Fatigue analyses of buildings with viscoelastic dampers

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Journal of Wind Engineering

and Industrial Aerodynamics 94 (2006) 377–395

0167-6105/$ -

doi:10.1016/j

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Fatigue analyses of buildings withviscoelastic dampers

Alessandro Palmeria,�, Francesco Ricciardellib

aDepartment of Civil Engineering, University of Messina. Vill. Sant’Agata 98166, Messina, ItalybDepartment of Mechanics and Materials, University of Reggio Calabria. Feo di Vito 89060, Reggio Calabria, Italy

Available online 13 February 2006

Abstract

Viscoelastic damping devices are effective in mitigating the buffeting response of medium- to high-

rise buildings. Their use has the effect of limiting displacements and accelerations, as well as of

reducing number and amplitude of fatigue cycles. The structural behaviour, however, is somehow

modified, and a standard Kelvin–Voigt model proves to be inaccurate in predicting the dynamic

response. For an accurate analysis, in fact, a model able to account for the viscoelastic memory is

needed. In this paper, the problem of estimating the fatigue life of structural components of tall

buildings provided with viscoelastic dampers is dealt with. A dynamic model of the building in the

modal space is established, able to account for the viscoelastic memory, as opposed to the classical

modal strain energy method. A cycle counting procedure is then summarised, based on the

separation of the dynamic response of the building into a quasi-static and a resonant part.

The fatigue life is then evaluated using the well-known Palmgren–Miner rule. An application to a

15-storey building is included, aimed at quantifying the inaccuracies arising when the memory effect

is neglected.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Tall buildings; Viscoelastic dampers; Modal analysis; Modal strain energy (MSE) method; Gust

buffeting; Cycle counting; Fatigue damage

1. Introduction

Fatigue life is one of the relevant parameters in the design of wind-exposed structures.As an example, long-span bridges that have performed well in wind for decades, both from

see front matter r 2006 Elsevier Ltd. All rights reserved.

.jweia.2006.01.005

nding author. Tel.: +39090 397 7170; fax: +39090 397 7480.

dresses: [email protected] (A. Palmeri), [email protected] (F. Ricciardelli).

www.elsevier.com/locate/jweia

ARTICLE IN PRESSA. Palmeri, F. Ricciardelli / J. Wind Eng. Ind. Aerodyn. 94 (2006) 377–395378

the serviceability and from the ultimate limit state points of view, have been recently foundto have accumulated fatigue damage, which results in cracking of the girder and otherstructural elements. In the case of medium- to high-rise buildings, the buffeting responsecan be mitigated and, as a consequence, the fatigue life extended, by providing the buildingwith damping devices. Among these, viscoelastic dampers have been widely used, and haveprovided a good performance.Viscoelastic dampers are characterised by a continuous constitutive law, which in many

cases can be accurately approximated through a linear functional. From the modellingpoint of view this is an advantage, as it brings linear equations of motion [1]. Thedrawback is in the fact that they exhibit a memory behaviour and, therefore, the equationsof motion are of the integro-differential type. Due to the difficulties arising with thenumerical solution of integro-differential equations, these are often transformed intodifferential equations, by adopting an equivalent Kelvin–Voigt model, the only viscoelasticmodel unable to account for the system memory. This, however, may result in a dramaticmodification of the system dynamics, bringing an inaccurate evaluation of some of theparameters of the structural response. In particular, when the fatigue life of the structuralcomponents is estimated, relevant parameters are the apparent oscillation frequency andthe upcrossing rates of the response envelopes, which prove to be largely affected by thesystem memory.In this paper, the problem of the assessment of the fatigue life of buildings subjected to

gust buffeting is dealt with, in the case in which these are provided with viscoelasticdamping devices. First, a dynamic model of the building in the modal space is established,which accounts for the viscoelastic memory of the devices, as opposed to the classicalmodal strain energy (MSE) method [2], which brings an equivalent Kelvin–Voigt model. Acycle counting method is then adopted, based on the separation of the building dynamicresponse into quasi-static and resonant parts. Doing so the total number of cycles isevaluated as the sum of the low-frequency (quasi-static) and high-frequency (resonant)cycles. Finally, the fatigue life is evaluated using the well-known Palmgren–Miner rule oflinear damage accumulation. An application to a 15-storey building is included in thepaper, aimed at quantifying the inaccuracies arising when the memory effect is neglected.

