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COMPARISON OF EIGENMODE BASED AND RANDOM FIELD BASED IMPERFECTION MODELING FOR THE STOCHASTIC BUCKLING ANALYSIS OF I-SECTION BEAM-COLUMNS

A. STAVREV, D. STEFANOV, D. SCHILLINGER and E. RANK Chair for Computation in Engineering, Department of Civil Engineering and Surveying, Technische Universit¨ at M¨ unchen, Arcisstr. 21, 80333 M¨ unchen, Germany {atanas.stavrev,dimitar.stefanov}@mytum.de, {schillinger,rank}@bv.tum.de The uncertainty of geometric imperfections in a series of nominally equal I-beams leads to a variability of corresponding buckling loads. Its analysis requires a stochastic imperfection model, which can be derived either by the simple variation of the critical Eigenmode with a scalar random variable, or with the help of the more advanced theory of random fields. The present paper first provides a concise review of the two different modeling approaches, covering theoretical background, assumptions and calibration, and illustrates their integration into commercial finite element software to conduct stochastic buckling analyses with the Monte Carlo method. The stochastic buckling behavior of an example beam is then simulated with both stochastic models, calibrated from corresponding imperfection measurements. The simulation results show that for different load cases, the response statistics of the buckling load obtained with the Eigenmode based and the random field based models agree very well. A comparison of our simulation results with corresponding Eurocode 3 limit loads indicates that the design standard is very conservative for compression dominated load cases. Keywords: Finite element analysis; Stochastic buckling analysis; I-section beam-columns; Eigenmode based imperfection modeling; Random field based imperfection modeling.

1. Introduction Thin-walled I-section beam-columns constitute one of the most frequently used construction elements in steel structures today1 . However, their open cross section made up of cantilevered thin plates causes a difficult failure process at ultimate strength, which usually combines both local plate buckling and global flexuraltorsional buckling2,3 . In addition, I-section beam-columns are very sensitive to small changes in geometry, which drastically reduce their ultimate load bearing capacity compared to the resistance of the perfect member4,5 . The most common approach to take into account geometric imperfections in the analysis of I-section beam-columns is a deterministic model based on conservative worst case assumptions1,2 . It relies on the fact that thin-walled structures are particularly sensitive to imperfections in the shape of their Eigenmodes. The imperfection model therefore assumes the critical Eigenmode of the I-beam, which is expected to be triggered at failure. It is a suitable tool for the application in structural engineering practice with a focus on deriving reliable upper bounds. 1

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Due to the randomness of geometric imperfections, resulting from arbitrary influences during industrial manufacturing, transportation and on-site assembly, buckling loads of a series of nominally equal I-section beam-columns exhibit a large scattering5,6 . This uncertainty is an important aspect, for example when economic design7 or the probability of failure8 is considered. Since experimental studies would involve buckling tests for very large numbers of I-sections, consuming prohibitively large financial and material resources, the analysis of the buckling load variability requires modern finite element (FEM) simulation9,10 in conjunction with a realistic stochastic model of geometric imperfections. Stochastic imperfection models for I-beams as reported in the literature can be generally classified into two categories: First, Eigenmode based imperfections can be extended by a multiplication with a scalar random variable, which introduces a stochastic variability in the standard modeling approach5,7,11,12 . This concept can be handled reliably with basic knowledge of statistics, but involves the Eigenmode assumption. Second, random fields are used to represent the spatial variability of geometric imperfections13,14,15,16 , which are thus capable of reproducing stochastic characteristics of experimental imperfection data without taking additional assumptions in terms of spatial shape or magnitude. Their accuracy and reliability has been demonstrated in many computational studies for a range of thin-walled structures17,18,19,20,21,22,23 , but their application requires some understanding of modern stochastic process theory. In this context, the present paper compares the two stochastic imperfection modeling concepts with respect to their theoretical background, assumptions and calibration. We then conduct a computational study for an example I-section beamcolumn, for which measurements of geometric imperfections from several nominally equal I-beam samples are provided in Ref. 24. First, a Gaussian random field based model is calibrated by deriving homogeneous and evolutionary power spectra from the geometric imperfection measurements with the help of periodograms25 and the method of separation26 . Using the spectral representation method27 , realizations of random imperfections are generated. Second, an Eigenmode based model is set up, which generates realization of imperfections with Gaussian random variables fitted to the measurements. Each stochastic imperfection model is able to generate an arbitrary number of imperfect I-section beam samples, which can be analyzed with nonlinear FEM. In the sense of the Monte Carlo method16,25 , the buckling load variability for each imperfection model is derived from the series of deterministic FEM results. Finally, the response statistics of both models are compared in detail in terms of histograms and statistic key parameters. The steps of the simulation process are illustrated in Fig. 1. Finally, we compare the buckling results for the example I-beam to corresponding limit loads given by Eurocode 3 (EC 3)1,28,29 . The paper is organized as follows: Section 2 provides a brief review of relevant elements of the theory of random fields. Section 3 introduces the example I-section beam-column and discusses in detail the background, the assumptions and the calibration of a random field based and an Eigenmode based stochastic imperfection model. Section 4 briefly illustrates some aspects of the discretization and the non-

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Figure 1: The four steps of the simulation process for the stochastic buckling analysis of I-section beam-columns.

linear analysis with the commercial FE software “MSC Nastran”30 . In section 5, the simulation results obtained with each imperfection model are presented and their stochastic characteristics are compared and assessed in detail. Section 6 briefly describes the derivation of corresponding limit loads by EC 3 and compares them to the simulation results. Section 7 provides a summary and some conclusions.

