 45 Cacuate how the sef-weight varies with x g d.45 x 45 g f G d Sef-weight per square meter g f x 45 g d.45 x x g f x x 45 g f x 45 70

85 gd45(x) V 45.uni x 45 f 45.soi.uni x 45 g b 45.uni d.45 x 45 x 45 b 45.uni b 45.uni f 45.soi.uni g d.45 x 45 b 45.uni x 45 b 45.uni f 45.soi.uni g d.45 x 45 b 45.uni f 45.soi.uni g d.45 x 45 b 45.uni f 45.soi.uni g d.45 x 45 Cacuate gravity center of gravity tp x x 45 F c F zd F c 45 d s 45 d s 45 x 45 F zd F c F t d s x d s x and the actua moment of sef-weight G d.45 x 45 71

86 tp x x 45 G d.45 x x 45 x 45 g d.45 x 45 x g d x x x 45 x 45 x x x x x x g f x 45 tp x x 45 x 45 M 45.uni x 45 3 f 45.soi.uni x 45 G b 45.uni 6 d.45 x 45 x 45 b 45.uni b 45.uni b 45.uni f 45.soi.uni x 45 G 3 d.45 x 45 b 45.uni x 45 f 45.soi.uni b 45.uni b 45.uni x 45 3 G d.45 x d s F c x 45 b 45.uni b 45.uni f 45.soi.uni x 45 G 3 d.45 x d s F zd 45 F c x 45 x 45 b 45.uni b 45.uni f 45.soi.uni x 45 3 G d.45 x d s F c x 45 F zd d s x 45 F t x d s 45 x d s d s 45 d s x 45 x

87 Shear force diagram 10 Vuni V45uni Shear force [MN] Lenght [m] 0 Bending moment diagram Bending moment [MN] Muni M45uni Length [m] Concusion: The moment distribution is simiar independent of oading situation. Its important to extend the reinforcement in order to achieve required capacity in the corners. Since the reinforcement is crossed the capacity is satisfied aso in the diagona direction. The design wi be based on the oading with unormed soi pressure and wind direction perpendicuar to the foundation. Its a common assumption to assume unormed soi pressure in ULS cacuation. The trianguar soi pressure gives however sighty higher positive moment. B.3 Sign convention 73

88 74 The figure above shows the sign convention. Moments resuting in tensie stresses at the bottom of the foundation is defined as positive. Observe that the diagrams shows negative downwards.

89 C Design in utimate imit state C.1 Sections Check in four dferent section, se figure. bottom U and top O reinforcement. x starts from the embedded stee ring edges. The interna forces is checked in four dferent sections, section 0-3. x d s m M Edu and M Edo are the positive and negative moments in the four dferent section. 3 x x x x x section1 x section x d s x d s 4 x d s 4 x d s 3. x 4 T T section1 section1 section section h m_section 3 h m h m x T h m_section h m_section 4 x h m 4 x x h m 4 75

90 Mean height of section 15.5 x 00.01m 15.5m size x 0.01 h m_section h m_mean engthh m_section m Mean height of the four sections h m ( x) x size x 1.65 m Tota mean height Choose bar diameter ϕ o c soi c 5mm ϕ u 5mm 50mm 100mm Top and bottom reinforcement Concrete cover to soi Concrete cover to tempate πϕ o A sio 4 πϕ u A siu mm mm Reinforcement area for one bar top Reinforcement area for one bar bottom Cacuate mean distance to reinforcement for top and bottom reinforcement d mu and d mo Definition of d in the four dferent sections.d mu is the mean distance from the top edge to the first ayer bottom reinforcement. d mo is the mean distance from the bottom edge to the first ayer of top reinforcement. i 0 3 d mui h m section1i d mu m ϕ u c soi ϕ u d mo m d moi h m section1i c ϕ o ϕ o 76

91 Assume a idea pastic behavior of reinforcement, i.e. no tension stfening and no strain imit Shows dferent materia modes for reinforcement bars. Assume horizonta top branch without strain hardening and strain imit. Curve (B) C. Design of bending reinforcement M uni section1i M uni sectioni M Edui M Edoi M Edu MNm m M Edo MNm m Preiminary Reinforcement area and bars per meter in the dferent sections M Edo 1 A m M Edu so A f yd 0.9d mo m su f yd 0.9d mu A so n A su o n A sio m u A siu Minimum spacing m m m 77

92 a o n o mm a u n u a oreq min mm n a ureq min 1 o n u Choose spacing: mm mm Required spacing with regard to bending a o 150mm a u 110mm Bottom reinforcement OBS This spacing is chosen with regard to crack width imitation se D. Crack widths SLS Cacuate utimate capacity for positive moment, i.e. bottom reinforcement is in tension i 0 3 d d mu m ϕ o d' c i soi ϕ o Top ( A' s ) and bottom (A s ) reinforcement amount 1 A s A a siu m 1 A' s u a o A sio m m m b 1 1m Width of the section i 0 3 α r 0.81 ε cu x sx 0.001m x si root α r f cd b 1 x sx x sx d' i ε x cu sx E s A' s f yd A s x sx x s mm d x i si ε si ε x cu si Size of the compressed zone of the dferent sections ε s Check that compression faiure in concrete have not occurred 78

93 f yd ε sy ε E s ε sy s x si d' i ε' si x si β r ε cu ε' s ε' s ε sy i 0 3 M Rd_posi α r f cd b 1 x si d β i r x si M Rd_pos MNm M Edu m E s ε' si A' s d d' i i MNm UR b.u M Edu m M Rd_pos % Degree of utiization of bottom reinforcement Top reinforcement Cacuate utimate capacity for negative moment, i.e. top reinforcement is in tension ϕ d d u mo m d' c 1.51 i soi ϕ u A' s A s A a a sio m u o m m b 1 1m x sx 0.51m i 0 3 α r 0.81 x si root α r f cd b 1 x sx x sx d' i ε x cu sx E s A' s A siu m f yd A s x sx 79

94 x s m d x i si ε si ε x cu si x si d' i ε' si ε x cu si β r Size of the compressed zone of the dferent sections ε s ε sy ε s ε sy ε' s ε' s ε sy M Rd_negi α r f cd b 1 x si d β i r x si i 0 3 M Rd_neg M Edo m UR b.o M Rd_neg MNm M Edo m % E s ε' si A' s d d' i i MNm Degree of Utiisation of top reinforcement C.3 Star reinforcement inside embedded stee ring M Rd_neg knm M Edo knm m d sr 4m a a o 150mm ϕ o 5mm Diameter of stee ring Same spacing as top reinforcement Bar diameter of top reinforcement 80

95 A s_req d sr A sio a M Edo M Rd_neg0 1 m mm n 56 ϕ star 5mm There is 56 hoes in the anchor ring, one bar in each hoe Bar diameter of star reinforcement φ 360deg n 6.49deg Bars inside a 90 deg ange 90deg n 90 n n φ i n 90 1 The reinforcement is paced in dferent direction towards the center of the anchor ring. Cacuate equivaent area A siring ϕ star π 4 A s_eqv A siring n i cos φn i mm UR b.star A s_req A s_eqv m % Utiisation degree of star reinforcement Layout of star reinforcement 81

96 C.4 Min and max reinforcement amounts [EN : ] Minimum reinforcement Contro of top reinforcement (esser than bottom reinforcement) b 1 1m d mo d m engthd mo d t d m b 1 used for cacuations per meter width 1.56 m f ctm A smin max0.6 b f 1 d0.001d t d yk Mean vaue of d of the four dferent sections m a mino 1m A smin A siu mm Maximum reinforcement Contro of bottom reinforcement (greater than top reinforcement) A cm h m_mean b 1 Average area of concrete cross section. A smax 0.04A cm m b 1 A s A a siu m u b 1 A' s A a sio m o Area of bottom reinforcement Area of top reinforcement A' s A s A smin 1 A smax 1 OK! OK! The chosen reinforcement amounts are within the imits C.5 Shear capacity Unreinforced capacity Check shear reinforcement is needed [EN : ] Check the maximum shear force in the four dferent sections 8