2. Alongwind response of buildings with viscoelastic dampers

2.1. Modal analysis

Under the assumption of a linear structural behaviour, the equations governing thealongwind vibration of a building, modelled as a Nx–DoF system, in the case in which thisis provided with viscoelastic dampers (Fig. 1a), can be written as

M €xðtÞ þ ½Cs þ CaðU ref Þ� _xðtÞ þ K xðtÞ þXNd

‘¼1

LT‘ L‘ j‘ðtÞ � _xðtÞ

¼ DðU ref Þ þD0ðU ref ; tÞ, ð1Þ

where Nd is the number of dampers, xðtÞ ¼ ½x1ðtÞ . . . xNxðtÞ�T is the array of the

Lagrangian coordinates, M is the mass matrix, K and Cs are the stiffness and viscousdamping matrices of the building without dampers, respectively, j‘ðtÞ is the relaxationfunction of the ‘th viscoelastic damper, L‘ ¼ ½L‘;1 . . . L‘;Nx

� is the associated array of

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Fig. 1. Building model in Lagrangian coordinates (a); kth modal Kelvin–Voigt oscillator for the MSE method (b);

kth modal viscoelastic oscillator (c).

A. Palmeri, F. Ricciardelli / J. Wind Eng. Ind. Aerodyn. 94 (2006) 377–395 379

influence coefficients, and where the asterisk indicates the convolution product, i.e.,

XNd

‘¼1

LT‘ L‘j‘ðtÞ � _xðtÞ �

XNd

‘¼1

LT‘ L‘

Z t

0

j‘ðt� tÞ _xðtÞdt,

in which, it is assumed that the system is at rest for to0. In Eq. (1), DðU ref Þ and D0ðU ref ; tÞare the arrays of mean and fluctuating drag forces, whose elements depend on the referencewind velocity U ref , i.e. the mean wind velocity at height zref ¼ 10 m in open country:

DiðU ref Þ ¼12rAiCD;iUðU ref ; ziÞ

2; D0iðU ref ; tÞ ¼ rAiCD;iUðU ref ; ziÞu0ðU ref ; zi; tÞ, (2)

finally, CaðU ref Þ is the matrix of the aerodynamic damping:

CaðU ref Þ ¼ r diag½CD;1A1UðU ref ; z1Þ . . . CD;NxANx

UðU ref ; zNxÞ�: (3)

In Eqs. (2) and (3) r is the air density, CD;i and Ai are the drag coefficient and the influencearea associated with the ith DoF, zi is the height of the centre of Ai, and UðU ref ; zÞ andu0ðU ref ; z; tÞ are the mean value and the turbulent component of the wind velocity at heightz, respectively, for the given value of U ref .


Since the viscoelastic forces are given by convolution integrals, Eq. (1) represents a set ofNx coupled integro-differential equations, whose numerical solution is quite cumbersome.As a consequence of this difficulty, in the engineering practice the dynamic analysis ofviscoelastically damped linear structures is usually performed in a reduced modal space byusing the MSE method, bringing an approximate modal response [2]. This methodoperates under the assumptions that: (i) the motion of the modal oscillators is decoupled,i.e. the inherent structural damping, the additional damping provided by the viscoelasticdampers and, for wind-exposed structures the aerodynamic damping too are proportionalto the distribution of mass and/or stiffness; and (ii) the total modal damping is purelyviscous, so neglecting the memory effects associated with the viscoelastic behaviour of thedampers. It has been shown that, for engineering purposes, the first assumption is metwhen the distribution of the viscoelastic dampers in the primary structure is almosthomogeneous; on the contrary, the second assumption may lead to unacceptableinaccuracies, depending on the dynamic characteristics of the building, of the viscoelasticdampers, and of the excitation [3].The MSE method is based on the classical coordinate transformation:

xðtÞ ¼XNq

k¼1

fkqkðtÞ, (4)

where NqpNx is the number of modes retained in the analysis, qkðtÞ is the kth modalcoordinate, and fk is the associated modal shape. The latter has to be evaluated, togetherwith the modal circular frequency ok through the iterative solution of the eigenproblem:

o2kMfk ¼ Kþ ok

XNd

‘¼1

LT‘ L‘

Z þ10

j‘ðtÞ sinðoktÞ dt

" #fk; fT

kMfk ¼ 1,

where the summation in the left-hand side term accounts for the additional stiffnessprovided by the viscoelastic dampers when the structure vibrates in the kth mode, whichdepends on the unknown kth natural frequency. Once the eigenvector fk and theeigenvalue ok are computed, the equivalent viscous damping ratio of the kth modaloscillator can be evaluated as

zkðU ref Þ ¼1

2ok

fTk Cs þ CaðU ref Þ þ

XNd

‘¼1

LT‘ L‘

Z þ10

j‘ðtÞ cosðoktÞ dt

" #fk.