2. Some elements of the theory of random fields The notion of a random field and its spectral representation are briefly reviewed. Readers interested in a more details are referred to Refs. 25, 31 and 32. 2.1. Random fields and their description A random field, also known as a stochastic process, is an ensemble of functions that can be characterized stochastically. A Gaussian random field f (x) is completely defined by a mean function µ(x), a standard deviation σ(x) and an autocorrelation function25,31,32 . It is called homogeneous, if µ(x), σ(x) and the autocorrelation func-

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tion do not change along x. If they do, we call the random field evolutionary26,31 . Considering the spectral representation of these parameters, we can estimate them from measurements as follows. 2.2. Power spectrum estimation A random physical phenomenon can be described by a series of m measurements that are interpreted as realizations h(i) (x), i = 1, 2, ..., m of the underlying random field h(x)25,31 . Measurements h(i) (x) are first divided into a deterministic mean µ(x) and zero-mean components f (i) (x)26 . If the zero-mean field f (x) can be assumed to be homogeneous, the corresponding power spectrum Sh (ω) can be estimated by the so-called periodogram25,31,32  Z 2  L 1 · f (i) (x) · e−Iωx dx  S˜h (ω) = E  (2.1) 2πL 0

where the term in absolute value is the Fourier transform of f (i) (x), E [ ] denotes the operator of mathematical expectation, L is the length of f (i) (x) and I is the complex unit. If the zero-mean part f (i) (x) of the measurements are evolutionary and can be assumed to be approximately separable, the corresponding power spectrum S(ω, x) can be estimated by the method of separation, recently introduced by Schillinger and Papadopoulos26,33   S˜h (ω) (i) 2 ˜ (2.2) S(ω, x) = E f (x) · R ∞ 2 0 S˜h (ω)dω

The left hand side of Eq. (2.2) denotes the estimated mean square; the right hand side represents a normalization of the periodogram based homogeneous estimate S˜h (ω) from Eq. (2.1). Due to the decoupling into a spatial and a frequency part, which simultaneously allows an accurate resolution in space and an optimum localization in frequency, the method of separation Eq. (2.2) is especially suitable for the robust estimation of strongly narrow-band power spectra, as they are typical for geometric imperfection measurements. It is shown analytically and numerically in Refs. 26 and 33 that for separable spectra, the estimate of Eq. (2.2) converges to the true spectrum for an infinite number of input samples and that the method of separation yields considerably better estimation results for narrow-band imperfection samples than standard evolutionary estimation techniques. Evolutionary power spectrum estimation based on the method of separation has been successfully adopted for engineering applications in Refs. 16 and 34. 2.3. Spectral representation of a random field If the power spectrum Sh (ω, x) of f (x) is known, an arbitrary number m of corresponding random samples can be generated by the spectral representation

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method27,35,36 , which reads for a one-dimensional univariate zero-mean Gaussian random field f (i) (x) =

−1   √ NX An cos ωn x + φ(i) 2 n

(2.3)

n=0

with

p

An =

2 · S(ωn , x) · ∆ω

(2.4a)

ωn = n · ∆ω

(2.4b)

∆ω = ωup /N

(2.4c)

A0 = 0 ∨ S(ω0 , x) = 0

(2.4d)

where i = 1, 2, ...m and n = 0, 1, 2...(N − 1). The parameter ωup is the cut-off frequency, beyond which the power spectrum is assumed to be zero, the integer N determines the discretization of the active frequency range, and φ(i) n denotes the th i realization of N independent phase angles uniformly distributed in the range [0, 2π]. To obtain samples of the original random field h(x) , the deterministic mean µ(x) has to be superposed to Eq. (2.3).

3. Stochastic imperfection modeling of an imperfect I-beam We consider the example of a typical I-beam and review how to set up and calibrate corresponding random field based and Eigenmode based imperfection models.

B = 175

δ3

δ2

δ1

δ9

Axis of Symmetry

85

t=5

θ

δ4

y,u

600

D = 260

t=5

z,v

x

250

Plate

665

x'

z,v Stiffeners

w = 5.9

(a) Cross-section dimensions.

δ5

δ6

δ7

δ8

(b) Displacement transducers for imperfection measurements.

(c) Length dimensions, additional stiffeners and plates.

Figure 2: The I-section test member with total length of 4000 and free length of 3330 mm.

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3.1. The example I-section beam-column The report Ref. 24 contains extensive imperfection measurements and corresponding experimental buckling loads for a series of six 4 m long I-beams, which are representative of doubly symmetric welded steel I-sections with typical levels of geometric imperfections. Their nominal geometry is illustrated in Fig. 2. For geometric imperfection measurements, displacement transducers were placed at nine cross-sectional locations δ1 to δ9 as shown in Fig. 2b (for more details on the measurement procedure, see Refs. 15, 16, 24). There are five types of local imperfections that were directly measured at positions δ1 , δ3 , δ4 , δ5 and δ7 . The three global imperfections u, v and θ were determined from the rest of the local measurements as follows: δ8 + δ9 (3.1) u= 2 v=

δ2 − δ6 2

(3.2)

δ9 − δ8 (3.3) 600 mm The parameters u, v and θ denote global cross-sectional deviations in weak and strong axis directions and rotation about the cross-sectional center of gravity. Some of the measurements are illustrated in Fig. 3. θ=

Figure 3: Examples of geometric imperfection measurements (from Ref. 24).

3.2. Conceptual modeling of an imperfect I-beam We follow the conceptual imperfection model developed by Schillinger et al. in Refs. 15 and 16, which fully accommodates all available measurements. The total geometric imperfection profile is composed of five local components λ1 to λ5 and

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λ1

λ2

7

δ3

Fixed Junctions Parabolic

γ1

γ3

γ2

λ5

Linear

λ3

λ4

Figure 4: The eight different components of the total geometric imperfection: local imperfections λ at the flange endges and the web center and global imperfections γ.

three global components γ1 to γ3 as illustrated in Fig. 4. Local imperfections denote local geometric deviations perpendicular to the flange and web plates in the crosssectional plane. They are imposed onto the perfect outer flange edges and the web center, while the web-flange junctions remain perfect. Intermediate imperfections in flanges and web are interpolated linearly and parabolically, respectively. Global imperfections γ1 to γ3 describe deviations from perfect alignment, i.e. weak and strong axis translations and cross-sectional rotation, which are superposed onto the locally imperfect geometry.

