Steel Moment-resisting Frames Research Articles

Optimal design of chevron braced friction damper for mainshock–aftershock vulnerability control

This study proposes the optimum design of chevron braced friction dampers (CBFDs) for the vulnerability control of steel moment–resisting frames (SMRFs) under mainshock-aftershock (MS-AS) sequences. First, the optimum parameters of the CBFD systems installed in the stories of an inelastic SMRF are found through minimizing the maximum damage index of stories averaged over seven scaled earthquake excitations. The uniform distribution of the story damage along the height of the controlled SMRF is adopted as constraint in the optimization procedure. The damage index is calculated based on the Park-Ang damage model which is expressed based on a linear combination of deformation, moment, and absorbed hysteretic energy of structural elements imposed by an earthquake excitation. Results reveal that the optimized CBFDs-equipped SMRF exhibits better distribution of the story damage than that of the uncontrolled SMRF. Finally, the vulnerability assessment of the optimized CBFDs-equipped SMRF under MS and MS-AS ground motions is investigated by the fragility curves. The fragility assessment emphasize that the optimized CBFDs effectively mitigate the seismic vulnerability of the controlled SMRF under MS-AS sequences at different damage states.

Design procedure of a shear fuse system for steel moment-resisting frames

In steel moment-resisting frames (MRFs), seismic energy is generally dissipated through the formation of flexural plastic hinges at beam ends. In ensuring such a mechanism, current code provisions limit the utilization of beams with small span-to-length ratios in MRFs. This study presents the design procedure of a novel shear fuse system that integrates replaceable shear links with non-prismatic beams to enhance the performance and repairability of MRFs with small beam span-to-length ratios. The proposed design strategically weakens the beam at its midspan by incorporating a replaceable link, prioritizing shear yielding in the link over flexural yielding in the beam outside the link, thus ensuring adequate frame ductility and protecting the critical beam-to-column connections. It can function as an efficient replaceable shear fuse system, as the damaged fuse can be substituted with a new one following a severe earthquake. In fulfilling the study’s purpose, first, several formulas are developed to provide a comprehensive design procedure. Next, example design calculations are presented for an MRF with a small beam span-to-depth ratio. Ultimately, a numerical investigation is provided to examine the proposed system’s performance compared to conventional MRFs. The results indicate that the proposed shear fuse system provides a more consistent cyclic behavior than the conventional prismatic beam design, exhibiting stable strain-hardening behavior up to the 5 % story drift ratio without noticeable strength degradation. It appropriately shifts the plastic hinges away from the beam-to-column connections while increasing the frame’s overall ductility with 1.14 times greater energy dissipation in the final loading cycles.

Modification of wavelet-based downsampling method to reduce computational error in nonlinear dynamic analysis

One of the significant obstacles in conducting linear and non-linear time history analysis is its time-consuming nature. In this article, a new downsamlping method based on discrete wavelet transform (DWT) and smoothing is proposed to overcome this problem. In order to assess the precision of this approach, 50,000 linear and non-linear dynamic analyses of single degree of freedom (SDOF) systems and 300 nonlinear dynamic analyses of frame structures have been performed. One hundred Fema440 records were utilized to generate approximate waves up to the third level and the outcomes of this method were then contrasted with those of DWT. It has been demonstrated that the third-level approximate wave produced by DWT, previously considered dependable in other research, generates significant errors in results and the average error (absolute error percentage of the acceleration spectrum) of its third-level approximate wave is approximately 17.5%. On the other hand, the proposed method generated approximate waves with an average error of less than 4.5% across all behavior coefficients and periods and the error rate decreases as the period and behavior coefficient increase. Analysis of steel moment-resisting frames indicated that the lowest error in both methods is achieved for the base shear and across different engineering demand parameters, the average error rate for the proposed method was below 7.5%. Furthermore, caution must be exercised when employing the proposed method for structures with periods shorter than 0.5 s.

