This study presents a machine learning-based approach for sensitivity analysis to examine how pa- rameters a/uniFB00ect a given structural response while accounting for uncertainty. Reliability-based sensitivity analysis involves repeated evaluations of the performance function incorporating uncertainties to esti- mate the in/uniFB02uence of a model parameter, which can lead to prohibitive computational costs. This chal- lenge is exacerbated for large-scale engineering problems which often carry a large quantity of uncertain parameters. The proposed approach is based on feature selection algorithms that rank feature impor- tance and remove redundant predictors during model development which improve model generality and training performance by focusing only on the signi/uniFB01cant features. The approach allows performing sen- sitivity analysis of structural systems by providing feature rankings with reduced computational e/uniFB00ort. The proposed approach is demonstrated with two designs of a two-bay, two-story planar steel frame with di/uniFB00erent failure modes: inelastic instability of a single member and progressive yielding. The feature variables in the data are uncertainties including material yield strength, Young’s modulus, frame sway im- perfection, and residual stress. The Monte Carlo sampling method is utilized to generate random realiza- tions of the frames from published distributions of the feature parameters, and the response variable is the frame ultimate strength obtained from /uniFB01nite element analyses. Decision trees are trained to identify important features. Feature rankings are derived by four feature selection techniques including impurity- based, permutation, SHAP , and Spearman’s correlation. Predictive performance of the model including the important features are discussed using the evaluation metric for imbalanced datasets, Matthews cor- relation coe/uniFB03cient. Finally, the results are compared with those from reliability-based sensitivity analysis on the same example frames to show the validity of the feature selection approach. As the proposed ma- chine learning-based approach produces the same results as the reliability-based sensitivity analysis with improved computational e/uniFB03ciency and accuracy, it could be extended to other structural systems. /one.pnum Introduction The use of structural analysis that incorporates the e/uniFB00ect of inelastic material and geometric properties, which is referred to as advanced analysis, inelastic analysis, and geometrically and materially nonlinear analysis with imperfections (GMNIA), has increased with signi/uniFB01cant advances in computerized structural analysis. Although there are now various sophisticated structural analysis software used by structural en- gineers, it is di/uniFB03cult to predict the actual performance of a steel frame with certainty due to ever-present uncertainties in material and geometric properties, and structural loads. For example, the shape and mag- nitude of geometric imperfections have a signi/uniFB01cant in/uniFB02uence on the response of a structure, and hence need to be modeled accurately when determining the load carrying capacity of a frame building. Residual stress and elastic modulus may have a signi/uniFB01cant e/uniFB00ect on strength by reducing the frame sti/uniFB00ness and in/uniFB02uence the system reliability. In order to accurately capture structural responses, it must be understood how the uncertainties in all the factors a/uniFB00ecting a structural system in/uniFB02uence system performance. Design by advanced analysis is permitted in existing steel design speci/uniFB01cations including Section /five.pnum of Eu- rocode /three.pnum [/one.pnum], Appendix /one.pnum of AISC /three.pnum/six.pnum/zero.pnum [/two.pnum], Appendix D of AS /four.pnum/one.pnum/zero.pnum/zero.pnum [/three.pnum], and Annex O of CSA S/one.pnum/six.pnum [/four.pnum]. The advanced analysis allows the direct modeling of system and member imperfections in a numerical model. The analysis shall consider geometric nonlinearities including geometric imperfections ( P-andP-), and 1 sti/uniFB00ness reductions including partial yielding of the cross section accentuated by the presence of residual stresses. It is demonstrated that inelastic analysis can capture accurate results including the complex in- teractions between members of a large structural system, and can capture the bene/uniFB01cial system e/uniFB00ect of load redistribution after the initial formation of plastic hinges [/five.pnum–/nine.pnum]. Previous studies have investigated the system reliability of steel frames for system-based design by the inelastic method with a consideration of uncertainties in material and geometric imperfections [/one.pnum/zero.pnum–/one.pnum/three.pnum]. Taras and Huemer [/one.pnum/four.pnum] investigated the impact of di/uniFB00erent load sequences on the reliability of three resistance functions including plastic cross- sectional resistance, in- and out-of-plane buckling resistance of beam-column, and resistance of a planar portal frame structure subjected to various load combinations. The study considered uncertainties in struc- tural