MASTER'S THESIS. Design of Wind Turbine Foundation Slabs

Transcription

1 MASTER'S THESIS 2008:128 CIV Design of Wind Turbine Foundation Slabs Pekka Maunu Luleå University of Technology MSc Programmes in Engineering Civil and mining Engineering Department of Civil and Environmental Engineering Division of Structural Engineering 2008:128 CIV - ISSN: ISRN: LTU-EX--08/128--SE

2 Design of Wind Turbine Foundation Slabs Pekka Maunu

3 Acknowledgements This thesis, submitted for the Degree of Master of Science at Luleå University of Technology, is carried out at the Institute of Concrete Structures at Hamburg University of Technology. I would like express my utmost gratitude to my supervisor Prof G. Rombach for all the help and good will, and for providing me the opportunity to prepare the thesis at the Institute. My sincere thanks also go to Mr S. Latte for his invaluable guidance and expertise in the field of reinforced concrete; the same goes for the examiner of the thesis, Prof J.-E. Jonasson from Luleå University of Technology. I would also like to direct special thanks to Prof L. Bernspång for always being there to guide me through my studies in Luleå. Thanks also for the comments regarding this work! Finally, thanks to my family and friends for making all this possible and even enjoyable! Hamburg, ii

4 Abstract In this study the structural behaviour of wind turbine foundation slabs is analysed with various numerical and analytical models. The studied methods include models suitable for hand-calculations, finite element models with plate elements resting on springs as well as three dimensional models of both the foundation slab and the soil. Linear elastic as well as nonlinear behaviour including cracking of concrete and the complex load transfer from the tower into the foundation through a steel ring is considered in the study. The elastic analyses show, for example, that whereas in a concentrically loaded foundation slab a significant part of the load is carried through diagonal compression struts thus resulting in less flexure than what was found with the FE-models, the largest section forces and moments in a slab subjected to large overturning moment are obtained with a three-dimensional FE-model of both the slab and the underlying soil; i.e. the section forces increase together with the accuracy of the model. An important issue when designing members according to nonlinear analyses is to consider proper choice of material parameters. The results of a nonlinear plate element analysis verify the assumption that considerable redistribution of the section forces takes place due to flexural cracking of concrete. However, because of the large amount of simplifications of a simple plate element model no major conclusions of the structural behaviour should be made. A three-dimensional elastic analysis of a typical wind turbine foundation slab considering the complex load transfer through a steel ring reveals that the global flexural behaviour of the structure can be modelled sufficiently well by simpler models. This model, however, yields the largest section forces and moments; this has to be considered when simplifications are made. Additionally, the high local stress concentrations and the relative movement of the steel ring anchorage have to be taken into consideration when designing the reinforcement. A complete, three-dimensional nonlinear analysis of the foundation slab shows that the steel ring anchorage in the slab is the most critical part of the structure. iii

5 Contents Chapter 1 Introduction General Objective of Study Scope of Thesis... 3 Chapter 2 Background Wind turbine foundation slabs Structural design principles for foundation slabs Soil structure interaction Limit state verifications... 9 Chapter 3 Elastic analysis of foundation slab Foundation slab subjected to concentric load Analysis assuming uniform soil pressure distribution Finite element analysis with plate elements Design with strut and tie models Foundation slab subjected to large overturning moment Analysis assuming linear soil pressure distribution Finite element analysis with plate elements Three-dimensional finite element analysis Summary of results Summary of Chapter Chapter 4 onlinear behaviour of reinforced concrete Material model for reinforced concrete Concrete Reinforcement steel Model verification Design methods to nonlinear analyses iv

6 4.3 Nonlinear analysis of the foundation slabs Summary of Chapter Chapter 5 Three-dimensional analysis and design of a typical wind turbine foundation slab Steel ring concrete slab interaction Three-dimensional model of the structure Results of elastic analysis Nonlinear analysis Material model Discrete modelling of reinforcement Results Particularities concerning crack width limitation Summary and conclusions References v

7 Chapter 1 Introduction 1.1 General The utilisation of wind as an energy resource has been gaining popularity among decision makers for the last years not least due to the ever growing demand of sustainable development. Over the past decade wind energy was the second largest contributor to new power capacity in the EU; this translates into some 30% share of the net increase in capacity. /14/ As with all developing technologies, also wind turbines have gone a long road up until now regarding nominal capacity and consequently the size of the facility itself. (fig. 1) From a structural point of view this means that the acting loads on the system have increased in par thus requiring more thought in how the required structural safety can be provided. It is, naturally, most likely that this development will continue still. Figure 1. Development of wind turbine size and nominal capacity from 1980 to /15/ 1

