Articles
Amir M. Alani*
Morteza Aboutalebi
DOI: 10.1002/suco.201100043
Analysis of the subgrade stiffness effect
on the behaviour of ground-supported
concrete slabs
This paper confirms that the structural behaviour of groundsupported slabs is a non-linear function of the structural properties of slabs as well as the supporting soil. The findings reported
emphasize that the suggested equations used in design codes
pay insufficient attention to the effect of the supporting ground
stiffness within the context of the mechanical behaviour of slabs
as far as ductility is concerned. The results presented demonstrate that ground stiffness has a significant effect on the ductility
of ground-supported slabs. It also pays particular attention to the
possibility of determining the ductility limit of slabs.
Keywords: stiffness, ground slab, failure, ductility, behaviour
1
Introduction
The increase in the capacity of a structure is not essentially equivalent to improved structural behaviour. For
around two decades, the leading engineering codes have
guided designers towards more ductile design procedures.
These procedures do not always end in structures with a
greater loadbearing capacity, but may well end up with the
structure’s energy-dissipating ability being increased.
Almost all industries use ground slabs as a vital base
for production lines or storage areas. The vast areas covered by ground-supported slabs are usually subjected to
both predicted and unpredicted stationary and moving
loads. These mostly transverse loads on slabs may cause
failure in the form of settlement, punching shear, shear
fracture, bending failure or bearing crush. In the UK, the
Concrete Society has published the 3rd edition of the
TR34 report [1], which is currently under review. It has
been noted both in this report and in the National Cooperative Highway Research Program (NCHRP) Report 372
[2] “Support under Portland cement concrete pavements”
that the subgrade stiffness has a minor effect on the required slab thickness and the flexural stresses built up by
the transverse loads. This conclusion in both references is
based on the yield-line method analysis in which the effect
of the support reaction has not been considered.
Although in most construction cases engineering the
subgrade layer is not considered an economically feasible
* Corresponding author: m.alani@gre.ac.uk, am50@gre.ac.uk
Submitted for review: 01 September 2011
Revised: 10 February 2012
Accepted for publication: 21 February 2012
102
option, the sub-base layer is often compacted to a certain
degree. Potentially, the subgrade stiffness can be altered to
the desired value by some compaction efforts. In this, the
modulus of subgrade reaction, as an independent factor,
can be considered as a design parameter in most practical
cases.
The issue of bearing stress distribution in the soil
supporting the ground slab was introduced by Westergaard more than 80 years ago [3]. Westergaard assumed
the slab to be a homogenous, isotropic elastic solid sustained by the reactions from an elastic subgrade which are
only vertical and proportional to the deflections of the
slab. He introduced the term “modulus of subgrade reaction k”, which represents the behaviour of the subgrade
based on the load per unit area causing unit deflection.
This behaviour is based in principle on the Winkler model, which is traditionally preferred for slab-on-ground design. In this research, the authors have based the soil representing Winkler springs on the elastic isotropic
constitutive model of behaviour in the same way as Westergaard proposed.
Researchers worldwide have reported different results based on both experimental and theoretical studies
into the effect of the k factor on the first crack and final
bearing capacity of ground-supported slabs [4, 5]. Although the design codes of practice commonly used, e.g.
the 3rd edition of the TR34 report [1] and the 1995
NCHRP Report 372, disregard the effect of the supporting
soil stiffness variation, some researchers do consider it to
be a significant influencing factor. Moreover, there is no
wide-reaching agreement about the trend of the effects of
changing the modulus of resilience of the soil layer. Irving,
as part of his work, reported that “increasing the compaction effort gives a slab an increased first cracking load
and an increased failure load” [4]. He also concluded from
his experimental studies that “an increase in k factor results in improved post-cracking ground slab behaviour”.
Opposing that, Kearsley has concluded that the increase
in ductility of fibre-reinforced concrete ground slabs is
more remarkable if the supporting soil is not so stiff [5].
