Design of Wind Turbine Foundation Slabs

2008:128 CIV

MASTER'S THESIS

Design of Wind Turbine Foundation Slabs

 Pekka Maunu

Luleå University of Technology MSc Programmes in Engineering  Civil and mining Engineering Department Department of Civil and Environmental Environmental Engineering Division of Structural Engineering 2008:128 CIV - ISSN: 1402-1617 - ISRN: LTU-EX--08/128--SE

Design of Wind Turbine Foundation Slabs

Pekka Maunu

Design of Wind Turbine Foundation Slabs

Pekka Maunu

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, 23.5.2008

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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.

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Contents Chapter 1 Introduction .................................................................................................................... 1

1.1 General.................................................................................................................... 1 1.2 Objective of Study .................................................................................................. 2 1.3 Scope of Thesis....................................................................................................... 3 Chapter 2 Background..................................................................................................................... 4

2.1 Wind turbine foundation slabs................................................................................ 4 2.2 Structural design principles for foundation slabs ................................................... 6 2.2.1 Soil – structure interaction............................................................................... 6 2.2.2 Limit state verifications ................................................................................... 9 Chapter 3 Elastic analysis of foundation slab .............................................................................. 12

3.1 Foundation slab subjected to concentric load....................................................... 12 3.1.1 Analysis assuming uniform soil pressure distribution................................... 13 3.1.2 Finite element analysis with plate elements .................................................. 16 3.1.3 Design with strut and tie –models ................................................................. 20 3.2 Foundation slab subjected to large overturning moment...................................... 22 3.2.1 Analysis assuming linear soil pressure distribution ...................................... 22 3.2.3 Finite element analysis with plate elements .................................................. 27 3.2.4 Three-dimensional finite element analysis .................................................... 29 3.2.5 Summary of results ........................................................................................ 35 3.3 Summary of Chapter 3.......................................................................................... 36 Chapter 4 onlinear behaviour of reinforced concrete .............................................................. 37

4.1 Material model for reinforced concrete ................................................................ 37 4.1.1 Concrete......................................................................................................... 37 4.1.2 Reinforcement steel ....................................................................................... 39 4.1.3 Model verification ......................................................................................... 40 4.2 Design methods to nonlinear analyses.................................................................. 43

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4.3 Nonlinear analysis of the foundation slabs........................................................... 45 4.4 Summary of Chapter 4.......................................................................................... 50 Chapter 5 Three-dimensional analysis and design of a typical wind turbine foundation slab 51

5.1 Steel ring – concrete slab interaction.................................................................... 51 5.2 Three-dimensional model of the structure............................................................ 57 5.3 Results of elastic analysis ..................................................................................... 60 5.4 Nonlinear analysis ................................................................................................ 67 5.4.1 Material model............................................................................................... 67 5.4.2 Discrete modelling of reinforcement ............................................................. 70 5.4.3 Results ........................................................................................................... 73 5.5 Particularities concerning crack width limitation................................................. 77 Summary and conclusions ........................................................................................... 80 References ..................................................................................................................... 82

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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 2005. /15/

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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.

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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.

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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. (www.energiewerkstatt.at)

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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. (www.noturbinesin.saddleworth.net)

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

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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 2.2.1 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.

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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 1885. /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 0 min/ max = V  / A ± M  / W  ) results in a linear soil pressure distribution that depends only 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

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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

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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.

2.2.2 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

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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

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

11

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. Idealised column c=4/4m

4025 kN N k =  

Concrete Ecm = 29 GPa; v  = 0,20

b

d avg  = 342 cm (252 cm) h = 3,5 m (2,6 m)

d avg  b = 17,7 m

Figure 7. System for the analysis.

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γ  = 1,35 for applied dead and live loads (ULS)

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  M  Ed  =

  d b 8

=

1,35 × 4025 × 17,7 8

= 12023  kNm.

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

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 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 ,h =3,5; shear  =

  d   A

(b / 2 − c / 2 − d avg ) =

5434 17,7 2

(17,7 / 2 − 4 / 2 − 3,42) = 59,5 kN/m,

and similarly for the thinner slab as v Ed ,h = 2, 6;shear  =

5434 17,7 2

(17,7 / 2 − 4 / 2 − 2,52) = 75,1 kN/m.

The shear force to represent punching is calculated at a peripheral section 1,5d   away from the face of the column (u1,5d ), with a subtraction of 50% of the upward soil  pressure acting in the area within the perimeter ( A1,5d ) as prescribed in the German code DIN1045-1 (2001) /11/: (fig. yyy)

v Ed , h=3, 5; punching ;1,5 d  =

  d  − 0,5 ×

  d   A

u1,5 d  5434 − 0,5 ×

v Ed , h= 2, 6; punching ;1,5 d  =

× A1,5 d 

5434

17,7 2 39,75

5434 − 0,5 ×

5434

17,7 2 48,23

=

× 180,76 = 80,2 kN/m;

× 121,37 = 110,2 kN/m.

This representation of punching check in DIN1045-1 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.

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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

v Ed ,h =3,5; punching  =

  d  −

  d 

 A u1, 0 d 

5434 − v Ed ,h = 2, 6; punching  =

× A1,0 d 

5434 17,7 2 31,83

5434 −

=

5434 17,7 2 37,49

× 107,5 = 95,2  kN/m;

× 76,3 = 129,1  kN/m.

Tributary reaction for beam-action shear 

 A1,5d 

1,0d 

1,5d 

u 1,5d 

u 1,0d 

33,7  45  °

°

Figure 8. Critical sections for beam-action shear and punching design.

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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/m3 is assumed for this analysis. As noted in ch. 2.2.1 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

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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.

0

�,��

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10,0 ��2,� � ���,� �

1�,0

������� ������������

1�,� 20,0

2�,2 2�,0

�0,0

2�,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

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100

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 b)

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.

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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.

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m Ed  [kNm/m]

v Ed;1,0d  [kN/m]

978 (126%)

129,1 (99%)

1900 (245%)

130,6 (100%)

775 (100%)

130,5 (100%)

Loaded area 6,6x6,6 m

520 (67%)

124,1 (95%)

Coupling of elements

589 (76%)

120,7 (92%)

978 (119%)

95,2 (98%)

1950 (237%)

96,5 (100%)

822 (100%)

96,8 (100%)

Loaded area 7,5x7,5 m

493 (60%)

92,4 (95%)

Coupling of elements

589 (72%)

93,3 (96%)

Method

Calculation by hand

Point load

h = 2,6 m

Loaded area 4x4 m

Calculation by hand

Point load

h = 3,5 m

Loaded area 4x4 m

Table 1. Summary of analysis results.

3.1.3 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

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  F t  =

5434   1

1 1 1   + + +   = 1727 kN. 3 × 3  tan 40 tan 50 tan 65 tan 70 

Assuming an effective width of beff  = 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 d /3

d 

40  50  °

°

65 

°

70 

°

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) 160-267 kN/m and 229-342 kN/m, respectively for the 3,5 m- and 2,6 mthick slabs. Hence it seems that all the studied load transfer models produce results that lie more or less on the conservative side.

21

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. Idealised column c=4/4m

4025 kN N k =  

M y,k  = 93345 kNm

b

Concrete Ecm = 29 GPa; v  = 0,20

d avg  = 342 cm (252 cm) h = 3,5 m (2,6 m)

d avg 

γ = 1,35 for applied dead and live loads (ULS)

y   x 

b = 17,7 m

Figure 14. System for the analysis – large uniaxial overturning moment.

3.2.1 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

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. 3.1.1; that is  M SYMM  = 1,35 × 4025 × 17,7 / 8 = 12023 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  ASYMM  = 1,35 × 93345 / 2 = ±63008  kNm. 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  EG = −63008 + 12023 = −50985  kNm;  M  POS  = 63008 + 12023 = 75031  kNm.

