An efficient framework for the elasto-plastic reliability assessment of uncertain wind excited systems

Structural Safety 58 (2016) 69–78

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An efficient framework for the elasto-plastic reliability assessment of uncertain wind excited systems Pietro Tabbuso a, Seymour M.J. Spence b,⇑, Luigi Palizzolo a, Antonina Pirrotta a, Ahsan Kareem c a

Department of Civil, Environmental, Aerospace, Materials Engineering (DICAM), University of Palermo, Viale delle Scienze, 90128 Palermo, Italy Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109, USA c NatHaz Modeling Laboratory, Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN 46556, USA b

a r t i c l e

i n f o

Article history: Received 11 June 2015 Received in revised form 2 September 2015 Accepted 7 September 2015

Keywords: Elasto-plastic structures Dynamic shakedown Wind loads Dynamic wind effects Reliability analysis Subset Simulation

a b s t r a c t In this paper a method to efficiently evaluate the reliability of elastic-perfectly plastic structures is proposed. The method is based on combining dynamic shakedown theory with Subset Simulation. In particular, focus is on describing the shakedown behavior of uncertain elasto-plastic systems driven by stochastic wind loads. The ability of the structure to shakedown is assumed as a limit state separating plastic collapse from a safe, if not elastic, state of the structure. The limit state is therefore evaluated in terms of a probabilistic load multiplier estimated through solving a series of linear programming problems posed in terms of the responses of the underlying linear elastic model and self-stress distribution. The efficiency of the proposed procedure is guaranteed by the simplicity of the mathematical programming problem, the underlying structural model solved at each iteration, and the efficiency of Subset Simulation. The rigor of the approach is assured by the dynamic shakedown theory. The applicability of the framework is illustrated on a steel frame example. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The unavoidably aleatory and uncertain nature of the environment in which building systems are constructed, as well as the inevitable epistemic and knowledge uncertainties involved in describing such an environment, implies the necessity of using probabilistic approaches for assessing the performance of structural systems. This realization was the driving force behind the development of reliability-based approaches in civil engineering [1,2], and the subsequent development of design codes based on reliability theory [3,4]. It is also the basis on which state-of-theart performance-based design is founded [5,6]. Recently there has been significant interest in developing specific reliabilitybased procedures for assessing the performance of wind excited structures [7–17]. However, as in classic reliability analysis, the limit states indicating structural failure are associated with first yield of the structural system. The situation is somewhat different in seismic engineering due to the importance of the post-yield behavior in defining adequate performance. For this reason, specialized methods, such as incremental dynamic analysis (IDA), ⇑ Corresponding author. Tel.: +1 734 764 8419; fax: +1 734 764 4292. E-mail addresses: [email protected] (P. Tabbuso), [email protected] (S.M.J. Spence), [email protected] (L. Palizzolo), [email protected] (A. Pirrotta), [email protected] (A. Kareem). http://dx.doi.org/10.1016/j.strusafe.2015.09.001 0167-4730/Ó 2015 Elsevier Ltd. All rights reserved.

have been developed with the aim of bringing together probabilistic design principles and step-by-step non-linear analysis [18]. In the design of wind excited structures, on the other hand, engineers generally do not explore the behavior of the structural system beyond the elastic limit. Probably the main reason for this can be found in societys intolerance towards damage of buildings due to wind storms. The downside of this is that the structural systems of many buildings are designed with no knowledge of their inelastic behavior, potentially leaving them exposed to undesirable collapse scenarios or at least unknown post-yield behavior. From a research perspective, the problem of understanding and modeling the inelastic behavior of wind excited structures has been the subject of a number of works [19–26] including the application of pushover analysis [27]. The main difficulty in defining a modeling procedure that can account for this effect is the extremely long duration of wind storms. Indeed, this characteristic practically eliminates the possibility of using methods such as IDA as they require non-linear dynamic integration of the entire load history, which constitutes a computationally daunting task [28,29]. An alternative approach for engineered building systems is presented by the well-established plastic theorem approaches [30–32]. Indeed, recent computational advances make these methods an attractive alternative for rapidly assessing the general behavior of ductile structures such as steel and concrete frames [33–35]. These methods also provide a complete picture of the post-yield behavior

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of the system, indicating for example when the system is in shakedown or low cycle fatigue (plastic shakedown) which can be of importance for structures subject to long duration cyclic loads such as extreme wind excitation. Shakedown analysis is classically carried out under a quasistatic loading scenario. Under these conditions, the goal is to understand whether an elasto-plastic structure subject to loads varying within a specified domain will eventually respond in a purely elastic manner after a finite amount of plastic deformation and is based on the well-known Bleich-Melan and Koiter theorems. In the last decades, many applications have been treated with this approach [35–38], including recent applications where both the loads and the strength parameters have been considered as uncertain [39,40]. When dynamic effects are important, dynamic shakedown analysis becomes necessary. This concept was first introduced by Ceradini [41] through the development of a lower bound theorem which took the form of conditions under which an elasto-plastic solid subject to an infinite dynamic load history will shakedown. The main difference between quasi-static and dynamic shakedown analysis is that in the first the loading history is defined by a convex domain in which the loads may be repeated indefinitely, while in the second the load history must be fully specified. In addition, unlike the quasi-static theory, in dynamic shakedown the solution will in general depend on the initial conditions of the fictitious elastic response. This situation generally makes the application of Ceradini’s theorem considerably more computationally involved compared to the quasi-static scenario. An important special case that significantly facilitates dynamic shakedown analysis, and which will be exploited in this work, is when the forcing functions are infinite and periodic [30,42–44]. The aim of this paper is to define a framework based on the aforementioned concept of dynamic shakedown that can be used for efficiently assessing the shakedown limit behavior of uncertain elastic perfectly plastic structures subject to long duration stochastic wind loads. 2. The dynamic shakedown problem for elastic perfectly plastic frames 2.1. Mechanical model In order to introduce the formulation of interest to this study, it is first convenient to consider a discrete elastic perfectly plastic plane frame defined by nb Euler–Bernoulli beam elements and nN free nodes in a small displacement and deformation regime (the generalization to 3D frames is immediate). Consider indicating with u and F the vectors of dimensions nf collecting the displacements and external loads of the free nodes, and with q and Q the vectors of dimensions nd collecting the generalized strains and stresses where nf ¼ 3  nN and represents the total number of free degrees of freedom of the system while nd ¼ 6  nb represents the total number of stress and strain parameters at the ends of each beam element. The equilibrium of the aforementioned system can be expressed as:

