Gusset Plate Connections Research Articles

Numerical hybrid simulation assessment of E‐BRBF and E‐FBF systems for mitigating P‐delta effects and enhancing post‐earthquake performance

AbstractThis article investigates the seismic stability of E‐BRBF and E‐FBF systems previously proposed as stability‐enhanced alternatives to conventional BRBF and FBF systems. The E‐BRBF system is a buckling restrained braced frame with an inverted‐V brace configuration in which one of the two braces is a conventional brace designed to remain elastic during a strong earthquake. When the buckling restrained brace (BRB) yields, an unbalanced vertical force develops at the brace‐to‐beam connection, which imposes flexural demand on the beam and creates a positive post‐elastic storey shear stiffness. The E‐FBF system is identical except that the buckling restrained braces are replaced by conventional bracing members constructed with an end friction connection designed and detailed to slip at a predetermined load. At every storey, the beam and the elastic braces are designed such that this post‐elastic stiffness is sufficient to cancel the negative storey shear stiffness resulting from P‐delta effects and introduce a re‐centring mechanism to achieve an enhanced performance in steel buildings subjected to interface subduction earthquakes. Using distributed plasticity numerical models, past studies demonstrated the effectiveness of E‐BRBF and E‐FBF systems in mitigating P‐delta effects, emphasizing the importance of a stable and predictable response of the beam, which is a critical element in the scheme that provides the post‐elastic stiffness of the system. This beam is subjected to axial and flexural demands and must remain near elastic to ensure the global seismic stability of the structure. The response of the beam can be affected by localized yielding and local instability effects resulting from residual stresses, stress concentration near gusset plate connections and geometric imperfections. Employing a refined nonlinear finite element model of the beam element, this article assesses the performance of 10‐storey E‐BRBF and E‐FBF systems utilizing multi‐platform numerical hybrid simulations. The evaluation involves static‐monotonic and dynamic response history analyses, incorporating seismic excitations representing Vancouver, BC's seismic hazard (Crustal, In Slab, and Subduction Interface). In the hybrid simulation model, the most critical member for the proposed system (i.e., first‐storey beam) is sub‐structured in ABAQUS using solid elements, while the remaining frames are integrated using OpenSees using fiber‐based sections. Numerical hybrid simulation dynamic nonlinear response history analyses demonstrate the adequacy of the E‐BRBF and E‐FBF systems in mitigating P‐delta effects with residual drifts within repair limits, unlike conventional systems where instability or excessive drifting occurred. Monotonic Pushover hybrid simulations reveal a more ductile beam behavior compared to standalone models, in which frame members are entirely modeled in OpenSees. The analysis also indicates a ductile plastic hinging beam failure mechanism with no instability failure modes at extreme drifts.

Lateral Force Distributions in Braced-Moment Frames

Braced frames intended to resist wind and seismic loads traditionally have been analyzed and designed as trusses with all joints modeled as pins, such that only the braces provide lateral force resistance. However, frames with gusset plate connections create a rigid joint zone between frame beams and columns, effectively resulting in moment frame behavior, particularly at larger drift angles when braces have yielded or buckled. Described herein are the force distributions for buckling-restrained braced frames (BRBF) subjected to story drift angles, where the lateral resistance of the frame comprises both brace and moment frame action.

Seismic collapse safety assessment of suspended zipper-braced frames

Seismic evaluation of suspended zipper-braced frames, which are an alternative to inverted-V-braced frames to improve their seismic behavior, is of greatest significance to determine the level of confidence in this type of seismic system during severe earthquakes. The arrangement and design parameters of these frames are mentioned in some references, but there is no probabilistic assessment of collapse risk based on various collapse uncertainties. To evaluate the probability of collapse and margin of safety, eighteen suspended zipper-braced frames with different geometry parameters in the most severe seismic design category (D max) have been designed. The designed frames were modeled in OpenSees software by considering the effect of gusset plate connections and evaluated by performing more than 15,800 dynamic and nonlinear static pushover analyses using FEMA P695 methodology. Total collapse uncertainty is considered in the evaluation of the probabilistic behavior of frames. The results show that the adjusted collapse margin ratio (ACMR) of designed frames by considering the total collapse uncertainty of 0.726 and 0.529 is 27% and 64% higher than the acceptance criteria, respectively. The results also indicate that a response modification coefficient of much more than 6 can be used for the economic design of long-period suspended zipper-braced frames.

