On the modelling of initial geometric imperfections of steel frames in advanced analysis

Journal of Constructional Steel Research 98 (2014) 167–177

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Journal of Constructional Steel Research

On the modelling of initial geometric imperfections of steel frames in advanced analysis Shabnam Shayan ⁎, Kim J.R. Rasmussen, Hao Zhang School of Civil Engineering, The University of Sydney, Sydney 2006, NSW, Australia

a r t i c l e

i n f o

Article history: Received 18 August 2013 Accepted 25 February 2014 Available online 18 April 2014 Keywords: Initial geometric imperfection Steel structures Elastic buckling analysis Inelastic analysis

a b s t r a c t Steel structural members and frames always indicate imperfections to various degrees. These include initial outof-straightness and initial out-of-plumb due to manufacturing and erection tolerances. In general, the shape and magnitude of geometric imperfections may have a significant influence on the response of a structure, and hence need to be modelled accurately when determining the load carrying capacity of a steel frame by advanced structural analysis. Most conveniently, geometric imperfections can be introduced in structural models as scaled eigenmodes obtained a priori from an elastic buckling analysis. However, it remains unanswered how many eigenmodes need to be incorporated and how to choose the scaling factors of each mode. This paper presents a study of how the strength of steel frames varies with the number and magnitudes of eigenmodes. Frames with random geometric imperfections are produced using the statistics of measurements of out-of-plumb and member imperfections, and analysed using advanced geometric and material nonlinear analysis. The imperfections are then resolved into eigenmodes and a second set of advanced analysis is carried out using a finite number of modes to represent the imperfections. Conclusions are drawn about the appropriate number and magnitudes of eigenmodes to use in advanced structural analyses of steel frames. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Steel structural members are not perfectly straight due to manufacturing and erection tolerances. In general, two types of initial geometrical imperfections should be taken into account in advanced (second-order inelastic) analysis: (i) the member out-of-straightness (bow imperfection) and (ii) the frame out-of-plumb (sway imperfection). In global frame analysis, the pattern of initial imperfections is often chosen to be the worst case scenario to maximize their destabilizing effects under the applied loads. Nevertheless, the worst case scenario of imperfections may be overly conservative. In reality, both initial outof-straightness and initial out-of-plumb are random, and a rational modelling of geometric imperfections can only be achieved by using probabilistic methods. The modelling of geometric imperfections is much more complicated for a frame than for a single column because not only the magnitude but also the pattern (shape and the direction) of the imperfection affects the overall response of the frame. Although extensive research has been conducted on advanced analysis for steel structural systems [1,2], a rational method of modelling initial geometric imperfection in advanced analysis has yet to be developed.

⁎ Corresponding author. Tel.: +61 450522922. E-mail address: [email protected] (S. Shayan).

http://dx.doi.org/10.1016/j.jcsr.2014.02.016 0143-974X/© 2014 Elsevier Ltd. All rights reserved.

There are a number of approaches to consider the effects of geometric imperfections in advanced analysis of steel frames. The common approaches include: scaling of elastic buckling mode (EBM), notional horizontal forces (NHF) method, reduction of member stiffness, and the direct modelling of initial geometric imperfections [3]. In the EBM method, a linear elastic buckling analysis of the perfect structure is first performed. The first buckling mode is then scaled to represent the imperfect geometry of the frame [4]. The assumption of the EBM method is that the first buckling mode represents the most critical imperfection geometry, which is similar to the deformation of the frame at collapse [5]. However, as a result of plastic deformations, the final failure mode of the frame may be different from the critical elastic buckling mode. An alternative approach was proposed by Alvarenga and Silveria [5] in which an inelastic second-order analysis is performed first to obtain the final collapse configuration, which is then used to model the imperfect geometry of the frame. This method, however, may be overly conservative. In the NHF method [6], artificial horizontal forces are introduced at each storey to account for the effect of frame out-of-plumb. This method is permitted in a number of steel specifications, e.g., AISC [7] and BS5950-1 [8]. For instance, AISC [7] stipulates the notional load as 0.2% of the gravity loads. Note that the value 0.2% is the maximum tolerance for out-of-plumb in steel structures as given in AISC [7]. The NHF method permits the use of straight elements in the structural model. The NHF method can also model imperfections of individual members by applying distributed lateral force along the

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member or a concentrated force at the middle of the member [3]. The degradation of member stiffness to model the effect of geometric imperfection was introduced by Kim [9]. In this method the elastic modulus E is reduced to 0.85E to account for geometric imperfections. The factor of 0.85 was determined by a calibration using plastic zone analyses and verified for a wide range of frames and columns. The model is applicable to both braced and unbraced frames with the same reduction factor of 0.85. This method has the advantage of eliminating the tedious work of explicit imperfection modelling or notional load application. However, the reduction factor 0.85 has not been verified by a probabilistic approach. Another approach is the direct modelling of initial geometric imperfections by offsetting the coordinates of the relevant nodes in the FE model from their original positions. The main difficulty associated with most of the aforementioned methods is that no information is provided about the pattern of the imperfections. The designer has either to guess or consider many possible combinations to find the worst case scenario, a difficult task for a real structure. On the other hand, an incorrectly defined initial geometric imperfection may be beneficial to the system strength rather than being detrimental. In addition, the extension of these methods to three dimensional frame analysis is not straightforward. The present study is concerned with developing a new method for modelling initial geometric imperfections in second-order inelastic analyses as a linear superposition of several scaled buckling modes. The statistical data of initial geometric imperfections are obtained in the literature and used in a probabilistic study to find a suitable number of buckling modes to be incorporated as well as the scaling factor for each buckling mode. For validation purposes, the performance of the proposed model is demonstrated by a number of case studies. The suggested procedure can be readily implemented into frame finite element analysis models and extended to 3D models. 2. Statistical data for initial geometric imperfections The methods for modelling geometric imperfections can also be classified as deterministic or random. For deterministic modelling, the maximum amplitude of an initial geometric imperfection is typically determined from a steel structural specification. The pattern of the initial out-of-straightness is often assumed to be a half-sine wave and the frame out-of-plumb follows a linear pattern with all columns leaning in the same direction. In probabilistic modelling, the initial geometric imperfections (both shape and the magnitude) are treated as random variables. The probabilistic modelling requires statistical information for the geometric imperfections, such as distribution type, mean, and standard deviation. Ideally, the probabilistic models should be established on the basis of sufficient experimental data. This section summarises the statistical information for the geometric imperfections, which is needed in developing the proposed methods.

