srikage &creep
TRANSCRIPT
Proceedings fib Symposium PRAGUE 2011
ISBN 978-80-87158-29-6 Session 3-4: Modelling and Design
1
NUMERICAL ANALYSIS OF CREEP AND SHRINKAGE IN
HIGH-RISE CONCRETE OR STEEL-CONCRETE BUILDINGS
Mario Alberto Chiorino Carlo Casalegno Claudia Fea Mario Sassone
Abstract
The expansion of concrete construction in high-rise buildings has made these structures sensitive to
the effects of creep and shrinkage. Initial and time dependent strains of concrete generate both
absolute and relative displacements in vertical elements that cannot be neglected.
The problem is further complicated by the sequential character of high-rise construction,
involving continuous sequences of loading steps and changes of structural configurations.
If these effects are not adequately understood and analyzed in design and construction phases,
several serviceability concerns may arise, affecting structural members as well as non structural
components. Ultimate safety may be influenced as well, as a result of delayed increases of axial
loads in vertical elements.
The paper presents a computational approach for the evaluation of these effects based on the
combination of a finite element discretization of the structural configuration, with account for its
evolving character and the sequence of loading steps, and of the recursive numerical algorithm for
the solution of the hereditary integral problem induced by the adoption of a linear aging
viscoelastic constitutive relation for creep. The proposed analysis procedure is applied to the
time-dependent analysis of a case study relative to a multi-storey building.
Keywords: High-rise, Creep, Shrinkage, Structural effects, Construction sequence
fib Symposium PRAGUE 2011 Proceedings
Session 3-4: Modelling and Design ISBN 978-80-87158-29-6
2
1 Introduction
Concrete and steel-concrete frame structures represent economical and effective solutions for the
realization of high-rise buildings. Over the years the height of these structure has dramatically
increased, due to the development and the availability of high-strength concrete and the capabilities
of modern computerized structural analysis. With the increasing height of the structures, vertical
supporting elements, as columns and walls, are subjected to successive load increments due to the
construction of the overlying floors, showing as a consequence significant axial shortenings. These
elastic deformations increase in time due to creep and shrinkage of concrete. In reinforced concrete
structures up to 30 stories or 120 m the effects of the delayed deformations are normally
disregarded without serious consequences [5]. In higher structures, as well as in composite or
hybrid structures, ignoring the effects of creep and shrinkage can lead to undesirable service
conditions, and in some cases also the stability of the building can be put at risk.
On the one hand, axial shortening of vertical supporting elements can negatively affect the
service behaviour of the building, causing damage of partitions, façade clothing, gutters, plumbing,
buckling of elevators guide rails, misalignment of the stops with respect to the floors. It can also
induce additional forces in horizontal stiffening members, like bracings, putting at risk the
structural safety itself [8-10-12-13-16-17].
On the other hand, the differences in the stress levels, in the characteristics of the elements
and in the environmental conditions can produce a differential shortening of the vertical supports,
with consequent negative effects on the service behaviour of the building, as the arise of cracking
phenomena and the development of a slope in the slabs [5-10-12-15-21].
The differences in the axial deformations can also induce the growth of bending moments in
beams and slabs and a redistribution of the loads between adjacent vertical members, which, in
turn, affects the following evolution of the viscous phenomena. Moreover, the stress distribution is
further affected by concrete creep, which causes the relaxation of bending moments induced by
differntial shortening of vertical members. The time-dependent load redistribution between vertical
elements can cause an increase in the axial loads and induce instability phenomena [5-10-12-15-
22].
In composite or hybrid buildings, made up of an internal concrete core and an external steel
frame, the differential shortening can be particularly evident, because concrete elements undergo
time-dependent deformations, due to creep and shrinkage, while steel elements do not [5-10].
In the case of cast-in-place structures, shortenings of vertical members occurring before the
realization of the slabs are less important, because slabs formworks are levelled at the time of
casting, while it is important to predict the shortenings that follow, in order to eventually cast the
slabs with an initial slop, with the aim of compensating the future differential shortenings. In the
case of composite structures with steel columns, instead, also the shortenings that occur before the
realization of the slabs must be predicted and compensated, since steel columns are fabricated with
predefined dimensions [10].