Regarding these rather contradictory reports and similar
beliefs within the community, the authors of this paper
recognized that there is ample scope to investigate the
state of the effect of subgrade and sub-base layers on
the behaviour of ground slabs from the ductility point of
view.
© 2012 Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin · Structural Concrete 13 (2012), No. 2
A. M. Alani/M. Aboutalebi · Analysis of the subgrade stiffness effect on the behaviour of ground-supported concrete slabs
The subject of “distribution of the bearing stresses in
supporting soil media beneath a slab under vertical load”
has been extensively investigated both theoretically and
experimentally [6–13]. The most important shortcoming
of the investigations was the presumed elastic behaviour of
the slab prior to Meyerhof’s reported work. He studied the
phenomenon of the concrete slabs and their ability to experience plastic behaviour under higher loading conditions. Meyerhof concluded that the theory developed
would lead to considerably more economic slab design.
The distinctive feature of this work was that the equations
drawn up were based on equating the reduction in potential energy as the load was imposed to the amount of
strain energy absorbed by the formation of cracks.
The strength and deformation behaviour of plain and
steel fibre-reinforced concrete ground slabs has also been
investigated by Falkner et. al [8]. In that study, experimental results for two plain concrete and four steel fibre-reinforced concrete ground slabs under central loading were
used to calibrate and validate the finite element (FE) models presented. Based on those results, a rationalized design
principle applied to plain and steel fibre-reinforced concrete ground slabs was introduced. A study by Meda et. al
[9] on the fracture behaviour of steel fibre-reinforced concrete ground slabs was performed by using non-linear fracture mechanics FE analyses. In this study, the non-linear
fracture mechanics FE analyses compliantly predicted experimental results for the crack development, ultimate
load and collapse mechanism of the ground-supported
slabs.
Chou applied the finite element method (FEM) to
demonstrate that the slab panel size has a significant effect on the build-up of stresses and also the settlement of
the slab [7]. Further, Shentu demonstrated in another
study that the subgrade reaction under a circular-loaded
slab has a non-linear inverse relationship with the extent
of the slab [10]. Tests of slabs have also been conducted by
Hadi in which the use of synthetic fibres has once again
been compared to steel fibres using slab specimens of
820 × 820 mm supported at four corners and subjected to
a central point load [11].
In addition, numerous experimental investigations,
particularly in the UK, have led to a much clearer understanding of the mechanical behaviour of slabs. In the late
1980s, tests were carried out by Beckett establishing the
effect of steel fibre type and size within this domain [12,
13]. It was established by Beckett that the aspect ratio of
the fibres played an integral part in the load at which first
cracking occurred in slabs. Subsequently, the most recent
investigations have been performed (and are being performed) at the University of Greenwich by Alani et al, in
which tests on larger slabs (6 × 6 m) have led to a better
understanding of the mechanical behaviour of slabs. Early
results have revealed that the effect of corner uplift can be
resolved by considering large-size slabs. As emphasized
before, this paper is a part of wider studies that Alani and
Beckett are currently carrying out at the University of
Greenwich. These long-term, substantial studies of large
ground-supported slabs are being carried out in specially
developed, state-of-the-art facilities dedicated to this field.
These studies have concentrated on size and types of reinforcement used (steel, synthetic and fabric fibres as well as
Fig. 1. Ground slab test facilities at the University of Greenwich
plain concrete) for ground-supported slabs for industrial
applications. Fig. 1 illustrates certain aspects of a steel fibre-reinforced concrete slab under investigation.
The subject of ground-supported slabs can be considered within the context of three interlinked elements: the
supporting soil, the slab and their mutual effects on the
failure phenomenon. A recent paper by Tchrakian highlights the inadequacies of investigations into the structural
performance of ground-supported slabs when the interaction between the slab and the subgrade is disregarded or
oversimplified [14]. This finding supports the idea of the
inclusion of the supporting soil characteristics as well as
the slab properties in any in-depth study of the subject in
the future. The equations by Meyerhof [6] are empirical in
nature and since the innovation of FEM many researchers
have set upon the task of predicting the plastic response of
slabs more accurately. The development of versatile and
powerful FE packages such as ANSYS and improvements
in computing hardware such as computers with multiple
CPUs has made it possible to investigate such phenomena
in more detail and with more precision.
