Design of Wind Turbine Foundation Slabs

8/10/2019 Design of Wind Turbine Foundation Slabs

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2008:128 CIV

M A S T E R ' S T H E S I S

Design of Wind TurbineFoundation Slabs

Pekka Maunu

Lule University of Technology

MSc Programmes in Engineering

Civil and mining EngineeringDepartment of Civil and Environmental Engineering

Division of Structural Engineering

2008:128 CIV - ISSN: 1402-1617 - ISRN: LTU-EX--08/128--SE


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Design of Wind Turbine Foundation Slabs

Pekka Maunu


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Acknowledgements

This thesis, submitted for the Degree of Master of Science at Lule 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 thethesis, Prof J.-E. Jonasson from Lule University of Technology.

I would also like to direct special thanks to Prof L. Bernspng for always being there to

guide me through my studies in Lule. 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................................... 133.1.2 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 5Three-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 briefbackground 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 modellingsimplifications 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 re-

calculating 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)
http://www.energiewerkstatt.at/http://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 slabs

compression zone. Finally, alternative c) presents a connection through a pre-stressed
http://www.noturbinesin.saddleworth.net/http://www.noturbinesin.saddleworth.net/


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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.

WMAV //maxmin/0

= ) 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/

M

V

0max

0min

V

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 wholeinteraction between the structure and the soil is simply spring stiffness per unit area (so

called modulus of subgrade reaction; cs), i.e. the soil pressure is linearly proportional to

the settlement ( scs=0 ).

This method completely ignores the interplay between neighbouring soil elements and

therefore doesnt 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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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


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

reinforcement.


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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.

Nk= 4025 kN

b

b = 17,7 m

h = 3,5 m(2,6 m)

Concrete

= 29 GPa; = 0,20

= 342 cm (252 cm)

E v

d

cm

avg

= 1,35 for applied dead andlive loads (ULS)

Idealised columnc = 4 / 4 m

davg

Figure 7. System for the analysis.


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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

120238

7,17402535,1

8=

==

bM dEd 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 beam-

action 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 1,0daway 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

5,59)42,32/42/7,17(7,17

5434)2/2/(

2;5,3, ==== avg

dshearhEd dcb

A

v kN/m,

and similarly for the thinner slab as

1,75)52,22/42/7,17(7,17

54342;6,2,

=== shearhEdv 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)

2,8023,48

76,1807,17

54345,054345,0 2

5,1

5,1

5,1;;5,3, =

=

==d

dd

d

dpunchinghEdu

AA

v kN/m;

2,11075,39

37,1217,17

54345,05434

2

5,1;;6,2, =

== dpunchinghEdv 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 1,0daway 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

2,9549,37

5,1077,17

54345434

2

0,1

0,1

;5,3, =

=

==d

dd

d

punchinghEdu

AA

v kN/m;

1,12983,31

3,767,17

54345434

2

;6,2, =

== punchinghEdv kN/m.

Tributary reaction for beam-action shear

A1,5d

u1,5d

1,0d1,5d

33,7 45

u1,0d

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 cs= 50 MN/m3is 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. Poissons ratio for concrete is taken as 0,20.

There are several ways of modelling the concentrated load transfer from a column into aslab. /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.

2,0

2,2

1,

10,0

1,0

20,0

2,0

0,0

0 , 1,

2,

,

Figure 10. Soil pressure distribution (kPa) resulting from the column load under a

cut along the slabs (4 x 4 m distributed load).


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a)

22

20

20

0

200

00

00

00

1000

1200

100

0 , 1,

,,

, 2,

, 2,

,, , 2,

, 2,

b)

,

2,,

10,

12,1120,

0

20

0

0

0

100

120

10

0 , 1,

,,

, 2,

, 2,

,, , 2,

, 2,

Figure 11. a) Bending moment distribution (km/m) near the column; b) principal

shear force (k/m) across a section 1,0dfrom 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 beam-

action 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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Method mEd[kNm/m] vEd;1,0d[kN/m]

Calculation by hand 978 (126%) 129,1 (99%)

Point load 1900 (245%) 130,6 (100%)

Loaded area 4x4 m 775 (100%) 130,5 (100%)

Loaded area 6,6x6,6 m 520 (67%) 124,1 (95%)

h= 2,6 m

Coupling of elements 589 (76%) 120,7 (92%)

Calculation by hand 978 (119%) 95,2 (98%)

Point load 1950 (237%) 96,5 (100%)

Loaded area 4x4 m 822 (100%) 96,8 (100%)

Loaded area 7,5x7,5 m 493 (60%) 92,4 (95%)

h= 3,5 m

Coupling of elements 589 (72%) 93,3 (96%)

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 doesnt 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


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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

172770tan

1

65tan

1

50tan

1

40tan

1

33

5434=

+++

=tF kN.