97 1 V Ed max max max V uni x d s V uni x 3 x V uni 4 x V uni 4 max V uni x 4 V Rd.c max C Rd.c k 100ρ f ck V uni x d s V uni x d s x 4 x V uni x d s 4 3. x kn m 1 3 k 1 σ cp b w dv min k 1 σ cp b w d 00 k min 1 i.0 d mui b 1 1m k mm k Area of tensioned reinforcement that reach at east ( bd d) away from current section Definition of tensioned reinforcement that reach at east ( bd d) away from current section, in this case A s is equa to the bottom reinforcement b 1 A s A a siu m u A s A s m ρ i min N Ed 0 A s 0.0 b 1 d mui Norma force ρ

98 A ci h m_sectioni b 1 Concrete area σ cpi min N Ed 0.f A cd ci 0.18 C Rd.c 0.1 γ mc 3 V min 0.035k f ck MPa 1... MPa f ck V Rd.ci maxc Rd.c k 100ρ i i MPa N V Rd.c V Rd.c kn m m V Rd.c V Ed UR shear.vrdc V Ed m V Rd.c 1 3 σ cp b d 1 mui σ cp k 1 V MPa mm mm mini k 1 MPa V kn Ed m Shear reinforcement ony needed around anchor ring (U-bows) for but due to assembing, minimum reinforcement is used Contro of concrete crushing [EN : ] m% f ck ν V 50MPa Ed b 1 0.5b 1 d mu νf cd Shear reinforcement Design of shear reinforcement [EN : ] A sw V Rd.s zf s ywd cot( θ) α cw b w zν 1 f cd V Rd.max cot( θ) tan( θ) 1 cot( θ) b d 1 mui mm mm 84

99 θ 45deg Choose ange f ck ν MPa α cw 1 No prestressing -> α cw =1 f ywd f yd ϕ w 5mm Reduction due to shear cracks Size of shear reinforcement bar A swi z ϕ w π 4 0.9d mu Area of one shear reinforcement bar V Ed b 1 s x 0.5m s root i s shear_req s s kn Guessed reinforcement spacing A swi z f s i ywd cot( θ) V Edi b 1 s x x m m Cacuate the required spacing Required spacing with regard to shear forces Minimum spacing of shear reinforcement according to turbine manufacturer. V Rd.si A swi z f s i ywd i cot( θ) V Ed m V Rd.s V Rd.s kn V Ed b kn ν 1 ν α cw b 1 zν 1 f cd V Rd.max kn V cot( θ) tan( θ) Rd.max V Rd.s

100 V Rdi min V Rd.maxi V Rd.si UR shear V Ed m V Rd % V Rd Utiisation ratio of shear in the dferent sections. N Shear reinforcement spacing 500mm diameter 5mm C.6 Loca effects and shear reinforcement around stee ring [EN : ] Contro of U-bow reinforcement The U-bow reinforcement is ocated around the embedded stee ring and wi both t up the compressive stresses and pu down the tensie stresses acting on the fange of the embedded stee ring. 86

101 Detaiing around embedded stee ring F xyd 880kN t u 1150mm h s 1750mm M xyd 56.7MNm t 1 400mm M da F xyd h 4 M xyd MNm ϕ Ubow 5mm Diameter of U-bow 87

102 Check concrete compression (crushing) Cacuate area moment of inertia of an annuus The stress distribution in the embedded stee ring is cacuated under the assumption of inear eastic theory with Navier's formua 1 I 0 4 π r 4 r 1 d sr 4m d 1 340mm r d sr d 1.17 m r 1 d sr d m I 0 π 4 4 r 4 r m 4 I 0 W annuus 3.966m 3 r Stresses under the fange of the embedded stee ring σ max.pos σ max.neg F zd M da kpa πd sr d 1 W annuus F zd M da kpa πd sr d 1 W annuus UR cc.ring σ max.pos σ max.neg f cd % Utiisation ratio of compression strength. No risk of crushing Contro shear reinforcement around anchor ring Assume a shear stress is transferred by the U-bows (V Ed > V Rdc ). Cacuate maximum mean stress on the fange 88

103 a ubow 0.1m ϕ Ubow 5mm σ mean.pos F zd πd sr d 1 M da r I 1 0 F zd πd sr d 1 M da r I MPa Max stress σ mean.neg F zd πd sr d 1 M da r I 1 0 F zd πd sr d 1 M da r I MPa Min stress σ Ubow σ mean.pos a ubow d 1 πϕ Ubow 4 σ mean.neg a ubow d 1 πϕ Ubow 4 1 σ Ubow f yd 1 σ Ubow UR shear.ubow f yd % Contro shear punching SS-EN : MPa 89

104 d 1.55m σ cp 0 3 V Rd.c C Rd.c k 100ρ f ck 1 k 1 σ cp v min k 1 σ cp k 1 00 d k 1 mm ρ C Rd.c v min 0.035k f ck MPa 0.5 MPa f ck V Rd.c.punch C Rd.c k100ρ 1 MPa V Rd.c.punch v min MPa 1 3 MPa 0.731MPa d sr V Rd.punch V Rd.c.punch dπ d( 500mm 4d) kn V Rd.punch σ mean.pos d 1 100mm 1 F c kn F c 0.77 V Rd.punch π V Rd.punch V Rd.c.punch d d sr kn V Rd.c.punch dπd sr N F c 1.18 V Rd.punch 90

105 D D.1 Loads Crack widths serviceabiity imit state SLS oads equiibrium M xysls F xysls M zsls G dsls MNm 800kN 5.863MNm 1.575MN g dsls kn m Sectiona forces M xysls F xysls h 4 e ss F zsls G dsls m Minimum eccentricity for soi pressure b ss 3 e ss G dsls kn 1.34 m Width of soi pressure F zsls G dsls f soi.ss b ss.381 MN m Sef weight and weight from soi eveny distributed over the ength of the foundation G dsls g dsls kn m M xysls F xysls h 4 F zsls F c.ss ds 4 M xysls F xysls h 4 F zsls F t.ss ds MN MN Compressive force from moment and vertica force Tensie force from moment and vertica force F c.ss F t.ss g dsls b ss f soi.ss F zsls N Check of goba equiibrium x 00.01m 15.5m x F soi.ss ( x) f soi.ss f 3 soi.ss x b ss 6 91

106 x M ss ( x) F soi.ss ( x) g dsls d s x x d s F soi.ss ( x) g dsls F c.ss x x d s F soi.ss ( x) g dsls F c.ss x x d s F soi.ss ( x) g dsls F c.ss x F zsls x d s F t.ss x d s x F zsls x b ss b ss x d s f soi.ss x g 3 dsls F c.ss x F zsls x d s F t.ss x d s x b ss d s x b ss x 1 Moment diagram M.ss(x) Moment [MNm/m] D. Crack width imitation Lenght [m] Check of aowabe crack width [EN : ] b 1 1m Thickness of the section a u 110mm a o 150mm Spacing to fufi crack requirement 9

107 M ss x 3 x M ss 4 b 1 M ssu x M ss 4 b 1 M sso x M ss 4 M ss x d s x M ss x d s 4 x M ss x d s 4 3. x M ss x d s knm knm b 1 A su A a siu m b 1 A' su A u a sio m o b 1 A so A a sio m b 1 A' so A o a siu m u i 0 3 d mu α x IIu 0.m m Guess d mo x IIu x IIui rootb 1 ( α 1) A' su x IIu d' ui x IIo 0.m x IIo x IIoi rootb 1 ( α 1) A' so x IIo d' oi x 0.44 IIo m x 0.07 IIu m m ϕ o d' oi c ϕ o ϕ u d' ui c soi ϕ u αa su x IIu d mui αa so x IIo d moi x IIu x IIo Height of compressive zone 93

108 3 b 1 x IIu x IIu I IIu b 1 1 x IIu ( α 1) A' su x IIu d' u 3 b 1 x IIo x IIo I IIo b 1 1 x IIo ( α 1) A' so x IIo d' o αa su d mu x IIu αa so d mo x IIo I IIo m 4 z ui d mui x IIui I IIu m 4 z oi d moi x IIoi Stee stress M ssui σ sui α z I ui IIui M ssoi σ su MPa σ soi α I IIoi z oi σ so MPa Maximum aowed crack width according to EN :005 NA with regard to L50 and XC3 w k.max 0.4mm α e E s E cm k t 0.4 Depending on oad duration, kt for ong term oad f ct.eff f ctm 3.8MPa A s m h m_mean m Effective area for a one meter thick section A c.effui min.5 h m_sectioni d mui A c.effoi min.5 h m_sectioni d moi h m_sectioni x IIui 3 h m_sectioni x IIoi 3 b 1 b 1 ξ 1 0 A' p 0 No pre- or post tensioned reinforcement 94

