Nonlinear Structural Analysis Research Articles

Identification of Nonlinearity in Bolted Beam using ResponseControlled Testing and ForceControlled Testing

Nonlinear phenomena such as super-and sub-harmonics and unstable branches in frequency response functions (FRFs) often occur in slender structures, which are common in the aerospace, civil and wind power domains. The widely used methods of linear modal analysis are invalid for the identification of these phenomena, which play an important role in nonlinear structural analysis as they can significantly affect the behaviour and stability of structures under various excitations. This paper proposes the methodology of Response-Controlled Stepped Sine Testing (RCT) and Force-Controlled Stepped Sine Testing (FCT) to identify nonlinear phenomena in a bolted beam. The bolted beam was excited with two different sets of predefined displacement and force amplitudesto measure the Nonlinear Frequency Response Functions (NLFRFs) of the bolted beam. The measurements were performed with RCT and FCT, using Stepped Sine signals to control the displacement and force amplitudes. The Harmonic Force Surface (HFS) technique was used as means of obtaining the NFRFs. It was found that RCT with HFS successfully identifiedthe nonlinearity that is very similar to the identification results obtained with FCT. In addition, RCT was computationally faster than FCT in identifying nonlinear bolted beams and avoided dealing with jump phenomena. These advantages of RCT and HFS can provide engineers and researchers with a great technique to accurately and efficiently identify nonlinearities in structures.

Nonlinear Dynamic Analysis Using the Force Analogy Method and the Ibarra–Medina–Krawinkler Model

ABSTRACTA methodology to integrate the Ibarra–Medina–Krawinkler (IMK) hysteretic model with the force analogy method (FAM) is proposed. FAM is a powerful tool for performing nonlinear structural analysis that reduces analysis time, addressing the main limitation of nonlinear analysis. One area of opportunity for FAM lies in its access to material nonlinearity models, which enable the reproduction of inelastic behavior in structures. The IMK model, utilized in nonlinear analysis for updating the stiffness matrix, incorporates various rules of material nonlinear behavior and considers cyclic deterioration of stiffness and strength. By combining the IMK model with FAM, the capability to model material nonlinear behavior in a computationally efficient manner is enhanced. The proposed methodology is validated through four numerical examples, demonstrating its effectiveness in replicating experimental behaviors and enhancing FAM capabilities in inelastic structural analysis.

Experimental and numerical investigation of failure modes in RC frame structures designed using the equivalent linearization method

The codes of many countries currently employ the column-to-beam flexural strength ratio (CBFSR) based on frequent earthquake internal forces to achieve the expected strong-column-weak-beam (SCWB) ductile failure mode. However, real earthquake damage indicates a general failure to achieve SCWB. The design method based on frequent earthquake internal forces struggles to consider factors such as internal force redistribution and variable axial forces under rare earthquakes. Therefore, developing a seismic design method directly based on rare earthquake internal forces is crucial. The equivalent linearization method is a practical approach for obtaining the internal forces of RC frame structures under strong earthquakes without extensive nonlinear computations. In this paper, nonlinear finite element analyses of RC frame structures with expected SCWB failure modes under various structural parameters were conducted to obtain ductility coefficients under rare earthquakes. Using the equivalent linearization method based on secant stiffness, the equivalent stiffness and damping of beams and columns were calculated. Simplified parameters were determined for engineering applications to facilitate seismic design based on rare earthquake internal forces. Two RC frame structures were designed using both the equivalent linearization method and the CBFSR code method, and then subjected to pseudo-static low-cycle reciprocating loading tests. The study investigated failure modes and hysteretic energy dissipation, as well as overall stiffness and bearing capacity under earthquake conditions. Results showed that the equivalent linearization method can effectively direct RC frame structures to achieve the expected strong earthquake failure mode.The dynamic time history analysis was performed on finite element models of different heights, and the results indicated that the equivalent linearization method significantly reduced plastic hinge occurrences and ductility coefficients of column ends, resulting in improved seismic performance compared to the CBFSR code method.

