Performance-based topology optimization for wind-excited tall buildings: A framework

Engineering Structures 74 (2014) 242–255

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Performance-based topology optimization for wind-excited tall buildings: A framework Sarah Bobby, Seymour M.J. Spence ⇑, Enrica Bernardini, Ahsan Kareem NatHaz Modeling Laboratory, Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN 46556, United States

a r t i c l e

i n f o

Article history: Received 21 March 2013 Revised 16 May 2014 Accepted 27 May 2014 Available online 20 June 2014 Keywords: Performance-based design Topology optimization Aerodynamic loads Fragility models Tall buildings

a b s t r a c t Topology optimization methods were originally developed in a deterministic setting notwithstanding the inherently uncertain environment in which the final structural systems must exist. Indeed, most design applications are affected by uncertainties in material properties, model idealization etc. as well as loads that are inherently aleatory in nature. In addition, the responses of the systems of interest to this study are generally affected by a significant amount of dynamic amplification therefore complicating the governing equations. While the performance-based assessment and optimization of fixed-topology structures set in the aforementioned environment has been the focus of a number of studies, the possibility of performing topology optimization within this setting has yet to be fully investigated. This paper presents a performance-based topology optimization framework developed for windsensitive tall buildings that rigorously accounts for the time-variant stochastic nature of the aerodynamic loads while considering additional time-invariant uncertainties describing the state/knowledge of the system parameters defining the mechanical properties of the structure. The framework is based on decoupling the performance-based assessment from the topology optimization problem through the definition of a number of approximate sub-problems. The successive resolution of these results in a final design that is not only optimum but also satisfies the user-defined performance targets. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Performance-Based Design (PBD) is an approach to design that has seen a vast expansion in the recent past, especially in the field of seismic engineering. It consists of introducing a set of desired performance objectives that must be satisfied by the structure. These requirements can be given in terms of probabilistic measures of acceptable performances, which are essentially established based on requests from the building’s stakeholders or, more generally, on societal needs and which may be measured according to different parameters (which do not have to be directly related to engineering quantities, for example monetary cost). PBD is usually seen as an alternative to the traditional prescriptive approach to design. In prescriptive design precise requirements must be satisfied in terms of structural response parameters and construction details; these requirements are established by codes and therefore tend to be simple, conservative and, in general, inflexible. Also, the traditional design methodology, if not simply deterministic, is semi-probabilistic partly because code provisions encompass ⇑ Corresponding author. Tel.: +1 574 631 2539; fax: +1 574 631 9236. E-mail addresses: [email protected] (S. Bobby), [email protected] (S.M.J. Spence), [email protected] (E. Bernardini), [email protected] (A. Kareem). http://dx.doi.org/10.1016/j.engstruct.2014.05.043 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

different kinds of structures and this leads to the necessity of more general and less sophisticated approaches to risk assessment [1] and partly due to the mathematical challenges posed by the development of a fully probabilistic approach. The semi-probabilistic approach consists in the use of safety factors to acknowledge the presence of uncertainties in the problem parameters but does not provide a quantitative measure of the structural reliability [2]. It should be observed that prescriptive design is not in principle in contrast with the PBD concept if the structural performance requirements are fixed by the code in order to achieve predefined performance objectives [3]. Nevertheless, the probabilistic PBD framework has a higher potential of quantifying the structural reliability by offering a much more thorough acknowledgment of all uncertainties present in the design problem and by dealing with different risks in a consistent and balanced way [4]. For these reasons, progress in the definition of PBD approaches is of fundamental importance if truly reliable, more efficient and economical structural designs are to be obtained. In the context of structural optimization, the probabilistic nature of the design approach and the inclusion of the uncertainties assume particular relevance. Indeed, fixed-topology optimization techniques with explicit modeling of the uncertainties in the structural performance, often termed reliability-based

S. Bobby et al. / Engineering Structures 74 (2014) 242–255

optimization methods, have been studied over the years and are now well-established [5–10, e.g.]. On the other hand, in the case of topology optimization, which aspires to determine the optimum material layout of a structure and has been widely used in the aerospace, mechanical, and civil engineering fields for structures and mechanical parts [11–15, e.g.], the explicit consideration of the uncertainties inherent in the problem at hand has gained attention only relatively recently [16,17]. In fact, in the last few years, works have been dedicated to the development of both Reliability-Based Topology Optimization (RBTO) methods, that is to the study of topology optimization with constraints requiring that the structure meets a predetermined minimum failure probability or measure of failure with respect to a performance measure [18– 20, e.g.], and of Robust Topology Optimization (RTO), where the objective is to minimize the influence of the uncertainties on the performance of the structure [21,22, e.g.]. In the development of the RBTO methods both component failure—which describes the probability of failure through the failure of an individual component in the structure [23]—and system failure—where at least one critical failure mode occurs [24,23,20]—have been considered. Nevertheless, the explicit inclusion of uncertainties concerning the inherently time-varying stochastic nature of most environmental loads (e.g. wind and seismic excitation) and the dynamic amplification that these loads generally cause in the response of the structural system has yet to be considered. For example, topology optimization methodologies have been developed imposing frequency constraints while considering exclusively time-invariant material uncertainties [25], and methods have been developed that attempt to ensure that the maximum displacement of the dynamic system under deterministic time-varying loads is within a userdefined limit [26,27]; however topology optimization explicitly considering time-variant/invariant uncertainties has not been performed in a truly dynamic setting. This is most likely due to the significant complications that arise under the aforementioned circumstances due to the governing second-order differential equations replacing the convenient and simple static equilibrium conditions. The objective of this paper is to propose a novel framework, defined specifically for the design of wind-excited tall buildings, in which the topology optimization of these structures can be carried out in a PBD setting. This framework will consider the generally time-invariant uncertainties characterizing the system parameters and model idealization, together with the time-variant uncertainty caused by the stochastic nature of naturally occurring environmental loads and the dynamic amplification that these will in general cause in slender structures.