8 Wind turbines are subjected to loads and stresses of very specific nature. On one hand, the wind itself acts in an unpredictable and varying manner thereby creating an environment prone to material fatigue. This applies also to wave loads induced by swell, ice loads etc. for off-shore wind turbines. On the other hand, as the facilities grow larger they also become more affected by a complex aeroelastic interplay involving vibrations and resonances creating large dynamic load components on the structure. /20/ From this load spectrum develops also the problematic of designing the foundation structure of a wind turbine. Hub heights of more than 100 metres, say, transfer a major eccentric load to the foundation due to a massive overturning moment and in relation a small vertical force (as the most common type of turbine tower is a light-weight steel tube). On-shore wind turbines are typically founded on massive cast-in-situ reinforced concrete slabs, in which the present study is concentrated, or alternatively, in the case of poor soil conditions, on combined slab and pile systems. For off-shore facilities the aforementioned additional load cases due to wave and ice forces, for example, place even harder requirements for the foundation structure. Common foundation types for off-shore wind turbines are the so called Monopile (steel tube driven into the ground), the gravity foundation made primarily of reinforced concrete, and the Tripod foundation whose three legs support the tower, as the name implies. /15/; /23/ 1.2 Objective of Study The design of slab foundations for wind turbines is mostly done manually using several simplifications and assumptions. Illustrating to the problematic is, for example, the fact that, say, 2500 ton foundation slab supporting a wind turbine is traditionally designed using the same methods and suppositions as a simple column footing which needs to resist a loading of a completely different nature. Typically, the soil stiffness as well as the thickness of the slab is neglected in an analysis; moreover the complex load transfer from the tower into the concrete foundation through a steel ring is not considered at all. The main purpose of this study, therefore, is to estimate the forces in flexural and shear reinforcement of typical foundation slab based on linear elastic behaviour as well as nonlinear behaviour due to the steel ring concrete interaction and cracking of concrete. 2

9 1.3 Scope of Thesis The remainder of this thesis is divided into four main chapters. In Chapter 2 a brief background information of wind turbine foundation slabs regarding design and construction is presented. The fundamentals of modelling the soil structure interaction are given, and the required limit state verifications are discussed briefly. Chapter 3 compares the results of various numerical and analytical methods to calculate member forces in typical slab foundations. Two slabs with a different thickness are considered in the analysis; first the slabs are subjected to concentric normal force only, after which a more realistic extreme load case is addressed. Several modelling simplifications are made; e.g. the complex load transfer from the tower into the foundation slab is idealised by a rectangular loaded area. Furthermore only elastic material behaviour is considered in the analysis. As an introduction to physical nonlinearity of reinforced concrete, Chapter 4 provides a material model used for concrete and reinforcing steel. The model is tested first by recalculating a documented experiment done with a simply supported beam; afterwards it is applied in a practical analysis of the aforementioned foundation slabs. Chapter 5 presents a complete, three-dimension model of the slab and the steel ring interface. Both elastic and nonlinear behaviour of reinforced concrete is considered in the analysis. Based on the results a design for the reinforcement is proposed; additionally, crack width calculations are carried out for supplementary surface reinforcement due to hydration-induced restraint common for a massive foundation slab. 3

10 Chapter 2 Background 2.1 Wind turbine foundation slabs Slab foundations for wind turbines are usually rectangular, circular or octagonal in form. The advantage of circular or octagonal slabs comes from the design of main flexural reinforcement; at least four reinforcement layers in the bottom surface can be provided which follow the principal bending moments better than an orthogonal reinforcement mesh. A downside is the more involved construction including many reinforcement positions and complex formwork. Therefore it is often found more economic to build a simple rectangular slab. Figure 2 shows such a wind turbine foundation slab in construction stage. Figure 2. Reinforcement in a wind turbine foundation slab. ( 4

11 The global dimensions of a wind turbine foundation slab are above all governed by normative regulations regarding safety against overturning /15/; as a rule, the foundation slabs are always subjected to extremely eccentric loading and have to be designed as such. Other soil stability related issues, such as substantial pore water pressure under the foundation, can also emerge as governing factors regarding the dimensions of the slab. Figure 3 presents a case where the rapidly increasing soil contact pressure due to the eccentric loading has resulted in subgrade failure and consequently in overturning of the whole facility. Figure 3. Fallen wind turbine facility. ( Special consideration has to be given to the connection between a steel tower and the foundation to ensure proper load transfer between the tower and the slab foundation. Figure 4 illustrates three commonly used construction variants. /15/ The alternative a) presents a so-called double flange joint, where a massive I-girder bent to form a ring is cast inside the concrete. The steel tower is then attached to a special connection flange with pre-stressed bolts. Variant b) shows a similar type of construction, which comes to question with very thick foundations. Here care has to be taken in designing the required suspension reinforcement in order to transfer the forces to the slab s compression zone. Finally, alternative c) presents a connection through a pre-stressed 5

12 anchor bolt cage. A steel flange is embedded in the slab before concreting, and on top of the foundation another ring-shaped T-girder is placed; the bolts are then stressed against both flanges. Fastening of the steel tower follows in the same manner as with the previous variants. Careful execution of construction of the tower foundation joint has to be carried out; the joint has to provide the assumed fixity in horizontal and rotational directions used in the tower calculations. This means that relatively small allowable construction tolerances are to be used. Figure 4. Typical construction variants for the load transfer from tower into foundation. /15/ 2.2 Structural design principles for foundation slabs Soil structure interaction The structural design of a foundation slab is above all governed by the distribution of soil pressure under it. As the purpose of a foundation slab is to distribute the more or less concentrated load into a larger area so that the soil can carry it without extreme negative consequences (e.g. bearing failure of the soil, excessive settlement etc.) it is the resulting soil pressure i.e. contact pressure that causes the bending moments and shear forces in the slab. The form of the pressure distribution therefore has a decisive impact on the magnitude of the internal forces of the structure. 6