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

moment; for instance, considering the 2,6 m thick slab, the moment caused by its self weight resisting the uplift is  M SLAB = 25 × 2,6 × 17,7 × 8,85 2 / 2 = −45055  kNm. 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  Ed , x , max = 75031 + (50985 − 45055) = 80961  kNm. In this case the minimum moment is caused by the fully utilised self weight:

 M  Ed , x , min = −45055  kNm. It is then assumed that the positive flexure is carried by a substitute beam with a breadth of beff  = 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 Ed , x , max = 80961 /( 4 + 2 × 2,52) = 8956  kNm/m; m Ed , x , min = −45055 /(2 × 4) = −5632  kNm/m. 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

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. 3.1.1 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 DIN1045-1 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

8956

-5632

802

h = 3,5 m

6922

-6373

590

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

Tributary reaction for shear 

1,0d 

Critical section u crit 

45 

°

σ 0 

σ max  σ u,crit 

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 ucrit : u crit  = 2 × ( 2 + 2,52) = 9,0  m Soil pressure resulting from the applied loads at different sections (see fig. 15): σ max = 1,35 × 4025 / 17,7 2 + 1,35 × 93345 × 6 / 17,7 3 = 154  kPa

σ u ,crit  = (8,85 + 2 + 2,52) × (119 + 154) / 17,7 − 119 = 87  kPa σ 0 = 8,85 × (119 + 154) / 17,7 − 119 = 18  kPa Tributary soil reaction for shear:

26

 R =

18 × 17,7 2 4

+

(154 − 18) × 17,7 2 6

−

18 × 9,0 2 4

−

(87 − 18) × 9,0 2 6

= 7215  kN

Shear force acting along the critical section: v Ed  = 7215 / 9,0 = 802  kN/m 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.

3.2.3 Finite element analysis with plate elements The system parameters are the same as in Ch. 3.1.3 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

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. ��000 �������� �� ��������� ��2,� � ���� ���� ��� ������ ��� �� ��2,� �

��000

����� ��1��

���� ���� ��� ������ �,���,� �� ��2,� �

��1�2 �����

��000

�������� �� ����� ��� �������� ��������� ��� �

�2��� �2�2�

�2000

�������� ��������� �,���,� �

0

�,��

1�,�

0

2000

��12 �000 �102 ���0 ����

�000

�0�� �1��

�000

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

 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. 1�00 12�� 1200

11�� 10��

1000 ��� �2� ���

�00

�00

�������� �� ��������� ��2,� � ���� ���� ��� ������ ��� �� ��2,� �

�00

�������� ��������� �,���,� �� ��2,� � �������� �� ��������

200

�������� ��������� ��� � �������� ��������� �,���,� �

0 0

�,��

1�,�

Figure 18. Principal shear force (k/m) across a lateral section 1,0 d  away from the face of the column.

3.2.4 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

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

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

1�,�0 0

�0

100

1�0

200 ���� �� ������ ��������� ���,� �

2�0

���� �� � ������� ���,� �

�00

 b)

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

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: − z 0

m x =

∫ σ   z dz .  x

h − z 0

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  = -5043 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 = -5495 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

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: 0

v=

∫

2 2 + σ  yz  σ  xz  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%.

228 

-245 

540 

-105 0

5400

-6670 

-1740  

-1270 

2830 

1980 

-644

683

Figure 22. Principal stress field and distribution of horizontal stresses (with top and bottom surface stresses in kPa) in the foundation slab.

33

��000 ��0�� ��000

�2000 0

�,��

1�,�

0

2000

�000

��� ���,� �

�000 ����

�000

a)

1200 1000

���

�00 �00 �00

��� ���,� �

200 0 0

�,��

 b) 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

It can thus be concluded that the more realistic representation of the soil – structure interaction and the nonlinearity of the stress and strain distributions in a thick foundation slab can be approximated sufficiently well with a plate element model resting on a compression-only surface spring support. Considering practical design, the differences between a soil behaviour idealised by springs and by volumetric elements do not seem to be large enough as to judge the greater computation and modelling effort to be acceptable. Same applies for plate elements versus three-dimensional solid elements; with an appropriate loading model the time-consuming stress integrations can  be avoided, as the differences in member forces will be minor.

3.2.5 Summary of results The results from the analysis of a foundation slab subjected to large overturning moment are presented in table 3 below. The differences are marked with respect to the FE-model with a loaded area corresponding to the idealised column dimensions. m Ed,x ,max [kNm/m]

m Ed,x,min [kNm/m]

v Ed  [kN/m]

Calculation by hand

8956 (125%)

-5632 (109%)

802 (63%)

Loaded area 4x4 m; Soil as springs

7188 (100%)

-5185 (100%)

1279 (100%)

Loaded area 6,6x6,6 m; Soil as springs

4102 (57%)

-2566 (49%)

1095 (86%)

Coupling of elements; Soil as springs

5673 (79%)

-3674 (71%)

1163 (91%)

Calculation by hand

6922 (98%)

-6373 (116%)

590 (68%)

Loaded area 4x4 m; Soil as springs

7039 (100%)

-5495 (100%)

868 (100%)

Loaded area 7,5x7,5 m; Soil as springs

3312 (47%)

-2328 (42%)

759 (87%)

Coupling of elements; Soil as springs

5540 (79%)

-4172 (76%)

827 (95%)

Method

h = 2,6 m

h = 3,5 m

Table 3. Summary of analysis results.

35

3.3 Summary of Chapter 3 It has been demonstrated how the design of foundation slabs can be verified against aggravatingly varying member forces when different methods are used for the analysis, even though the models itself are essentially the same. Design of flexural reinforcement is generally somewhat uncritical for slabs as the  bending moment is effectively redistributed as the flexural cracking propagates. This issue is studied further in the following chapter. Most conservative flexural design for a foundation slab subjected to a large overturning moment is obtained with a simple hand analysis; however, as a three-dimensional FEanalysis with volumetric elements shows, it seems to be not that far from reality. Correspondingly, a FE-analysis with plate elements yields the most accurate results, when the loading is applied as a pressure trapezoid over the actual column area. Contradictory to a foundation slab subjected to purely concentric normal force, a load spread further to the mid-plane of a plate element model appears to result in too low member forces. Regarding shear design, different difficulties stir up than with flexural design. The disagreement of the principal shear force used for design is not as great between the different numerical models as what is the case with bending moment. However, a traditional hand calculation method seems to notably underestimate the critical shear force, suggesting that the main problem is to interpret the actual mechanism of shear failure in foundation slabs subjected to eccentric loading. Finally, it can be assumed that the quality of soil – structure interaction represented by one-dimensional springs is acceptable regarding structural analysis purposes, as the differences in member forces with regard to a more complex volumetric soil model are not major.

36

Chapter 4 onlinear behaviour of reinforced concrete In this chapter the effects of nonlinear behaviour of reinforced concrete on the resulting member forces in the foundation slabs are studied. This nonlinearity is caused primarily  by cracking of the concrete in tension and yielding of the reinforcement steel or crushing of the concrete in compression. Furthermore, factors such as dowel-action of reinforcement over a crack, concrete aggregate interlocking and the bond conditions  between reinforcement and intact concrete, as well as time-dependent effects of creep and shrinkage contribute to the nonlinear response of a member. This chapter starts with defining and verifying a material model for reinforced concrete, after which it is used in analysing the foundation slabs presented in the previous chapter. As the aim of this analysis is to estimate the resulting member forces in a slab, a finite element analysis with plate elements is considered. The nonlinearity in this analysis is thereby caused solely by flexural cracking of the slabs.

