Probabilistic progressive collapse analysis of steel-concrete composite floor systems

Journal of Constructional Steel Research 129 (2017) 129–140

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Journal of Constructional Steel Research

Probabilistic progressive collapse analysis of steel-concrete composite floor systems Yang Ding, Xiaoran Song, Hai-Tao Zhu ⁎ School of Civil Engineering, Tianjin University / Key Laboratory of Coast Civil Structure Safety (Tianjin University), Ministry of Education, Tianjin 300072, China

a r t i c l e

i n f o

Article history: Received 18 June 2016 Received in revised form 21 October 2016 Accepted 4 November 2016 Available online xxxx Keywords: Steel-concrete composite floor Progressive collapse Component-based connection model Probabilistic analysis Fragility curve

a b s t r a c t The paper presents a probabilistic analysis of steel-concrete composite floor against progressive collapse considering uncertainties in strength and ductility of steel connections. Using component-based connection model, an analytical framework for developing probabilistic connection models is proposed. The connection models developed are further introduced in probabilistic structure analysis against progressive collapse. Tornado diagram-based sensitivity analysis is performed to determine the influential random variables for structural resistance capacity. Using Latin Hypercube sampling of both random structure variables and external loads, random realizations of structures are generated and progressive collapse analysis is carried out using pseudo-static pushdown method. The proposed framework is applied to study the vulnerability of composite floor. Fragility curves corresponding to three limit states are developed. Discussions on the influences of steel connection and slab continuity on collapse vulnerability are given. Finally, results from the present probabilistic method are compared with those from deterministic approach. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Civil structures might be subjected to incidental accidents during their service time, such as explosion and impact. These incidents may cause damage to structure members and may even trigger the collapse of the building [1]. Although the failure probability of building structures is low, the consequence is catastrophic [2]. Therefore, progressive collapse resistant design of structural buildings is critical to the safety of structures. Steel frame building is a common prototype building for moderate and high seismic regions. Because steel connections provide integrity for structure under column loss scenario, they are recognized as the most vulnerable part of typical steel buildings [3]. Therefore, extensive experimental studies [3–5] have been carried out to assess the rotation capacity resulting from catenary effect. Different failure modes were observed including bolt shear failure, plate bearing failure etc. [6]. With the advancement of computer resources, numerical simulations are typically utilized to evaluate the rotation capacity of steel connections in design practice. Compared with detailed micromodels, component-based model is a practical and simplified way to characterize the connection behavior [7]. It has been widely used by a number of researchers [8– 10] and is proofed to be an effective methodology for capturing the axial force-moment interaction of connections resulting from catenary action effect. ⁎ Corresponding author. E-mail address: [email protected] (H.-T. Zhu).

http://dx.doi.org/10.1016/j.jcsr.2016.11.009 0143-974X/© 2016 Elsevier Ltd. All rights reserved.

Compared with researches on steel connections, studies on the robustness of steel-concrete composite floor systems are scarce. Research from Alashker [11] highlighted the 3D modeling technique and membrane effect from floor system. Although concrete slab contribute significantly to collapse resistance, computational studies from Alashker et al. [12] and Sadek et al. [13] indicated that composite floor systems with shear tab connection were susceptible to collapse under column removal scenario. Main [14] developed a simplified numerical model for steel composite floor system and proposed a new tie force demand formulation. Johnson et al. [15] conducted a 1:2 scale experimental test of a 3 × 3 bay composite floor system subjected to four difference column removal scenarios. As described above, the previous discussed progressive collapse analyses were carried out with deterministic parameters. The progressive collapse assessment procedures in current design guidelines, such as UFC 4-023-03 [16], GSA [17] and EC1-1-7 [18], are also deterministic in nature. Large load combinations are used to compensate for structural safety and unknown factors. While such method may seem to provide a simple and seemingly conservative means of treating uncertainty, it cannot provide the probabilities that structures meet the performance objectives. In reality, random variability of material strength and geometry section sizes inevitably introduce fluctuations to the collapse resistance capacity. So far, only limited studies have focused on quantifying failure probability of building structures under column removal. Park and Kim [19] conducted fragile analysis of steel moment frames. The probabilities of failure against steel frame with various moment connections were obtained using FOSM method. Xu and Ellingwood [20]

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for developing probabilistic steel connection models is presented. The connection models above are further introduced in probabilistic analysis of steel-concrete composite floor subjected to column loss. Fragility curves with respect to three limit states are obtained. Discussions on the influences of steel connections and slab continuity on structure collapse vulnerability are provided. Finally, results from probabilistic analysis are compared with those from deterministic approach.

Material

Developing probabilistic steel connection models

Geometry section

Steel structure model

Concrete floor, deck, beam, column...

Progressive collapse analysis

Pseudo-static pushdown

Damage state evaluation

Define limit state

2. Probabilistic assessment methodology

Sensitivity analysis

Tornado diagram

Influential design Variable : Capacity

External load: Demand

Fig. 1 presents the probabilistic assessment framework. As shown in Fig. 1, the framework contains two levels of uncertainty propagation. Firstly, an analytical framework for developing probabilistic steel connection models is proposed and detailed in Section 2.1. After that, the probabilistic connection model is introduced in stochastic analysis of structure systems. The flowchart of probabilistic structure analysis is presented in Section 2.2. 2.1. Development of probabilistic steel connection models

Monte Carlo simulation Develope fragility curves Fig. 1. Framework of probabilistic analysis considering uncertainties from steel connections.

