Reliability calibrations for the design of cold-formed steel portal frames by advanced analysis

Engineering Structures 182 (2019) 164–171

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Reliability calibrations for the design of cold-formed steel portal frames by advanced analysis Francisco Sena Cardoso, Hao Zhang, Kim J.R. Rasmussen, Shen Yan

T

⁎

The University of Sydney, Australia

A B S T R A C T

The steel industry is developing a design-by-advanced analysis specification for cold-formed steel construction. This effort provides an opportunity to utilize the latest nonlinear structural analysis (advanced analysis) to design steel structures based on their overall system behaviour. This paper concerns the system reliability calibrations of this design-by-analysis method, with a particular focus on cold-formed steel portal frames. Four typical portal frames are considered. The system reliability assessment takes into account all important random variables. A limit-state design criterion is developed which is consistent with a desired level of system safety.

1. Introduction Several steel structures design specifications permit design by advanced nonlinear structural analysis. Advanced analysis has received considerable attention from the research communities for several decades. Fully nonlinear analyses have been developed capable of accurately capturing the true behaviour of steel structures subject to complex buckling modes. In professional practice, as early as 1998, the Australian Standard AS4100 [1] included the design by advanced analysis method. It allows internal actions to be obtained by nonlinear structural analysis. However, connection and section capacities are still required to be evaluated and checked. The scope of using advanced analysis in AS4100 is limited to fully braced compact cross-sections. More recently, the American Steel Specification AISC 360-10 [2] significantly broadened the scope of advanced analysis (referred to as “inelastic analysis” in the Specification). It permits the design of a steel frame to be based directly on the nonlinear structural analysis without the need for checking capacities of individual members. For this reason, the design by advanced analysis method is termed as the Direct Design Method (DDM) in this paper. In AISC 360-10, the advanced analysis method can be applied to non-compact cross-sections and members not fully braced, provided the relevant limit states are captured in the analysis. The recently revised version of the Australian/New Zealand Standard for Cold-formed Steel Structures AS/NZS4600 [3] now also features provisions for the DDM by advanced analysis. The Direct Design Method is not to be confused with the Direct Strength Method, which is permitted in Australia/New Zealand [3], and North America [4]. The Direct Strength Method is a design method for thin-walled members (columns and beams) [5]. The Direct Strength

⁎

Method is to determine the strength of a member as opposed to the system strength given by the DDM. Moreover, the current Direct Strength Method is based on computational member elastic buckling stability analysis, while the DDM requires a fully nonlinear system analysis. The Direct Design Method must satisfy certain modelling/analysis requirements articulated in the standards, as well as structural reliability requirements. Appendix I of AISC360-10 [2] requires that the DDM must account for the natural variabilities in system, member and connection resistance, and provide a structural reliability for the frame no less than the current member-based design method. To fulfill this requirement, a resistance factor at the system level can be combined with the nominal frame strength predicted by the advanced analysis. The design equation is given by:

ϕs Rn ⩾

∑i γi Qni

(1)

where ϕs and Rn are the resistance factor and nominal resistance of the system, respectively, and Qni and γi denote the structural load and the corresponding load factor. The resistance Rn is the ultimate load-carrying limit of the frame at incipient instability of the frame. Rn is computed using the nominal values of structural parameters. The structural failure risk due to the uncertainties in structural resistance is controlled through the use of ϕs. The current resistance factors stipulated in the specification, e.g. 0.9 for flexural members, were developed for individual member safety check, as opposed to the system-level check in the design-by-analysis method. The system resistance factor needs to be determined by a structural reliability calibration procedure conducted at the system level. The system reliability implications of the DDM have been examined

Corresponding author. E-mail address: [email protected] (S. Yan).

https://doi.org/10.1016/j.engstruct.2018.12.054 Received 13 August 2018; Received in revised form 16 December 2018; Accepted 17 December 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

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F. Sena Cardoso et al.