Weak form quadrature shell elements based on absolute nodal coordinate formulation

Weak form quadrature elements for moderately thick shells with arbitrary initial configurations are developed under the framework of continuum mechanics and the absolute nodal coordinate formulation (ANCF). Locking problems of shell analysis are discussed. Nonlinear analysis of various shell structures is conducted. The joint constraint equations for shells with discontinuous slopes are established. Five examples encompassing static and dynamic shell analysis, post-buckling analysis of shells, as well as analysis of shells with discontinuous mid-surface slopes are examined to assess the performance of the proposed elements. Satisfactory results are obtained, validating the efficacy of the proposed elements.

Evaluation of Load-Bearing Capacity in Japanese Arch Steel Aqueduct Bridge through Structural Redundancy Assessment

Structural redundancy assessment of steel aqueduct bridges is made by the analysis of a case study using the Musota Aqueduct bridge structure: a simply supported steel arch bridge erected in 1973 and a seven- span continuous steel aqueduct bridge. In this paper, as a case study, after the validation of the model, the structural redundancy of the Musota aqueduct bridge in Wakayama City with respect to its load-carrying capacity after the failure of hanging components due to corrosion was investigated. The conventional procedure for the assessment of redundancy makes use of static nonlinear structural analysis. A three-dimensional finite-element model of this bridge was developed to simulate its behavior. The results from the linear analysis are compared with those from the nonlinear analysis to investigate the appropriateness of the former in the evaluation of redundancy. A detailed nonlinear static finite element study is carried out into the hangers' components of the arch bridge in order to clarify the implications involved in the failure of redundancy. Finally, recommendations for prudent bridge maintenance methods are presented based on findings from the investigation.

Study on the buckling behavior of aluminum alloy sheets - before and after repaired with composite patches

In this paper, the compressive buckling behavior of aluminum alloy plates with an elliptical hole of various sizes is investigated. In order to improve the stability of these thin plates, T700/QY8911 composite laminate is used as a patch to repair the hole. The study included an analysis of the critical and post-critical behaviour using experimental and numerical methods. Experiments focus on buckling loads, post-buckling behavior and the relationship between sizes of holes and buckling load. Meanwhile, the buckling load and buckling mode are determined by finite element analysis, using linear analysis of eigenvalue problems modes, and then, the nonlinear analysis of structures with initiated geometrically imperfection is carried out, studying its post-buckling behavior, damage behavior and transfer of load. The results show that the buckling load of the open-hole specimen is related to size of opening. The existence of patch has a significant influence on stress distribution, and the buckling capability of repaired specimens is noticeably improved to the plate without a hole. And the compression experimental results are consistent with the numerical results, revealing that the developed finite element model of the structure is correct.

Enhanced Multi‐Objective Optimization Model for Bridge Performance Assessment and Prediction, Based on Improved PCA, K‐Means Clustering, and Kaplan–Meier Survival Algorithm

ABSTRACTThe research proposes a hybrid algorithm model that combines model‐driven and data‐driven approaches for the direct application of bridge health monitoring technology in bridge management. This comprehensive study encompasses a series of analytical techniques and methodologies to build a multi‐objective optimization model for bridge performance assessment and prediction. It focuses on the processing of multi‐source heterogeneous data, selection of key sub‐parameters using Principal Component Analysis (PCA), enhanced K‐means clustering analysis, determination of structural component target thresholds, time‐dependent survival probability analysis, regression fitting, and timing prediction of the bridge system for both the components of double‐layer truss arch bridge and the bridge system. The initial phase of the study concentrates on the diversification and decentralization of monitored data from various sources, integrating and cleaning data obtained from different sources to ensure data quality and consistency. PCA technique is applied to identify key sub‐parameters that have significant impacts on the performance of structural components. Enhanced K‐means clustering analysis is carried out to effectively group and classify the identified key sub‐parameters. Numerical simulations, including structural nonlinear analysis, are conducted to determine the target thresholds of bridge structure, providing important benchmarks for performance evaluation. Finally, a multi‐parameter regression model is used to evaluate and update the performance of the bridge structure, taking into account survival probability (using the Kaplan–Meier method), maintenance history, and material deterioration to estimate the most critical time for the bridge system. A case study is conducted to validate the suggested comprehensive algorithms for a double‐layer truss arch combination bridge, which contributes to enhancing performance evaluation and predicting the most critical time for structural components and bridge system in the bridge management and maintenance practices. It should not be ignored that, the accuracy and reasonability of bridge structure system performance evaluation and prediction depend largely on the selection of target thresholds.