13 V V a) b) Figure 5. Soil pressure distributions under a rigid foundation. a) Small applied vertical load V, b) redistribution after soil plasticizing. For extremely rigid foundation slabs with an axisymmetric and relatively small load the soil pressure distribution can be assumed to be concave in form, with stress peaks at the foundation edges (fig. 5a). This distribution is valid only if the soil is assumed to have an elastic, isotropic behaviour, i.e. the soil is modelled as elastic, isotropic half-space, as first presented by Boussinesq in /7/ However when the load increases, the soil under the foundation edges plasticizes, thus being able to take gradually less and less stress as the plasticizing advances. This results in pressure concentration closer to the applied load, and therefore the soil pressure distribution takes a convex form as the load reaches the bearing capacity of the soil, according to Prandtl-Buisman (fig. 5b). /21/ However, modelling the complex elastic-plastic behaviour of the soil is often times too elaborate for structural design purposes and thus simplifications are made. LINEARLY VARYING SOIL PRESSURE DISTRIBUTION A simple model (and therefore suitable for hand calculations) of describing the distribution of soil pressure under a foundation slab is to assume that no interaction between the structure and the soil occurs. Use of the theory of elasticity for beams (e.g. σ = V / A M / W ) results in a linear soil pressure distribution that depends only 0 ± min/ max on the magnitude of the applied loads and on the surface area of the foundation. (fig. 6a) For smaller and in proportion somewhat stiff foundations (e.g. ordinary column 7

14 footings) this method is nevertheless a rather good approximation. For larger, flexible foundations under concentrated loads the linear soil pressure distribution leads to a conservative design, as the soil pressure concentrations under loads (and therefore the smaller resulting internal forces) are neglected. On the other hand, the linearity can also be on the dangerous side regarding design, for instance in the case of rigid, deep founded slabs and some continuous slab systems. /3/; /8/; /23/ V V M σ 0min σ 0max a) b) Figure 6. a) Model assuming linear soil pressure distribution; b) model based on the subgrade reaction modulus. MODULUS OF SUBGRADE REACTION One widely used method for a simple approximation of the structure soil interaction is to prescribe an elastic spring foundation underneath a foundation, which means, in mechanical sense, that the soil is represented by a series of vertical springs independent from each other (also known as the Winkler type spring foundation after the formulator). /19/; /34/ (fig. 6b) Hence the single parameter that describes the whole interaction between the structure and the soil is simply spring stiffness per unit area (so called modulus of subgrade reaction; c s ), i.e. the soil pressure is linearly proportional to the settlement ( σ 0 = c s s ). This method completely ignores the interplay between neighbouring soil elements and therefore doesn t result in a realistic soil deformation in many cases, although in the case of a single concentrated load acting on a footing the results agree quite well with more sophisticated methods. /30/ Moreover, it should be noted that the modulus of 8

15 subgrade reaction is not something that is purely determined by soil properties but depends on the whole system: magnitude and type of loading, dimensions of the foundation, stiffness of the soil etc. /23/ Therefore one can never fundamentally state a certain value for the modulus of subgrade reaction for a given type of soil. All the previous considered, problematic is then the determination of the modulus of subgrade reaction itself. The choice of the soil stiffness is a factor of importance in the design of a foundation; it is obvious, for example, that the bending moments resulting in a centrically loaded flexible slab resting on stiff springs can be considerably smaller than when softer springs had been evaluated, thus resulting in unsafe design. Anyhow, there exists numerous formulae in the literature (see e.g. /4/) for approximating the modulus of subgrade reaction; they are usually based on the stiffness modulus of the soil medium in question and the dimensions of the foundation. DISCRETE MODELLING OF SOIL BY THE FINITE ELEMENT METHOD The finite element method provides a means to model the behaviour of soil more accurately than the two previous models; instead of just issuing a one-dimensional stiffness for the soil, the soil medium itself can be modelled with discrete elements. Even if just elastic, isotropic soil behaviour is assumed (the parameters thus being the Young s modulus and the Poisson s ratio which, on the contrary to the bedding modulus, can be considered as soil characteristics) the structure soil interaction can be described more realistically than with the modulus of subgrade reaction. For instance, a foundation slab under a uniform load will not result in any member forces with an above introduced spring foundation, as the deformation of each individual spring will be the same; however a soil layer modelled with finite elements will take the continuity of the soil medium into consideration and consequently resulting in nonuniform deformation behaviour Limit state verifications In the ultimate limit state (ULS) slab foundations have to be verified against structural failure under extreme static loads; a dynamic analysis including fatigue calculations for both concrete and steel (sometimes referred to as fatigue limit state) has not been traditionally required even in the case of wind turbine foundations, which are subjected 9

