Nonlinear Structural Analysis Research Articles

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 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 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.

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.

Geometrically nonlinear static analysis of multi-component structures through variable-kinematics finite elements

Geometrically nonlinear static analysis of multi-component structures through variable-kinematics finite elements

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.

Simulation of soil-structure interaction using inelasticity-separated finite element method

The soil–structure interaction (SSI) effect is nonnegligible during the nonlinear seismic response analysis of large-scale or underground structures. However, the computation of nonlinear response of a structure considering SSI effect usually is a time-consuming process because the numerical model should involve the structure itself, a wide-range soil domain and the soil–structure interface, especially for large-scale structure. This study aims to incorporate the inelasticity-separated finite element method (IS-FEM), which is an algorithm recently developed for efficient structural nonlinear analysis, to improve the efficiency of the soil-structure interaction system. To this end, an inelasticity-separated contact element model for modeling the nonlinear behavior of soil and structure interface is developed in this study. To derive the inelasticity-separated governing equation of this model, a reference elastic stiffness is defined so that the normal and shear deformation of any point pair in the presented contact element is decomposed into reference linear and reference nonlinear parts according to this reference elastic stiffness. As a result, the problem of inconsistency between the requirement of IS-FEM for constant initial linear elastic stiffness and the existence of inflection point for the stress versus relative displacement relationship of contact behavior can be solved. Then, the Goodman element formulation for depicting the nonlinear contact behavior is introduced and an interpolation scheme to establish the reference nonlinear deformation field in an element is incorporated. By combining the proposed model with existing inelasticity-separated elements, a global analytical model of the soil–structure interaction system can be established within the framework of IS-FEM. Because the nonlinearity usually occur in some local regions of the global model and the use of the IS-FEM only require performing these operations for a small scale matrix representing local nonlinearity, the computational efficiency of nonlinear structural response analysis considering SSI effect can be improved significantly. The applicability and efficiency of the proposed method is finally verified by two numerical examples.

Nonlinear static analysis of masonry structures with mortar joints and cracking units by optimization‐based rigid block models

AbstractA rigid block model with elasto‐plastic softening interfaces is developed for nonlinear static analysis of masonry structures subject to monotonic loading. Cracking, crushing, and shear failures are taken into account at interfaces, following a simple micro‐modeling approach. An optimization‐based formulation is used for the solution of the equation systems governing the behavior of the rigid block assemblage. A simple incremental solution procedure is implemented to take into account the material softening behavior and the effects of large displacements on equilibrium conditions. The interface models are validated against tension and shear tests on bi‐block prisms from the literature. Applications to numerical and experimental out‐of‐plane loaded masonry walls as well as to circular arches with mortar joints are presented to evaluate the effects of tensile strength and the accuracy of the developed model against responses involving P‐Δ effects. Comparisons with experimental tests on shear walls also involving cracking of the units in the failure mechanisms are finally reported to discuss the potentialities and limitations of the proposed modeling approach.

Koiter–Newton Reduced-Order Method Using Mixed Kinematics for Nonlinear Buckling Analysis

The Koiter–Newton method improves the computational efficiency of nonlinear buckling analysis; however, the construction of reduced-order models using fully nonlinear kinematics is still a tedious and time-consuming work. In this paper, the Koiter–Newton reduced-order method using mixed nonlinear kinematics is presented for the geometrically nonlinear buckling analysis of thin-walled structures. Strain energy variations up to the fourth order were achieved using mixed kinematics for the improved Koiter theory. Corotational kinematics, which is inconvenient for high-order variations, was applied to calculate the first- and second-order variations for the internal force and tangent stiffness, respectively, whereas the third- and fourth-order strain energy variations were facilitated by explicit algebraic formulations using updated von Kármán kinematics. A reduced-order model with 1+m degrees of freedom was established, of which m perturbation loads were considered to make the method applicable for buckling problems. The geometrically nonlinear response was traced using a predictor–corrector strategy by combining the nonlinear prediction solved by the reduced-order model and the correction using Newton iterations. Numerical examples of structures with various buckling behaviors demonstrate that the performance of the proposed method is not obviously affected by using simplified kinematics, and sometimes it even exhibits a superior capability for path-following analysis.

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.

Novel approach for precise identification of vibration frequencies and damping ratios from free vibration decay time histories data of nonlinear single degree of freedom models

The significance of Single Degree of Freedom (SDOF) systems lies in their ability to serve as foundational elements for modeling more complex dynamic problems. By capturing essential dynamic behavior with simplicity, SDOF models enable efficient analysis and comprehension of complex systems, justifying the investigation of these simplified models. In nonlinear scenarios, SDOF models result in time series data wherein vibration frequencies vary over time. Classically, time–frequency or Hilbert transforms applied to temporal responses are frequently used to identify the evolution of frequencies and damping ratio over time. These techniques provide results that reflect the spectrum composition achieved for the analyzed time window and present difficulties in precisely determining the magnitude and the exact instant of an effective frequency or damping ratio variation. In this sense, this work introduces a new methodology capable of accurately identifying the vibration frequency as a function of time, i.e., the instantaneous frequency, along with the instantaneous damping ratio. At this initial stage, the focus is on validating the methodology by comparing its performance with the classical approach based on time–frequency transforms. The initial results obtained from synthetic free vibration decay responses of SDOF nonlinear models highlight the accuracy of our findings compared to those obtained from time–frequency transforms. The presented methodology holds promise for further advancement, with potential impacts including structural damage identification, modal identification and nonlinear dynamic analysis.