Finally, delayed deformations of concrete can produce a growth in time of the lining of the
structure, due for example to the presence of eccentric loads, to the yielding of foundations or to
the presence of concrete with different properties [2], with the risk of putting on risk the stability of
the building itself.
Over the years, with the increasing in the height of concrete buildings, the attention of
researchers and engineers in the effects of concrete time-dependent behaviour on this kind of
structures has progressively increased.
The first studies on the argument date back to the end of 1960s, due to the work of Fintel and
Khan [11-12]. The authors proposed a simplified procedure for the prevision of time-dependent
columns shortenings due to elastic and time-dependent deformations.
The early works by Fintel and Khan have been resumed by many authors in the following
years, as in [8-10-13-16-17-22-23].
Proceedings fib Symposium PRAGUE 2011
ISBN 978-80-87158-29-6 Session 3-4: Modelling and Design
3
A much more refined approach, adopted by several authors in recent years, consists in a finite
element formulation of the problem, with concrete creep modelled by the use of rheological models
chains (rate-type approach) [9-14-15].
A finite element approach is used in [21] also, in association to the Age-adjusted Effective
Modulus Method [4] to take into account creep effects. The use of the AEMM is suggested in [5]
and [18] also.
The effects of time-dependent deformations of concrete have been considered in the design of
some recent super-tall buldings, as the Taipei 101 Tower [24] and the Burj Dubai Tower [2].
In the following, a computational procedure is presented, based on the coupling of the
traditional methods for the numerical approximation of the viscoelastic integral constitutive
equations with the finite element method, which allows the rigorous solution of general viscoelastic
problems. Structures like high-rise building, characterized by significant non-homogeneities and
complex construction sequences, can be effectively analyzed for time-dependent effects through
the described procedure, as shown in the numerical example that follows.
2 Proposed analysis procedure
Let eq. (1) be the local matrix relationship between nodal forces and nodal displacements in
a Bernoulli beam element, referred to the local coordinate system:
{ } [ ]{ }sKf = (1)
In order to introduce the integral viscoelastic constitutive law expressed through the creep function,
it is necessary to rewrite eq. (1) in the inverse form:
{ } [ ] { }fKs1−= (2)
The introduction of the viscoelastic constitutive relation leads to the following expression for the
nodal displacements at time t:
( ){ } [ ] ( ) ( ){ }'tdf't,tJEKts
t
c ∫−=
0
1 (3)
To come back to the fundamental relation (1) it is necessary to invert again eq. (3), in order to
express the nodal forces as a function of the nodal displacements. This inversion is not possible in
an analytical way, because of the integral form of the expression. To go on with the discussion it’s
then necessary to introduce an integration numerical algorithm.
The numerical computation of this type of integrals can be developed replacing the hereditary
integral with a finite sum, using the trapezoidal rule ore the rectangular rule [3].
Subdividing time t into discrete times t1, t2, …, ti, …, tk and using the trapezoidal rule, the
general recursive formula is obtained for the nodal displacements increment at the generic time tk ,
after some manipulations [6-19-20]:
{ } [ ] [ ]{ } [ ]
[ ]{ }i
kk
t
k
i
ikikikik
ctkkkkct
f)t,t(J)t,t(J)t,t(J)t,t(J
EKf)t,t(J)t,t(JEKs
∆
∆∆
∑−
=−−−−
−−
−
−−+
++=
1
0
1111
1
1
1
2
1
2
1
(4)
The nodal displacements increment { }s∆ at time tk depends on the stresses increment { }f∆ at time
tk and on the stresses increments at previous steps ti. The expression is hence composed of two
terms: an unknown term represented by the stresses increment at the last step t=tk and a known
term represented by the sum of the previous history of stresses until time t=tk-1.
fib Symposium PRAGUE 2011 Proceedings
Session 3-4: Modelling and Design ISBN 978-80-87158-29-6
4
The discretized constitutive equation (4) can now be inverted, assuming as known the stress
increments occurred at previous time steps:
{ }( )
[ ]{ }
[ ]{ }i
kk
t
k
i
ikikikik
kkkk
t
ckkkk
t
f)t,t(J)t,t(J)t,t(J)t,t(J
)t,t(J)t,t(JsK
E)t,t(J)t,t(Jf
∆
∆∆
∑−
=−−−−
−−
−−+⋅
⋅+
−+
=
1
0
1111
11
12
(5)
Expression (5) represents a general extension of relation (1) to the viscoelastic beam finite element.