Post-earthquake repairability-based methodology for enhancing steel MRFs

Self-centering beam-to-column connections (SBCs) were widely investigated to enhance conventional steel moment-resisting frame’s (MRF’s) post-earthquake repairability by reducing residual displacements. The peak displacement-targeted design methodology was adopted to develop the design guidelines for enhancing MRFs with the emerging self-centering members, where the desired post-earthquake repairability of the enhanced MRF needs to be achieved with inevitable iterative dynamic analyses. This paper intends to develop a novel post-earthquake repairability-based methodology for enhancing MRFs with SBCs, where residual and peak displacements are both targets. To this end, artificial neural network (ANN) models were developed to estimate the peak and residual displacements of the enhanced MRFs with SBCs under given seismic intensities. Peak and residual displacement-based design (PRBD) procedures were developed based on the proposed post-earthquake repairability-based methodology and established ANN models. Four design cases were designed through the developed PRBD procedure and analyzed. The design and analysis outcomes demonstrate that the improved MRFs can attain the intended post-earthquake repairability and meet the peak and residual displacement objectives without necessitating iterative design and dynamic analyses. This confirms the effectiveness and precision of the proposed post-earthquake repairability-based approach and PRBD procedure. Moreover, the enhanced MRFs with partially self-centering behavior achieved nearly zero residual displacements (lower than 0.2%) under maximum considered earthquakes. This finding shows that the partially self-centering MRF is a more cost-efficient solution for developing seismic resilient MRF than fully self-centering MRF to achieve excellent post-earthquake repairability with a lower requirement of self-centering capacities.

Probabilistic alternate path analysis of steel moment-resisting frames

This research presents a probabilistic assessment of progressive collapse in intermediate Steel Moment-Resisting Frames (SMRFs) subjected to different levels of column damage. Damage levels were represented by gradual reductions in column stiffness to the extent that tensile forces are created in columns sustaining large deformations, and columns were allowed to enter completely plastic zone. To this aim, low- to mid-rise structures with 4, 8, and 12 stories were examined. The effect of composite slabs on the vertical displacement response of SMRFs was taken as a variable. A number of 11 column damage scenarios were defined with either one or two damaged columns, located at different positions in the plan, which were applied to all floor levels. Incremental dynamic analysis was conducted for each damage scenario using the finite element OpenSees framework. Moreover, a state-of-the-art approach was employed in fragility analysis. The results showed that higher floors are more sensitive to partial damage due to less structural components involved in progressive collapse. However, in case of large damages, lower floors prove to be more critical due to greater deal of gravity loads. Shorter and taller structures perform better in large and partial damages to column, respectively. Moreover, the failure probability of SMRFs reduced by considering the composite slab stiffness. Subjected to single-column damage scenarios, SMRFs reach life safety limit state once tensile forces are created in the damaged column. In contrast, all performance levels are met in double-column damages when the column has still its compressive capacity.

Numerical investigation on stress concentrations of circular steel X joints

The most important reasons for the usage of steel as a building material are its high strength and ductile behavior, which can be defined as inelastic deformation or displacement capacity under a certain load. Steel frames are required to sustain the horizontal loads during an earthquake. The steel frames are classified as moment-resisting steel frames, concentric steel braced frames, eccentric braced steel frames, and buckling restrained braced frames. These steel frames can be designed with a high ductility level frame, limited ductility level frames, and ductility level mixed frames. In this study, stress concentrations due to the maximum compression force that may occur during an earthquake have been examined by the finite element method at the connection point of the circular cross-sectional braces of the central X-type braced steel frame with a high ductility level. Stress concentrations in the X-type brace were investigated depending on the variation of the cut surface required for the connection plate due to the change in the angle between the braces and the differences in the weld thickness applied in this region. In the models, the angle between the braces is designed as 60°, 75°, and 90° and the weld thicknesses are defined as 3.5 mm, 5 mm, and 7 mm. The results are obtained by applying the same compressive and tensile forces to the braces under the same boundary conditions and compared for these different models. It is obtained that the highest stress values occur along the direction of the tensile force in the tension profile and nearly at a distance of 5 mm after the cut. The maximum stress values decrease when the intersection angle between the braces increases from 60° to 90°.