109 A su ξ 1 A'p A so ξ 1 A'p ρ p.effui ρ A p.effoi c.effui A c.effoi Δε ui f ct.eff σ sui k t 1 α ρ e ρ p.effui p.effui E s Δε oi f ct.eff σ soi k t 1 α ρ e ρ p.effoi p.effoi E s σ sui σ soi Δε ui max Δε ui 0.6 Δε E oi max Δε oi 0.6 s E s Δε u Δε o k k 1 For reinforcement bars with good interactive properties For reinforcement in tension ϕ u ϕ o k 3u k c 3o soi c According to EC NA k Recommended vaue k 1 k k 4 ϕ u s r.maxui k 3u c s ρ r.maxoi k 3o c p.effui k 1 k k 4 ϕ o ρ p.effoi w kui s r.maxui Δε ui w koi s r.maxoi Δε oi Crack width w ku mm w ko mm Crack width for the dferent sections w kui w k.max w koi w k.max OK! Cacuated crack width ess then the aowed

110 Utiisation degree of crack width w ku w UR ko crack.width.u % UR w k.max 8.33 crack.width.o w k.max % 96

111 E Fatigue cacuations with equivaent oad cyce method E1. Loads and sectiona forces Instead of using the fu oad spectra one equivaent oad width is cacuated from the oad spectra. See Appendix I Mean ampitudes from appendix ΔMmean kNm ΔF xymean 18kN ΔF z 0kN Mean oads F xmean 316kN F ymean 4kN F xymean F xmean F ymean kN M xmean 1888kNm M ymean 193kNm M xymean M xmean M ymean knm F zmean 47kN min/max fatigue oad ΔMmean M df1 M xymean ΔMmean M df M xymean 14.85MNm 7.901MNm ΔF xymean F xydf1 F xymean ΔF xymean F xydf F xymean 07.05kN 45.05kN ΔF z F zdf1 F zmean ΔF z F zdf F zmean Equiibrium M df1 F xydf1 h 4 M df F xydf h 4 e f m e F zdf1 G f d F zdf G d.148 m Min/max eccentricity 97

112 b f1 3 e f m b f 3 b f1 1 b f 1 e f m Width of soi pressure The fatigue oads are sma and the soi pressure is spread over the fu ength. The distribution can be soved, two equations and two unknowns. Min oad (oad 1) f 11 f 1 expicit f 1 = G d F zdf1 sovef 1 The gravity center must be equa to the eccentricity f 11 G d F zdf1 G d F zdf1 f 11 G d F zdf1 f 11 f 11 f 11 6 Expression for the distance to the gravity center 3 f 11 6 G d F zdf1 Simpied expression f 11 3 f 11 = e 6 G d F zdf1 f1 f m kn f 1 G d F zdf1 f 11 Max oad (oad ) m kn expicit sovef 11 e f1 6 6F zdf1 6G d 98

113 f 1 3 f 1 = e 6 G d F zdf f f m kn f G d F zdf f m kn expicit sovef 1 e f 6 6F zdf 6G d Min compressive and tensie resutant M df1 F xydf1 h 4 F cf1 d s F zdf MN Compressive force from moment and vertica force M df1 F xydf1 h 4 F tf1 d s F zdf MN Tensie force from moment and vertica force F cf1 F tf1 g d f 11 f 1 F zdf1 0N Check of goba equiibrium Max compressive and tensie resutant M df F xydf h 4 F cf d s F zdf MN Compressive force from moment and vertica force M df F xydf h 4 F tf d s F zdf MN Tensie force from moment and vertica force F cf F tf g d f 1 f F zdf 0MN Bending moment distribution fatigue oading x 00.01m 15.5m F 11 ( x) f 11 f 1 x f 11 f 1 x 3 6 Check of goba equiibrium 99

114 x x M f1 ( x) F 11 ( x) f 1 g d d s x x x d s F 11 ( x) f 1 g d F cf1 x x x d s F 11 ( x) f 1 g d F cf1 x x x d s F 11 ( x) f 1 g d F cf1 x d s F tf1 x F 1 ( x) f 1 f x f 1 f x x M f ( x) F 1 ( x) f g d x 3 6 d s x x x d s F 1 ( x) f g d F cf x x x d s F 1 ( x) f g d F cf x x x d s F 1 ( x) f g d F cf x d s F tf x d s x F zdf1 x F zdf1 x d s x F zdf x F zdf x d s x d s x d s x d s x 0.5 Fatigue oading max/min moment Mf1(x) Mf(x) [MNm/m] [m] 100

115 Minimum moment in section 0-3, fatigue M f1 x 3 x M f1 4 b 1 b 1 M 1u x M 1o M f1 4 x M f1 4 Maximum moment in section 0-3, fatigue M f x 3 x M f 4 b 1 b 1 M u x M o M f 4 x M f 4 M f1 x d s x M f1 x d s 4 x M f1 x d s 4 3. x M f1 x d s 4 M f x d s Shear force distribution fatigue oading V f1 ( x) f 11 x f 11 x f 11 x f 11 x V f ( x) f 1 x f 1 x f 1 x f 1 x f 11 f 1 f 11 f 1 f 11 f 1 f 11 f 1 f 1 f f 1 f f 1 f f 1 f x x x x x x x x g d x g d x g d x g d x g d x g d x g d x g d x x M f x d s 4 x M f x d s 4 3. x M f x d s 4 d s x F cf1 F zdf1 F cf1 d s x F zdf1 F cf1 F tf1 d s x F cf F zdf F cf d s x F zdf F cf F tf d s x d s d s x d s x x 101

116 0.4 Fatigue oading max/min shear force V.f1(x) V.f(x) Shear force [MN/m] [m] Minimum shear force in section 0-3, fatigue V f1 x 3 x V f1 4 b 1 b 1 V f1_pos x V f1_neg V f1 4 x V f1 4 Maximum shear force in section 0-3, fatigue V f x 3 x V f 4 b 1 b 1 V f_pos x V f_neg V f 4 x V f 4 V f1 x d s x V f1 x d s 4 x V f1 x d s 4 3. x V f1 x d s 4 E. Fatigue contro bending moment V f x d s x V f x d s 4 x V f x d s 4 3. x V f x d s 4 Check top and bottom reinforcement and compressive concrete.use Navier's formua to cacuate stresses, determine neutra axis and moment of inertia. Assume fuy cracked member (stage II). According to EC compressive stresses must be checked as we.for concrete ony compressive stresses is checked. 10

117 Stress range for bottom reinforcement and compressed concrete Fatigue due to positive moment (bottom reinforcement in tension) b 1 A s A a siu m b 1 A' s A u a sio m o d d mu m d' c ϕ i o 1.43 ϕ o x II 0.3m Guess x II x IIi rootb 1 ( α 1) A' s x II d' i x II mm b 1 x II x II I II b 1 1 x II ( α 1) A' s x II d' I II m Stee stress top (o) reinforcement z d' x II αa s x II d x i II αa s d x II M 1ui M ui σ s1posoi α z σ I i sposoi α z IIi I i IIi Min and max stresses σ s1poso MPa σ sposo MPa Concrete stress top (o) (check top fibre on safe side) z x II M 1ui M ui σ c1posoi z σ I i cposoi z IIi I i IIi Min and max stresses 103

118 σ c1poso MPa σ cposo Stee stress bottom (u) reinforcement z d x II MPa M 1ui M ui σ s1posui α z σ I i sposui α z IIi I i IIi Min and max stresses σ s1posu MPa σ sposu MPa Star reinforcement on the top (o) d sr M u0 m d sr M 1u m knm d sr A s A a siu 0.018m A' s A s_eqv m u d d mu m d' c soi ϕ o ϕ o x II 0.3m Guess x II x II rootd sr ( α 1) A' s x II d' αa s x II d x II x II mm 3 d sr x II I II 1 x II d sr x II ( α 1) A' s x II d' αa s d x II z d x II d sr M 1o0 m σ s1ring.pos α z σ I sring.pos α II d sr M o0 m I II z σ s1ring.pos Pa σ sring.pos Pa Stress range for top reinforcement and compressed concrete 104