q¼eþp

ð3Þ

while C is a compatibility matrix depending only on the geometry of the system. The elasticity equations for this system may be expressed as:

Q ¼ De þ Q 

ð4Þ

where D is the block diagonal matrix containing the elastic stiffness matrices of the nb beam elements defining the structure while Q  is the vector collecting the perfectly clamped element generalized stresses. The generalized stresses at each cross section of the structure cannot lie outside of the yield surface, therefore the vector Q must respect the following inequality:

u ¼ NT Q  R 6 0

ð5Þ

where u is the piece-wise linearized yield vector, N is a block diagonal matrix of unit external normals to the piece-wise linear convex yield surface while R is the plastic resistance vector. When at least one of inequalities of Eq. (5) is an equality, plastic strain can occur according to the following plastic flow rule:

_ p_ ¼ N k;

k_ P 0;

uT k_ ¼ 0; u_ T k_ ¼ 0

ð6Þ

in which k represents the vector of plastic multipliers. Eqs. (1)–(6) together with the initial conditions

u ¼ u0 ;

u_ ¼ u_ 0 ;

p ¼ p0 ;

for t ¼ 0

ð7Þ

govern the dynamic analysis problem of elastic perfectly plastic plane frames. It is worth noting that the external stiffness matrix of the plane frame is given by K ¼ C T DC. This problem is usually solved by means of a step-by-step procedure set in a deterministic environment. To the authors’ knowledge, no exact solution exists. When the response of an elasto-plastic structure subject to dynamic loading becomes purely elastic, after a first phase of finite duration in which some plastic deformations are produced, the structure is said to have adapted to an elastic state and ‘‘dynamic shakedown” has occurred. In other words, a finite field of timeindependent plastic strains has formed that allows the structure to respond in a purely elastic regime. The term shakedown implies the finite nature of the plastic deformations that in general are to be considered modest even though their exact amount is unknown as is the exact time at which the plastic phase ends and the purely elastic one begins. If the structural response exceeds the shakedown limit, the structure is exposed to a sort of inadaptation collapse, characterized by the uncontrolled growth of the plastic deformation during the load history with the possibility of failure due to the excessive accumulation of plastic strains (incremental collapse), or by inverting plastic strains in a cyclic loading scenario, with the possibility of failure by fatigue (alternating plasticity collapse). 2.2. Dynamic shakedown

follows, the over-dot indicates the time derivative while ðÞT indicates the transpose of the relevant quantity. Geometric compatibility between strains and displacements of the nodes can be imposed through:

A criterion for dynamic shakedown, for a structure subject to a fully specified loading history from t ¼ 0 to t ¼ þ1, was given first by Ceradini [41,43] and is as follows: a necessary and sufficient condition for dynamic shakedown is that there exists a finite time r P 0, and some initial conditions in terms of displacements u0 , velocities u_ 0 and plastic strains p0 , such that the purely elastic stress response (fictitious) to the given load history with these ini^ ðtÞ proves to be inside the yield surface at any subtial conditions Q

q ¼ Cu

sequent time t P r:

€  V u_ C T Q ¼ F  Mu

ð1Þ

where C T is the equilibrium matrix while M and V are the mass and damping matrices. Furthermore, in Eq. (1) and in what

ð2Þ

where q represents the sum of an elastic (e) and a plastic (p) strain:

u ¼ N T Q^ ðtÞ  R < 0; 8t P r:

ð8Þ

P. Tabbuso et al. / Structural Safety 58 (2016) 69–78

Considering that the initial conditions, in terms of displacements
 and velocities u0 ; u_ 0 , give rise to free-motion stress fields Q F ðtÞ and that the initial plastic strains p0 can be represented by a selfstress distribution q, with a change of time variable (t ¼ r þ s), an alternative and more general form of the dynamic shakedown theorem can be given [31,45]: a necessary and sufficient condition for dynamic shakedown is that there exists a finite time r P 0, a free-

motion stress field Q F ðsÞ and a time independent self-stress distribution q such that the sum of these stresses with the elastic stress

response to the loads backward truncated at r, namely Q E ðr þ sÞ, proves to be inside the yield surface at any time s P 0:

h

i

u ¼ N T Q E ðr þ sÞ þ Q F ðsÞ þ q  R < 0; 8s P 0:

ð9Þ

It is worth observing that from a physical standpoint the freemotion stress field Q F ðsÞ is representative of the dynamic effects produced by the plastic strains generated by the structure’s motion during the transient phase up until adaptation, while the analogous static effects are represented by q. As stated before, in Ceradini’s approach [41,43], these effects are simulated by means of fictitious initial conditions fixed at remote time t ¼ 0. In order to obtain information such as the safety factor with respect to inadaptation (i.e. non-shakedown) of the structure, let the forcing function vector be defined through a scalar multiplier s > 0 so that F s ðtÞ ¼ sFðtÞ. For this time-dependent loading system, the shakedown safety factor is that particular value of s, indicated with sp , such that shakedown occurs for all s < sp while shakedown does not occur for all s > sp . Under these conditions the shakedown safety factor can be estimated by first solving, for a fixed value of r, the maximization problem (see e.g. [43–46]):

sðrÞ ¼ max s

ð10aÞ

s;Q F ;q

subject to

h

i

u ¼ N T sQ E ðr þ sÞ þ Q F ðsÞ þ q  R 6 0; 8s P 0

ð10bÞ

CT q ¼ 0

ð10cÞ

and then looking for the asymptotic value of the non-decreasing function sðrÞ:

sp ¼ lim sðrÞ ¼ max sðrÞ: r!1

rP0

ð11Þ

Eq. (10c) represents the self-equilibrium condition. The function sðrÞ is non-decreasing as an increase of r will reduce the number constraints in the maximization problem outlined in Eqs. (10) [45]. The evaluation of the safety factor sp , through the resolution of Eqs. (10) and (11), is in general a challenging task. In the case of periodic actions the problem can be considerably simplified. For this particular but important case, it has been demonstrated [42–45] that the maximum safety factor, i.e. Eq. (11), can be estimated by dropping the free-vibration stress and setting r ¼ 0 (and so t ¼ s). In other words, in the case of periodic actions it is always possible to find some particular initial conditions (u00 ; u_ 00 ) which make the purely (fictitious) elastic response of the system coincident at all times with its ‘‘forced vibration” counterpart, while the shakedown multiplier associated with these initial conditions coincides with sp . Under these circumstances the

sp ¼ max s s;q

71

ð12aÞ

subject to

 S ¼ max N T Q S ðtÞ Q

ð12bÞ

u ¼ sQ S þ N T q  R 6 0

ð12cÞ

CT q ¼ 0

ð12dÞ

06t6T

 S is the elastic envelope stress vector defined for each yieldwhere Q ing mode as the maximum of the plastic demand in time [47–49], Q S ðtÞ is the purely elastic steady-state response of the structure while T is the period of the forcing function. 3. Proposed reliability-based assessment framework for elastic perfectly plastic systems 3.1. The probabilistic shakedown safety factor With the aim of defining a probabilistic shakedown safety factor for uncertain structural systems subject to stochastic wind loads, consider the illustrative wind velocity time history of Fig. 1. In particular, it is common in wind engineering to take the central portion of the storm, of duration T, as a finite realization of a stationary stochastic process, or at least a weakly stationary processes. This stationary segment of maximum wind speeds can last several hours and is generally considered responsible for causing the bulk of wind-related damage to structural systems. What is of interest here is to define an approach for estimating the safety of an uncertain structural system against fatigue failure or incremental plastic collapse during the stationary segment of an extreme wind storm. Because of the finite length of wind storms, and so of the external loading process, the criterion introduced in Section 2.2 for identifying whether a structure will shakedown or not cannot be directly applied [44,45]. Indeed, the aforementioned approach is based on discriminating between shakedown and nonshakedown on the basis of the bounded or non-bounded nature of the total plastic work done by the structure during the infinite load history. A simple solution to this dilemma is to consider the wind storm of duration T infinitely repeated, as illustrated in Fig. 2. This artificially generated wind storm will have an infinite duration, as will the associated loading history, and be periodic with period T. This simple extrapolation allows the particularly favorable form of the shakedown criterion, given at the end of Section 2.2, to be used for determining whether the structure will

elastic backward truncated stress response Q E ðr þ sÞ coincides with the steady-state response Q S due to the periodic action. Furthermore, the shakedown multiplier sp is the same for any r, as the function sðrÞ is constant, and hence Eq. (11) becomes unnecessary (see e.g. [30,43–45]). Therefore, in the case of periodic actions, the shakedown safety factor can be evaluated through the following reduced linear programming problem:

Fig. 1. Illustration of a typical wind event where the highlighted maximum intensity segment of duration T may in general be considered stationary.

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Fig. 2. Illustration of how the stationary segments are appended therefore defining a periodic excitation of infinite duration starting at t0 .