Fragility analysis of tubular structures based on local-buckling driving variables

Performance-Based Earthquake Engineering (PBEE) is computationally demanding, due to the multiple high-fidelity nonlinear dynamic structural response analyses required to compute fragility curves. Local buckling of tubular steel structures is not properly characterized by typical Engineering Demand Parameters (EDPs) such as story drifts or plastic rotation angles. Targeting the two issues above, in this manuscript we propose using state variables based on Lumped Damage Mechanics (LDM) to characterize Local Buckling (LB) in PBEE. Hence, we propose an efficient and innovative procedure for the fragility analysis of complex tubular structures prone to fail due to local buckling. Moreover, local buckling produces a loss of stiffness, with loads transferred to intact or to less-damaged elements. Eventually, the structure forms a global collapse mechanism. Herein, we show how to identify the most likely global collapse mechanism in non-symmetrical tubular structures subjected to random seismic loading. This involves evaluating damage indices in different elements and their correlation, as well as identifying the combination of LB failures that are more likely to form a global collapse mechanism. Fragility curves characterizing the onset of LB at individual elements, and the most likely global collapse mechanism, are constructed. A simple frame structure is addressed, where the accuracy of the LB-LDM model is checked against experimental results. Another case study involving a non-symmetric tubular wharf illustrates the search for the most likely global collapse mechanism, and the derivation of its fragility function.

A study on solid-shell finite element formulations applied to nonlinear thermoelastic analysis of thin-walled structures

The solid-shell elements have shown great advantages in three-dimensional (3D) finite element (FE) simulation of thin-walled structures. To overcome the numerical lockings for 3D element with a large span-thickness ratio, the assumed natural strain (ANS) method combined by either the hybrid stress (HS) or enhanced assumed strain (EAS) formulations are commonly used. In this work, two types of finite element formulations of the solid-shell elements are developed for nonlinear thermoelastic analysis of thin-walled structures, which are termed as the “ANS+HS” and “ANS+EAS” formulations. Accordingly, two eight-node solid-shell elements (CSSH8 and CSSE8) are developed based on the “ANS+HS” and “ANS+EAS” formulations, respectively. The Green–Lagrange displacement–strain relation is applied to involve the geometrical nonlinearities, and both the thermal expansion and temperature-dependent material properties are considered. Nonlinear thermoelastic equilibrium equations are constructed in the framework of the two types of finite element formulations using the Hellinger–Reissner and conventional variational principles, respectively. The proposed method can easily realize three different coupling analyses for thermal–mechanical loads by modifying the parameters of nonlinear thermoelastic equilibrium equations. Numerical examples demonstrate that the proposed method using CSSH8 and CSSE8 elements traces the nonlinear thermoelastic response of the isogrid stiffened panel as accurately as the shell, solid-shell, and solid elements of ABAQUS. Furthermore, the path-following capability of the proposed method using the CSSH8 with “ANS+HS” formulation is slightly superior to that using the CSSE8 with “ANS+EAS” formulation, but both better than ABAQUS.

A novel iterative algorithm for nonlinear inelastic dynamic analysis of truss structures; MCB-Spline+Sp time integration method

The focus of this work is on the development of a novel algorithm for the nonlinear dynamic analysis of structures. In the sake of a general and easy-to-implement solution method, the tangential stiffness matrix is calculated by combining the Cauchy-Riemann equations derived from the differentiation of complex functions and using the nodal displacement perturbation vector. An implicit direct time integration method based on a cubic B-Spline is integrated with the incremental-iterative method using a quadrature rule based on the interpolation of spline functions, which is referred to as the MCB-Spline+Sp time integration method. A step-by-step algorithm for developing a computer code to implement this method is provided. The effectiveness of the proposed approach is evaluated through several nonlinear time history analyses, which demonstrate its accuracy and efficiency by observing less computational efforts and fast convergence rate. Moreover, a comparison is made between the developed method and well-established techniques such as the Newmark and Standard-Bathe time integration methods in combination with the iterative Newton-Raphson method.