119 Fatigue due to negative moment (top reinforcement in tension) b 1 A s A a sio m b 1 A' s A o a siu m u d d mo m d' c 1.51 i soi ϕ u ϕ u x II 0.3m Guess x II x IIi rootb 1 ( α 1) A' s x II d' i x II mm αa s x II d x i II 3 b 1 x II x II I II b 1 1 x II ( α 1) A' s x II d' I II m 4 αa s d x II Stee stress top (o) reinforcement z d x II M 1oi σ s1negoi α z σ I i snegoi α IIi M oi I IIi z i σ s1nego MPa σ snego Stee stress bottom (u) reinforcement z d' x II MPa M 1oi σ s1negui α z σ I i snegui α IIi M oi I IIi z i 105

120 σ s1negu MPa σ snegu MPa Concrete stress bottom (u), check bottom fibre on safe side z x II M 1oi M oi σ c1negui z σ I i cnegui IIi I IIi σ c1negu MPa σ cnegu Star reinforcement on the top (o) z i MPa d sr M o0 m d sr M 1o m knm M o0 M 1o kNm d sr A s A s_eqv A' s A a siu 0.018m u d d mo m x II 0.3m Guess d' c ϕ o ϕ o x II x II rootd sr ( α 1) A' s x II d' αa s x II d x II x II mm 3 d sr x II I II 1 x II d sr x II ( α 1) A' s x II d' αa s d x II z d' x II d sr M 1o0 m σ s1ring.neg α z σ I sring.neg α II d sr M o0 m I II z σ s1ring.neg Pa σ sring.neg Pa 106

121 Fatigue verication reinforcement stress range top reinforcement Δσ soi max σ snegoi σ s1negoi σ sposoi σ s1posoi Δσ so MPa stress range bottom reinforcement Δσ sui max σ snegui σ s1negui σ sposui σ s1posui Δσ su MPa The bottom reinforcement amount is constant. The reinforce under the anchor ring "between" section zero and zero is incuded in this check γ s.fat 1.15 For straight reinforcement bars γ F.fat 1.0 Δσ Rsk 16.5MPa Δσ Rsk γ s.fat MPa Stress range star reinforcement Δσ sring max σ sring.pos σ s1ring.pos σ sring.neg σ s1ring.neg Δσ sring MPa Δσ Rsk 16.5MPa Δσ Rsk γ s.fat MPa Verication of fatigue from equivaent oad [EN : ] maxδσ su γ F.fat maxδσ so γ F.fat Δσ Rsk 1 γ s.fat Δσ Rsk 1 γ s.fat 107

122 UR fat.b.u UR fat.b.o Δσ su γ F.fat Δσ Rsk γ s.fat Δσ so γ F.fat Δσ Rsk γ s.fat % % Utiisation degree of bending reinforcement bottom (u) and top (o) (fatigue) Δσ sring γ F.fat Δσ Rsk 1 γ s.fat U fat.star Δσ sring γ F.fat Δσ Rsk % Utiisation degree of star reinforcement ( fatigue) γ s.fat Fatigue verication of concrete Fatigue in compressed concrete, concrete stress range [EN : ] Δσ co σ cposo σ c1poso Δσ cu σ cnegu σ c1negu 1.551MPa 1.745MPa t 0 8 s c 0.5 Assumed concrete age when fatigue oading starts Depending on cement type CEM 4.5 N β cc 8 exps c 1 t β cc 1 0 f cm β cc f ck 8MPa f cd Pa k 1 1 For N=10^6 cyces f cd.fat k 1 β cc f cd 1 f ck MPa 50 f cd.fat 4.6MPa σ cd.min.equ.o σ c1poso σ cd.min.equ.u σ c1negu σ cd.max.equ.o σ cposo σ cd.max.equ.u σ cnegu E cd.min.equ.o σ cd.min.equ.o f cd.fat Lowest compressive stress eve in a cyce 108

123 E cd.max.equ.o E cd.min.equ.u E cd.max.equ.u σ cd.max.equ.o f cd.fat σ cd.min.equ.u f cd.fat σ cd.max.equ.u f cd.fat Highest compressive stress eve in a cyce Lowest compressive stress eve in a cyce Highest compressive stress eve in a cyce E cd.min.equ.oi R equ.oi Stress ratio R E equ.o cd.max.equ.oi E cd.min.equ.ui R equ.ui E Stress ratio R cd.max.equ.ui equ.u E cd.max.equ.oi R equ.oi 1 E cd.max.equ.ui R equ.ui E cd.max.equ.oi R equ.oi E cd.max.equ.ui R equ.ui UR fat.c.ui E cd.max.equ.ui R equ.ui UR fat.c.oi E cd.max.equ.oi R equ.oi UR fat.c.u % UR fat.c.o E3. Fatigue contro of oca effects Compressed concrete around the embedded stee ring F xydf1 F xydfat F xydf M M df1 F zdf1 dsfat M df F dfat F zdf M dafat F xydfat h 4 M dsfat knm % Utiisation ratio of compressed concrete 109

124 σ mean.fc.fat σ mean.ft.fat F dfat πd sr d 1 F dfat πd sr d 1 M dafat r I 1 0 M dafat r I 1 0 F dfat πd sr d 1 M dafat 1 r I 0 F dfat M dafat r πd sr d 1 I 0 1 Fatigue in compressed concrete at fange, concrete stress range [EN : ] Stress under and over the fange σ mean.fc.fat0 σ anchor_fc σ mean.fc.fat1 σ anchor_ft σ mean.ft.fat0 σ mean.ft.fat Δσ c_bottom σ anchor_fc1 σ anchor_fc0 Δσ c_top σ anchor_fc1 σ anchor_fc0 MPa MPa 3.179MPa 3.179MPa MPa MPa Δσ c_bottom Δσ c_top Min/max stress Min/max stress t 0 8 s c 0.5 Assumed concrete age when fatigue oading starts Depending on cement type CEM 4.5 N β cc 8 exps c 1 t β cc 1 0 f cm β cc f ck 8MPa f cd Pa k 1 1 For N=10^6 cyces f cd.fat k 1 β cc f cd 1 f cd.fat 4.6MPa f ck MPa 50 E cd.max.equ E cd.min.equ σ anchor_ft f cd.fat σ anchor_ft f cd.fat Highest ratio in a cyce Lowest ratio in a cyce R equ E cd.min.equ E cd.max.equ 110

125 E cd.max.equ R equ 1 1 UR fat.cc.ring E cd.max.equ R equ UR fat.cc.ring % U-bows Fatigue U-bow [EN : ] a ubow 100mm σ Ubow.max Spacing of U-bows σ mean.fc.fat0 a ubow d 1 πϕ Ubow 4 σ mean.fc.fat1 a ubow d 1 πϕ Ubow Utiisation ration for fatigue in compressed concrete under embedded stee ring 4 Δσ st σ Ubow.max1 σ Ubow.max0 Δσ st MPa MPa D 600mm ϕ w 5mm Bending diameter Diameter U-bow ζ D ϕ w Reduction factor due to bent reinforcement bars Δσ Rsk 16.5MPaζ MPa γ F.fat 1 γ s.fat 1.15 Δσ st MPa Verication of fatigue from equivaent oad Δσ Rsk γ s.fat MPa Δσ st γ F.fat Δσ Rsk γ s.fat U fat.ubow Δσ st γ F.fat Δσ Rsk % Utiisation ration for fatigue in U-bow reinforcement γ s.fat 111