Table 1 Statistical characteristics of initial out-of-straightness of hot-rolled I-sections in literature. Mean (μ)

Standard deviation (σ)

Number of measurements

Reference

0.00160 (1/625) 0.000204 (1/4910) 0.00079 (1/1266) 0.00050 (1/1996) 0.00008 (1/12500) 0.00025 (1/4000)

0.000600 0.000160 0.000326 0.000433 0.000053 0.002000

7 9 208 437 75 350

[20] [18] [19] [19] [21] [22]

the length of the member are required to obtain the statistics of initial out-of-straightness. For a member in compression the buckling modes are assumed to take the form of sin (iπx) where i = 1, 2, 3, … and x ∈ [0, 1] is the non-dimensional coordinate along the length of the member (L) (see Fig. 1). In general, the initial out-of-straightness of the member can be expressed in terms of a linear combination of a given number of these buckling modes: m

dx ¼ Σi¼1 ai sinðiπxÞ x∈ ½0; 1

in which dx is the initial out-of-straightness at location x, ai is the scale factor for the ith mode and m is the number of buckling modes included. In the following, it is assumed that a sample of N members is available and that for each member, the out-of straightness at m locations along the length of member is measured. This study is based on the initial out-of-straightness measurements of nine IPE 160 columns carried out at the University of Politecnico di Milano [10] and published by ECCS Committee 8.1 [11]. Although the number of sample is limited (only 9), the reported data are very valuable as they give the geometric imperfection measurements at mid-point and also quarter points. As three readings of out-of-straightness are available for each sample, the out-of-straightness can be expressed as a linear combination of the first three buckling modes as shown in Fig. 1. The scale factors, or contributions of each mode (ai, i = 1, 2, 3), can be determined by solving a set of three equations for each member (Eq. 1) and subsequently the statistical information of the scale factors a1, a2 and a3 (mean and coefficient of variation (COV)) can be obtained. As shown in Table 1, the mean of the absolute values of measured outof-straightness at mid-point, as reported in [11], is equal to 0.000204 (1/4910). This value appears to be very small compared to those reported in other similar studies, e.g., Fukumoto and Itoh [12]. Thus, while the COVs of the scale factors are based on these nine samples, the mean values are scaled up by a factor of 2.62 to match with the mean (1/1996) reported in [12], which is based on a much larger sample (437 measurements) of the initial out-of-straightness at the mid-span of the members. The scale factor of 2.62 was obtained based on the fact that if the first three buckling modes are used to model the initial

2.1. Initial out-of-straightness Although a great number of experimental results on column strength can be found in the literature, very few studies report the detailed measurements of initial imperfections along the length of the member. In most studies, out-of-straightness is assumed to follow a half-sine shape and only the value at mid-span is reported which does not provide sufficient information about the contribution of higher order buckling modes with multiple half-waves. The statistical data for the out-ofstraightness at mid-height of steel I-section members are summarised in Table 1. The presented results in this table show that a significant difference exists between measured imperfections from different regions. It appears that on average, Japanese sections have smaller initial out-of-straightness compared to those from Europe and North America. In this study, both the shape and magnitude are treated as random variables. Thus, detailed measurements of initial imperfections along

ð1Þ

P

P L

Mode 1

0

Mode 2

0

Mode 3

0

1 1 1

sinπ χ χ

sin2π χ χ

sin3π χ

χ

Fig. 1. First three buckling modes of simply supported axially loaded column.

S. Shayan et al. / Journal of Constructional Steel Research 98 (2014) 167–177

geometric imperfection of a single member, at mid-span only the first and third modes contribute (see Fig. 1). Therefore, the mean of the non-dimensional initial out-of-straightness at mid-span of a single member may be calculated as the difference between the mean of the scale factors corresponding to mode one and three μ a1 −μ a3 which is 0.00048 (1/2000) after scaling by factor of 2.62. The final statistical characteristics, i.e. mean (μ), standard deviation (σ) and COV of the scale factors, are summarised in Table 2. The distribution of the scale factors was found to be approximately normal.

2.2. Initial out-of-plumb The out-of-plumb can also be treated as a random variable and modelled as all columns leaning in the same direction or as each column leaning in its own direction. Horizontal displacements in both in-plane and out-of-plane directions of a multistorey steel frame have been reported in [13]. Beaulieu and Adams [14] measured the out-of-plumb of 916 columns in two horizontal directions and reported the mean as almost zero and the standard deviation as 0.00162. Beaulieu and Adams [15] reported about 725 out-of-plumb measurements on two high rise buildings in Canada. More than 900 out-of-plumb measurements were conducted in Germany by Lindner and Gietzelt [16]. The total 1760 measurements recorded in Canada and Germany suggest a mean of 0.00002 and a standard deviation of 0.000173. The histogram of these data is plotted in Fig. 2 and appears to be normally distributed. Since these data are based on the measured out-of-plumb of excising structures in various parts of the world, they represent realistic values and are used in the present study. It is assumed that all columns lean in the same direction and a single value of random out-of-plumb is applied to the whole frame.