Design-focused Interpretable Machine Learning Models for Compressive Capacity Prediction of Gusset Plate Connections

Gusset plates can experience large compressive loads when used to connect bracing systems to beam-column joints in steel frames and truss bridges. The strength of a gusset plate is therefore vital to allow the bracing members to reach their ultimate capacity without a brittle failure of the gusset plate connection. Due to the large complexity of the stress distribution mechanism, simplified approaches commonly adopted for design can lead to inaccurate and highly conservative results. The present study aims at developing a simple yet efficient gusset plate compressive strength prediction model that does neither rely on crude assumptions nor require complicated calculations. A meticulously compiled database consisting of 68 experimental and 184 numerical datasets was deployed to aid in developing a high-performance machine learning (ML) model selected amongst 10 machine learning techniques studied. The superiority of the CatBoost and XGBoost models was confirmed with an accuracy of 95% and 96%, respectively significantly outperforming current design practices and widely used empirical formulas. SHapley Additive exPlanations (SHAP) and Taylor diagram were used to explain the feature importance and model performance. The developed interpretable ML models can instill trust in a model that explains its decisions, as opposed to other black box approaches. It was found that CatBoost was the most efficient in predicting the compressive strength of gusset plates with lowest mean absolute error of 50.76 kN and Nash-Sutcliffe model efficiency coefficient of 0.984, compared to the most accurate predictions observed among the existing methods (modified Thornton method) with mean absolute error and Nash-Sutcliffe model efficiency coefficient of 227.22 kN and 0.872, respectively. Finally, a reliability analysis was performed according to the LRFD approach to identify the level of safety of predictions obtained by the CatBoost and XGBoost models. The CatBoost model exhibited remarkably low dispersion with a safety factor of 0.753 thereby providing an adequate level of safety for the design of gusset plates.

Nonlinear Analysis of the Gusset Plate Connection for Braced Frames Subjected to Monotonic and Cyclic Loading

Nonlinear Analysis of the Gusset Plate Connection for Braced Frames Subjected to Monotonic and Cyclic Loading

Experimental investigation on steel bracing members with bolted gusset plate connections

In concentrically braced frames, low-cycle fatigue failure may occur in the brace or bolted connection due to accumulated damage from multiple earthquakes. However, the current research on the hysteretic behavior and fatigue life of H-shaped bracing members with bolted gusset plate connections is lacking, and there is a notable omission regarding the impact of bolt slippage and plastic deformation within the connection. The objective of this study is to address the aforementioned limitation through pseudo-static tests of 8 steel bracing members with bolted gusset plate connections under different clearances, connection coefficients (akin to Ry used in brace design under AISC 341-16 Seismic Provisions), and loading patterns. Results showed that not all loading patterns were suitable for such tests due to bolt slippage. The strength capacity, energy dissipation capacity and deformability of specimens with five types of clearances displayed small variation in tests. It has been demonstrated that utilizing a connection coefficient (1.03) lower than the prescribed value (1.3) can still result in a favorable hysteretic performance. The effective length factor of the bracing member under double bolted gusset plate connections is estimated to be 0.79, which can provide valuable information for optimizing the brace design. Additionally, a method for predicting the fatigue life of test specimens is proposed. The fatigue performance and failure mode of the bracing members under bolted connections with clearances of 0t, -2t, -4t and -6t (where t represents the thickness of the gusset plate) are superior to those under welded connections.

Remaining fatigue endurance of gusset plate connections from a dismantled railway bridge

The replacement of an existing steel railway bridge in Sweden opened up for fatigue testing connections with a history of about 70 years of in-service loading. One decisive detail in the bridge was the gusset plates welded to the flange edge of I beams. This has low fatigue resistance in the governing standards and was deemed exhausted in theoretical assessments. To estimate the remaining fatigue endurance, 20 similar samples of these were cut out from the bridge and tested. During service, they were exposed to alternating compression and tension stresses. The testing was designed to determine their category for constant amplitude fatigue loading and to evaluate the influence of compressive stresses. It is shown that the compressive stresses have an influence on the fatigue endurance and a correction function has been fitted to the results. Despite many years of loading, they had a fatigue endurances significantly greater than that estimated theoretically. A detail category of 63 was derived, which, together with the correction function, provides a significant increase in fatigue life, which should be possible to use for similar details in other existing steel structures.