The approach outlined in this section requires the characterization of the time-dependent wind loads acting on the structure due to the wind storm. In particular, given an event described by a wind speed history, it is necessary to describe the aerodynamic response of the structure being investigated. To this end, various methods including Computational Fluid Dynamics (CFD), wind tunnel testing, as well as quasi-steady modeling, can be used. In this work a quasi-steady model is adopted for simulating the stochastic wind loads acting on the structure during the stationary segment of the wind storm. Details of the model can be found in Appendix A. Even though a quasi-steady model was used in this work, it should be observed that the approach presented is this work is not affected by the particular model used to simulate the time dependent wind loads and can therefore be coupled with any wind loading model deemed appropriate by the analyst. 3.2. Failure domain and solution strategy

shakedown, and so respond in a purely elastic manner, at some point during the infinite load history. The practical significance of this lies in the fact that, if it is found that the structure does indeed shakedown during the artificially infinite loading history, then it cannot have failed due to plastic fatigue or incremental plastic collapse during the actual finite loading event of duration T. It is this observation that is exploited in this work for characterizing the safety against plastic collapse of wind excited structural systems. In particular, once the steadystate stress response Q S ðtÞ of the structure to a realization of the stationary segment of the non-amplified load is determined, it is possible to estimate the corresponding shakedown safety factor, sp , by solving the problem outlined in Eqs. (12). In particular, if sp P 1 the structure will shakedown, if sp < 1 it will not shakedown. By simulating over a large number of samples, a probabilistic description can be given to sp (indicated in the following as Sp to distinguish it from its deterministic counterpart), and therefore to the capacity of the structure to resist plastic collapse. With respect to the approach presented here, some observations should be made: (1) The satisfaction of Eqs. (12) for a particular realization of the artificial infinite and periodic load history does not necessarily mean that the structure will shakedown before the end of the effective load process of duration T. What it does imply is that the structure, within the limits of first-order plasticity theory, will not suffer plastic fatigue failure or incremental plastic collapse. (2) The non-satisfaction of Eqs. (12) does not necessarily mean that the structure will suffer plastic collapse. Indeed, it is possible that the real storm (of finite duration T) ends before enough time has elapsed for plastic collapse to occur. This implies that the proposed approach is conservative albeit only marginally as the typical duration of extreme wind storms (several hours) makes the possibility of a structure surviving in a condition that does not imply eventual shakedown remote. (3) Although the approach outlined here is presented considering as a driving excitation the stationary segment of the wind storm, the approach could equally be applied to the entire wind storm (which would then be repeated indefinitely for defining the artificial infinite and periodic loading histories) and therefore to any non-stationary wind event such as hurricanes or thunderstorm downbursts [50,51]. In other words, the stationary segment is considered here as it is generally accepted that for classic synoptic wind events, i.e. straight winds, the rampup and attenuation part of the storm do not participate in causing damage and can therefore be ignored.

As discussed in Section 3.1, in this paper dynamic shakedown under a wind storm of artificially infinite duration has been chosen as a limit state between acceptable and unacceptable plastic regimes. If the structure exceeds this limit, it could suffer from potentially dangerous phenomena such as incremental collapse (excessive accumulation of plastic strain), alternating plastic collapse (production of plastic strain in each cycle leading to possible failure for fatigue) or even instantaneously plastic collapse. In this framework, a structure is safe when the scalar load multiplier s is not greater than its dynamic shakedown multiplier sp . To better understand this failure domain, it is useful to consider the different structural behaviors that an elastic perfectly plastic structure can have when responding in a steady-state through a Bree-like diagram as shown in Fig. 3. With reference to this diagram and indicating with s and s0 the dynamic load multiplier and the fixed time-independent vertical load multiplier respectively, it is possible to distinguish five different structural behaviors: purely elastic (E), elastic shakedown (S), low cycle fatigue or plastic shakedown (F), incremental collapse or ratcheting (R) and instantaneous collapse (I). The presence of the fixed/quasi-static load multiplier does not affect the considerations and observations so far made. Fig. 3 clearly illustrates how the proposed approach gives a full description of the failure domain of a structure modeled as elastic perfectly plastic. While in this work the material behavior is limited

Fig. 3. Bree-like diagram illustrating the failure domain in terms of the vertical and horizontal load multipliers.

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P. Tabbuso et al. / Structural Safety 58 (2016) 69–78

to elastic perfectly plastic, it should be observed that shakedown theory has been extended to a number of more sophisticated material models [47,48] and therefore the approach here may also be extended to these material models. The aim of this work is the study of the elasto-plastic behavior of uncertain systems subject to stochastic external excitation. With this in mind, it is convenient to define the vector h 2 Rn containing the uncertain parameters with respect to both the structural model as well as the stochastic wind loads. Let q : Rn # ½0; 1Þ be the joint probability density function (PDF) characterizing the uncertain parameters h ¼ ½h1 ; h2 ; . . . ; hn . Without loss of generality these parameters can be assumed as independent (i.e. it is assumed that an appropriate transformation can be used in order to eliminate any dependency between the parameters) so that Q qðhÞ ¼ nj¼1 qj ðhj Þ, where qj : Rn # ½0; 1Þ are the marginal PDFs of the various components hj . Under these conditions, the failure probability can be written in the following form:

Z PðFÞ ¼

Z IF ðhÞqðhÞdh ¼

"