The expression can be rewritten in a more compact form:
{ } [ ] { } { }kkkk tttt
fsK~
f Ψ∆∆ −= (6)
The [ ]kt
K~
matrix is the "tangent" stiffness matrix at each time step tk and the { }fΨ represents the
global effect at time tk of the stress history in the element.
If also shrinkage is taken into account relation (6) results:
{ } [ ] { } { } { }kkkkk tshtttt
ffsK~
f ∆Ψ∆∆ −−= (7)
where { }shf∆ represents the vector of the increments between time tk-1 and tk of the nodal axial
loads equivalent to the effect of shrinkage.
The local stiffness matrices are then assembled, as well as the nodal forces vectors and the
vectors representing the effect of the stress history, and the global matrix and vectors are obtained.
The global matrix and vectors are then partitioned as usual in order to take into account the external
restraints:
{ } [ ] { } [ ] { } { }kkkkkk tFtRtFRtFtFFtF fsK
~sK
~f Ψ∆∆∆ −+= (8)
{ } [ ] { } [ ] { } { }kkkkkk tRtRtRRtFtRFtR fsK
~sK
~f Ψ∆∆∆ −+= (9)
When the incremental displacements are calculated, the stresses in the elements can be obtained by
applying eq. (6) again.
The solution of the viscoelastic problem is hence obtained incrementally as a sequence of
elastic analyses, in which the elastic modulus is updated at the actual value and the effect of the
stress history is taken in account at each step as external nodal actions, calculated on the basis of
the solutions of previous steps.
In the case of structures composed of elements with different viscoelastic properties, or
comprising elastic elements, the different behaviour of the members is taken into account simply
assigning different properties to the singular finite elements.
Successive changes in the static scheme can be considered as well, due changes in the
restraint conditions or to the introduction or subtraction of elements, through modifications of the
dimensions or the partitioning of the stiffness matrix and the vectors at the different time steps.
The described algorithm has been implemented in the Matlab 7 programming environment,
in association with the commercial finite element analysis software TNO Diana 9.4. The external
Matlab procedure is devoted to the management of the time-dependent terms
( ) ( )( )1,,/1 −+ kkkk ttJttJ ,{ }kt
fΨ and { }ktshf∆ of eq. (7) and to the storing of the results relative to
the different time steps, while the procedures of the finite element analysis are managed by Diana,
used as an elastic finite element solver at each time step.
At each time step, this data are computed and written to a Diana input file for the batch
execution of the analysis, together with the data relative to the finite element model (geometry,
materials, loads, supports, etc.).
Proceedings fib Symposium PRAGUE 2011
ISBN 978-80-87158-29-6 Session 3-4: Modelling and Design
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Term ( ) ( )( )12 −+ kkkk t,tJt,tJ/ of eq. (5) is entered as the value of the elastic modulus, while
terms { }kt
fΨ and { }ktshf∆ are introduced as loads applied to the nodes of each viscoelastic element,
in the local coordinate system.
A simple elastic analysis is then executed in Diana, and an output file is generated, containing
the results of the analysis in terms of nodal forces and displacements (and any other desired result).
Finally, in Matlab environment, the output file of the analysis is read and the results are
stored.
If the structure being analyzed is subjected to a change in the static scheme at a generic time
tk, the input data for the finite element analysis are modified from time tk to the time of the
successive change.
The described analysis procedure allows a rigorous approach to general viscoelastic
problems, as no approximation is introduced at the level of the constitutive relation (only
a numerical approximation is introduced, which can be made negligible through the adoption of an
appropriate time discretization), and simple or complex problems (as problems characterized by the
presence of non-homogeneities and problems of sequential constructions) can be dealt with.