yielding behaviour of the steel [13]. The nominal value of the yield stress is 550 MPa, with a modulus of elasticity of 200 GPa. For cold-formed steel sections, coiling, uncoiling, cold bending to shape, and straightening of the formed member lead to a complicated set of initial stresses in the section. Likewise, as a result of the manufacturing process, the yield stress of the section corners is typically enhanced, a phenomena commonly referred to as corner strength enhancement. Regarding the modelling of residual stresses and corner strength enhancement in finite element analysis, three main approaches have been followed: (i) model the actual manufacturing process, by assuming the stress-strain curve of virgin steel, followed by the modelling of the actual coiling, uncoiling, roll-forming or press-braking processes, e.g., in Pastor et al. [14], (ii) model residual stresses and corner strength enhancement independently, e.g., in Crisan et al. [15], and (iii) simply ignore simultaneously both effects assuming that the favourable effect of corner strength enhancement is cancelled by the unfavourable effect of residual stress, e.g., in Sarawit [16], Sena Cardoso and Rasmussen [17]. The present study adopts the third approach, assuming that the enhanced yield stress in the highly worked corner compensates the effect of residual stress. As observed from joint component tests [18], the moment-rotation curves of typical bolted portal frame joints can be reasonably described by a multi-linear curve. Thus, a bi-linear curve in Fig. 2 is adopted to model the partial rigidity of the joints. The bi-linear moment-rotation curve is defined by four parameters, M1, M2, K1 and K2. The present study does not consider joint failure. It is assumed that at the ultimate state of the frame, the joints have not reached their ultimate moment capacities or their ductility limits. Three types of initial geometric imperfections are relevant for a cold-formed frame, i.e., imperfections at frame, member and crosssection levels. The member imperfections of a space member include strong axis out-of-straightness (camber imperfection), weak axis out-ofstraightness (bow imperfection), and twist imperfection. Fig. 3 demonstrates the sectional imperfections, including the local and distortional imperfections. It is assumed that the camber, bow and twist imperfections along the member length is described by a half sinusoidal (single wave) curve, i.e.,

in a limited number of case studies, e.g., [6–9]. Buonopane and Schafer [6] investigated the system reliabilities of Load and Resistance Factor Design (LRFD) and the DDM using 16 planar two-storey two-bay gravity frames. The uncertainties in steel yield stress, dead and live loads were considered. The uncertainties in other parameters, e.g., Young’s modulus and sectional properties, were ignored. Zhang et al. [9] examined the system reliabilities of a number of simple structures, including a continuous beam and four frames with different levels of redundancy and capability of load redistribution. The study discussed the reliability implications of the LRFD and the second-order inelastic method in AISC360-10. A similar work has been conducted by Thai et al. [7] for two low-rise planar frames with partially restraint connections. A general reliability framework for assessing the system reliabilities of steel frames was developed in [10]. Using the reliability framework in [10], systematic reliability calibrations were conducted for the DDM for typical planar steel frames [11] and space steel frames comprising members of locally stable hollow sections [12]. All these aforementioned studies focused on structures comprising compact members with full lateral bracing. The present study concerns the derivation of system resistance factors suitable for the limit state design by advanced analysis of coldformed steel portal frames comprising members of locally unstable sections. Section 2 introduces the four baseline frames used for the reliability calibration, the nonlinear finite element (FE) structural models and the system failure modes. The four baseline frames are chosen to represent typical constructions of cold-formed portal frames. Section 3 presents the framework of the reliability calibration, including the probabilistic models of random variables, and the system reliability assessment tool. In Section 4, the reliability calibration results are presented and discussed. System resistance factors consistent with a desired system reliability level are suggested. 2. Baseline frames and FE models 2.1. Descriptions of the frames Wind design of four portal frames are considered to derive the system resistance factor for the advanced analysis. The frames are denoted by WF1 to WF4. Series WF1 and WF2 have a relatively short span of 8 m, comprising members made from single channels. Series WF3 and WF4 have a large span of 14 m, comprising members made from back-to-back channels. Fig. 1 shows the layout of the frames. These frames represent the common construction of cold-formed portal frames with typical values of span, heights and pitch angles as used in practice. The geometries and cross-section sizes of WF1-WF4 are summarized in Table 1. The frames are selected to cover a variety of failure modes; this will be discussed in some details in Section 2.3. The steel is considered as elastic-perfectly plastic. Sensitivity analyses suggested that the common failure modes of portal frames with cold-formed steel sections are not particularly influenced by the post-

π·z ⎞ αi (z ) = Ai ·sin ⎛ ⎝ L ⎠

(2)

in which αi represents the ith member imperfection (camber, bow, and twist), z is the coordinate along the member, L represents the length between end restraint points, and Ai is the amplitude of the ith member imperfection. For the cross-section distortional imperfection, the magnitude (δ in Fig. 3) varies along the member by:

π·z ⎞ δdis (z ) = Adis ·sin ⎛ ⎝ Ld ⎠ ⎜

⎟

(3)

in which δdis (z ) represents the distortional imperfection magnitude at location z, Ld is the buckling half-wavelength for the distortional imperfection, and Adis is the maximum amplitude (at the middle of Ld). Similarly, the variation of the local imperfection, δloc (z ) , along the member is given by

π·z ⎞ δloc (z ) = Aloc ·sin ⎛ ⎝ Ll ⎠ ⎜

⎟

(4)

in which Ll is the buckling half-wavelength for the local imperfection, and Aloc is the maximum amplitude. The nominal parameters for the four frames are given in Table 2, including the connection stiffness parameters and the initial geometric imperfections. In particular, the design (nominal) value of frame out-ofplumbness angle is 1/500 (as specified in AS4600 [3] and AISI S100-10 [4]). Design value of member out-of-straightness (camber and bow imperfections) is L/1000. Note that the nominal frame models do not

Fig. 1. Geometry of the portal frame and wind load. 165

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Table 1 Dimensions and section sizes of the frames (unit: mm).