Doing so, the three sources of damping are accounted for through the viscous modaldamping ratios zkðU ref Þ, and the dynamic behaviour of the kth modal oscillator isapproximated by a Kelvin–Voigt model (Fig. 1b), the only viscoelastic model unable toaccount for memory effects, whose motion is governed by the differential equation

€qkðtÞ þ 2zkðU ref Þok _qkðtÞ þ o2kqkðtÞ ¼ fT

k ½DðU ref Þ þD0ðU ref ; tÞ�. (5)

As an alternative to the MSE method, Palmeri et al. [3] have recently proposed a method inwhich the modal oscillators are SDoF dynamic systems with viscoelastic memory (Fig. 1c).The coordinate transformation is again that of Eq. (4), but in the latter case the modalshapes fðkÞ and the modal circular frequencies oðkÞpok are evaluated without iterations,

ARTICLE IN PRESSA. Palmeri, F. Ricciardelli / J. Wind Eng. Ind. Aerodyn. 94 (2006) 377–395 381

via the ordinary eigenproblem

o2ðkÞMfðkÞ ¼ Kþ

XNd

‘¼1

LT‘ L‘j‘ð1Þ

" #fðkÞ; fT

ðkÞMfðkÞ ¼ 1, (6)

where j‘ð1Þ stands for the equilibrium modulus of the ‘th viscoelastic damper, i.e. itselastic stiffness

j‘ð1Þ ¼ limt!1

j‘ðtÞ,

and where in order to make distinction between the modal quantities of the latter approachand those evaluated through the MSE method, the k subscript is written in parentheses.

The integro-differential equation of motion of the kth modal oscillator can be written as

€qðkÞðtÞ þ 2zðkÞðU ref ÞoðkÞ _qðkÞðtÞ þ o2ðkÞqðkÞðtÞ þ GðkÞðtÞ � qðkÞðtÞ

¼ fTðkÞ½DðU ref Þ þD0ðU ref ; tÞ�, ð7Þ

in which the viscous damping ratio

zðkÞðU ref Þ ¼1

2oðkÞfTðkÞ½Cs þ CaðU ref Þ�fðkÞpzkðU ref Þ

accounts only for the structural and aerodynamic damping, but does not include theadditional viscoelastic damping, as the effects of the dampers are directly included inthe modal equations of motion through their relaxation functions. Even though thecomputational effort to obtain the time-domain solution of Eq. (7) is in principle muchlarger than that required to solve Eq. (5), it can be effectively reduced by using anexpedient state–space approximation [1,4] of the modal relaxation function, given by

GðkÞðtÞ ¼XNd

‘¼1

fTðkÞL

T‘ L‘fðkÞ½j‘ðtÞ � j‘ð1Þ�:

Once the modal response has been evaluated, as solution of either Eq. (5) or Eq. (7), thenEq. (4) allows evaluating the array xðtÞ, which describes the motion of the structure in theLagrangian space. Finally, any quantity of interest, yðtÞ ¼ aTxðtÞ, can be evaluated aslinear combination of the modal coordinates:

yðtÞ ¼XNq

k¼1

bkqkðtÞ; bk ¼ aTfk,

where a ¼ ½a1 . . . aNx�T is the array listing the coefficients to be used for combining the

Lagrangian coordinates in order to obtain the selected response yðtÞ, while bk is thecorresponding kth coefficient for combining the modal coordinates. When the proposedmodal analysis of Eqs. (6) and (7) is used, the subscripts in the above expressions have tobe considered as in parentheses.

2.2. Alongwind response

When the wind action is modelled as a stationary Gaussian random process, the meanresponse and the power spectral density (PSD) of the dynamic part of the response are


evaluated as

yðU ref Þ ¼XNq

k¼1

bkHkðU ref ; 0ÞfTk DðU ref Þ;

SyyðU ref ;oÞ ¼XNq

h¼1

XNq

k¼1

bhbkH�hðU ref ;oÞHkðU ref ;oÞfThSDDðU ref ;oÞfk; ð8Þ

where the asterisk indicates the complex conjugate, SDDðU ref ;oÞ is the matrix whoseelements are the auto- and cross-PSDs of the fluctuating drag forces D0iðU ref ; tÞ, andHkðU ref ;oÞ is the frequency response function (FRF) of the kth modal oscillator, whichdepends on the way the oscillator is defined.When the MSE method is applied, the modal FRF is that of a system with viscous

damping:

HkðU ref ;oÞ ¼ fo2k � o2 þ jo½2zkðU ref Þok�g

�1, (9)

where j ¼ffiffiffiffiffiffiffi�1p

is the imaginary unit. On the other hand, when the proposed approach isused, the modal FRF takes the more complicated form

H ðkÞðU ref ;oÞ ¼ o2ðkÞ � o2 þ jo 2zðkÞðU ref ÞoðkÞ

"(

þXNd

‘¼1

fTðkÞL

T‘ L‘fðkÞðFhj‘ðtÞi � j‘ð1ÞÞ

#)�1, ð10Þ

where Fh�i stands for the Fourier transform operator. In many cases the dynamicbehaviour of the viscoelastic dampers can be effectively described through simplerheological models, for which a closed-form characterisation is given in the frequencydomain through the Fourier transform of the relaxation function. As an example, whenthe standard solid model (Fig. 2a) is used, the ‘th viscoelastic damper is described by anelastic spring, representing the equilibrium modulus j‘ð1Þ, in parallel with a Maxwellelement (made of an elastic spring of stiffness k‘ in series with a viscous dashpotof constant c‘); accordingly, the relaxation function (Fig. 2b) and its Fourier transform

Fig. 2. Standard solid model of the ‘th viscoelastic damper (a); relaxation function (b) and its Fourier

transform (c).


