The machine learning-based feature selection approach is presented to estimate the e/uniFB00ect of un- certainties and identify failure modes of structures that incorporate a low failure probability and high- dimensional uncertainties. As structures are designed to have few failures, a dataset classi/uniFB01ed based on the failure status becomes imbalanced, which poses a challenge for the predictive modeling of ma- chine learning classi/uniFB01ers. Moreover, in order to improve the accuracy and e/uniFB03ciency of the model per- formance, it is necessary to determine the critical factors and redundant factors, especially for a large feature set. This study benchmarks the novel method for sensitivity analysis by using datasets that ex- acerbate the problems involved in class imbalance and large input features. This study investigates two planar steel frames with spatially uncorrelated properties between structural members. Geometric and material properties are considered as uncertainties, such as material yield stress, Young’s modulus, frame sway, and residual stress. Six feature importance techniques including ANOVA, mRMR, Spearman’s rank, impurity-based, permutation, and SHAP are employed to measure the feature importance and identify parameters germane to the prediction of structural failures. Logistic regression and decision tree mod- els are trained on the important feature set, and the predictive performance is evaluated. The use of the feature importance approach for structures with a low probability of failure and a large number of uncertain parameters is validated by showing identical results with the reliability-based sensitivity study and appropriate predictive accuracy. /one.pnum Introduction A reliability-based sensitivity analysis estimates how uncertainty in the input parameters a/uniFB00ects system per- formance by analyzing the dependence of the failure probability on the inputs, which requires repeated evaluation of the performance function, resulting in signi/uniFB01cant computational cost and time. This chal- lenge is exacerbated for large-scale engineering problems which often carry a large quantity of uncertain parameters. Researchers have made e/uniFB00orts to improve the computational e/uniFB03ciency of sensitivity analysis. For example, Wu [/one.pnum] proposed the adaptive importance sampling approach, which improves computational e/uniFB03ciency by minimizing oversampling in the safe region of the limit-state surface. The score function ap- proach proposed by Rubinstein and Kroese [/two.pnum] estimates all sensitivities by the gradient and derivative of parameters. The score function method does not require additional simulations for reliability sensitivity analysis. Torii et al. [/three.pnum] applied polynomial expansions to the performance function and its derivatives for the probability of failure sensitivity analysis. Proppe [/four.pnum] introduced the local reliability sensitivity analysis using the moving particles method, which estimates the failure probability based on the new locations of the moved data points in the design space. In recent years, the interest in arti/uniFB01cial intelligence has been growing in the /uniFB01eld of structural engineering because it provides e/uniFB03cient solutions to the problems in this /uniFB01eld relative to traditional computational techniques. The use of arti/uniFB01cial intelligence in steel structural design has been focused on arti/uniFB01cial neural networks for design of steel members or connections such as compression members [/five.pnum], steel panels [/six.pnum], steel connections [/seven.pnum, /eight.pnum], and cold-formed steel channels [/nine.pnum]. For reinforced concrete (RC) members or systems, machine learning techniques are implemented to predict structural responses such as the shear capacity of /uniFB01ber RC beams [/one.pnum/zero.pnum], structural response of RC deep beams [/one.pnum/one.pnum] and RC slabs [/one.pnum/two.pnum]. Data-driven 1 machine learning approaches were used for fragility, risk, and vulnerability assessment of a special steel moment resisting frame building [/one.pnum/three.pnum] and RC building frames [/one.pnum/four.pnum]. Also, machine learning approaches were used to identify failure modes and rank the signi/uniFB01cant factors a/uniFB00ecting the failure mode of RC members [/one.pnum/five.pnum], RC frames [/one.pnum/six.pnum], and steel frames [/one.pnum/seven.pnum]. Regarding the assessment of structural members of systems using machine learning techniques, RC structures have been mainly investigated rather than steel structures. In particular, studies on steel structures have used neural networks, which is a speci/uniFB01c technique of machine learning, and have focused more on structural members than systems. Koh and Blum [/one.pnum/seven.pnum] introduced the machine learning-based feature selection framework to be used