Collapse fragility assessment of steel roof framings with force limiting devices under transient wind loading

Abstract

Steel structural frame is a popular structural form to cover large-span roof space and under high winds. Either part of the roof enclosure or the entire roof structure can be lifted off a building, particularly for low sloped roofs subject to wind-induced suction force. Collapse of roof could cause severe economic loss and poses safety risk to residents in the building. The buckling of members in a steel roof frame structure, which may lead to progressive collapse, may be dynamic in nature. This paper presents a fragility analysis of the collapse of steel roof frame structures under combined static and transient wind loading. Uncertainties associated with wind load change rate and member imperfections are taken into account in this study. A numerical example based on a Steel Joist Institute (SJI) K series joist was used to demonstrate the use of force limiting devices for collapse risk mitigation. For the presented fragility assessment of steel roof collapse, a Monte Carlo method combined with response surface approach was adopted, which greatly reduces the computation time and makes the Monte Carlo simulation feasible for probabilistic collapse analysis of steel roof frame structures.

Keywords

collapse / dynamic response / fragility analysis / Monte Carlo simulation / wind load

Introduction

Steel structural framing offers a popular structural form to cover the large roof space of gymnasiums, industrial facilities and transportation terminals, with potential use as shelter structures for a disastrous event such as hurricane. However, under high winds, either part of the roof enclosure or the entire roof structure can be lifted off a building, especially for low sloped roofs subject to wind-induced suction force. Collapse of roof could cause severe economic loss and poses safety risk to building residents. Furthermore, with recent emergence of building integrated renewable energy production system such as solar panels integrated with building facades and roofs, damage inflicted on the solar panels by excessively large deflection or collapse of roof would cause economic loss and power disruption to building residents.

The collapse of a steel roof frame structure can be initiated by the buckling of a few members, as a result of load redistribution causing a subsequent progressive overstress condition in other members and thus its load carrying capacity is usually limited by the failure of first member or set of members to fail. In an attempt to alter the brittle collapse behavior of steel roof frame structures under extreme wind loading, force limiting devices (FLD) can be used for robustness enhancement. These devices are fitted to critical compression members, and designed to provide a purely plasto-elastic behavior with a long plateau of member ductility (e.g., as exhibited in buckling-restrained braces). Since collapse is initiated by only a few critical members, the use of force limiting devices can be limited to a small selection of the most critical compressive chord or web diagonal members. The feasibility of using force limiting devices for controlling the behavior of space truss was validated in a research conducted by El-Sheikh [1]. The principle behind using these devices is to introduce artificial ductility in truss compression members, which would otherwise possess brittle post-buckling characteristics involving a loss of both stability and strength upon buckling.

Considering the fact that steel roof structures are susceptible to collapse under extreme wind loadings, this research aims to investigate the dynamic collapse risk of steel roof frame structures under transient wind loads and corresponding robustness enhancement techniques for collapse prevention. A nonlinear dynamic analysis that accounts for both material and geometrical nonlinearities was carried out for this simulation based study. Emphasis is given to effects of the dynamic member failure due to buckling, and wind load change rate. A numerical example based on a Steel Joist Institute (SJI) K series joist was used to demonstrate the use of force limiting devices for collapse risk mitigation. For the presented fragility assessment of steel roof collapse, a Monte Carlo method combined with response surface approach was adopted, which greatly reduces the computation time and makes the Monte Carlo simulation feasible for probabilistic collapse analysis of steel roof frame structures.

Dynamic collapse simulation

To accurately simulate the dynamic collapse of steel roof framing structures subject to combined gravity and transient wind load, nonlinear time history analysis was performed in this study. The finite element model, which considers dynamic member buckling, yielding of tension members, and geometric nonlinearity due to large deflection of the roof structures undergoing collapse, is established using the OpenSees simulation platform [2].

A progressive collapse refers to a structural failure that is initiated by localized structural damage and subsequently develops, as a chain reaction, into a failure involving a major portion of the structural system [3]. Several mechanisms might contribute or lead to the progressive collapse of steel roof frame structures including: buckling of compression member, yielding of a tension member, facture of a tension member or connector, and nodal instability. The collapse of a steel frame structure can be initiated by the buckling of a few members [4], and its load carrying capacity is usually limited by the failure of first member or set of members to fail.