4.1 Material model for reinforced concrete 4.1.1 Concrete Abaqus/Standard offers several models to describe the nonlinear behaviour of concrete; in this study the smeared cracking and damaged plasticity models are used. (See /1/ for a detailed description) In the compression zone the uniaxial stress – strain behaviour of concrete is modelled as trilinear. (fig. 24) Range of elasticity is taken as 60% of the ultimate compressive strength: at stress levels between 50-70% of the ultimate strength cracks at nearby aggregate surfaces start to bridge in the form of mortar cracks and other bond cracks

37

continue to grow slowly. /25/ Under biaxial compression the concrete exhibits increased ultimate strength; here a typical assumption of 1,16 f c is used. A more versatile, parabolic stress – strain curve (e.g. /11/; /27/) is not needed in this study, as flexural cracking dominates the structural behaviour of the models at design loading and in typical massive slabs in bending the compressive stresses stay by far in the elastic region.

σ c  f c  0,85f c  0,60f c 

ε c1

0,0035  ε 

Figure 24. Idealised behaviour of concrete in uniaxial compression.

The tension zone is modelled linearly elastic up to the cracking stress. The cracking stress is determined (if not otherwise dictated by normative clauses) according to /33/ from the relation  f ct  = 2,12 × ln(1 + f c / 10) . [ MPa ] There exists a cohesive force in plain concrete in a region in front of a stress-free crack (in the so called Fracture Process Zone); as a result, a discontinuity in displacements is  present, but not in the stresses, whose magnitude is dependent of the crack opening (or the tensile strain, for that matter). /31/ In numerical simulations the post peak softening  behaviour is usually calibrated to follow a trend obtained by experimental results. This  poses a problem for practical design purposes, as many reinforced concrete structures are unique regarding reinforcement configuration, dimensions etc.; there is not necessarily experimental research done to act as reference.

38

Two models for the strain softening branch in tension are used. These are the linear fracture

energy

–based

model

used

with

the

smeared

cracking

model

of

Abaqus/Standard, and the bilinear fracture energy –based model used with the damaged  plasticity model. (fig. 25) The fracture energy required to propagate a tensile crack of unit area is calculated from the linear relation (/33/ ) G F  = 0,0307 × f ct  .

[ Nmm/mm2 ]

This equation gives somewhat higher values than the one found in CEB/FIP Model Code 90, (/16/) for example. This is however justified in the sense that the resulting increase in stiffness can be used to describe the so called tension stiffening effect: the intact concrete between cracks continues to carry tension transferred through the reinforcing bars.

σ t 

σ t  f ct 

f ct 

G f 

G f 

1/3f ct 

a)

u 0 

u 

2/9u 0 

 b)

u 0 

u 

Figure 25. Idealised strain softening behaviour of concrete. a) Linear and b) bilinear stress – crack opening relation.

4.1.2 Reinforcement steel A linearly elastic – linearly plastic stress – strain relationship is used to describe the reinforcement steel. (Fig. 26) The ratio between the stress at a strain of 0,025 and the stress at first yield is taken usually as 1,05 (1,08 if it can be assumed that high-ductile steel is used; this depends naturally on pre-determined conditions regarding the design  problem at hand).

39

σ s f t  f y 

0,025 

ε 

Figure 26. Idealised stress – strain behaviour of reinforcing steel.

4.1.3 Model verification The established model for reinforced concrete is tested by re-calculating a simply supported beam loaded with a concentrated load at mid-span, as presented in /25/. (fig. 27) The behaviour of the beam is characterised by flexural cracking and the yielding of reinforcement; it suits therefore well for testing the material model. The beam was originally tested by Burns and Siess in 1962 /6/, and was referred to as specimen ‘J-4’ in that experiment.

P  20 cm

46 cm

51 cm

3,66 m

Figure 27. Beam ‘J-4’.

The principal material parameters used in the numerical models are as follows: (adopted from /25/)  f c = 33,2 MPa;

40

 f  y = 310 MPa;  E c = 26,2 GPa;  E  s = 203 GPa;

 ρ = 0,99%. Two models with different tension softening branches are done using four-node plane stress elements for the concrete part and two-node truss elements for the reinforcement. The load – deflection behaviour of the numerical models is very satisfactory with regard to the measured results: both models predict the yielding load quite accurately. (fig. 29a) The somewhat stiffer response can be attributed to many things; e.g. bond slip, mesh sensitivity and the idealisation of the tension softening behaviour. Considering structural design using the established material model, however, the stiffer response does not necessarily mean that an unsafe design would be obtained. When the design is  based on member forces resulting from a non-linear analysis, a greater stiffness of a statically indeterminate reinforced concrete structure means less ductility and consequently less stress redistribution; hence the resulting maximal member forces will  be greater in magnitude and the design on the safe side. Fig. 29b shows the stress in reinforcement at mid-span of the beam in relation to the deflection. Rapid increase in the stress is observed as cracking advances, and ultimately the steel yields as the failure load is achieved. Finally, figures 28a and b illustrate the cracking of the beam at two load levels. The behaviour is characterised by diagonal flexural cracking, as expected.

a)

b)

Figure 28. Principal cracking strains in beam ‘J-4’ under a total load of a) 64 k and b) 128 k. (Bilinear tension softening model)

41

1�� 1�0 12�    � 100     �     �    �    �    �

��

�������� �������� ������� � ��������

�0

������ ������� ���������

2� 0 0

2

� � ���������� ��

�

a)

�00 2�0    � 200    �    �    �    �    � 1�0    �    �    �

�������� ������� ���������

100 ������ ������� ���������

�0 0 0

2

�

�

�

���������� ��

 b)

Figure 29. a) Load-deflection behaviour and b) stress in reinforcement at mid-span of beam ‘J-4’.

42

4.2 Design methods to nonlinear analyses The traditional “design of a critical section”  per se  is not required with nonlinear analyses, as the behaviour of the system is depicted quasi-realistically through the nonlinear material laws. It is, therefore, in many cases possible to calculate a maximum capacity load for a system, and to compare it to the magnitude of the relevant design load combination. This procedure is often combined with a unified safety factor concept for the resistance capacity, as in the DIN1045-1, for example. /11/ It means, in essence, that once a nonlinear analysis is carried out using expectable mean values of the material parameters, the resulting maximum capacity load Rk  which the system is able to carry is reduced by a safety factor γ R. Then a comparison against the relevant design load combination is performed:  E d  = γ  g Gk  ⊕ γ q Qk  ≤

 Rk  γ  R

= Rd  .

This works quite well for typical static systems in building construction, such as flat floor slabs. /17/ Even though the superposition of different load cases is no more allowed due to the dependence of the calculations on the stiffness of the system, it is still sufficient to analyse such systems with the total load on all spans: the load carrying capacity will be more or less completely utilised both at supports and at spans through moment redistribution as the flexural cracking forms plastic hinges at the supports. In the case of the foundation slabs studied in this work, on the other hand, the above mentioned procedure is not so straightforward to use. The dimensions of such foundation slabs are above all governed by normative requirements of sufficient safety against overturning and other stability related issues. Therefore a maximum structural  capacity load is difficult to evaluate as the system would have to be changed when the loading would increase too much in relation to the stability requirements. As a result the concept of unified safety factor can be used to apply it to each and every material parameter, after which the capacity of the chosen system configuration against design loading can be checked. Alternatively the nonlinear analysis can be used for finding out the member forces at a prescribed design load level, and then design the critical cross sections as usual. Using the latter procedure, it would make sense to use

43

unfactored mean values of the material parameters in the analysis to find out the member forces according to realistic deformation behaviour of the system; the design of the critical cross sections is anyhow performed with the required safety (see e.g. /30/ for related discussion). Due to the direct linkage of the amount of reinforcement and the stiffness of a system, the nonlinear design process has to be carried iteratively. (fig. 30) For each reinforcement configuration there is a unique maximum capacity load, which is, according to DIN1045-1, defined when one or more of certain critical states is reached:

•

εc ≥ 3,5 mm/m

•

εs ≥ 25 mm/m

•

System reaches kinematic state; i.e. the calculation is no more stable.

There are generally two ways to proceed with the design of the structure. First option is to perform a linear elastic analysis and use the resulting reinforcement as a first guess in a nonlinear analysis, and iteratively find the configuration with which the ultimate limit state still can be verified; the other possibility is to start with a minimum reinforcement governed by allowable crack width etc. and from that way iteratively arrive to the required capacity. Member Forces

Stiffness

Section Design

Reinforcement 

Figure 30. Dependence between member forces and reinforcement.