A fast search method of buckling load for telescopic boom structure

The telescopic boom structures are extensively utilized in engineering applications involving large mobile cranes, aerial work platforms of significant height, and similar mechanical devices. These are predominantly fabricated using solid-webbed box types, latticed truss designs, or a combination thereof. Under load, they exhibit pronounced geometric nonlinear effects, with the relationship between load and displacement demonstrating marked nonlinear characteristics. This paper focuses on the geometric nonlinear modeling of slender multi flexible beam structures. The overall structural system is divided into several substructures, and the concept of multi flexible system modeling is borrowed to establish a follow-up connected body foundation on each substructure. By decomposing the node displacement and rotation in the substructure, the large displacement and large rotation of the node are decomposed into rigid body motion of the connected base and small displacement and small rotation relative to the connected base, effectively representing the large displacement and large rotation of the substructure as rigid body motion of the connected base. A modeling method for the geometric nonlinear analysis of slender and flexible multibody beam structures is proposed. By decomposing the nodal displacement and rotation within the substructures, the large displacement and rotation of the nodes are broken down into the rigid body motion of the body-fixed coordinate and small displacement and rotation relative to the body-fixed coordinate. This approach establishes the application conditions for calculating the structure’s virtual power of deformation using traditional beam elements with linear strain. A method for solving the instability load of slender and flexible multibody structures is proposed. This method integrates the characteristics of the system’s nonlinear equilibrium equations with respect to load, by deriving the system equations with respect to a load increment control parameter. This paper converts nonlinear equilibrium equations into first-order ordinary differential equation (ODE). The method proposed in this paper provides a more efficient solution for the buckling load of the telescopic boom. It can be used to systematically and quickly calculate the structure of the telescopic boom.

Quantitative Property of MF-Discrepancy and Efficient Point-Selection Strategy for the Nonlinear Stochastic Response Analysis of Structures with Random Parameters

The response analysis of high-dimensional and strongly nonlinear systems with random parameters remains a significant challenge in stochastic computational mechanics. To address this challenge, some methods based on the high-efficacy point sets have been developed, in which efficient global-point-set methods represented by low-discrepancy are of paramount importance in generating representative point sets. Several discrepancies including the extended F-discrepancy (EF-discrepancy) and the generalized F-discrepancy (GF-discrepancy) have been introduced to assess the uniformity and the efficacy of a representative point set. In such context, a maximal marginal EF-discrepancy (MF-discrepancy), which is an extended form of the GF-discrepancy, is proposed in this paper and then the properties of the MF-discrepancy are studied in detail. The probability distribution of the MF-discrepancy is derived, including a rigorous proof for random point sets and a model based on an assumption for some generic point sets. A generalized Koksma-Hlawka inequality is established accordingly to govern the worst error estimate. The lowest bound of the MF-discrepancy is given, and two intuitive quantitative indices are proposed to measure the goodness of the MF-discrepancy. Based on the lowest bound, an enhanced point-selection strategy with a unified theoretical framework for minimizing the MF-discrepancy is proposed. In this framework, locally minimizing the MF-discrepancy yields the two-step point-selection method, and a new point-selection strategy is proposed based on the global minimization of the MF-discrepancy, which is verified to be efficient and robust, especially in high-dimensional cases. Several numerical examples, including a 2-story shear frame, a 10-story shear frame, and a 10-story reinforced concrete frame structure modeled by the finite element method, are studied, verifying the efficiency and the robustness of the proposed point-selection strategy.

Effect mechanism of low-velocity impact loading rate on the failure modes transition of steel-UHPC-steel composite plates

This study delved into the transition mechanism of loading rate on the failure mode transition in steel-ultra-high performance concrete-steel (SUHPCS) composite plates under concentrated low-velocity impact loads. A series of drop-weight tests elucidated that, as impact velocity escalated, the composite plate shifts from local punching failure, characterized by circumferential cracks, to overall flexural failure, marked by radial cracks resembling yield lines. Given the constraints in capturing strain data in drop-weight test, a 3D nonlinear structural analysis using Abaqus software was conducted. This numerical simulation method was validated by drop-weight test results in terms of crack patterns and impact force time histories. Moreover, structural dynamic punching and flexural loads at a specific loading rate were obtained with the assistance of the dynamic increase factor (DIF). Failure mode could be judged through comparison between the two critical loads. The discrepancy in strain sensitivity exhibited by steel plates and UHPC accounted for the transition in failure modes of the SUHPCS composite plates. DIFs for UHPC compressive strength and steel plate strength were lower than DIF for UHPC tensile strength, as these factors were around 1.5, 1.7 and 2.6, respectively. Parameter analysis showed that the critical transition impact velocity for a composite plate with a core thickness of 80 mm was 11 m/s, surpassing the 13 m/s threshold for a composite plate with a 100 mm-thick core. Moreover, changes in DIFs of UHPC and steel strengths were less than 2 % with various impact masses and the composite plate maintained the same failure mode.