2. Performance-based design methodology Within the framework of PBD, researchers at the Pacific Earthquake Engineering Research (PEER) Center have developed the so-called PEER Equation based on the Total Probability Theorem. The PEER Equation allows the estimate of the mean annual rate of exceedance of a specified performance level, which represents the risk associated with the structure. Although originally developed for seismic engineering applications, the PEER Equation is general and has been adopted in other research areas, such as fire, blast and wind engineering [28], with the intent of extending the PBD approach to these fields. The PEER Equation is particularly advantageous from a practical point of view because it decomposes the complex process of risk assessment into several sub-tasks, independent of each other and to be carried out sequentially, involving experts in different fields of engineering as well as end users and decision makers [29]. Indicating with kðaÞ the mean annual rate of exceedance of the event A ¼ a (where capital letters

243

indicate variables while lower case letters indicate their realizations) and with GðajbÞ the Complementary Cumulative Distribution Function (CCDF) of random variable A given B ¼ b, the PEER Equation can be written as:

kðdv Þ ¼

Z dm

Z

Z

edp

Gðdv jdmÞ  jdGðdmjedpÞj  jdGðedpjimÞj  jdkðimÞj

im

ð1Þ where dv indicates the decision variable corresponding to the performance objective (for example, monetary cost); dm is the damage measure indicating the state of damage of structural and/or non-structural parts (e.g. the plastic deformation accumulated in an element or the loss of function of a structural/non-structural element); edp is the engineering demand parameter, which is the value assumed by the structural response parameter that is linked to the damage occurrence (e.g. the rotation of a joint or the interstory drift); and im is the measure of the intensity of the event of interest (wind, earthquake, etc.). The quantities dv ; dm; edp and im can be vectors, in which case the integrals of Eq. (1) become multifold. By writing the mean annual rate as in Eq. (1), the choice has been implicitly made that the parameters dv ; dm; edp and im are defined so that dm conditioned on edp is independent of im, and dv conditioned on dm is independent of both edp and im [30]. As such the various terms that appear in the triple integral of Eq. (1) are independent of each other and can be obtained through separate analyses, referred to as hazard analysis (providing kðimÞ), structural analysis (for GðedpjimÞ), damage analysis (for GðdmjedpÞ) and loss analysis (for Gðdv jdmÞ). It is of particular interest to consider only the two sub-tasks of structural analysis and damage analysis (this is made possible by the decoupling allowed by the formulation of Eq. (1)), and to express the CCDF of the damage measure for a given intensity of the hazard as:

GðdmjimÞ ¼ PðDM > dmjimÞ Z GðdmjedpÞ  jdGðedpjimÞj ¼ edp Z GðdmjedpÞ  pðedpjimÞ  dedp ¼

ð2Þ

edp

where pðajbÞ indicates the probability density function (PDF) of the random variable A given B ¼ b, while PðA > aÞ indicates the probability that A exceeds a. If the structural performance is expressed in terms of a damage value, i.e. the failure of the structure corresponds to the achievement of a specified value of damage, LSDM , the exceedance probability PðDM > LSDM jimÞ represents the fragility of the structure. This can be assessed, for any value of im, through Eq. (2), in what is commonly referred to as fragility analysis. Also, the curve representing the values of PðDM > LSDM jimÞ as a function of im is termed the fragility curve, and is widely adopted in the design of civil structures as a means to describe/study the structural behavior with reference to a certain performance level. Being only a component of the complete risk analysis expressed by Eq. (1), the fragility analysis is by definition less complete than this last [29]; however, it should be observed that the establishment of acceptable probabilities of exceedance of damage values given specified hazard intensities (i.e. a fragility-based design), though simpler than a full risk analysis, may be efficient in guiding the performance of the designed structure. This is also important in light of the fact that a full probabilistic description of the hazard is often unavailable. 3. Topology optimization Topology optimization has traditionally been used to determine the optimum material layout for a structure. The aforementioned

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optimum is in general defined as that which minimizes an objective function while satisfying a number of designer-imposed constraints. A common objective is the minimization of the material volume of the structure while ensuring a number of designer-imposed stiffness constraints (often in terms of limits on the maximum deflection of the structure or strain energy, i.e. compliance). These problems are generally formulated in a deterministic setting under static load distributions notwithstanding the inherently uncertain and time-dependent environment in which the structures are generally set. For a discretized domain, this traditional setting can be formally written as follows:

Find q ¼ fq1 ; . . . ; qn gT n Z X to minimize VðqÞ ¼ e¼1

ð3Þ

Xe

qe dX

ð4Þ

C j ðqÞ ¼ kTj ðf s  KðqÞuðqÞÞ þ KTj uðqÞ  LSj

subject to:

C j ðqÞ ¼ KTj uðqÞ  LSj 6 0 j ¼ 1; . . . ; N

approximations such as finite differences may be used, when solving large-scale problems it is more desirable to compute the gradients directly, if possible. For quantities that are explicit functions of the design variables (e.g. the objective function of Eq. (4)) this is straightforward. For quantities that are implicit functions of the design variables, such as the displacements, derivatives can be found using the direct differentiation method [32]. Alternatively, if the number of constraints is significantly less than the number of design variables, as in most practical cases of interest, it is more efficient to use the adjoint method. In this approach, for each constraint a zero function is added before taking the derivative. A popular and convenient choice of such a function is that represented by a multiple of the static equilibrium conditions of Eq. (6), yielding:

ð5Þ

KðqÞuðqÞ ¼ f s

ð6Þ

0 6 qe 6 1 e ¼ 1; . . . ; n

ð7Þ

ð10Þ

where kTj is an arbitrary constant. Taking the derivative of the constraint with respect to the design variables therefore gives:

@C j ðqÞ @uðqÞ @KðqÞ ¼ ðKj  KðqÞkj ÞT  kTj uðqÞ @ qe @ qe @ qe

ð11Þ

where q is the element-wise normalized material density design variable vector; n is the total number of elements composing the discretized structure; Xe denotes the domain of element e; V is the volume of material in the design domain of the structure; C j is the jth displacement or compliance constraint; N is the total number of displacement or compliance constraints; LSj is the maximum allowable value of the displacement response or compliance; K is the stiffness matrix; u and f s denote the static displacement and loading vectors, respectively; Kj is a vector of constants extracting the displacement (or combination of displacements, e.g. inter-story drift) to be constrained or, in the case of compliance constraints, coincides with f s . A desirable solution to the problem outlined in Eqs. (3)–(7) is represented by the design variable vector q taking on the componental binary solution qe ¼ 0 or qe ¼ 1 where qe ¼ 0 indicates a void element while qe ¼ 1 indicates a solid element. A small minimum value for the design variables, qmin , is typically used to represent the void state in order to prevent numerical instabilities. A popular way to achieve a binary solution is to adopt the Solid Isotropic Material with Penalization (SIMP) model [31]. In this strategy, in order to force the design towards a primarily binary solution, the intermediate-valued design variables are penalized using the following scheme:

where kTj is chosen as the solution to KðqÞkj ¼ Kj to eliminate @uðqÞ=@ qe . Thus the derivative is simply given by:

De ðqe Þ ¼ qpe D0

4.1. Problem formulation

ð8Þ

where De represents the constitutive matrix over the element e; D0 represents the constitutive matrix of the material in the solid phase i.e. qe ¼ 1; and p is the penalization factor, where p P 1. Within this setting, the global stiffness is expressed as: n X KðqÞ ¼ Ke ðqe Þ ¼ e¼1

n Z X e¼1

Xe

BT De ðqe ÞBdX

ð9Þ

where Ke is the element stiffness matrix and B is the strain– displacement matrix of the shape function derivatives. 3.1. Sensitivities with respect to the design variables Topology optimization problems are typically characterized by design variable vectors of high dimensions in order to guarantee an adequate resolution of the discretized structure. This makes the adoption of gradient-based optimization algorithms extremely desirable due to their rapid convergence rates. However, for these methods to be efficient, objective function and constraint sensitivities must be calculated in a timely manner. Although

@C j ðqÞ @KðqÞ ¼ kTj uðqÞ @ qe @ qe

ð12Þ

Obviously in the case where the compliance measure is constrained, kTj is taken to be uT . In the case of a continuum design domain a filtering or projection technique may be required to prevent instabilities [33–35, e.g.]. Although this was not indicated in the sensitivity calculations given above, if these methods are used their effect on the structure during the optimization process must obviously be considered in the sensitivity calculation. 4. Performance-based topology optimization: a framework The previous sections introduced a general framework for the performance-based assessment of wind-sensitive structures and illustrated the concept of traditional deterministic topology optimization. This section will present a strategy for performing topology optimization within the performance-based design framework presented in Section 2.

The topology optimization problem of interest to the present work may be posed as:

Find q ¼ fq1 ; . . . ; qn gT n Z X to minimize VðqÞ ¼ e¼1

Xe

ð13Þ

qe dX

ð14Þ

subject to:

Pðg j ðq; ; Y; IMÞ 6 0jimj Þ  Pobj 6 0 j ¼ 1; . . . ; Nc j

ð15Þ

MðqÞ€z þ Cðq; Þz_ þ KðqÞz ¼ fð; Y; IMÞ

ð16Þ

0 6 qe 6 1 e ¼ 1; . . . ; n

ð17Þ

where M and C denote the mass and damping matrices, respectively; z denotes the displacement vector where the obvious dependency of z on q has not been indicated for the sake of compactness; f is the external loading vector;  is a vector of time-invariant random variables describing uncertainties that remain constant from event to event, e.g. uncertainties in damping

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S. Bobby et al. / Engineering Structures 74 (2014) 242–255

or epistemic uncertainties in the model idealization; Y is a vector of stationary stochastic processes modeling the time-variant aleatory nature of the wind loads; Nc is the number of performance constraints; Pobj is the jth exceedance probability performance j objective while g j is the jth limit state function, which for the purpose of this paper is given by:

g j ðq; ; EDPj ; IMÞ ¼ LSEDPj  EDPTj ðq; ; EDPj ; IMÞ

from the master DOF therefore allowing truly global systems to be defined that are independent of the particular selection of master DOF. It should also be observed that the amount of stiffness coming from the floors that can be used by the global lateral load resisting system is simply defined by how the secondary system is modeled. This obviously allows any local concentrated effects due to load transfer to be properly modeled.

ð18Þ 4.2. The proposed solution strategy

where LSEDPj is the limit state capacity of the jth performance constraint, EDPTj is the largest peak occurring during the stationary response of duration T of EDPj caused by an event of intensity IM while EDP j is a random peak occurring during T. The minimization of the volume of material in the design domain was chosen as a simplistic objective function in this study in order to focus on the main contribution of this paper: the development of a topology optimization framework in the context of performance-based design. However, this objective function is closely related to more practical potential objective functions for tall building topology design, including the building cost, which can be seen as a function of the material volume, among other things. The class of problems that are of interest to this study are characterized by a vertical cantilever form with mass located at a limited number of Degrees Of Freedom (DOF) of the structure. Furthermore, the stationary external loading vector, f, is characterized by having non-zero components only at a limited number of DOF. The N dof DOF having either non-zero mass or non-zero external load component or both are termed master DOF and belong to the finite element model of the floor system (secondary system), which is superimposed to the finite element model of the design domain, therefore defining what will be termed the complete system, a schematic of which is shown in Fig. 1. According to the definition of the complete system as superposition of the design domain and the secondary system, the aerodynamic and inertial loads enter the system at the floor levels, which has been widely documented to accurately reproduce in an unbiased fashion how the loads enter tall building systems [36–38] and in general how inertial loads enter multistory building systems [39,5,40,41]. Then, the loads can be transferred to the design domain through the secondary system and therefore the nodes that belong to both systems (Fig. 1). This ensures that the design domain is decoupled

4.2.1. The reduced system Due to the concentrated nature of the mass and external loading vector of the complete system, as schematized in Fig. 1, the dynamic response of the aforementioned system may be estimated in the master DOF from the following reduced set of equilibrium equations:

~ q; Þ~z_ þ Kð ~ qÞ€~z þ Cð ~ qÞ~z ¼ ~fð; Y; IMÞ Mð

ð19Þ

~ is the reduced mass matrix obtained directly from M by where M simply eliminating the rows and columns of M pertaining to the ~ is the reduced order damping degrees of freedom with zero mass; C z is the displacement vector describing the response of the matrix; ~ complete system in the master DOF; ~f is the reduced forcing function vector obtained from f by simply eliminating the components (by definition zero-valued) of f pertaining to the degrees of freedom of the complete system not included in those of the master DOF; ~ is the reduced order stiffness matrix directly estimated while K from the complete stiffness matrix K through static condensation. It should be observed that if the damping of the complete system is assigned directly to its N dof vibration modes then the use of the same modal damping for the N dof vibration modes of the reduced system will ensure that the response estimated in the master DOF by solving Eq. (19) will be identical to the response estimated by solving the complete system defined in Eq. (16). 4.2.2. Fragility constraints The optimization problem outlined in Eqs. (13)–(17) is characterized by the fragility-based performance constraints of Eq. (15) on limit state functions that are in terms of the time-variant random variable EDPj as well as the time-invariant random variables . As briefly outlined above, what is of interest is the probabilistic description of the largest peak, EDPTj , of the stationary response process of the jth constrained engineering demand parameter occurring during an observation period T conditional on the hazard intensity. To evaluate the aforementioned probability, it is convenient to initially consider the following exceedance probability, conditional on  ¼ e and IM ¼ im, using the Total Probability Rule over the random variable EDP where the subscript j has been dropped for simplicity: Z P ðEDP  > edpje;imÞ ¼ PðEDP > edpje;edp;imÞpðEDP  je;imÞdEDP EDP 

ð20Þ where pðEDP  je; imÞ is the conditional probability density function of EDP . If it is assumed that the peaks EDP in T are independent and identically distributed, their arrivals can be modeled as a Poisson process in time with mean crossing rate mPðEDP > edpje; imÞ where m is the mean rate of arrival of the peaks EDP during T. Under these assumptions, the probability that the largest peak occurring during T is greater than edp given e and im may simply be written as:

P ðEDPT > edpje; imÞ ¼ 1  emTPðEDP Fig. 1. Schematic of the class of problems for which the optimum topology is desired.



>edpje;imÞ

ð21Þ

The unconditioning over the time-invariant variables  is carried out by using the result obtained in Eq. (21) and applying once again the Total Probability Rule yielding:

246

PðEDP T > edpjimÞ ¼

S. Bobby et al. / Engineering Structures 74 (2014) 242–255

Z e

PðEDPT > edpje; imÞpðejimÞde

ð22Þ

where pðejimÞ is the joint conditional probability density function of

. In general the random variables  will be independent of IM therefore allowing pðejimÞ to be substituted with pðeÞ. By estimating the conditional probabilities PðEDP T > edpjimÞ following the sequential application of Eqs. (20)–(22) the inherent difference between the time-variant and time-invariant uncertainties is explicitly taken into account. Indeed, if  was also time-variant, its random nature should be included in the modeling of the mean rate of the Poisson model defined by Eq. (21). 4.3. Approximate sub-problem As already mentioned in Section 3.1, the problems of general interest to topology optimization are characterized by design variable vectors q of dimensions commonly on the order of several thousands making the adoption of gradient-based optimization algorithms particularly attractive. However, while the calculation of the gradients of the objective function in Eq. (14) does not present any particular difficulty, the calculation of the gradients of the performance-based fragility constraints (Eq. (15)) is not straightforward due not only to the governing second order differential equations, but also to their probabilistic nature. To overcome this difficulty, decoupling strategies, which separate the inherently nested probabilistic analysis from the optimization loop by replacing the original optimization problem with an approximate sub-problem, are of particular interest. The basic idea behind decoupling strategies is to first perform a reliability analysis (probabilistic performance-based assessment) in the current design point, from which information is gleaned for defining an approximate optimization problem, which is then fully solved [42]. The solution of the approximate optimization problem yields the new design point, which is used to build and solve a subsequent approximate optimization problem. This two-step process is repeated until convergence. The focus of this paper is on defining a decoupling approach that can be used in reliability-based topology optimization of tall buildings systems. In particular, a method will be described that is based on defining, during the probabilistic performance assessment, a number of fragility-based equivalent static loading distributions that will be used to define the approximate optimization problem. With the aforementioned objective in mind, it is convenient to recast the fragility constraints of Eq. (15) in the following inverse reliability form:

particular, alongside the aforementioned loads and for  ¼ e and IM ¼ imj , the following load distribution may be defined in terms of EDPj (also estimated in the reduced system):

! EDPj  lEDPj ~ ~r ðe; imj Þ ~EDP ðe; EDP ; imj Þ ¼ Fðe; im F F Þ þ j j EDP j

rEDPj

ð23Þ

d  is the threshold for which the largest peak occurring where EDP Tj during T has an exceedance probability equal to Pobj while g^j is j the corresponding value of the limit state function. The inversion of sign in writing Eq. (23) is simply in order to satisfy the optimization convention that safety is represented by a negative value of the constraint. The satisfaction of the aforementioned constraints obviously ensures the satisfaction of the original constraints. The desirable property of these constraints compared to the original ones is that they are in terms of peak response functions instead of probabilities. It will be seen that this alternative setting will allow the definition of an appropriate approximate sub-problem. To this end, it is convenient to consider the following total quasi-static loads defined through the reduced system of Eq. (19):

~ q; Þ~z_ ðtÞ ¼ Kð ~ ; Y; IMÞ ¼ ~fð; Y; IMÞ  Mð ~ qÞ~€zðtÞ  Cð ~ qÞ~zðtÞ Fð

ð24Þ

The significance of these loads is that if they are quasi-statically applied to the reduced system they will reproduce the same response as the original dynamically amplified system. In

ð25Þ

 ~ is the expected value of F; ~ l where F EDP j and rEDP j are the mean and standard deviation of EDPj respectively for given values of  and IM, ~r while F is given by: EDP j