Assuming an effective width of dcbeff 2+= 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.

40

N /3d

50 65 70

c

d

Figure 13. A strut and tie model of the foundation slabs.

The tensile forces in reinforcement from the bending moments resulting from the FE-

models 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 dz 9,0 yields values ranging from (excluding the pointload models) 160-267 kN/m and 229-342 kN/m, respectively for the 3,5 m- and 2,6 m-

thick slabs. Hence it seems that all the studied load transfer models produce results that

lie more or less on the conservative side.


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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.

Nk= 4025 kN

b

b = 17,7 m

h = 3,5 m(2,6 m)

Concrete= 29 GPa; = 0,20

= 342 cm (252 cm)

E v

d

cm

avg

= 1,35 for applied dead and liveloads (ULS)

Idealised columnc = 4 / 4 m

My,k= 93345 kNm

davg

x

y

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.


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FLEXURAL ANALYSIS

The bending moments mEd,xcan 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

120238/7,17402535,1 ==SYMMM 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 mustbe led equally to both halves of the foundation:

630082/9334535,1 ==ASYMMM 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.

509851202363008 =+=EGM kNm;

750311202363008 =+=POSM kNm.

+

N

Line of zero moment

Fictitious soil pressurefrom asymmetric load case

Soil pressure fromsymmetric load case

M

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


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moment; for instance, considering the 2,6 m thick slab, the moment caused by its self

weight resisting the uplift is

450552/85,87,176,2252 ==SLABM kNm.

It has to be pointed out that the design action of the slabs 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 MPOS

determined previously; i.e. the maximum bending moment in the 2,6 m thick slab will

be

80961)4505550985(75031max,, =+=xEdM kNm.

In this case the minimum moment is caused by the fully utilised self weight:

45055min,, =xEdM kNm.

It is then assumed that the positive flexure is carried by a substitute beam with a breadth

of bdcbeff += 2 where cmeans 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:

8956)52,224/(80961max,, =+=xEdm kNm/m;

5632)42/(45055min,, ==xEdm 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.


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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 1,0dfrom 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 vRd,ctof 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.

mEd,x,max[kNm/m] mEd,x,min[kNm/m] vEd[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)


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1,0d

Tributary reaction for shear

Critical section ucrit

45

maxu,crit

0

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:

0,9)52,22(2 =+=critu m

Soil pressure resulting from the applied loads at different sections (see fig. 15):

1547,17/69334535,17,17/402535,1 32max =+= kPa

871197,17/)154119()52,2285,8(, =+++=critu kPa

181197,17/)154119(85,80 =+= kPa

Tributary soil reaction for shear:


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27

72156

0,9)1887(

4

0,918

6

7,17)18154(

4

7,1718 2222=

+

=R kN

Shear force acting along the critical section:

8020,9/7215 ==Edv 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 columnregion. (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.


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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.

1

1

2

102

12

0

0

22

12

000

000

000

2000

0

2000

000

000

000

0 , 1,

2,

2,

,, 2,

,,

Figure 17. Bending moment mx(km/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


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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.

11

12

10

2

0

200

00

00

00

1000

1200

100

0 , 1,

2,

2,

,, 2,

,,

Figure 18. Principal shear force (k/m) across a lateral section 1,0daway 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.


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With a Youngs modulus of 200 MPa and a Poissons ratio of 0,30 the elastic soil half-

space 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.

10 m

5b

b=17,7m2,5b

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.


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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)

b)

0

0

100

10

200

20

00

2,00 1,0

,

,

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.


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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 armzfrom the neutral axisz0over

the cross section height yields the bending moment acting in the corresponding

direction:

dzzm

z

zh

xx

=0

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 mEd,x,max= 7568 kNm/m, and

the minimum mEd,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% (mEd,x,max = 7039kNm/m and mEd,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


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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:

dzvh

yzxz +=0

22 .

Across the width of the slab at a distance 1,0daway from the edge of the loaded area a

shear force distribution as presented in fig. 23b is then found. The resulting peak of vEd

= 958 kN/m is again best represented by the plate element model with 4 m x 4 m loaded

area for the pressure trapezoid (vEd = 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%.

540

-105 0 -1740

5400 -6670

2830

-1270

1980

-644

683

228

-245

Figure 22. Principal stress field and distribution of horizontal stresses (with top

and bottom surface stresses in kPa) in the foundation slab.


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a)

0

000

000

2000

0

2000

000

000

000

0 , 1,

,

b)

0

200

00

00

00

1000

1200

0 ,

,

Figure 23. a) Bending moment mx(km/m) and b) principal shear force (k/m)

across a lateral section 1,0daway from the face of the column in the 3,5 m thick

slab. (Three-dimensional modelling of structure and soil)


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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.