Heave Hapex Span Column (H × W × t) Rafter (H × W × t)

WF1

WF2

WF3

WF4

4800 6000 8000 C(302 × 96 × 1.5) C(352 × 108 × 2.4)

6300 7500 8000 C(203 × 76 × 2.4) C(352 × 108 × 3.0)

7800 9900 14,000 2C(203 × 76 × 2.4) 2C(352 × 108 × 3.0)

4800 6900 14,000 2C(254 × 76 × 1.2) 2C(352 × 108 × 3.0)

H × W × t = height × width × thickness Table 2 Connection properties and initial geometric imperfections (nominal values).

Fig. 2. Moment-rotation curve of semi-rigid joints.

WF1

WF2

WF3

WF4

Eave joint M1 [kNm] M2 [kNm] K1 [kNm/rad] K2 [kNm/rad

64 76 6400 400

64 76 6400 400

76 92 7600 400

76 92 4800 300

Apex joint M1 [kNm] M2 [kNm] K1 [kNm/rad] K2 [kNm/rad

56 80 1400 1500

56 80 1400 1500

68 96 17,000 1750

68 96 10,500 1125

Column base M1 [kNm] M2 [kNm] K1 [kNm/rad] K2 [kNm/rad Out-of-plane purlin

60 100 6000 450 spaced 1 m

60 100 6000 450 spaced 0.6 m

60 100 6000 450 spaced 0.7 m

spaced 0.76 m 1/500

spaced 0.763 m 1/500

60 100 6000 450 spaced 0.7 m spaced 0.76 m 1/500

1/1000, 0.763 1/1000, 6.1

1/1000, 0.76 1/1000, 7.6

1/1000, 0.78

1/1000, 0.6 1/1000, 3.6

1/1000, 0.7 1/1000, 6.8

1/1000, 0.7 1/1000, 4.8

Out-of-plane girts Frame Out-ofplumb

Column initial geometric imperfections Bow imp., L (m) 1/1000, 0.76 Camber imp., L (m) 1/1000, 4.6 Rafter initial geometric imperfections Bow imp., L (m) 1/1000, 1 Camber imp., L (m) 1/1000, 3.6

spaced 0.78 m 1/500

1/1000, 4.6

L = length between restraint points

while the nominal structural models do not include the twist or sectional imperfections, the structural analysis models used in the reliability calibrations do incorporate the local, distortional and twist imperfections. Therefore, possible influence of random twist and sectional imperfections on system safety are reflected in the derived ϕs factor. 2.2. Advanced FE analysis models Structural modelling and analyses are carried out using the software ABAQUS [19]. Rafters and columns are modelled using shell-elements (S4R element in ABAQUS). For back-to-back cold-formed channels, the interactions between two webs are simulated by creating a TIE constraint between the webs. Four elements are typically used cross the width of each flat plate. The geometric imperfections are incorporated in the structural models directly by placing the relevant FE nodes at the defined imperfect geometry. The most critical combination of the directions of each type of geometric imperfections is determined and used for the nominal structural model. For the design of a main-wind-forceresisting frame, external wind load pressure is applied to the windward wall, leeward wall and the roof, as shown in Fig. 1. It is assumed that the gravity load has already included the self-weight of the structure. The ultimate limit states of the frames are determined from the loaddisplacement (apex drift) response curves computed by static pushover analyses. If a peak point exists in the load-displacement response, the

Fig. 3. Sectional initial geometric imperfections for single channel and double channels, (a) distortional imperfections, and (b) local imperfections.

include the twist imperfection or the sectional imperfections. Sensitivity analyses have shown that the sectional imperfection mainly affects the failure modes involving sectional buckling, with a reduction of system ultimate strength typically no greater than 3% [13]. For other failure modes involving no cross-sectional buckling, the influence of sectional imperfection is even smaller [13]. It should be noted that 166

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parameters (yield stress, elastic modulus and cross-section thickness) have been previously studied in the reliability calibrations of earlier versions of cold-formed seel design codes [21,22]. Other uncertain parameters are discussed in some details herein.

corresponding load is the ultimate load the frame can support. In the case where the load-displacement response has no clear descending part, the point where the stiffness of the curve reduces to 5% of the initial stiffness is taken as the limit point [20]. In addition, a displacement-based criterion is also imposed, i.e., the system limit point cannot exceed the point when the lateral displacement (at apex) reaches 5% of the frame height. As will be discussed subsequently, in most cases the ultimate strengths of the frames are defined by the peak points in the load-displacement responses, except for a few cases in which the displacement criterion governed the frame strength limit. Details of the nonlinear finite element models can be found elsewhere [13].