Seismic behaviour of a steel moment resisting frame structure featuring hourglass double split tee joints

The occurrence of brittle fractures in welds of full-strength joints of steel moment-resisting frames (MRFs) during earthquakes in Northridge (1994) and Kobe (1995) marked a turning point in seismic design philosophies for steel structures. This led to the development of new strategies focusing on enhancing structural resilience and energy dissipation. While Reduced Beam Sections (RBS) were traditionally favoured, recent approaches include partial-strength joints and replaceable dissipative fuses, in some cases also equipped with self-centring components. These systems often employ friction or yielding dampers, ensuring that the weakest joint component comprises well-designed dampers for optimal ductility and energy dissipation.In this context, the present study evaluates the performance of a large-scale structure featuring MRFs with partial-strength double-split tee (DST) joints and hourglass dampers. These joints are conceptually similar to ADAS dampers and the Simpson Strong Tie moment connection prequalified by AISC 358–16. The research involves conducting a pseudo-dynamic test campaign on a two-storey steel structure designed according to EC8 provisions. Findings from experiments and a numerical model developed using OpenSees are discussed. Incremental Dynamic Analyses (IDAs) compare structures with hourglass DST joints to those with full-strength joints, showing enhanced self-centring capacity in DST-equipped structures despite lacking dedicated self-centring components. Additionally, structures with hourglass DST joints demonstrate significant low-cycle fatigue life, potentially reducing the need to replace dissipative components after severe seismic events.

Surrogate model‐aided global sensitivity analysis framework for seismic consequences estimation in buildings

AbstractSeismic consequences estimation for individual buildings is valuable for various stakeholders, including government entities, building owners, and insurers. The robustness of estimation results in the presence of incomplete input information can typically be investigated through sensitivity analysis. However, the estimation process's complexity and the sensitivity analysis's computational burden hinder its practical application, which requires a more efficient procedure to facilitate broader use. This paper proposes a novel framework for sensitivity analysis of seismic consequences estimation to improve the efficiency and reliability of such analysis. The proposed approach encompasses three key components: (1) stochastic ground motion modeling (SGMM)‐based seismic consequences estimation to evaluate the economic, environmental, and social consequences given specific buildings by considering different hazard levels, (2) the training of surrogate model (Gaussian process model) for structural analysis to reduce the computational cost of the evaluation process, and (3) variance‐based global sensitivity analysis to investigate the importance of parameters of concern in the estimation process. The entire procedure is implemented in Python, adhering to object‐oriented programming, and does not rely on external software. Then, the proposed methodology is applied to two distinct three‐story steel moment‐resistant frames (SMRFs) subjected to four different hazard levels to demonstrate its effectiveness. The SGMM method can generate specific ground motions for each hazard level, mitigating the potential for result bias from using ground motions with unrealistic characteristics. Furthermore, the SGMM method is particularly suitable for automated analysis processes reducing the laborious task of screening ground motions from the database. Comparative analysis with surrogate‐free estimation reveals that the surrogate‐based analysis delivers reliable results with significantly reduced computational cost. The results of analyzing different structures under varying hazard levels reflect the variability in sensitivity analysis of consequences estimation, highlighting the necessity of the proposed flexible and efficient framework. Furthermore, the proposed framework's advantages, limitations, and future research needs are discussed.

Seismic response of high strength steel frames equipped with energy dissipation bays subjected to seismic sequences

Most of the seismic codes currently used are based on isolated earthquakes, whereas most earthquakes are seismic sequences acting repeatedly. In contrast to isolated earthquake scenarios, seismic sequences involve a series of seismic events, including a mainshock followed by aftershocks, suggesting that seismic sequences can reflect the structural damage accurately. This paper investigated the seismic performance of steel moment-resisting frames equipped with energy dissipation bays (EDBs) subjected to seismic sequences. It was demonstrated that the EDBs perform well in protecting main frames from isolated earthquakes of a certain magnitude, and the seismic sequences did not worsen the main frames' plastic damage. However, when the magnitude exceeded a certain threshold, the EDBs began to lose some of their protective capabilities, and the main frames' plastic damage from seismic sequences significantly increased. For the high strength steel frames equipped with EDBs, an approach that is more appropriate for estimating the residual damage of main frames in finite elements was suggested. The comparison results demonstrated the method's improved accuracy.

GNN-LSTM-based fusion model for structural dynamic responses prediction

With the rapid growth of deep learning technology, the potential for its use in structural engineering has substantially increased in recent years. This study proposes an innovative deep-learning fusion network architecture based on the graph neural network (GNN) and long short-term memory (LSTM) network. The proposed fusion model can accurately predict nonlinear floor acceleration, velocity, and displacement responses of typical steel moment-resisting frames (SMRF) with 4 through 7 stories subjected to strong ground motions. The fusion model framework overcomes a major drawback in existing deep learning models as it can predict the dynamic responses of various structures. This has widened the range of potential applications of structural surrogate models generated through deep learning technology on the design and analysis of building structures with different geometries. Additionally, this paper presents two LSTM-optimized learning strategies, namely packing padded sequences and sequences compression strategies. These strategies improve the model’s performance significantly without modifying its architecture. Finally, this study reveals that the model’s internal graph embedding is highly correlated with certain critical structural parameters, such as the first natural period and the height of a building. This shows that the proposed fusion model is interpretable and has the ability to extract vital information that influences structural dynamic responses.