3.3. The random field based modeling concept In the random field based model, local as well as global imperfections are assumed to be fully uncorrelated, and can thus be modeled by eight independent one-dimensional random fields along the longitudinal axis of the beam. Following Refs. 15 and 16, the five random field representations λk , k = 1, ..., 5 for local components are assumed to be zero-mean and homogeneous, so that corresponding power spectra S˜k (ω) can be estimated by inserting measurements δ1 , δ3 , δ4 , δ5 and δ7 into the periodogram Eq. (2.1). The random field representation for global components γl , l = 1, 2, 3, consist of mean functions µl (x) evaluated from the corresponding series of processed measurements u, v and θ, and of the corresponding zero-mean evolutionary random fields, whose power spectra S˜l (ω, x) are estimated by inserting the zero-mean parts of u, v and θ, into the method of separation Eq. (2.2). An arbitrary number of local and global random field samples λk and γl can then be

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generated by spectral representation as (i)

λk (x) =

−1   √ NX Ak,n · cos ωn x + φ(i) 2 n

(3.4a)

n=0

Ak,n = (i)

q

2 · S˜k (ωn ) · ∆ω

γl (x) = µl (x) +

(3.4b)

−1   √ NX Al,n · cos ωn x + φ(i) 2 n

(3.5a)

n=0

Al,n =

q

2 · S˜l (ωn , x) · ∆ω

(3.5b)

where ∆ω = 3 · 104 rad/mm and parameters i, n and φ are defined in Eq. (2.3). However, the power spectra obtained from the experimental data of Section 3.1 are inaccurate, because imperfection measurements comprise both local and global components. We follow the solution proposed by Schillinger et al. 15,16 , which separates the imperfection measurements in the frequency domain ω into two distinct parts that contain smaller local wave-lengths and larger global wave-lengths, respectively. The global buckling modes of the perfect beam correspond to Euler modes that include at most two half-waves in longitudinal direction, whereas the lowest local buckling mode consists of several half-waves (see for example Fig. 7). Hence the frequency domain is partitioned into two distinct frequency parts ωglobal = (0.0000 ; 0.0024) [rad/mm]

(3.6a)

ωlocal = (0.0024 ; 0.0150) [rad/mm]

(3.6b)

The transition point corresponds to 2.5 half-waves, global frequencies beyond the upper cut-off frequency of ω = 0.015 rad/mm are not taken into account. Figure 5 shows examples of a homogeneous spectrum for local imperfections, an evolutionary spectrum for global imperfections, and the corresponding mean function, all estimated from the corresponding measurements in the six I-beams reported in Ref. 24. Evolutionary power spectra are released from spurious highfrequency oscillations in spatial direction, which are a consequence of the small number of input measurements, with the spectral smoothing procedure suggested in Ref. 26. The complete geometric profile at each longitudinal position x is obtained by mapping the initial perfect cross-sectional geometry (y, z) to the imperfect crosssectional geometry (Y, Z) in the form     "  1−z2 #         λ5 (D−t)2 y cosγ3 −sinγ3 γ1 Y y 0 · + + (3.7) = + 2y + sinγ3 cosγ3 z γ2 Z z λk B 0 {z } {z } | | local components

global components

where B , D and t denote flange width, section height and plate thickness of the I-section according to Fig. 2. The flange index in λk , k = 1, ..., 4 has to be chosen according to the current flange position of (y, z). Imperfection samples generated

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(a) Evolutionary power spectrum of the first global imperfection γ1 .

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(b) Homogeneous power spectrum of the first local imperfection λ1 .

Overall imperfection [mm]

0.1 0 −0.1 −0.2 −0.3 −0.4 0

500

1000

1500 2000 2500 Beam longitudinal axis [mm]

3000

3500

4000

(c) Mean function of the first global imperfection γ1 . Figure 5: Examples of evolutionary and homogeneous power spectra and mean function.

from the random field based model are illustrated in Fig. 6, consisting of local homogeneous flange and web imperfections, global non-homogeneous translations and cross-sectional rotation as well as the complete geometric imperfection profile.

3.4. The Eigenmode based modeling concept The Eigenmode based approach for the stochastic modeling of geometric imperfections is an extension of the standard deterministic concept. For I-section beamcolumns, design standards and regulations1,2 define the critical global Eigenmode as a half sine wave, so that global imperfection samples γl read   π (3.8) γl (x) = Aglob,l · sin x L0 where the scalar Aglob,l denotes the amplitude at mid-span, l = 1, 2, 3 is a counter

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(a) Flange imperfection λ1 - λ4 .

(b) Web imperfection λ5 .

(c) Weak axis translation γ1 .

(d) Strong axis translation γ2 .

(e) Cross-sectional rotation γ3 .

(f ) Complete imperfection Eq. (3.7).

Figure 6: I-section sample with local and global geometric imperfections generated by the random field based approach (magnified by a factor of 100 for better visibility).

denoting again weak and strong axis translations and rotations, respectively, and L0 is the free length of the beam. The critical local Eigenmode is assumed here in the form of the lowest local Eigenmode of the perfect structure, which can be determined for example by a FEM based linear buckling analysis10,30 for the load case under consideration. As an example, Figure 7 shows the lowest local Eigenmode

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of the perfect I-section beam-column for the load case of axial compression. A scalar local imperfection field floc (x, y, z) is obtained by subtraction of the perfect geometry from the local mode shape and normalization with respect to its largest absolute value. In analogy with Eq. (3.8), it can be used to derive a deterministic local imperfection model in the form λ(x, y, z) = Aloc · floc (x, y, z)

(3.9)

which can be modulated by a suitable scalar amplitude Aloc . The introduction of uncertainty in the Eigenmode based global and local imperfection models of Eqs. (3.8) and (3.9) can be simply achieved by considering the scalar amplitudes Aglob,l and Aloc as Gaussian random variables Aglob,l = µglob,l + σglob,l · Z

(3.10a)

Aloc = µloc + σloc · Z

(3.10b)

where µglob,l and σglob,l with l = 1, 2, 3 denote the means and standard deviations of the global weak and strong axis translations and the global rotation, respectively, and µloc and σloc are the corresponding quantities of the local imperfection profile. Parameter Z denotes a zero-mean random variable that follows the standard Gaussian probability distribution25,32 . The free parameters of the simplified Eigenmode based imperfection model are the mean and standard deviations of Eqs. (3.10a) and (3.10b), for which a variety of calibration approaches could be applied. For example, a conservative calibration determines µglob,l from the maximum values that occur in the global measurements u, v, and θ of Eqs. (3.1) through (3.3). In the present case, we are interested in a calibration, which accommodates most of the ideas of the random field based model, so that the comparability is not further biased beyond the Eigenmode assumption. Therefore, we calibrate the parameters of the Eigenmode based model by using the information of the random field based model in the following way:

Figure 7: Failure mode of the perfect I-section obtained by linear buckling analysis with “MSC Nastran”. Displacements are magnified by a factor of 80 for better visibility. For details of the FE discretization, see Section 4.