44

4.3 onlinear analysis of the foundation slabs As the system and its loading are principally identical as in the preceding chapter, the linear elastic analysis is used to determine the statically required flexural reinforcement needed for the first iteration of the nonlinear analysis. The soil in this analysis is modelled with nonlinear compression-only springs, as in the elastic analysis. Similar  plate element models for the foundation slabs are as well used. The column normal force and overturning moment are applied as a pressure trapezoid over the 4 m x 4 m column area. The choice applicable material parameters used for concrete in nonlinear analyses for determining the member forces is still an issue of great uncertainty. /30/ The DIN10451 prescribes the compressive strength of concrete to be factored as  f cR = α  × 0,85 × f ck  , where α  is generally to be taken as 0,85. For a C30/37 used in the foundation slabs would hence result f  result  f cR cR = 21,7 MPa. As the aim of this analysis is to study the member forces in the slabs due to nonlinear behaviour of reinforced concrete, the mean value f  value f ctm ctm = 38 MPa is used instead. As explained in the previous chapter, the required structural safety can be applied afterwards when designing the reinforcement for the member forces obtained from a nonlinear analysis. The tensional cracking strength of concrete is a subject where other reasoning has to be thought of. The use of the mean value f  value  f ctm ctm would probably be too optimistic especially when considering massive structures, where various restraint effects (e.g. uneven temperature gradient due to hydration) induce cracking before the structure is even loaded. /30/ On the other hand, no tensional strength at all generally results in numerical  problems, which consequently leads to uneconomical design as the amount of reinforcement has to be increased in order to provide the stabilising stiffness. This analysis is therefore done assuming  f ct  = 0,5 f ctm , which equals to 1,45 MPa for a C30/37. The contribution of concrete in tension between the cracks (tension stiffening effect) is modelled with a linear stress – strain relation for the tension softening branch

45

of the concrete: the cracking strain at which the tensile strength of concrete is completely exhausted is taken as 10-times the maximum elastic strain. The material strengths used in the analysis are summarised in the following:  f c = f cm = 38 MPa;  f ct  = 0,5 f ctm = 0,5 × 2,9 = 1,45 MPa;

 f  y = 1,1 f yk  = 1,1 × 500 = 550 MPa. For simplicity, the required reinforcement to cover the maximum bending moments is spread throughout the slabs orthogonally. In reality, the top layer reinforcement in such foundation slabs as the ones studied here would require special consideration because of the tower connection through a steel ring; radial and tangential reinforcement would have to be provided due to constructional requirements. The design of statically required top and bottom flexural reinforcement according to the linear elastic analysis is carried out according to DIN1045-1 in the ultimate limit state for the 2,6 m-thick slab in the following, exemplarily.  f cd  =

 f  yd  =

α  f ck  γ c  f  yk  γ  s

= 0,85 × 30 / 1,5 = 17,0 MPa;

(C30/37)

= 500 / 1,15 = 435 MPa;

(BSt500)

Bottom layer: µ  =

m Ed   f cd d 2

=

7188 × 10 3 17,0 × 100 × 252 2

= 0,0666 ;

ω  = 1 − 1 − 2 µ  = 1 − 1 − 2 × 0,0666 = 0,069 ; 2 a s ,rqd  = df cd  / f yd  = 0,069 × 100 × 252 × 17,0 / 435 = 67,95 cm /m.

Top layer: µ  =

5185 × 10 3 17,0 × 100 × 252 2

= 0,0480 ;

= 0,049 ;

46

2 a s , rqd  = 0,049 × 100 × 252 × 17,0 / 435 = 48,26 cm /m.

The requirement for minimum reinforcement according to clause 13.1.1 (1) of DIN1045-1 can be ignored for massive foundation structures such as the slabs in question /10/; it is obvious that the redistributing soil pressure would provide for a ductile structural failure for a foundation structure. Other minimum reinforcement requirements, such as the limitation of crack width due to various restraint effects (such as the flowing off of hydration heat during the concrete hardening process, as mentioned above), should, on the other hand, be considered. However, in this analysis they are omitted for simplicity. Figure 31a shows how the peaks of the sagging bending moments diminish in both slabs as they distribute laterally while the concrete cracks in top and bottom surfaces. Figures 32a and b illustrate the flexural cracking strains in top and bottom surfaces for both slabs. A plot of the bending moment distributions under design loading shown in fig. 33 clearly illustrates the phenomenon of bending moment redistribution: after cracking has  been initiated and the plastic zone propagates, a bending moment can increase only a small amount in that region. The effect is less pronounced in the negative, hogging moments; less plasticity occurs in the top surfaces of the slabs. A new design according to the bending moments from the nonlinear analysis would result in approximately 80% of the bottom reinforcement required by the linear elastic analysis for both foundation slabs. For a massive foundation slab this means a considerable saving. Shear verification can as well be done against a notably smaller design shear force (14% and -8% for the 2,6 m- and 3,5 m thick slabs respectively) compared to the linear elastic calculation. (fig. 31b) Also here lateral redistribution takes place due to cracking of the concrete. Whereas the member forces decrease when considering flexural cracking, the opposite is true for settlements. Reduced flexural stiffness of the cracked structure means that the applied loads are led directly to the soil in larger extent; hence the soil pressure and the settlements will increase.

47

��000

����� ����� ��1�� �����

��000

�2000 0

� ,� �

1 �,�

0

2000 ������� ��2,� � �������� �� ��2,� ��2,� �

�000

������� ���,� � �������� �� ���,� ���,� �

���� ���0

�000

�0�� �1�� �000

1�00 12�� 1200 10�� 1000 ��� �00

�02

�00 ������� ��2,� �

�00

���������� ��2,� � ������� ���,� �

200

���������� ���,� �

0 0

�,� �

1 � ,�

Figure 31. a) Design bending moment m x  along the slabs. b) Principal shear force across a section 1,0d  1,0 d  away  away from the column.

48

a)

b)

Figure 32. Principal cracking strains in the bottom surface of the foundation slabs. a) h = 2,6 m; b) h = 3,5 m.

a)

b)

Figure 33. Qualitative distribution of bending moment m x  in a) elastic and b) cracked foundation slab under equal loading. Blue colour denotes bending moment causing tension in the bottom surface; red colour denotes bending moment causing tension in the top surface.

49

4.4 Summary of Chapter 4 Although the material model for reinforced concrete used in this chapter seems to reflect the load – deflection response of a real flexural specimen more than adequately, it is nevertheless a cruel fact that the behaviour of a foundation slab with massive dimensions and restraint-induced and dynamic real-life loading differs from a laboratory-tested simply supported beam. Therefore great care should be taken when first choosing the ingoing material parameters and when afterwards assessing the results. A nonlinear flexural analysis of typical massive foundation slabs has demonstrated the redistributing behaviour of the member forces. The decrease in maximum bending moment in the studied case is approximately 20%; for the shear force the decrease is around 10%. A corresponding design with less reinforcement can consequently be carried out. It has to be nevertheless remembered that the serviceability limit state must also be verified; in the case of extreme redistribution of the elastic bending moments other requirements, such as crack width limitation due to restraint-induced actions, might become governing regarding design.

50

Chapter 5 Three-dimensional analysis and design of a typical wind turbine foundation slab The present chapter deals with a three-dimensional modelling of a real wind turbine foundation slab. The flow of forces in the slab is analysed with elastic models, and a design proposal is made from the results. Questions intended to be answered with the help of three-dimensional models are the load transfer through a massive steel ring and the related problematic with anchorage of the forces in the uplift-case, as well as the validity of the previous model assumptions regarding practical design of such structures.