3. Modelling of initial geometric imperfection by linear combination of eigenmodes Since current methods for modelling initial geometric imperfections in global frame analysis are overly conservative and present difficulties such as guessing the worst direction of imperfection or manually offsetting node coordinates, a new procedure is introduced in this study. This method is based on the superposition of a finite number of frame buckling modes which play the important role in triggering instability of the structure. If the buckling load of the first mode is not close to the ultimate load, this mode may not represent the critical shape of the initial imperfection [17]. Thus, a combination of several eigen buckling modes is a more appropriate approach. For cold-formed steel members, studies have been carried out to model local and global perturbations in the geometry as a linear combination of buckling modes with scale factors calculated on the basis of experimental measurements [17,18]. Theoretically, this methodology can also be applied to hot-rolled steel frames, provided we have the knowledge about how many elastic buckling modes to include and the scaling factor for each mode. A probabilistic framework is proposed here for determining a suitable number of buckling modes and their magnitudes for modelling initial geometric imperfections of steel frames.

3.1. Amplitudes of buckling modes In order to derive a general method for modelling imperfections as a superposition of scaled eigenmodes, the amplitude or the contribution of each mode needs to be determined. The general procedure for developing the scale factors can be summarised in seven essential steps: (1) For each member of a given frame, a random member-ofstraightness is generated as a superposition of three sine functions with 1, 2 and 3 half-waves and scale factors randomly determined using the statistical information presented in Table 2. (2) An additional frame out-of-plumb is randomly generated according to the statistical information given in Fig. 2. (3) Steps 1 and 2 are repeated n times to create n frames with random geometric imperfections. (4) For each frame, an elastic buckling analysis is performed to obtain the buckling shapes of the first m modes. (5) Error minimization is then performed between the n randomly generated imperfect frames in Step 3 and the linear combination of m eigenmodes. This results in n values of scale factors for each mode, denoted by Xkj (k = 1, …, n and j = 1, …, m). (6) The absolute values of the n scale factors are non-dimensionalised and their statistical characteristics are obtained. The mean values are denoted by x j (j = 1, …, m). (7) Steps 1 to 6 are repeated for a range of frame layouts with different geometries and loading conditions. The mean values of the nondimensionalised scale factors obtained from the different frame layouts are presented as the finial values of the scale factors, denoted as SFj, and can be implemented into advanced analysis to model initial geometric imperfections (both out-of-straightness and out-of-plumb). Twenty braced frames and twenty-three unbraced frames (Fig. 3) were chosen in the present study to derive the scaling factors of buckling modes for modelling geometric imperfections. The frames were chosen to represent typical low-to-mid rise moment frames with regular and irregular configurations, and were subjected to gravity loads. Detailed information about member sizes and applied loads for the frames can be found in [19]. Using the statistical characteristics obtained for initial out-ofstraightness (Table 2), for each frame a sample of n (n = 200) sets of random scale factors (ai, i = 1, 2, 3) are generated. Since the absolute values of scale factors are considered in finding the statistical characteristics, a random sign is generated and assigned to each scale factor. These values are subsequently substituted into Eq. 1 to generate a sample of 200 random imperfections for each member of the frame. Additional random out-of-plumb imperfections are superimposed to the whole frame (Fig. 2). Latin Hypercube Sampling (LHS), which is a highly efficient sampling method, is used to generate random variables. Fig. 4(a) shows a series of portal frames with random imperfections generated using the approach described in which Δri represents the randomly generated imperfection at node i including both out-ofstraightness and out-of-plumb. For each frame, an elastic frame buckling analysis is run using the FE software ABAQUS to obtain the buckling shapes for the first m elastic frame buckling modes. The scale factors (Xkj , k = 1,2,…200 and j = 1, …,m) are calculated using error minimization between the randomly generated shapes of imperfection and a linear combination of scaled buckling modes. The error for the kth frame is defined as:

2 k no kr m k kj Er ¼ Σi¼1 Δi − Σ j¼1 X j δi

Table 2 Statistical characteristics of scale factors. Statistics

a1

a2

a3

Mean (μ) Standard deviation (σ) COV Distribution

0.000556 0.000427 0.768 Normal

0.000139 0.000071 0.511 Normal

0.000073 0.000078 1.068 Normal

169

ð2Þ

in which no is total number of frame nodes, Δkr i is the randomly generated imperfection at node i of the kth frame, m is the number of modes included, Xkj is the scale factor for mode j corresponding to the kth frame, and δji is the deformation of node i in mode j corresponding to the kth

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n

n 1000

500

μ = 0.00002 σ = 0.000173 N = 1760

400 300

μ = 0.0013 σ = 0.00114 N = 1760

800 600

COV = 0.88

400

200

(a)

100

-40 -20

0

20

40

(b)

200

ψ. 104 (rad)

10

20

30

40

50

ψ. 104 (rad)

Fig. 2. Out-of-plumb statistics reported by Lindner and Gietzelt [16] (a) with sign and (b) absolute.

frame (Fig. 4(b)). The scale factors for all braced and unbraced frames appear to be normally distributed. To enable application to steel frames in general, the scale factors are non-dimensionalised by dividing by H (total frame height) or L (member length) depending on whether the corresponding mode is a sway or a non-sway mode, xkj = Xkj /(H, L). In most FE software like ABAQUS the buckling eigenmodes are normalized such that the maximum displacement component is unity. In this study, sway modes are defined as those for which the maximum deformation (unity displacement) occurs at the top of the frame while those modes with maximum displacement occurring within the members are classified as non-sway. The average (x j ) of the 200 non-dimensional scale factors, xkj (k = 1,…,200 and j = 1,…,m), is then calculated for the 20 braced frames and the 23 unbraced frames. These factors appear to be very similar for different frames. To find a single value for the scale factor of each mode to be implemented into advanced analysis, the average of the x j values of all studied frames is calculated and denoted as SFj. Obviously, this average may vary depending on the considered frames' configurations, applied loads and member cross-sections. However since in this study a wide range of low-to-mid-rise frame is considered, the derived scale factors are deemed to be applicable to typical steel frames. To obtain the contribution of each mode to the modelling of the imperfection, the average non-dimensional scale factors (SFj) can be normalized and expressed as the product of the proportion of each mode (Pj) and a single factor (F), which are shown in Table 3 for unbraced frames and Table 4 for braced frames, i.e. SFj = Pj × F, where Pj is normalized scale (participation) factor determined from Eq. 4 and m is number of modes.     SF j   P j ¼ Xm    SF j  j¼1