for struc- tural sensitivity analysis. This framework measures the feature importance of all parameters and ranks them to determine the important or redundant parameters for the prediction of system failure. Two pla- nar steel frames were investigated with the consideration of uncertainties that a/uniFB00ect steel frame behaviors, such as yield stress, Young’s modulus, frame sway, and residual stress. The frames have di/uniFB00erent failure modes and the ultimate frame strength obtained from /uniFB01nite element analysis was used as the response variable. The feature rankings derived by four feature importance techniques showed the identical order of factors resulting in the largest failure probability obtained from the conventional sensitivity studies. It is demonstrated that the general procedure of the proposed feature importance method can be used for sensitivity analysis. The approach is e/uniFB03cient because all feature importances are estimated from a single training. Moreover, variable space can be reduced by removing irrelevant parameters to improve both computational e/uniFB03ciency and accuracy. This study benchmarks the feature importance approach on the datasets that incorporate a larger set of uncertainties, more data points, and a high imbalanced ratio. The importance of benchmark studies has been emphasized in the machine learning community [/one.pnum/eight.pnum,/one.pnum/nine.pnum]. Machines carry out tasks based on learning from a given dataset, therefore the best algorithm will not be the same for all the datasets [/two.pnum/zero.pnum]. Therefore, benchmarking of the feature importance approach must be accomplished to draw conclusions for the use of the approach in a wide range of structural systems. The steel frames investigated in this study have the same layout as for Koh and Blum [/one.pnum/seven.pnum] but have a dif- ferent spatial correlation scenario. Unlike the correlated scenario that considers all columns (or beams) to have the same properties, this study applies the uncorrelated scenario where all structural members have di/uniFB00erent properties. Sensitivity studies on structural systems with uncorrelated properties inform which speci/uniFB01c structural member largely in/uniFB02uences the frame failure. However, the increased number of uncertainties poses additional challenges in performing reliability sensitivity studies, which require re- peated evaluations of the performance function. Regarding the feature importance approach, a large set of uncertainties would make the feature importance method di/uniFB03cult to derive consistent rankings between various techniques. In addition, the uncorrelated scenario contains a fewer number of failures than the cor- related case, therefore the class-imbalanced ratio increases. Machine learning classi/uniFB01ers are sensitive to imbalanced data because the prediction is biased towards the majority class [/two.pnum/one.pnum, /two.pnum/two.pnum]. As structural failures should always occur rarely in structural design, it is critical to overcome the class imbalance problem. In summary, this study aims to examine how the feature importance approach works on structural sensitivity analysis when /uniFB01tting high-dimensional and extremely class-imbalanced data, which presents challenges in model training for structural engineering problems. This study implemented six existing feature importance methods to measure the importance score. There are typically two categories in feature importance methods: (/one.pnum) data analysis techniques that directly ana- lyze the data without model /uniFB01tting to measure the feature importance and (/two.pnum) model analysis techniques that identify important features based on predictions from trained models [/two.pnum/three.pnum]. (/one.pnum) ANOVA (Analysis of variance), (/two.pnum) mRMR (minimal-redundancy-maximal-relevance) [/two.pnum/four.pnum], and (/three.pnum) Spearman’s rank correlation coe/uniFB03cient [/two.pnum/five.pnum] are utilized as data analysis techniques. This study used a decision tree classi/uniFB01er [/two.pnum/six.pnum] to measure the feature importance by using model analysis techniques. Model analysis techniques include two feature importance methods, which are (/four.pnum) impurity-based importance and (/five.pnum) permutation impor- tance, and (/six.pnum) SHAP (SHapley Additive exPlanations) [/two.pnum/seven.pnum]. Based on the measured feature importance, logistic regression [/two.pnum/eight.pnum] and decision tree [/two.pnum/six.pnum] models are /uniFB01tted to predict whether a steel frame fails. The predictive performance is evaluated by speci/uniFB01city, recall, and the Matthews correlation coe/uniFB03cient [/two.pnum/nine.pnum]. The results are compared with the reliability-based sensitivity analysis results to validate the feature im- portance framework. Finally, the best feature importance technique