3 Numerical example
The multi-storey building represented in Fig. 1, ten storey high, is considered as a case study. Each
storey is 3 m high, for a total height of 30 m. The structural system consists of a central concrete
core and a mesh of external columns. The structure is supposed to be built sequentially, with a rate
of 7 days for storey. The construction of the core and of the external frame is supposed
contemporaneous. In the two-dimensional finite element the construction sequence is simulated.
A vertical strip of the building is considered in the analyses, comprehending one half of the central
core and two external columns. The central core is modelled as a beam element with a stiffness
equal to the one of the portion of the core considered. In a first option the external columns are
considered made of concrete, while in a second one they are considered made of steel, in order to
investigate the differences in the time-dependent behaviour of these two building types. The beams
connecting the columns to the central core are considered made of concrete. The dimensions of the
structural elements are summarized in Tab. 1.
Fig. 1 The structure object of study: plan view and finite element models relative to the different
construction stages.
Although the modelling of the structure and of the construction sequence is quiet simplified, and
the height of the building considered is not very significant, the results of the time-dependent
analyses show some interesting aspects of the behaviour of the structural types being considered.
fib Symposium PRAGUE 2011 Proceedings
Session 3-4: Modelling and Design ISBN 978-80-87158-29-6
6
Tab. 1 Geometrical characteristics of the structural members.
Floors Concrete columns
size [cm]
Steel columns
section
Beams dimensions
(b x h) [cm]
Central core
thickness [cm]
1-2 45x45 HEA 400 30x60 25
3-4 40x40 HEA 340 30x60 25
5-6 35x35 HEA 300 30x60 25
7-8 30x30 HEA 240 30x60 25
9-10 25x25 HEA 200 30x60 25
The results relative to the time-dependent analysis of the structure with the external concrete
columns are represented in Figs. 2 to 5. In what concerns the deformations of the vertical members,
the maximum long-term shortening is obtained in correspondence of the 10th floor, of about 1.5÷2
cm (depending on the prevision model) in correspondence of the columns, and about 1.2÷1.6 cm in
correspondence of the central core (Fig. 2). The floors are supposed to be levelled at the time of
casting, by compensating the previous differential shortenings of the underlying floors, in order to
obtain an initial horizontal configuration of the slabs. The maximum long-term differential
shortening between the columns and the central core results about 3÷5 mm, in correspondence of
the 9th floor (Fig. 3). The initial step increments visible in the diagram are due to the application of
the loads relative to the construction of the upper floors. The difference in the displacements at the
end of the construction of about 1 mm, due to the different stress levels in the two vertical
elements, is increased in time due to creep and to the difference in the shrinkage strains. The
central core, in fact, has a higher 2Ac/u ratio with respect to the external columns, thus resulting in
lower long-term shrinkage and creep strains. Although in the structure considered the differences in
the long-term shortening appear quiet limited (4 to 5 mm), it must be noted that such a difference
over ten storeys can become a difference of several centimetres in a one hundred storeys structure.
In Fig. 4 the bending moment at the joint between the beam and the column, in
correspondence of the 9th floor, is represented. After the initial step variations, a gradual decrease
of the bending moment is observed, followed by an increase until long-term, except for the model
B3 prediction, which shows a further slight decrease from time t equal to about 40 years. The shape
of the diagrams, characterized by successive changes in the general trend, is due to the differences
in the creep and shrinkage rates between the external columns and the central core, due to the
different values of the volume to surface ratio, which cause forces redistributions between the
members. The values obtained at the end of the construction are increased up to about three to four
times at long term, depending on the creep prediction model being adopted, thus putting in
evidence the importance of considering the effects of the time-dependent properties of concrete in
the design of tall buildings.
In Fig. 5 the evolution of the axial force in the column of the ground floor is represented. The
diagram shows a not significant stress transfer between the central core and the column.