IF ðhÞ

# n Y qj ðhj Þ dh

ð13Þ

j¼1

where F  Rn is the failure domain in the space of the basic random variables collected in h and identified as the space where the probabilistic shakedown load multiplier Sp is less than unity (i.e. F ¼ fh : Sp ðhÞ < 1g) while IF : Rn # f0; 1g is an indicator function defined as IF ðhÞ ¼ 1 when h 2 F and IF ðhÞ ¼ 0 otherwise. It should be observed that the integral of Eq. (13) is in general of high dimensions (when considering the stochastic excitation the integral will have a total dimension in the order of thousands) and therefore cannot be efficiently solved using classical analytical approaches, such as first-order reliability methods [52]. 3.2.1. Subset Simulation Monte Carlo simulation has been widely used for solving integrals of the type shown in Eq. (13) (see e.g. [53–55]) due to its robustness and ability to solve problems with complex failure regions. However, this approach becomes inefficient when applied to problems characterized by small failure probabilities, such as the applications of interest to this work, as the number of samples that are required in order to achieve a predetermined level of accuracy is proportional to the ratio 1=PðFÞ. Couple this with the fact that, for each sample, an optimization problem must be solved and it is not difficult to recognize that a more efficient approach is needed. In this work Subset Simulation [56] is considered for overcoming these hurdles. This method is based on the idea that a small probability of failure can be expressed as a product of larger conditional failure probabilities by introducing appropriate intermediate failure events. In particular, given a failure domain, let F 1  F 2      F m ¼ F be a decreasing nested sequence of failure regions so that F k ¼ \ki¼1 F i ; k ¼ 1; 2; . . . ; m. The probability of failure PðFÞ can be represented as the probability of falling in the final subset F m . From the definition of conditional probability the following holds: m1 m1 PðFÞ ¼ PðF m Þ ¼ Pð\m i¼1 F i Þ ¼ PðF m j\i¼1 F i ÞPð\i¼1 F i Þ

¼ PðF m jF m1 ÞPð\m1 i¼1 F i Þ ¼    ¼ PðF 1 Þ

m1 Y

PðF iþ1 jF i Þ

ð14Þ

i¼1

Eq. (14) shows that the probability of failure can be calculated as the product of a number of conditional probabilities. Since the failure region is defined as F ¼ fh : Sp ðhÞ < 1g, then the intermediate failure regions can be defined as F i ¼ fh : Sp ðhÞ < ^si g. In this way, the probability of failure can be rewritten as:

m1 Y

PðFÞ ¼ PðSp < 1Þ ¼ PðSp < ^s1 Þ


 P Sp < ^siþ1 jSp < ^si

ð15Þ

i¼1

where ^s1 > ^s2    > ^sm ¼ 1 is a decreasing sequence of threshold levels. These levels are generated adaptively using information from simulated samples so that the conditional probabilities are approximately equal to a common specified value p0 (experience shows that p0 ¼ 0:1 is a prudent choice [57]). To calculate the probability of failure expressed by Eq. (15), PðSp < ^s1 Þ and PðSp < ^siþ1 jSp < ^si Þ for i ¼ 1; . . . ; m  1 must be calculated. The first is the unconditional probability of failure that can be estimated directly using Monte Carlo simulation. The second represents the conditional probability that can be simulated through a Markov chain Monte Carlo simulation based on the modified Metropolis–Hastings algorithm [56]. The overall Subset Simulation procedure can be summarized as follows: 1. generate N samples of h by direct Monte Carlo simulation such that they are independent and identically distributed as the original PDF qðhÞ; 2. choose a value of the shakedown multiplier ^s1 such that ½ð1  p0 ÞN responses lie outside the subset F 1 while p0 N samples belong to F 1 ¼ fSp < ^s1 g, i.e. to ‘‘conditional level 1”; 3. starting from each sample in ‘‘conditional level 1”, generate, using the modified Metropolis–Hastings algorithm, ð1  p0 ÞN additional conditional samples so that level 1 has a total of N samples. 4. repeat steps 2 and 3 for higher conditional failure levels until N samples at conditional level m have been generated. 3.3. Overall procedure The overall procedure defined by the proposed simulationbased elasto-plastic framework is illustrated in the flowchart of Fig. 4. It should be observed that the overall efficiency of the procedure is due to: (1) the linear nature of the dynamic problem solved for each sample hi ; (2) the structure of the optimization problem of Eqs. (12a)–(12d) which constitutes a linear programming problem; (3) the integration with Subset Simulation that minimizes the number of samples necessary for estimating the distribution of Sp . 4. Case study 4.1. Description As a case study, the five-floor two-span plane steel frame shown in Fig. 5 is considered. The geometry of the frame is fully described by the span lengths (here assumed as L1 ¼ 600 cm, L2 ¼ 400 cm) and by the inter-story height (H ¼ 400 cm). The material is assumed having an elastic perfectly plastic constitutive relation, as shown in Fig. 6, and is therefore completely described by the yield stress ry and the Young’s modulus E. These two parameters are considered as uncertain. In particular a log-normal distribution is assigned both for the yield stress (mean 355 MPa and standard deviation 15 MPa) and the Young’s modulus (mean 210 GPa and standard deviation 10 GPa). Obviously, this choice causes both the stiffness and the resistance of the structure to become aleatory. All the elements of the frame have box cross sections (see Fig. 7a) with width b ¼ 200 mm, height h ¼ 200 mm and constant web thickness t ¼ 5 mm. Furthermore, as shown in Fig. 5, two rigid perfectly plastic hinges are located at the extremes of each element with one additional hinge positioned at the center of all beams. The convex yield domain of a plastic hinge depends on the cross section being investigated. In particular, it can be observed that the domain is piece-wise linear. This should be seen as a limitation as in general any yield domain can be easily linearized without

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P. Tabbuso et al. / Structural Safety 58 (2016) 69–78

Fig. 5. The five-floor two-span steel frame of the case study: geometry and load conditions.