depending on the failure modes is discussed. /two.pnum /two.pnum Machine learning techniques for the feature importance approach The primary task of this study is to estimate the e/uniFB00ect of uncertainties on failure modes of steel frames using the machine learning-based feature importance approach. In a high-dimensional dataset, it is im- portant to determine the relevant features and remove the redundant features in an e/uniFB00ort to prevent over/uniFB01tting of a model and reduce training time, thereby improving the model performance and computa- tional e/uniFB03ciency. This study measures the feature importance and compares the critical features between the feature selection methods. Once the feature rankings are derived from the feature importance tech- niques, a classi/uniFB01cation model is employed to predict the failure status of a structure by increasing the size of the feature set used for training. The model accuracy of each feature importance method is measured to determine how many features are informative or irrelevant for identifying structural failures. The fea- ture ranking as well as the model performance are considered when evaluating the feature importance techniques for structural sensitivity analysis. /two.pnum./one.pnum Feature importance techniques /two.pnum./one.pnum./one.pnum ANOVA The ANOVA test compares the relationship between features and response variables based on the value of F-statistic. The feature importance score JANOV A is equal to the value of F-statistic. The score can be calculated by Eq. /one.pnum: JANOV A (xi) =MP m=1Nm(x(i) mx(i))=(M1) MP m=1(Nm1)s2m=(N1)(/one.pnum) whereNm= the number of instances that y=m,x(i) m= the sample mean of feature xifor classm,s2 m = the sample variance of feature xifor classm,x(i)= the grand mean of feature xi,M= the number of classes,M= /two.pnum in a binary dataset, N= the total number of instances. The importance score is the ratio of between-group variance to within-group variance, thus this technique assesses the di/uniFB00erence between the mean values of the corresponding feature xibetween the classes. A higher value of the F-statistic indicates a larger di/uniFB00erence between the mean values among the classes, thus the feature has a signi/uniFB01cant e/uniFB00ect on the classes. Note that ANOVA is always positive because it is based on variance, which is always positive. /two.pnum./one.pnum./two.pnum mRMR The mRMR technique [/two.pnum/four.pnum] ranks features by mutual information, which considers both relevance and re- dundancy of features. The feature relevance indicates a correlation with the response variable, and the feature redundancy represents the information duplicated between features. As the dataset is discrete not continuous, the mutual information di/uniFB00erence (MID) is used as the mRMR criterion, which can be estimated by Eq. /two.pnum: JmRMR (xi) =I(xi;y)1 jSjX xj2SI(xi;xj) (/two.pnum) wherejSj= the feature set size (number of features), S= a feature set, xj= a feature not selected in the setS, andI= the mutual information. The /uniFB01rst term represents the relevance of the feature xiabout the response variable y. The relevance is determined from the outcome variable prediction. The second term estimates the redundancy, which is measured within the selected features xiandxj. By intuition, a feature with a negative JmRMR value has a small relevance and large redundancy, therefore, including it in model training would decrease the predictive performance of the model. /two.pnum./one.pnum./three.pnum Spearman’s rank The Spearman’s rank correlation coe/uniFB03cient [/two.pnum/five.pnum] measures a monotonic nonlinear relationship between two variables, a feature xiand the response variables y. The measured coe/uniFB03cient varies between -/one.pnum as the perfect negative correlation and +/one.pnum as the perfect positive correlation. A feature with the largest absolute value is considered the most important. /three.pnum /two.pnum./one.pnum./four.pnum Impurity-based importance Impurity-based importance considers the node impurity in a tree to estimate the importance. A node con- taining instances of one class only is pure while a node with greater than or equal to two classes is impure. The impurity-based feature importance can be computed by Eq. /three.pnum [/three.pnum/zero.pnum], which measures the impurities at a nodejbefore and after splitting and then averages the impurity decrease by N(i), which is the number of nodes in a tree split based on xi. A negative feature importance value ( Jimpurity ) indicates that including it in model training would decrease the predictive performance of the model. Jimpurity (xi) =P i2N(i)(Impurity before node kImpurity after node k) jN(i)j(/three.pnum) /two.pnum./one.pnum./five.pnum Permutation importance Permutation importance measures the importance by removing a single feature column in a