The following section presents the assumptions and
the techniques used to develop the FE models with the
help of ANSYS 12.1.
2
Theoretical background
A concrete slab resting on an elastic subgrade can be represented by an elastic slab bearing on a dense liquid or
elastic solid by the conventional theory of elasticity [15].
The theory of elasticity predicts the behaviour of these
slabs for loads at or below initial yield. The theory of elasticity becomes less accurate for higher loads [6]. For that
reason, a plastic approach might be a feasible option for
analytical purposes. The yield-line theory can also be
adopted in order to evaluate the ultimate bearing capacity
of reinforced concrete slabs, which is based on assuming a
rigid-plastic slab resting on an elastic subgrade.
The case studied in this paper concerns a load applied at the centre of a flat slab resting on the ground. The
mode of failure considered is a circular negative failure on
the top surface and positive radial failure lines from the
Structural Concrete 13 (2012), No. 2
103
A. M. Alani/M. Aboutalebi · Analysis of the subgrade stiffness effect on the behaviour of ground-supported concrete slabs
load application point to the perimeter of that circle on
the bottom surface. Fig. 2 illustrates this case more clearly.
The governing equation for the vertical deflection
w(x, y) of a plate subjected to a point load P at the centre
of the slab and a spread support reaction q(x, y) is given by
Timoshenko [16]:
( )
( ) ( )
D∇ 2∇ 2w x, y = Pδ 0 , 0 − q x, y
(1)
In this equation, δ is the Dirac function, which is equal to
zero apart from in the vicinity of the origin of the system
of coordinates where the load is applied. Based on this differential equation, the displacement of a fixed circular
plate of radius r with a central concentrated load P is calculated by Timoshenko [16] as:
∆=
Pr 2
16π D
(2)
where:
D=
Eh3
12 1 − v 2
(
)
(3)
and h stands for the slab thickness and v is Poisson’s ratio.
The elastic stiffness is therefore equal to:
KS =
16π D
r2
(4)
In this case the collapse load without soil support PCo is
found to be:
PCo =
(
)
2π M + M′
(5)
2a
1−
3r
where M and M' are the positive and negative moment capacities respectively, and a is the radius of the load application area [17]. It is also possible to prove that the collapse load of the soil support PP with soil stiffness K is
equal to [15]:
P=
(
)
2π M + M′




4

2a 
γ Kπ r
1−
1 +

 2a  
3r 
3D 1 −

3r  

(6)
This in turn demonstrates the considerable effect of the
soil on the ultimate capacity of the ground-supported slab.
As mentioned before, based on this equation, the loadbearing capacity is a non-linear function of the slab and
the supporting soil properties. It is shown later that the
ductility of the slab is not only a function of the slab
properties but also depends on the supporting soil condition.
It has been proved that as slab plan area increases,
settlement decreases due to increased bending action, and
the built-up stresses rise as well [7]. It is interesting to note
that in Chou’s investigation, a limit has been found for the
104
Fig. 2. Cone-shaped failure mechanism
Structural Concrete 13 (2012), No. 2
slab plan size at which this relationship breaks down. Two
distinct types of behaviour were observed by Chou in his
study: a) “large slab effect”, and b) “small slab effect”. In
his studies of large slabs, slab plan size was noted to have
limited effect on the variation of resulting stresses and deflections under loading conditions.