3.1. Semi-rigid joint properties Joints of cold-formed portal frames often use bolted connection systems. Although test data of cold-formed joints exist in the literature, e.g., [23–28], there are only very limited test results of nominally identical joints to estimate the stochastic variability of joint properties. Component tests of cold-formed portal frame joints were reported in [24]. The connections were formed by connecting the webs and flanges of back-to-back channels with gusset plates using bolts. Four nominally identical apex joints and two eave joints were tested, and the joint moment-rotation responses were fitted to a multi-linear curve. Based on the four apex joint tests, the COV’s of M1, M2, K1 and K2 were found to be 0.087, 0.086, 0.232 and 0.042, respectively [24]. The results suggest that the initial stiffness K1 has a larger uncertainty than the other parameters. As the statistical data for cold-formed joints are very scarce, the present paper assumes that cold-formed joints have similar uncertainties as typical hot-rolled steel joints, for which a relatively large supporting database is available, e.g. [29,30]. Typically, the COV of the initial stiffness of hot-rolled steel connections is on the order of 0.20–0.30. Adding engineering judgement, the present study assumes that K1 has a relatively large COV of 0.30, K2 has a COV of 0.15, M1 has a COV of 0.10 and M2 has a COV of 0.15. It is further assumed that the means of these parameters are equal to their nominal values. All these parameters are modelled by lognormal distributions.

2.3. Failure behaviours of the frames The four frames under given values of gravity loads are first examined to investigate their failure modes. In all analyses, the given gravity loads are first fully applied, then the wind load is applied incrementally until the failure of the system. Frame WF1 (under a gravity load of 1.2 kN/m): At a wind load of 3.04 kN/m, distortional buckling and yielding occur near the eave and base of the right column. Frame sway instability occurs when the wind load increases to 4.57 kN/m. At the ultimate limit state, spatial plastic hinges formed near the eave and the base of the right column. Frame WF2: under a constant gravity load of 2.4 kN/m, distortional buckling occurs near the eave of the right column when the applied wind load is about 5.36 kN/m. When the wind pressure is further increased to 8.74 kN/m, both columns start yielding and lateral-torsional buckling occurs in the left column. The frame is at a state of incipient collapse when the wind load reaches 10.91 kN/m. The failure mode of frame WF3 (under a gravity load of 1.8 kN/m) is excessive drift. The 5% drift criterion is reached when the wind load increases to 7.63 kN/m. For frame WF4 (under a gravity load of 2.4 kN/m), yielding starts near the base of the right column when the wind load is about 5.79 kN/ m. At a wind load of 8.40 kN/m, the eave of the right column develops distortional buckling. Frame sway instability occurs when the wind load increases to 9.81 kN/m. At that time spatial plastic hinges develop near the base-plate and eave of the right column.

3.2. Initial geometric imperfections Survey data of out-of-plumbness of cold-formed frames do not exist in the archival literature. Instead, the present study uses the database of the out-of-plumbness of general steel structures [31,32]. A lognormal random variable can describe the out-of-plumbness angle, with a zero mean and a standard deviation of 1/610 [13]. Table 4 summarizes the statistics of the amplitudes of member imperfections reported in the literature. The results are quite scattered, e.g., the mean major axis out-of-straightness varies between 1/1340 and 1/4794. The weighted average values (weighting by the number of data in each source) of the mean major and minor axes member out-ofstraightness are 1/2490 and 1/2249, respectively. These two values are significantly smaller than the typical design value of member out-ofstraightness in main stream specifications (1/1000 in AS4600 [3] and AISI S100-10 [4]). The weighted average values in the last row of Table 4 are used as the statistics of the member imperfection

3. System reliability calibration Table 3 presents the statistics of the uncertain variables, including elastic modulus, yield stress, cross-section thickness, geometric imperfections, and joint properties. For comparison, the design (nominal) values are also listed. The nominal values are used in the design stage, while the statistics are used for the probabilistic study and reliability calibration. The stochastic characteristics of some of the uncertain

Table 4 Statistics of the amplitudes of member imperfections.

Table 3 Nominal values and statistics of the uncertain parameters.

Frame out-of-plumb Section thickness t Bow imp. Camber imp. Twist (degree/m) Distortional imp Local imp E Fy M1 M2 K1 K2

Ref.