dataset. First, a tree-based algorithm is /uniFB01tted to obtain the baseline model performance. After training the model, a sin- gle feature column is randomly shu/uniFB04ed to remove the association between the feature and the response variable. The performance of the permuted model is evaluated and compared with the baseline model performance. The di/uniFB00erence in accuracy is considered to be the importance score as shown in Eq. /four.pnum. The feature that results in the largest Mean Decrease in Accuracy (MDA) is the most important. The permu- tation method provides a negative score when a feature has no e/uniFB00ect and shu/uniFB04ed data are shown to be more accurate. Jpermutation (xi) = accuracy for dataset without permutation accuracy for permuted dataset of xi(/four.pnum) /two.pnum./one.pnum./six.pnum SHAP The SHAP algorithm [/two.pnum/seven.pnum] identi/uniFB01es how much each feature contributes to the response variable based on the predictions for linear models trained on all feature subsets. The di/uniFB00erence of the predictions from the modelfS[figtrained on a feature subset Sincluding a feature xiand another model fSexcludingxiis interpreted as the e/uniFB00ect of xi. The SHAP importance score is a weighted average of all possible di/uniFB00erences, as shown in Eq. /five.pnum: JSHAP (xi) =X SFnijSj!(jFjjSj1)! jFj![fS[fig(xS[fig)fS(xS)] (/five.pnum) whereF= the set of all features, and S= all feature subsets without xi. Since all possible subsets are used to measure the SHAP score, the computation time of the SHAP algorithm is expensive because the model is repeatedly trained on all possible feature subsets. SHAP measures the in/uniFB02uence of features in terms of the prediction of either positive (minority class) or negative (majority class) outcomes. /two.pnum./two.pnum Classi/uniFB01cation-based techniques /two.pnum./two.pnum./one.pnum Logistic regression Logistic regression [/two.pnum/eight.pnum] is trained using the top- kfeatures obtained from the feature importance methods, wherek= the number of important features. Logistic regression is used for the classi/uniFB01cation problems, which uses the logistic sigmoid function and transforms the output into a probability value between /zero.pnum and /one.pnum as follows in Eq. /six.pnum: P(Y=m) =1 1 +exp((w0+P i2nwixi))(/six.pnum) whereP(Y=m)is the probability of presence of Class m; Class /zero.pnum represents no failure and Class /one.pnum represents failure in this study, wi= regression coe/uniFB03cients, xi= input features, and n= the number of input features. /four.pnum A linear function is embedded in the logistic regression model, which is given as the natural logarithm of the ratio ofP(Y= 1) toP(Y= 0) . Logistic regression estimates the regression coe/uniFB03cients by minimizing the value of the ratio shown in Eq. /seven.pnum. log(P(Y= 1) 1P(Y= 1)) =log(P(Y= 1) P(Y= 0)) =w0+X i2nwixi (/seven.pnum) Feature scaling was performed on the datasets used for training a logistic regression classi/uniFB01er, which is sensitive to the location of data points. A decision tree is scale-invariant because it trains the model based on decision rules. Therefore, the datasets for the logistic regression algorithm are transformed to a stan- dardized scale, which has a mean value = /zero.pnum and standard deviation = /one.pnum. /two.pnum./two.pnum./two.pnum Decision tree A tree-based classi/uniFB01er, decision tree [/two.pnum/six.pnum], is used to measure the performance in addition to estimate the feature importance score of the model analysis techniques including impurity-based, permutation, and SHAP . Decision tree continuously splits the data according to a certain parameter such as impurity or entropy. Decision rules for splitting and the leaf nodes are the /uniFB01nal outcomes of the decision tree. The decision tree algorithm used in this study splits the nodes based on entropy (Eq. /eight.pnum) until all leaves are pure: H(S) =cX m=1pmlog2pm (/eight.pnum) wherepm= the probability of Class moccurring in the data. /two.pnum./three.pnum Evaluation metric After training a model with a training set, the model performance is evaluated by using a test set, which was not involved in training. A confusion matrix provides a visualization of the model performance and it is used as a performance measurement for a machine learning classi/uniFB01cation problem. The confusion matrix for binary classi/uniFB01cation consists of the four di/uniFB00erent cases as shown in Eq. /nine.pnum: M=TP FN FP TN (/nine.pnum) where True Positive (TP) = the number of actual positives that are correctly predicted positives, True Nega- tive (TN) = the number of actual negatives that are correctly predicted negatives, False Negative (FN) = the number of actual positives that are incorrectly predicted negatives, and False Positive (FP) = the number of actual negatives that are incorrectly predicted