Probabilistic Assessment of Global Damage Index for Steel Moment-Resisting Frames Based on Local Responses

ABSTRACT A new scenario-based method is proposed to determine the performance range of structural systems. The advantage of this method is using inter-storey drift as a damage index, while exploiting local damage indices of the main structural members without the need for detailed analytical models and computationally expensive non-linear dynamic analyses. A new damage function is proposed by linking the local (plastic hinge rotations) and global (inter-storey drifts) damage levels. The accuracy of the proposed function is evaluated and compared with the indices in ASCE41–06, ATC-13, and HAZUS for a set of steel moment-resisting frames subjected to earthquakes of varying intensities.

Effect of Beam-Column Connection Types on the Response Modification Factors of Steel Frames

A key component of steel structural design is the careful selection of structural modeling joints in steel structures, as the behavior of the joints affects the structure's strength and displacement characteristics. According to the capacity for transferring moment, connections in the analysis of frames are categorized as rigid, semi-rigid, or pinned. Also, the response modification factor (R-factor) is an effective parameter used in the seismic design of structures. However, the influence of the beam-column connection's stiffness factor on the response modification factor did not seem to have been considered in seismic design codes. Consequently, the R-factor under static pushover and dynamic loading is being studied for moment-resisting steel frames (MRSFs) with 3-, 6-, and 12-story using three different forms of beam-column connections depending on the connections' stiffness (m). The rigidities of the connections are taken 20, 10, and 5 for rigid, stiff semi-rigid, and flexible semi-rigid connections, respectively. The MRSFs are subjected to ten records with varying frequency contents and ground motion durations. The ductility reduction factor (Rμ), the over-strength reduction factor (Ro), and the R-factor were determined. The results indicated that the beam-column connection rigidity factors affected the R0, Rμ, and R-factors. Also, the R-factors were more affected by the rigidity factors for the beam-column connections and the number of story frames.

Probabilistic model updating of steel frame structures using strain and acceleration measurements: A multitask learning framework

This paper proposes a multitask learning framework for probabilistic model updating by jointly using strain and acceleration measurements. This framework can enhance the structural damage assessment and response prediction of existing steel frame structures with quantified uncertainty. Multitask learning may be used to address multiple similar inference tasks simultaneously to achieve a more robust prediction performance by transferring useful knowledge from one task to another, even in situations of data scarcity. In the proposed model-updating procedure, a spatial frame is decomposed into multiple planar frames that are viewed as multiple tasks and jointly analyzed based on the hierarchical Bayesian model, leading to robust estimation results. The procedure uses a displacement–stress relationship in the modal space because it directly reflects the elemental stiffness and requires no prior knowledge concerning the mass, unlike most existing model-updating techniques. Validation of the proposed framework by using a full-scale vibration test on a one-story, one-bay by one-bay moment resisting steel frame, wherein structural damage to the column bases is simulated by loosening the anchor bolts, is presented. The experimental results suggest that the displacement–stress relationship has sufficient sensitivity toward localized damage, and the Bayesian multitask learning approach may result in the efficient use of measurements such that the uncertainty involved in model parameter estimation is reduced. The proposed framework facilitates more robust and informative model updating.