ð4Þ

The imperfection amplitude (Aj) to be incorporated into finite element analysis can be calculated as, 

A j ¼ P j  F  H For sway modes : A j ¼ P j  F  L For non‐sway modes

ð5Þ

Based on the results presented in Tables 3 and 4, it can be seen that the scale factors for different modes are very similar for braced frames where all buckling modes represent out-of straightness. The scale factor of the first mode for unbraced frames, which represents out-of-plumb, is much higher than the scale factors for higher order modes. The COVs of the scale factors (SFj) for unbraced and braced frames are presented in Tables 5 and 6, respectively. As will be shown in subsequent sections, despite the relatively large variance in imperfection magnitudes, the variance of the capacity of steel frames resulting from

stochastic imperfections is a small fraction of the variance of the SFj in Tables 5 and 6.

3.2. Numbers of eigenmodes Evidently, the derived scale factors vary with the number of eigen buckling modes, which therefore needs to be determined. Theoretically, it is expected that greater accuracy will be achieved by combining more buckling modes but at the same time this number should be reasonable and optimal. To find the appropriate number of eigenmodes, three typical frames have been studied (Fig. 5). For all frames, the span length is 6 m while the storey height is 4 m and the same between all levels. A total of 200 random geometric imperfections have been produced for each frame following the methodology presented in Section 3.1. The generated member and sway imperfections must then be multiplied by the member length (L) and frame height (H) respectively to obtain the actual magnitude, since the statistical characteristics in Table 2 and Fig. 2 are non-dimensional. Those imperfect frames are then imported into ABAQUS and analysed under equal vertical loads (P = 1000 kN) applied at the top of each column thus providing the ultimate load factor of each frame (λ(actual)k, k = 1, …, 200). Second-order twodimensional inelastic (advanced) analysis is used to determine the frame ultimate load factor, accounting for all material and geometric nonlinearities. To model the material nonlinearity, a 2D plastic-zone beam-column element is used to trace the spread of plasticity through the cross-section and along the member length. The arc-length technique is used to obtain the complete load-deflection response. The material is modelled as elastic-perfectly plastic with the elastic modulus (E) and yield stress equal to 200 GPa and 320 MPa respectively. All cross-sections are 150UB14 which is fully compact (Fig. 5). A mesh convergence study is performed and one element per 200 mm length is used for all members. So as to investigate the influence of imperfections on the frame ultimate strength as the only trigger of second-order effects, residual stress is not taken into account. Elastic buckling analyses are performed to obtain the buckling deformations of each frame for the first ten modes. Using Eq. 2, a set of scale factors (Xkj , k = 1,…,200 and j = 1,…,m) considering that a finite number of buckling modes (m) are evaluated for each frame by error minimization as explained in Section 3.1. Incorporating buckling modes scaled by these factors into finite element models, a second set of advanced analysis is carried out and the ultimate load factors are obtained (λmk ), where k refers to the kth imperfect frame. Note that in this set of simulations, the frame is modelled with the perfect geometry and the imperfection is applied to the model as scaled eigenmodes using the *IMPERFECTION option of ABAQUS. To be able to compare the results of the three different frames, the ratio of λmk/λ(actual)k is calculated in which λmk is the ultimate load factor

S. Shayan et al. / Journal of Constructional Steel Research 98 (2014) 167–177

171

Fig. 3. Steel frame layouts to determine appropriate scale factors.

of the kth frame considering the linear combination of m scaled modes and λ(actual)k is the ultimate strength of frame k obtained from an advanced analysis of the frame with randomly generated imperfections.

This ratio is called the “bias” for ease of reference. The mean and COV of the bias are plotted in Fig. 6(a) and (b) for the three frames. The values of absolute error are also calculated for all 200 frames of each type, as

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S. Shayan et al. / Journal of Constructional Steel Research 98 (2014) 167–177

Fig. 4. (a) Examples of randomly generated shapes of initial imperfection for a simple portal frame. (b) Example of buckling mode (j) for a simple portal frame.

Table 3 Proportion of each mode to model initial geometric imperfection, unbraced frames. Number of modes

P1

P2

1 2 3 4 5 6 7 8 9 10

1 0.782 0.623 0.522 0.441 0.386 0.350 0.316 0.283 0.266

0.218 0.161 0.121 0.103 0.090 0.079 0.071 0.064 0.060

P3

P4

0.216 0.178 0.157 0.138 0.124 0.112 0.098 0.091

P5

0.178 0.155 0.138 0.127 0.110 0.099 0.089

P6

P7

P8

P9

P10

F

0.143 0.120 0.109 0.103 0.091 0.084

0.128 0.113 0.104 0.091 0.082

0.098 0.091 0.085 0.077

0.093 0.089 0.083

0.099 0.090

0.077

0.001228 0.001566 0.001838 0.002147 0.002504 0.002817 0.00309 0.003413 0.003782 0.004030

P5

P6

P7

P8

P9

P10

F

0.073

0.001228 0.001566 0.001838 0.002147 0.002504 0.002817 0.003090 0.003413 0.003782 0.004030

Table 4 Proportion of each mode to model initial geometric imperfection, braced frames. Number of modes