(Fig. 2c) take the forms

j‘ðtÞ ¼ j‘ð1Þ þ k‘ exp �t

t‘

� �; Fhj‘ðtÞi ¼ j‘ð1Þ þ

k‘

t�1‘ þ jo

where t‘ ¼ c‘=k‘ is the damper relaxation time, a measure of the time scale of the deviceresponse.

More complicated spring-dashpot chains [5], fractional derivative models [6], or theLaguerre polynomial approximation technique [1] can be applied when improvedmathematical representations of the viscoelastic dampers are needed.

3. Cycle counting

Once the structural response of interest yðtÞ ¼ yþ y0ðtÞ has been probabilisticallycharacterised for a given value of the reference wind velocity U ref , the assessment of thefatigue life of the related structural components requires knowledge of the expectednumber of cycles with mean value m ¼ ðyþ �yÞ=2 and amplitude s ¼ ðy� �yÞ=2, y and �ybeing the peak and the valley in a generic cycle, respectively (Fig. 3). Since the statistics ofthe random process yðtÞ depend on the value of U ref in the time interval DT over which themeans are taken, all the quantities involved in any cycle counting are functions of thereference wind velocity. However, in order to simplify the notation, this dependence is notexplicitly indicated in this section.

Given the random nature of the wind-induced response, cycle counting can beperformed by using two alternative approaches. The first one is the classical Monte Carlosimulation method, which requires the following three steps: (i) the numerical simulationof an adequate number of samples of the structural response yðtÞ, (ii) the application toeach sample of a deterministic time-domain cycle counting method, e.g. the Rainflowcounting method [7], and (iii) the evaluation of the statistics involved in the assessment ofthe fatigue life of the structural components. Even though of general applicability, inpractical situations this first approach may result excessively time consuming. In order to

upcrossings

relative minimum,or valley y

amplitude s

s

relative maximum,or peak y

mean value �

time t

mean value y

structural response y(t) = y + y′(t)

∨

∧

Fig. 3. Response cycle counting for the fatigue damage evaluation.


reduce the computation effort, probabilistic tools can be used to evaluate the expectednumber of cycles in the random process yðtÞ, directly from the knowledge of its statistics. Inthis section an effective procedure for the probabilistic cycle counting, following such asecond approach, is summarised.In principle, a full probabilistic approach requires the evaluation of the joint probability

density function (PDF) pmsðm; sÞ of mean value and amplitude of the cycles. This becausethe mean value m in a generic cycle of the structural response yðtÞ is not necessarily equal tothe mean value y over the entire response process (see Fig. 3). With the aim of simplicity, inthe following the deterministic condition that m � y is assumed, i.e. the probability mass ofthe m variable is lumped at y. Accordingly, the mean value m is given by the first of Eq. (8),and only the marginal PDF of the cycle amplitude psðsÞ has to be evaluated. The latter isrelated to the expected number of cycles nðsÞ in each time interval DT with amplitude in theinterval s to sþ ds

nðsÞ ds ¼ npDTpsðsÞ ds, (11)

where np is the expected rate of occurrence of peaks in the random process yðtÞ.Eq. (11) holds for any random process yðtÞ, but the quantities np and psðsÞ depend on the

spectral characteristics of yðtÞ, as well as on the selected cycle counting method. This isbecause different distributions of peaks and valleys in the structural response areassociated with the different cycle counting methods [8], and a method which is veryaccurate in one case may excessively under- or over-estimate the cycle count in othercircumstances. In particular, when yðtÞ is a narrowband process, the generic sample is asinusoidal time history with slowly varying amplitude and phase. In this special case, npand psðsÞ are given by

np ffio2p; psðsÞ ffi

s

~y2exp �

s2

2 ~y2

� �, (12)

where ~y ¼ffiffiffiffiffil0p

and o ¼ffiffiffiffiffiffiffiffiffiffiffiffil2=l0

pare the standard deviation and the apparent circular

frequency of yðtÞ, respectively. These quantities can be evaluated when the spectral zero-and second-order moments l0 and l2 are known, which are defined as

li ¼ 2

Z oC

0

oiSyyðoÞ do;

oC being a cut-off circular frequency, i.e. a frequency such that for o4oC the energycontent of the process yðtÞ is negligible.For broadband processes no rigorous formulations are available in the literature. The

simplest approach is the so-called narrowband approximation, in which Eq. (12) is stillused, so neglecting the effects on the cycle counting of the shape of the PSD SyyðoÞ. In spiteof its simplicity, this approach is to be avoided in practical applications, since it proves tobe too conservative when the bandwidth of yðtÞ increases.In order to improve the results for the alongwind vibration of slender vertical structures,

it has been elucidated [9,10] that the fluctuating part y0ðtÞ of the structural response can beviewed as a bi-modal random process, given by the sum of two statistically independentrandom processes (Fig. 4):

y0ðtÞ ¼ y0BðtÞ þ y0RðtÞ,

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Syy(log scale)