~r ðe; imj Þ ¼ F EDP j

1

rEDPj ðe; imj Þ

CF~ ðe; imj ÞCEDPj

ð26Þ

where CF~ ðe; imj Þ is the covariance of the total quasi-static loads of Eq. (24) while CEDPj is the vector of influence functions whose kth component (k ¼ 1; . . . ; N dof ) represents the response EDPj in the complete system when a unit load acts in the kth master degree of freedom of the reduced system. The loads defined in Eq. (25) represent a load distribution that if statically applied to the master DOF of the complete system will yield a response level for EDP j equal to EDPj for given values of IM and . The validity of the load distribution defined in Eq. (25) may be confirmed by the fact that, in the presence of negligible dynamic amplification, it has been shown that this load distribution is the most probable in causing the peak value EDP j [43,44,38]. As for the random peaks EDPj , the arrival times of the vectors ~ FEDPj in T may be modeled as a Poisson process with mean rate equal to that of the peaks EDPj . Under these circumstances the vector associated with the largest peak occurring during the obser~EDP , will be a perfectly correlated random vector vation period T; F Tj with kth conditional marginal distribution given by:

Pð e F kEDP > ~f kedpj je; imj Þ ¼ PðEDP Tj > edpj je; imj Þ

ð27Þ

Tj

where ~f kedpj is the kth component of the threshold calculated by substituting into Eq. (25) the value edpj in place of EDP j , i.e. ~f ~ ~ edpj ¼ FEDPj ðe; edpj ; imj Þ. If the random vector FEDPTj is now considered conditioned only on hazard intensity, IM ¼ imj , the following fragility-based equivalent static load distribution may be defined:

~  ~ ~ F c  ðimj Þ ¼ FEDPTj ðimj Þ þ gj ðimj ÞFrEDP ðimj Þ EDP Tj

ð28Þ

Tj

 ~EDP is the expected value of the loads F ~EDP ; F ~r  is a load where F EDP Tj Tj Tj distribution that if statically applied to the structure would yield a response in EDP j equal to rEDPTj and is given by:

~r  ðimj Þ ¼ F EDP Tj

d  ðq; ; EDP ; imj Þ  LSEDP 6 0 g^j ðq; ; EDPj ; imj Þ ¼ EDP Tj j j

j

1

rEDPTj ðimj Þ

CF~EDP ðimj ÞCEDPj

ð29Þ

Tj

~EDP while g is where CF~EDP is the covariance matrix of the loads F j Tj given by: Tj

gj ðimj Þ ¼

d  ðimj Þ  l  ðimj Þ EDP EDP Tj Tj

ð30Þ

rEDPTj ðimj Þ

where lEDP and rEDPTj are the mean and standard deviation of EDPTj Tj and can be assessed from the solution of Eq. (22). Now consider the ( )T following load distribution F c  ¼ EDP

8
Tj

F 1c  ; . . . ; F dc  EDP Tj

EDP Tj

where:

ik2N ð31Þ iXk2N

where N is the set of master DOF while i ¼ 1; . . . ; d where d indicates the total number of degrees of freedom of the complete system. The significance of the fragility-based equivalent static loads defined in Eq. (31) is that if they are statically applied to the complete system d  and are based on they will cause a response in EDP j equal to EDP Tj

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S. Bobby et al. / Engineering Structures 74 (2014) 242–255

the second order description of the time-variant/invariant uncertainties affecting the problem. The fragility-based equivalent static load distribution F c 

Definition of Initial Complete System and Uncertain Parameters

EDP Tj

obviously depends on the design variable vector q (Eq. (23)). However, if it is assumed that for relatively small changes around the current design point the load distribution F c  is independent of EDP Tj q, then the following approximate sub-problem can be defined:

Find q ¼ fq1 ; . . . ; qn gT n Z X to minimize VðqÞ ¼ e¼1

Design Cycle = 1 Design Cycle = i+1

Define Reduced System

ð32Þ Performance-Based Assessment Xe

qe dX

ð33Þ Penalty Update

Construction of Approximate Sub-Problem

subject to:

KTj uðqÞ  LSEDPj 6 0 j ¼ 1; . . . ; Nc

ð34Þ

KðqÞuðqÞ ¼ F c 

ð35Þ

EDP Tj

j ¼ 1; . . . ; Nc

0 6 qe 6 1 e ¼ 1; . . . ; n

Gradient-Based Topology Optimization to Solve Sub-Problem

ð36Þ

The approximate sub-problem outlined in Eqs. (32)–(36) has not only decoupled the probabilistic performance-based design assessment from the successive topology optimization problem, but it has also reduced the probabilistic optimization problem with second order governing equations (Eqs. (13)–(17)) to a standard static problem allowing the design gradients to be estimated by simply applying the adjoint method of Section 3.1. Also, because F c  is defined EDP Tj

from the reduced system, the non-zero loads are applied only to the master DOF therefore avoiding difficulties that can arise when solving topology optimization problems where all DOF are loaded [45], a situation that would have arisen if the proposed procedure were defined directly on the complete system. The approximate subproblem may thus be solved using established techniques for static topology optimization problems. To account for the dependency of F c  on the design vector, after the convergence of a sub-problem,

Maximum SIMP Penalty Value Attained?

No

Yes Loads Converged?

No

Yes Final Design Fig. 2. Flowchart of the proposed performance-based topology optimization procedure.

EDP Tj

the fragility-based equivalent static load distribution is updated and a new sub-problem is defined and solved until there is negligible difference between successive load updates. Each load update and sub-problem resolution is termed a design cycle. 4.4. Solution algorithm To solve the performance-based topology optimization problem outlined in Eqs. (13)–(17) using the proposed framework the procedure outlined in Fig. 2 is adopted. In particular, the successive resolution of the sub-problems is coupled with the continuation method [34] i.e. the penalty p used in the parameterization scheme of Eq. (8) is increased at each design cycle. Obviously, final convergence will be achieved only when the penalty has arrived at its final value and the loads F c  have remained constant between EDP Tj

two successive sub-problems. For the implementation of the performance-based design assessment on the reduced system, the following two step Monte-Carlo algorithm can be adopted (where for initial input imj is fixed, and the probabilistic characteristics of the timeinvariant random random vector  are assumed known, as are the characteristics of the vector valued stochastic process Y):

Fig. 3. Schematic showing location of planar system within the 3D building.