126 G Fatigue verication with the fu oad spectra Fatigue cacuations for the fu oad spectra given by the wind turbine suppier. G1. Loads and sectiona forces Loads Ampitudes from oad spectra: Input of oad spectra from exce, Appendix I: ΔM_input ΔF xy_input Mxy.xs Fxy.xs n number of cyces.xs ΔM ΔM_inputkNm ΔF xy ΔF z 0kN Mean oads F xmean ΔF xy_input kn 316kN F ymean 4kN F xymean F xmean F ymean M xmean 1888kNm kN M xymean M xmean M ymean knm F zmean 47kN Min/max fatigue oad rows( ΔM_input) 80 k 0 rows( ΔM_input) 1 Tota 80 oads Due to technica functionaity in Mathcad the oads initiay can not be on vector form. ΔM 0 M df1 M xymean ΔM 0 M df M xymean ΔF xy0 F xydf1 F xymean ΔF xy0 F xydf F xymean 11

127 ΔF z F zdf1 F zmean ΔF z F zdf F zmean Sectiona forces M df1 F xydf1 h 4 M df F xydf h 4 e f1 e F zdf1 G f d F zdf G d Depending on oad-magnitude the soi pressure wi distribute trianguar over the fu ength or part of the fu ength. Smaer oad resut in a sma eccentricity and the distribution is as foows: The fatigue oads are sma and the soi pressure is spread over the fu ength. The distribution can be soved, two equations and two unknowns. The foowing index system is used: f 11 - Max soi pressure (eft side in figure above) and min fatigue oad (oad 1) f 1 - Max soi pressure (eft side in figure above) and max fatigue oad (oad ) f 1 - Min soi pressure (right side in figure above) and min fatigue oad (oad 1) f - Min soi pressure (right side in figure above) and max fatigue oad (oad ) The gravity center must be equa to the eccentricity G d F zdf1 f 11 G d F zdf1 f 11 G d F zdf1 f 11 f 11 f

128 3 f 11 6 G d F zdf1 Simpied expression Equiibrium (for oad 1) f 11 3 f 11 = e 6 G d F zdf1 f1 expicit sovef 11 e f1 6 6G d 6F zdf1 f 11 e f1 F zdf1 f 11 f 1 expicit f 1 = G d F zdf1 sovef 1 G d F zdf1 f 1 e f1 F zdf1 Max oad (oad ) e f1 6 6G d 6F zdf1 f 1 expicit f 1 = e 3 6 G d F zdf f sovef 1 e f 6 6G d 6F zdf f 1 e f F zdf G d F zdf f e f F zdf f 1 G d F zdf f e f F zdf e f 6 6G d 6F zdf For arger oads the soi pressure is spread over a smaer part of the ength. When the soi pressure is ess then the fu ength, the width is a function of the eccentricity. 114

129 The width of the soi pressure assuming trianguar distribution is: 3 b f1 e f1 e f1 3 b f e f Where b f1 is for oad 1 and b f for oad e f The soi pressure can be cacuated: f soi1 b f1 e f1 F zdf1 f soi b f e f F zdf F zdf1 G d. => b f1 e f1 F zdf G d b f e f. => f soi1 e f1 F zdf1 f soi e f F zdf F zdf1 G d 3 e f1 F zdf G d 3 e f The moment distribution can be cacuated as a function of the eccentricity and oads Moment distribution x 00.01m 15.5m Moment distribution when the soi pressure is spread over the fu ength F 11 xe f1 F zdf1 is the moment from soi pressure f 11 e f1 F zdf1 f 1 e f1 F zdf1 F 11 xe f1 F zdf1 x f 1 e f1 F zdf1 f 11 e f1 F zdf1 x

130 F 11 xe f1 F zdf1 f 1 e f1 F zdf1 M f1 xe f1 F zdf1 F cf1 F tf1 F 1 xe f F zdf F 11 xe f1 F zdf1 x x g d f 1 e f1 F zdf1 x d s g d F cf1 x F 11 xe f1 F zdf1 f 1 e f1 F zdf1 x d s g d F cf1 x F zdf1 x f 1 e f F zdf f e f F zdf M f xe f F zdf F cf F tf x x F 11 xe f1 F zdf1 f 1 e f1 F zdf1 x x g d d s F zdf1 F cf1 x x d s F tf1 x x F 1 xe f F zdf f e f F zdf F 1 xe f F zdf f 1 e f F zdf d s x d s x d s x d s x f e f F zdf x x g d f e f F zdf x d s g d F cf x F 1 xe f F zdf f e f F zdf x d s g d F cf x F zdf x F 1 xe f F zdf f e f F zdf x d s g d F cf x d s F tf x x x d s x d s x x F zdf x d s x x 3 6 d s x 116

131 Moment distribution when the soi pressure is spread over part of the ength f soi1 e f1 F zdf1 F soi1 xe f1 F zdf1 x f soi1 e f1 F zdf1 b f1 e f1 x 3 6 moment from soi pressure x F soi1 xe f1 F zdf1 g d M' f1 xe f1 F zdf1 F cf1 F tf1 f soi e f F zdf F soi xe f F zdf x g d F soi1 xe f1 F zdf1 x g d F soi1 xe f1 F zdf1 d s F cf1 x x g d F soi1 xe f1 F zdf1 d s F cf1 x d s F tf1 x d s x d s F cf1 x F zdf1 x F zdf1 x f soi1 e f1 F zdf1 b f1 e f1 x 3 x d s g d F cf1 x F zdf1 x d s F tf1 x x f soi e f F zdf b f e f x 3 6 b f1 e f1 d s x d s x d s x b f1 e f1 x b f1 e f1 117

132 x F soi xe f F zdf g d M' f xe f F zdf F cf F tf x g d F soi xe f F zdf x g d F soi xe f F zdf d s F cf x x g d d s x d s F cf x F zdf x F soi xe f F zdf d s F cf x F zdf x d s F tf x f soi e f F zdf b f e f b f e f x 3 x d s g d F cf x F zdf x d s F tf x Use correct moment distribution, i.e. depending on soi pressure distribution M f1 xe f1 F zdf F cf1 F tf1 b f1 e f1 M' f1 xe f1 F zdf F cf1 F tf1 b f1 e f1 F zdf F cf F tf M f xe f F zdf F cf F tf b f e f b f e f M fat1 xe f1 F zdf F cf1 F tf1 M fat x e f M' f xe f F zdf F cf F tf Cacuate moments in four dferent sections in order to cacuate stress variation ΔM ΔM M df1 M xymean M df M xymean ΔF xy ΔF xy F xydf1 F xymean F xydf F xymean ΔF z ΔF z F zdf1 F zmean F zdf F zmean M df1k F xydf1k h 4 F M zdf1 dfk F xydfk h 4 F cf1k F d s 4 cfk d s M df1k F xydf1k h 4 F M zdf1 dfk F xydfk h 4 F tf1k F d s 4 tfk d s F zdf 4 F zdf 4 d s x d s x d s x b f e f x b f e f 118

133 M df1k F xydf1k h 4 M dfk F xydfk h 4 e f1k e F zdf1 G fk d F zdf G d Largest moment ampitude Contro fatigue in dferent sections The fatigue contro is performed in the section described in C.1. Since reinforcement shoud be contro for both tension and compression both negative and positive moment is considered. Minimum moment in section 0-3, fatigue M 10uk M fat1 x e f1k F zdf1 F cf1k F tf1k b 1 3 x M 11uk M fat1 e 4 f1k F zdf1 F cf1k F tf1k x M 1uk M fat1 e 4 f1k F zdf1 F cf1k F tf1k x M 13uk M fat1 4 e f1 F k zdf1 F cf1k F tf1k b 1 b 1 b 1 b 1 M 10ok M fat1 x d s e f1k F zdf1 F cf1k F tf1k x M 11ok M fat1 x d s 4 e f1k b 1 F zdf1 F cf1k F tf1k x b 1 M 1ok M fat1 x d s 4 e f1 F k zdf1 F cf1k F tf1k 119