developed a J-integral formulation of fracture demand model for preNorthridge steel moment connections. Le and Xue [21] proposed a multi-scale numerical model and conducted a probabilistic analysis of RC frames under column loss. Due to the uncertainties in material properties, geometry section and workmanship, ultimate strength and rotation capacity of steel connections under catenary actions inevitably fluctuate, as indicated from experimental test [22]. However, studies on quantifying variations in steel connections and probabilistic analysis of steel-concrete composite floor systems considering uncertainties of steel connections have not been found in the previous literatures. Motivated by these limitations, this paper presents a probabilistic analysis of steel-concrete composite floor system against progressive collapse considering uncertainties in the ultimate strength and ductility of steel connections. Within the framework of component-based connection, an analytical procedure

Step 1

Define component-based steel connection models

Because experimental tests on steel connections under monotonic axial loading have only begun to be carried out, experimental data are scarce. Therefore, in this paper, probabilistic connection models are developed analytically. Within the framework of component-based connection model, the Monte Carlo simulation method is adopted to propagate the uncertainties from material parameters and geometry sizes to the strength and ductility of steel connection. The procedure is presented in Fig. 2 as the following steps. (1) Set up the component-based connection model according to the type of beam-column joint analyzed [23]. (2) Generate random variables such as material strength and geometry section sizes. Because the Monte Carlo simulation method does not provide a prior estimate of the sample size at a certain confidence level, convergence tests are conducted to determine the minimum sample size required in Monte Carlo simulations [24]. (3) Calculate the axial force-displacement curve of each component according to the material properties and connection configurations generated in step (2). (4) Individual component springs at the same bolt level are assembled in series to create a single "effective spring" to capture all of the related deformations [7].

Step 2

Generate random variables

Calculate R- δ of each component Monte R Carlo Simulation

R

Step 3

δ

δ

Estimate statistics

R

Step 6

δ

Assemble R- Δ

R

R

δ

Step 4

Simplify R- δ curve

Step 5

Fig. 2. Flowchart for developing probabilistic connection models.

δ

MC results

Y. Ding et al. / Journal of Constructional Steel Research 129 (2017) 129–140

varying u0, load-displacement curve considering dynamic effects (IDA curve) FD-u is obtained.

General R- δ curve Ultimate (δ u, Ru)

R Yield (δ y, Ry)

Bilinear R- δ curve “ Equal area”

δ Fig. 3. Bilinear idealized axial force-displacement curve.

(5) Simplify the axial force-displacement behavior of the effective spring. The area under force-displacement represents energy absorption of steel connections up to fracture under monotonic loading. So energy equivalent principles are adopted to simplify the nonlinear curve into a bilinear relationship, as shown in Fig. 3. Therefore, the probabilistic connection model can be described with four parameters, namely, δy, Ry, δu and Ru. (6) Estimate the sample mean value, standard deviation and correlation coefficient matrix from the MC results. Using K-S test and Chi-square test, determine the distribution laws leading to the best fit of the random variables. The force-displacement relation of the effective spring is introduced in numerical model of composite floor systems subjected to column loss. Because fracture of connection is directly resulted from catenary effect and failure in compression force has not been observed in experimental tests [3,5], random variability in the force-displacement relation of compression zone is considered to be deterministic. Research from [25] also indicated that compression zone has a minor influence on structural collapse resistance. 2.2. Probabilistic structure analysis framework 2.2.1. Pseudo-static pushdown Probabilistic progressive collapse analysis is carried out based on nonlinear finite element analysis combined with the Monte Carlo simulations. It will be computationally intensive to combine MCS with incremental dynamic analysis (IDA) because each sample structure requires a series of nonlinear dynamic time-history analyses [8]. Therefore, an approximate pseudo-static pushdown methodology proposed by Izzuddin [26] is used in this study instead of IDA. Compared with IDA, only a single static pushdown analysis is required. The approximate IDA curves are developed as:

F D ðu0 Þ ¼

1 u0

Z

u0 0

131

F S ðuÞdu

ð1Þ

where FS(u) is the load-displacement curve obtained from static analysis; is the; u0 is the peak vertical displacement of composite floor. By

2.2.2. Limit state Table 1 shows the limit states in this study. These definitions are similar to those specified in FEMA-356 [27]. Based on the rules, the floor collapse resistance capacities FDk (k = IO, LS, CP) are traced from FD-u curves. The IO connection rotation capacity of the shear connection with slab from FEMA-356 [27] is chosen to bound the IO limit state. However, the rules for LS and CP are not chosen as the associated limit states. The criteria from FEMA are based on test results under seismic loads. The differences are small at IO while at large deformations, connection rotation capacities under monotonic loads are two or three times larger than the number suggested by FEMA [6]. Instead, the fracture of steel connection is selected as the LS indicator. The structure steps into nonlinear condition after connection fracture, but capacity is not fully exhausted [29]. With the increasing of external load, composite floor undergoes large vertical deformation. The rules for CP limit state are based on the study in [28]. At this level of deformation, the floor is intended to collapse with the fracture of steel deck [28]. This criterion was also utilized in UFC 4-023-03 [16] to derive the required tie strength. 2.2.3. Sensitivity analysis methodology The variability of the structural collapse resistance capacity due to the randomness of design variables is evaluated using Tornado diagram analysis (TDA) method. The TDA method begins with a deterministic analysis with all of the random variables set to their nominal values [24]. Pseudo-static pushdown analysis is carried out, and structural collapse resistance capacities corresponding to three performance levels are traced from load-displacement curves. Next, each random variable is set to the upper and lower bound, and deterministic analysis is conducted twice with two extreme values, while all of the other random variables are set to their nominal values. In this study, the lower and upper bound are set to the mean ± standard derivation of random variables. The absolute difference between structure resistances can be referred to as a measure of sensitivity. Finally, random variables are further ranked in descending order based on the absolute of their difference values. The most influential random variables to structural capacities are determined. 2.2.4. Development of fragility curve A fragility curve is a function that defines the probability of a structure reaching or exceeding a certain limit state for a given intensive measure. If capacity and demand are expressed in terms of intensity measure, the fragility curve can be simplified into cumulative distribution function of the IM-value capacity [30]. In this study, the overload factor α is chosen as IM, which is defined as the ratio of structure capacity to gravity load acting on the floor systems [8]. The fragility curves can be expressed as lognormal distributions [30]:   ln ðα=μ k Þ ðk ¼ IO; LS; CPÞ P k ðLS≥lsjIM ¼ α Þ ¼ Φ βk