16 to an extremely cyclic load spectrum. /20/ This repetitive nature of loading may increase the damage induced in a structure by accelerating crack propagation or the degradation of stiffness. /31/ Fatigue in reinforced concrete is a relatively new topic, and therefore not yet anchored in the practice. /24/ The research on fatigue has nevertheless been gaining interest in recent years, and one can only expect that fatigue assessment will become a standard verification in the near future. The most essential detail verifications in the ULS are Flexural resistance of both concrete and reinforcement Shear resistance with or without shear reinforcement (including punching) Examination of concentrated stresses anchorage, tensile splitting, local crushing etc. Detailed numerical analyses of problems where a suitable, simplified analytical model cannot be found The structure needs to as well be verified against adequate performance in the serviceability limit state (SLS). Typical verifications include Crack width limitation Settlement control as well as a deflection analysis in general Limitation of stresses to ensure sufficient durability of the structure Of these the limitation of crack width is usually most problematic to verify, as the magnitude of stresses induced by restraint due to hydration, for example, is relatively large for massive foundation slabs hence requiring often uneconomic amounts of supplementary reinforcement. Besides the pure limit state verifications, detailed design of reinforcement with corresponding reinforcement layouts is in many cases the most time consuming part of the design. Here a multitude of different issues have to be considered. These include adequate lap lengths and proper anchorage of the reinforcement (including shear reinforcement), consideration of allowable bends in the case of thick bars, as well as a 10

17 number of regulations concerning constructive (i.e. theoretically not required) reinforcement. 11

18 Chapter 3 Elastic analysis of foundation slab The aim of the present chapter is to compare various conventional analytical and numerical methods to calculate member forces in typical wind turbine foundation slabs. This analysis is based on linear elastic behaviour of construction materials and soil. At first the foundation slabs loaded only with a concentric normal force are inspected; this serves to establish the various methods of analysis, as well as pointing out some fundamental assumptions. After that, the actual problem of a large overturning moment in comparison to the magnitude of the normal force is introduced. 3.1 Foundation slab subjected to concentric load In reality the structure has a column with a circular, tubular cross section; however in this analysis it is idealised to a rectangular one (4 m x 4 m). Two slab alternatives with different thicknesses are studied. The slabs represent typical square foundations for some 100 m tall wind turbine tower. The system is presented in figure 7. The foundation is loaded with a concentric normal force, which corresponds to the design dead load from the wind turbine tower. h = 3,5 m (2,6 m) Idealised column c = 4 / 4 m b N k = 4025 kn d avg Concrete E = 29 GPa; v = 0,20 d cm avg = 342 cm (252 cm) γ = 1,35 for applied dead and live loads (ULS) b = 17,7 m Figure 7. System for the analysis. 12

19 3.1.1 Analysis assuming uniform soil pressure distribution The hand calculations are done according to the well established procedure presented in numerous design guides (e.g. /4/; /26/); this means that a uniform soil pressure distribution according to the theory of elastic beams independent of the soil properties is assumed. Furthermore, the thickness of the foundation slab has absolutely no effect on the magnitude or the distribution of the member forces; that is, the slab is assumed to be rigid. FLEXURAL ANALYSIS The total bending moment in one direction can be calculated from equilibrium conditions as db 1, ,7 M Ed = = = knm. 8 8 Lateral distribution of the bending moment can be done with a strip method of choice (see e.g. /18/) keeping in mind that the moment is concentrated mostly under the column region; for example, the maximum bending moment per unit width in this case will be 978 knm/m. It must be noted that the above calculation does not take into account the fact that a significant portion of the applied normal force is carried at the corners of a rectangular column (or at the perimeter of a circular one) (/13/) hence resulting in a smaller acting bending moment. SHEAR ANALYSIS A foundation slab supporting a concentrically placed column can theoretically fail like a wide beam (i.e. the critical section extends in a plane across the entire width of the slab) as well as through punching out a cone around the column. /26/ The so called beamaction shear failure is seldom governing the design; nevertheless it should be checked. Punching, on the other hand, is a complex phenomenon and the mechanism of failure is not involving merely shear transfer. Depending of loading and construction the failure can, apart from the tension strength of concrete being exceeded, develop from a failure of the compression zone, from a local bond failure in the flexural reinforcement or 13

20 because of inadequate anchorage of punching (shear) reinforcement. /9/ The design is therefore carried by evaluating a semi-experimentally determined equivalent shear force in particular critical peripheral sections. The critical beam-action shear force (fig. 8) is located at a section 1,0d away from the face of the column and it is assumed to spread uniformly across the whole width of the slab, as it would do in a wide beam. The shear force per unit width along the section is calculated as v Ed d 5434, h= 3,5; shear = ( b / 2 c / 2 d avg ) = (17,7 / 2 4 / 2 3,42) = 2 A 17,7 and similarly for the thinner slab as ved 5434, h= 2,6; shear = (17,7 / 2 4 / 2 2,52) = 2 17,7 75,1 kn/m. 59,5 kn/m, The shear force to represent punching is calculated at a peripheral section 1,5d away from the face of the column (u 1,5d ), with a subtraction of 50% of the upward soil pressure acting in the area within the perimeter (A 1,5d ) as prescribed in the German code DIN (2001) /11/: (fig. yyy) v 5434 d 0,5 A ,5 180,76 1,5d 2 A 17,7 = u 48,23 d Ed, h= 3,5; punching;1,5 d = = 1,5d ved ,5 121, ,7 = 39,75, h= 2,6; punching;1,5 d = 110,2 kn/m. 80,2 kn/m; This representation of punching check in DIN is derived from the equivalent check for flat floor slabs. Yet it has been shown that in the case of thick foundation slabs the inclination of the conical failure surface is much steeper than a critical section at 1,5d away from the face of the column would suggest (see e.g. /9/; /21/). The provision of allowable subtraction of only 50% of the favourable soil reaction under the punching cone is derived from this fact; i.e. to approximate the steeper crack inclination. 14