134 M 13ok M fat1 x d s 3 x b 1 4 e f1 F k zdf1 F cf1k F tf1k Maximum moment in section 0-3, fatigue M 0uk M fat x e fk F zdf F cfk F tfk b 1 3 x M 1uk M fat e 4 fk F zdf F cfk F tfk x M uk M fat e 4 fk F zdf F cfk F tfk x M 3uk M fat 4 e f F k zdf F cfk F tfk b 1 b 1 b 1 b 1 M 0ok M fat x d s e fk F zdf F cfk F tfk x M 1ok M fat x d s 4 e fk b 1 F zdf F cfk F tfk x b 1 M ok M fat x d s 4 e f F k zdf F cfk F tfk 3 x b 1 M 3ok M fat x d s 4 e f F k zdf F cfk F tfk Stress variation due to moment Fatigue due to bending. Check top and bottom reinforcement and compressive concrete Use Navier's formua to cacuate stresses, determine neutra axis and moment of inertia. Assume fuy cracked member (stage II). According to EC compressive stresses must be checked as we. For concrete ony compressive stresses is checked. Bar spacing : Bar diameter: Concrete cover: a o 150mm ϕ u 5mm c soi 0.1 m a u 110mm ϕ o 5mm c 50mm α γ s.fat 1.15 For straight reinforcement bars γ F.fat 1 Δσ Rsk 16.5MPa [EC : ] 10

135 G. Fatigue in bending reinforcement Fatigue due to positive moment (bottom reinforcement in tension) i 0 3 πϕ o A sio 4 πϕ u A siu mm mm b 1 A s A a siu m b 1 A' s A u a sio m o d' c ϕ i o ϕ o d mu m d d mu 1.43 x II 0.3m Guess x II x IIi rootb 1 x II mm ( α 1) A' s x II d' i 3 b 1 x II x II I II b 1 1 x II ( α 1) A' s x II d' I II m 4 Stee stress top (o) reinforcement z d' x II section 0 αa s x II d x i II αa s d x II Δσ SO0posk M 0uk M 10uk α z I 0 II0 11

136 14.438MPa max Δσ SO0pos N O0posk Δσ Rsk γ s.fat γ F.fat Δσ SO0posk Δσ Rsk γ s.fat γ F.fat Δσ SO0posk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SO0posk γ F.fat Δσ SO0posk d O0posk n k... N O0posk D Opos0 d O0posk k section 1 Δσ SO1posk M 1uk M 11uk α z I 1 II MPa max Δσ SO1pos N O1posk Δσ Rsk γ s.fat γ F.fat Δσ SO1posk Δσ Rsk γ s.fat γ F.fat Δσ SO1posk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SO1posk γ F.fat Δσ SO1posk d O1posk n k... N O1posk D Opos1 d O1posk k section Δσ SOposk M uk M 1uk α z I II 4.315MPa max Δσ SOpos 1

137 N Oposk Δσ Rsk γ s.fat γ F.fat Δσ SOposk Δσ Rsk γ s.fat γ F.fat Δσ SOposk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SOposk γ F.fat Δσ SOposk d Oposk n k... N Oposk D Opos d Oposk 0 k section 3 Δσ SO3posk M 3uk M 13uk α z I 3 II MPa max Δσ SO3pos N O3posk Δσ Rsk γ s.fat γ F.fat Δσ SO3posk Δσ Rsk γ s.fat γ F.fat Δσ SO3posk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SO3posk γ F.fat Δσ SO3posk d O3posk n k... N O3posk D Opos3 d O3posk 0 k Stee stress bottom (u) reinforcement z d x II section 0 Δσ SU0posk M 0uk M 10uk α z I 0 II0 13

138 MPa max Δσ SU0pos N U0posk Δσ Rsk γ s.fat γ F.fat Δσ SU0posk Δσ Rsk γ s.fat γ F.fat Δσ SU0posk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SU0posk γ F.fat Δσ SU0posk d U0posk n k... N U0posk D Upos0 d U0posk k section 1 Δσ SU1posk M 1uk M 11uk α z I 1 II1 7.79MPa max Δσ SU1pos N U1posk Δσ Rsk γ s.fat γ F.fat Δσ SU1posk Δσ Rsk γ s.fat γ F.fat Δσ SU1posk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SU1posk γ F.fat Δσ SU1posk d U1posk n k... N U1posk D Upos1 d U1posk k section Δσ SUposk M uk M 1uk α z I II MPa max Δσ SUpos 14

139 N Uposk Δσ Rsk γ s.fat γ F.fat Δσ SUposk Δσ Rsk γ s.fat γ F.fat Δσ SUposk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SUposk γ F.fat Δσ SUposk d Uposk n k... N Uposk D Upos d Uposk k section 3 Δσ SU3posk M 3uk M 13uk α z I 3 II MPa max Δσ SU3pos N U3posk Δσ Rsk γ s.fat γ F.fat Δσ SU3posk Δσ Rsk γ s.fat γ F.fat Δσ SU3posk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SU3posk γ F.fat Δσ SU3posk d U3posk n k... N U3posk d U3posk n k... N U3posk D Upos3 d U3posk k Fatigue due to negative moment (bottom reinforcement in tension) b 1 A s A a sio m b 1 A' s A o a siu m u 15

140 d mo m d d 1.51 mo d' c i soi ϕ u x II 0.3m Guess ϕ u x II x IIi rootb 1 ( α 1) A' s x II d' i x II mm b 1 x II x II I II b 1 1 x II ( α 1) A' s x II d' I II m Stee stress top (o) reinforcement z d x II αa s x II d x i II αa s d x II section 0 Δσ SO0negk M 0uk M 10uk α z I 0 II MPa max Δσ SO0neg N O0negk Δσ Rsk γ s.fat γ F.fat Δσ SO0negk Δσ Rsk γ s.fat γ F.fat Δσ SO0negk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SO0negk γ F.fat Δσ SO0negk d O0negk n k... N O0negk D Oneg0 d O0negk k 16

141 section 1 Δσ SO1negk M 1uk M 11uk α z I 1 II MPa max Δσ SO1neg N O1negk Δσ Rsk γ s.fat γ F.fat Δσ SO1negk Δσ Rsk γ s.fat γ F.fat Δσ SO1negk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SO1negk γ F.fat Δσ SO1negk d O1negk n k... N O1negk D Oneg1 d O1negk k section Δσ SOnegk M uk M 1uk α z I II 47.63MPa max Δσ SOneg N Onegk Δσ Rsk γ s.fat γ F.fat Δσ SOnegk Δσ Rsk γ s.fat γ F.fat Δσ SOnegk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SOnegk γ F.fat Δσ SOnegk d Onegk n k... N Onegk D Oneg d Onegk k section 3 17

142 Δσ SO3negk M 3uk M 13uk α z I 3 II MPa max Δσ SO3neg N O3negk Δσ Rsk γ s.fat γ F.fat Δσ SO3negk Δσ Rsk γ s.fat γ F.fat Δσ SO3negk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SO3negk γ F.fat Δσ SO3negk d O3negk n k... N O3negk D Oneg3 d O3negk k Stee stress bottom (u) reinforcement z d' x II section 0 Δσ SU0negk M 0uk M 10uk α z I 0 II0 9.09MPa max Δσ SU0neg N U0negk Δσ Rsk γ s.fat γ F.fat Δσ SU0negk Δσ Rsk γ s.fat γ F.fat Δσ SU0negk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SU0negk γ F.fat Δσ SU0negk d U0negk n k... N U0negk D Uneg0 d U0negk k section 1 18

143 Δσ SU1negk M 1uk M 11uk α z I 1 II1 5.56MPa max Δσ SU1neg N U1negk Δσ Rsk γ s.fat γ F.fat Δσ SU1negk Δσ Rsk γ s.fat γ F.fat Δσ SU1negk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SU1negk γ F.fat Δσ SU1negk d U1negk n k... N U1negk D Uneg1 d U1negk 0 k section Δσ SUnegk M uk M 1uk α z I II.68MPa max Δσ SUneg N Unegk Δσ Rsk γ s.fat γ F.fat Δσ SUnegk Δσ Rsk γ s.fat γ F.fat Δσ SUnegk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SUnegk γ F.fat Δσ SUnegk d Unegk n k... N Unegk D Uneg d Unegk 0 k section 3 19

144 Δσ SU3negk M 3uk M 13uk α z I 3 II3 0.73MPa max Δσ SU3neg N U3negk Δσ Rsk γ s.fat γ F.fat Δσ SU3negk Δσ Rsk γ s.fat γ F.fat Δσ SU3negk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SU3negk γ F.fat Δσ SU3negk d U3negk n k... N U3negk D Uneg3 d U3negk 0 k Star reinforcement on the top (o) A s_eqv mm b 1 A s A s_eqv A' s A a sio m o d d mo m d' c ϕ o ϕ o x II 0.3m Guess x II x II rootb 1 ( α 1) A' s x II d' αa s x II d x II x II 85.16mm 3 b 1 x II x II I II b 1 1 x II ( α 1) A' s x II d' Stee stress star reinforcement z d' x II αa s d x II Δσ SSTARk α M 0ok M 10ok I II z 130