~ definition of the fragility-based equivalent static loads F . In Ec DP  Tj

1. Generate a sample, e, of the random vector .  2. Conditionally on e generate a sample peak edpTj (Eq. (21)) ~ together with the associated vector FedpTj (Eq. (27)).

step 2, classic results from the study of random dynamics can be adopted to solve for PðEDPTj > edpj je; imj Þ. For instance, if the response of EDPj ðtÞ in T is considered Gaussian, then the aforementioned probability may be estimated as:

The repetition of steps 1 and 2 defines the Monte-Carlo algorithm yielding the solution of Eq. (22) and therefore an estimate of the d  and ultimately g that is used in Eq. (28) for the threshold EDP

PðEDPTj > edpj je; imj Þ ¼ 1  eT m~ðedpj ;e;imj Þ

Tj

j

~ is the crossing rate of the threshold edpj given by: where m

ð37Þ

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Fig. 4. Initial ground structure and optimal lateral load resisting system for case study 1: (a) the complete structure (design domain); optimum topologies for (b) rf ¼ 0:0025, (c) rf ¼ 0:0050, (d) rf ¼ 0:0075.

Fig. 5. Aerodynamic load: (a) time history; (b) power spectral density.

2

1 edpj  lEDPj ðe; imj Þ m~ðedpj ; e; imj Þ ¼ m~0 exp 4 2 rEDPj ðe; imj Þ

!2 3 5

5. Case studies

ð38Þ

~0 is the zero-crossing rate given by: where m

m~0 ¼

1 2p

rEDP _ ðe; imj Þ j rEDPj ðe; imj Þ

ð39Þ

where rEDP is the standard deviation of the derivative of the process _ j EDPj ðtÞ.

The case studies in this section aim to demonstrate the applicability of the framework developed in Section 4 to wind-sensitive tall buildings. In particular, the objective of all case studies of this section is the definition of the optimum topology for a planar lateral load resisting system envisaged as part of a 3D building as illustrated in Fig. 3. The following uncertainties will be considered for all examples. The time-invariant uncertainty affecting the problems, , is taken as uncertainty in the modal damping properties of the structures. The stationary vector-valued stochastic process

S. Bobby et al. / Engineering Structures 74 (2014) 242–255

Fig. 6. Histograms of the uncertain modal damping ratios:

249

lf ¼ 0:02, (a) rf ¼ 0:0025; (b) rf ¼ 0:0050; (c) rf ¼ 0:0075.

the continuation method was used for all examples where p was increased from 2 to 5 using a step size of 0.5. 5.1. Stationary stochastic process Y The pressure measurements used in the case studies were taken from wind tunnel tests performed at the boundary layer wind tunnel of the CRIACIV (Inter-University Research Centre on Building Aerodynamics and Wind Engineering) in Prato, Italy. Wind tunnel studies were performed on a 1:500 scale rigid model of the 3D building envelope shown in Fig. 3 [46–48]. A suburban terrain with a power law exponent of the mean wind profile of 2/9 was assumed. Synchronous multi-pressure measurements were taken using 126 pressure taps at a sampling frequency of 250 Hz for 30 s of recorded data.

Fig. 7. Material volume convergence history for case study 1.

modeling the time-variant and aleatory nature of the exciting functions, Y, is taken to be pressure measurements from wind tunnel studies performed on rigid building models. The intensity measure IM is taken to be the mean wind speed at the top of the building in the X-direction, V H , as shown in Fig. 3. In particular, the forces fðY; IMÞ were calculated using the pressure measurements while considering a hazard intensity im ¼ 37 m=s for all case studies. The topology of the inner planar frame of the structure was optimized for all case studies considering a single fragility constraint, with T = 3600 s, on the engineering demand parameter EDP defined as the displacement at the center of the top of the frame in the Xdirection, d, with limit state LSEDP ¼ LSd ¼ 0:75 m. The exceedance probability performance objective of the aforementioned fragility constraint was taken as Pobj ¼ 10% for all case studies. In addition,

5.2. Case study 1 The first case study investigates the optimal topology of a cantilevered structure comprised of frame elements. The complete system (represented by a ground structure) and boundary conditions for the structure are shown in Fig. 4a. The master DOF of the complete system were defined as the X-direction displacements of the nodes of the complete system. The masses assigned to each degree of freedom were simply obtained from their relative influence areas. To model the horizontal stiffness offered by the floor systems, released (i.e. only axial stiffness is considered) non-designable W24  176 beams connect all nodes in the horizontal direction. The external load associated with this case study, fðY; imÞ, was determined by integrating and appropriately scaling the pressure on the top half of the structure and distributing the resulting load among the master DOF at the roof level. The external loading components at the remaining master DOF were set to zero.

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Fig. 8. Components of the fragility-based equivalent static loads corresponding to the master DOF belonging to the vertical axis of symmetry for case study 1: (a) rf ¼ 0:0025; (b) rf ¼ 0:0075.