ð2Þ

Table 1 Limit states and performance levels of composite floor under column loss. Damage level

Description

No damage

Intact

Insignificant damage Moderate damage Collapse

Fully-operational, reparable Non-operational, reparable Non-reparable

dbg: depth of the connection bolt group (unit in inch); Lspan: span length.

← ← ←

Limit state

Quantitative measure

Reference

Immediate occupancy Life safety Collapse prevention

u N (0.014–1.0e−4dbg)Lspan Connection failure u N 0.20Lspan

[27]

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Y. Ding et al. / Journal of Constructional Steel Research 129 (2017) 129–140

Fig. 4. Layout of the steel composite floor and details of three connections. (a) Layout of the composite floor system [14]. (b) Details of ST connection [31]. (c) Details of WB connection. (d) Details of EP connection.

where P k (LS ≥ ls | IM = α) is the probability of the structure exceeding limit state k for a given IM = α; Ф(.) is the standard Gaussian cumulative distribution function; μ k and β k are median value and lognormal dispersions of the fragility curve, respectively. Based on Monte Carlo simulation results, μ k and βk

are determined by the point estimate method [30]:

μk ¼

N 1X ln α k;i N i¼1

ð3aÞ

Y. Ding et al. / Journal of Constructional Steel Research 129 (2017) 129–140

FPBR: fin plate under bearing BS: bolt under shear BWBR: beam web bearing CWC: column web under compression

ANB: angle-bolt bending BT: bolt under tension ABBR: angle under bearing CFB: column flange bending

133

WT: weld under tension EPB: endplate bending EFFS:” effective” spring

FPBR BS BWBR

ANB BS BWBR BT ABBR

CFB

BT

WT

EPB

EFFS

FPBR BS BWBR

ANB BS BWBR BT ABBR

CFB

BT

WT

EPB

EFFS

FPBR BS BWBR

ANB BS BWBR BT ABBR

CFB

BT

WT

EPB

EFFS

CW C

CW C

(a) ST

CW C

(b) WB

(c) EP

CWC

(d) Effective spring

Fig. 5. Component-based models of three connections. (a) ST (b) WB (c) EP (d) Effective spring

βk ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX  2 u ln α k;i −μ k u t i¼1 N−1

ð3bÞ

where N is the sample size in stochastic finite element simulations and αk , i = FDk , i/(DLi + LLi) represents the IM value of the ith sample structure at limit state k; DL i and LL i are random dead load and live load acting on the floor systems, respectively. 3. Case study 3.1. Prototype structure building The proposed probabilistic evaluation framework is applied to typical steel-concrete composite floor systems designed by NIST [31]. Fig. 4 presents the layout of the floor systems. The 4 × 4 bay floor system is also considered to study the influence of floor continuity on collapse vulnerability. The 4 × 4 bay floor system represents potion of the gravity floor system of two prototype 10-story buildings [31]. Since this study focuses on collapse risk of gravity frames, no steel moment frames are considered. The composite floor system is single-story and the steel columns extend half-story height above and below the floor. The composite floor is composed of 83 mm concrete slabs sitting on 76 mm deep steel deck with a thickness of 0.91 mm. W1.4 × 1.4 is utilized as anticrack reinforcement with the cross section area of 9.0 mm2/m, and the grid space of welded-reinforcement is 6 in × 6 in. ASTM A992 steel is used in all of the steel beams and columns. The design dead load acting on the floor consists of the 2.2 kN/m2 self-weight and an additional

1.44 kN/m2 dead load. The design live load is assumed to be 2.4 kN/m2 [31]. Further design details of the prototype structure can be found in [31]. In this study, steel columns, beams and girders are connected with simple connections. Typical simple connections include shear tab (ST) connections, web cleat (WB) connections and partial endplate (EP) connections [3]. Fig. 4(b) shows the details of the ST connection designed by NIST [31]. To study the influence of steel connections on collapse resistance capacity, additional WB and EP connections, are designed according to the AISC specification [32]. The connection configurations of WB and EP are similar to those tested in [3]. Load combination 1.2DL + 1.6LL is used to check the required shear resistance of the simple connections. 3.2. Steel connection model 3.2.1. Component-based connection model Fig. 5 shows the component-based connection models of three simple connections. Components in series are assembled into a single “effective spring” to represent the component deformation at the same level. A compression-only component (CWC) is added to represent the possible contact behavior between beam flange and column flange. Because no shear failure has been observed in the experiment [3], a rigid shear spring is used to transfer shear forces. Analytical equations from EC3-1-8 [23] and previous studies are used to determine the force-displacement relation of each component according to the connection configuration and material parameters. Table 2 lists mechanical model of each component.