Fig. 4. Flowchart of the proposed simulation-based elasto-plastic reliability assessment framework.

introducing any untoward approximations. As shown in Fig. 7b, for the case study considered in this section, the yield domain is defined in terms of the interaction between the bending moment M and the axial stress N. It is worth observing that the axial yield stress can be written as N y ¼ ry A while the bending yield moment as M y ¼ ry M p , where A and Mp are the area and the plastic section modulus of the relevant cross section. In order to describe the dynamic response of the frame, a floorwise lumped mass model was adopted. In particular, the mass of each floor was calculated as M ¼ lq0 ðL1 þ L2 Þ=g, where g is the gravitational acceleration, q0 ¼ 30 kN/m and ðL1 þ L2 Þ are the dead loads and the length of each floor respectively while l is a lognormal random variable (mean 0.8 and standard deviation 0.04) modeling the uncertainty in the mass. The horizontal displacements of the floors have been chosen as dynamically significant degrees of freedom. By using static condensation, this leads to the following equations of motion:



M tt

0

0

0



  €t C tt u þ €r 0 u

0 0



  K tt u_ t þ K rt u_ r

K tr K rr



ut ur



 ¼

FðtÞ



0 ð16aÞ

where ut is the vector of horizontal floor displacements, ur is the vector collecting the remaining displacements, C tt is the damping

Fig. 6. Constitutive relation of the elastic perfectly plastic material model.

matrix of the reduced system estimated by directly fixing the modal damping ratios fj (j ¼ 1; . . . ; 5), K tt ; K tr ; K rt ; K rr are suitable submatrices obtained from the external stiffness matrix K of the complete system, FðtÞ is the stochastic forcing function while M tt is the lumped mass matrix. Through simple mathematical manipulation, Eq. (16a) can be reduced to the following set of differential and algebraic equations:

P. Tabbuso et al. / Structural Safety 58 (2016) 69–78

75

 2 are the mean wind speeds at heights z1 and z2 where v 1 and v respectively, Dz ¼j z1  z2 j while C z is a constant that can be set equal to 10 for design purposes [63].

4.2. Results and discussion

Fig. 7. (a) Box section of the beam and column elements. (b) The piece-wise linearized rigid perfectly plastic yield domain of the plastic hinges.

 t ¼ FðtÞ € t þ C tt u_ t þ Ku M tt u

ð16bÞ

ur ¼ K 1 rr K rt ut

ð16cÞ

 ¼ K tt  K tr K 1 K rt is the condensed stiffness matrix. In this where K tt way the complete vector of displacements u can be obtained from the vectors ut and ur . For the structure of this case study, the modal damping ratios fj were taken as log-normal random variables with mean 0.05 and standard deviation 0.005 [58,59]. In order to define the stochastic wind loads, the quasi-steady model with simulated wind time histories as outlined in Appendix A is adopted [60]. In particular, the following values are assumed for the model parameters: q ¼ 1:25 kg/m3, C j ¼ 1:3 and Aj ¼ 24 m2 for j ¼ 1; 2 . . . ; 5. Furthermore, the mean wind speed at each floor was determined from the following power law:

 z a

v j ¼ v 10 b

j

ð17Þ

10

where v 10 is the reference wind speed at 10 m above ground, zj is the height of the jth floor, b ¼ 0:65 and a ¼ 1=6:5. In particular, v 10 is to be considered representative of the largest wind speeds to occur on an annual basis for a location with significant wind exposure. Therefore, v 10 is defined here by a Type II distribution with mean value 30 m/s and standard deviation 3.5 m/s For the power spectral density (PSD) function of the longitudinal wind velocity fluctuations over the height of the building Sjj ðxÞ ¼ Sj ðxÞ; j ¼ 1; 2; . . . ; 5, the model proposed by [61] was used:

Sj ðxÞ ¼

where

1 200 2 zj v h 2 2p  v j

1 xz

1 þ 50 2pvj j

i5=3 ;

j ¼ 1; 2; . . . ; 5

ð18Þ

v  is the shear velocity of the flow expressed by:

v  ¼ v 10 b

ka   log 10 z0

ð19Þ

where ka ¼ 0:4 is the Von Kármán’s constant while z0 is the ground roughness height taken here to be uniformly distributed between 0.01 m and 0.03 m. The cross spectral density was defined as:

Sjk ðxÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sj ðxÞSk ðxÞcjk ðxÞ j; k ¼ 1; 2; . . . ; 5; j – k

ð20Þ

where cjk is the coherence function between v j ðtÞ and v k ðtÞ. The model suggested by Davenport [62] was used for the coherence function between the velocity fluctuation at the heights z1 and z2 :

"