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• In analogy with the previous section, we assume in Eq. (3.10a) a non-zeromean distribution of each global amplitude Aglob,l and in Eq. (3.10b) a zero-mean distribution of the local amplitude Aloc with µloc = 0. • For the global translations and rotation, the means µglob,l are the values of the mean functions µglob,l (x = L0 /2) of Section 3.3 at mid-span. • Using the fact that the standard deviations can be obtained from an integration of the power spectrum over the frequency range ω (see e.g. Refs. 25, 26 or 32 for details), each global standard deviation σglob,l can be determined from the corresponding estimated evolutionary power spectrum S˜glob,l (ω, x = L0 /2) of Section 3.3 at mid-span by Z 0.0024 2 S˜glob,l (ω, x = L0 /2) dω (3.11) σglob,l = 2 0.0000

where the integral takes into account the frequency range of Eq. (3.6a). • Using the same identity, the standard deviation σloc of the local imperfection profile is chosen as the maximum of the five integrals of the estimated homogeneous power spectra S˜loc,k (ω, x = L0 /2) of Section 3.3 as  Z 0.0150 2 σloc = max S˜loc,k (ω) dω (3.12) k

0.0024

where the integrals take into account the frequency range of Eq. (3.6b). With respect to the random field based model of the previous section, the local imperfection field floc (x, y, z) can be interpreted as being composed of fully correlated single local flange and web imperfections, which again underlines the conservative character of the Eigenmode based model. The complete geometric imperfection profile (Y,Z) of the I-section at longitudinal position x is obtained again by mapping the imperfections onto the perfect geometry (y,z) as follows            y Y cosγ3 −sinγ3 y γ1 · (3.13) + = + λ+ sinγ3 cosγ3 z Z γ2 z Figure 8 illustrates the local and global components and the complete imperfection profile of an Eigenmode based I-section sample. 4. Stochastic buckling analysis using a commercial FE program We briefly describe the integration of our geometric imperfection models into a commercial FE software and the Monte Carlo method, illustrating that buckling load variabilities can be determined without the availability of specialized software or an expert knowledge in advanced stochastic methods. 4.1. Finite element discretization of an I-section beam-column All finite element computations are performed with the commercial FE program “MSC Nastran”30,37 . We discretize the beam geometry with quadrilateral elements

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(a) Local imperfection profile λ.

(b) Weak axis translation γ1 .

(c) Strong axis translation γ2 .

(d) Cross-sectional rotation γ3 .

13

(e) Complete imperfection Eq. (3.13). Figure 8: I-section sample with local and global geometric imperfections generated by the Eigenmode based approach (magnified by a factor of 100 for better visibility).

of type “CQUAD4”, which are standard 4-node bilinear shell elements, taking into account membrane and bending as well as transverse shear (thick shell theory)37 . Random field based and Eigenmode based imperfections can be simply incorporated into the finite element discretization by adjusting the nodal positions according to the imperfect geometry described by Eqs. (3.7) and (3.13), respectively. A corresponding discretization of the I-section beam is shown in Fig. 10a and takes into

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Figure 9: Reaction frame for beam-column tests under combined major axis bending and compression loads (from Ref. 24).

account only the free-length part of the I-beams in the reaction frame of Fig. 9 with appropriate boundary conditions. Due to the perfect pins at the member ends, rotations about the major y-axis and translations along the x-axis are left unconstrained. Rotations about the minor z-axis are constrained. Due to additional stiffeners and plates (see Fig. 2c and Fig. 9), rotations about the longitudinal axis and translations along the y-axis are prevented. The central web points at both boundaries are constrained against z-axis translations, and the central web point at one boundary against x-axis translation. The additional axial stiffness and moments of inertia due to the welds at the flange-web junctions are compensated by slightly increasing the thicknesses of shell elements as illustrated in Fig. 10b. Experiments show that web-flange junctions remain unaffected by local buckling deformations7 . Following Schillinger et al.15,16 ,

18.11

8.4 6.156

25.98

5.578

5.0

(a) Discretization of a perfect beam-column with quadrilateral shell elements, beam and truss elements.

(b) Model of the flange-web junctions.

Figure 10: Finite element model of the free-length member.

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this is ensured by a triangle of three very stiff truss elements that connect the outer junction nodes (see Fig. 10b) and prevent a relative deformation within the junction. Note that triangles of truss elements do not introduce spurious bending stiffness and do not interact with each other along the longitudinal axis. Since the support conditions in the reaction frame prevent out-of-plane distortion of the cross-section, corresponding warping constraints are imposed by beam elements at both ends of the I-section member (see Fig. 10a). Their stiffness components required to constrain out-of-plane distortion, i.e. bending stiffness around the global z-axis in the flanges and torsional stiffness in the web, are increased by a factor of four orders of magnitude with respect to the stiffness of steel.

4.2. Finite element based buckling analysis At both beam ends, force boundary conditions resulting from the compression and bending jacks of the reaction frame of Fig. 9 are transferred to equivalent stresses in x-direction. The corresponding buckling load of each geometrically imperfect Ibeam sample is determined by a geometrically nonlinear FE analysis, which takes into account large deformation effects37,38 . The arc-length procedure is applied to track the equilibrium path, which simultaneously increases all applied forces, until a bifurcation or limit point is found, associated with buckling failure38 . We assume that the failure of the beam occurs at a deformation state, which is purely elastic, and no plasticity and residual stress effects are taken into account during the simulation. For a more involved computational model that takes into account plasticity and residual stresses as well as random field based thickness and geometric imperfections at the same time, see e.g. Refs. 16 and 20. A suitable mesh density for imperfect I-section beam-columns is determined by a convergence study, which results in a compromise between accuracy and computational efficiency with a mesh of 2,1672 nodes, 21,266 shell elements and 108,360 degrees of freedom (see Fig. 10a).

4.3. The Monte Carlo method The geometric imperfection models and the FEM based buckling analysis provide us with the tools to determine the stochastic variability of the I-section beam-column under consideration with a Monte Carlo simulation. In the following, we generate m imperfect I-beams for each load case, using either the random field based or Eigenmode based model, and determine the corresponding m individual buckling i loads Pcrit , i = 1, 2, ...m by repeated deterministic FEM computations. From the m individual FEM results, the buckling load variability is characterized in terms of histograms and stochastic key parameters mean µ, standard deviation σ, skewness and coefficient of variation Cov = σ/µ. For corresponding definitions, we refer to Refs. 25 and 32 or any other statistics textbook.