Table 2 Summary of component mechanical behaviors. Component

Scheme of component

Reference

Bolt bearing on plate (FPBR, BWBR, ABBR)

[33]

T-stub (CFB & EPB)

[34]

Bolt in shear (BS)

[35]

Weld in tension (WT)

[36]

Angle bolt bending (ANB)

[37,38]

Bolt tension (BT)

[23]

Column web compression CWC

[23]

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Table 3 Statistical quantities for probabilistic connection. Source of uncertainty

Nominal COV Distribution Reference value

Diameter of the bolt, thickness of the endplate, fin plate and angle Effective welded throat Yield strength of A36 steel (Mpa) Ultimate strength of A36 steel (Mpa) Yield strength of A992 steel (Mpa) Ultimate strength of A992 steel (Mpa) Ultimate failure elongation of steel material Ultimate strength of bolt (Mpa) Elastic modulus of steel material (Mpa)

–

5%

Lognormal

[39]

– 248 400 344 450 0.3

15% 11% 5.1% 11% 5.1% 10%

Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal

[39] [40] [40] [40] [40] [41]

827 2.06e5

7% 3%

Lognormal Lognormal

[39] [39]

3.2.2. Random variables for probabilistic connection model Table 3 lists basic statistical quantities of random variables used to develop probabilistic steel connection models. It should be mentioned that geometry section sizes of steel structures are equal to the mean values with a COV of 5%, except for the welded throat for which COV is 15%, according to the studies by Ellingwood [39]. The larger variation in sizes of welded throat is attributed to the possible welded deficiency. Based on the studies above, the COV for the thickness of the shear tab, angle and endplate and the diameter of the bolt is taken as 5%, while 15% is taken for the sizes of welded throat. 3.3. Composite floor systems 3.3.1. Numerical model of composite floor Commercial finite element software LS-DYNA [42] is used in numerical analysis. Connections are modeled using component-based connection model as illustrated in Section 3.2 to capture axial force-moment interaction effect. Composite floor is modeled as a multi-layer shell element in the numerical simulations. The material model MAT_EC2 (MAT_172) [42] is used for concrete simulation. With this material model, concrete cracking under tension and crush failure under compression can be modeled. Alternative thicknesses are assigned to shell elements representing strong and weak strips of the concrete slab. Weld reinforcements in the slab are modeled as a single layer of shell element with the corresponding mechanical behavior. Steel deck is represented by beam element with the corresponding material behaviors [11]. The equivalent beam elements share the same nodes with shell element representing concrete slab. Rigid links are set up to model the composite behavior. The upper and lower ends of columns are modeled as pinned representing inflection point in steel column. The material model MAT_PIECEWISE_LINEAR_PLASTICITY (MAT_24) [42] is used for all of the steel components. 3.3.2. Random variables for probabilistic structure analysis In addition to uncertainties from strength and ductility of steel connections, material uncertainties associated with concrete slab, steel deck and welded reinforcements shown in Table 4 are also considered in sensitivity analysis. Based on the results from TDA, the influential random variables respect to structural capacity is taken into account in

generating random sample structures and developing fragility curves. Because the IM is defined as the ratio of structural capacity to demand, load randomness (variation of demand) should be considered. The external load consists of dead load (DL) and live load (LL). According to studies from Ellingwood [39], DL is modeled as normal distribution with normal to mean value of 1.05 and COV of 10%. LL is modeled as extreme values-I distribution with normal to mean value of 0.30 and COV of 60%.

4. Results and discussion 4.1. Results of probabilistic connection model 4.1.1. Monte Carlo convergence test The MC sample size is first set to 10,000 in this study. Fig. 6 presents the convergence test results of the mean and standard deviation of four axial force-displacement parameters of ST connections. The vertical axis from Fig. 6(a)–(b) is normalized by the mean and standard deviation from simulation with a sample size of 10,000. Fig. 6(c)–(d) shows the coefficient of variability (COV) of the associated parameters in Fig. 6(a)–(b), respectively. The minimum sample size is determined with a predefined error tolerance, namely 5% of COV. Convergence tests of WB and EP connection are also conducted. It is found that a sample size of 10,000 is large enough to satisfy the predefined error tolerance.

4.1.2. Statistic of probabilistic connection models Table 5 lists statistic properties of the axial force-displacement parameters for the three connections analyzed. Comparison studies on the mean value of Ru and δu indicate that WB and EP connections provide superior strength and ductility over ST connection. On average, WB and EP connections have ductility 3.0 and 2.1 times that of ST connections and have an ultimate strength 1.5 and 1.4 times that of ST connections. In addition to mean values, the standard deviation and COV are also provided, representing the evitable fluctuation of connection model parameters. Generally, the COV of parameters associated with yield point is smaller than those with ultimate point and the COV of ductility related parameters are larger than those of strength related ones. It is also observed that parameters of axial-force displacement relations from different connections share different COV values. The COV of δu for ST, WB and EP are 8.0%, 15.7% and 37.8%, respectively. Because an effective spring can be regarded as a series system, the ultimate strength and ductility of steel connections rely on the weakest component. EP connection with nominal parameters has high ductility because ductile components control the failure load, as shown in Fig. 7. However, considering the randomness of design parameters, failure of the connections may be dominated by brittle component. As a consequence, the overall behavior of connection is brittle with limited ductility. The variations of design parameters influence the force-displacement of each component. Furthermore, they also have an impact on the dominant failure component. If a steel connection consists of both ductile and brittle components, the randomness of variables results in a high probability that the failure mode is controlled by the brittle component with limited ductility.

Table 4 Random variables considered in sensitivity analysis. Source of uncertainty

Random variable

Nominal value

COV

Distribution

Reference

Concrete compressive strength (Mpa) Yield strength of steel deck (Mpa) Ultimate strength of steel deck (Mpa) Fracture strain of steel deck (Mpa) Yield strength of welded reinforcement (Mpa) Ultimate strength of welded reinforcement (Mpa) Fracture strain of welded reinforcements

fc fyd fud εud fyw fuw εuw

20.7 230 310 0.18 414 620 0.07

10% 10% 8% 10% 5% 5% 15%

Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal

[43] [39] [39] [39,44] [43] [43] [43]

Y. Ding et al. / Journal of Constructional Steel Research 129 (2017) 129–140

135

Fig. 6. Convergence test results of Monte Carlo simulation for ST connection. (a) Normalized mean value. (b) Normalized standard deviation. (c) COV of the mean value. (d) COV of the standard deviation.