� (logscale)

broadband (quasi-static) response

resonant (narrowband) response

�C�T

Fig. 4. Power spectral density function of the response, with quasi-static and resonant components.


where y0BðtÞ and y0RðtÞ are the quasi-static (broadband) and resonant (narrowband) parts ofy0ðtÞ, respectively. Aim of this decomposition is to independently count (i) the numbernLðsÞ ds of low-frequency cycles, which are associated with the so-called pseudo-envelope[11] of y0ðtÞ:

eðtÞ ¼ y0BðtÞ þ aRðtÞ,

where aRðtÞX0 is the envelope of y0RðtÞ, and (ii) the number nHðsÞ ds of the high-frequencycycles, which are directly associated with y0RðtÞ.

In order to show the role of the quantities introduced earlier in this section a sample ofthe structural response yðtÞ is plotted in Fig. 5, along with the associated envelope aRðtÞ

and pseudo-envelope eðtÞ. In the first plot the resonant part of the response, y0RðtÞ, appearsas a high-frequency fluctuation around the slowly varying value, given by the sum of themean value, y, and of the quasi static response, y0BðtÞ. In the second plot the envelope aRðtÞ

is shown, connecting the peaks of y0RðtÞ. aRðtÞ is much smoother than y0RðtÞ and without itsperiodic oscillation. These properties allow evaluating the amplitude of high-frequencycycles. In the third plot the pseudo-envelope eðtÞ is shown, connecting the peaks of thetime-varying part of the response y0ðtÞ, which allows evaluating the amplitude of low-frequency cycles.

The total number nðsÞ ds of cycles is evaluated as the sum of the two contributions:

nðsÞ ds ¼ nLðsÞ dsþ nHðsÞ ds. (13)

Repetto [9] has recently proposed two approximate expressions which allow evaluating in adiscrete form the quantities in Eq. (13), leading to a more accurate result than thenarrowband approximation. One can show that the continuous counterpart of theseexpressions can be written as

nLðsÞ ds ¼ �dnþe ðsÞds

DTU �dnþe ðsÞds

� �ds,

nHðsÞ ds ¼oR

2pDTpaR

ðsÞ ds, ð14Þ

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t

t

t

y(t) = y + y′(t)y + y ′ (t)

envelope of the resonant response

pseudo-envelope of the response

structural response

mean value y

amplitude of ahigher-frequency cycle

amplitude of alower-frequency cycle

B

y ′(t)e(t)

αR (t)y′ (t)

R

Fig. 5. Quasi-static y0BðtÞ and resonant y0RðtÞ components of the response, envelope aRðtÞ of the resonant response

and pseudo-envelope eðtÞ of the response.

A. Palmeri, F. Ricciardelli / J. Wind Eng. Ind. Aerodyn. 94 (2006) 377–395386

where U½�� denotes the unit step function, paRðsÞ is the PDF of the Raleigh-distributed

envelope aRðtÞ:

paRðsÞ ¼

s

l0;Rexp �

s2

2l0;R

� �

and nþe ðsÞ is the mean upcrossing rate of the deterministic threshold s by the pseudo-envelope eðtÞ:

nþe ðsÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2;B þ q2

Rl2;R2p

rdPeðsÞ

ds(15)


in which PeðsÞ is the cumulative distribution function (CDF) of eðtÞ:

PeðsÞ ¼ Fsffiffiffiffiffiffiffiffil0;B

p !

�l0;R~y2

exp �s2

2 ~y2

� �F

ffiffiffiffiffiffiffiffil0;Rl0;B

ss

~y

!,

where Fð�Þ is the CDF of a standard Gaussian variable, having zero mean and unitvariance. In the above expressions oR and 0pqRp1 are the apparent circular frequencyand the bandwidth parameter of the resonant part of the structural response, respectively:

oR ¼

ffiffiffiffiffiffiffiffil2;Rl0;R

s; qR ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

l21;Rl0;Rl2;R

s.

Finally, li;B and li;R are the ith spectral moments of the random processes y0BðtÞ and y0RðtÞ,respectively:

li;B ¼ 2

Z oT

0

oiSyyðoÞ do; li;R ¼ li � li;B

where oT, with 0ooTooC, is the transition circular frequency defining the boundarybetween the quasi-static and resonant parts of y0ðtÞ (see Fig. 4).

Comparison between Eqs. (11) and (13) allows deriving approximate expressions of theexpected rate of occurrence of peaks, np, and of the PDF of the cycle amplitude, psðsÞ:

np ffi1

DT

Z þ10

½nLðsÞ þ nHðsÞ� ds; psðsÞ ffinLðsÞ þ nHðsÞ

npDT(16)

For the sake of simplicity, the counting of high-frequency cycles given by the secondexpression of Eqs. (14) does not include the correction proposed by Repetto [9], in order toavoid the re-counting of cycles associated with the low-frequency fluctuations. Numericalsimulations demonstrated, in fact, that the effects of such refinement are negligible for thepurpose of comparing performances of the modal analyses dealt with in the previoussection.