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5. Comparison and assessment of simulation results In the following, we compare the response statistics resulting from Monte Carlo simulations with the random field based and the Eigenmode based geometric imperfection models. The examined load cases include pure axial compression with end force N , pure major axis bending with end moment M and four combinations with ratios M/N = 0.050, 0.125, 0.250 and 0.500 m. 5.1. Simulated buckling modes of the examined I-beam The failure modes of imperfect I-section beam-columns consist of local flange and web buckling, global flexural or lateral-torsional buckling, or a combination of some of these1,3,16 . Which kind of buckling mode occurs is predominantly determined by the shape and size of the geometric imperfections and the loading applied at the ends. Some of the buckling modes obtained from the random field based and Eigenmode based simulations are illustrated in Figs. 11 and 12, respectively. Under compression dominated loads, the most frequent failure mode is local plate and flange buckling in combination with global lateral-torsional buckling, involving global translation along the weak axis with a cross-sectional rotation. Other failure modes include pure flexural or pure torsional buckling, which involve just a global translation along the weak axis or global rotation around the I-section center, respectively. A very rare mode is pure local flange and web buckling, which occurred only once in an almost perfect I-section beam-column. Under bending dominated loads, one flange is in tension and therefore stabilized. The most frequent failure mode in this case consists of combined local plate buckling in the flange under compression and a global rotation of the compressed part of the I-section. These observations show that both geometric imperfection models are able to trigger a range of different failure modes. The dominance of global flexural-torsional buckling is in accordance with engineering experience for this kind and length of I-beams and corresponds well with the observations reported in Ref. 24. 5.2. Buckling load variability of the random field based approach The buckling load variabilities of all six load cases obtained with the random field based imperfection model are shown in the histograms of Figs. 13a through 18a, which approximate the probability distribution of the buckling load for each load case. The coefficient of variation Cov represents a normalized measure of stochastic dispersion independent of the absolute value of the buckling load, which thus allows an objective comparison of buckling load variabilities among varying load cases or different structures. For pure compression in Fig. 13a, the current 3.33 m long I-section beam-column exhibits a Cov = 6.6%, which is significantly higher than the Cov = 1.3% observed in a 1.33 m long beam-column of the same crosssection reported in Ref. 16. This demonstrates the dramatic increase of the buckling load variability with the length of the I-beam. With increasing moment, the mean

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(a) Lateral torsional buckling, load case pure compression.

(b) Pure local flange and web buckling, load case pure compression.

(c) Lateral torsional buckling, load case M/N = 0.050. Figure 11: Various failure modes of beam samples computed by the Monte Carlo random field based simulation. The dotted and the red lines denote the original and deformed configurations of the displayed cross-sections.

value of the maximum normal force decreases constantly from µ = 691kN under pure compression to µ = 151.6kN under a force moment ratio of M/N = 0.5 m. In addition, for bending dominated load cases, the scattering of buckling loads dramatically increases to coefficients of variations up to Cov = 30% in Fig. 18a.

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(a) Lateral torsional buckling, load case pure compression.

(b) Torsional buckling, load case pure compression.

(c) Lateral torsional buckling, load case pure major axis bending. Figure 12: Various failure modes of beam samples computed by the Monte Carlo Eigenmode based simulation. The dotted and the red lines denote the original and deformed configurations of the displayed cross-sections.

Comparing the histograms of the different load cases, we can observe that for compression dominated load cases, the probability distribution is symmetric and similar to a normal distribution. However, with increasing bending moment, the probability distribution of the buckling load becomes more skewed. Since the bulk

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300

300 Mean = 691.03 kN StDev = 46.02 kN Cov = 6.66 % Skew = 959.15 kN

200

250 Frequency [−]

250 Frequency [−]

19

150 100 50

200

Mean = 6%7.91 kN StDev = 34.61 kN Cov = 5.34 % Skew = 383.74 kN

150 100 50

0 500

600 700 800 Ultimate Buckling Load [kN]

0 500

900

(a) Histogram obtained with random field based simulation.

600 700 800 Ultimate Buckling Load [kN]

900

(b) Histogram obtained with Eigenmode based simulation.

Figure 13: Buckling load distribution of the load case of pure compression.

200 150

250 Mean = 619.99 kN StDev = 52.61 kN Cov = 8.49 % Skew = 971.55 kN

100 50 0

Frequency [−]

Frequency [−]

250

200 150

Mean = 560.$2 kN StDev = (9.78 kN Cov = 7.00 % Skew = 997.93 kN

100 50

500 600 700 800 Ultimate Buckling Load [kN]

(a) Histogram obtained with random field based simulation.

0

500 600 700 800 Ultimate Buckling Load [kN]

(b) Histogram obtained with Eigenmode based simulation.

Figure 14: Buckling load distribution of the load case M/N = 0.050.

of the values lies to the left of the mean and a tail on the right side occurs, the histogram resembles more a log-normal distribution. This is also confirmed by the positive skewness parameter shown in Figs. 17a and 18a, which provides an objective measure of the asymmetry of the probability distribution25 . 5.3. Buckling load variability of the Eigenmode based approach In analogy with the previous section, Monte Carlo simulations are conducted with random Eigenmode based geometric imperfections. The resulting buckling load variabilities are displayed in Figs. 13b through 18b, side by side with the corresponding random field based results. A comparison of the histograms indicates that the two different imperfection models lead to very similar results. First, the probability distribution changes its shape with increasing moment from an approximately normal

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Frequency [−]

200 150

Mean = 451.78 kN StDev = 52.06 kN Cov = 11.52 % Skew = 213.46 kN

100

200 Frequency [−]

20

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50 0

150

Mean = 396.14 kN StDev = 4 .11 kN Cov = 12.14 % Skew = 1450 kN

100 50 0

300 400 500 600 Ultimate Buckling Load [kN]

(a) Histogram obtained with random field based simulation.

300 400 500 600 Ultimate Buckling Load [kN]

(b) Histogram obtained with Eigenmode based simulation.

Figure 15: Buckling load distribution of the load case M/N = 0.125.

300

200 150 100 50 0

250 Frequency [−]

Frequency [−]

250

300 Mean = 303.68 kN StDev = 44.71 kN Cov = 14.72 % Skew = 1020.86 kN

200

Mean = 262.62 kN StDev = 34.44 kN Cov = 13.11 % Skew = 1447.25 kN

150 100 50

200 300 400 500 Ultimate Buckling Load [kN]

(a) Histogram obtained with random field based simulation.