Nominal

mean

Std Dev

Dist. type

1/500 tn L/1000 L/1000 0 0 0 En Fyn M1n M2n K1n K2n

0 tn L/2250 L/2500 0.26 0.81t 0.375t En 1.1Fyn M1n M2n K1n K2n

1/610 0.05tn L/3500 L/3300 0.2 0.62t 0.34t 0.06En 0.11Fyn 0.1M1n 0.15M2n 0.3K1n 0.15K2n

Normal LogNormal Normal Normal Normal Normal Normal LogNormal LogNormal LogNormal LogNormal LogNormal LogNormal

[43] [44] [45] [46] [47] [48] [49] [50] [51] Weighted Average

Note: the subscript n represents the nominal value.

# of data

32 12 31 4 89 23 29 210 24

Camber (×L)*

Bow (×L)*

Twist ratio

Mean

Std Dev

Mean

Std Dev

Mean [o/m]

Std Dev [o/m]

1/4794 – 1/3673 1/2510 1/1564 1/4578 1/1340 1/3477 1/1427 1/2490

1/11470 – 1/3984 1/2829 1/1937 1/8403 1/3797 1/5643 1/2422 1/3829

1/1912 1/2359 1/2715 1/2462 1/2685 1/4444 1/1148 1/2242 1/2457 1/2249

1/4646 1/1842 1/4475 1/6655 1/4475 1/14190 1/4342 1/3054 1/5596 1/3716

– – 0.1312 0.1083 – – 0.3609 0.3740 – 0.2590

– – 0.0984 0.0492 – – 0.1969 0.2428 – 0.1968

* L = length between restraint points. 167

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95 were studied in [33]. It was found that Wmax has a mean of 0.90W50 and a COV of 0.35, in which W50 represents the design wind load with a 50-year return period wind speed. An Extreme Type I distribution can fit Wmax. This wind load statistic has been used extensively in the development of LRFD criteria in the United States. Note that the wind directionality factor was estimated based on a simple geometric analysis at a time when there were no data on directionality effects. Thus in [33], it was assumed that the directionality factor had a mean of 0.85 (which is equivalent to its nominal value). Recent experimental research has revealed that this assumption was conservative [34]. The mean value of the directionality factor is closer to 0.72 (which is 0.835 of its nominal value) for non-hurricane regions. Using the updated statistics of the directionality factor, the mean of Wmax in ASCE7-16 decreases from 0.9W50 to 0.75W50, with the COV unchanged [34], as shown in Table 6. The wind load statistics for AS/NZS1170.2 were examined in [35–38], and are summarized in Table 6. It is estimated that Wmax has a mean of 0.68W50 and a COV of 0.39. These values are not dissimilar to the updated American wind load statistics. In the aforementioned studies, the means of Wmax were expressed using W50 since the loading standards traditionally defined the design (nominal) wind loads using the 50-year-return period wind speeds. However, in current American and Australian wind load standards, the nominal wind speeds are defined using a high-return-period (i.e., 500–700 years) as opposed to the traditional 50-year return period to eliminate the need for the “cyclone factor” in earlier loading standards. The wind load for T-year return period, denoted by WT, is approximately related to W50 as [34]:

Table 5 Survey of local imperfection results. Ref.

Shape of cross-section

[43] Channels and hats [44] Channels [44] – [52] Channels [53] Channels [53] Channels [45] Channels [46] Channels [54] Channels [47] Channels [51] Channels [55] Channels Weighted average

# of data

41 12 45 45 11 33 31 4 24 88 24 30

Local

Distortional

Mean

StDev

Mean

StDev

0.420t 0.400t – – 0.550t – 0.480t 0.224t – 0.280 t 0.306t 0.480t 0.375t

0.720t 0.330t – – 0.220t – 0.150t 0.119t – 0.070 t 0.282t 0.170t 0.34t

– – 2.140t 0.790t – 0.680t 0.470t 0.750t 1.360t 0.260 t 0.350t 0.890t 0.81t

– – 1.290t 0.270t – 0.300t 0.170t 0.220t 0.350t 0.490 t 0.850t 0.210t 0.62t

Note: t = section thickness.

amplitudes, assuming normal distributions. In the probabilistic studies (simulations), the amplitudes of the member imperfections are sampled randomly, and then randomly assigned a direction. Eq. (2) is then used to determine the member imperfections at any location along the member length. The statistical data of the local imperfections from the literature are summarized in Table 5. Results from different sources are combined to obtain the weighted average values. It was found that the local imperfection has a mean value of 0.375t, and a standard deviation of 0.34t; the mean of the distortional imperfection is 0.81t with a standard deviation of 0.62t. These statistics are adopted in the present study.