4.1.3. Non-parameter test Results from the K-S test and Chi-square test indicate the lognormal distribution lead to a better fit of the sample. Histogram plot and lognormal distribution fit of each parameter for WB connection are presented in Fig. 8. It should be mentioned that all of the parameters are truncated at zero so that random variables are physically reasonable.

4.2. Results of probabilistic composite floor analyses 4.2.1. Sensitivity analysis Fig. 9 presents sensitivity analysis results for 4 × 4 bay structure with WB subjected to column D4 removal. The collapse resistance capacity of the composite floor systems is defined by the vertical distributed load. TDA results from other column loss scenarios are similar. Generally, fc has great impact on collapse resistance capacity across all of the limit

states. At small deformations, the structural capacity to resist external load stems from the flexural strength of concrete slabs. At large deformations, the concrete slab provides a compressive ring to equilibrate the in-plane tensile forces resulting from membrane action. In addition to fc, parameters related to steel connections and steel decks also influence the structural capacity, but the impact varies across different limit states. (1) IO limit state. As shown in Fig. 9(a), Ry and δy have a large impact on the collapse resistance. Catenary action has not been activated and collapse resistance is determined by connection parameters at small displacement. (2) LS limit state. It can be observed that the collapse resistance is

Table 5 Statistic properties of parameters for ST, WB and EP connections. TYPE Parameter Statistic values

ST

WB

EP

δy (mm) Ry (kN) δu (mm) Ru (kN) δy (mm) Ry (kN) δu (mm) Ru (kN) δy (mm) Ry (kN) δu (mm) Ru (kN)

Correlation matrix

Mean

Std

COV (%) δy

0.691 80.536 13.180 109.390 1.352 31.031 27.392 183.002 0.812 82.358 17.919 152.154

0.047 6.639 1.059 9.673 0.147 5.702 4.290 24.308 0.122 12.069 6.782 25.035

6.854 8.243 8.034 8.843 10.879 18.376 15.662 13.283 14.972 14.654 37.846 16.454

1.000 0.343 0.541 0.596 1.000 0.226 0.502 0.414 1.000 0.597 −0.822 −0.603

Ry

δu

1.000 0.640 0.917

1.000 0.606 1.000

1.000 0.093 0.903

1.000 0.392 1.000

Ru

1.000 −0.497 1.000 0.213 0.668 1.000

Fig. 7. Influence of fluctuation on the ultimate strength and ductility of steel connection. (a) Deterministic (b) Probabilistic

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Y. Ding et al. / Journal of Constructional Steel Research 129 (2017) 129–140

Fig. 8. Histograms of probabilistic connection models: WB connection.

more sensitive to the random variables δu and Ru. This result is expected because the definition of the LS limit state is directly related to connection failure. It is also observed that random variables related to steel deck and welded reinforcements have limited impact on collapse resistance at LS level. (3) CP limit state. As shown in Fig. 9(c), δu and Ru are the influential variables for composite floor in addition to fc. At large deformations, the majority of collapse resistance comes from steel deck so fyd is also sensitivity parameter. It is also observed that structural capacities are not sensitive to fys, fud, fus, εus and εud. So the five parameters are regarded as deterministic parameters in developing fragility curves. It should be mentioned that the composite floors in this study are lightly reinforced which does not meet the tie force requirement in UFC 4-023-03. Structures reinforced with thicker steel deck and more reinforcements might be more sensitive to the related design parameters.

4.2.2. Fragility curve Fig. 10 shows the normalized IDA curves from Monte Carlo simulations. Collapse resistance capacities corresponding to IO, LS and CP limit states traced from IDA curves, are also illustrated in Fig. 10. For brevity, only results of 4 × 4 bay structure with D4 column loss are presented. IDA curves are further summarized to produce the 50%, 16% and 84% fractile of the normalized collapse resistance. Based on the IDA results above, median values and dispersions of fragility curves are determined and presented in Table 6. To investigate the influence of steel connection on structure vulnerability, Fig. 11 presents the fragility curves for 4 × 4 bay floor system subjected to column D4 loss. Because the median values μ reflect structural capacities on

average, it can be shown that the influences of the connection strength and ductility on collapse resistance vary across different limit states. (1) At the IO limit state, the influence of the connection has only an insignificant effect on structure vulnerability. WB connection has the smallest Ry and δy, so composite floor with WB has the smallest collapse resistant capacity. (2) At the LS limit state, because WB and EP connection can sustain larger rotation capacities, composite floor with WB and EP connections are less vulnerable to connection failure. The median values are 1.43 and 1.26 times that with ST on average. (3) At the CP limit state, collapse resistance of composite floor with WB and EP connections are 1.23 and 1.20 times that with ST connections. The ultimate strength of EP connection increases by 50.2% compared with ST connection while only 20% increase in collapse resistance can be observed at CP level. Therefore, connection strength becomes less influential at large deformations. To investigate the influence of slab continuity on collapse resistance capacity, Fig. 12 shows the fragility curves for 2 × 2 bay and 4 × 4 bay composite floors with ST connections. The contributions of slab continuity effect are more profound at large deformations. For CP limit state, the slab continuity effect increases the median value μ by 15.4% for 4 × 4 bay floor systems under loss of D4 column and by 37.5% for 4 × 4 bay floor systems under loss of C3 column loss compared with 2 × 2 bay floor system. While median values μ reflect structure capacities on average, dispersions β from fragility curves represent randomness of the collapse resistance capacity. As shown in Table 6, dispersions from the LS limit state are the largest among the three limit states. This might not be the traditional case because dispersions usually increase with the damage intensity. As mentioned above, LS limit state is directly related to connection failure and parameters related to connection failure are modeled as random variables. So a higher scatter of LS capacity is