5.1 Steel ring – concrete slab interaction As was stated in Ch. 3.1.1 in reality the studied wind turbine slab foundation type supports a circular, hollow steel tower. This tower is attached to the slab through a steel ring, which is cast inside the concrete. (See Ch. 2.1) The steel ring has an I-shaped cross section; hence the bond between the ring and the concrete is provided by contact through the flanges as well as by friction at the whole interface. Geometry of the steel ring – slab connection is illustrated in fig. 34. To introduce the problem of the interaction between the steel ring and the concrete slab first a loading consisting of only the concentric normal force is considered. (See Ch. 3.1.1) It is thereby sufficient to build a rotation symmetric model of the structure; however the applied normal force has to be adjusted to account for the smaller contact  pressure area of a circular axisymmetric slab.

51

D=440 

11

190 

5,2 

350 

11

Soft layer 

50 

160 

Figure 34. Steel ring – concrete slab connection. (Dimensions in cm)

The elastic material parameters being used for concrete are the same as in previous analyses: Young’s modulus is taken as 29 GPa (corresponding to a C30/37) and Poisson’s ratio as 0,20 (corresponding to elastic behaviour of the material). The steel ring is assumed to be made of ordinary structural steel with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0,30. A soft layer under the bottom flange of the steel ring, which is intended to prevent a local punching failure from occurring, is taken into account by leaving a 1 cm thick empty space between the bottom flange and the concrete; hence no stresses will transfer between the bottom steel flange and the concrete surface underneath. Soil under the slab is modelled as springs with a modulus of subgrade reaction of 50 MN/m3, as in the previous analyses. The tower normal force is applied as a uniform pressure on the top flange of the steel ring over an area that corresponds to the ring web cross section. The steel ring interacts with the concrete slab via an Abaqus/Standard surface contact algorithm; the interaction with this approach can be modelled to handle both compression (i.e. contact normal to the surfaces) and frictional shear (i.e. tangential contact). It is assumed that besides the direct anchorage through the flange only frictional bond exists in the interface between the steel ring and the concrete slab; any

52

adhesive bond would certainly be destroyed already in early stages of the loading history. The frictional behaviour is modelled through the basic Coulomb friction model, where the shear stress carried across the interface before slipping occurs (so called sticking region) is defined as a fraction of the contact pressure at the interface (i.e. τ crit  =  µp). (fig. 35a) There is, however, some “elastic slip” allowance made in the stick region (fig. 35b); this helps the solver to find a converging solution. /1/ An ideal behaviour is assumed for the friction – slip rate relation regarding static and kinetic friction (i.e. the friction coefficient that opposes the initiation of slipping is the same as the friction coefficient that opposes already established slipping).

  s   s   e   r    t   s   r   a   e    h    S

  s   s   e   r    t   s   r   a   e    h    S

τ crit 

Slipping friction

τ crit 

 µ  Stick region

Sticking friction

Total slip

Contact pressure

a)

 b)

Figure 35. Friction model for steel ring – concrete interface.

The influence of friction between the steel – concrete interface was found to be very minor regarding the behaviour of the structure. The difference in peak contact pressure at the top flange of the steel ring is less than 20% between a completely frictionless contact and an unrealisticly rough contact with  µ  = 2,75. (fig. 36) Therefore a reasonable value for the friction coefficient of µ = 0,7 is chosen for the steel – concrete interface.

53

 p

 µ  = 0,7  µ = 0

 µ  = 2,75

Figure 36. Qualitative distribution of contact pressure in the steel – concrete interface.

Figure 37 shows the flow of principal stresses as well as the distribution of the horizontal stress component over selected cross sections in the concrete slab. The nonlinearity in the stress distributions can be recognised clearly. An integration of the horizontal stresses times the lever arm z   from neutral axis yields a maximum bending moment of ca. 362 kNm/m in the region where the load is applied; this is approximately 73% of the smallest bending moment resulting from the simplified  plate-element analysis in Ch. 3.1.2 (loaded area 7,5 x 7,5 m; m = 493 kNm/m). A strut and tie model corresponding to the axisymmetric foundation slab is presented in fig. 38. Assuming a compression strut inclination of 45° the resulting radial tension force can be calculated as  f t  = 5434 / π  /( 4,4 + 2 × 3,42) = 154 kN/m.

54

By integrating the horizontal tension stresses a tensional force of about 164 kN/m is obtained, which agrees well with the strut and tie model. A considerable portion of the applied load is hence transferred through diagonal compression in addition to flexure.

-118 kPa

-165 kPa

-151 kPa

210 kPa  x = 0 

1,50 m

-33 kPa

209 kPa

109 kPa

3,00 m

28 kPa 5,00 m

Figure 37. Principal stress field and distribution of horizontal stresses in the concrete slab.

55

4,4 m

N d 

d ≈ 342 cm

Figure 38. Strut and tie model for a rotation symmetric system.

Contact opening 

Concrete

Bottom flange of steel ring 

Soft layer 

Figure 39. Contact opening at the steel – concrete interface.

Figure 39 shows the deformed mesh in the bottom region of the steel – concrete interface. It confirms that the contact formulation is working as expected; i.e. the bottom steel flange departs from the concrete surface as the steel ring is compressed downwards  by the applied load. The contact interface causes also the tensional flexural stresses in the concrete immediately under the steel ring (see fig. 37) as the sides of the bottom steel flange compress against the concrete.

56

The soil pressure distribution resulting from the 3D-axisymmetric analysis differs slightly from a corresponding analysis with a plate element model. (Fig. 40) The disagreement in soil pressure under the centre of the slab is approximately 5%. A plate element model thereby underestimates the true stiffness of the structure to some extent.

0

�,��

10,0 12,0 1�,0

�� � ����������� ����� ��������

1�,0

�������

1�,0 20,0 22,0 2�,0

Figure 40. Soil pressure distribution (kPa) resulting from the applied normal force.

5.2 Three-dimensional model of the structure The FE-mesh of the foundation slab is presented in fig. 41a. Due to partial symmetry of the applied load case (i.e. uniaxial overturning moment together with normal force from the tower) only one half of the structure is modelled. Because of the relative complexity of the meshing, the concrete slab is assembled together from several parts with mesh-tie constraints to make the structure continuous. Using the mesh-tie constraint also allows the different parts to have different element densities; a relatively fine mesh is required for example in the part in contact with the steel ring due to the complex stress field as well as due to the contact interaction model. As the steel ring has to be modelled with fairly many elements, in order to represent realistic load transfer through flange and web  bending in addition to pure normal force, the mesh of the concrete region in contact must be at least equally fine so that no penetration of the contact surfaces will occur.

57

The discretization of the structure is done with linear 8-noded volumetric elements. Total number of elements in the model is 47 753 with 165 984 degrees of freedom. For a realistic representation of the loads which the wind turbine tower transfers into the foundation slab a two meter tall segment of the tubular steel tower is modelled with shell elements. The shell elements are then coupled to the solid elements of the ring. (Fig. 41b)

a)

 b)

Figure 41. a) Three-dimensional model of the foundation slab. b) Mesh of the steel ring and part of the tower.

58

Figure 42. Reference point, coupled with the tower section, through which the loading is applied.

Loading is applied through a reference point which is coupled with the tower section, as shown in fig. 42. The studied load combination considers the factored applied design loads for verification of the ultimate limit state, which are, as in the previous analyses   d  = 1,35 × 4025 = 5434  kN and

 M d , y = 1,35 × 93345 = 126016  kNm. The design overturning moment M d,y includes both the bending moment caused by the thrusting horizontal force due to wind loading and the bending moment due to horizontal shear force at the tower – foundation interface. Additional load cases are the self weights of concrete and steel, which are assumed to be unfactored at 25 kN/m3 and 78 kN/m3, respectively. Soil is represented with similar compression-only springs as in the previous plate element models. The stiffnesses of the individual springs are adjusted with respect to face dimensions of the elements in the bottom surface of the slab so that the global soil stiffness will correspond to approximately 50 MN/m3.