x C Dz cjk ðDz; xÞ ¼ exp  1 z 2p 2 ½v 1 þ v 2 

# ð21Þ

As already outlined in Section 4.1, the uncertain parameters concerning the system are the damping ratios fj , the yield stress ry , the Young’s modulus E, and the mass through l. In particular, fj ; ry , E and l were considered to be independent random variables while all 5 vibration modes were considered in the response (i.e. j ¼ 1; . . . ; 5). This resulted in a total of 10 independent lognormal random variables describing the system uncertainties. With respect to the loads, the reference wind speed, v 10 , was taken as a Type II distribution while the roughness height, z0 , was taken as a uniform distribution. The stationary segment of the storm, T, was taken as having a duration of 16,085 s. In order to generate realizations of FðtÞ following the model of Appendix A with the aforementioned duration and with a sampling frequency of 1:27 Hz, a total of 10,240 independent and uniformly distributed random numbers in ½0; 2p are required [64]. A typical realization of the 5th floor forcing function, F 5 ðtÞ, is shown in Fig. 8(a) while Fig. 8(b) shows the corresponding target and simulated PSD function. The total size of the random vector h is therefore 10,252 which constitutes a truly high-dimensional reliability problem. In estimating the failure probability of the system, a total of 3700 simulations were used in running the Subset Simulation algorithm (1000 realizations for four conditional levels were generated while considering a conditional failure probability for levels 2, 3 and 4 of p0 ¼ 0:1). In calibrating the modified Metropolis–Hastings algorithm, the proposal PDFs for each uncertain parameter were chosen as uniform PDFs centered at the current sample point and with width equal to twice the standard deviation of the relevant component of h. For each realization of h, the linear programming problem of Eqs. (12) was solved therefore yielding a realization of the probabilistic shakedown safety factor Sp . In order to compare the results of the proposed framework with a more traditional elastic design criterion, the probabilistic elastic safety factor Se (i. e. the factor by which the external loads FðtÞ must be multiplied in order to bring the structure to its elastic limit) was also calculated for each realization of h. By running the proposed framework, the annual probability of the structure suffering collapse due to low cycle fatigue or incremental plastic collapse was estimated to be PðSp 6 1Þ ¼ 0:1125 while the annual probability of the structure exiting the elastic regime was estimated to be PðSe 6 1Þ ¼ 0:9985. The large difference between these values attests to the significant plastic reserves of the frame under investigation. It should be noted that in calculating these values the external loads are not amplified, therefore the stochastic excitation rigorously retains the spectral characteristics of the loading model. By observing how Subset Simulation may also be used to estimate the distributions of the simulated response parameters, Fig. 9 reports the distributions of Se and Sp . These distributions, which are calculated at no extra computational cost to the methodology, provide invaluable information on the sensitivity of the probabilities associated with suffering plastic collapse—or exiting the elastic regime—due to amplifications in the external excitation. Indeed, although the load multipliers only modify the amplitude of FðtÞ, i.e. the spectral characteristics of the excitation are not rigorously updated as the loads are modified, the distributions of Se and Sp provide information closely related to how the structure would respond to increases/decreases in the wind excitation. Also, any approximations in the spectral characteristics of the loads vanish as the multipliers tend to unity, and therefore to the actual stochastic wind

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(a)

(b)

Fig. 8. A realization of the top floor stochastic forcing function F 5 ðtÞ: (a) time history of total length T ¼ 16; 085 s; (b) corresponding PSD function.

the name suggests, in order to make the problem more tractable, the system is simplified as a single-degree-of-freedom elastoplastic oscillator.

Fig. 9. Distributions of the elastic, Se , dynamic shakedown, Sp , quasi-static (QS) shakedown, Sq , and limit analysis (LA), Sl , safety factors.

4.2.1. Comparisons In order highlight the role of dynamic amplification, Fig. 9 also reports the distribution of the shakedown multiplier Sq while considering a strictly quasi-static response (i.e. the contribution of inertial and damping forces is neglected during the calculation of the response of the system). As can be seen, dynamic amplification makes a significant contribution to the probability of plastic collapse. Indeed, the shakedown multiplier results to be around 10% lager if dynamic amplification is neglected. With the aim of illustrating the benefits of the proposed approach, in a quasi-static setting, over other simplified incremental static analysis methods, the proposed framework was compared to a simplified probabilistic limit analysis. In particular, the limit analysis was defined in terms of the following equivalent static wind loads (ESWLs): ðESWLÞ

Fj excitation acting on the structure. Within this context, the distributions of Se and Sp provide a full picture of the system-level plastic capacity of the structure. In particular, the availability of Se allows a direct comparison with a classic linear elastic design philosophy and provides invaluable information about the post-yield behavior of the structure over a full range of load intensities. For example, looking at Fig. 9 it can be seen that the plastic reserve of the case study structure is relatively insensitive to the load intensity with an elastic overload of around 50% causing failure due to low cycle fatigue or incremental plastic collapse. In other words, if the structure has a certain probability of exiting the elastic regime, a load increase in the order of 50% is needed to have the same probability of failure due to low cycle fatigue or incremental plastic collapse. It is foreseen that this type of information could significantly increase the attractiveness of designing wind excited structures to safely enter into a post-yield state during extreme wind events. For instance, the structural system of the case study could be designed with a 10% elastic overload under extreme loading conditions knowing that collapse is expected to occur at 50% overload. It should also be pointed out that by using this approach the limit behavior of the structure is estimated at a system level while the mechanical description is given at a local level through the plastic hinges. This is in contrast to the classic single-degreeof-freedom non-linear dynamic integration techniques where, as

" !# 1 rðz¼12Þ ; qC j Aj v 2j 1 þ 2g v 2 v z¼12

¼B

j ¼ 1; 2; . . . ; n

ð22Þ

ðz¼12Þ where rv and v z¼12 are the standard deviation and mean of the wind speed at height z ¼ 12 m, g is a peak factor to be taken equal to 3.4 while B ¼ 0:85 is a coefficient taking into account the lack of correlation that exists in the fluctuating wind loads [65]. By defining an ESWL for each sample, hj , of h, a corresponding collapse mulðjÞ

tiplier, sl , can be estimated by carrying out a classic static limit analysis. The distribution of the sampled multipliers is shown in Fig. 9. The significant difference between the proposed approach and the simplified probabilistic limit analysis is clear and demonstrates the benefits of the proposed approach, even in a quasistatic setting. In particular, the differences seen between the two approaches may be attributed to the following: (1) the fact that limit analysis does not account for the possibility of plastic collapse due to low cycle fatigue or ratcheting (regions F and R of Fig. 3) which, due to the significant duration of extreme wind events as well as the presence of a mean response component in the alongwind direction, are likely to play an important role in defining the plastic collapse of wind excited structures; and (2) the approximate nature of the ESWLs of Eq. (22). While the relative contribution of the above inconsistencies to the difference seen in Fig. 9 is difficult to quantify, it can be observed that a simplified limit analysis will in general introduce significant errors in the estimation of the plastic capacity of wind excited structures.