In this study, a force-based element with corotational formulation and fiber discretization of the cross section is adopted for the simulation of buckling, post buckling, and hysteretic responses of pinned and fixed members in steel roof frames. In this force-based corotational element, both material and geometric nonlinearities are considered. Through an element state determination procedure [5], the force-based corotational elements are implemented based on the direct stiffness method. The formulation of force-based elements is based on interpolation functions for the internal force variation [6]. Neuenhofer and Filippou [7] formulated a force-based element for geometrically nonlinear analysis of plane frame structures, with linear constitutive behavior and small rotations. Following that, de Souza [8] extended the material nonlinear force-based element proposed by Neuenhofer and Filippou [5] to include geometrically nonlinearity through deriving the transverse displacements from the curvatures using Lagrangian polynomial interpolation. The adopted kinematics is based on the assumption of moderately large deformations along the element, thus rigid body displacements and rotations can be arbitrarily large.

Geometric nonlinearity due to large displacement can be considered stiffness matrix of basic system transforms via updatable transformation matrix in every instance in a corotational formulation. The element formulations take place in a basic system or corotating frame of reference that is free of rigid body displacement modes. The corotational formulation separates rigid-body modes from local deformations. This approach to the element formulation separates material nonlinearity inside the basic system from geometric nonlinearity of the corotating frame [9]. Equilibrium equations are solved in the deformed shape and large displacement effects are thus accounted for accurately. The corotational formulation is also employed in the present simulation-based roof collapse analysis.

In the analysis of roof frame elements, the material nonlinearity is considered by integrating the material stress-strain relations defined for each fiber over the section area, which is commonly referred to as “fiber discretization.” Distributed plasticity is obtained by integration of the section force-deformation response over the member length in the corotating frame of reference. As shear effects are neglected, a uniaxial stress-strain relation is employed at the material point. The effect of spreading of plastic deformation along the member axis is considered in this study.

The standard simulation approach for dynamic collapse, as well as other nonlinear structural dynamic problems, is to time discretize the governing equations of motion by Newmark time integration then solve them via the Newton-Raphson algorithm [10]. The basic Newmark constant acceleration method (β= 0.25 and γ= 0.50) is adopted for nonlinear dynamic analysis here. The iteration must be performed at each time step in order to satisfy equilibrium.

Force limiting device: mechanism and modeling

To minimize the likelihood of disproportional structural failures such as progressive collapse, many modern building codes require robustness of the structures and provide strategies and methods to obtain robustness [11]. Robustness describes the property of structures to resist an unforeseen or extraordinary event according to specified design objectives. In an attempt to alter the brittle collapse behavior of steel roof frame structures under extreme wind loading, force limiting devices (FLDs) can be used for robustness enhancement.

This section presents a comparative study of steel roof framing structures with and without force limiting devices (e.g., buckling-restrained brace with modification for roof structure) for reducing the likelihood of dynamic collapse of steel roof framing structures.

Modeling of force limiting device

Structural redundancy defines the small number of members whose loss is sufficient to trigger an overall collapse in a highly statically redundant spatial truss structure. In order for the statical redundancy to be translated into structural redundancy, a ductile overall behavior must be dominant. To achieve this type of behavior, truss compression members must be effectively prevented from buckling.

FLDs which are designed to provide a pure elastic-plastic behavior with a long plateau of member ductility (e.g., as exhibited in buckling-restrained braces) can be used for robustness enhancement by fitting them to substitute critical compression members. The fundamental concept behind FLD is to restrain low-order member buckling modes, creating full and stable hysteretic loops under tension-compression cyclic load. For example, with the use of such FLDs as buckling restrained braces [12], member buckling can be held off till a fairly large compressive strain without significant strength degradation.

A model for FLD has been developed on the basis of modeling work done for buckling restrained braces. In numerical modeling of FLD, a simple, one-dimensional, pin-ended truss element is used to have the appropriate uniaxial force-displacement properties. The hysteresis loop of the FLD model calibrated with the experimental data reported by Clark et al. [13] are shown in Fig. 1, in contrast to that of the corresponding buckled bracing model.