Design and cyclic testing of a gusset plate connection for precast concrete buckling-restrained braced frames

A design method for precast concrete buckling-restrained braced frames with traditional gusset plate connections has not yet been codified. In addition, experimental data for this novel system are lacking. The primary objectives of this research were to design and experimentally test a partial system under representative seismic loads and to determine the applicability of the uniform force method (UFM) for connection interface force distribution. A quasi-static cyclic test was performed on a scaled, partial system. Experimental results showed that the UFM alone does not accurately predict interface forces for this system because the method does not account for frame action. Results also showed that there is some change in column base fixity as the frame undergoes larger horizontal displacements. This was the first U.S. laboratory test on a scaled, partial version of this system. Development of a codified design method would require further testing to determine an appropriate interface force distribution and quantify the change in column base fixity.

Cold Formed Steel Type C On Beams With Connection Variations

The disadvantages of using cold rolled steel (cold formed steel) include a thin profile and low stiffness of the connecting parts. Stiffness research of various types of joints is needed to study element failure, machine error or connection failure which causes the collapse of cold rolled steel structures. Analysis is needed for cantilever structures, because the maximum deflection that occurs in cantilever structures is greater than that of simply supported beam structures. The shape and type of cold rolled steel joints greatly affect the stiffness and stability of cold rolled steel structures so that the strength of several types of joints must be studied. The purpose of this study was to analyze the stiffness of the test results under the influence of bending loads on each type of cold rolled steel beam-column connection. Cold steel C channel profile with a thickness of 0.75 mm is used as the object in this study. The beam-column structure is assembled with various kinds of connections, including direct screw connections, screw-gusset plates, screw-angle plates, and screw-angle plate-gusset plates. This research shows addition of the elbow plate increases the strength of the test object by means of a screw connection and gusset plate to the load. However, the results of the maximum load in the experiment still have not reached the maximum load of the analysis results. In the screw connection the damage that occurs is tearing in the connecting plate (tear out), in the gusset plate connection the damage that occurs is the collapse of the gusset plate, in the screw-angled plate connection there are 2 damages, where the damage starts with a tear in the connecting plate ( tear out) and followed by damage to the screw that is damaged (pull out).

Behavior of Gusset Plates Connecting Angle Sections in Compression

This paper contains experimental and numerical study on the behavior of the gusset plate in compression. Gusset plate connections are usually used for the transfer of load from one member to another, they are generally used in bridge trusses and steel braced frames. However, due to lack of technical information regarding the behavior of the gusset plate under compression, these have been under research focus. It has been seen that the gusset plate shows a buckling failure by swaying on one side as there is no out of plane restraint. In this paper, a study is initiated aimed at the gusset plate connected with the angle section on a single side. Key items that were investigated in the study were strain, strength, and displacement. A total of 5 specimens were used for the experimental investigation. The parameters which were considered for the study, the number of bolt lines and the thickness of the gusset plate. Based on the experimental results of load and strain, a numerical model was developed and compared with experimental load-carrying capacity of plate and displacements. After going through the study unpredicted behavior was observed for the two-bolt line and suggestions were given accordingly.

Experimental and Numerical Investigation of Joints for a Pultruded Fiber-Reinforced Polymer Truss.

Bolted connections usually govern the structural rigidity and load-carrying capacity of pultruded glass fiber-reinforced polymer (GFRP) truss structures. In this study, a novel bolted integrated gusset plate (IGP) connection is proposed to enhance the stiffness and capacity of GFRP truss structures. Nine double-lap shear tests of GFRP joints and numerical simulation were conducted to investigate the influence of variable design parameters of the bolted GFRP joints (number of bolts, width and thickness of GFRP, edge distance of bolts, and the employment of adhesive). Three full-scale GFRP truss joints were tested under static loading to study the response of a typical bolted connection, a bolted gusset plate connection, and the proposed IGP connection. The nine double-lap shear tests showed that the bolted-bonded mixed connection has 50% higher shear stiffness and 27% higher ductility compared with bolted joints, and bearing failure dominated the capacity of most specimens, which agreed well with numerical simulation results. Tests on the three full-scale GFRP truss joints showed that the bolted gusset plate can substantially reduce the number of cracks and improve the initial stiffness, but the maximum bearing capacity of the joints did not increase because the shear fracture of pultruded GFRP webs governs the capacity. The proposed IGP substantially increased the stiffness and capacity compared with the bolted connection and typical bolted gusset plate connection. The full-scale GFRP joint test is suggested to be used together with direct shear tests to study the performance of joints of the GFRP truss.