Fig. 9. Sensitivity of collapse resistance capacity: 4 × 4 bay floor with WB connection. (Note: μv and σv represent mean and standard derivation of design variable, respectively.) (a) IO (b) LS (c) CP

Y. Ding et al. / Journal of Constructional Steel Research 129 (2017) 129–140

137

Fig. 10. IDA curves of 4 × 4 bay system under column D4 removal sceneries. (a) IDA curves: ST connection. (b) 16%, 50% and 84% fractile of IDA. (c) IDA curves: WB connection. (d) 16%, 50% and 84% fractile of IDA. (e) IDA curves: EP connection. (f) 16%, 50% and 84% fractile of IDA.

expected. It also can be observed that variations of structure capacity from composite floor with EP and WB connections are larger than those with ST connection. Although connection strength has limited influence on collapse resistance at large deformation, the uncertainties in strength and ductility of connections dominate the random variations of collapse resistance.

Table 6 Median values and dispersions of fragility curves. Column loss scenarios

2 × 2 bay

4 × 4 bay: D4 column loss 4 × 4 bay: C3 column loss

Limit state

ST μ

β

μ

WB β

μ

EP β

IO LS CP IO LS CP IO LS CP

0.372 0.647 0.934 0.444 0.783 1.077 0.528 0.971 1.283

0.131 0.139 0.131 0.132 0.138 0.129 0.132 0.138 0.129

0.283 0.986 1.219 0.332 1.120 1.332 0.378 1.276 1.538

0.139 0.151 0.139 0.139 0.147 0.135 0.141 0.148 0.136

0.393 0.859 1.151 0.464 0.982 1.292 0.552 1.156 1.466

0.135 0.238 0.164 0.135 0.212 0.152 0.136 0.196 0.148

4.2.3. Discussion: compare with deterministic analysis The results from probabilistic analysis are compared with those using deterministic parameters according to the design guidelines. Neglecting variations of parameters, structures robustness αdet is measured using the ratio of capacity to demand [8]. αdet larger than 1.0 implies that the potential of progressive collapse is low. Table 7 lists load combinations adopted by different progressive collapse design guidelines. Because structure capacity is evaluated using deterministic parameters, αdet depends on load combinations. Using probabilistic results, the reliability levels for αdet according to the guidelines are calculated and presented in Table 8. The reliability level is expressed as probability of nonexceedance. The overload factor αpro corresponding to reliability level 0.50 and 0.95 are also determined based on probabilistic results. Fig. 13 shows the probability distribution of α from 2 × 2 bay system with ST connection corresponding to CP limit state. Deterministic results αdet according to different guidelines are also presented in Fig. 13. As shown in Table 8, the load combination from EC1-1-7 [18], GSA [17] and UFC 4-023-03 [16] results in reliability levels of 0.51–0.66, 0.25–0.38 and 0.91–0.98, respectively. The higher reliability level from UFC 4-023-03 is expected because the largest load factors are used in

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Fig. 11. Fragility curves of 4 × 4 bay floor system under column D4 removal scenarios. (a) IO (b) LS (c) CP

Fig. 12. Fragility curves of composite floor with ST connection under different column loss. (a) IO (b) LS (c) CP

5. Conclusion Table 7 Load combinations from different design guidelines. Guideline

Load combination

EC1-1-7

1.05 (DL + LL / 3 + W / 3) 1.0DL + 0.25LL 1.2DL + 0.5LL + 0.2W

GSA-2003 UFC 4-023-03

This paper presents a probabilistic analysis of steel composite floor system considering uncertainties in the ultimate strength and ductility of steel connections. An analytical framework for developing probability connection models is proposed. The connection models developed are further introduced into probabilistic analysis of steel composite floor against progressive collapse. The following conclusions can be drawn from this study:

DL: dead load; LL: live load; W: wind load.

the progressive collapse assessment. Take the 4 × 4 bay composite floor with ST connection subjected to loss of D4 column as an example. According to the UFC code, the overload factor of CP limit states is 0.85, which means the potential of structure reaching CP limit state is high. However, the reliability levels with respect to αdet for UFC code is 0.975 on the conservative side, which means that there is 97.5% certainty that the overload factors of structure are larger than this number. As shown, the 50% percentile overload factor is 1.069. The results indicate that load combinations used in UFC code are conservative so that the robustness αdet are far away from the median value of stochastic simulation. On the contrary, robustness assessment using GSA guideline may lead to non-conservative design. In most of the cases analyzed, probability of nonexceedance levels range between 0.25–0.38.