Location and weakening of force limiting devices

Typically, a compromise has to be struck between the improvement to structural behavior associated with these devices and the cost addition caused by their use. Therefore, the use of FLDs is limited to a small selection of the most critical compression chord members in this study. This includes a strategy for selectively replacing key compressive members in steel roof structures with FLDs to prevent collapse or at least alter the collapse sequence to a favorable collapse mechanism.

The critical members in its progressive collapse path under specified wind load were first identified, which are later replaced with force limiting devices in a comparative study of robustness enhancement techniques. Simple linear finite element analysis of the original steel roof frame structure without FLDs was conducted to determine the internal forces in all members. The locations of the FLDs are chosen based on the axial force from this analysis.

For a different application on seismic response mitigation, Reinhorn et al. [14] and Viti et al. [15] introduced the concept of weakening structures (e.g., reducing strength) to reduce simultaneously total accelerations and inter-story drifts. The main features of this procedure were to reduce maximum acceleration and associated forces in buildings subjected to seismic excitation by reducing their strength (weakening).This concept was employed in this study to improve the structural behavior under dynamic wind loading. Note that the yield force of FLD was designed to be equal to the buckling force (i.e., critical load) of a member with imperfection. The purpose of member weakening in the roof framing is to make the selected weakened member as a fuse element where favorable ductile damage of the overall structure is concentrated. This concentrated structural weakening strategy could also bring other benefits such as ease in damage inspection and rapid structural repair with member replacement post extreme events.

Numerical simulation study

Wind load profile

The effect of downburst on structures has recently been gaining attention due to the collapse of lightweight steel structures such as transmission towers and roof structures [16,17]. For example, the collapse of steel joist roof structures reported at Dallas was attributed to microburst-induced uplift force during a thunderstorm event in February 2001 [18]. Microburst is a small concentrated downburst that produces an outward burst of damaging winds at the surface that are short-lived, lasting only 5-10 min, with maximum wind speed up to 168 mph (270.4 kph). Extreme wind load could also occur in hurricanes or typhoons. To study the effect of transient wind load on the collapse behavior of steel roof framing structures, a representative wind load profile, which is reported by Fujita [19], is adopted in this study. This wind load history is plotted in Fig. 2.

Prototype roof frame

A common construction method for commercial and public buildings is the use of lightweight metal roof decking supported by steel bar joists. This kind of roof structural system is very economical for gravity load carrying; however, when subjected to extreme transient wind load such as induced by downburst, it could potentially be damaged [18].

A typical roof joist- 26K9 (K-series) in the SJI catalog (SJI standard specification 2005) is selected for this study and out-of-plane displacement is restrained. In this type of joist, the end web members are very slim compared to the chord members in the same joist and the buckling of this end web members controls the collapse load of the joist during the uplift induced by suction wind load. This is supported by the observations from a real roof collapse (involving the same type of K-series joist) caused by downburst induced uplift as reported by Nelson et al.[18]. A 2-dimensional analysis was thus performed with all out-of-plane displacements restrained. The selected 26K9 joist is approximately 50 ft (15.24 m) in the span length, 26 inch (0.66 m) in its depth, and spaced 12.5 ft (3.81 m) apart on center. The design of chord sections for the K-series joists is based on a yield strength of 50 ksi (345 MPa) while the design of the web sections for the K-series joists is based on a yield strength of 36 ksi (250 MPa).

The configuration of the 26K9 joist and a modified version with FLDs are schematically shown in Fig. 3. Both the top and bottom chord are made from two 2 ×2 ×3/16 inches (5.08 cm× 5.08 cm× 0.476 cm) steel angles back to back while webs are 15/16 inches (2.38 cm) diameter rounds. The dead loads applied to the structure include the self-weight of the joist, and roofing materials involving solar panels were applied to the model by assigning lumped mass to the nodes of the top chords of the joist. The roofing also includes building integrated photovoltaic panels (BIPVs), and the weight of each 1.028-m by 0.629-m BIPV panel is calculated based on the data supplied by a leading BIPV manufacturer. The total dead load was thus determined to be 10.6 psf (0.508 kPa). The dead loads for the interior joists spaced 12.5 ft (3.8 m) apart as well as wind load were applied as concentrated loads acting at the upper chord nodes following a spatial load profile in Fig. 3(a).