loads, strength and sti/uniFB00ness, cross-sectional dimensions, and out-of-plumbness. Cardoso et al. [/one.pnum/five.pnum] developed system reliability-based criteria for the design of steel rack frames, including randomness of ge- ometric and material properties. The study derived system resistance factors for rack frames based on /uniFB01ve series of rack frames subjected to gravity loads. To estimate how the uncertainties in/uniFB02uence system performance, sensitivity analysis can be conducted and several studies have proposed approaches for reliability-based sensitivity analysis. Bjerager and Krenk [/one.pnum/six.pnum] considered the derivatives of the Hasofer-Lind index as local sensitivity of the failure probability based upon the /uniFB01rst-order reliability method. Garnier et al. [/one.pnum/seven.pnum] estimated the sensitivities by evaluating the gradient of the probability of normal random variables obtained from Monte Carlo tools. Buonopane [/one.pnum/eight.pnum] presented reliability-based sensitivity analyses of two planar steel frames designed using /uniFB01rst-order and second-order analysis. However, the reliability-based sensitivity analysis involves repeated evaluations of the performance function to estimate the in/uniFB02uence of the incorporated uncertainty, which can lead to prohibitive computational resources. Many e/uniFB03cient reliability-based sensitivity analysis methods to reduce the computational e/uniFB00ort have been proposed. Rubinstein and Kroese [/one.pnum/nine.pnum] introduced the application of the score function method that measures the partial derivatives of each parameter without additional simulations for the estimation of all sensitivities. The line sampling based sensitivity method was proposed by Lu et al. [/two.pnum/zero.pnum] to measure the sensitivity from the partial derivatives of the failure probability in the score function approach. Proppe [/two.pnum/one.pnum] applied the moving particles method [/two.pnum/two.pnum] to local sensitivity analysis, which moves samples to new locations in the design space and counts the number of moves reaching the failure region for the calculation of the failure probability. Consideration of the large set of parameters in reliability-based sensitivity analysis is challenging due to the repeated evaluation of the performance function. Moreover, high-dimensional data increases the curse of dimensionality [/two.pnum/three.pnum], which refers to the phenomenon arisen from too many feature variables that increase sparsity in data, storage space, and computational costs. The high-dimensional data is often resorted to feature selection in order to avoid potential problems. Feature selection techniques are broadly used in the /uniFB01eld of data analysis and machine learning. Several papers investigated various feature selection methods and compared their performances [/two.pnum/four.pnum–/two.pnum/seven.pnum]. Although structural design requires considering several un- certainties, the application of feature selection techniques to structural design problems is limited. Some papers developed the machine learning algorithms for developing seismic fragility curves for frame build- ings [/two.pnum/eight.pnum], predicting the capacity of reinforced concrete beams [/two.pnum/nine.pnum] and cold-formed steel channels [/three.pnum/zero.pnum], compressive strength of concrete [/three.pnum/one.pnum], failure mode of RC bridge columns [/three.pnum/two.pnum] using the experimental or /uniFB01eld observation data. Hwang et al. [/three.pnum/three.pnum] investigated the e/uniFB00ect of uncertainties on the seismic collapse risk of RC building frames using the data generated by Monte Carlo simulation. Although these papers ex- amined the relative importance of input features, further studies may need to examine how the important features are determined depending on the failure modes and structural responses. As the application of machine learning on steel structures is fewer than other materials, it is meaningful to explore the e/uniFB00ect of uncertainties related to the behavior of steel frames. Moreover, it is important to reduce the number of features to improve the model e/uniFB03ciency and accuracy in machine learning, but there is a lack of study in using feature selection techniques to structural design problems. In this paper, the feature importance methods using machine learning technique are implemented to ex- amine the e/uniFB00ects of random properties on the strength of two planar steel frames: one frame fails by inelastic instability of a single member and the second frame fails by progressive yielding. Both frames have limited capability to redistribute load. The design frame strengths were designed using the inelastic method speci/uniFB01ed in AISC /three.pnum/six.pnum/zero.pnum [/two.pnum]. Uncertainties of yield strength, elastic modulus, sway imperfection (out- of-plumbness), and residual stress were considered. To collect the database, /uniFB01nite element analyses were conducted in OpenSees and the Monte Carlo sampling method was utilized. The four feature importance methods are employed including impurity-based, permutation, SHAP [/three.pnum/four.pnum], and Spearman’s correlation co- e/uniFB03cient [/three.pnum/five.pnum]. The feature importance methods generate the ranking of features based on the importance /two.pnum score, and the features with the highest scores are considered more likely to in/uniFB02uence structural failure. In addition, the class imbalance issue in structural design problems is examined. The model performance is evaluated by using the Accuracy and the Matthews correlation coe/uniFB03cient (MCC) [/three.pnum/six.pnum], which is appropriate for the imbalanced data. Reliability-based sensitivity studies of steel frames were conducted not only to validate the proposed approach but also to provide a further understanding of how randomness in material and geometric properties a/uniFB00ect the system strength depending on the failure mode. The aim of this paper is to (/one.pnum) propose the feature importance strategy based on machine learning techniques for examining how uncertainties a/uniFB00ect the structural failures and provide prospects in the application of feature importance techniques to sensitivity analysis of structures, (/two.pnum) describe the feature selection approach that improves model accuracy and e/uniFB03ciency in prediction of structural failures, (/three.pnum) address how to develop and evaluate the model with the class imbalanced data, and /uniFB01nally (/four.pnum) provide undemanding solutions for using machine learning techniques to structural engineers. /two.pnum Proposed framework for sensitivity analysis of structural systems The traditional sensitivity analysis of structures is based on reliability analysis that computes the reliabil- ity index or probability of failure in respect of a factor under consideration to examine the e/uniFB00ect of the factor on the structures. However, the reliability-based sensitivity analysis requires repeated evaluations of the performance function and additional simulations. If a structure to be considered has complexity such as nonlinear structural behavior or a large quantity of structural members, computational costs will increase and thereby decreasing computational e/uniFB03ciency. In order to avoid repeated evaluations of the performance function and to reduce the computational e/uniFB00ort, this study proposes a framework using ma- chine learning techniques which can be used as an alternative to the reliability-based sensitivity analysis of structures with high e/uniFB03ciency and accuracy. The machine learning-based feature selection methods are employed to estimate feature importance that can be ranked based on the importance score. A feature with the maximum score is considered the most important feature. Once the rank of features is obtained, a machine learning classi/uniFB01er is implemented to measure the model performance when the model carries only the top- kfeatures, where k= the number of important features. The model performance is evaluated by varying the number of features considered in the feature set. The least important features that decrease the model performance or result in the same performance can be removed from the feature set to increase the accuracy and e/uniFB03ciency of the model. The proposed method is demonstrated by comparing the results obtained from reliability-based sensitivity analysis. /two.pnum./one.pnum Applied structural system Two designs of a two-bay, two-story non-symmetric planar steel frame based on those of Ziemian [/three.pnum/seven.pnum] have been selected as examples, which are designed according to AISC /three.pnum/six.pnum/zero.pnum [/two.pnum]. Fig. /one.pnum shows the geome- try, applied loads, and support conditions for the frames. The member sizes and loads of each frame are summarized in Table /one.pnum. The frames are subjected to the gravity load combination of /one.pnum./two.pnum D+ /one.pnum./six.pnum L, with the live-to-dead load ratio assumed to be L=D = 1:5, which is a typical value [/three.pnum/eight.pnum]. 4.57m 6.10mPo0.467Po 6.10m 14.63mC4 C5 C6 C1 C2 C3B3 B4 B1 B2 Figure /one.pnum: Frame layout Finite element (FE) analyses of the two frames were performed with OpenSees [/three.pnum/nine.pnum], including material and geometric nonlinearity. An elastic-perfectly-plastic material model was used as the steel material prop- erty to improve simulation performance. Each element was subdivided into /one.pnum/six.pnum displacement-based and /three.pnum Table /one.pnum: Member sizes and loads for example frames 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 /uniFB01ber-type elements. Connections were assumed to be fully-rigid to disregard any potential /uniFB02exibility of the connections. The frames were given sway imperfections of h//five.pnum/zero.pnum/zero.pnum, where h= the frame height, by displacing the relevant nodes from their nominal locations. All cross-sections contain the Galambos and Ketter residual stress model [/four.pnum/zero.pnum] which has tensile stress along the entire web, tension at the web-to- /uniFB02ange intersection, and peak compression at /uniFB02ange tips equal to /zero.pnum./three.pnum FynwhereFyn= nominal material yield strength. The presence of residual stresses