Srikage &Creep

Proceedings fib Symposium PRAGUE 2011

ISBN 978-80-87158-29-6 Session 3-4: Modelling and Design

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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

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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].



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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.



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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.).



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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.



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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



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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



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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.

References

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Mario Alberto Chiorino � Politecnico di Torino

Dipartimento di Ingegneria Strutturale e

Geotecnica

Viale Mattioli 39

10125 Torino, Italy

� + 39 0115644864

☺ [email protected]

Carlo Casalegno � Politecnico di Torino


Geotecnica

Viale Mattioli 39

10125 Torino, Italy

� + 39 0115644874


Claudia Fea � Politecnico di Torino


Geotecnica

Viale Mattioli 39

10125 Torino, Italy

� + 39 0115644880


Mario Sassone � Politecnico di Torino


Geotecnica

Viale Mattioli 39

10125 Torino, Italy

� + 39 0115644867


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Srikage &Creep