145 15.071MPa max Δσ SSTAR N STARk Δσ Rsk γ s.fat γ F.fat Δσ SSTARk Δσ Rsk γ s.fat γ F.fat Δσ SSTARk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ SSTARk γ F.fat Δσ SSTARk d STARk n k... N STARk D star d STARk k Damage resuts : D Upos D Uneg % % D Opos D Oneg % % D star % 131

146 G.3 Shear force distribution Shear force distribution fatigue oading f 11 e f1 F zdf1 x f 11 e f1 F zdf1 f 1 e f1 F zdf1 V f1 xe f1 F zdf1 F cf1 F tf1 g d x x x d s f 11 e f1 F zdf1 x f 11 e f1 F zdf1 g d x F cf1 f 1 e f1 F zdf1 x d s x f 11 e f1 F zdf1 x f 11 e f1 F zdf1 F zdf1 g d x F cf1 f 1 e f1 F zdf1 f 11 e f1 F zdf1 x f 11 e f1 F zdf1 F zdf1 g d x F cf1 F tf1 f 1 e f1 F zdf1 x x d s x d s x 13

147 f soi1 e f1 F zdf1 V' f1 xe f F zdf F cf F tf V f xe f F zdf F cf F tf f soi1 e f1 F zdf1 x b f1 e f1 f soi1 e f1 F zdf1 f soi1 e f1 F zdf1 x b f1 e f1 g d x F cf1 f soi1 e f1 F zdf1 f soi1 e f1 F zdf1 x b f1 e f1 F zdf1 g d x F cf1 f soi1 e f1 F zdf1 x f soi1 e f1 F zdf1 x b f1 e f1 F zdf1 g d x F cf1 F tf1 f soi1 e f1 F zdf1 b f1 e f1 F zdf1 F cf1 F tf1 g d x f 1 e f F zdf x f 1 e f F zdf f e f F zdf f 1 e f F zdf x f 1 e f F zdf g d x F cf f e f F zdf x x x g d x d s x x x d s x d s b f1 e f1 x g d x x d s x b f1 e f1 d s x d s x f 1 e f F zdf x f 1 e f F zdf F zdf g d x F cf f e f F zdf f 1 e f F zdf x f 1 e f F zdf F zdf g d x F cf F tf f e f F zdf x x d s x d s x 133

148 f soi e f F zdf V' f xe f F zdf F cf F tf V fat1 xe f F zdf F cf F tf f soi e f F zdf x b f e f f soi e f F zdf f soi e f F zdf x b f e f g d x F cf f soi e f F zdf f soi e f F zdf x b f e f F zdf g d x F cf f soi e f F zdf x f soi e f F zdf x b f e f F zdf g d x F cf F tf f soi e f F zdf b f e f F zdf F cf F tf g d x x x x g d x d s x V f1 xe f F zdf F cf F tf b f e f b f e f V fat xe f F zdf F cf F tf V' f1 x e f F zdf F cf F tf V f xe f F zdf F cf F tf b f e f b f e f V' f x e f F zdf F cf F tf d s x d s b f e f x x d s x b f e f Minimum shear force in section 0-3, fatigue 134

149 Shear force [MN/m] Fatigue oading max/min shear force V.f(x) V.f1(x) [m] G.4 Fatigue in U-bows Fatigue U-bow [EN : ] a ubow 100mm d sr 4m d 1 340mm ϕ Ubow 5mm 1 I 0 4 π r 4 r 1 d sr 4m 135

150 r d sr d 1.17 m r 1 I 0 d sr d m π 4 4 r 4 r m 4 I 0 W annuus 3.966m 3 r σ mean.pos.fat1 σ mean.pos.fat σ mean.neg.fat1 F zdf1 πd sr d 1 F zdf πd sr d 1 F zdf1 πd sr d 1 M df1 r I 1 0 M df r I 1 0 M df1 r I 1 0 F zdf1 πd sr d 1 M df1 1 r I 0 F zdf M df r πd sr d 1 I 0 F zdf1 M df1 r πd sr d 1 I 0 F zdf M df F zdf M df σ mean.neg.fat r πd sr d 1 I 1 r 0 πd sr d 1 I 0 σ mean.pos.fat σ mean.pos.fat1 a ubow d 1 Δσ Ubow.pos πϕ Ubow 4 σ mean.neg.fat σ mean.neg.fat1 a ubow d 1 Δσ Ubow.neg πϕ Ubow max max Δσ Ubow.pos Δσ Ubow.neg D 600mm Bending diameter D ζ ϕ Ubow MPa MPa Δσ Rsk 16.5MPaζ MPa γ F.fat 1.0 γ s.fat Reduction factor due to bent reinforcement bars 136

151 N Ubowk Δσ Rsk γ s.fat γ F.fat Δσ Ubow.posk Δσ Rsk γ s.fat γ F.fat Δσ Ubow.posk 5 9 Δσ Rsk γ s.fat Δσ Rsk γ s.fat γ F.fat Δσ Ubow.posk γ F.fat Δσ Ubow.posk N Ubowk n k... N Ubowk D Ubow N Ubowk k 137

152 H Utiisation degree Utiisation degree of bending reinforcement top (o) and bottom (u) UR b.u % UR b.o Utiisation degree of shear capacity without shear reinforcement UR shear.vrdc m% Utiisation degree of shear capacity with shear reinforcement spacing 500mm UR shear % Utiisation degree of bending in star reinforcement UR b.star m % % Utiisation degree of U-bow reinforcement, tensie side (t), compressive side (c) Compressive: UR shear.ubow % Tensie: UR shear.ubow % Utiisation degree of crack width in the dferent sections top (o) and bottom (u) UR crack.width.u % UR crack.width.o % Utiisation degree of compressed concrete under the stee ring UR cc.ring % 138

153 Utiisation degree of bending reinforcement top (o) and bottom (u) for fatigue oading Equivaent oad UR fat.b.u UR fat.b.o % % ac damage summation D Upos D Oneg % % Utiisation degree of compressed concrete top (o) and bottom (u) for fatigue oading UR fat.c.u % UR fat.c.o % Utiisation degree of compressed concrete under stee ring for fatigue oading UR fat.cc.ring % Utiisation degree of shear reinforcement cosest to the stee ring for fatigue oading Equivaent oad ac damage summation U fat.ubow % D Ubow % Utiisation degree of star reinforcement for fatigue oading Equivaent oad U fat.star % ac damage summation D star % 139

154 I Fatigue Loads Load nr Neq= 10^7 m= 7 Sr,Mi= (sum(ni*sr,mi^ m/10^7) Sr,Fi= =(sum(ni*sr,fi ^m/10^7) Sr,Fi=Fxy [kn] Sr,Mi=Mx y [knm] ni n acc ,7656E+6 8,49655E ,1459E+6 8,85165E ,1361E+6 4,06934E ,4883E+6 3,85034E ,8319E+6 1,8176E ,8644E+6 1,3171E ,56779E+6 9,15875E ,3690E+5 1,91501E ,353E+5,143E ,77597E+6 9,048E ,71335E+6 6,5117E ,48747E+5 9,58743E ,08719E+5 4,4597E ,49677E+6 6,711E ,91613E+6 5,15046E ,71889E+5 1,60464E ,807E+6 6,73553E ,98617E+5 7,13174E ,07834E+5,57535E ,303E+6 3,7435E ,34748E+5 1,73363E ,3439E+6 4,1464E ,9695E+5 5,37849E ,8118E+6 5,59996E ,8067E+6 5,58414E ,7953E+5 3,95477E ,4643E+5 3,301E ,97797E+5 9,3874E ,34893E+6 4,83595E ,69007E+5 1,50768E ,5998E+5 3,33508E ,1095E+6 3,61685E ,47165E+5 7,97504E ,7667E+6 5,5719E ,57608E+5 8,3101E ,5555E+4 1,460E ,33761E+6 4,93105E ,1487E+6 3,71657E ,99845E+5 3,70671E ,79759E+6 5,947E ,85875E+6 6,14957E ,63179E+5,0568E