P1

P2

1 2 3 4 5 6 7 8 9 10

1 0.528 0.380 0.293 0.229 0.200 0.168 0.150 0.133 0.124

0.472 0.323 0.261 0.217 0.188 0.159 0.141 0.128 0.119

P3

P4

0.297 0.228 0.178 0.155 0.124 0.112 0.102 0.096

0.217 0.177 0.150 0.129 0.113 0.102 0.096

0.199 0.172 0.151 0.133 0.114 0.106

0.134 0.116 0.107 0.098 0.087

0.154 0.137 0.126 0.117

0.106 0.099 0.088

0.099 0.093

all considered frames. At the same time, the COV decreases from the average of 4.32% for all three frames when only one mode is included to 2.32% when including six modes. It is interesting to observe that although increasing the number of modes can provide better representation of the initial imperfection, including only the first mode does not result in a significant error from the actual mean. For the case of including only the first mode, the mean values of bias for Frame 1 and Frame 3

defined by Eq. 3. The mean and maximum values of absolute error in percent (%) are plotted in Fig. 6(c) and (d).     k Er ¼ 1−λmk =λðactualÞk  ð3Þ Based on this study, the following observations can be made: (1) From Fig. 6(a) and (b), it can be seen that by including an increasing number of modes the bias is approaching to unity for

Table 5 COV of scale factors of each mode, unbraced frames. Number of modes

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

1 2 3 4 5 6 7 8 9 10

0.749 0.752 0.754 0.754 0.755 0.753 0.753 0.754 0.756 0.755

0.790 0.796 0.828 0.831 0.832 0.813 0.804 0.801 0.799

0.819 0.814 0.802 0.787 0.794 0.793 0.783 0.791

0.806 0.813 0.811 0.805 0.803 0.800 0.801

0.832 0.835 0.832 0.815 0.825 0.826

0.805 0.806 0.801 0.803 0.810

0.829 0.826 0.800 0.799

0.827 0.817 0.809

0.804 0.817

0.802

S. Shayan et al. / Journal of Constructional Steel Research 98 (2014) 167–177

173

Table 6 COV of scale factors of each mode, braced frames. Number of modes

SF1

SF2

SF3

SF4

SF5

SF6

SF7

SF8

SF9

SF10

1 2 3 4 5 6 7 8 9 10

0.868 0.854 0.803 0.803 0.812 0.808 0.816 0.817 0.821 0.830

0.887 0.901 0.904 0.892 0.894 0.899 0.896 0.900 0.900

0.846 0.830 0.830 0.841 0.839 0.840 0.844 0.840

0.855 0.852 0.869 0.867 0.878 0.874 0.867

0.832 0.824 0.788 0.787 0.788 0.785

0.836 0.838 0.849 0.836 0.840

0.804 0.813 0.786 0.782

0.825 0.822 0.815

0.827 0.824

0.829

P

P

75 mm

P

P

P

P

P

P

P

P

P

H

H 150

h

5 mm

h

h

mm

7 mm

L

L

Frame 1

Frame 2

L

L

Frame 3

Fig. 5. Steel frame layouts.

are 0.982 and 1.017 respectively, while the maximum COV is 5% (in Frame 1). (2) As to be expected, the mean and maximum values of absolute error reduce for all three frames by considering more modes (Fig. 6(c, d)). It should be noticed that incorporating only the first mode can reasonably model the initial imperfection with the mean value of error less than 5%. The maximum error among all 200 simulations is 15.2% considering only one mode,

corresponding to Frame 1, and decreases to less than 10% for all frames when six modes are included. (3) It appears that one, three and six are “good” numbers of buckling modes to be used for modelling initial geometric imperfections. Using the first mode is easy and can produce reasonable results. There are noticeable reductions in the mean and maximum values of error in changing from 2 to 3 modes. Thus, three modes can be a better alternative compared to one mode.

Fig. 6. (a) Mean value of bias, (b) COV of bias, (c) mean of absolute error (%), and (d) maximum of absolute error (%), for 200 simulations.

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S. Shayan et al. / Journal of Constructional Steel Research 98 (2014) 167–177

5m

4m

Frame 1

200UB29 P

P

200UB29 P

200UB29

200UB29 P

P

P

200UB29

200UB29

200UB29

6m

6m

6m

200UB29 w 200UB29 w 200UB29

200UB29 w

200UB29 w

16 KN

8 KN

200UB29

200UB29

6m

6m

200UB25 32 KN

16 KN

64 KN

310UB40

2×5m

Frame 4

Frame 3

Frame 2

w

6m

32 KN

16 KN

w

2×5m

200UB29 P

200UC52 150UC30

200UB29 P

200UB29 w

150UB14 150UB14

P

150UB14

250UB37

250UB37

8m 5m

IPE220

IPE220

IPE 270

P

150UB14

360UB56 P=500KN P

P

KN4×4

P=500KN P

P=100 KN

3×4m

150UB14 150UB14 150UB14

w=30 KN/m

Frame 5

31.7 KN/m

200UB29 P

200UB29

200UB29

4m

6m

20.44 IPE360 20.44 KN IPE400

49.1 KN/m 49.1 KN/m

w1

w1

IPE 200

IPE 270

IPE 270

IPE 200

w2

w2 =72.58 KN/m

IPE 220

2×6m

4m

Frame 7

Frame 6

2×4m

P

w1 =28.27 KN/m

w1

IPE 330

6m

6m

HEB 120

IPE330

200UB29

49.1 KN/m

HEB 140

P

20.44 KN

HEB 140

200UB29 P

IPE300 P

HEB 140

P

200UB29

49.1 KN/m

HEB 140

200UB29 P

IPE300 20.44 KN

HEB 120

P

P=100 KN

5.8 KN/m

200UB29 P

200UB29

6×3.75 m

P

4×4 m

150UB14 150UB14 150UB14 150UB14

200UB29 P

49.1 KN/m

HEB220 HEB220 HEB220 HEB220 HEB160 HEB160

IPE240 20.44 KN

HEB260 HEB260 HEB240 HEB240 HEB200 HEB200

10.23 KN

4m

Frame 8

Fig. 7. Steel frame layouts, un-braced frames.

Although 10 modes have the smallest mean and maximum errors, insignificant reduction in error is achieved by increasing the number of modes from 6 to 10. It can be concluded that six modes may predict the actual shape of imperfection and strength of a steel frame accurately.

in Tables 3 and 4. The ultimate load factor for each frame is then obtained and denoted as λ1n, λ3n and λ6n, considering one, three and six modes respectively. These ultimate load factors are compared with the mean of the ultimate load factor of the 200 randomly generated shapes for each frame, denoted as λ. The ultimate load factor distribution is found to be normal.