H∞ optimum design and nonlinear seismic performance assessment of tuned-mass-damper-inerter (TMDI) supported by an auxiliary structure

Tuned Mass Damper Inerter (TMDI) and its simplified form, Tuned Inerter Damper (TID), are highly effective in mitigating seismic responses in buildings, particularly when rigidly connected to the ground. However, practical constraints often limit the feasibility of this ideal setup. This study introduces a novel configuration: the AS-type TMDI, which connects the TMDI to an auxiliary structure (AS) adjacent to the main building to simulate a ground connection. To design the AS-type TMDI, this research builds upon insights from studies on TMDIs in adjacent buildings and develops an analogous 3-DOF model representing the TMDI-linked adjacent-building system. This model incorporates two buildings of different heights, allowing the TMDI to be connected at any floor level. Utilizing fixed-point theory, the study derives an analytical solution for the optimal design of TMDI systems within this novel configuration. Additionally, a method is introduced to determine the optimal stiffness and design procedure for the AS itself. The performance of the proposed AS-type TMDI is evaluated through comprehensive nonlinear seismic analysis of a benchmark nine-story steel frame structure, accounting for material and geometric nonlinearities. Results demonstrate that the AS-type TMDI performs comparably to a virtually grounded TMDI in effectively reducing the seismic responses. Furthermore, this innovative configuration outperforms both a conventional retrofitting technique, where the building is rigidly connected to the AS, and a system where a viscous damper alone connects the main building to the AS. This highlights the AS-type TMDI's potential to significantly improve the building’s seismic performance.

Uniaxial compressive stress-strain behavior of steel fiber reinforced geopolymer concrete confined by stirrups: Experimental study and analytical model

The compressive stress-strain relationship of confined steel fiber reinforced geopolymer concrete (SFRGC) is a basis for its structural nonlinear analysis and engineering design. This paper presents an experimental study on the stress-strain behavior of stirrup confined-SFRGC under uniaxial compression, with emphasis on the effects of stirrup spacing, stirrup strength and steel fiber (SF) volume fraction. The failure modes, stress-strain curves, energy absorption capacity and post-peak ductility were analyzed. The results demonstrated that due to the stirrup constraint and SF fiber crack-bridging effects, the integrity of specimens was significantly improved, showing notable ductile failure mode. The stress-strain behavior of core SFRGC was notably affected by the constraint level of stirrups, however, the effect of SF was mainly reflected in post-peak stress-strain branches. Reducing the stirrup spacing, improving the stirrup strength, and increasing the SF content can effectively enhance the strength and deformation characteristics of confined-SFRGC, as well as ductility. Moreover, synergetic effect of SF and stirrups on improving the deformation capacity of SFRGC was observed, whereas the peak stress and corresponding strain changed insignificantly with the SF content. In addition, the stirrups can reach yield around the peak stress point. Based on the analysis of test results, formulae for characterizing the peak stress, peak strain, elastic modulus as well as stress-strain responses of confined-SFRGC were proposed, with the effective confinement index of stirrups and fiber reinforcing factor introduced. The model predicted curves were in great agreement with the test curves.

Static analysis using flexibility disassembly perturbation for material nonlinear problem with uncertainty

Nonlinear finite element analysis of large-scale structures usually requires a lot of calculation cost, because it is necessary to repeatedly inverse the modified stiffness matrix caused by nonlinearity in the calculation process. When considering the uncertainty in materials, the calculation of the nonlinear analysis will be more unbearable. To improve the computational efficiency, this work develops a new method for nonlinear analysis with material uncertainty based on flexibility disassembly perturbation (FDP) approach. The FDP is an algorithm that can quickly calculate the inverse of a stiffness matrix. The basic idea of the proposed method is to introduce the FDP formula into Newton-Raphson iteration method to accelerate the nonlinear iterative calculation. Three numerical examples, one statically determinate structure and two statically indeterminate structures, are used to verify the accuracy and efficiency of the proposed method. The results show that the calculation time of the proposed method is far less than that of the existing complete analysis and combined approximation algorithms. In terms of computational accuracy, for statically determinate structures, the proposed algorithm can obtain exact solutions that are identical to the complete analysis results, while for statically indeterminate structures, the proposed algorithm can obtain approximate solutions that are very close to the complete analysis results.