155 ,04505E+6 3,0033E ,93556E+5 1,13743E ,94119E+5 5,6107E ,6304E+5 1,53706E ,13694E+6 6,3041E ,58496E+5,8763E ,6117E+5 1,06549E ,916E+6 5,66893E ,64911E+5 1,66653E ,61703E+5,4708E ,4781E+5 1,0451E ,98747E+5,0481E ,933E+6 4,957E ,77576E+6 4,5533E ,6116E+6 1,18493E ,1576E+4,9687E ,59749E+5 4,0961E ,445E+5,46366E ,6313E+6 5,9509E ,9153E+6 6,60744E ,89418E+6 7,6373E ,9349E+5 1,56649E ,13474E+6 4,808E ,5486E+5 1,0617E ,83873E+6 4,79937E ,71413E+6,03054E ,13495E+6 6,45668E ,38877E+6 4,000E ,73541E+6 1,1969E ,41111E+6 1,0316E ,4614E+5 3,76867E ,79E+6 1,13781E ,10784E+5 8,757E ,978E+6 5,537E ,169E+6 1,61956E ,393E+5 1,95898E ,0598E+6 1,33694E ,85631E+6 1,8381E ,10675E+6 5,60407E ,0579E+6 3,16797E ,51897E+6 6,69563E ,59996E+6 4,583E ,31519E+6,47606E ,9544E+6,968E ,0497E+6 8,04065E ,7317E+6 1,5366E ,80369E+6 4,79437E ,4019E+6 1,05394E ,51398E+4 1,98573E ,8487E+4 4,88561E

156 ,3784E+6 1,00714E ,5494E+7 4,1914E ,8994E+6 1,874E ,3307E+7 3,3495E ,09198E+4 1,86751E ,37681E+6 6,45637E ,7368E+6 5,5536E ,59735E+7 3,6909E ,39915E+6 1,7103E ,6747E+6 7,6084E ,89789E+6 7,86664E ,65999E+5 5,31441E ,66593E+7 4,63009E ,8430E+6,45773E ,09149E+6 1,97093E ,55483E+6,65556E ,5393E+6 3,5991E ,5909E+6 5,30405E ,69339E+6 4,7058E ,4956E+7 6,75155E ,07383E+6 1,40998E ,3706E+6,03613E ,7193E+6 7,8443E ,905E+6,53603E ,648E+5 5,5197E ,87E+7 4,37843E ,56091E+6,9145E ,77675E+6 3,3608E ,9659E+6 1,1308E ,53517E+6 1,55077E ,48747E+6,903E ,63368E+7 5,5865E ,141E+6 1,4105E ,5854E+5 1,1144E ,97086E+6,46036E ,43598E+6,716E ,33598E+6 9,98945E ,18989E+7 5,08836E ,70318E+6,43888E ,7331E+6,3909E ,966E+5 1,16993E ,793E+7 7,45513E ,3353E+6 5,77787E ,40449E+6 1,8043E ,78883E+5,53787E ,1443E+7 5,15331E ,16743E+6 9,76074E ,5546E+6 1,6006E ,61393E+5 9,6875E ,4594E+7 6,66794E+14

157 ,8845E+6 1,31696E ,797E+5 3,07473E ,808E+6,3563E ,7167E+4,7691E ,66114E+6,7031E ,17133E+6 3,56477E ,531E+6 6,085E ,1674E+6 1,53717E ,4918E+6,14963E ,1647E+6 9,566E ,03797E+6 5,8968E ,50134E+6,5041E ,44908E+5 3,6370E ,6148E+5 4,9343E ,19448E+5 8,3518E ,0991E+6 3,80E ,0909E+5 8,70678E ,41417E+6 3,18141E ,886E+6 1,0074E ,48388E+4 8,13861E ,3359E+6 1,84377E ,8958E+6 1,331E ,1158E+6,317E ,6061E+6 8,97307E ,574E+6 5,56646E ,48074E+6 1,5411E ,15167E+6 1,81696E ,76873E+6,1769E ,3997E+5 1,70005E ,3536E+6 7,59183E ,8877E+5 5,46188E ,7537E+6,88789E ,07704E+6 5,1316E ,38044E+4 3,4105E ,97037E+6 1,8883E ,6688E+6 1,05747E ,5891E+5 5,57344E ,05074E+6 7,15656E ,70198E+6 1,84031E ,1915E+6 5,85318E ,4367E+4 1,7468E ,711E+5 8,99845E ,6465E+6,56595E ,09444E+5,89E ,4536E+5 9,5755E ,8737E+5 7,1408E ,39583E+6 1,0731E ,50365E+5 4,87113E ,13167E+6 6,4854E ,9444E+6 1,56684E

158 ,56914E+4 1,6443E ,6453E+5,4631E ,7833E+5,96447E ,7133E+6 1,5133E ,44736E+4 9,33917E ,04896E+5 5,19365E ,56717E+5 3,35971E ,6815E+5 8,1771E ,19555E+5 1,55389E ,46034E+5 3,15678E ,0196E+6 1,07594E ,95584E+4 1,05095E ,31188E+6 9,7704E ,13694E+5 8,46737E ,77098E+5 3,15667E ,393E+5,65343E ,76554E+5 8,13106E ,5068E+4 3,968E ,08E+5 4,19535E ,1637E+ 1,19833E ,98534E+5 7,10973E ,80683E+5 3,3894E ,47381E+5,1606E ,31115E+3 1,3366E ,3466E+5 4,4704E ,45978E+5 5,56587E ,9574E+4 6,7848E ,67675E+4 5,48E ,511E+5,81739E ,98475E+5 4,4517E ,49847E+4 3,18508E ,5731E+5 1,8995E ,6006E+4 6,94138E ,45653E+5 4,840E ,38999E+4 1,13651E ,17954E+5 1,7749E ,14454E+4 1,387E ,3795E+5,0796E ,7887E+5 1,14394E ,318E+4,8978E ,3833E+5,41375E ,68319E+4 6,4677E ,0044E+4 5,43443E ,558E+4 1,35879E ,01063E+4,8638E ,39088E+3 5,40804E ,80539E+4 3,47699E ,53036E+4 1,89673E ,03416E+4 1,405E ,64994E+4 7,48554E+1

159 ,04888E+4 3,30498E ,0389E+4 4,4888E ,764E+3,56179E ,05434E+4 8,45069E ,06841E+4,14795E ,84383E+3 1,68068E ,05E+08 8,15438E+3 4,81917E ,13E+08 6,19071E+3 3,65866E ,16E+08,34849E+3 7,04598E ,E+08 3,5865E+3 1,05867E ,38E+08 4,666E+3 4,57758E ,44E+08 1,3998E+3 1,5031E ,48E+08 7,78816E+ 3,45501E ,55E+08 1,7131E+3 7,63615E ,56E+08 4,07665E ,76E+08 9,88461E+ 8,88531E ,85E+08 4,06093E+ 3,65039E ,9E+08 3,38405E+ 3,04194E ,96E+08,69756E+1 1,81415E ,17E+08 1,3539E+ 9,105E ,9E+08 7,86559E+1 1,66564E ,5E+08 1,3888E+,94101E ,51E+08,8971E ,66E+08 5,5837E+0,01488E ,86E+08 7,84074E+0,893E ,98E+08 4,6316E ,5E+08 1,01571E+1 1,1386E ,58E+08 1,637E+1 1,41569E ,63E+08 1,9311E ,67E+08 1,14559E ,9E+08 7,4897E ,8E+08 1,0515E ,81E+08 1,5837E ,1E+08 1,16411E ,51E+08 8,9113E ,55E+08 3,06986E E+08 1,3396E ,37E+08 1,08089E ,8 sum Sr,Xi= 6,44495E+8,3446E+16 Mxyeq=sum(S.Mi)^(1/m)= 13049,76938 Fxyeq=sum(S.Fi)^(1/m)= 18,

160 J Sections of strut-and-tie mode 1 A-A B-B C-C D-D E-E 146

161 F-F G-G H-H I-I

162 C* B* A A* B* C* 148