A thorough study of how different factors such as
slab size, modulus of subgrade resilience and rebar ratio
affect the behaviour of the slab under different types of
failure modes, e.g. ductile flexural and/or brittle punching
shear failure, is under investigation by the authors of this
paper. However, the focus here is on a better understanding of how modulus of subgrade reaction affects the failure characteristics of ground-supported slabs. Although
when studying the behaviour of ground slabs it is traditionally accepted that the first yield line mechanism
demonstrates the ultimate load of the system [17], if the
failure mode tends to become a more ductile one, then
there is a strong possibility that the first crack line becomes tolerable when the load is not permanent. Moreover, the idea of rehabilitating the slabs under a ductile
failure condition before they are permanently damaged
will save considerable effort and resources, whereas the
brittle modes of failure do not allow such attributes to be
exercised.
3
Finite element models
It was considered that the use of finite element (FE) methods is a viable way of studying how subgrade stiffness affects the ductility of ground-supported slabs. No doubt
this approach supersedes any form of practical and experimental study which will require a relatively substantial
amount of resources and time.
The non-linear capabilities of ANSYS for modelling
crack formation and patterns of concrete structures/elements make it particularly suitable for investigating the
ductility of ground-supported slabs. It also caters for possible elimination of tension causing upward deflection of
the edges.
The FE models in this study consisted of 3 × 3 m
square panels 20 cm deep with 5 cm long Link10 elements
representing the elastic soil based on the Westergaard theory and the Winkler spring method [3]. In order to achieve
A. M. Alani/M. Aboutalebi · Analysis of the subgrade stiffness effect on the behaviour of ground-supported concrete slabs
Fig. 4. Loading pattern on top surface of slab
Fig. 3. Solid and link elements in FE model
consistent results, the no-tension property of the Link10
element was used. Fig. 3 shows the slab and the link elements representing the supporting soil. Solid65 concrete
solid elements capable of modelling the reinforced concrete slabs have been used, for which the concrete material has been modelled as precisely as possible. The Solid65
element is capable of modelling cracking in tension and
crushing in compression. In the ANSYS 12.1 models, the
failure criteria of the concrete were described via the
Willam and Warnke model [18] given the following constants:
– the shear transfer coefficient was set for an open and a
closed crack with values of 0.2 and 0.8 respectively
– uniaxial crushing stress ( ƒc) was set to 20 MPa
– uniaxial tensile cracking stress was set to 2 MPa
The above constitutive model is capable of considering the
tensile cracking of concrete. The slab modelled in this
study consisted of several layers of elements in the FE
models. This will allow cracks to develop in the most vulnerable layer with respect to the applied load which in
turn can spread to the other layers. The programme developed was written such that the effect of cracks on the stiffness of the slab is taken to account according to each load
increment. This was achieved by modifying the stiffness
matrix in each previous loading condition within the yielded solution. It is also useful to note that the reduced ability of concrete to transfer shear across cracks is considered
in ANSYS.
The reinforcement was defined in three perpendicular directions for which a 0.001 rebar ratio was assumed
for all directions based on the random reinforcement
alignment for steel fibres. The reinforcing material is a
multi-linear representation of ordinary rebar material.
The load was applied to a set of nine central nodes
(the load applied on the central node was twice that on
the other eight nodes) at the upper slab surface as shown
in Fig. 4. Spreading the load over the nine central nodes is
understood to increase the chance of convergence of the
FE models. By preventing a crush failure in the centre of
the slab, this would allow a better understanding of the to-
tal effects within the slab rather than restricting the final
step to the effects of local crushing under the applied load.
The amount of load applied to the slab was far greater
than the anticipated capacity of the slab, so cracking
and/or crushing is guaranteed before the analysis finishes.
This way, the number of results saved was increased to
20 000 prior to the termination of the analyses.
The supports of the system were placed at the bottom of each link element separately to confine the movement in all three perpendicular main directions. It should
be noted that the stiffness produced by each of the Link10
elements is the function of three factors: the modulus of
elasticity of the material, the cross-sectional area and the
inverse of their length. In order to calculate the resulting
stiffness of the subgrade, this should be multiplied by the
number of link elements in each unit area. In this particular work, the variation in the modulus of subgrade reaction was controlled by the area of the link elements and
the other factors were kept constant. The soil was assumed to range from a very soft non-impact layer to a theoretically almost infinitely stiff one. A total of 25 different
FE models were analysed, and the most important numerical results are presented. Loss of contact between the slab
and the soil is accounted for by using the gap property of
the Link10 element of ANSYS. If the no-tension contact
between soil and structure is not modelled, the unreal tension between the soil and the slab in the vast areas where
the deflection of the slab is upward will cause considerable moments and shears in the slab and consequently the
structural results will be unreliable.