0

200 300 400 500 Ultimate Buckling Load [kN]

(b) Histogram obtained with Eigenmode based simulation.

Figure 16: Buckling load distribution of the load case M/N = 0.250.

distribution to a skew distribution with a tail on the right. Second, with respect to the minimum and maximum buckling loads that occur in each pair of Monte Carlo simulations, the results of both imperfection models are in surprisingly good accordance throughout all load cases. Taking a more detailed look at the stochastic key parameters, we can corroborate the strong correlation between the simulations based on the two different imperfection models. First, with increasing moment, the mean value of the maximum normal force decreases constantly within the same range in the random field based as well as the Eigenmode based simulations. Second, the surge in scattering with increasing end moments observed for the random field based results is reproduced by the Eigenmode based results. This is illustrated by an increase in the corresponding coefficients of variation from Cov = 5.3% for pure compression

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300

300

200

Mean = 151.57 kN StDev = 45.63 kN Cov = 30.11 % Skew = 1526.25 kN

150 100

250 Frequency [−]

Frequency [−]

250

200

Mean = 139.11 kN StDev = 56.24 kN Cov = 40.43 % Skew = 3061.0+ 

150 100 50

50 0 0

21

0 0

100 200 300 400 Ultimate Buckling Load [kN]

(a) Histogram obtained with random field based simulation.

100 200 300 400 Ultimate Buckling Load [kN]

(b) Histogram obtained with Eigenmode based simulation.

400 350 300 250 200 150 100 50 0 0

Mean = 98.56 kNm StDev = 31.28 kNm Cov = 31.73 % Skew = 735.29 kNm

100 200 300 400 Ultimate Buckling Moment [kNm]

(a) Histogram obtained with random field based simulation.

Frequency [−]

Frequency [−]

Figure 17: Buckling load distribution of the load case M/N = 0.500.

400 350 300 250 200 150 100 50 0 0

Mean = 86.44 kNm StDev = 30.77 kNm Cov = 35.60 % Skew = 1256.1 kNm

100 200 300 400 Ultimate Buckling Moment [kNm]

(b) Histogram obtained with Eigenmode based simulation.

Figure 18: Buckling load distribution of the load case pure major axis bending moment.

in Fig. 13b up to Cov = 40% in Fig. 17b. Third, the increase in skewness that confirms the changing shape of the distribution occurs in the random field based as well as the Eigenmode based simulations. However, the latter lead in general to slightly lower means throughout all load cases and to higher standard deviations and coefficients of variation for the bending dominated load cases. These results show that for the example I-section beam-column, both imperfection models trigger buckling loads with equivalent statistical response characteristics. The more conservative character of the Eigenmode based variabilities can be explained by the difference in frequencies between the two imperfection models. While the random field based model can contain all frequencies that are present in the power spectra, the Eigenmode based model deterministically contains a combination of only those imperfection modes, which the I-section beam-column under consideration can be expected to be most sensitive to.

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6. Comparison of simulated buckling loads to EC 3 limit loads Finally, we compare our simulation results for the example I-beam to corresponding limit loads derived according to Eurocode 3 specifications1,28 . To facilitate the reconstruction of the limit loads for the interested reader, we briefly illustrate the EC 3 formulas first. 6.1. Derivation of limit loads for I-beams according to EC 3 The buckling resistance of thin-walled members under combined compression and bending is ensured, if the following expressions are satisfied (see formulas (6.61) and (6.62) in the EC 3 standard28 ): • Failure around the y-axis My,ed + ∆My,ed Mz,ed + ∆Mz,ed Ned + kyy + kyz ≤ 1.0 My,rk M Nrk χ y γM 1 χLT γ χLT γz,rk M1

(6.1)

M1

• Failure around the z-axis

Ned My,ed + ∆My,ed Mz,ed + ∆Mz,ed + kzy + kzz ≤ 1.0 M M χz γNMrk1 χLT γy,rk χLT γz,rk M1

(6.2)

M1

Equations (6.1) and (6.2) require that the interactions of the design loads Ned , My,ed and Mz,ed relative to the corresponding resistances Nrk , My,rk and Mz,rk always lead to a factor not larger than one, where the k values denote so-called interaction factors along the corresponding directions. The resistances are calculated from the yield stress and a so-called effective cross-section (cross-section class 4), where ∆My,ed and ∆Mz,ed denote additional moments that result from the corresponding cross-section reduction. To account for uncertainties, a variety of load and resistance factors are introduced that affect all quantities in Equations (6.1) and (6.2)28,29 . Factor γM 1 =1.1 reduces the resistances. Factors χy , χz and χLT that are all smaller than 1.0 account for the influence of imperfections on bending and lateral torsional failure, respectively. Furthermore, the design loads Ned and Med are obtained from the actual loads N and M by multiplication with a load factor of α=1.35 (static loading). With the compression to moment ratios applied by the reaction frame24 , one can solve Equations (6.1) and (6.2) to obtain the actual load limits allowed by EC 3 specifications. The results for the six load cases examined here are listed in Tab. 1. 6.2. Simulated buckling load variabilities vs. EC 3 limit loads In the framework of the present study, a comparison of the stochastic simulation results with the EC 3 limit loads can only be of qualitative nature, since the simulations only account for the influence of random geometric imperfections, while the

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EC 3 formulas account for all random effects that may arise for example also from uncertain material parameters or loading conditions. The influence of combined loading on the ultimate strength of I-section members is typically illustrated by interaction curves1,2,39 that plot the ultimate axial load N vs. the ultimate end moment M for different fixed ratios M to N . Figure 19a compares the interaction curves obtained with the mean values of the Monte Carlo simulations based on the random field and Eigenmode models to the interaction curve obtained from the corresponding EC 3 limit loads. Additionally, it shows an experimental interaction curve, which is derived from six experimental buckling loads that were tested in the reaction frame of Fig. 9 and reported along with the geometric imperfection data bank in Ref. 24. The simulated mean interaction curves are in good accordance with the experimental interaction curve, which underlines the plausibility of the present computational results. In particular, the simulated curves exhibit a convex shape, which is confirmed in Ref. 39. One can observe that the mean values of the simulation are by more than a factor two smaller than the corresponding EC 3 limit loads. Figure 19b shows the worst case simulation results in comparison to the EC 3 curve for each load case. For the interested reader, we also tabulated the corresponding values in Tab. 1. The results indicate that for compression dominated load cases, the EC 3 design specifications are very conservative. For bending dominated load cases, the lowest failure loads of the simulations come much closer to the EC 3 limit, and therefore seem to be much more well tuned to the actual behavior of the structure. This observation is in accordance with recent efforts in structural design research, which are targeted at improving design standards to more economically use the resistance of I-section members under certain loading conditions (see for example Ref. 7 and the references therein). The results of Fig. 19 also illustrate that stochastic simulations in the present state of development can constitute a reliable and valuable tool in design related research, which can support and supplement experimental tests and deterministic simulations.