FINITE ELEMENT ANALYSIS OF GUSSET PLATE CONNECTION DESIGN FOR COLD-FORMED STEEL FRAMES

The application of cold-formed steel sections has been extended from secondary members to primary structural members in recent years. This increases the use of gusset plate connections in cold-formed steel since it is a common connection used in steel structures. However, current design codes on connection design do not have a comprehensive method to consider the effects due to the buckling of the thin cold-formed steel sections. Therefore, it is important to develop a more accurate model to predict the capacity of cold-formed steel connections. This paper aims to propose an equation for gusset plate beam-to-column connection using finite element models. Finite element models have been developed and compared with existing test results. The failure mode and ultimate strength of the numerical models are similar to the experimental results. The validated finite element models are then used to study the effects of gusset plate thickness, effects of cold-formed steel section depth and thickness, and the effects of bolt size and spacings. Elastic and plastic stiffnesses are obtained from the developed models. The connection behavior followed a typical elastic-plastic curve according to the connection ductility and failure mode. An empirical model is developed from the finite element models to predict the joint behavior of gusset plate beam-to-column connection for cold-formed steel structures. AS 4600 may have underestimated the initial stiffness of the connection.

Low-cycle fatigue performance and damage sequence investigation of I-shaped steel braced systems and braced frames

Simulations of 12 braced systems (I-shaped steel (Q355B) brace with the gusset plate connection) and 5 braced frames (single diagonal) are conducted. The results show that the low-cycle fatigue life of the braced system and the braced frame is determined by the minimum value of the fatigue life between the gusset plate connection and the brace. For the gusset plate connection, small value of clearance distance and large value of loading amplitude lead to small fatigue life. The ability for ductility property and energy dissipation of the braced system is optimal when the fatigue life of the gusset plate connection and brace is similar. The critical plane method (LZH model) and ductile damage method are applied to predict the fatigue life and investigate the fatigue damage of the specimens. These two methods have good applicability not only on fatigue life prediction but also the damage sequence of each component (including the gusset plate connection and the brace). Based on experimental data from related literature, fatigue parameters that are used in life prediction methods of steel Q355B (including the base metal and welding material) are calibrated.

Practice-Accessible Methodology for Nonlinear Refined Analysis of Gusset Plate Connections for Steel Truss Bridges

A new approach is proposed for estimating structural capacity of gusset plate connections of steel truss bridges. The approach involves nonlinear refined analysis of a truss model, consisting of a single truss made of shell elements for two gusset plates at a subject connection and frame elements for truss members. Connectors (bolts or rivets) are excluded from the proposed model in order to simplify the model, with the knowledge that their associated failure modes can be addressed by simplified design calculations. Only yielding and buckling failure modes of gusset plates are considered. The new approach is calibrated by comparison with laboratory test data from NCHRP Project 12-84. The primary intent of this paper’s approach is to reduce the complexity of the refined analysis developed under this National Cooperation Highway Research Program project and make it more user-friendly to load rating engineers seeking accurate estimation of gusset plate capacity. Further, an imperfection sensitivity curve characterizing reduction in capacity as a function of lateral imperfection (gusset plate out-of-flatness) in gusset plate geometry is introduced as an added tool for the load rating engineer, providing more confidence in capacity estimation. Based on a limited case study, the proposed refined analysis provides an estimate of gusset plate capacity that is approximately equal to capacity calculated by the truncated Whitmore method of the AASHTO Manual for Bridge Evaluation (2018). In other words, the proposed approach validates the truncated Whitmore method as compared to the more conservative partial shear method of the AASHTO Manual.