21 An alternative method in general more conservative but nevertheless straightforward would be simply to take the critical perimeter at 1,0d away from the face of the column, and to allow a 100% subtraction of the acting soil pressure within the resulting area. This approach has been proposed in recent research (/21/) as well. The resulting force is then equivalent to the principal shear force acting along the peripheral section allowing direct comparisons with numerical analyses as well, without the need of complicated and inaccurate integrations of the soil reaction. Having said the above, the punching shear force at 1,0d away from the face of the column equals to 5434 d ,5 d A1,0 d 2 17,7 v Ed, h= 3,5; punching = A = = 95,2 kn/m; u 37,49 1,0d ,3 2 17,7 v Ed, h= 2,6; punching = = 129,1 kn/m. 31,83 Tributary reaction for beam-action shear A 1,5d 1,5d 1,0d u 1,5d u 1,0d 33,7 45 Figure 8. Critical sections for beam-action shear and punching design. 15

22 3.1.2 Finite element analysis with plate elements The foundation slabs are modelled in Abaqus/Standard as linear elastic plate structures. The finite element mesh consists of rectangular 4-node plate elements with an approximate side length of 0,35 m. A spring surface support with a modulus of subgrade reaction of c s = 50 MN/m 3 is assumed for this analysis. As noted in ch the determination of a true value for the subgrade modulus is impossible as there exists no such thing; however the assumed value could represent dense sand under the slabs in question. Poisson s ratio for concrete is taken as 0,20. There are several ways of modelling the concentrated load transfer from a column into a slab. /30/ At first, one could just apply a point load to the centre node of the slab. Another method is to spread the concentrated load into an equivalent surface pressure, either over the column sectional area or under 45 to the mid-plane of the slab. Finally, a more or less rigid link can be created through kinematic coupling of a reference node (to which the point load is applied) and the surface that represents the column sectional area. (fig. 9a-d) a) b) c) d) Figure 9. Different ways of applying the column load. a) Point load; b) 4 x 4 m distributed load; c) under 45 distributed load; d) coupling of elements in the column region. FLEXURAL ANALYSIS Resulting bending moment distributions from the various models are presented in figures 11a. It can be immediately noted that a single point load should not be used in analysing a slab, as it gives a singularity peak in the bending moment distribution. Distributing the load over the column sectional area more than halves the aforementioned peak; a further 16

23 load distribution to the mid-plane of the slab reduces the bending moment even more. The coupling model situates in between the two load distribution methods. Regarding shear, the differences between the various load transfer models are somewhat negligible. On the contrary to the beam theory the finite element method produces different member forces for the two slabs due to differences in bending stiffness. For example, the peak bending moment with 4 m x 4 m pressure load is about 6% smaller in the 2,6 m thick slab (775 knm/m) than in the 3,5 m thick slab (822 knm/m). This means that because the flexural stresses in the thinner slab can carry a smaller amount of the applied load a greater amount is led directly into the supporting soil springs at the column region; i.e. the soil pressure distribution will be more concentrated under the column region. (fig. 10) The tendency is the same with the coupling model even though the peak values are equal in both slabs. These peaks are but singularities occurring at the corner nodes of the loaded area and in general should not be considered in design. 10,0 15,0 0 8,85 17,7 h=2,6 m h=3,5 m Uniform distribution 17,3 20,0 25,0 23,2 30,0 29,0 Figure 10. Soil pressure distribution (kpa) resulting from the column load under a cut along the slabs (4 x 4 m distributed load). 17

24 0 0 8,85 17, DAfStb Heft Point load 1000 Loaded area 4x4 m Loaded area 7,5x7,5 m 1200 Coupling of elements Point load, h=2,6 m 1400 Loaded area 4x4 m, h=2,6 m Loaded area 6,6x6,6 m, h=2,6m a) Coupling of elements, h=2,6 m ,5 124,1 120, ,8 93,3 92,4 b) Point load Loaded area 4x4 m Loaded area 7,5x7,5 m Coupling of elements Point load, h=2,6 m Loaded area 4x4 m, h=2,6 m Loaded area 6,6x6,6 m, h=2,6m Coupling of elements, h=2,6 m 0 8,85 17,7 Figure 11. a) Bending moment distribution (k m/m) near the column; b) principal shear force (k /m) across a section 1,0d from the face of the column. 18

25 The loaded area in the mid-plane of the slab under 45 is naturally smaller when the slab is thinner; thus the pressure and consequently the bending moment with the associated model will be larger (11%). SHEAR ANALYSIS While the hand calculation method assumes a constant shear force along a lateral section, the FE-analysis gives considerably higher local values in the middle of the section, whereas close to the edges the shear is almost negligible. (fig. 11b) This implicates evidently that the shear is not carried only by one-way action, but is distributed in a ring around the column; see fig. 12b. The distribution of principal compression stresses in the top surface is analogous to the shear force; there exists a compression ring around the column. (fig. 12a) It is obvious that the slabs would fail in punching rather than as a wide one-way spanning beam. Designing the slabs for beamaction shear (considering the slabs as a series of narrower strips of arbitrary width) against the local shear force peaks resulting from a FE-analysis, therefore, can not be recommended. A summary of results from the different analyses is presented in table 1. a) b) Figure 12. a) Distribution of principal compression stress and b) principal shear force in the top surface in a concentrically loaded foundation slab. 19