(1) Steel connections composed of brittle components present larger variation. Connection design using deterministic parameters neglects possible fluctuations of strength and ductility of steel connections. (2) The compressive strength of concrete, yield strength of steel deck, and ultimate strength and ductility of steel connections are the most influential random variables on collapse resistance capacities based on the sensitivity analysis results. (3) Median values from fragility curves suggest that steel connections that can develop catenary actions after large rotations improve the collapse resistance capacity of composite floor system. However, these impacts become less effective at larger deformations. Variations of composite floor collapse resistance capacity are largely dominated by fluctuations resulting from

Table 8 Comparison between probabilistic and deterministic analysis: CP limit state. Column loss scenarios

2 × 2 bay

4 × 4 bay: D4 column loss

4 × 4 bay: C3 column loss

Connection type

ST WB EP ST WB EP ST WB EP

αdet

αpro

Reliability level

50%

95%

UFC

EC1

GSA

UFC

EC1

GSA

0.924 1.214 1.140 1.069 1.326 1.282 1.274 1.531 1.453

0.767 0.980 0.898 0.889 1.078 1.025 1.059 1.246 1.170

0.746 1.013 0.927 0.852 1.085 1.029 1.011 1.248 1.159

0.891 1.209 1.107 1.017 1.296 1.229 1.208 1.490 1.385

0.980 1.330 1.217 1.118 1.425 1.351 1.328 1.638 1.522

0.968 0.917 0.919 0.973 0.944 0.948 0.975 0.949 0.958

0.611 0.511 0.568 0.651 0.569 0.605 0.658 0.583 0.629

0.331 0.251 0.346 0.358 0.294 0.367 0.372 0.310 0.375

Y. Ding et al. / Journal of Constructional Steel Research 129 (2017) 129–140

Fig. 13. Comparison between deterministic and probabilistic analysis: 2 × 2 bay floor with ST connection.

steel connections. (4) Result from deterministic progressive collapse analysis with large load combinations does introduce a safety margin. Robustness evaluation results according to UFC 4-023-03 are rather conservative with probability of nonexceedance ranging from 0.91 to 0.98. On contrary, using load combinations from GSA may lead to a non-conservative design with the probability of nonexceedance ranging from 0.25 to 0.38.

Results from probabilistic analysis presented in this study highlight the importance of modeling uncertainties in steel connections and its impact on collapse resistance capacity. Fragile analysis provides the probabilities of structures reaching certain performance levels and can be further extended to assess the cost and benefit of strengthening measures against progressive collapse. It should be noted that the incidental accidents also will play a major role in the failure of structures. In this paper, we do not consider the actual extra loads leading to the collapse of the floor system. The study simply removes a column from the floor system to study the failure probability given some local damages. The removed column can be caused by some incidental accidents, such as impact and blast. The randomness feature of incidental accidents that lead to the local damage is not considered herein. The results in this study combined with the studies on the annual risk of incidental accidents and failure risk of structural components lead to a comprehensive framework in predicting the collapse risk of building under incidental accidents.

Acknowledgement The authors wish to acknowledge the financial supports from the National Key Research and Development Program of China (grant no. 2016YFC0701105, 2012BAJ07B05), the National Natural Science Foundation of China (grant no. 51238007, 51178306) and the Natural Science Foundation of Tianjin, China (grant no. 13JCZDJC35200). The second author gratefully acknowledges the selfish help from Prof. Sherif El-Tawil and Dr. Hong-hao Li (University of Michigan, Ann Arbor) in carrying out this study.

Reference [1] S. El-Tawil, H. Li, S. Kunnath, Computational simulation of gravity-induced progressive collapse of steel-frame buildings: current trends and future research needs, J. Struct. Eng. 140 (8) (2013), A2513001.