4. Verification and illustrative examples 4.1. Unbraced frames The application of the proposed amplification factors (Aj) is verified by means of a probabilistic approach. Different regular and irregular, braced and unbraced frames are studied. Advanced analysis is run for each frame considering one, three and six eigenmodes to model initial geometric imperfections using the proposed amplitude factors provided

A total of eight unbraced frames are chosen to investigate the effect of initial geometric imperfection on the frame ultimate strength. Fig. 7 shows the frame configurations and loading. Four frames are adopted from the literature and represent practical cases (Frame 1 [23],

Table 7 Verification results, unbraced frames.

Elastic modulus (GPa) Yield strength (MPa) λ λ1n λ3n λn6 Er1 (%) Er3 (%) Er6 (%) μ (λk/λ6n) COV (λk/λ6n) Min (λk/λ6n) Max (λk/λn6)

Frame 1

Frame 2

Frame 3

Frame 4

Frame 5

Frame 6

Frame 7

Frame 8

210 300 1.7702 1.7897 1.7811 1.7725 1.0896 0.612 0.1298 0.9987 0.0432 0.8782 1.0873

200 320 2.339 2.37 2.351 2.348 1.321 0.506 0.4 0.996 0.0800 0.8019 1.1725

200 320 1.598 1.43 1.472 1.475 10.52 7.875 7.7 1.083 0.0702 0.8851 1.2265

200 320 1.326 1.348 1.362 1.341 1.617 2.636 1.096 0.989 0.0321 0.9168 1.0534

200 250 1.311 1.282 1.282 1.283 2.235 2.212 2.166 1.022 0.0031 1.013 1.0314

200 320 1.1884 1.0497 1.0781 1.0813 11.671 9.2814 9.0121 1.099 0.0744 0.8765 1.2692

205 235 1.13 1.118 1.122 1.12 1.062 0.752 0.894 1.009 0.0133 0.9633 1.0282

210 275 1.084 1.062 1.064 1.065 2.011 1.836 1.79 1.018 0.0159 0.9745 1.0679

S. Shayan et al. / Journal of Constructional Steel Research 98 (2014) 167–177

(a)

(b) 500 KN

500 KN

than unity is also investigated in this study. Frame 2 with the highest COV is considered and redesigned with the values of column slenderness (λc) equal to 0.88 and 1.37 respectively. The member crosssections and material properties can be found in Fig. 8 and the simulation results are presented in Table 8. It can be concluded from the results shown in Table 8 that changing the column slenderness from unity to 0.88 and 1.37 decreases the COV to 6.3% and 5.4% respectively. This shows that when a diverse range of column slenderness values are considered, the frame is less sensitive to imperfection. For most cases, using the imperfection amplitudes presented in Table 3 can provide an excellent agreement between the finite element analysis of frames with actual random imperfect shapes and linear combination of frame buckling modes, with the mean error less than 10%.

500 KN

4m

fy =235 MPa, E =210 GPa

250UB37

8m

150UB14

250UB37

250UB37

8m

360UB56

250UB37

500 KN

4m

fy =320 MPa, E =200 GPa

Fig. 8. Frame 2, (a) column slenderness parameter λc = 0.88 and (b) column slenderness parameter λc = 1.37.

Frame 5 [1], Frame 7 [24] and Frame 8 [25]). The other four are designed such that the column slenderness parameter (λc) takes the value of unity for most columns, since this is the value for which the squash load and elastic buckling load coincide and produce the greatest sensitivity to initial imperfection [1]. The column slenderness can be calculated as qffiffiffiffiffiffiffiffiffiffi
λc ¼ kL=πrÞð f y =E :

175

ð4Þ

Material properties including elastic modulus and yield stress as well as the ultimate load factors (λ1n, λ3n and λ6n) of different frames are summarised in Table 7. These ultimate load factors are then compared with the mean of the ultimate load factor of the 200 randomly generated shapes for each frame (λ). The errors considering different numbers of modes are reported in Table 7 (Er1, Er3 and Er6). As expected, the values of absolute error reduce for most frames by considering more modes. Based on the presented results, it can be seen that the highest errors are related to those frames which are designed to have the column slenderness parameter equal to unity for most columns (Frame 3 and Frame 6). These frames fail by instability of the whole system. The maximum error is about 11.7% if considering only one mode, corresponding to Frame 6, and decreases to 9% when six modes are included. It appears that although considering six modes can reduce the error to be less than 10% for all frames, including only the first mode does not result in a significant error from the actual mean, especially for more practical frames which are not very sensitive to imperfections. To investigate the variation of the 200 simulations from the ultimate load factor using six modes (λ6n), the ratio between the λk, k = 1,…, 200, and λ6n is calculated. The mean, COV, and maximum and minimum values of λk/λ6n are shown in Table 7 and represent the imperfection modelling error. This may be used to assess the reliability of ultimate strengths provided by advanced analysis. It should be noticed that the mean values of λk/λ6n for all frames are around unity which shows that the proposed model by applying the scale factors provided in Table 3 can accurately predict the mean. The maximum COV is 8% corresponding to Frame 2. Although the COVs of the derived scale factors vary between 75% and 85%, which indicate a significant scatter, this results in only 8% COV in the ultimate strength. Again, it can be observed that the highest values of COV, approximately 7% to 8%, are associated with those frames which are sensitive to imperfections and fail by frame instability. In reality, it is highly unlikely that all columns of a frame have a slenderness parameter of about unity, so the statistics corresponding to these frames can be considered as upper bounds. For more practical frames the COV can decrease to as low as 0.3%. The variation of the COV of the ratio between the ultimate loads (λk/λ6n) when the column slenderness parameter is less than or greater