positives. Several statistical rates can be computed based upon the values given in the confusion matrix. For ex- ample, Accuracy is de/uniFB01ned as the ratio of the correctly predicted instances to all the instances, ( TP+ TN)/(TP+TN+FP+FN). F/one.pnum-score represents the harmonic mean of precision and recall, where precision isTP=(TP+FP)and recall is TP=(TP+FN). Accuracy and F/one.pnum-score are the most popular metrics, but they lead to the overoptimistic in/uniFB02ated measures especially on imbalanced datasets because several classi/uniFB01ers learn towards the majority class [/two.pnum/one.pnum, /two.pnum/two.pnum]. For instance, when a dataset has /zero.pnum./zero.pnum/one.pnum %mi- nority class and a model predicts all minority classes incorrectly, i.e., all data points are classi/uniFB01ed as the majority class, the model accuracy is /nine.pnum/nine.pnum./nine.pnum/nine.pnum %, which is nearly perfect. In structural engineering practice, however, it is critical to identify structural failures (the minority class) rather than safe structures (the ma- jority class), thereby necessitating using the right metric that can correctly predict both classes in a binary classi/uniFB01cation. When a structure is designed to have little load redistribution, the target reliability index Tis /two.pnum./seven.pnum [/three.pnum/one.pnum], which corresponds to the failure probability Pfof /three.pnum./five.pnum103, derived from =1(Pf), where 1= inverse standard normal cumulative density function. Moreover, ASCE /seven.pnum [/three.pnum/two.pnum] suggests using a target relia- bility index between /three.pnum./zero.pnum and /four.pnum./zero.pnum for structural components subjected to dead, live, and other loads except earthquake loads, depending on the risk category from I through IV where category I represents the lowest level of risk to human life. Likewise, a classi/uniFB01cation dataset for structural design problems will be severely /five.pnum imbalanced, which is indicated from the values of TandPf. As structural design problems often include an imbalanced dataset, it is challenging to select an adequate statistical metric that provides informative and truthful results. This study employed three statistical measures to evaluate the performance including speci/uniFB01city, recall, and the Matthews correlation coe/uniFB03cients (MCC) [/two.pnum/nine.pnum]. For imbalanced class distributions, the majority class is typically referred to as the negative outcome and the minority class is assigned to the positive outcome. Therefore, structural failure is the positive outcome and no failure is the negative outcome. Speci/uniFB01city is the probability that an actual negative will test negative and is calculated by TN=(TN+FP), which is the true negative rate. Speci/uniFB01city refers to how well a model identi/uniFB01es the frames which have no failure. Recall, also called the true positive rate, is the ratio of correct positive predictions to the total positive examples and is computed by TP=(TP+FN). Recall informs how many positive predictions are missed from the prediction. The MCC is a reliable measure for imbalanced classi/uniFB01cation problems because it takes into account the ratio between positive and negative outcomes, which are not considered in both speci/uniFB01city and recall. The MCC is independent of the class imbalance, thus can reduce misleading results on imbalanced datasets [/two.pnum/two.pnum]. The value of MCC varies between -/one.pnum and /one.pnum, similar to other correlation coe/uniFB03cients. The score is high only when all four categories in the confusion matrix are generated correctly. The MCC is computed by: MCC =TPTNFPFNp (TP+FP)(TP+FN)(TN+FP)(TN+FN)(/one.pnum/zero.pnum) /three.pnum Reliability-based sensitivity analysis Existing steel design speci/uniFB01cations [/three.pnum/three.pnum–/three.pnum/six.pnum] provide guidance for inelastic analysis, also referred to as ad- vanced analysis and GMNIA, which directly considers geometric and material nonlinearities and includes uncertainty in system, member, and connection strength and sti/uniFB00ness. Estimating structural performance with certainty is challenging because of the inherent uncertainty in a structural system which a/uniFB00ects system performance. Reliability-based sensitivity analysis estimates the e/uniFB00ect of an input variable by evaluating the structural performance with the variable under consideration as random while all other variables are at their nominal values. After repeated simulations for each property under consideration, the probability of failurePfis estimated by n=N [/three.pnum/seven.pnum] wheren= number of simulations which resulted in failure and N= total number of simulations. The system reliability index is computed based on Pf. Unlike a reliability analysis which takes into account multiple random variables simultaneously, a sensitivity