Discrete differential geometry-based model for nonlinear analysis of axisymmetric shells

In this paper, we propose a novel one-dimensional (1D) discrete differential geometry (DDG)-based numerical method for geometrically nonlinear mechanics analysis (e.g., buckling and snapping) of axisymmetric shell structures. Our numerical model leverages differential geometry principles to accurately capture the complex nonlinear deformation patterns exhibited by axisymmetric shells. By discretizing the axisymmetric shell into interconnected 1D elements along the meridional direction, the in-plane stretching and out-of-bending potentials are formulated based on the geometric principles of 1D nodes and edges under the Kirchhoff–Love hypothesis, and elastic force vector and associated Hessian matrix required by equations of motion are later derived based on symbolic calculation. Through extensive validation with available theoretical solutions and finite element method (FEM) simulations in literature, our model demonstrates high accuracy in predicting the nonlinear behavior of axisymmetric shells. Importantly, compared to the classical theoretical model and three-dimensional (3D) FEM simulation, our model is highly computationally efficient, making it suitable for large-scale real-time simulations of nonlinear problems of shell structures such as instability and snap-through phenomena. Moreover, our framework can easily incorporate complex loading conditions, e.g., boundary nonlinear contact and multi-physics actuation, which play an essential role in the use of engineering applications, such as soft robots and flexible devices. This study demonstrates that the simplicity and effectiveness of the 1D discrete differential geometry-based approach render it a powerful tool for engineers and researchers interested in nonlinear mechanics analysis of axisymmetric shells, with potential applications in various engineering fields.

A reduced-order finite element formulation for the geometrically nonlinear dynamic analysis of viscoelastic structures based on the fractional-order derivative constitutive relation

A reduced-order finite element formulation for the geometrically nonlinear dynamic analysis of viscoelastic structures based on the fractional-order derivative constitutive relation

Exploring the impact of epoxy coated 3D-Printed polymers on surface roughness and mechanical behavior: An experimental and numerical study

The present study used polymers such as PLA and ABS filaments for the 3D printing parts with varying layer thicknesses. The mechanical properties and surface finish of the as-printed and epoxy resin-coated samples were investigated. Noteworthy findings indicate that when the PLA material is printed at a layer thickness of 0.06 mm followed by no coating one can achieve superior tensile strength of approximately 39 MPa as compared to 20 MPa of ABS material. The ultimate tensile strength of the coated materials averaged approximately 15 MPa for both PLA and ABS. The surface roughness of PLA material printed with a 0.06 mm layer thickness showed a 95.13 % reduction after coating. Despite ABS samples being harder than PLA material, the layer thickness emerged as the most influential factor in enhancing the hardness. The dominance of lower layer thickness over other conditions is particularly evident in the modulus of elasticity. In addition to the experimental investigation, a non-linear structural analysis of the uniaxial tensile behaviour of the coated and uncoated 3D prints was also carried out. The fracture modes and deformation behaviours in numerical modelling and experimental results across various parametric conditions were found to be in good agreement. Specifically, the numerical results at lower loading conditions for a layer thickness of 0.06 mm PLA and ABS samples, in the uncoated conditions, mirrored the experimental outcomes.

A cable finite element formulation based on exact tension field for static nonlinear analysis of cable structures

This paper presents a numerically exact cable finite element model for static nonlinear analysis of cable structures. The model derives the exact expression of the tension field using the geometrically exact beam theory coupled with the fundamental mechanical characteristics of cables. The equations for the cable elements are formulated by addressing the equilibrium conditions at the element boundaries and ensuring compatibility within the elements. Unlike previous studies that typically provide explicit expressions for cable models, this study develops a formulation that emphasizes numerical precision and broad applicability. It achieves this by deriving linearized equations with implicit expressions incorporating integrals. The present model accurately computes internal forces and deformation states, and determines the unstrained length of the cable. Additionally, it accounts for the variability in cross-sectional stiffness along the cable's length. The paper discusses solution implementations using the complete tangent matrix and element internal iterations. The effectiveness of the proposed cable elements is demonstrated through numerical examples.