4
Numerical results
The structural response factors monitored in the FE models consist of the deflections, stresses and cracking pattern
representative of the plastic energy dissipation.
To validate the FE results, as implied before, the experimental data for 6 × 6 × 0.15 m slabs as the result of a
separate study was used. To that effect, a similar FE model
with the same dimensions as in the test was developed
and the deflections of the slab in the test were compared
with those of the FE model. The comparison between results is shown in Fig. 5. The results demonstrate a satisfac-
Structural Concrete 13 (2012), No. 2
105
A. M. Alani/M. Aboutalebi · Analysis of the subgrade stiffness effect on the behaviour of ground-supported concrete slabs
Table 1. Effect of the support stiffness on the structural response of the
slabs
Fig. 5. Comparative experimental and FE modelling results for the deflection of a 6 × 6 m slab under central load
k (N/mm3)
Bearing capacity
(kN)
Extent of cracked
zone (m)
0.1
111
0.6
0.08
102
0.9
0.06
88
1.8
0.0575
85
1.8
0.055
83
2.1
0.05
77
2.1
0.0475
74
2.1
0.045
71
2.1
0.04
66
2.4
0.035
63
2.4
0.03
61
2.4
0.025
55
2.4
0.02
50
2.7
0.0196
50
2.7
0.0185
49
2.7
0.0175
47
3
tory degree of compatibility between the experimental and
the FE results.
The short-term settlement of the slab is inversely proportional to the modulus of subgrade reaction. As shown
in Fig. 6, the maximum deflection of the slab under a specific amount of load at the centre of the slab increases almost linearly as the amount of subgrade resilience modulus decreases. It is an obvious and predictable result, but
what is very important and must be emphasized is the rate
of change of deflection, especially in the central region of
the slab. As seen here, the gradient of deflection is directly
related to the stiffness of the subgrade. This leads to the
belief that as the stiffness of the soil increases, the main resisting part of the slab is more and more confined to a
smaller central region of the slab.
These results could lead to two major consequences.
Firstly, increasing the stiffness of the sub-base material
causes a more locally activated flexural region in the slab
which will decrease the slab’s energy dissipation capability. Secondly, it will increase the crack width in the central
part of the slab, which can subsequently increase the possibility of rebar deterioration and excessive crack opening.
The concentration of the flexural behaviour of the
slab in the central region is also evident tracing the cracking zone of the slab as subgrade resilience modulus is increased. Fig. 7 illustrates the cracked zones of the slab at
the ultimate loading condition for selected k values. It
should be noted that the amount of the ultimate load is
not the same for these cases. This will be discussed later.
The confinement of the cracked zone as the k value increases is an endorsement of the disadvantage of a higher
value of subgrade stiffness claimed previously.
(a)
(b)
0.015
46
3
0.0125
44
3
0.01
0.005
41
37
3
3
Fig. 6. Deflection of a 3 × 3 m slab under similar loads: (a) for k = 0.020 N/mm3, (b) for k = 0.080 N/mm3
106
Structural Concrete 13 (2012), No. 2
A. M. Alani/M. Aboutalebi · Analysis of the subgrade stiffness effect on the behaviour of ground-supported concrete slabs
(a) for k = 0.020 N/mm3
(b) for k = 0.040 N/mm3
(c) for k = 0.0475 N/mm3
(d) for k = 0.060 N/mm3
Fig. 7. Crack formation in a 3 × 3 m slab under similar loads with different k values
Table 1 indicates the results of the finite element
models.