The finite element model for the joist consists of a total of 43 corotational nonlinear beam-column elements to account for the large displacement effect potentially experienced during roof collapse. The top and bottom chords are modeled as a continuous member with no moment release at the chord nodes, while the web diagonals in the joist are pin-connected to both the top and bottom chords. The left support of the joist is assumed to be pin supported while the right is a roller support. The lumped mass assigned to the top chord nodes are determined based on the corresponding dead loads. The natural frequencies of this roof joist are calculated to be: 3.85, 14.1, 26.3, 30.3, 43.5, 58.8, and 71.4 Hz for the first seven vibration modes. The viscous damping ratio of the joist model is assumed to be 2%.

A two-element model with initial camber is used to simulate the buckling of compression member in the prototype joist without FLDs (labeled as no FLD) subject to the wind load scaled from the time history given in Fig. 2. To study the effect of the FLDs on the collapse behavior of steel roof framing structures, a case with FLD is defined in Fig. 3(c). When a compression member is replaced with a FLD, its behavior is assumed to be ductile with increasing deformation capacity under the same buckling critical load. The locations of the critical members replaced by FLDs are determined from a preliminary analysis, in which two end web members were found to buckle during up-lift phase. In the cases with FLDs (labeled as FLD), these critical compression members are replaced with FLDs.

Results and discussion

Figure 4(a) shows the displacement time histories of node L3, which has the largest deflection values and thus is used as a reference point. Case 1 (labeled as “No FLD”) refers to the prototype joist structure without any FLDs. Case 2 (labeled as “w/ FLD”) refers to the modified joist structure with the two end web diagonals being replaced with FLDs. In Case 1, roof joist failure occurred due to excessively large deflection. However, in Case 2, the roof frame did not collapse as the FLD employed for this structure can maintain its resistance without strength degradation after its critical load is reached. In Case 1, there is a sharp upward turn in the displacement response curve, signifying the occurrence of member failure shortly before the peak wind pressure is reached at 200 s. Actually member U1L1 was found to buckle at this time instant. In Case 2, member U1L1 was replaced by FLD and thus would not buckle. This modification dramatically reduces the up-lift deflection of the roof frame and thus decreases the potential energy that would be converted to kinematics energy in the un-loading process. For this reason, both the up-lift deflection and residual displacement of Case 2 are smaller than those in Case1. Figure 5 shows that after reaching critical force, the forces in FLDs remained unchanged and the axial deformation of U1L1 in Case 2 are much smaller than that in Case 1. The use of FLDs in critical compression members eliminate sudden upward movement (upward overshoot) of the joist structure due to member buckling and enhance its collapse resistance. It is thus clear that incorporating FLDs into the steel roof framing can enhance its robustness in resisting roof collapse under transient wind loading. It is worth noting that only a limited number of critical members in the original steel joist need to be replaced with FLDs.

Fragility assessment of roof framing collapse

The objectives of a roof system are: life safety of the occupants, prevention of excessive economic losses due to damage and repair, and other goals such as supporting and avoiding the loss of expensive renewable energy production equipment located on the roof. In fragility assessment, the hazard is a potentially harmful event, action or state of nature. In modern structural engineering practice, the hazard usually is quantified by its annual probability or mean rate of occurrence, as in ASCE Standard 7 (2005). The consequences of the hazard – local damage, building collapse, personal injury, loss of life, economic losses, or damage to the environment – must be measurable [3].

In the fragility assessment of steel roof collapse presented in this section, the uncertainties associated with wind load rate and member imperfection of critical compression member are accounted for. Among these two parameters, one is associated with the wind load history and the other is associated with the modeling uncertainties. No correlation was specified as these two events are normally considered independent of each other. A Monte Carlo simulation method combined with response surface approach was used to evaluate the collapse fragility function.