26 Method m Ed [knm/m] v Ed;1,0d [kn/m] Calculation by hand 978 (126%) 129,1 (99%) Point load 1900 (245%) 130,6 (100%) h = 2,6 m Loaded area 4x4 m 775 (100%) 130,5 (100%) Loaded area 6,6x6,6 m 520 (67%) 124,1 (95%) Coupling of elements 589 (76%) 120,7 (92%) Calculation by hand 978 (119%) 95,2 (98%) Point load 1950 (237%) 96,5 (100%) h = 3,5 m Loaded area 4x4 m 822 (100%) 96,8 (100%) Loaded area 7,5x7,5 m 493 (60%) 92,4 (95%) Coupling of elements 589 (72%) 93,3 (96%) Table 1. Summary of analysis results Design with strut and tie models As said, the hand calculations were based on the assumptions of beam theory, and the finite element analysis was performed using plate elements. These simplifications denote linear stress and strain states across the thickness of the slab an assumption which actually doesn t hold true for such massive structures as the foundation slabs in question. It is pointed out in /30/ that the column load is not carried only by flexure but also by diagonal compression stresses. Regarding the foundation slabs as wide beams a strut and tie model as illustrated in fig. 13 can be devised, for example. /28/ The column load is transferred to the ground 20

27 through compression struts at varying (to some extent arbitrary) angles. It follows then that the resultant tensile force in the bottom reinforcement in one direction in the column region equals to Ft 5434 = tan tan tan = 1727 kn. tan 70 Assuming an effective width of b eff = c+ 2d for the slabs, the tensile forces per unit length in one reinforcement direction will be 159 knm/m and 191 knm/m, respectively for the 3,5 m- and 2,6 m-thick slab. c N /3 d d Figure 13. A strut and tie model of the foundation slabs. The tensile forces in reinforcement from the bending moments resulting from the FEmodels are not at all explicit to determine, as the design of cross section is anyhow carried out assuming a cracked state and consequently the internal lever arm will not be fixed. However, assuming z 0, 9d yields values ranging from (excluding the point load models) kn/m and kn/m, respectively for the 3,5 m- and 2,6 m- thick slabs. Hence it seems that all the studied load transfer models produce results that lie more or less on the conservative side. 21

28 3.2 Foundation slab subjected to large overturning moment For this analysis the system is fundamentally the same as in the previous chapter; however a large overturning moment is introduced to combine with the column axial force. (Fig. 14) The loading represents the type of which a large wind turbine tower transfers into its foundation in extreme cases. For simplicity, only uniaxial bending is considered. The magnitude of the moment means that the dead weight of the slab has to resist the uplift of the base and consequently the overturning together with the column normal force thus contributing to the flexure. h = 3,5 m (2,6 m) y Idealised column c = 4 / 4 m x b b = 17,7 m N k = 4025 kn M y,k = knm d avg Concrete E = 29 GPa; v = 0,20 d cm avg = 342 cm (252 cm) γ = 1,35 for applied dead and live loads (ULS) Figure 14. System for the analysis large uniaxial overturning moment Analysis assuming linear soil pressure distribution There exist some methods suitable for hand calculations for the design of eccentrically loaded foundations. For example, the required member forces can be calculated assuming a linear, trapezoidal soil pressure distribution, or by approximating a constant soil pressure acting in a reduced contact area, see e.g. /22/ or /33/ for more details. Difficulties may arise when only part of the slab base has contact with underlying soil, i.e. a partial uplift occurs. This means that the soil pressure under the area in contact increases overproportionally. Consequently top reinforcement is also needed to resist the arising negative moment causing tension at the top surface of the slab. 22

29 FLEXURAL ANALYSIS The bending moments m Ed,x can be calculated by treating separately the symmetric load case, which is the column normal force creating a uniform soil pressure distribution under the foundation slab, and the asymmetric load case, which is the overturning moment resulting in a fictitious, trapezoidal soil pressure distribution. /33/ The bending moments from the symmetric part can be calculated as presented in Ch ; that is M SYMM = 1, ,7 / 8= knm. Because of the asymmetry of the second load case there exists a line of zero moment (i.e. hinge) in the centre of the slab. (Fig. 15) Therefore the overturning moment must be led equally to both halves of the foundation: M = 1, / 2= ± knm. ASYMM After adding the bending moments resulting from the two load cases there will appear a positive as well as a negative bending moment; the latter is needed to resist the fictitious tension created between the soil and the foundation. M = = knm; EG M = = knm. POS N M Line of zero moment + Soil pressure from symmetric load case Fictitious soil pressure from asymmetric load case Figure 15. Determining the bending moments in a foundation slab subjected to eccentric loading. For a foundation slab without piles the only entity that can create the required moment to resist the fictitious tension is the self weight of part of the slab behind the line of zero 23