analysis consid- ers only one random variable per simulation set to examine how the random variable a/uniFB00ects the system behavior, therefore the strength distribution might have a smaller COV compared to that from a reliability analysis. The normal probability plot [/three.pnum/eight.pnum] is used to estimate Pfand when no failure cases occur. Researchers have investigated the system reliability of various steel structures estimated by considering uncertainties in the systems. Buonopane [/three.pnum/nine.pnum] conducted a reliability sensitivity study on two steel frames by considering uncertainties in yield strength, elastic modulus, residual stress, and sway and bow imperfec- tions. Szyniszewski [/four.pnum/zero.pnum] investigated the e/uniFB00ect of random geometric imperfections on progressive collapse propagation by analyzing /three.pnum-D steel framed buildings with uncorrelated geometric imperfections between structural members. Shayan et al. [/four.pnum/one.pnum] presented a probabilistic study regarding modeling random geomet- ric imperfections on regular and irregular sway and braced planar steel frames. Thai et al. [/four.pnum/two.pnum] evaluated the system reliability of steel frames with semi-rigid connections. Uncertainties in gravity loads, material properties, cross-sectional properties, and connection properties were included in the reliability analysis. Zhang et al. [/four.pnum/three.pnum] examined the system reliability of /uniFB01ve steel structures including a beam, a portal frame, and three low-rise frames. Randomness considered in the analysis includes gravity loads, material proper- ties, cross-sectional properties, and sway imperfection. Cardoso et al. [/four.pnum/four.pnum] calibrated the system reliability of cold-formed steel portal frames with uncertain parameters in material properties, cross-section thick- ness, joint properties, and geometric imperfections. /six.pnum /four.pnum Data collection /four.pnum./one.pnum Structural system Two example steel frames designed according to AISC /three.pnum/six.pnum/zero.pnum [/three.pnum/four.pnum] are analyzed in this study, which have the same layout but di/uniFB00erent member sizes and loads, adopted from [/four.pnum/five.pnum]. Fig. /one.pnuma and Table. /one.pnum summarize the information of the example frames including geometry and applied loads. The frames are modeled in OpenSees [/four.pnum/six.pnum] with displacement-based and /uniFB01ber-type elements. All cross-sections contain the residual stress with the Galambos and Ketter pattern [/four.pnum/seven.pnum] and the /uniFB01ber distribution as shown in Fig. /two.pnum. The nominal peak compressive residual stress value is /zero.pnum./three.pnum Fyn, whereFyn= nominal material yield strength of /two.pnum/four.pnum/eight.pnum MPa (/three.pnum/six.pnum ksi). Initial sway imperfection of h//five.pnum/zero.pnum/zero.pnum was given to all columns in the frame, where his the story height. The inelastic /uniFB01nite element (FE) analyses are conducted with all nominal properties to /uniFB01nd the nominal value of the ultimate load ratio , which is the ratio of ultimate to factored design loads. The ultimate load ratios obtained from the analyses are /one.pnum./zero.pnum/eight.pnum for both frames, i.e., they have a limited capacity for load redistribution. Fig. /one.pnumb and c illustrate the location of the most highly yielded zones for Frames /one.pnum and /two.pnum, respectively. Frame /one.pnum fails by the instability (inelastic buckling) of a ground /uniFB02oor column C/two.pnum and at failure members C/two.pnum and B/two.pnum are partially yielded. Frame /two.pnum fails from a gradual sequence of yielding, thereby multiple members have highly yielded zones where a location along the member yielded greater than /seven.pnum/five.pnum %, which are B/one.pnum, B/two.pnum, B/three.pnum, B/four.pnum, C/five.pnum, and C/six.pnum. 4.57m 6.10mPo0.467Po 6.10m 14.63m 53.5%80.9% 93.8% 95.8% 85.8%81.7% 63.1% 91.4%92.5% 82.5% C4 C5 C6 C1 C2 C3B3 B4 B1 B2(a) (b) (c) C2B2C5 C6B3 B4 B1 B2 Figure /one.pnum: (a) Frame layout; location of highly yielded zones for (b) Frame /one.pnum; (c) Frame /two.pnum Table /one.pnum: Member sizes and applied loads Element Frame /one.pnum Frame /two.pnum C/one.pnum W 620 W1214 C/two.pnum W 1482 W1499 C/three.pnum W 1468 W1482 C/four.pnum W 68:5 W1012 C/five.pnum W 14145 W14109 C/six.pnum W 14145 W14109 B/one.pnum W 30132 W2784 B/two.pnum W 36182 W36135 B/three.pnum W 2455 W1840 B/four.pnum W 30116 W2794 Loads ( Po) /one.pnum/one.pnum/one.pnum./eight.pnum/six.pnum kN/m /one.pnum/zero.pnum/nine.pnum./four.pnum/five.pnum kN/m /four.pnum./two.pnum Uncertainty The Monte Carlo sampling method is used to generate samples of the uncertainties in material yield strength Fy, modulus of elasticity E, sway imperfection, and residual stress. Table /two.pnum summarizes the statistical infor- mation and