Collapse fragility assessment procedure

The collapse capacity of the roof structure is assessed based on the incremental dynamic analysis (IDA) technique. In IDA, the numerical model for nonlinear analysis should be capable of capturing both the material and geometry nonlinearities in the structure. Compared to the IDA developed for seismic application, the IDA procedure presented here is similar, with the only difference being the wind load instead of earthquake ground motion. The peak wind load pressure is used as the wind intensity measure (IM). For each wind load record, the nonlinear time-history analysis is repeated each time by increasing the scaling factor on the wind load until the one leading to roof collapse occurs. In this study, roof collapse is defined as the mid-span deflection exceeding 2% of the roof span length or initiation of any member buckling. This procedure has been applied to assess the performance of both cases with or without FLDs. To cut down the computation time, only the middle portion of the wind load history (i.e., from 12 to 16 min) in Fig. 2 was used in the IDA of this study. The wind load in the first 12 min is lower than the peak wind load in the middle portion. The structure remains elastic within this time period. The wind load after 16 min is much lower than the largest peak attainted between 12 to 16 min. Numerical test shows that collapse could only happen near the first peak of the middle portion of the wind load history. Since the aim of IDA is to determine the collapse load of the roof structure, only the middle portion of the wind load history (i.e., between 12 min and 16 min in Fig. 2) is used for the IDA study. The sampling interval for the numerical simulation is 0.02 s.

In the fragility assessment procedure, the structural design parameters, such as loadings, material parameters and geometry, can be chosen as the set of basic random variables X which determine the probabilistic response of structural systems. The failure condition is defined by a deterministic limit state function,

(1)G(\mathrm{X})=G(x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_n)\le \mathrm{0},

G(X) is the limit state, which is the collapse of the steel roof frame in this study. Fragility can be defined as the conditional probability of failure of the steel roof frame system for a specified limit state, as follows,

(2)

where D = random variable describing the intensity of the demand on the system. The conditional probability denotes the fragility. Failure at the roof frame support (i.e., bearing that supports the roof frame) due to uplift can be an issue as well, but this failure mode is excluded from this study. The fragility of a structural system is commonly modeled by a lognormal cumulative distribution function (CDF), as follows,

(3)F_R(y)=\mathrm{\Phi }[\ln (y/{\mu }_R)/{\varsigma }_R],

in which Φ [⋅] = standard normal CDF, μ_R = median capacity, ξ_R = logarithmic standard deviation.

The Monte Carlo method has been widely used to study the effects of modeling uncertainties on the fragilities for structure response. Using the Monte Carlo method, one can generate realizations of each random variable, and the model is then analyzed to determine the structure collapse capacity. One drawback that limits the utilization of Monte Carlo method is the analysis process which needs to be repeated hundreds and thousands times to get the statistical distributions of collapse capacity results associated with the input random variables. This issue is prevailing especially when the analysis involves material or geometric nonlinearities, or both, which are very computationally intensive.

To quantify the effects of modeling uncertainties on the collapse of steel roof structures, a Monte Carlo approach was employed in this study. To reduce the enormous computational demand required by a full Monte Carlo simulation, a response surface based method was adopted. A response surface is a simplified functional relationship between the input random variables and collapse capacity of roof structure. In the response surface based method, sensitivity analyses are first used to probe the effects of modeling variables on the median collapse capacity of the system [20]. The results of the sensitivity analysis provide the inputs to regression analysis used to create the response surface, which represents the median collapse capacity as a function of model random variables. The loss of accuracy in the estimation of collapse capacity depends on the degree to which the highly nonlinear predictions of structure response can be accurately represented by the simplified surface. For example, the response surface can be idealized into a second-order polynomial functional form, which approximately represents the nonlinear limit states and interactive effects between the model random variables in certain applications.

In the response surface method, the true limit state function, G(X), is approximated by a simple and explicit mathematical expression G′(X), which is typically a k-th order polynomial, with undetermined coefficients. By fitting the response surface to a number of designated sample points of the true limit state, an approximate limit state function can be constructed. The selection of the form of the approximated limit function G′(X) should be based on the shape and nonlinearity of the true limit state function. Intensive studies have been conducted on the selection of G′(X) and the most common one is the quadratic polynomial [21]. After the response surface has been constructed, as the approximated limit state function is explicit, Monte Carlo simulation can be performed more efficiently since the realization of the limit state values from the response surface function requires much less computational effort [22]. A more realistic approach is to choose some variable points other than the sample points by interpolation, then compare their corresponding limit state values on G′(X) and G(X). If the difference is greater than a pre-specified tolerance value, these points would be incorporated into the set of sample points and the response surface would be updated using the new set of sample points. If the differences are within the tolerance, the response surface can be considered sufficiently accurate and no further adjustment is necessary.