Subassemblage tests and analysis of buckling-restrained braced reinforced concrete frames with various gusset connections

The buckling-restrained brace (BRB) is an effective energy dissipation damper, which can effectively improve the lateral stiffness, strength and energy dissipation capacity of reinforcement concrete (RC) structures. However, few studies have been conducted on the influences of gusset plate connection type on BRB rotations and seismic performance of braced RC frames. In this study, subassemblage quasi-static cyclic test was performed to five buckling-restrained braced RC frame subassemblages (BRBRCFs) with various gusset plate connections, including bolted gusset plate connection, welded gusset plate connection, pinned gusset plate connection, bolted-pinned gusset plate connection and welded-pinned gusset plate connection. The effects of various gusset connections on failure modes, ductility, stiffness degradation, energy dissipation capacity, gusset plate and BRB end zone rotation deformations of the subassemblages were evaluated and discussed. The test results indicated all specimens exhibited stable hysteretic behavior and excellent ductility with negligible degradation of stiffness and strength at the same loading level. The rotation responses of the gusset plate were generally greater than those of BRB end zone. Additionally, the finite element (FE) model of BRBRCFs with various gusset plate connections was established and verified using experimental results. This study facilitates applications of BRBs with various gusset plate connections in RC frame structures.

Failure pattern and load distribution mechanism of stiffeners in the tube–gusset plate connection: Testing, numerical modelling and theoretical analysis

In tube–gusset plate connection (TGPC), the deformation stiffness of the stiffener plate is much greater than that of chord wall. The local instability and material failure in the stiffener plate are the primary failure modes of TGPC. In this article, an experimental, numerical and theoretical study on both the failure pattern and internal force distribution mechanism of the stiffener ring plates in the TGPC was carried out. An experimental study including the material tensile coupon tests and twelve TGPC tests was initially conducted, which detailedly presented the failure modes of the stiffener plates under either compression or tension. The failure of the stiffener plates was dominated by local instability when compression and was controlled by material failure when tension. Subsequently, a numerical investigation using finite element models (FEMs) of KT– and X–type joints was performed. The FEMs were validated against the present test results and those available in literature. Thereafter, in order to obtain the additional data of TGPC with multi–stiffener ring plates and verify the accuracy of the calculation model proposed below, the FEMs were then extended for a further investigation of the TGPCs containing three, four, and five ring stiffener plates with different gusset plate lengths. With considering the influence of the bending deformation in the plane of gusset plates on the distributed loads in stiffener ring plates, an elastic beam–spring bearing (EBSB) model was developed based on the conjugate beam theory. Finally, the numerical results were compared with those obtained using the Japanese code and EBSB model. It was seen that the influence of the bending deformation of the gusset plate on distributed loads in the stiffener ring plates became significant, and the proportion of the shear deformation of the gusset plate between the stiffeners to the total deformation was decreased, when the length of the gusset plate was increased. The distributed load in each stiffener ring plate predicted by the EBSB model was found to be more accurate than that obtained using the Japanese code. The proposed EBSB model can be used as the guidelines for the design of joints in actual transmission line engineering projects.

Fracture prediction for square hollow section braces under extremely low cycle fatigue

This paper examines the extremely low cycle fatigue (ELCF) fracture of concentrically loaded square hollow section (SHS) braces subjected to cyclic loading. Numerical analyses are presented for both individual bracing members and bracing members integrated into concentrically braced frames (CBFs). The behaviour of the individual members was predicted using solid finite element (FE) simulations that employed a ductile fracture model and a nonlinear damage evolution rule. The solid FE model, which was validated using data from experiments, could adequately predict both the hysteretic response and the ELCF fracture cracking process. The coupled effects of instabilities (i.e. local and global buckling) and fracture on the ELCF performance of the braces were assessed, and the rotation capacity prior to fracture was quantified. This quantified rotation capacity was then incorporated into fibre-based FE models of CBFs as a member-level fracture criterion. The structure-level simulations were able to accurately capture the complex interactions between the frame components, i.e. the columns, beams, brace–gusset–plate​ connections and beam-to-column connections, and hence replicate the overall behaviour of CBFs, specifically, two-storey chevron braced frames. The influence of cross-section and member slenderness was evaluated and the importance of considering both in the development of cross-section slenderness limits was highlighted. The combined member- and structure-level simulation approach is proposed as an accurate and efficient means of assessing the seismic performance of CBFs.