Hybrid-self-centring steel frames: Insights and probabilistic seismic assessment

This paper comprehensively discusses the seismic performance of steel moment-resisting frames with hybrid-self-centring connections (SMRF-HSCC) employing energy dissipation sequences by probabilistic seismic assessment. The hysteretic model of the HSCC was first developed based on experimental results, and a simplified nonlinear-spring-based model to simulate the behaviour of the connection was proposed and verified by physical test data. A series of codified prototype structures were developed, considering the fracture behaviour of post-tensioned (PT) steel strands in the connections. The seismic performance of prototype structures was assessed in terms of nonlinear static analyses and nonlinear dynamic analyses. The results indicated that the hysteretic response of SMRF-HSCC exhibited trilinear hysteretic features with the expected energy dissipation sequences. The influence of the high-mode effects on SMRF-HSCC was mitigated with the increase in energy dissipation and post-yield stiffness of the structure. Finally, the fragility analyses of the prototype structures were further performed, and the risk assessment was also made considering a 50-year service period. The collapse probability and the exceedance probability of target residual drift over a 50-year service period of SMRF with conventional self-centring connections showing flag-shaped hysteresis were higher than that of SMRF-HSCCs. The results also confirmed that the collapse probability and exceedance probability of target residual drift over a 50-year service period of SMRF-HSCC can be adjusted by reasonably modulating the arrangement of the HSCCs.

Displacement Control of Existing Steel Moment-Resisting Frame Structures Using Buckling-Restrained Braces

ABSTRACT Large displacements are expected to occur in existing code-compliant steel moment-resisting frames during a severe earthquake, affecting the functionality of buildings. This study investigated the use of buckling-restrained braces for controlling the displacement response and ensuring an occupiable performance level of the structures. A design procedure for the braces based on the energy balance concept is proposed, where the hysteretic energy demand of the frame before and after attaching the braces is equated. Time-history analysis results show structures designed using the proposed procedure satisfied the targeted performance level, and no significant change in peak floor acceleration response was observed.

Collapse hybrid simulation for testing steel building columns subject to boundary condition changes

AbstractThis paper develops a hybrid simulation to experimentally evaluate the ultimate behavior of steel columns subject to boundary condition changes and bending moment redistribution due to the progressive seismic collapse of building structures. The proposed experimental method combines sub‐structuring test techniques and refined finite element (FE) analysis methods in a hybrid scheme. The inelastic behavior of steel moment‐resisting‐frames (MRF) is computationally simulated while ground‐floor columns are physically tested online with the FE frame analysis. Boundary conditions at the interface node between the column test and the frame analysis are reproduced with the aid of three actuators that impose axial loads and bending deformations to the column test. Two square tubular steel columns are tested as part of a five‐storey five‐bay steel MRF designed to employ two different heights of the ground floor (i.e., 4.6 and 5.1 m, respectively). The hybrid simulation revealed that individual failures of the peripherical frame members significantly alter the distribution of the bending moment along the height of the column test, thus changing the boundary conditions at its top end. The imposed moment distribution was further influenced by local buckling initiated at the base of the column test and ground‐floor beams and could cause the top end of the upper‐storey columns of the frame to buckle. In the MRF with 4.6 m‐height ground‐floor, this resulted in a global collapse mechanism that involved the ground floor and the second storey. In the MRF with 5.1 m‐height ground‐floor, a weak‐storey collapse mechanism was formed within the ground floor. Compared to a corresponding isolated column tested as cantilever using conventional methods, the columns tested with the proposed hybrid simulation exhibited a more favorable inelastic behavior primarily due to the more realistic loads and changes of boundary conditions which are ignored in conventional test methods.

The importance of hazard-consistency when estimating seismic residual drifts in steel moment frames

Residual drifts are routinely used to assess the post-earthquake safety of buildings. Despite their importance, studies on the probabilistic assessment of residual drifts in multi-storey buildings with Steel Moment Resisting Frames (SMRF) are far less common than those dealing with their transient peak drift counterpart. More seriously, although residual drift prediction models have been developed with a particular broad ground-motion type or soil condition in mind, all models proposed to date remain oblivious to the central issue of hazard consistency. Hence, missing the causal connection between the seismic hazard level and the ground-motion suite used and potentially compromising the meaningfulness of their results. This oversight is expected to introduce a bias in the estimation of residual drifts but its magnitude has not yet been properly evaluated. In this paper, we evaluate and quantify the significance of these effects. To this end, we use the Conditional Scenario Spectra framework, which provides a set of realistic earthquake spectra with assigned rates of occurrences based on their spectral shape and intensity, thus preserving the critical relationship of hazard consistency. Nonlinear Response History Analyses (NRHA) of 24 deteriorating SMRFs under 816 ground-motion records are performed to construct a database of residual drift demands. These NRHA results are used to examine the residual drift trends and to construct benchmark residual drift hazard curves. An extensive feature selection process employing several Machine Learning (ML) algorithms precedes the development of hybrid data-driven predictive models. Finally, we compare our hazard-consistent predictions with currently available models and quantify the massive under- and over-estimations associated with previously proposed non-hazard-consistent assumptions.