Method mEd,x,max[kNm/m] mEd,x,min[kNm/m] vEd[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%)

h= 2,6 m

Coupling of elements;

Soil as springs5673 (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%)

h= 3,5 m

Coupling of elements;Soil as springs

5540 (79%) -4172 (76%) 827 (95%)

Table 3. Summary of analysis results.


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3.3 Summary of Chapter 3

It has been demonstrated how the design of foundation slabs can be verified againstaggravatingly 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 FE-

analysis 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.


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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


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continue to grow slowly. /25/ Under biaxial compression the concrete exhibits increased

ultimate strength; here a typical assumption of 1,16fcis 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.

0,0035

c

fc

0,85fc

0,60fc

c1

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

)10/1ln(12,2 cct ff += . [ 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.


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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/ )

ctF fG = 0307,0 . [ 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 resultingincrease 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.

a)u0 u

t

fct Gf

b)u0 u

t

fct Gf

1/3fct

2/9u0

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).


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0,025

s

ftfy

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 wasoriginally tested byBurnsand Siessin 1962 /6/, and was referred to as specimen J-4 in

that experiment.

P

3,66 m

46 cm 51 cm

20 cm

Figure 27. Beam J-4.

The principal material parameters used in the numerical models are as follows: (adopted

from /25/)

fc= 33,2 MPa;


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fy= 310 MPa;

Ec= 26,2 GPa;

Es= 203 GPa;

= 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)


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a)

0

2

0

100

12

10

1

0 2

b)

0

0

100

10

200

20

00

0 2

Figure 29. a) Load-deflection behaviour and b) stress in reinforcement at mid-span

of beam J-4.


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4.2 Design methods to nonlinear analyses

The traditional design of a critical section per se is not required with nonlinearanalyses, 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 loadRkwhich the system is able

to carry is reduced by a safety factor R. Then a comparison against the relevant design

load combination is performed:

d

R

kkqkgd R

RQGE ==

.

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


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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:

c 3,5 mm/m

s 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

Section Design

Reinforcement

Stiffness

Figure 30. Dependence between member forces and reinforcement.


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4.3 onlinear analysis of the foundation slabs

As the system and its loading are principally identical as in the preceding chapter, thelinear 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 fordetermining the member forces is still an issue of great uncertainty. /30/ The DIN1045-

1 prescribes the compressive strength of concrete to be factored as

ckcR ff = 85,0 ,

where is generally to be taken as 0,85. For a C30/37 used in the foundation slabs

would hence resultfcR= 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 fctm

= 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 fctmwould 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 ctmct ff 5,0= , 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


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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:

38== cmc ff MPa;

45,19,25,05,0 === ctmct ff MPa;

5505001,11,1 === yky ff 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.

0,175,1/3085,0 ===c

ckcd

ff

MPa; (C30/37)

43515,1/500 ===s

yk

yd

ff

MPa; (BSt500)

Bottom layer:

0666,02521000,17

107188

2

3

2

=

==

df

m

cd

Ed ;

069,00666,0211211 === ;

95,67435/0,17252100069,0/, === ydcdrqds fdfa cm2/m.

Top layer:

0480,02521000,17

1051852

3

=

= ; 049,0= ;


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26,48435/0,17252100049,0, ==rqdsa cm2/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.


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1

1

0

0

000

000

2000

0

2000

000

000

000

0 , 1,

2,

2, ,

,

12

10

02

0

200

00

00

00

1000

1200

100

0 , 1,

2,

2,

,

,

Figure 31. a) Design bending moment mxalong the slabs. b) Principal shear force

across a section 1,0daway from the column.


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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 mxin 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.


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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.


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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.


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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. crit= p).

(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).

a)Contact pressure

Shearstress

Stick region

crit

b)Total slip

Shearstress

Slipping friction

Sticking friction

crit

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 = 2,75. (fig. 36) Therefore a

reasonable value for the friction coefficient of= 0,7 is chosen for the steel concrete

interface.


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= 0

p

= 0,7

= 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 zfrom 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

154)42,324,4/(/5434 =+= tf kN/m.


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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.

x = 0 1,50 m 3,00 m 5,00 m

210 kPa 209 kPa 109 kPa 28 kPa

-118 kPa -165 kPa -151 kPa -33 kPa

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

concrete slab.


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Nd

4,4 m

d 342 cm

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

Soft layer

Contact opening

Bottom flange of steel ring

Concrete

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.


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The soil pressure distribution resulting from the 3D-axisymmetric analysis differs

slightly from

Design of Wind Turbine Foundation Slabs

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