Fig. 5a shows the resulting time history of the component of the total load applied at the roof level f ðY; imÞ. The damping of the complete system was defined by directly assigning the modal damping ratio of the first mode. Three examples are given, in which the damping ratio, f, is modeled as a time-invariant lognormal random variable with a mean value lf ¼ 0:02 and standard deviations rf ¼ 0:0025; rf ¼ 0:0050, and rf ¼ 0:0075, as shown in Fig. 6a–c for Examples 1 to 3, respectively. 5.2.1. Results Fig. 4b–d shows the optimum topology for Examples 1, 2, and 3, respectively. The material volume history shows rapid and steady convergence for all three examples, as shown in Fig. 7. It should be observed that the initial increase in volume shown in Fig. 7 is simply due to the continuation method, as the penalization parameter increases from 2 at Design Cycle 1 to 5 at Design Cycle 8. After the maximum penalization parameter is reached the volume of material converges without increasing. Fig. 8a and b shows the original, converged, and final components of the fragility-based equivalent static loads corresponding to the master DOF belonging to the vertical axis of symmetry of Examples 1 and 3, respectively. In particular, the insensitivity of the method to the increasing uncertainty in the damping ratio of the reduced system is

Fig. 9. Peak displacement history for the master degree of freedom subject to the performance constraint for case study 1.

encouraging as it provides an initial indication of the robustness of the proposed performance-based topology optimization strategy. It is also apparent that the robustness of the structure

S. Bobby et al. / Engineering Structures 74 (2014) 242–255

Fig. 10. Fragility curves for optimal structures in case study 1: (a)

r ¼ 0:0025; (b) r ¼ 0:0050; (c) r ¼ 0:0075.

Fig. 11. Resulting topologies for various sizes of beams comprising secondary floor system.

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Fig. 12. Schematic of design domain.

increased with increasing amounts of damping uncertainty. The volume of material in the structure increased as the amount of uncertainty in f increased as expected, as can be seen by comparing the volume history for all three examples in Fig. 7. The initial natural frequency of the frame structure was 0.1392 Hz, and the natural frequencies of the optimal structures are 0.1438 Hz, 0.1470 Hz, and 0.1494 Hz for Examples 1, 2, and 3, respectively. When comparing these values to the load spectrum in Fig. 5b it is apparent that the dynamic nature of the structures should be considered in the optimization problem. Finally, the maximum top displacement limit of LSd ¼ 0:75 m was met for the converged solutions (shown in Fig. 9), and corresponds to a final exceedance probability of 10% for a hazard intensity of im ¼ 37 m=s as desired and shown in Fig. 10a–c for Examples 1, 2 and 3, respectively. It is interesting that the chevron bracing scheme produced in case study 1 was obtained when modeling the floors using nonrigid W24  176 members, despite the fact that chevron bracing is commonly associated with ‘‘rigid’’ floors, e.g. [49]. In order to investigate this further, trials were repeated for Example 2 (rf ¼ 0:005) using three different sizes for the beams of the secondary floor system: W8  21, W14  26, and W18  65. These sizes were chosen in order to understand how a reduction in axial stiffness of the beams of the secondary system affects the global topology. The resulting structures are illustrated in Fig. 11. It can be seen that as the stiffness of the secondary floor system reduces, the lateral load-resisting system becomes more dominated by Xbracing, which is a typical bracing scheme found for structures in which either no floor system or a weak floor system is modeled [50,20, e.g.]. These results reinforce the importance of realistically modeling the size of the members comprising the secondary

Fig. 13. Case study 2: (a) initial design domain (initial complete structure); (b) optimal design for Example 1; (c) optimal design for Example 2 (the locations of the floors are shown at each possible outrigger level).

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This structural design domain is divided into three main portions: the core and skin of the structure, indicated by C, are optimized to have the minimum possible volume; an outrigger domain, denoted by K, connects the core to the skin at optimal locations to efficiently add lateral resistance to the structure and is intended optimized to a specific material volume; and floors, which are part of a non-design zone, are denoted by x. This case study considers the optimization of the planar lateral load resisting system defined in terms of the aforementioned domain. The resulting complete structure is shown in Fig. 13a and was modeled using both beam and shell elements. A density filter [35] was used to prevent instabilities in the continuum design domain during the solution of the sub-problem.

Fig. 14. Material volume convergence history for case study 2.

system during the optimization process, as the resulting topology will be affected. 5.3. Case study 2 The second case study serves to apply the framework developed in Section 4 to the more general design domain shown in Fig. 12.

5.3.1. Problem formulation In order to impose design constraints on different portions of the design domain (e.g. prescribe a specified volume of material for the K domain), the problem formulation for this case study has been modified as follows:

Find q ¼ fq1 ; . . . ; qn gT nC Z X to minimize V C ðqÞ ¼ e¼1

Xe

ð40Þ

qe dX

ð41Þ

Fig. 15. Components of the fragility-based equivalent static loads corresponding to the master DOF belonging to the vertical axis of symmetry for case study 2: (a) Example 1; (b) Example 2.

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integration of the scaled pressures. The uncertainty in the modal damping ratio of the complete system was modeled as a lognormal random variable with mean value lf ¼ 0:02 and standard deviation rf ¼ 0:0050 as shown in Fig. 6b and was defined by directly assigning the damping ratio given by the distribution to the first mode. To model the horizontal stiffness offered by the floor systems, released non-designable W24  176 beams connect all nodes present at each possible outrigger level. In estimating the resonant response of both examples, the common assumption that the first mode will dominate the aforementioned response of tall buildings was used. Thus, modal analysis was performed and the ~ and F were determined by using the first mode load vectors F bd T bd T in the calculation of the resonant contribution to the response of the system.

Fig. 16. Peak displacement history for the degree of freedom subject to the performance constraint for case study 2.

subject to:

KTj uðqÞ  LSEDPj 6 0; KðqÞuðqÞ ¼ F c  ; nK Z X h¼1

Xh

EDP Tj

j ¼ 1; . . . ; N c j ¼ 1; . . . ; Nc

qh dX ¼ V K

0 6 qe 6 1 e ¼ 1; . . . ; nC ;

ð42Þ ð43Þ ð44Þ

0 6 qh 6 1 h ¼ 1; . . . ; nK

ð45Þ

where V C is the volume of material in the portion of the design domain belonging to C; nC is the number of elements in C; nK denotes the number of elements in K, and V K is the designer-prescribed volume of material in the K domain. It may be desired by the designers to optimize the location of a number of outriggers in the final design. This can be achieved by the optimization of material in the outrigger domain K, where the elements of each outrigger are associated with the same design variable. The placement of the outriggers in the K domain is then optimized simultaneously with the volume of material in the C domain. Two examples were considered using two distinct loading scenarios. The master DOF of the complete system for both examples were defined as the X-direction displacement of five nodes (one at the center of mass, two at the edge of the core and two at the extremes of each outrigger) at 39 heights representing the various possible outrigger levels. The values of the masses were simply obtained from their relevant influence areas. The first example considered a scenario with the external loading vector defined in an analogous fashion to that in the first case study (Section 5.2). For the second example, non-zero components of the external loads were calculated for all 39 levels through appropriate