Load case

Loading ratio

EC 3 limit loads

1 2 3 4 5 6

M =0.0 M =0.05N M =0.125N M =0.25N M =0.5N N =0.0

N =305.37 N =226.53 N =154.27 N =102.07 N =60.74 M =37.36

Worst case simulated buckling load random fields Eigenmodes N =569.95 N =488.27 N =302.06 N =207.40 N =84.08 M =46.32

N =544.77 N =472.72 N =295.02 N =203.48 N =70.02 M =48.83

Table 1: Comparison of limit loads allowed by EC 3 specifications with worst case buckling loads obtained from the random field based and Eigenmode based stochastic simulations. Compressive loads N and moments M are given in kN and kNm, respectively.

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800

Experiments from Ref. 24. Random field based simulation Eigenmode based simulation Eurocode 3 limit loads

700 600

Ultimate Buckling Load [kN]

Ultimate Buckling Load [kN]

800

500 400 300 200 100 0 0

20

40

60

80

100

600 500 400 300 200 100 0 0

20

40

60

80

100

Ultimate Buckling Moment [kNm]

Ultimate Buckling Moment [kNm]

(a) Mean values of the simulated stochastic interaciton curves.

Experiments from Ref. 24. Random field based simulation Eigenmode based simulation Eurocode 3 limit loads

700

(b) Minimal values of the simulated stochastic interaciton curves.

Figure 19: Comparison between simulated stochastic interaction curves, experimental interaction curve and Eurocode 3 limit loads.

6.3. The stochastic interaction diagram Finally, we provide a comprehensive overview of our results by so-called stochastic interaction diagrams as proposed by Schillinger et al.15,16 . The three-dimensional plots extend the two axes of the two-dimensional deterministic interaction curve, i.e. applied axial load and end moment, by a third axis for the frequency, which approximates the probability density of the distribution. Corresponding diagrams derived from the results of the random field based and the Eigenmode based imperfection models are plotted in Figs. 20 and 21, respectively, and compared to the deterministic EC 3 limit loads. The stochastic interaction diagram thus summarizes all relevant information of the response behavior of the I-section beam-column, illustrating for example the increase in scattering and skewness for bending dominated load cases, the location of outliers in both directions, and the conservative character of the EC 3 limit loads for compression dominated load cases. 7. Summary and conclusions The analysis of the buckling load variability in nominally equal I-beams requires FEM simulation in conjunction with a realistic stochastic model of geometric imperfections. In this context, the present paper compares two stochastic imperfection models and their ability to simulate the buckling load variability of a typical Isection beam-column. The first modeling concept is based on Gaussian random fields, which are able to represent the spatial variability of imperfections along the beam axis. The second modeling concept extends the standard deterministic approach, varying geometric imperfections in the form of the critical Eigenmode by a Gaussian random variable. The simulation results show that both imperfection

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Frequency

Eurocode 3 limit loads

400 200 0 0 Ul tim 50 ate Bu ck 100 lin g M 150 om en t [ 200 kN m ]

800 600 400 250

0

ng L

i 200 uckl ate B Ultim

oad

[kN]

Figure 20: Stochastic interaction diagram obtained from the random field based simulation.

Frequency

Eurocode 3 limit loads

400 200 0 0 Ul tim 50 ate Bu 100 ck lin g 150 M om en 200 t[ kN m ] 250

800 600 400 200 0

Ultim

ing uckl

Load

[kN]

ate B

Figure 21: Stochastic interaction diagram obtained from the Eigenmode based simulation.

models lead to buckling load variabilities, which are in surprisingly good accordance. The corresponding response statistics such as histograms, coefficient of variation and skewness exhibit an equivalent behavior throughout all examined load cases. Due to the Eigenmode assumption, the results of the Eigenmode based simulations are slightly more conservative. The quality of the simulation results are corroborated by comparing the simulated mean interaction curve to an experimental interaction curve, which agree very well in size and shape. A comparison to corresponding EC 3 limit loads indicate that the design standard is very conservative for compression dominated load cases. The present study thus confirms that for I-section beam-columns, the stochastic imperfection model based on the Eigenmode assumption constitutes a valid basis for

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the simulation of the uncertainty in buckling loads. It also illustrates that stochastic simulations in the present state of development can be considered reliable and mature. They are thus ready to support design related research efforts by providing valuable stochastic information in addition to standard experimental tests and deterministic simulations.

Acknowledgments This publication is based on work supported by Award No. UK-c0020, made by King Abdullah University of Science and Technology (KAUST). Furthermore, the authors acknowledge support from the Munich Center of Advanced Computing (MAC) and the International Graduate School of Science and Engineering (IGSSE) of the Technische Universit¨ at M¨ unchen. Extensive research reports related to buckling experiments in I-sections have been kindly provided by Prof. Kim Rasmussen from the University of Sydney and Dr. Andreas Lechner from the Technical University of Graz. Their assistance is also gratefully acknowledged.