WT = W50 [0.36 + 0.1 ln(12T )]2

In ASCE7-16 [39], the nominal wind load Wn (for Risk Category II structures) corresponds to a return period of 700 years, thus W700/ W50 = 1.6. Using W700 as the nominal wind load Wn, the mean of Wmax is expressed as 0.75W50 = 0.75W700/1.6 = 0.47Wn. In AS/NZS1170.2 [40], the design wind speed Wn is for a return period of 500 years (for common structures with Importance Level 2 and a 50-year service life). Therefore, W500/W50 = 1.5. Accordingly, the mean of Wmax is rewritten as 0.68W50 = 0.45W500 = 0.45 Wn. It can be seen that the wind loads in ASCE7-16 and AS/NZS1170.2 have similar statistics, i.e., the mean of wind load Wmax is about 0.45Wn ∼ 0.47Wn, with a COV approximately 0.35–0.40. In the present paper, the wind load Wmax is modelled as an Extreme Type 1 largest distribution. The mean and COV of Wmax are taken as 0.47Wn and 0.40, respectively. This wind load probabilistic model covers both the wind loads in ASCE7-16 and AS/NZS1170.2.

3.3. Wind load statistics Wind loading is the major influence in cold-formed structural design. It is therefore important to evaluate wind load statistics carefully. In a general form, the wind load, W, on a structure may be written as

W = c·Cp·E ·G·D·V 2

(5)

in which c is a constant, V = wind speed, Cp = pressure coefficient, E = exposure factor, G = gust factor, and D = directionality factor. In design, the wind load of interest is the maximum wind load, denoted by Wmax, corresponding to the maximum wind speed, Vmax, to occur in 50 years (typical service lifetime). All the parameters in Eq. (5) (except for the constant c) are random. The estimation of the statistics of these random parameters is a very difficult task requiring extensive supporting data. In certain cases, the Delphi questionnaire method (expert judgement) has been used to estimate the wind load statistics [33]. As more data becomes available, the wind load probabilistic models are updated [34]. Table 6 summarises the statistics for the wind pressure parameters in the American loading standards ASCE7-95 and ASCE7-16 and the Australian standard AS/NZS1170.2-11. The wind load statistics for the American loading standard ASCE7-

3.4. System reliability analysis method In general, wind loads govern the design of light gauge steel frames. Therefore, the ϕs for cold-formed portal frames is calibrated considering wind load combination. The reliability calibration of gravity loads only is not presented in this paper, but can be found elsewhere [13]. The safety of a structure can be quantified by its failure probability, Pf. Direct Monte Carlo simulation is often used for evaluating structural reliability [41]. However, this method is computationally prohibitive for the present study, as capturing the rare event of structural failure requires a considerable amount of trails (on the order of 105 or larger), each trail requiring a nonlinear shell element-based analysis. Thus, the simplified reliability analysis method introduced in [10] is used in the present study. In this method, the stochastic characteristics of the lateral resistance of the frame is first estimated using random sampling techniques, and then compared with the wind loads to approximate the failure probability. Wind-resistance design must satisfy Eq. (7), i.e.,

Table 6 Wind load statistics for ASCE7 and AS/NZS1170.2. ASCE7-95

Cp E G D Vmax/V50 Wmax/W50

ASCE7-16

AS/NZS 1170.2

Mean/ nominal

COV

Mean/ nominal

COV

Mean/ nominal

COV

0.91 0.97 0.96 1.0 1.0 0.90

0.15 0.15 0.12 0.12 0.12 0.35

0.91 0.97 0.96 0.835 1.0 0.75

0.15 0.15 0.12 0.12 0.12 0.35

0.95 0.90 0.95 0.81 1.0 0.68

0.15 0.20 0.10 0.1 0.12 0.39

(6)

ϕs Rn = 1.2Dn + Wn

(7)

where Dn = dead load, and Wn = wind load. To evaluate the system 168

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reliability, the limit state function corresponding to Eq. (7) can be expressed as:

g ( ) = Rw − Wmax

Table 7 Statistics of lateral resistance and system reliability index of WF1.

(8)

in which Rw represents the lateral capacity of the frame (simultaneously subjected to the dead load Dn). The system fails if Rw is smaller than Wmax. The core of the reliability analysis is to obtain the stochastic model for Rw. This can be achieved using the random sampling technique through the following steps: 1. Generate a frame with randomly sampled geometric, material and stiffness properties, and loaded with the dead load Dn. Note that the dead load is treated deterministically herein since its uncertainty is significantly smaller than that of wind load. 2. Carry out an advanced analysis to compute the frame’s lateral resistance Rw of the frame. 3. Repeat Steps 1 and 2 to obtain samples of the lateral resistance Rw, and compute its empirical statistics (sample mean, standard deviation, and distribution type).