As shown in Fig. 8, the extent of the plastic zone,
where most of the input energy is dissipated by plastic deformation of the cracks, decreases as the modulus of resilience of the supporting soil increases. This very important result has never been experimentally investigated
because setting up experimental models for different soil
types with various controlled sub-base stiffness conditions
is not only very expensive but also very time-consuming.
Furthermore, Fig. 9 demonstrates that the loadbearing capacity of the slab is a non-linear function of the soil
stiffness. The ultimate slab capacity increases as the k factor increases, but it is not linearly proportional, and the increase in capacity is not considerable with respect to the
increase in k.
It has also been observed that the capacity of the
slab increases to some extent as the stiffness of the supporting soil layer increases. Basically, structural engineers
are required to provide the minimum essential value of resilience modulus of the subgrade and sub-base layers within the context of ground-supported slabs. They are also required to provide the predicted short- and long-term
settlements and bearing capacity as well as practical prerequisites such as the integrity of the sub-base under construction traffic. To that effect, the findings of this investi-
Fig. 8. Extent of the cracked zone plotted against soil stiffness
Fig. 9. Ultimate plastic bearing capacity of slab plotted against soil stiffness
Structural Concrete 13 (2012), No. 2
107
A. M. Alani/M. Aboutalebi · Analysis of the subgrade stiffness effect on the behaviour of ground-supported concrete slabs
gation suggest that it would be useful to consider that using stiffer soils under the ground-supported slabs can reduce the area of the slab in which the maximum stresses
are built up and the energy is dissipated by cracking and
crushing. Hence, the increase in the modulus of subgrade
stiffness does not essentially improve the behaviour of the
slab.
5
Conclusions
No particular specifications are presented in the widely
available design codes for the modulus of resilience of the
subgrade reaction and hence ground slabs could be designed regardless of the subgrade soil characteristics.
This paper sets out to investigate how subsoils with
different stiffness values affect the mechanical behaviour
of ground-supported slabs. This aim was fulfilled by a set
of finite element (FE) analyses carried out by applying the
ANSYS 12.1 package to 3 × 3 × 0.2 m slabs made up of
Solid65 elements with a 0.001 volumetric rebar ratio and
Link10 supporting elements which represented the supporting soil layer with various stiffness values.
This work demonstrated that the increase in the
modulus of resilience will increase the rate of deflection of
the slab in a smaller area under the applied point load
with an overall decrease in the maximum deflection of the
whole slab. This subsequently leads to the area in which
concrete cracking occurs being confined to a smaller
zone, which in turn can be interpreted as restricting the
area affected by the energy dissipation. This will cause
both a decrease in the ductile behaviour and an increase
in the crack width and depth of the slab.
It can be concluded that although a minimum value
of ground stiffness is necessary to provide the required capacity, greater stiffness will avoid a desirable ductile behaviour which besieges a considerable portion of the slab
in the load-conveyance process. Therefore, it is of paramount importance to stress the fact that the effect of the
modulus of resilience on the behaviour of ground slabs
cannot be ignored from a ductility point of view, which
contrasts with the predictions of the design codes commonly used.
It is considered that the findings of this paper could
potentially be used in the design process of any groundsupported concrete slabs in order to provide the minimum
soil stiffness required to avoid excessive deflections. It can
be recommended that in designing ground-supported concrete slabs, designers establish a minimum value for the
subgrade modulus of resilience, based on the deflection
and bearing capacity requirements, and then design the
slab. This will no doubt provide sufficient information to
engineer the soil accordingly.
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108
Structural Concrete 13 (2012), No. 2
Prof. Amir M. Alani
Head of Dep. of Civil Engineering
and the Bridge Wardens’ Chair in
Bridge & Tunnel Engineering, University of Greenwich, Chatham, UK
m.alani@gre.ac.uk
Morteza Aboutalebi
Post-doctoral Fellow
Department of Civil Engineering
University of Greenwich, Chatham, UK
aboutalebi@greenwich.ac.uk