30 moment; for instance, considering the 2,6 m thick slab, the moment caused by its self weight resisting the uplift is 2 M = 25 2,6 17,7 8,85 / 2= knm. SLAB It has to be pointed out that the design action of the slab s self weight is taken with a partial safety factor of 1,0; it is considered as a favourable action as it effectively reduces the eccentricity of the applied loads. As the self weight of the slab is not enough to counter the tension, the difference has to be carried in the other half of the foundation slab in addition to the moment M POS determined previously; i.e. the maximum bending moment in the 2,6 m thick slab will be M = ( ) knm. Ed, x,max = In this case the minimum moment is caused by the fully utilised self weight: M = knm. Ed, x,min It is then assumed that the positive flexure is carried by a substitute beam with a breadth of b eff = c+ 2d b where c means the width of the column and d the average effective depth of the slab. This corresponds to approximately 45 distribution of the forces inside the slab. For the negative flexure, it is suggested in /33/ that an effective width of two- to three-times the column width can be used. Looking again at the 2,6 m thick slab the following bending moments are finally obtained: m = 80961/(4+ 2 2,52) 8956 knm/m; Ed, x,max = m = 45055/(2 4) = 5632 knm/m. Ed, x,min Calculations for the 3,5 m thick slab are performed analogously; it follows then that the bending moments are as presented in table 2. 24

31 SHEAR ANALYSIS When such a large moment is being transferred from the column into the slab it is questionable if punching as presented in the case of concentrically loaded foundation slab is something that is worth looking into. There exists no more a continuous compression ring around the column as is the case with smaller eccentrities of the applied loads; therefore also the multi-axial stress conditions resulting in a higher resistance to failure are missing. Based on this statement it seems reasonable to design the foundation slabs against beam action shear and not against punching. Firstly, the design shear force acting along a section at a distance 1,0d from the face of the column could be calculated analogously to Ch keeping in mind that now the soil pressure distribution is trapezoidal (see fig. 8); i.e. this model would assume that the shear force distributes uniformly across the breadth of the slab. This assumption results in a design shear force of 522 kn/m in the 2,6 m thick slab and 437 kn/m in the 3,5 m thick slab. The shear resistance v Rd,ct of a cross section without shear reinforcement according to DIN would be around 530 kn/m and 700 kn/m for the 2,6 m and 3,5 m thick slabs, respectively, for a C30/37 concrete and for a longitudinal reinforcement ratio of 0,15%. There would thus be no need for shear reinforcement in the slabs. m Ed,x,max [knm/m] m Ed,x,min [knm/m] v Ed [kn/m] h = 2,6 m h = 3,5 m Table 2. Member forces in the slabs assuming linearly varying soil pressure distribution. Alternatively a so-called sector model can be used for the shear design of foundation slabs. /12/; /13/; /29/ In such a model it is assumed that the shear force occurring in the most stressed sector of the slab governs the failure mechanism; i.e. it is assumed that the shear force is not uniform across the breadth of the slab. (fig. 15) 25

32 Tributary reaction for shear 1,0d Critical section u crit 45 σ 0 σ u,crit σ max Figure 15. Sector model for punching shear analysis after /13/. Critical shear force according to the sector model as in fig. 15 is calculated exemplarily for the 2,6 m thick slab in the following. Length of the critical section u crit : u = 2 (2+ 2,52) = 9,0 m crit Soil pressure resulting from the applied loads at different sections (see fig. 15): 2 3 σ = 1, /17,7 + 1, /17,7 154 kpa max = σ = (8, ,52) ( ) /17, kpa u, crit = σ = 8,85 ( ) /17, kpa 0 = Tributary soil reaction for shear: 26

33 ,7 (154 18) 17,7 18 9,0 (87 18) 9,0 R = + = 7215 kn Shear force acting along the critical section: v = 7215 / 9,0= 802 kn/m Ed With analogous calculations for the 3,5 m thick slab the shear force equals to 590 kn/m. Compared with the uniform distribution of shear force across the whole breadth of the slabs it is clear that now the thinner slab would require some amount of transversal reinforcement. However, the 3,5 m thick slab could still be verified without reinforcement, although the sector model results in some 35% larger design shear force Finite element analysis with plate elements The system parameters are the same as in Ch except for the loading. The total column load including the overturning moment is applied in three different ways: As an equivalent trapezoidal pressure over the column sectional area; as an equivalent trapezoidal pressure spread further to the mid-plane of the slab under 45 ; and finally as a point load and a point moment with kinematic coupling of the elements in the column region. (fig. 16) In addition, the soil springs are defined to be very soft in tension, thus allowing the possible uplift to occur realistically without the springs taking any significant amount of tension. Loaded area Figure 16. Methods to apply the loading. 27

34 FLEXURAL ANALYSIS The resulting bending moments along the slabs are shown in figure 17. Differences between the two slabs are somewhat small; the 2,6 m thick slab tends to gather a slightly larger maximum moment than the 3,5 m thick slab, with consequently smaller minimum bending moment peak. The exception are the models where it is assumed that the acting loads spread to the mid-plane of the slabs, with which also the minimum moment is greater in the thinner slab. This is explained by the smaller area of the pressure trapezoid Coupling of elements; h=2,6 m Pressure trapezoid 4x4 m; h=2,6 m Pressure trapezoid 6,6x6,6 m; h=2,6 m Coupling of elements Pressure trapezoid 4x4 m Pressure trapezoid 7,5x7,5 m 0 0 8,85 17, Figure 17. Bending moment m x (k m/m) along the foundation slabs. SHEAR ANALYSIS Regarding shear force, the different loading models give this time significantly varying results. (fig. 18) The distribution of shear force across the breadth of the slabs is not uniform, as regardless of the overturning moment acting in only one direction the slabs 28