Continuum damage mechanics based ductile fatigue-fracture prediction in buckling steel braces

A Continuum Damage Mechanics criterion (the Stress Weighted Ductile Fracture Model - SWDFM) is used to predict Ultra Low Cycle Fatigue fracture in buckling steel braces. The focus is to examine the efficacy of this approach for predicting fracture in large-scale structural components with localization, when the constitutive and fracture parameters have been calibrated only at the coupon scale. A dataset consisting of 11 two-thirds scale, cyclically loaded, hollow brace specimens, and corresponding material coupon specimens is utilized for assessment of the fracture prediction approach. In these tests, fracture occurs either at the center where cyclic local buckling amplifies the strains, or at the gusset-plate connections. Particle Swarm Optimization is used to calibrate material constitutive parameters based on load-deformation response of the coupon tests, whereas the SWDFM parameters are calibrated using their fracture response. These parameters are applied within Continuum Finite Element simulations of the brace tests. The results indicate that the approach predicts fracture in a satisfactory manner. The error in fracture predictions is most correlated with the error in predicting the event of localization (i.e., local buckling). There does not appear to be a strong relationship between the error in the constitutive model calibration (based on the coupon tests) and the prediction of either local buckling or fracture. This suggests that in themselves, the small-scale coupon specimens are unable to calibrate the constitutive model with the required accuracy to capture localization. Commentary is provided for calibration and application of the CDM-based approach to predict fracture.

Tension and compression stiffness of ring stiffener in tube–gusset plate connections

Tube–gusset plate joints with multiple ring stiffeners are commonly used in steel tube towers. The load on the gusset plate can be distributed to each stiffener according to its Tension and Compression (T&C) stiffness. The ultimate state of joints with multi-stiffener plates can be equivalent to that of the stiffener plate. In this study, an experimental investigation and theoretical analysis of the failure mode and T&C stiffness of ring stiffener under symmetrical loading were performed. The experimental programme included material tests and 10 groups of static loading tests of the ring stiffener plates. Then, finite-element models were developed and validated against the test results. The validated numerical models were utilised to perform numerical investigations of 280 different single-ring stiffener plate specimens with D/t=32.3–55.9 and R/tr = 6–33.3. Using the experimental and numerical results, the parameters affecting the T&C stiffness of the ring stiffener plates were identified. According to the deformation characteristics of the ring stiffener plates, a T-shaped ring beam calculation model was established, along with the corresponding calculation formula. Calculation results based on the formula were compared with the numerical analysis results, and the results indicated that the theoretical model can accurately calculate the T&C stiffness of the ring stiffener plate. Thus, the proposed calculation method for the T&C stiffness of a ring stiffener can be used to guide the design of joints in actual engineering projects.

A seismic design method for one-piece gusset plate connections

In multi-story Special Concentrically Braced Frames (SCBFs), there are bracing nodes where two braces associated with two sequential adjacent stories of a similar bay intersect each other at beam-to-column joints. In these bracing nodes, under loading, one of the braces is in tension and the other is in compression. Consequently, maximum vertical force is imposed on the column. Note that in these bracing nodes, force distribution at a gusset plate affects the force distribution at the other gusset plate. Generally, two separate gusset plates are considered for the bracing nodes with two braces at the beam-to-column joints and designed according to Uniform Force Method (UFM). While the UFM proposed for corner gusset plates with one brace does not consider the force distribution effects at both gusset plates. To resolve this problem, the current study proposes a one-piece gusset plate for the bracing nodes with two braces at the beam-to-column joints. The one-piece gusset plate involves the gusset plate of the tension brace, the gusset plate of the compression brace, a part of the beam web whose length is equal to the interface of the gusset plate to the beam, and finger stiffeners. The finger stiffeners are considered in the connection region on both sides of the one-piece gusset plate to simulate the presence of the beam flanges removed in this region. Note that in this connection, the beam is extended to near the one-piece gusset plate and thereafter is connected to it. The aims of this study are as follows: (1) considering the force distribution effects on both gusset plates, (2) calculating the force and bending moment demands of the gusset plates in the bracing nodes with two braces at the beam-to-column joints via proposing a new method, and (3) determining the gusset plate dimensions in the bracing nodes with two braces at the beam-to-column joints through presenting a new procedure. Five one-bay, two-story SCBFs with the brace angles of 37−37, 45−45, 59−59, 37−59 and 59−37° (the first number is the lower story brace angle and the second number is the upper story brace angle) were designed in accordance with the method and procedure proposed in this study. These specimens were simulated in ABAQUS and subjected to a monotonic loading. The accuracy of the proposed method was confirmed via comparing the results of ABAQUS and UFM with those of the proposed method.