4. Cumulative damage and fatigue life

4.1. Damage accumulation

The most popular among damage accumulation rules is the well-known Palmgren–Miner linear rule, in which the cumulative damage D is deterministically given by

D ¼X

i

di,

where the sum has to be extended to the whole structural life, and di ¼ dðmi; siÞ is thefraction of damage that, in the selected structural component, is associated with the ithcycle fmi; sig of the structural response yðtÞ. The latter coincides with the inverse of themean value Nðmi; siÞ of the number of cycles of constant mean mi and amplitude si required

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number of cycles N

cycle amplitude s

s1

sj

sj+1

N0,1N0,jN0,j+1

mj

Fig. 6. S–N curve for fatigue life estimation.


to cause the fatigue failure in the structural component:

dðmi; siÞ ¼1

Nðmi; siÞ.

Usually, the so-called S–N curve, relating the cycle amplitude s to the value of N is given ina bi-logarithmic graph by a piecewise-linear function (Fig. 6):

Nðm; sÞ ¼ N0ðseqÞ ¼N0;j

sj

seq

� �mj

; sjpseqosjþ1; j ¼ 1; 2; . . .

þ1; seqos1;

8><>: (17)

in which the dependence on the mean value m is taken into account through the equivalentamplitude seq, i.e. the amplitude of the zero-mean cycle that produces in the selectedstructural component the same fraction of damage as the actual cycle fm; sg:

seq ¼ seqðm; sÞ ¼s

1� m=su,

where su is the corresponding ultimate strength associated with the structural response ofinterest. Moreover, in Eq. (17) s1 is the cut-off amplitude, i.e., no fraction of damage isproduced by cycles with equivalent amplitude smaller than s1; N0;j is the mean value of thenumber of cycles with equivalent amplitude sj that causes the fatigue failure, and thecontinuity condition of the S–N curve is satisfied when N0;jþ1 ¼ N0;jðsj=sjþ1Þ

mj ; finally,mj41 is the jth S–N index, defining the slope of the S–N curve in the interval ½sj ; sjþ1�, andexperimental investigations prove that mjþ1omj.

4.2. Fatigue life

Given the randomness of the structural responses under wind forces, the damageaccumulation too is a random process, and the fatigue life of each structural component is a

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amplitude s

reference wind velocity Uref

∆Uref

∆s

k, l -

th bl

ock

sl = (l- 1 ) ∆s2

Uref,k = ∆Uref(k- 1 )2

Fig. 7. Discrete procedure for the evaluation of the expected fatigue life.


random variable. In many engineering situations, the estimate of the expected fatigue life,TF, is an exhaustive design parameter, as the prediction of higher-order statistics is verydifficult to carry out. With the assumption that damage accumulation is a stationary randomprocess, the expected fatigue life can be evaluated through the deterministic relationship:

TF ¼1

D1

,

D1 being the so-called damage intensity, i.e. the mean value of the damage yearlyaccumulated in the selected structural component.

According to Repetto and Solari [12,13], an approximate evaluation of D1 can be carriedout as the sum of the contributions resulting from subdivision of the domain of definitionof each random variable involved in the fatigue analysis in a number of intervals of equallength. In this case, the random variables to be considered are the reference wind velocity,U ref , and the cycle amplitude, s. Hence, the damage intensity is given by

D1 ¼ N t

XNu

k¼1

XNs

‘¼1

d1;k;‘

in which N t is the number of time intervals DT in one year; Nu and Ns are the number ofthe intervals used in subdividing the domains of U ref and s, respectively; d1;k;‘ is the meanvalue of the k; ‘th contribution to damage in the time interval DT , given by

d1;k;‘ ¼npðU ref ;kÞDTpsjU ref

ðs‘jU ref ¼ U ref ;kÞpU refðU ref ;kÞDsDU ref

N½yðU ref ;kÞ; s‘�, (18)

where U ref ;k ¼ ðk �12ÞDU ref and s‘ ¼ ð‘ � 1

2ÞDs define the centre of the k; ‘th block, DU ref

and Ds being its dimensions (Fig. 7). Moreover, in Eq. (18) npðU ref Þ and psjU refðsjU ref Þ are

the same quantities as those given in Eq. (16), for which dependence on the reference windvelocity has been explicitly indicated; finally, pU ref

ðU ref Þ is the PDF of U ref , for which theRaleigh distribution can be used:

pU refðU ref Þ ¼ 2

Cu

U ref ;d

� �2

U ref exp �CuU ref

U ref ;d

� �2" #

; Cu ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� log

DT

TR

� �s, (19)

where U ref ;d is the design value of the reference wind velocity for the given return time TR.