Dn

ϕs

Wn

R¯ w

VR

β

1.2

0.9 0.8 0.7

3.927 3.414 2.902

4.575

0.127

2.43 2.79 3.21

0.8

0.9 0.8 0.7

4.169 3.654 3.139

4.687

0.147

2.29 2.63 3.20

0.45

0.9 0.8 0.7

4.377 3.857 3.335

4.915

0.145

2.29 2.64 2.99

R¯ w = mean of Rw, VR = COV of Rw.

Step 4. Repeat Step 2 and Step 3 for each combination of Dn and ϕs. For each Dn, a ϕs versus βs curve is plotted. The average curve for the three values of Dn is used to represent the relationship between ϕs and βs for the frame. This procedure is demonstrated using the example of WF1. Table 7 presents the values of Wn associated with the three values of Dn and three values of ϕs, i.e. a total of nine combinations of Dn and ϕs. For each pair of (Dn, Wn, ϕs), the frame is at its ultimate limit state according to the design Eq. (7). Next, a sample distribution of the lateral resistance of WF1 under the (deterministic) nominal dead load Dn is generated using 200 LHS simulations. Since the system lateral resistance is generally influenced by the gravity loads acting on the frame, the probabilistic analyses of the system lateral resistance need to be conducted for all three values of Dn. Fig. 4 plots the histograms of the lateral resistance (Rw) of WF1 under three values of Dn (1.2 kN/m, 0.8 kN/m and 0.45 kN/m). The mean values and COVs of Rw for WF1 are listed in Table 7. It suggests that the COVs of Rw under different Dn are quite similar, in a narrow range of 0.13 and 0.15. Rw can be modelled as a lognormal. With the probabilistic information of Rw and Wmax, the reliability indices of WF1 for each case of ϕs and Dn are computed using the direct Monte Carlo technique and listed in the last column of Table 7. For a prescribed target βs, the required ϕs can be directly obtained from Table 7. For example, to achieve a target βs of 2.5, ϕs needs to be 0.855 for WF1. Tables 8–10 summarize the loading information, lateral resistance models, and reliability analysis results of the other three baseline frames. It can be seen that the uncertainty of WF3’s ultimate strength is relatively low, about 0.05–0.06. Similar observation is also made in the case of WF2 under the particular dead load Dn of 3.6 kN/m. This is because the failures of WF3 and WF2 (with Dn of 3.6 kN/m) are controlled by the deformation criterion. This failure mode is less influenced by the uncertainty of material strength (Fy). Also, the randomness in material stiffness (Young’s modulus) is significantly less than the variability associated with steel yield stress. On the other hand, the failure of other frames are affected more by the randomness in both material strength and stiffness. The COVs for the ultimate strengths of these frames vary between 0.11 and 0.18, with a typical value of 0.15. Table 11 summarizes the average values of βs (at three levels of dead load) for the four frames considering three possible system resistance factors (ϕs = 0.7, 0.8 and 0.9). For a given value of ϕs, the reliability indices for the four frames are comparable. For instance, for a ϕs value of 0.90, the reliability indices of the four frames vary between 2.34 and 2.56, with an average value of 2.41. Alternatively, the required ϕs for achieving a certain reliability index can be readily obtained from Table 11. Hot-rolled steel members subject to dead and live loads have a target reliability index approximately 2.6–2.8, and about 2.5 under wind loads in the existing AISC LRFD [21,22,42]. Historically, the

Step 1 can be performed using the standard random sampling technique, or more efficient techniques such as Latin Hypercube Sampling (LHS). This study typically used 200 LHS simulations to cover the probability space of Rw. Once the statistics of Rw is obtained, the system failure probability is equivalent to evaluating the probability that Rw is smaller than Wmax. This represents the simplest form of reliability evaluation, and can be computed easily [41]. Limit state structural design has traditionally used the reliability index, β, to quantify the structural reliability. The probability of failure Pf is related to β by β = Φ−1(1 − Pf), and Φ denotes the cumulative distribution of a unit normal [22,41]. In this paper, the system reliability index is denoted by βs, with the subscript “s” to emphasise it is a “system” reliability. 4. Reliability calibration results Using the reliability analysis method presented in Section 3, the system reliability index βs of a given system can be computed, and the relation between βs and ϕs can be established. Obviously, the required value of ϕs for achieving a given target reliability index may not be the same for different frames. Even for the same frame, the required resistance factor depends on the loading scenario, notably the wind-togravity load ratio. Since wind loads are more uncertain than the dead load, the required resistance factor would decrease as the wind load becomes more dominant. However, for design purposes, a single resistance factor is desirable. To achieve this goal, the reliability calibration considers three different levels of design dead load for each of the four baseline frames. The three values of dead load represent (relatively) heavy, medium and light dead loads. Accordingly, the design wind load that the frame can support becomes more dominant as the applied dead load is reduced. Under each design dead load, three values of ϕs (0.7, 0.8 and 0.9) are considered to establish the link between ϕs and system reliability index βs. For a structural system designed according to Eq. (7), the linkage relating βs and ϕs is determined using the following four steps. Step 1. Establish a “nominal” FE structural model using the nominal values of frame properties. For a prescribed design dead load Dn and a given system resistance factor ϕs, choose the nominal wind load Wn such that the frame satisfies Eq. (7). Step 2. Perform a probabilistic analysis (simulations) to estimate the statistics of the lateral resistance Rw of the frame designed in Step 1 (see Section 3.4). Step 3. Evaluate the reliability associated with the limit state: g () = Rw − Wmax. 169