was considered by applying residual stress to each /uniFB01ber in the cross section. A reduction factor of /zero.pnum./nine.pnum was applied to FyandEof all members in order to design members by the inelastic method as speci/uniFB01ed in Appendix /one.pnum of AISC /three.pnum/six.pnum/zero.pnum [/two.pnum]. The structures were analyzed using second-order inelastic analysis. The load-displacement curves and the location of highly yielded zones from the inelastic analysis are pre- sented in Fig. /two.pnum. If a section has at least /seven.pnum/five.pnum %of cross-section area yielded, it was considered highly yielded. Analyses of the frames were performed with all properties at nominal values. The ultimate load ratio , the ratio of ultimate to factored design loads, is /one.pnum./zero.pnum/eight.pnum for both frames, indicating a limited capacity of inelastic load redistribution for both frames. It is observed that Frame /two.pnum has a clear yield point while Frame /one.pnum exhibits a rounded curve around the yield point due to their di/uniFB00erent failure modes. There was no highly yielded zone in Frame /one.pnum but elements of B/two.pnum and C/two.pnum located close to their connection were partially yielded with yield ratios of /six.pnum/three.pnum./one.pnum %and /five.pnum/three.pnum./five.pnum %. Frame /one.pnum collapsed due to the instability (global buckling) of the slender column, C/two.pnum. The failure mode of Frame /two.pnum is progressive yielding with six highly yielded members: B/one.pnum, B/two.pnum, B/three.pnum, B/four.pnum, C/five.pnum, and C/six.pnum. The FE model used in this study has the same load ratio and failure modes with the previous studies [/one.pnum/zero.pnum, /one.pnum/three.pnum], therefore verifying the modeling technique used herein. 53.5% 82.5% 85.8% 95.8%80.9% 93.8%81.7%63.1% 91.4%92.5%C2B2 B3 B4 B1 B2C5 C6 0 50 100 150 Roof displacement (mm)00.40.81.2Load ratio, λ = 1.080 50 100 150 Roof displacement (mm)00.40.81.2Load ratio, λ = 1.08(a) (b) Figure /two.pnum: Load-displacement curve and location of highly yielded zones: (a) Frame /one.pnum; (b) Frame /two.pnum /four.pnum /two.pnum./two.pnum Uncertainty As it is di/uniFB03cult to predict the actual performance of a steel frame with certainty due to ever-present un- certainties in material and geometric properties, therefore several steel design codes [/one.pnum, /two.pnum, /four.pnum] require the engineer to consider the uncertainties in the analysis even with requiring advanced structural analysis. This study considered the uncertainties in yield strength ( Fy), elastic modulus ( E), sway imperfection, and residual stress. Statistical information for these uncertainties is summarized in Table /two.pnum. Yield strength was modeled as a lognormal distribution with a mean value /one.pnum./one.pnum/zero.pnum Fynand a COV of /zero.pnum./zero.pnum/six.pnum, where Fyn= nominal yield strength of /two.pnum/four.pnum/eight.pnum MPa (/three.pnum/six.pnum ksi) [/four.pnum/one.pnum]. Elastic modulus was modeled as a lognormal distribution with a mean value of Enand a COV of /zero.pnum./zero.pnum/four.pnum, where En= nominal elastic modulus of /two.pnum/zero.pnum/zero.pnum GPa (/two.pnum/nine.pnum,/zero.pnum/zero.pnum/zero.pnum ksi) [/four.pnum/one.pnum]. After random samples of FyandEare randomly drawn from each distribution, /zero.pnum./nine.pnum Fyand /zero.pnum./nine.pnum Eare used as material properties in the analysis in order to conduct the inelastic analysis according to Appendix /one.pnum of AISC /three.pnum/six.pnum/zero.pnum [/two.pnum]. The magnitude of sway imperfection was modeled as a lognormal distribution with a mean value of /one.pnum//seven.pnum/seven.pnum/zero.pnum and a COV of /zero.pnum./eight.pnum/seven.pnum/five.pnum [/four.pnum/two.pnum]. After a random magnitude of imperfection has been determined, the random value is multiplied by the height of the frame to consider the imperfection of the overall frame. The /uniFB01rst story and the second story have the same direction of sway imperfection; i.e., multistory columns have no changing direction of sway imperfection therefore the magnitude of sway imperfection accumu- lates continuously at each story. The direction of imperfection was distributed to have equal probability in each direction (left or right). The scale factor of residual stress ( X) was modeled as a normal distribution with a mean value of /one.pnum./zero.pnum/six.pnum/four.pnum and a COV of /zero.pnum./two.pnum/seven.pnum [/four.pnum/three.pnum], which is based on the experimental data of Galambos and Ketter [/four.pnum/zero.pnum]. The random scale factor Xis multiplied to /zero.pnum./three.pnum Fynto compute the peak compression stress. Once the peak compressive residual stresses are determined, the rest of the residual stresses in the cross section were set based on the residual stress pattern [/four.pnum/zero.pnum]. The frames are assumed as spatially correlated, which means that all columns or beams have the same random properties. This assumption was consistent with common construction practice [/one.pnum/three.pnum]. Table /two.pnum: Description of feature variables Variable Mean COV Distribution