A hybrid buckling-restrained brace for enhancing the seismic performance of steel moment resisting frames

Buckling-restrained braces (BRBs) are effective and widely used devices to enhance the seismic performance of steel moment resisting frames (SMRF), but they might fail to prevent significant structural damage or structural collapse under unexpectedly high seismic loads. To address this issue, a novel hybrid BRB (HBRB) composed of a conventional BRB and a friction damper in series is proposed, where the activated force/displacement of the friction damper could be adjusted to protect the HBRB from failure. The paper begins by providing a detailed explanation of the configuration and working principle of the proposed HBRB. The cyclic behavior of the HBRB is thoroughly analyzed both theoretically and numerically. Building on this, a performance-based design approach is developed, focusing on enhancing the seismic response of SMRF using the HBRB. To assess the effectiveness of the proposed HBRB and design method, two typical SMRFs, one with 9 stories and the other with 20 stories, are enhanced with the HBRB. Comprehensive numerical models are established, considering the bare SMRFs, HBRB-enhanced SMRFs, and conventionally BRB-enhanced SMRFs. Nonlinear static and dynamic analyses are conducted to evaluate the seismic performance of these steel frames. Special attention is given to the performance comparison between HBRB-enhanced SMRFs and conventionally BRB-enhanced SMRFs, specifically under the extremely rare earthquake (ERE) scenario. The results highlight that the HBRB can achieve comparable energy dissipation capacity to conventional BRB while displaying superior energy dissipation efficiency. Moreover, the proposed design method achieves the predefined seismic performance objectives for the enhanced SMRFs in a straightforward and time-efficient manner. Furthermore, the HBRB proves effective in eliminating the failure risk associated with conventional BRB under ERE conditions, effectively mitigating structural damage and preventing collapse.

Uniform deformation distribution of structures at different seismic hazard levels using endurance time method

This paper evaluates the application of the endurance time method in the optimum performance design of structures using the uniform deformation theory. The structures used in this study include shear-building systems with 5, 10, and 15 stories, and steel moment-resisting frames with 3, 7, and 12 stories. Initially, shear-building systems are optimized to have uniform story ductility ratios at low, moderate and high seismic hazard levels separately using three sets of ground motion records and a compatible series of endurance time acceleration functions. The effectiveness and accuracy of the endurance time method are assessed by comparing lateral force distribution obtained from this method with its corresponding attained from ground motion records. Moreover, as the number of stories and target ductility increases, the compatibility of these two methods improves. However, it is found that the optimum structure at one seismic hazard level does not necessarily lead to the optimum structure at other seismic hazard levels. Next, by adding dampers with gap to the systems, a procedure is suggested to optimize these structures at different seismic hazard levels simultaneously. These dampers are activated at moderate and high seismic hazard levels. Finally, similar optimization algorithm is used for steel moment-resisting frames to make their story drift ratios or plastic hinge rotations uniform along the height at different seismic hazard levels. Results show that combining endurance time method and uniform deformation theory leads to a promising optimization procedure that can significantly reduce computational costs with reasonable error.

Probabilistic investigation of the pounding effect in steel moment resisting frames with equal and unequal heights

In this research, the influence of the pounding phenomenon of two adjacent steel moment resisting frames (SMRFs) with 2 and 5-story with an intermediate ductility is investigated by considering equal and unequal heights under seismic excitations. In this way, two scenarios are applied. The first and second scenarios are related to the pounding of two SMRFs with an equal height (both frames with 2-story) and an unequal height (2-story and 5-story frames), respectively. In the following, at first, sensitivity analysis is conducted by the Monte Carlo simulation (MCS) method. Then, the collapse performance of the studied SMRFs is evaluated by incremental dynamic analysis (IDA), and fragility curves under 14 far-fault records and also, the failure probability is predicted by the Kriging meta-model. The results of sensitivity analysis indicate that the yield strength of cross-sections and dead load were the greatest effect on failure probability computation. Also, in the statistical level of 50%, the collapse probability of SMRFs with an equal height is 17% compared to the SMRFs with an unequal height. Also, the results of the Kriging meta-model show that the failure probability is increased by 65% in an unequal height versus an equal height.