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Table 9 Statistics of lateral resistance and system reliability index of WF3. Dn

ϕs

Wn

R¯ w

VR

β

1.8

0.9 0.8 0.7

6.792 6.002 5.212

7.41

0.056

2.38 2.71 3.10

1.2

0.9 0.8 0.7

6.895 6.107 5.317

7.498

0.048

2.35 2.69 3.13

0.8

0.9 0.8 0.7

6.963 6.175 5.387

7.569

0.054

2.38 2.7 3.05

R¯ w = mean of Rw, VR = COV of Rw.

Table 10 Statistics of lateral resistance and system reliability index of WF4. Dn

ϕs

Wn

R¯ w

VR

β

2.4

0.9 0.8 0.7

8.637 7.495 6.430

11.388

0.161

2.66 2.98 3.37

1.8

0.9 0.8 0.7

8.830 7.813 6.745

11.697

0.18

2.60 2.88 3.24

1.0

0.9 0.8 0.7

9.640 8.567 7.283

11.838

0.172

2.41 2.72 3.11

R¯ w = mean of Rw, VR = COV of Rw.

Table 11 Reliability indices βs of four frames (average of three levels of Dn). ϕs

WF1

WF2

WF3

WF4

Average

0.9 0.8 0.7

2.34 2.69 3.14

2.38 2.65 2.94

2.37 2.70 3.09

2.56 2.86 3.24

2.41 2.73 3.10

target reliabilities of cold-formed members are somewhat lower than those of hot-rolled steel members, partly due to the higher live-to-dead load ratio in cold-formed steel members. Under gravity loads only, coldformed steel structural members are calibrated to fulfil a target member reliability index of approximately 2.5 [3,4]. For the Direct Design Method, if a target reliability index for frame systems can be established, the ϕs versus βs relationship in Table 11 can assist the code writers to arrive at a suitable value of ϕs. For instance, to achieve a βs value of 2.5 under wind loads, the required ϕs for WF1 to WF4 would be 0.856, 0.856, 0.861 and 0.910, respectively. The average value of ϕs is 0.87. Considering that resistance factors in design standards are customarily rounded to the nearest 0.05, a ϕs of 0.85 may be adopted. The final choices of target system reliability indices for the advanced analysis method ultimately rest with the specification committee.

Fig. 4. Histograms of lateral resistance (Rw) of WF1. Table 8 Statistics of lateral resistance and system reliability index of WF2. Dn

ϕs

Wn

R¯ w

VR

β

3.6

0.9 0.8 0.7

12.223 11.135 9.918

13.328

0.071

2.36 2.63 2.91

1.8

0.9 0.8 0.7

9.245 8.400 7.568

10.602

0.105

2.44 2.70 2.97

1.0

0.9 0.8 0.7

8.580 7.717 6.853

9.546

0.107

2.34 2.63 2.95

5. Conclusion A reliability analysis has been presented for designing cold-formed steel portal frames by the Direct Design Method using advanced nonlinear structural analysis. The system reliability calibration is based on four frames representing typical constructions of cold-formed steel portal frames, with various failure modes. Through simulation techniques, the probabilistic characteristics of the ultimate lateral strengths of the baseline frames are determined. It is found that the ultimate strength of frame WF3 has a relatively small uncertainty, with a COV of about 0.05. This is because the failure of frame WF3 is controlled by the deformation criterion, thus less influenced by the uncertainty of

R¯ w = mean of Rw, VR = COV of Rw. 170

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material strength. The limit states of other frames are defined by the peak points of the load-displacement responses, and thus affected by the randomness in both material strength and stiffness. The ultimate strengths of these frames have coefficients of variation in the range between 0.11 and 0.18, with a typical value of 0.15. The derived values of ϕs for the four frames are comparable. If assuming a system reliability index of 2.5 for wind design, the values of ϕs of the four frames vary between 0.856 and 0.910; the averaged ϕs is 0.87. This system resistance factor may be used for cold-formed portal frames with similar failure modes to those examined in the present work. The variation of system reliability level (βs) as a function of the frame resistance factor ϕs developed in this paper can help the specification committees to arrive at the suitable ϕs for the Direct Design Method.

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