59

5.3 Results of elastic analysis The deformed system is shown in fig. 43. As easily can be observed, the steel ring tries to punch itself out of the concrete; consequently the top flange is detached from its embedment. This leaves the heel of the slab partly “hanging” from the bottom flange. The settlement curve and thereby also the calculated soil pressure distribution are akin to the model with plate elements, although a slight difference in stiffness between a three-dimensional and a plate element model can be seen. (Fig. 44) This behaviour was observed also in the analysis of the concentric load case previously.

Figure 43. The deformed system. (Scale factor 75,0)

60

0

�,��

1�,�

0

�0

100

1�0 �� 200

�� ��� ��������

2�0

Figure 44. Comparison of soil pressure distributions (kPa).

Figures 47a and b show distribution of minimum and maximum principal stresses in the concrete slab, respectively. Figures 48a-c depict tensor plots of the principal stress field as seen from a vertical section in the centre of the slab as well as in the top and bottom surfaces. A maximum compression stress of approximately 23,5 MPa occurs in the region where the top flange of the steel ring gets compressed against the concrete. A stress of this magnitude in the ultimate limit state is by no means critical, as in the steel ring – concrete interface region a multiaxial stress field is utilised. A massive amount of tension arises, as expected, in the zone where the bottom flange is anchored in the concrete due to the uplifting flange pressure. Maximum uplifting force along the flange resulting from the contact pressure is approximately 5000 kN/m, which is divided more or less equally on both sides of the flange. This force has to be transferred to the compression zone with vertical suspension reinforcement; i.e. the design could be carried out analogously to a beam with a dapped end (half joint). (fig. 46) Some horizontal ring reinforcement is also needed to carry additional tension in tangential direction. It is observable that the horizontal contact pressure along the steel web and the related friction interface has some influence in reducing the magnitude of

61

the flange contact pressure; although the friction effect is largely reduced by the fact that contact is almost completely lost on one side of the web. (Fig. 45) A simple check using standard principles of mechanics of elasticity, i.e. considering a hollow circular cross

section

subjected

to

combined

bending

and

normal

force,

as

per

σ max =   / A − M  / W  , results in this case in a peak uplifting force of around 7800 kN/m along the flange. Thereby some 35% of the overturning moment is transferred by the horizontal contact forces. Additionally, radial reinforcement bars will be required on the top surface of the slab in order to provide doweling in the heel of the slab and the concrete inside the steel ring: the “hanging” of the heel due to the loss of contact, as previously mentioned, will induce massive problems regarding local stresses resulting from the uplifting pressure once cracking of concrete is taken into account in the analysis, and consequently a part of the slab tries to punch out. The radial reinforcement has to be fully anchored in the concrete inside the steel ring; practically, this means that holes have to be provided in the steel ring.

Figure 45. Loss of contact at the steel – concrete interface.

62

Suspension reinforcement  Inclined crack surface in concrete

Uplifting force due to contact pressure

Radial reinforcement at the top surface

>                  l                    b                   

Figure 46. Steel ring anchorage with special reinforcement considerations.

Figure 48b shows the principal stress field in the top surface of the slab. Because of the discontinuity between the concrete slab and the steel ring, the governing principal tensional stresses act in tangential direction in relation to the ring. In the bottom surface (fig. 48c), on the other hand, the direction of the governing tensional stresses is oriented approximately parallel to the global x-axis. For the design of purely statically required reinforcement, therefore, it is sufficient to consider the negative, tangential bending moment for the top surface and the bending moment in  x-direction for the bottom surface. As it has to be assumed that the extreme overturning moment may act in any direction, the slab has to be provided with maximum amount of main reinforcement in each direction. The mentioned bending moment curves as well as the principal shear force curve obtained through integration of the relevant stress components at several sections along the slab are presented in fig. 49. A maximum bending moment causing tension in the bottom surface, according to this analysis, is m x,max  ≈ 7990 kNm/m. This is approximately 14% more than what was

63

 predicted in the plate element analysis with a simplified load transfer model (m x,max = 7039 kNm/m; plate element model with 4 m x 4 m loaded area). The difference with respect to the three-dimensional analysis with a simplified load transfer model, on the other hand, is only around 5% (m x,max  ≈ 7568 kNm/m). However, the trend is clearly moving towards a considerably larger elastic sagging bending moment as the accuracy of the model increases (three-dimensional modelling of soil; realistic modelling of the load transfer behaviour).

a)

 b)

Figure 47. a) Minimum and b) maximum principal stresses in the slab.

64

a)

 b)

c)

Figure 48. Principal stress field as seen from a vertical section in the centre of the slab (a) and in the top (b) and bottom (c) surfaces.

65

�� ���/� �  �� ��/�

�

��000 ��0�0 ���/� ��000

�1000 0

��,��

�,��

1000

�000 �0�0 ��/�

���� �� 

�000

� 

�000 ���0 ���/� �000

Figure 49. Elastic section forces in the foundation slab.

The tangential bending moment causing tension at the top surface of the slab reaches a minimum of mtan,min  ≈ -4060 kNm/m. The favourable difference in relation to the corresponding minimum bending moments in the  x-direction resulting from the  previous analyses is more than 20% (e.g. m x,min  = -5495 kNm/m in the plate element model with 4 m x 4 m loaded area). This can be explained by the fact that only partial fixity is present in the steel ring – concrete interface as the ring is free to detach from the concrete.

66

Regarding shear force, the critical maximum at approximately a distance of 1,0d   away from the steel ring in x-direction is found to be v Ed  ≈ 1010 kN/m. Again, this value is considerably higher than what the plate element analysis suggested (v Ed   = 868 kN/m; difference 17%). The design shear force obtained with the three-dimensional analysis with a simplified load transfer model agrees better with the current analysis. (v Ed  = 958 kN/m; difference 5%). It seems that also here the same trend applies as with the sagging  bending moment: as the accuracy of the model increases, so do the design member forces.

5.4 onlinear analysis 5.4.1 Material model The material model for the reinforced concrete used for a nonlinear analysis of the three-dimensional model follows the principles presented in Ch. 4. In this case, however, the softening branch of concrete in compression is taken into account due to the realistic modelling of the steel ring – concrete interface and thereby the increased local stress transfer; it is assumed that the compression strength is fully exhausted with a crushing strain of approximately 5 mm/m. It is well-known that one can ‘tune up’ a material model endlessly to gain quasi-perfect results when considering the purpose of numerically modelling real-life experiments; however, the amount of uncertainties in untested full-scale constructions must be taken into consideration and reasonable compromises have to be made regarding acceptability of obtained results and the computational effort required. One of the main issues influencing the nonlinear response of a structure, as also witnessed in Ch. 4, is the softening behaviour of concrete in tension. Given the large amount of degrees of freedom in the studied foundation slab model, a simple linearly regressing tensile softening branch is used in order to minimise the computational effort through an increased local stiffness of the concrete elements (fig. 50); it must be remembered, that for the reasons mentioned in Ch. 4 a factored value of the cracking strength of concrete (0,5 f ctm) will be used anyway so that the overall tensile behaviour must be considered tolerable. Nevertheless, to verify the global behaviour and possible failure modes of the type of structure under study, a punching failure of a concentrically

67

loaded column footing ‘DF2’ as experimented by  Hegger et al.  /22/ will be recalculated in the following. σ t  fctm=2 ,9 MPa

0,5f ctm

ε pl,0  = 2 mm/m

ε 

Figure 50. Model for concrete (C30/37) in tension used for the wind turbine foundation slab.

Parameters used for the damaged plasticity model of concrete are the same as the ones used for analysis of the wind turbine foundation slab later on. The material strengths and elastic parameters, as told in /22/, are:  f c = 22,0 MPa;  f ct  = 1,76 MPa;  f  y = 552 MPa;  E c = 22,6 GPa;  E  s = 200 GPa (an assumed value). Figure 51 shows the FE-mesh of the footing, where the double symmetry is utilised in order to reduce computational time. The real structure was a rectangular, 20 cm thick foundation slab with a side length of 90 cm. Flexural reinforcement consisted of Φ14 mm bars at a spacing of 10 cm in two orthogonal directions (with an average effective depth of 15 cm); in the FE-model the reinforcement is built discretely with truss elements. Column geometry of 15 cm x 15 cm results in a shear span of a/d  = 2,5.