The value of the true limit state function is evaluated at a number of samples of variables, X, to determine the unknown coefficients such that the error of approximation at the samples of X is minimized. Higher order polynomials are usually not preferred because ill-conditioned systems of equations may be encountered [23] and a large difference between G′(X) and G(X) may occur outside the domain of sample points [24].

To capture the nonlinearity of the true limit state more precisely, the following form of quadratic polynomial is used here,

(4)G\prime \left(\mathbf{X}\right)=a+{\sum }_{i=1}^n{b}_i{X}_i+{\sum }_{i=1}^n{c}_i{X}_i^2+{\sum }_{i=1}^{n-1}{\sum }_{j=i+1}^n{d}_{ij}{X}_i{X}_j,

where a, b_i, c_i, d_ij are coefficients to be determined by curve fitting. They can be obtained by standard regression analysis. The number of sample points need to be larger than or at least equal to the number of the polynomial coefficients. In Eq. (4), the number of coefficients is equal to 1+ 2n + n(n -1)/2. Therefore, if the number of random variables n is 2, at least 6 sample points would be needed to construct the response surface. In this study, sample points are chosen at a specified number of combinations of the mean values μ_i and μ_i±hσ_i, as illustrated in Fig. 6.

When the shape of the true limit state is not close to linear or quadratic, the parameter h, which controls the size of the sampling domain, plays a significant role in the accuracy of the response surface approximation. Since the form of the true limit state function and the failure probability are usually not known a priori and may vary with the application, no magic number for h can thus be suggested for general use. While an explanation of the sensitivity of the parameter h is possible, no clear guidelines are available on its proper choice [25]. While Bucher and Bourgund [26] obtained good results by choosing h = 3 in their numerical examples, Rajashekhar and Ellingwood [27] showed that using a constant value of h = 3 may not always yield good solutions, particularly when G(X) is highly nonlinear. For this reason, eight verification points are selected to evaluate the accuracy of the response surface. In this study, h =±1.7 and h =±1.0 are used for the selection of sample points. Accuracy assessment was carried out at the eight verification points, as listed in Tables 1 and 2. Following the creation of the response surface, the random variables are sampled 10,000 times in total. For each set of sampled value of the two random variables, the response surface is used to calculate the collapse wind pressure. After that, the fragility curve is determined using the simulation values obtained above.

As described above, two random variables are considered in the fragility analysis: wind load change rate and the initial imperfection rate of critical members (i.e., ratio of imperfection to member length). The wind load change rate is expressed in terms of the wind pressure increase rate of the rise peak, as schematically illustrated in Fig. 2 (dashed line, note the unit difference). The wind change rates remain constant during each IDA. The probabilistic distribution of the wind pressure change rate random variable is assumed to be Gaussian with its mean value μ₁ = 0.85 kPa/min and standard deviation σ₁ = 0.153 kPa/min. The initial imperfection rate of critical member follows Gaussian distribution with μ₂ = 0.5% and σ₂ = 0.235%. Nine sample points are selected to construct the response surface, as listed in Table 3. It is noted that each variable is sampled within the range of±1.7 σ away from the mean value.

Results and discussion

A variety of approaches to the formulation of measures for robustness or related characteristics were published [28], which can be divided into two categories: measures based on structural behavior and those based on structural attributes, with the first group prevailing. The measures based on the structural behavior – also called performance-based measures – can be further divided into deterministic and probabilistic measures (e.g., Refs. [29-31]. In this study, the failure probability of roof collapse is used as a robustness measure in studying the effectiveness of the FLDs to enhance the collapse resistance capacity of steel roof framing structures under transient wind loading. It is noted again that collapse is defined here as the first occurrence when the mid-span deflection of the roof frame exceeds 2% of its span length, or the initiation of any member buckling in the roof frame.