5. Numerical example

The approach presented in the preceding sections was applied to a 15-storey building,whose shear-type frame is shown in Fig. 8a. The overall height is zNx

¼ 52:50 m, with aninterstorey height of 3:50 m. The total mass is 2:813� 106 kg, and the floor lateral stiffnessis K ¼ 50� 106 N=m. The structural damping in the first mode is assumed to be zs ¼ 0:01.A number of viscoelastic dampers were added to the frame, as shown in Fig. 8a, whichwere modelled as standard linear solid elements having all the same properties, i.e. anequilibrium modulus j‘ð1Þ ¼ 0, a relaxation time t‘ ¼ t ¼ 0:3 s and a stiffness k‘ ¼ aK ,with a ¼ 0:3 being a non-dimensional parameter governing the reaction force exerted byeach device. In Fig. 8b the first modal shape f1 (dashed line) obtained through theapplication of the MSE method is compared to the first modal shape fð1Þ (solid line)obtained through the application of the proposed procedure. The negligible differencesbetween the two curves are the result of a distribution of the devices which is not exactlyproportional to the frame mass and/or stiffness distribution. The first natural frequenciesevaluated with the two methods are o1 ¼ 1:74 rad=s and oð1Þ ¼ 1:67 rad=s.The following data were used to evaluate the matrix of the aerodynamic damping (Eq.

(3)) and the forcing vectors (Eq. (2)): r ¼ 1:25 kg=m3, CD;i ¼ 1:3, Ai ¼ 87:5 m2 for i ¼

1; . . . ;Nx � 1 and Ai ¼ 43:75 m2 for i ¼ Nx. The PDF of the reference wind velocity (Eq.(19)) is characterised by Dt ¼ 600 s, U ref ;d ¼ 20 m=s and TR ¼ 50 years, bringing a modalvalue, i.e. the most recurrent value in the life of the structure, U ref ¼ 3:68 m=s. Alogarithmic wind profile in the form

UðU ref ; zÞ ¼U ref

lnðzref=z0Þln

z

zref

� �(20)

was used in the calculations, with z0 ¼ 1 m. The longitudinal turbulent fluctuations arecharacterised by the Kaimal spectrum:

SuðU ref ; z;oÞ ¼2po

u2�

100f

0:44þ 33fð Þ5=3; f ¼

o2p

z

UðU ref ; zÞ

0 0.0003 0.0006 0.00090

7

14

21

28

35

42

49

proposed MSE

z i [

m]

�1,i [m](b)(a)

Fig. 8. Schematic of the example tall building (a) and first modal shape (b).


in which the friction velocity is related to the reference wind velocity as

u� ¼kU ref

lnðzref=z0Þ,

where k ¼ 0:4 is the Von Karman constant. The vertical coherence of the longitudinalturbulent fluctuations is given by

CohðU ref ; z1; z2;oÞ ¼ expo2p

Czjz1 � z2j

½UðU ref ; z1Þ þ UðU ref ; z2Þ�=2

� �, (21)

with Cz ¼ 15.In Fig. 9a the moduli of the FRF of the first modal oscillator are shown, as obtained

using the MSE method (Eq. (9), dashed line) and the proposed method (Eq. (10), solidline), for a reference wind velocity U ref ¼ 3:4 m=s, close to the modal value. The maindiscrepancy between the two curves are concentrated in the low frequency range, while theresonant peak is well approximated by the MSE method; for purpose of comparison, alsothe FRF for the case in which no additional dampers are added is shown in the figure(dot–dashed line).

The mean building drift yðtÞ ¼ xNxðtÞ=zNx

is chosen as relevant response parameter, andin Fig. 9b the corresponding PSDs are shown, as evaluated accounting only for the firstvibration mode. It can be noticed that the error in the low-frequency range on the FRFevaluated through the MSE method is translated into the underestimation of the quasi-static wind induced response.

In Figs. 10a and b the standard deviation ~y and the apparent circular frequency o of theselected response are shown as a function of U ref , evaluated with a cut-off frequencyoC ¼ 10 rad=s. The introduction of the damping devices not only mitigates the amplitudeof the alongwind-induced vibration, as confirmed by the reduction of ~y, but also stronglymodifies the apparent oscillation frequency. These two quantities completely define thecycle number and distribution for the narrowband process approximation (Eq. (12)).

(a) � [rad/s]

0

0.5

1

1.5

2

[s-2

]H

1

0 4 51 2 3

(b) � [rad/s]

proposed

MSE

without dampers

0.01 0.1 11x10-15

1x10-13

1x10-11

1x10-9

S yy

[s]

Fig. 9. Modulus of the modal frequency response function (a) and power spectral density function of the chosen

response (b).

ARTICLE IN PRESS

(a) Uref [m/s]

y %

0 5 10 15 200

0.00015

0.0003

0.00045

0.0006

proposedMSEwithout dampers

(b) Uref [m/s]

� [

rad/

s]

0 5 10 15 200

0.5

1

1.5

~

Fig. 10. Standard deviation (a) and apparent circular frequency (b) of the structural response as a function of

Uref.

