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Proceedings ISBN 978-80-87158-29-6

fib Symposium PRAGUE 2011 Session 3-4: Modelling and Design

NUMERICAL ANALYSIS OF CREEP AND SHRINKAGE IN HIGH-RISE CONCRETE OR STEEL-CONCRETE BUILDINGS

Mario Alberto Chiorino

Carlo Casalegno

Claudia Fea

Mario Sassone

AbstractThe 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

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fib Symposium PRAGUE 2011 Session 3-4: Modelling and Design

Proceedings ISBN 978-80-87158-29-6

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, faade 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 [5-10-12-1522]. 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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Proceedings ISBN 978-80-87158-29-6

fib Symposium PRAGUE 2011 Session 3-4: Modelling and Design

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.

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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:

{ f } = [K ]{s}

(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:

{s} = [K ]1 { f }

(2)

The introduction of the viscoelastic constitutive relation leads to the following expression for the nodal displacements at time t:

{s(t )} = [K ]

1

E c J (t ,t' ){df (t' )}0

t

(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 its 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]:

{s}tk 1

k

= [K ] E c1

1 [J ( t k ,t k ) + J ( t k ,t k 1 )]{f }tk + [K ]1 Ec 2

1 2 [J ( t k ,t i ) + J ( t k ,t i 1 ) J ( t k 1 ,t i ) J ( t k 1 ,t i 1 )]{f }ti i =0

(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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fib Symposium PRAGUE 2011 Session 3-4: Modelling and Design

Proceedings ISBN 978-80-87158-29-6

The discretized constitutive equation (4) can now be inverted, assuming as known the stress increments occurred at previous time steps:

{f }tk 1 i =0

k

=

2 1 [K ]{s}tk (J ( t k ,t k ) + J ( t k ,t k 1 ))Ec J ( t k ,t k ) + J ( t k ,t k 1 )

(5)

[J ( t k ,t i ) + J ( t k ,t i 1 ) J ( t k 1 ,t