35 bend also in the perpendicular direction. Largest shear forces are obtained with the 4 x 4 m pressure trapezoid and lowest when the loading is spread into the mid-plane of the slabs. The critical shear forces according to the FE-models are up to 60% higher than what was obtained with the sector model in the previous chapter; therefore a design using the FE-results would certainly be more conservative Coupling of elements; h=2,6 m Pressure trapezoid 4x4 m; h=2,6 m Pressure trapezoid 6,6x6,6 m; h=2,6 m Coupling of elements Pressure trapezoid 4x4 m Pressure trapezoid 7,5x7,5 m 0 0 8,85 17,7 Figure 18. Principal shear force (k /m) across a lateral section 1,0d away from the face of the column Three-dimensional finite element analysis To answer the question of which of the previously studied plate element models best represents realistic behaviour of a massive foundation slab subjected to a large overturning moment, a three-dimensional model of the 3,5 m thick slab is analysed. In this analysis also the soil is modelled discretely with volumetric elements. The soil medium is modelled so as to allow the stresses to be distributed wide enough in it. 29

36 With a Young s modulus of 200 MPa and a Poisson s ratio of 0,30 the elastic soil halfspace results in settlements similar in magnitude as the previous soil spring model; these elasticity parameters are also reasonable regarding the previous assumption of dense sand forming the primary layer of soil. It is thus safe to assume that the system is comparable to the soil spring model. A schematic illustration of the model geometry with the FE-mesh is shown in fig. 19. Due to symmetry only half of the system needs to be modelled, thus saving computational time. 2,5b b=17,7m 10 m 5b Figure 19. Model geometry and FE-mesh. The interface between the slab and soil is modelled using surface contact interaction properties available in Abaqus/Standard. This allows the slab to lift up without tension being created at the interface; the slab is also free to displace in the horizontal direction. The loading is applied on top of the slab as a pressure trapezoid over the idealised column area. 30

37 The first thing to be observed with a volumetric soil model is the difference in soil pressure distribution compared with the soil spring model. (fig. 20a and b) The elastic soil half space results in pressure concentrations at the edges of the slab (see also Ch. 2.2). Furthermore, as the neighbouring soil elements interact with each other in all directions as opposed to the spring model, the soil outside the slab boundaries is also being affected by the settlement depression. Figure 21 shows the deformed mesh of the system. a) -2,00 17, Soil as volume elements; h=3,5 m Soil as springs; h=3,5 m b) 300 Figure 20. a) Distribution of soil pressure beneath the 3,5 m thick slab according to soil spring model (left) and volumetric soil model (right). b) Soil pressure distributions (kpa) under a cut along the slabs. 31

38 Figure 21. Deformed FE-mesh of the model. FLEXURAL ANALYSIS Figure 22 illustrates the flow of forces in the foundation slab with this simplified load transfer model. The nonlinear distribution of the horizontal stress component can also be seen. Integrating the stresses multiplied by lever arm z from the neutral axis z 0 over the cross section height yields the bending moment acting in the corresponding direction: m x = z 0 h z σ zdz. 0 x Along the slab a bending moment curve as shown in fig. 23a is then obtained. The maximum bending moment resulting from this model is m Ed,x,max = 7568 knm/m, and the minimum m Ed,x,min = knm/m. These values agree surprisingly well with bending moments from the plate element model using the same method of load transfer (i.e. 4 m x 4 m pressure trapezoid); differences are less than 10% (m Ed,x,max = 7039 knm/m and m Ed,x,min = knm/m). Greatest underestimation of the member forces clearly results when assuming that the column normal force and the overturning moment act through a pressure trapezoid distributed to the mid-plane of a plate element model; the bending moments are less than half of the ones obtained with this three-dimensional analysis. Load spread to the mid-plane should therefore not be used for designing a foundation slab subjected to a 32

39 large overturning moment even though for concentric loading it seems to best reflect the true behaviour. SHEAR ANALYSIS Analogously to the bending moments, also the shear force is obtained through an integration of the principal shear stress over a cross section height: v= σ xz + σ yz dz. h Across the width of the slab at a distance 1,0d away from the edge of the loaded area a shear force distribution as presented in fig. 23b is then found. The resulting peak of v Ed = 958 kn/m is again best represented by the plate element model with 4 m x 4 m loaded area for the pressure trapezoid (v Ed = 868 kn/m). The difference is also this time approximately 10%. Load transfer model with a pressure trapezoid spread further to the mid-plane of a plate element model underestimates the maximum shear force almost 25% Figure 22. Principal stress field and distribution of horizontal stresses (with top and bottom surface stresses in kpa) in the foundation slab. 33

40 ,85 17, D; h=3,5 m 6000 a) D; h=3,5 m 200 b) 0 0 8,85 Figure 23. a) Bending moment m x (k m/m) and b) principal shear force (k /m) across a lateral section 1,0d away from the face of the column in the 3,5 m thick slab. (Three-dimensional modelling of structure and soil) 34