68

Figure 51. Finite element mesh of column footing ‘DF2’.

Without further consideration on the actual subgrade conditions in the experiment (as a remark, the footing was founded on densely bedded sand), a surface spring support with a subgrade reaction modulus of 50 MN/m3, as used throughout this study, is applied. Figure 52 illustrates the obtained cracking strains at failure as well as the stresses in flexural reinforcement. Although the calculated ultimate load of approximately 640 kN is some 20% higher than what was measured in the experiment (V test  = 530 kN), there is no question of the validity of the acquired failure mode, i.e. concentric punching. The stress level in the longitudinal reinforcement shows also that the failure occurred significantly before the yielding strength of the steel was reached. As mentioned before, calibration of the tensile behaviour of concrete together with a realistic soil model (as done in the numerical analysis in /22/) would most likely result in better agreement with the calculated and tested ultimate load capacity. It is, finally, concluded that the rather simple but computationally efficient model introduced in Ch. 4 is sufficient enough for the purpose of the analysis of the wind turbine foundation slab. Nevertheless, it must still be pointed out that the actual structure and the associated potential failure modes have not been verified by real-life experiments, so that the results of the present analysis can only be considered as purely theoretical.

69

Figure 52. Principal cracking strains in column footing ‘DF2’ immediately prior to failure (above); Stresses in flexural reinforcement (below).

5.4.2 Discrete modelling of reinforcement Design of reinforcement of the foundation slab is based on the results from the elastic analysis. Only statically required reinforcement is considered in the analysis; additional serviceability reinforcement for crack width control due to restraint-induced residual stresses will be revisited in Ch. 5.5. The design of main flexural reinforcement considering the elastic section moments in the top and bottom layers is summarised in table 4. As already mentioned before, the maximum amount of reinforcement will be provided throughout the slab. It has to be noted that the concrete cover for the tangential reinforcement in the top surface is taken relatively large (15,0 cm) because of the thick top flange of the steel ring; this reinforcement is thought to make up one layer with the radial bars anchored in the

70

concrete inside the steel ring. An orthogonally assembled mesh of serviceability reinforcement will then be placed closer to the surface (see Ch. 5.5). For the bottom layer the nominal concrete cover is taken as 5,0 cm, as the slab is assumed to be cast directly on soil. In addition to the determination of required amount of reinforcement, care must be taken when considering the detail design, such as proper anchorage and lapping of the reinforcement. For such a massive slab, however, the end anchorage should not pose a problem as even relatively thick reinforcement bars can be bent with sufficient mandrel diameters. The slab is chosen to be reinforced quite heavily in the radial direction, but it has to be nevertheless remembered that the steel ring cannot be punched too densely with holes; the deformation of the steel ring in relation to the slab was found to be relatively large in the elastic analysis, so the through-going radial reinforcement will be somewhat exposed to corrosive action. A total of 52 Φ25 mm bars are chosen for the radial direction, so that the spacing of the holes in the steel ring will be 26,5 cm. m Ed

d avg

[kNm/m]

[cm]

Bottom; x-direction

7990

Bottom; y-direction Top; tangential

2

a s,rqd 

a s,prvd  [cm /m];

[cm2/m]

Reinf. configuration

 µ [-]

ω [-]

342

0,040

0,041

54,80

56,6;

Φ30-12,5

7990

342

0,040

0,041

54,80

56,6;

Φ30-12,5

-4060

334

0,021

0,022

28,72

32,7;

Φ25-15

Table 4. Main flexural reinforcement according to elastic section moments with  f cd  = 17,0 MPa and  f  yd  = 435 MPa.

Shear design of the foundation slab is carried out against the critical shear force of 1010 kN/m obtained from the elastic analysis. The capacity without shear reinforcement, calculated as v Rd ,ct  = 0,10κ (100 ρ l  f ck  )1 / 3 × d  = 0,10 × 1,24 × (100 × 0,00165 × 30)1 / 3 × 3420 = 722  kN/m is inadequate so that vertical shear reinforcement will be needed. The reinforcement is designed as for a beam strip of 1 m breadth, and then spread peripherally to account for

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the fact that the design action can occur in each direction, as for the flexural reinforcement. For a section with shear reinforcement the shear force carrying capacity of concrete is defined according to clause 10.3.4 (3) of DIN1045-1 as v Rd ,c = 0,24 f ck 

1/ 3

z  = 0,24 × 301 / 3 × 0,9 × 3420 = 2293  kN;

i.e. for the given case the capacity is significantly larger than the design shear force. It is, therefore, sufficient to design the reinforcement according to the minimum allowed compression strut inclination (cotθ   = 3,0). Thereby the required amount of transversal reinforcement is a sw, rqd  =

v Ed   f  yd  z cot θ 

=

1010 × 10 3 435 × 0,9 × 342 × 3,0

= 2,51  cm2/m/m.

This requirement is satisfied by providing vertical Φ16 mm stirrups at a spacing of 75 cm in the longitudinal direction and 100 cm in the lateral (peripheral) direction (a sw,prvd  = 2,68 cm2/m/m). Without further calculations a total of six perimeters of the stirrups will be provided; this will certainly be sufficient to cover the shear force envelope. The required suspension reinforcement due to the overturning moment is calculated by hanging the uplifting force of the steel ring to the compression zone of the concrete slab (a s = 2500/43,5 = 57,5 cm2/m). Thereby three perimeters of Φ20 mm stirrups at a lateral spacing of 15 cm on both sides of the steel ring are provided. In order for the suspension reinforcement to be as effective as possible, all the reinforcement perimeters should be  placed as close to the steel ring as possible. For the nonlinear analysis of the foundation slab, the above designed reinforcement is modelled discretely with truss elements embedded into the concrete. The embedment constraint significantly improves the computational efficiency of the model; however it also means that perfect bond conditions exist between concrete and reinforcement. Therefore the often critical anchorage of vertical reinforcement (see e.g. /13/, /32/) can not be assessed; instead it has to be assumed that this issue will be addressed by careful detail design. Finally, figure 53 shows the FE-model of the reinforcement configuration.

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Figure 53. FE-model of reinforcement in the foundation slab.

5.4.3 Results As can be seen from figures 54a and b, the structural behaviour of the system is completely different once cracking of the concrete is taken into account. Whereas a maximum compression stress of 23 MPa was obtained in the elastic analysis, now the stresses exceed 30 MPa; maximum uniaxial pressure is approximately 28 MPa where the bottom flange of the steel ring tries to push out from its position. Figure 55 shows the minimum principal plastic strains in the concrete. Along a part of the bottom flange of the steel ring the inclined crushing strains reach around 4,6 mm/m; this very local crushing started when approximately 85% of the design overturning moment was applied. It is a consequence of a failure in a local compression zone due to the ‘reverse’  punching-like behaviour and implies the critical nature of the anchorage of the steel ring. The initiation of flexural cracking in the top and bottom surfaces has limited the tensional stresses in concrete but nevertheless the crack widths are yet relatively small so that the tension carrying capacity is not at all exhausted. Hence the main flexure is still carried mostly by the concrete. Extremely wide cracks, on the other hand, occur due to the uplift of the steel ring, as predicted in the elastic analysis. (Fig. 56) Noteworthy is

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also that the main crack surfaces are inclined towards the top surface of the slab, as expected. Here the tension carrying capacity of concrete is almost totally exhausted; consequently the most stressed reinforcement is the vertical suspension reinforcement together with the radial top surface reinforcement. (Fig. 57) As a remark, this is the reinforcement that does not get attention at all in traditional bending and shear analyses of foundation slabs. Furthermore, considering the dynamic real-life loading of a wind turbine facility and thereby the fatigue limit state, one can expect this special reinforcement to become even more critical.

a)

 b)

Figure 54. a) Minimum and b) maximum principal stresses in the slab.