4.2. Braced frames A total of eight regular and irregular braced frames are chosen to investigate the effect of initial geometric imperfection on frame strength. Fig. 9 shows the frame configurations and loading. Four frames are adopted from the literature and represent practical cases (Frame 1 [26], Frame 2 [27], Frame 3 and Frame 4 [28]). The braces are considered perfect without any initial geometric imperfections. To avoid those modes in which the braces buckle, in the buckling analyses (but not in the inelastic analyses), the braces are modelled by applying a lateral restraint at each storey level. The results of the simulations are summarised in Table 9. The maximum error is about 6.6% considering only one mode, corresponding to Frame 4, and decreases to 1.65% when six modes are included. Generally, the absolute values of error for these eight braced frames are lower than those for the set of eight unbraced frames discussed in Section 4.1. The mean error is less than 5% for all frames incorporating six modes. The maximum COV is 2.47% corresponding to Frame 8. As it can be seen in Table 9, for some frames considering only the first mode can predict the ultimate load more accurately than considering 3 modes. For these frames the ultimate limit state deformation is similar to the first buckling mode, and hence second order effects can be captured fairly accurately by scaling the first buckling mode. Conversely, the first two buckling modes are similar but tend to be in opposite directions, and so the inclusion of both modes reduces the effect of imperfections. Thus, only if the first buckling mode does not include the buckling of those columns which participate in the failure of the frame, more modes are required to model the imperfection accurately. It can be concluded that for braced frames, although increasing the number of modes can provide a better representation of the initial imperfection, (i) including only the first mode does not result in a significant error from the actual mean, and (ii) if multiple modes are to be included, including six modes is preferable to including three modes. 5. Conclusion Initial geometric imperfections affect the nonlinear behaviour of structures and may have a considerable influence on the ultimate

Table 8 Simulation results for Frame 2 for different column slenderness values.

λ λ6n Er6 (%) μ (λk/λ6n) COV (λk/λ6n) Min (λk/λ6n) Max (λk/λ6n)

λc = 0.88

λc = 1

λc = 1.37

1.9348 1.9278 0.3617 1.0036 0.063 0.8355 1.1331

2.33859 2.34796 0.40061 0.99175 0.08001 0.80189 1.17246

1.4392 1.4389 0.0206 1.0002 0.054 0.8709 1.0848

S. Shayan et al. / Journal of Constructional Steel Research 98 (2014) 167–177

P

2 KN W12x96

P

4 KN

W14x48

7.315m

6.096m

Braces: 2L 50.8×50.8×3.175 mm

IPE 100

P

P IPE 140

P

P IPE 140

P

P IPE 140

P

Braces: 2L 65×65×9 mm

Frame 2

=15 KN/m

3m

3m

Braces : L 76.2×76.2×12.7 mm

Frame 1

P

P

2×3 m

W14x4

W21x50

P

2×3.657m

26.7 KN

P

IPE 100 HEB 120 HEB 120

2×3.657m

70 KN/m

P=370 KN

W14x48

W12x96

W14x4

W18x40

P IPE 140

250UB25

250UB25 =30 KN/m

250UB25

250UB25

250UB25

250UB25

250UB25

250UB25

250UB25

6m

6m

6m

Braces: L 150×150×12 mm

Braces: 2L 100×100×12 mm

Frame 3

150UB14 150UB14 150UB14

P=2000 KN

P

5×3 m 3×4m

52.5 KN/m 44.5 KN

P=370 KN IPE 140

HEB 220 HEB 220 HEB 220 HEB 220 HEB 220

176

Frame 4

Frame 5

6.1 m

5.3 m

6m

6.1 m

Braces: 2L 100×100×12 mm

6m

6m

IPE 360 =62 KN/m

IPE 360

IPE 360

IPE 360

4m

6m

6m

4m

Braces: 2L 100×100×12 mm

Braces: 2L 100×100×12 mm

Frame 7

Frame 6

IPE 360 8m

IPE 360

HEB120

=76 KN/m

HEB120

200UB29

=48 KN/m

HEB120 HEB120

200UB29

200UB29

200UB18

4.88 m 5.5 m

200UB29

200UB29

8m

W8x17

W18x40

W8x17

W8x3

W18x40

W18x40

W18x40

W8x31

W8x31

W18x40

W8x31

W8x17

200UB29 W18x40

=40 KN/m

6.8 KN/m

200UB29

4×4m

6.71 m

W18x40 W8x17

=40 KN/m

W8x31

W8x17

W18x40

200UB29

200UB18

=70 KN/m

200UB18 200UB18

=20 KN/m

Frame 8

Fig. 9. Steel frame layouts, braced frames.

Table 9 Verification results, braced frames.

λ λ1n λ3n λ6n Er1 (%) Er3 (%) Er6 (%) μ (λk/λ6n) COV (λk/λ6n) Min (λk/λ6n) Max (λk/λ6n)

Frame 1

Frame 2

Frame 3

Frame 4

Frame 5

Frame 6

Frame 7

Frame 8

1.6257 1.6358 1.6355 1.6354 0.6174 0.5992 0.5931 0.9941 0.0018 0.99 0.9981

1.119 1.116 1.103 1.118 0.304 1.413 0.134 1.001 0.0093 0.9758 1.016

1.088 1.066 1.064 1.081 1.982 2.184 0.631 1.006 0.0245 0.9289 1.0386

1.327 1.239 1.303 1.305 6.615 1.853 1.649 1.017 0.0127 0.9896 1.0889

1.183 1.23 1.208 1.194 4.026 2.166 0.983 0.99 0.0226 0.9368 1.0423

1.1459 1.1716 1.1671 1.1669 2.1915 1.8143 1.7975 0.982 0.0144 0.9465 1.0057

1.258 1.197 1.234 1.234 4.893 2.053 1.999 1.02 0.0217 1.0058 1.0581

1.071 1.058 1.022 1.066 1.199 4.587 0.517 1.005 0.0247 0.9128 1.039

strength. Thus, imperfections need to be modelled appropriately in advanced structural analysis. This study outlines a convenient method for modelling initial geometric imperfections as a linear combination of scaled eigenmodes. The method is easy to implement into finite Table 10 Recommended proportions of each mode to model initial geometric imperfection.

