foundation slab eurocode
TRANSCRIPT
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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 Engineering
Department of Civil and Environmental EngineeringDivision 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 the
thesis, 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................................... 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 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)
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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
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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 whole
interaction 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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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
Youngs modulus and the Poissons 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 non-
uniform 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
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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 and live 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 1,0d 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
5,59)42,32/42/7,17(7,17
5434)2/2/(
2;5,3,==== avg
d
shearhEd dcbA
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
d
d
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 1,0d 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
2,9549,37
5,1077,17
54345434
2
0,1
0,1
;5,3, =
=
==d
d
d
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/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. Poissons 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.
29,0
23,2
17,3
10,0
15,0
20,0
25,0
30,0
0 8,85 17,7
h=2,6 m
h=3,5 m
Uniform distribution
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)
822
493
589
775
520589
DAfStb Heft 240
0
200
400
600
800
1000
1200
1400
0 8,85 17,7
Point load
Loaded area 4x4 m
Loaded area 7,5x7,5 m
Coupling of elements
Point load, h=2,6 m
Loaded area 4x4 m, h=2,6 m
Loaded area 6,6x6,6 m, h=2,6m
Coupling of elements, h=2,6 m
b)
96,8
92,493,3
130,5
124,1120,7
0
20
40
60
80
100
120
140
0 8,85 17,7
Point load
Loaded area 4x4 m
Loaded area 7,5x7,5 m
Coupling of elements
Point load, h=2,6 m
Loaded area 4x4 m, h=2,6 m
Loaded area 6,6x6,6 m, h=2,6m
Coupling of elements, h=2,6 m
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 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.
-
20
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
-
21
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 point
load 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.
-
22
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 live loads (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.
-
23
FLEXURAL ANALYSIS
The bending moments mEd,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
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 must
be 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
-
24
moment; for instance, considering the 2,6 m thick slab, the moment caused by its self
weight resisting the uplift is
450552/85,87,176,225 2 ==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 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:
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.
-
25
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 vRd,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.
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)
-
26
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:
-
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 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.
-
28
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.
-3674
5673
-5185
7188
-2566
4102
-4172
5540
-5495
7039
-2328
3312
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
0 8,85 17,7
Coupling of elements; h=2,6 m
Pressure trapezoid 4x4 m; h=2,6 m
Pressure trapezoid 6,6x6,6 m; h=2,6 m
Coupling of elements
Pressure trapezoid 4x4 m
Pressure trapezoid 7,5x7,5 m
Figure 17. Bending moment mx (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
-
29
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.
1163
1279
1095
827868
759
0
200
400
600
800
1000
1200
1400
0 8,85 17,7
Coupling of elements; h=2,6 m
Pressure trapezoid 4x4 m; h=2,6 m
Pressure trapezoid 6,6x6,6 m; h=2,6 m
Coupling of elements
Pressure trapezoid 4x4 m
Pressure trapezoid 7,5x7,5 m
Figure 18. Principal shear force (k(/m) across a lateral section 1,0d 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.
-
30
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.
-
31
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
50
100
150
200
250
300
-2,00 17,70
Soil as volume elements; h=3,5 m
Soil as springs; h=3,5 m
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.
-
32
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 z0 over
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 = 7039
kNm/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
-
33
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 1,0d away from the edge of the loaded area a
shear force distribution as presented in fig. 23b is then found. The resulting peak of 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
-1050 -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.
-
34
a)
-5043
7568
-6000
-4000
-2000
0
2000
4000
6000
8000
0 8,85 17,7
3D; h=3,5 m
b)
958
0
200
400
600
800
1000
1200
0 8,85
3D; h=3,5 m
Figure 23. a) Bending moment mx (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)
-
35
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 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%)
h = 3,5 m
Coupling of elements;
Soil as springs 5540 (79%) -4172 (76%) 827 (95%)
Table 3. Summary of analysis results.
-
36
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 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.
-
37
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
-
38
continue to grow slowly. /25/ Under biaxial compression the concrete exhibits increased
ultimate strength; here a typical assumption of 1,16fc 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.
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.
-
39
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 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.
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).
-
40
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 was
originally tested by Burns and Siess in 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;
-
41
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)
-
42
a)
0
25
50
75
100
125
150
175
0 2 4 6 8
Deflection mm
Loa
d k
N
Measured
Bilinear tension softening
Linear tension softening
b)
0
50
100
150
200
250
300
0 2 4 6 8
Deflection mm
Str
ess
MP
a
Bilinear tension softening
Linear tension softening
Figure 29. a) Load-deflection behaviour and b) stress in reinforcement at mid-span
of beam J-4.
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43
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 R. Then a comparison against the relevant design
load combination is performed:
d
R
k
kqkgd RR
QGE ==
.
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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44
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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45
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 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 result fcR = 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 fctm 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 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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46
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
ck
cd
ff
MPa; (C30/37)
43515,1/500 ===s
yk
yd
ff
MPa; (BSt500)
Bottom layer:
0666,02521000,17
1071882
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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47
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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48
-5185
7188
-4785
5668
-5495
7039
-5399
5730
-6000
-4000
-2000
0
2000
4000
6000
8000
0 8,85 17,7
Linear; h=2,6 m
Nonlinear; h=2,6 m
Linear; h=3,5 m
Nonlinear; h=3,5 m
1279
1097
868
802
0
200
400
600
800
1000
1200
1400
0 8,85 17,7
Linear; h=2,6 m
Nonlinear; h=2,6 m
Linear; h=3,5 m
Nonlinear; h=3,5 m
Figure 31. a) Design bending moment mx along the slabs. b) Principal shear force
across a section 1,0d away from the column.
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49
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 mx 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.
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50
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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51
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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52
350
190
16050
5,2
11
11
Soft layer
D=440
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: Youngs modulus is taken as 29 GPa (corresponding to a C30/37) and
Poissons 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 Youngs modulus of 210
GPa and a Poissons 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
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53
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
Sh
ea
r s
tre
ss
Stick region
cri t
b)Total slip
Sh
ea
r s
tre
ss
Slipping friction
Sticking friction
cri t
Figure 35. Friction model for steel ring concrete interface.
The influence of fricti