From the above discussion, it is known that the response surface here is represented by a second order polynomial through regression analysis of the nine sample points. The goodness of fit for the response surface is evaluated by the statistical measure of R², termed as coefficient of determination. R² is defined as: R² = 1 - SS(regress)/SS(total), in which SS(regress) is the sum of the squared distances from sample points (true value determined from IDA) to the response surface; SS(total) is the sum of squared distances to the horizontal plane defined by the average of the collapse wind pressure of all nine sample points. For the regression of Case 1 (i.e., No FLD), R² equals to 0.99. For Case 2 (i.e., w/ FLD), this value is 0.99. The response surfaces for these two cases are shown in Figs. 7 and 8 respectively. Points 1 to 8, which are indicated as the circle points in Fig. 6, are used to check the accuracy of the response surfaces. The results are listed in Tables 2 and 3 for comparison purpose. It is seen that the accuracy of any point within the region defined by 1.7s from the mean value point for each variable is fairly good. The errors are less than 1%. However, for the extrapolation region outside this region, the error becomes greater, although still below 5%. The accuracy thus depends on the distance from the mean value of the two random variables and nonlinearities of the response surface.

Another point with an imperfection rate of 0.1% and wind change rate of 1.11 kPa/min (derived from the Fujita downburst wind record [19] is also used to evaluate the response surface. For the No FLD case, the calculated collapse wind pressure is 1.471 kPa and the estimated value from the response surface is 1.503 kPa. The error is about 2.2%. For the w/ FLD case, the calculated collapse wind pressure is 1.680 kPa and the estimated value from the response surface is 1.714 kPa. The error is about 2.0%.

After the creation of response surfaces, Monte Carlo sampling is carried out to assess the collapse probabilities that involve the effects of the two random variables. The random variables are sampled 10,000 times total, which follows normal distribution as assumed. The regressed response functions are used to calculate the corresponding collapse capacities for each set of random variable samples. The histograms for the two case studies are shown in Fig. 9. The fragility curves can then be generated from the Monte Carlo simulation results. The median, mean and standard deviation values of the collapse wind pressure are 1.329, 1.334 and 0.0367 kPa for the No FLD case, and 1.546, 1.548 and 0.0323 kPa for the w/ FLD case. Therefore, by adding FLDs on the roof frame (only two members are replaced with FLDs in this example), the median collapse pressure was increased by 16%. The standard deviation was decreased slightly. For the roof structure in this study, the wind speed corresponding to these two median values are 120 mph (54 m/s) and 140 mph (62 m/s) respectively. In the other words, for the 140 mph wind, adding FLD decreases the collapse probability to 50%. The 95% confidence level values are 1.4 kPa for the No LFD case and 1.61 kPa for the w/ FLD case, corresponding to 130 mph (58 m/s) and 145 mph (65 m/s) respectively.

The fragility curves for two cases are shown in Fig. 10, which illustrates the effects of modeling uncertainties. It is observed that the use of FLDs can shift the median point, which corresponds to the wind intensity at 50% collapse probability. Use of FLDs can have a favorable effect on the roof frame’s collapse resistance capacity by shifting the median point to the right.

Conclusions

A collapse risk analysis has been conducted to investigate the fragility of steel roof frame structures subject to transient wind load. Two cases were considered: one without any force limiting devices (FLDs), and the other with two regular members replaced by FLDs. To account for the modeling uncertainties in the nonlinear collapse analysis of roof framings, a Monte Carlo method in conjunction with response surface estimation is used, which has been proved to be expedient and provide reasonably accurate results. Using the response surface method, very few data points are needed to create the response surface instead of hundreds and thousands nonlinear simulations. The accuracy of the response surface depends on the nonlinearity and monotonicity of the effects of the random variables on the structure response of interest. Two random variables are considered in the fragility analysis - wind load rate and the initial imperfection rate of critical members (i.e., ratio of imperfection to member length). The random variables are assumed to be no correlative. It is found from this study that the estimates given by the response surface for both cases with and without FLDs are fairly accurate.

From the results of this study, it is observed that the utilization of FLDs is helpful in reducing the failure probability of roof collapse under transient wind load. This robustness enhancement could be explained as that unlike buckled members, FLD exhibits a more ductile behavior with no significant strength deterioration. When the rapid rise of wind pressure on the roof structure leads to its overshoot response due to buckling of certain critical compressive members, the FLDs provide sufficient axial force to support the roof structure and thus increase collapse wind pressure of the roof. The results from fragility analysis show that incorporating FLDs at the critical locations would shift the median collapse capacity higher.

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