Bibliography 1. N.S. Trahair, M.A. Bradford, D.A. Nethercot and L. Gardner, The Behaviour and Design of Steel Structures to EC 3, (Taylor and Francis, 2008). 2. T. Galambos, Guide to Stability Design Criteria for Metal Structures, (Wiley, 1998). 3. L.A. Godoy, Theory of Elastic Stability: Analysis and Sensitivity, (Taylor and Francis, 1999). 4. M.P. Blyfield and D.A. Nethercot, Material and geometric properties of structural steel for use in design, Structural Engineering 75(21) (1997) 1–5. 5. Z. Kala, Sensitivity analysis of the stability problems of thin-walled structures, Journal of Constructional Steel Research 61(3) (2005) 415-422. 6. I. Elishakoff, Uncertain buckling: its past, present and future, International Journal of Solids and Structures 37 (2000) 6869–6889. 7. R. Greiner, M. Kettler, A. Lechner, B. Freytag, J. Linder, J.-P. Jaspart, N. Boissonnade, E. Bortolotti, K. Weynand, C. Ziller and R. Oerder, Plastic Member Capacity of Semi-Compact Steel Sections - a more Economic Design, Final Report, (European Research Fund for Coal and Steel, 2008). 8. C. Rebelo, N. Lopes, L. Sim˜ oes da Silva, D. Nethercot and P. Vila Real, Statistical evaluation of the lateral-torsional buckling resistance of steel I-beams - Part 1: variability of the Eurocode 3 resistance model. Journal of Constructional Steel Research 65(4) (2009) 818–831. 9. K.J.R. Rasmussen and G.J. Hancock, Buckling analysis of thin-walled structures: numerical developments and applications, Progress in Structural Engineering and Materials 2(3) (2000) 359–368. 10. R. Kindmann and M. Kraus, Steel Structures: Design using FEM, (Wiley, 2011). 11. B.W. Schafer, M. Grigoriu and T. Pek¨ oz, A probabilistic examination of the ultimate strength of cold-formed steel elements, Thin-Walled Structures 31 (1998) 271–288. 12. V.K. Verma and B.N. Singh, Thermal buckling of laminated composite plates with random geometric and material properties, International Journal of Structural Stability and Dynamics 9(2) (2009) 187–211.

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13. J. Zhang and B. Ellingwood, Effects of Uncertain Material Properties on Structural Stability, Journal of Structural Engineering ASCE 121(4) (1995) 705–717. 14. B.W. Schafer and L. Graham-Brady, Stochastic Postbuckling of Frames using Koiter’s Method, International Journal of Structural Stability and Dynamics 6(3) (2006) 333– 358. 15. D. Schillinger, Stochastic FEM Based Stability Analysis of I-Sections With Random Imperfections., Diploma Thesis, (Universit¨ at Stuttgart, 2008). http://www.ibb.unistuttgart.de/publikationen/fulltext/2008/schillinger-2008.pdf 16. D. Schillinger, V. Papadopoulos, M. Bischoff and M. Papadrakakis, Buckling analysis of imperfect I-section beam-columns with stochastic shell finite elements, Computational Mechanics 46 (2010) 495–510. 17. C.A. Schenk and G.I. Schu¨eller, Buckling analysis of cylindrical shells with cutouts including random boundary and geometric imperfections, Computer Methods in Applied Mechanics and Engineering 196 (2007) 3424–3434. 18. V. Papadopoulos, D.C. Charmpis and M. Papadrakakis, A computationally efficient method for the buckling analysis of shells with stochastic imperfections, Computational Mechanics 43 (2008) 687–700. 19. M. Tootkaboni, L. Graham-Brady and B.W. Schafer, Geometrically non-linear behavior of structural systems with random material property: An asymptotic spectral stochastic approach, Computer Methods in Applied Mechanics and Engineering 198 (2009) 3173–3185. 20. V. Papadopoulos, G. Stefanou and M. Papadrakakis, Buckling analysis of imperfect shells with stochastic non-Gaussian material and thickness properties, International Journal of Solids and Structures 46 (2009) 2800–2808. 21. M. Broggi, A. Calvi and G.I. Schu¨eller, Reliability assessment of axially compressed composite cylindrical shells with random imperfections, International Journal of Structural Stability and Dynamics 11 (2011) 215–236. 22. G. Stefanou, Response variability of cylindrical shells with stochastic non-Gaussian material and geometric properties, Engineering Structures 33(9) (2011) 2621–2627. 23. G. Muscolino and A. Sofi, Response statistics of linear structures with uncertain-butbounded-parameters under Gaussian stochastic input, International Journal of Structural Stability and Dynamics 11 (2011) 775–804. 24. A.S. Hasham and K.J.R. Rasmussen, Member capacity of thin-walled I-sections in combined compression and major axis bending, Research Report No R746, (School of Civil Engineering, University of Sydney, 1997). 25. A. Papoulis, S.U. Pillai, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, 2002). 26. D. Schillinger and V. Papadopoulos, Accurate estimation of evolutionary power spectra for strongly narrow-band random fields, Computer Methods in Applied Mechanics and Engineering 199 (2010) 947–960. 27. M. Shinozuka and G. Deodatis, Simulation of stochastic processes by spectral representation, ASME Applied Mechanics Review 44 (1991) 191–203. 28. European Committee for Standardization (CEN), EN 1993 - Eurocode 3: Design of steel structures, (European Committee for Standardization, 2006). 29. A. Goris (Ed.), Schneider Bautabellen fuer Ingenieure, (Werner Verlag 2012) (in German). 30. MacNeal-Schwendler Corporation. General Purpose Finite Element Software MSC Nastran, Version 2008.r3. 31. E. Vanmarcke, Random Fields: Analysis and Synthesis, (MIT Press, 1983). 32. D.E. Newland, An introduction to Random vibrations, spectral and wavelet analysis,

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(Wiley, 1993). 33. D. Schillinger and V. Papadopoulos, The Method of Separation: A Novel Approach for Accurate Estimation of Evolutionary Power Spectra, in Computational Methods in Stochastic Dynamics, eds. M. Papadrakakis et al., (Springer, 2010). 34. M. Broggi and G.I. Schu¨eller, Efficient modeling of imperfections for buckling analysis of composite cylindrical shells, Engineering Structures 33 (2011) 1796–1806. 35. M. Grigoriu, On the spectral representation method in simulation, Probabilistic Engineering Mechanics 8 (1993) 75–90. 36. J. Liang, S.R. Chaudhuri and M. Shinozuka, Simulation of non-stationary stochastic processes by spectral representation, Journal of Engineering Mechanics ASCE 133 (2007) 616–627. 37. MacNeal-Schwendler Corporation. MSC Nastran Numerical Methods User’s Guide. 38. J. Bonet and R. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis, (Cambridge University Press, 2008). 39. A.S. Hasham and K.J.R. Rasmussen, Interaction curves for locally buckled I-section beam-columns, Journal of Constructional Steel Research 58 (2002) 213–241.

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