(a) s

�+ aR [

s-1]

1x10-8 1x10-7 1x10-6 1x10-5

0

0.02

0.04

0.06

0.08

(b) s

�+ e [s

-1]

1x10-8 1x10-7 1x10-6 1x10-5

0

0.02

0.04

0.06

0.08

proposedMSEwithout dampers

Fig. 11. Upcrossing frequency (a) of the aRðtÞ envelope and (b) of the eðtÞ pseudo-envelope.


More accurate results require to characterise the upcrossing rates of the aRðtÞ envelopeand of the eðtÞ pseudo-envelope from a probabilistic point of view. In Figs. 11a and b theexpected upcrossing rates nþe ðsÞ (Eq. (15)) and nþaRðsÞ ¼ oRqRpaR

ðsÞ=ffiffiffiffiffiffi2pp

are compared,associated with a reference wind velocity U ref ¼ 3:4 m=s. These quantities allowscharacterising the number of low- and high-frequency cycles for the estimation of theexpected fatigue life. The figures show the addition of dampers considerably reduces theupcrossing rate of the pseudo-envelope eðtÞ, increases the peak of the upcrossing rate of theenvelope aRðtÞ, and shifts it towards the lower amplitudes. In both cases the consequence isa reduction of the expected damage accumulation. Also in this case the discrepancy


between the results obtained with the MSE and the proposed methods are evident, whichinfluence the cycle counting and the estimate of the expected fatigue life.

To evaluate the cumulated damage associated with the selected response parameter, thefollowing quantities are used in the S–N curve: m1 ¼ 5, m2 ¼ 3, N0;1 ¼ 108, N0;2 ¼ 5� 106,s1 ¼ yðU ref Þ ¼ 4:58� 10�6, su ¼ yðU ref ;dÞ þ 3 ~yðU ref ;dÞ ¼ 0:00404. The selected amplitudesfor the blocks are DU ref ¼ U ref ;d=50 ¼ 0:40 m=s and Ds ¼ s1=20 ¼ 2:29� 10�7. InFigs. 12a and b the surfaces and the grey-scale maps describing the distribution ofdamage as a function of U ref and s (Eq. (18)) are shown, as evaluated with the MSEmethod (graphs on the right) and through the proposed procedure (graphs on the left). TheMSE method tends to underestimate the fatigue damage, which brings a longer fatigue lifeTF ¼ 215 years, as compared to TF ¼ 164 obtained through the application of the newprocedure, with an overestimation e ¼ 31%.

In Fig. 13 the expected fatigue life is plotted as a function of the device relaxation time t,for three different values of the a parameter, as evaluated through the MSE (dashed line)and proposed methods (solid line). In the plot the fatigue life TF ¼ 21 years of the buildingwithout dampers is also shown (dot–dashed line). Also in this case it is clear how the MSEmethod tends to overestimate the fatigue life for all the values considered of the deviceparameters.

Fig. 12. Distribution of the expected damage fractions in one year: (a) proposed method; (b) modal strain energy

method.

ARTICLE IN PRESS

0 0.2 0.4 0.6

0

100

200

300

400

proposed

MSEwithout dampers

TF

[yea

rs]

τ [s]

� = 0.5

� = 0.3

� = 0.1

Fig. 13. Influence of the device parameters on the expected fatigue life.


6. Conclusions

A model for the evaluation of the fatigue life of structural components of buildingsprovided with viscoelastic dampers subjected to gust buffeting has been presented inthis paper. The innovative feature of the model is that it allows accounting for thememory behaviour of the devices. In these buildings, indeed, the viscoelastic memorymay dramatically changes the wind-induced response and, as a consequence, theprocess of formation and accumulation of the damage in the structural componentscould be strongly modified by the viscoelastic devices. These effects can be measured alsowith the well-known Palmgren–Miner rule, although this is a memoryless damageaccumulation rule: the proposed approach, in fact, takes into account the memory in thestructural response, while the Palmgren–Miner rule neglects the memory in the damageaccumulation.The procedure has been applied to a medium-rise building, about fifty meters high, and

the results compared to those obtained through the application of the modal strain energymethod, unable to account for the viscoelastic memory. It was found that the memory ofthe devices significantly affects the results. In particular, inaccuracies were found in thefrequency response function, as well as in the standard deviation of the response and in itsapparent oscillation frequency, when the memory is neglected. That is, when a cyclecounting method was applied for the estimation of the structure fatigue life, it was foundthat neglecting the memory brings significant inaccuracies in the upcrossing rates of theresponse process, and globally an overestimation of the fatigue life. Based on these resultsit is concluded that appropriate dynamic models including memory have to be used whenestimating the fatigue life of a building provided with viscoelastic dampers, in case areasonable level of accuracy is required.


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Fatigue analyses of buildings with viscoelastic dampers

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