Central core
External columns
External columns
Central core
Proceedings fib Symposium PRAGUE 2011
ISBN 978-80-87158-29-6 Session 3-4: Modelling and Design
7
Fig. 2 Concrete columns. Long-term vertical
displacements of the different floors.
Fig. 3 Concrete columns. Maximum differential
displacements in correspondence of the 9th floor.
Fig. 4 Concrete columns. Bending moment at the joint
between the beam and the column, 9th floor.
Fig. 5 Concrete columns. Axial force in the
ground floor column.
The situation is significantly different in the case of the structure with the external steel columns. In
what concerns the deformations of the vertical members, the maximum long-term shortening
results about 1.1÷1.6 cm (Fig. 6), in correspondence of the top of the central core, while
a maximum long-term differential shortening of about 9÷13 mm is obtained in correspondence of
the same floor (Fig. 7). The higher difference is due to the fact that the central concrete core
continues to shorten with time due to creep and shrinkage, while the steel column does not. The
steel column shows a slight time-dependent increment of deformations, due to the stress transfer
between the central core and the columns, which is in this case very significant (Fig. 9). It must be
noted that also in this case the floors are supposed to be levelled at the time of construction, thus
neglecting the differential shortenings already occurred in the underlying floor. If also these
deformations are taken into account, the total long-term differential shortenings would be even
more significant.
Fig. 6 Steel columns. Long-term vertical
displacements of the different floors.
Fig. 7 Steel columns. Maximum differential
displacements in correspondence of the 10th floor.
External columns
External columns
Central core
Central core
fib Symposium PRAGUE 2011 Proceedings
Session 3-4: Modelling and Design ISBN 978-80-87158-29-6
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Fig. 8 Steel columns. Bending moment at the joint
between the beam and the column, 9th floor.
Fig. 9 Steel columns. Axial force in the ground
floor column.
In Fig. 8 the bending moment at the joint between the beam and the column, in correspondence of
the 10th floor, is represented. The evolution is in the sense of a significant development in time of
a negative bending moment, as a consequence of the time-dependent shortening of the central core
due to creep and shrinkage.
4 Conclusions
A computational procedure has been presented, which allows a rigorous approach to general
viscoelastic problems. The procedure has been applied to the evaluation of the long-term behaviour
of a multi-storey building. Despite the quiet simplified modelling of the structural system and of
the construction sequence, and the small height of the building considered, the results of the
analyses put in evidence the importance of taking into account the effects of the time-dependent
deformations of concrete in the design of tall buildings, in order to avoid long-term serviceability
concerns due to absolute and differential shortening of vertical members, as the damage of
partitions, façade clothing, plumbing, and other non-structural elements, and the arise of bending
moments and cracking phenomena in beams and slabs.
Stress transfers between adjacent vertical members could also represent a serious concern, in
particular in concrete-steel hybrid structures, where it can induce significant time-dependent load
increments in steel columns, which could induce instability phenomena.
The results of the analyses underline the necessity of a reliable approach to the evaluation of
time-dependent behaviour, able to properly describe the effects of structural non-homogeneities,
which result in significant stress redistributions between structural members and in complex
interactions between creep and shrinkage, which cannot be caught in the frame of a simplified
analysis.
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fib Symposium PRAGUE 2011 Proceedings
Session 3-4: Modelling and Design ISBN 978-80-87158-29-6
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Mario Alberto Chiorino � Politecnico di Torino
Dipartimento di Ingegneria Strutturale e
Geotecnica
Viale Mattioli 39
10125 Torino, Italy
� + 39 0115644864
Carlo Casalegno � Politecnico di Torino
Dipartimento di Ingegneria Strutturale e
Geotecnica
Viale Mattioli 39
10125 Torino, Italy
� + 39 0115644874
Claudia Fea � Politecnico di Torino
Dipartimento di Ingegneria Strutturale e
Geotecnica
Viale Mattioli 39
10125 Torino, Italy
� + 39 0115644880
Mario Sassone � Politecnico di Torino
Dipartimento di Ingegneria Strutturale e
Geotecnica
Viale Mattioli 39
10125 Torino, Italy
� + 39 0115644867