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Figure 55. Local crushing strains (εc > 3,5 mm/m) in concrete.

Figure 56. Principal cracking strains. Isometric contour view (top); top surface (below, left); bottom surface (below, right).

 No signs of ‘global’ shear cracking can be seen; neither does the shear reinforcement get activated significantly. It seems, therefore, that the beam-action shear design is well on the conservative side.

75

Regarding section forces the greatest difference with respect to the elastic analysis is obtained with the negative tangential moment; the nonlinear analysis results in mtan,min ≈ -3300 kNm/m (difference -19%). The tangential reinforcement in the top surface can therefore be provided with Φ25 mm bars at 20 cm spacing, for example. This represents about 25% less provided reinforcement than according to the elastic analysis (a s,prvd  = 24,5 cm2/m). For the bottom surface the nonlinear analysis yields an m x,max  ≈ 7450 kNm/m (difference -7%). A corresponding configuration of reinforcement could be

Φ32 mm bars at 15 cm spacing, i.e. the provided reinforcement amount is approximately 5% less than that obtained from the elastic analysis (a s,prvd   = 53,6 cm2/m). In the design shear force no difference worth mentioning is obtained. The response differs thereby quite significantly from the nonlinear analysis performed in Ch. 4; it is the opinion of the author that a nonlinear plate element analysis negelcts too many essentialities in the given case (i.e. stiffness effect of the structure, load transfer from the tower into the slab, shear cracking) in order to safely utilise the obtained theoretical increase in structural capacity.

Figure 57. Tensional stresses in reinforcement bars.

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5.5 Particularities concerning crack width limitation As already mentioned in Ch. 4.4, massive reinforced concrete structures are exposed to various restraint-induced effects, which can lead to severe cracking even before the member is subjected to the actual external loads. In this context it is most important to consider the concrete in its early age as a movement-restraining temperature gradient due to hydration process takes place. During the hydration process the concrete tries to expand due to the heat being generated; after reaching the maximum temperature the concrete starts to cool down and consequently wants to shorten. This free movement of the concrete is partly restrained  by the underlying soil through friction, and therefore tensional stresses will occur at a cross section of the structure. The distribution of these stresses is, in the case of thick slabs on grade, eccentric as the restraint takes place at the bottom surface. (fig. 58a) The cooling process results additionally in a temperature gradient over the depth of the slab causing curvature, which is resisted partly by the self-weight of the slab and partly by the soil; therefore a further restraint-induced moment occurs, causing tension on the top surface of the slab. (Fig. 58b) Further details on the topic can be found in /5/, for instance.

σ

σ 

a)

b)

Figure 58. Restraint-induced stresses due to a) subgrade friction and b) temperature gradient.

77

Φ20-10  d1 = 3,5+2+1 = 6,5 cm

Figure 59. Supplementary top surface reinforcement.

Calculation of the cracking due to hydration is carried out according to DIN1045-1 in the following for the studied foundation slab. It is intended to be verified, if an orthogonal surface reinforcement layer consisting of Φ20 mm bars at 10 cm spacing (see fig. 59) is sufficient in limiting the crack width. It is furthermore assumed for simplicity, that only this reinforcement is resisting the restraint-induced effects; i.e. the main flexural reinforcement in tangential and radial directions will not be considered. For a foundation slab in exposure class XC2 the allowable crack width is 0,3 mm. The section normal stress due to subgrade friction (with an assumed soil friction angle of 30°) is calculated as σ  s , Fric = 17,7 / 2 × 25 × 0,577 / 1000 = 0,13  MPa. Critical moment due to the temperature gradient can result in tension at the top surface of the slab that is, at the most, the cracking strength which has developed in the earlyage concrete. Following /5/, a cracking stress of σ Cr   = 1,34 MPa is used here for a C30/37. After superposition of both the concentric and the eccentric restraint effect the depth of the tensile zone is found to be approximately 1,94 m. The code prescribes a parameter k c to account for the favourable effects of any residual compression stresses within the above calculated tensile zone:

 

k c = 0,4 × 1 +

 

   = 0,46 . 2 / 3 × 1,34  0,13

Thus the force to be carried by the reinforcement will be  F  s = 0,46 × 1,34 × 1,94 / 2 = 0,60  MN/m,

78

and the corresponding stress σ  s =

0,60 × 10 4 31,4

= 191  MPa.

Special consideration has to be given when determining the depth of the effective reinforcement zone heff ; for thicker members this will be larger than 2,5d 1, which is  prescribed in the code. /9/ According to /11/ for the given structure (h/d 1 = 3,5/0,065 ≈ 54) a suitable value will be heff  = 4,75d 1  ≈ 31 cm, so that the effective percentage of reinforcement equals to ρeff  = 31,4/31/100 = 0,010. Maximum crack spacing is calculated as  s r , max =

d  s 3,6 ρ eff 

=

20 3,6 × 0,010

= 555 <

σ  s d  s 3,6σ Cr 

=

191 × 20 3,6 × 1,34

= 792 ,

and the difference in mean strains between concrete and reinforcement (with the ratio of Young’s modules taken as αe = 7,3) as σ  s − 0,6 × ε  sm − ε cm =

σ Cr   ρ eff 

(1 + α e ρ eff  )

=

 E  s

> 0,4

σ  s

 E  s

= 0,4 ×

191 − 0,6 ×

191 200000

1,34

× (1 + 7,3 × 0,010) 0,010 = 5,24 × 10 − 4 200000

= 3,82 × 10 − 4 .

It must be noted that the above formula differs slightly from the one in the code (Eq. 136). Instead of the factor 0,4 in the original equation, which takes into account an approximately 70% reduction in bond stiffness due to creep, here a factor 0,6 is used (0,4/0,7 = 0,6) as the early-age restraint is not a continuous, long-time effect. /5/ Correspondingly the factor 0,4 is brought to the threshold value. Finally, the calculated crack width is obtained as wk  = s r , max (ε  sm − ε cm ) = 555 × 5,24 × 10 −4 = 0,29  mm. Thus the requirement is satisfied. It is noteworthy that the amount of surface reinforcement required is relatively large for this massive foundation slab; the design should therefore not be overlooked.

79

Summary and conclusions Rapidly increasing use of wind turbine facilities as sustainable energy resources and the technological advancements resulting in increasing hub heights and turbine sizes  present the need for a detailed study of the reinforced concrete slab foundations used for these structures. Traditionally, the slab foundations have been designed identically to ordinary column footings, which are predominantly loaded with a concentric normal force. The true behaviour of these small-scale column footings is well known through numerous experiments with real structures; this means that an adequate design can be carried out with well established methods. This is not the case for massive, extremely eccentrically loaded foundation slabs required to support a wind energy facility. 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. The overall influence in results when modelling the elastic soil – structure interaction by springs or  by volumetric elements, however, was not found to be major. Thus it is sufficient for  practical design purposes to model the soil by compression-only springs. The topic of nonlinear behaviour of reinforced concrete is introduced with plate element analysis of the studied foundation slabs. An important issue when designing members according to nonlinear analyses is to consider proper choice of material parameters; as

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no unified regulations are anchored to the building codes so far. Additionally, the composition of a massive foundation slab can differ from a laboratory-tested small-scale specimen due to numerous issues, including residual stresses, restraints, etc. The results of the 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. Massive inclined cracking is encountered under design load; consequently the provided suspension reinforcement is highly stressed. Only minor flexural cracking occurs in the top and bottom surfaces and thus little stress redistribution takes place. Absolutely no signs of global shear cracking are witnessed; it is thereby concluded that the carried verification against a beam-action shear failure is well on the safe side for this type of structure. Any definitive conclusions can nevertheless only be made when the obtained, theoretical behaviour is verified by experiments with real structures.

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