Unbraced frames Braced frames

Number of modes

P1

3 6 6

0.60 0.20 0.20 0.002 0.40 0.10 0.15 0.15 0.10 0.10 0.003 0.20 0.20 0.15 0.15 0.15 0.15 0.003

P2

P3

P4

P5

P6

F

element analysis and obviates the difficulties of current methods such as offsetting nodes or guessing the worst imperfection shape. The study considers regular and irregular sway and braced planar frames. It may be extended to the 3D space frames. Based on the results of advanced analysis, the appropriate number and magnitudes of eigenmodes have been suggested in this study. It can be concluded that for unbraced frames, although six buckling modes may predict the actual shape of imperfection more accurately, including only first three modes does not result in a significant error. The maximum modelling error is about 8%. For braced frames, since in some cases the first two buckling modes are similar but in opposite directions, their effects may largely cancel out and three buckling

S. Shayan et al. / Journal of Constructional Steel Research 98 (2014) 167–177

modes may not be sufficient to model the imperfection accurately. Thus, for braced frames, including six modes is recommended. The associated maximum modelling error is about 2.5%. The recommended number of modes and rounded values of scaling factors to use for modelling imperfections of braced and unbraced frames is summarised in Table 10. Acknowledgements This research is supported by Australian Research Council under Discovery Project Grant DP110104263. This support is gratefully acknowledged. References [1] Clarke MJ, Bridge RQ, Hancock GJ, Trahair NS. Advanced analysis of steel building frames. J Constr Steel Res 1992;23(1–3):1–29. [2] Liew JYR, Chen WF, Chen H. Advanced inelastic analysis of frame structures. J Constr Steel Res 2000;55:245–65. [3] Chan SL, Huang HY, Fang LX. Advanced analysis of imperfect portal frames with semirigid base connections. J Eng Mech 2005;131(6):633–40. [4] Gu JX, Chan SL. Second-order analysis and design of steel structures allowing for member and frame imperfections. Int J Numer Methods Eng 2005;62(5):601–15. [5] Alvarenga AR, Silveria RAM. Second-order plastic-zone analysis of steel frames — part II:effects of initial geometric imperfection and residual stress. Lat Am J Solids Struct 2009;6(4):323–42. [6] Liew JYR, White DW, Chen WF. Notional-load plastic-hinge method for frame design. J Struct Eng ASCE 1994;120(5):1434–54. [7] AISC. Load and resistance factor design specification. 2nd ed. Chicago: AISC; 1993. [8] BS5950-1. Structural use of steelwork in buildings, part 1: code of practice for design. London: British Standards Institution; 2003. [9] Kim SE. Practical advanced analysis for steel frame design. [Ph.D. Thesis] West Lafayette, IN: Purdue University; 1996. [10] C.E.A.C.M. Laboratory report, commission 8. Politecnico di Milano, Istituto di Scienza delle Construzioni, Laboratorio Prove Materiali; 1966. [11] Sfintesco D. Fondement experimental des courbes Europeennes de flambement. J Constr Métal 1970;3:5–12.

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[12] Fukumoto Y, Itoh Y. Evaluation of multiple column curves using the experimental data-base approach. J Constr Steel Res 1983;3(3):2–19. [13] ECCS. Manual on the stability of steel structures. 2nd ed. Brussels, Belgium: European Convention for Constructional Steelwork; 1976. [14] Beaulieu D, Adams PF. A statistical approach to the problem of stability related to structural out-of-plumb. International Colloquium on Stability of Structures under Static and Dynamic Loads, Washington, D.C.; May 17-19, 1977. p. 114–21. [15] Beaulieu D, Adams PF. The results of a survey on structural out-of-plumbs. Can J Civ Eng 1978;5(4):462–70. [16] Lindner J, Gietzelt R. Imperfektionsannahmen für Stützenschiefstellungen (Assumptions for imperfecftions for out-of-plumb of columns). 1984;53(4):97–102 [in German]. [17] Rasmussen KJR, Hancock GJ. Geometric imperfections in plated structures subject to interaction between buckling modes. Thin-Walled Struct 1988;6:433–52. [18] Zeinoddini VM, Schafer BW. Simulation of geometric imperfections in cold-formed steel members using spectral representation approach. Thin-Walled Struct 2012; 60:105–17. [19] Shayan S, Rasmussen KJR, Zhang H. On the modeling of initail geometric imperfections and residual stress of steel frames. Research report, R.935. The University of Sydney; 2012. [20] Tebedge N, Marek P, Tall L. Column testing procedure. Fritz engineering laboratory report, no. 351.1. Bethlehem, Pennsylvania: Lehigh University; 1969. [21] Itoh Y. Ultimate strength variations of structural steel members. [PhD Thesis] Nagoya: Nagoya University; 1984. [22] Tomonaga K. Actually measured errors in fabrication of Kasumigaseki Building. 3rd Regional Conference Proceedings. Tokyo, Japan: Muto Institute of Structural Mechanics; 1971. [23] Kala Z. Sensitivity analysis of steel plane frames with initial imperfections. Eng Struct 2011;33:2342–9. [24] Vogel U. Calibrating frames. Stahlbau 1985;54:295–311. [25] Cabrero JM, Bayo E. Development of practical design methods for steel structures with semi-rigid connections. Eng Struct 2005;27:1125–37. [26] Hadianfard MA, Razani R. Effects of semi-rigid behavior of connections in the reliability of steel frames. Struct Saf 2001;25:123–38. [27] Lui EM, Chen WF. Behavior of braced and unbraced semi-rigid frames. J Solids Struct 1988;24(9):893–913. [28] Andreaus U, D'Asdia P, Iannozzi F. On the optimal choice of the shape of antiseismic bracing systems. Atti dell'8th W.C.E.E. San Francisco, U.S.A.: Luglio; 1984 459–66.