139

[2] M. Grant, M.G. Stewart, Probabilistic risk assessment for improvised explosive device attacks that cause significant building damage, J. Perform. Constr. Facil. 29 (5) (2014), B4014009. [3] B. Yang, K.H. Tan, Experimental tests of different types of bolted steel beam– column joints under a central-column-removal scenario, Eng. Struct. 54 (2013) 112–130. [4] S.A. Oosterhof, R.G. Driver, Behavior of steel shear connections under column-removal demands, J. Struct. Eng. 141 (4) (2014) 04014126. [5] J.M. Weigand, J.W. Berman, Integrity of steel single plate shear connections subjected to simulated column removal, J. Struct. Eng. 140 (5) (2013) 04013114. [6] B. Yang, K.H. Tan, Numerical analyses of steel beam–column joints subjected to catenary action, J. Constr. Steel Res. 70 (2012) 1–11. [7] F. Block, J. Davison, I. Burgess, R. Plank, Principles of a component-based connection element for the analysis of steel frames in fire, Eng. Struct. 49 (2013) 1059–1067. [8] K. Khandelwal, S. El-Tawil, Pushdown resistance as a measure of robustness in progressive collapse analysis, Eng. Struct. 33 (9) (2011) 2653–2661. [9] B. Yang, K.H. Tan, G. Xiong, Behaviour of composite beam–column joints under a middle-column-removal scenario: component-based modelling, J. Constr. Steel Res. 104 (2015) 137–154. [10] C. Liu, K.H. Tan, T.C. Fung, Dynamic behaviour of web cleat connections subjected to sudden column removal scenario, J. Constr. Steel Res. 86 (2013) 92–106. [11] Y. Alashker, H. Li, S. El-Tawil, Approximations in progressive collapse modeling, J. Struct. Eng. 137 (9) (2011) 914–924. [12] Y. Alashker, S. El-Tawil, F. Sadek, Progressive collapse resistance of steel-concrete composite floors, J. Struct. Eng. 136 (10) (2010) 1187–1196. [13] F. Sadek, S. El-Tawil, H. Lew, Robustness of composite floor systems with shear connections: Modeling, simulation, and evaluation, J. Struct. Eng. 134 (11) (2008) 1717–1725. [14] J.A. Main, Composite floor systems under column loss: collapse resistance and tie force requirements, J. Struct. Eng. 140 (2014), A4014003. [15] E.S. Johnson, J.E. Meissner, L.A. Fahnestock, Experimental behavior of a half-scale steel concrete composite floor system subjected to column removal scenarios, J. Struct. Eng. 142 (2015) 04015133. [16] Department of defense (DOD), Design of Buildings to Resist Progressive Collapse UFC-4-023-03, Department of Defense, Washington, DC, 2009. [17] General Services Administration (GSA), Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernization Projects, General Services Administration, Washington, DC, 2003. [18] European Committee for Standardization, Eurocode 1: Actions on Structures - Part 1–7: Accidental Actions, BS EN 1991-1-7:2006, British Standards Institution, UK, 2006. [19] J. Park, J. Kim, Fragility analysis of steel moment frames with various seismic connections subjected to sudden loss of a column, Eng. Struct. 32 (6) (2010) 1547–1555. [20] G. Xu, B.R. Ellingwood, Probabilistic robustness assessment of pre-Northridge steel moment resisting frames, J. Struct. Eng. 137 (9) (2011) 925–934. [21] J.-L. Le, B. Xue, Probabilistic analysis of reinforced concrete frame structures against progressive collapse, Eng. Struct. 76 (2014) 313–323. [22] S.L. Thompson, Axial, Shear and Moment Interaction of Single Plate “Shear Tab” Connections.Master's Thesis, Milwaukee School of Engineering, Milwaukee (US), 2009. [23] European Committee for Standardization, Eurocode 3: Design of Steel StructuresPart 1–8: Design of Joints, BS EN 1993-1-8: 2005, British Standards Institution, UK, 2005. [24] T.H. Lee, K. Mosalam, Probabilistic seismic evaluation of reinforced concrete structural components and systems, PEER Technical Report 2006/04, University of California, Berkeley, US, 2006. [25] K. Khandelwal, S. El-Tawil, F. Sadek, Progressive collapse analysis of seismically designed steel braced frames, J. Constr. Steel Res. 65 (3) (2009) 699–708. [26] B. Izzuddin, A. Vlassis, A. Elghazouli, D. Nethercot, Progressive collapse of multi-storey buildings due to sudden column loss—part I: simplified assessment framework, Eng. Struct. 30 (5) (2008) 1308–1318. [27] FEMA-356, Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Federal Emergency Management Agency, Washington, DC, 2000. [28] Y. Alashker, S. El-Tawil, A design-oriented model for the collapse resistance of composite floors subjected to column loss, J. Constr. Steel Res. 67 (1) (2011) 84–92. [29] H. Li, S. El-Tawil, Three-dimensional effects and collapse resistance mechanisms in steel frame buildings, J. Struct. Eng. 140 (8) (2013), A4014017. [30] D. Vamvatsikos, C.A. Cornell, Incremental dynamic analysis, Earthq. Eng. Struct. Dyn. 31 (3) (2002) 491–514. [31] X. Liang, Q. Shen, S. Ghosh, Assessing Ability of Seismic Structural System to Withstand Progressive Collapse: Seismic Design and Progressive Collapse Analysis of Steel Frame Buildings, National Institute of Standards and Technology, Gaithersburg, MD, 2007. [32] American Institute of Steel Construction, Load and Resistance Factor Design Specification for Structural Steel Buildings (LRFD), 1999 (Chicago, IL). [33] C.O. Rex, W.S. Easterling, Behavior and modeling of a bolt bearing on a single plate, J. Struct. Eng. 129 (6) (2003) 792–800. [34] S. Spyrou, J. Davison, I. Burgess, R. Plank, Experimental and analytical investigation of the ‘tension zone’ components within a steel joint at elevated temperatures, J. Constr. Steel Res. 60 (6) (2004) 867–896. [35] M. Sarraj, The Behaviour of Steel Fin Plate Connections in Fire, (Ph.D. thesis), The University of Sheffield, Sheffield, UK, 2007.

140

Y. Ding et al. / Journal of Constructional Steel Research 129 (2017) 129–140

[36] Y. Hu, B. Davison, I. Burgess, R. Plank, Component modelling of flexible end-plate connections in fire, Int. J. Steel Struct. 9 (1) (2009) 1–15. [37] J. Shen, A. Astaneh-Asl, Hysteresis model of bolted-angle connections, J. Constr. Steel Res. 54 (3) (2000) 317–343. [38] B. Yang, K.H. Tan, Robustness of bolted-angle connections against progressive collapse: mechanical modelling of bolted-angle connections under tension, Eng. Struct. 57 (2013) 153–168. [39] B. Ellingwood, T.V. Galambus, J.G. MacGregor, C.A. Cornell, Development of a Probability Based Load Criteria for American National Standard A58: Building Code Requirements for Minimum Design Loads in Buildings and Other Structures, National Bureau of Standards, Washington DC, 1980. [40] F.M. Bartlett, R.J. Dexter, M.D. Graeser, J.J. Jelinek, B.J. Schmidt, T.V. Galambos, Updating standard shape material properties database for design and reliability, Eng. J.-Am. Inst. Steel Construction 40 (2003) 2–14.

[41] J. Melcher, Z. Kala, M. Holický, M. Fajkus, L. Rozlívka, Design characteristics of structural steels based on statistical analysis of metallurgical products, J. Constr. Steel Res. 60 (3) (2004) 795–808. [42] J. Hallquist, LS-DYNA Keyword User's Manual (Version 971), Livermore Software Technology Corporation, Livermore (CA), 2007. [43] D.F. Wiśniewski, P.J. Cruz, A.A.R. Henriques, R.A. Simões, Probabilistic models for mechanical properties of concrete, reinforcing steel and pre-stressing steel, Struct. Infrastruct. Eng. 8 (2) (2012) 111–123. [44] L.S. da Silva, C. Rebelo, D. Nethercot, L. Marques, R. Simões, P.V. Real, Statistical evaluation of the lateral–torsional buckling resistance of steel I-beams, part 2: variability of steel properties, J. Constr. Steel Res. 65 (4) (2009) 832–849.