5.3.2. Results Fig. 13b and c show the final designs for Examples 1 and 2, respectively. The volume history, given in Fig. 14, shows rapid and steady convergence for both Example 1 and Example 2. The initial and final fragility-based load distributions corresponding to the master DOF belonging to the vertical axis of symmetry, ~ , for Examples 1 and 2 are shown in Fig. 15a and b, respectively. F bd T ~ at convergence is also shown in Fig. 15 for both Additionally, F bd T examples and matches well with the load corresponding to the final optimal structure. These results again show rapid and steady convergence, further validating the proposed framework. The initial and final natural frequencies for the structures in Example 1, 0.108 Hz and 0.119 Hz, and Example 2, 0.108 Hz and 0.114 Hz, respectively, make it apparent that the dynamic nature of these structures should be considered in the topology optimization problem. The displacement histories corresponding to the constrained deflection for both examples are shown in Fig. 16 and illustrate that the constraint was met for the final systems, as was the associated performance objective of 10% as can be seen in Fig. 17a and b for Examples 1 and 2, respectively. Finally, it is interesting to observe how the structural systems that emerge work in harmony with the floor systems with some cases of classic chevron bracing appearing. 6. Conclusions In this paper a framework was proposed for the optimal topology design of wind-sensitive tall buildings within a performance-based design setting that rigorously accounts for the time-variant aleatory nature of the aerodynamic loads while additionally considering time-invariant system uncertainties. The performance of the structure is described through a number of

Fig. 17. Fragility curves for optimal structure, case study 2: (a) Example 1; (b) Example 2.

S. Bobby et al. / Engineering Structures 74 (2014) 242–255

fragility-based constraints on the engineering design parameters. For solving the resulting probabilistic and dynamic topology optimization problem, a novel decoupling strategy is developed. The method centers on deriving a number of fragility-based equivalent static load distributions that are rigorously defined from the response statistics of the system. By assuming these loads independent of the design variables, a static sub-problem is defined that not only decouples the performance-based assessment from the topology optimization problem but also assumes a form that can be efficiently solved by any well-established gradient-based optimization algorithm. By updating the fragility-based static loads at the end of each converged sub-problem until load convergence, the performance requirements of the original problem are guaranteed for the final optimum topology. The applicability of the proposed strategy was illustrated through the application of the method to a number of case studies. The validity of the framework was demonstrated by the rapid and steady convergence seen in all cases. Acknowledgments Support for this work was provided by the NSF Grant No. CMMI1301008 and the Global Center of Excellence at Tokyo Polytechnic University, funded by MEXT. References [1] Ellingwood B. Probability-based codified design for earthquakes. Eng Struct 1994;16:498–506. [2] Kareem A. Reliability analysis of wind-sensitive structures. J Wind Eng Ind Aerodyn 1990;33:495–514. [3] Augusti G, Ciampoli M. Performance-based design in risk assessment and reduction. Prob Eng Mech 2008;23:496–508. [4] Ellingwood B. Probability-based codified design: past accomplishments and future challenges. Struct Saf 1994;13:159–76. [5] Jensen HA, Valdebenito MA, Schuëller GI. An efficient reliability-based optimization scheme for uncertain linear systems subject to general Gaussian excitation. Comput Methods Appl Mech Eng 2008;198(1):72–87. [6] Valdebenito MA, Schuëller GI. Efficient strategies for reliability-based optimization involving non-linear, dynamical structures. Comput Struct 2011;89:1797–811. [7] Spence SMJ, Gioffrè M. Efficient algorithms for the reliability optimization of tall buildings. J Wind Eng Ind Aerodyn 2011;99(6–7):691–9. [8] Spence SMJ, Gioffrè M. Large scale reliability-based design optimization of wind excited tall buildings. Prob Eng Mech 2012;28:206–15. [9] Jensen HA, Kusanovic DS, Valdebenito MA, Schuëller GI. Reliability-based design optimization of uncertain stochastic systems: gradient-based scheme. J Eng Mech 2012;138:60–70. [10] Spence SMJ, Kareem A. Data-enabled design and optimization (DEDOpt): tall steel buildings frameworks. Comput Struct 2013;134(12):134–47. [11] Liang QQ, Xie YM, Steven GP. Optimal topology design of bracing systems for multistory steel frames. J Struct Eng 2000;126(7):823–9. [12] Yin L, Ananthasuresh GK. Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct Multidisc Optim 2001;23:49–62. [13] Belblidia F, Bulman S. A hybrid topology optimization algorithm for static and vibrating shell structures. Int J Numer Methods Eng 2002;54:835–52. [14] Maute K, Allen M. Conceptual design of aeroelastic structures by topology optimization. Struct Multidisc Optim 2004;27:27–42. [15] Stromberg LL, Beghini A, Baker WF, Paulino GH. Application of layout and topology optimization using pattern gradation for the conceptual design of buildings. Struct Multidisc Optim 2011;43(2):165–80. [16] Kareem A, Spence SMJ, Bernardini E, Bobby S, Wei D. Using computational fluid dynamics to optimize tall building design. CTBUH J 2013(3):38–43. [17] Bobby S, Spence SMJ, Bernardini E, Kareem A. Performance-based topology optimization of wind-excited tall buildings. In: 3rd International conference on soft computing technology in civil, structural and environmental engineering (CSC2013), Varenna, Italy; 2013. [18] Maute K, Frangopol DM. Reliability-based design of MEMS mechanisms by topology optimization. Comput Struct 2003;81:813–24. [19] Kharmanda G, Olhoff N, Mohamed A, Lemaire M. Reliability-based topology optimization. Struct Multidisc Optim 2004;26:295–307.

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