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 F j ðtÞ ¼ gj ðv j þ v j ðtÞÞ2  gj v 2j þ 2v j v j ðtÞ ;

Finally, it should be observed that these results were obtained by running the proposed methodology on a typical laptop in a little over an hour, showing the strong efficiency of the proposed approach. This result is particularly meaningful if the computational effort required by the adoption of the IDA approach in place of the dynamic shakedown problem of Eqs. (12a)–(12d) is considered. Indeed, in this case, an IDA would have to be run for each sample point of the simulation. It should also be observed that every IDA requires multiple (between 10 and 20) non-linear dynamic analyses to be carried out [18], and that extreme wind events have, as in the case study, a typical duration in the order of hours. The use of IDA would therefore require the solution of tens of thousands of non-linear dynamic analyses for event durations in the order of hours. This, on a typical laptop, would take days–instead of hours.

cient equal to 0:5qC j Aj , where q is the air density, C j is an experimentally determined quasi-steady pressure coefficient while Aj the influence area of the jth degree of freedom in the direction of the mean wind. In this model the zero-mean fluctuating component of the wind field is described by an n-dimensional multivariate stochastic vector process v ðtÞ with components for N ! 1 given by [60]:

5. Conclusion

where Dx is the sampling frequency, xml is given by:

This paper focused on the development of an efficient framework for the post-yield analysis of uncertain elastic perfectly plastic framed structures subject to stochastic wind loads. The aim was to provide a tool that allowed for a full probabilistic description of the impending inelastic behavior of systems subject to dynamic loads of long duration that was not only rigorous from a theoretical standpoint but also capable of giving results in a timely fashion without having to resort to super-computing. To this end a method was proposed based on combining the theoretical rigor of dynamic shakedown analysis with the computational efficiency of advanced simulation-based reliability assessment. In particular, the resolution of the dynamic shakedown problem was reduced to solving a series of linear programming problems therefore ensuring the efficiency in defining safe plastic excursion from the insurgence of dangerous phenomena such as low cycle fatigue, incremental collapse and instantaneous collapse. The applicability of the proposed method was illustrated on a five-story two-bay elasticperfectly plastic steel frame with 60 possible plastic hinges and subject to uncertainties in the mechanical properties as well as long duration (over 4 h) stochastic wind loads. By applying the proposed method, not only were the elastic and plastic system-level collapse probabilities estimated, but also the distributions of the associated multipliers. Due to the efficiency of the procedure (the case study was solved in a less than an hour on a typical laptop), it is believed that these results could be extremely useful to the design process of typical wind excited structures. Future work will focus on extending the procedure to more complex material behaviors, such as materials with hardening, as well as to bounding techniques to ensure that the entity of the plastic deformations produced during the transient phase do not exceed assigned limits. Acknowledgments This research effort was in part supported by the National Science Foundation (NSF) under Grant No. CMMI-1462084 and Grant No. CMMI-1462076. This support is gratefully acknowledged. The authors are also grateful to Professor M. Di Paola for the fruitful discussions and for the collaboration with the University of Palermo, Italy. Appendix A. Stochastic model for the wind loads Of the many quasi-steady models available in the literature for modeling the stochastic wind loads (e.g. [66,67]), in this work a classic approach is used (e.g. [68,60]). The forcing function FðtÞ is described by the following relationship:

j ¼ 1; 2; . . . ; n

ðA:1Þ

 j is the mean wind velocity at the height zj while v j ðtÞ is the where v corresponding fluctuating component. Furthermore, gj is a coeffi-

v j ðtÞ ¼ 2

n X N X

j Hjm ðxml Þ j

pffiffiffiffiffiffiffiffi

Dx cos xml ðtÞ  hjm ðxml Þ þ Uml ;

m¼1 l¼1

j ¼ 1; 2; . . . ; n

xml ¼ ðl  1ÞDx þ

ðA:2Þ

m Dx; l ¼ 1; 2; . . . ; N n

ðA:3Þ

while Hjm ðxml Þ is a typical element of a matrix HðxÞ, which is obtained decomposing the cross spectral density matrix (defined in the main text) in the following form:

SðxÞ ¼ HðxÞH T ðxÞ

ðA:4Þ



where ðÞ indicates the complex conjugate. This decomposition may be performed using Cholesky’s method or eigenfunction expansion [60,68,69]. In Eq. (A.2), hjm ðxÞ is the complex angle given by:

hjm ðxÞ ¼ tan

1



Im Hjm ðxÞ

Re Hjm ðxÞ

ðA:5Þ





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2pn 2pnN ¼ Tb ¼ Dx xup

ðA:6Þ

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