A probabilistic distributed digital twins approach for short-term stability and islanding of smart grid

The simultaneous evaluation of voltage, frequency, and transient stability enhances the reliability of the power systems, due to the transient effect of stability aspects on each other. However, different quantitative and qualitative parameters in the aspect and nonlinear structural analysis of the system make simultaneous and accurate predictions difficult. For this reason, the simultaneous probabilistic prediction of the stability state of three aspects with high certainty can be considered the most accurate method. This study presents a simultaneous prediction method for the stability aspect states by means of the decision theory and the game theory matrix on the basis of distributed digital twins (DDTs). The decision matrix weighs the aspect states in each bus using the distributed structure measurement and prioritizes the weak buses. In addition, the stability of different bus aspects is predicted using the game matrix. Furthermore, the changes in the system stability are predicted with high probability and certainty in the digital twins (DTs) graph, and if required, islanding is performed on the basis of the weak buses' priorities. The effects of the three aspects on each other, and the use of real-time data of the DDTs, have contributed to the high accuracy and speed of the proposed method. Simulations were performed on IEEE 14-bus, 39-bus, and 118-bus systems by means of MATLAB and DIgSILENT. The results show the advantages of the proposed method over other methods.

Nonlinear Dynamic Analysis of High-Strength Concrete Bridges under Post-Fire Earthquakes Considering Hydrodynamic Effects

This study employed the linear interpolation method to ascertain the curve relationship between the elastic modulus and stress of high-strength concrete C60 with temperature, and the nonlinear dynamic analysis of high-strength concrete bridge structures subjected to post-fire earthquake action at varying water levels was subsequently evaluated. It was established that both the hydrodynamic effects and the temperature effects have a considerable impact on the structural dynamic response of bridges. The presence of water has been observed to increase the dynamic response of pier structures. At water levels of 0 m and 10 m, the temperature effect results in a reduction in the fundamental frequencies of acceleration and displacement responses by 73.68% and a decrease in the fundamental frequency of stress responses by 83.33%. At a water level of 20 m, the fundamental frequencies of the acceleration, displacement, and stress responses decrease by 53.49%. In consideration of the acceleration and displacement at the pier top and stress at the pier base at a water depth of 10 m, the superposition of temperature effects and hydrodynamic effects results in an increase of 59.06% in acceleration, 25.93% in displacement, and 49.53% in stress than combination effects, respectively. At a water depth of 20 m, the superposition of temperature and hydrodynamic effects results in an increase of 92.82%, 100%, and 127.85% in acceleration, displacement, and stress, respectively. The combined effects of hydrodynamic and temperature effects cannot be described merely as a linear superposition of the two single actions. The research findings provide a significant theoretical basis for understanding the impact of multiple disasters, such as fires and earthquakes, on bridge structures.

Bond performance and constitutive model of steel bar in G-UHPC under cyclic loading

The increasing embrace of the low-carbon and environmentally sustainable construction approach has opened exciting opportunities for using environmentally friendly geopolymer based ultra-high-performance concrete (G-UHPC) in engineering structures, drawing significant attention to its performance. Seismic performance is a crucial factor in structural design, and it is greatly affected by the bond behavior between steel bar and concrete. As a result, a thorough understanding of the bond behavior of steel bars to G-UHPC provides essentially theoretical support for the structural design and nonlinear finite element analysis of G-UHPC structures, which is vital for their widespread implementation in earthquake engineering. This study tested 54 specimens from 18 groups to examine the bond performance of G-UHPC and steel bars under cyclic loading. The influence of concrete and steel bar types, protective layer thickness, bond length, and steel fiber on critical bond performance indicators were evaluated, and the bond stress-slip hysteretic and skeleton curves were analyzed. Subsequently, based on the experimental data, a constitutive model was proposed to predict the bond stress-slip relationship of steel bars and G-UHPC under cyclic loading. The results indicated that the steel bars exhibited superior bond performance with G-UHPC as compared to NSC. The bond and residual strength between steel rebars and G-UHPC was approximately 1.73 times and 3 times greater, respectively, in comparison with that between NSC and steel rebars. Increasing the strength of steel bars only resulted in a limited 2 % improvement in bond strength. The steel bar diameter, protective layer thickness, bond length, and the dosage and length of steel fibers in G-UHPC significantly influenced the bond performance. The addition of irregular-shaped steel fibers to G-UHPC has demonstrated potential in improving energy consumption, with the most remarkable enhancement observed for the inclusion of twisted steel fibers. Specifically, the incorporation of twisted steel fibers signally enhanced the energy consumption to approximately 1.3 times the energy dissipation as compared to straight steel fibers with the same dosage and aspect ratio. In addition, the accuracy of the proposed constitutive model has been validated.

Research progress on freeze–thaw constitutive model of concrete based on damage mechanics

Abstract In cold areas, freeze–thaw damage seriously affects the long-term use and safe operation of concrete structures. The constitutive model is an important foundation for predicting deformation and strength characteristics of concrete materials and for the non-linear analysis of concrete structures. This study is based on the elaboration of methods for the constitutive model of damaged materials by using damage mechanics and others. This study focuses on the existing constitutive model results of concrete under the static axial compression load, dynamic load, and coupling environmental load, and analyzing the problems in existing studies. Research has shown that segmented models exhibit higher fitting accuracy of concrete freeze–thaw constitutive model under static axial compression loads. By defining coupled damage variables, it is possible to approach the actual freeze–thaw damage of concrete under environmental coupling, and attention should be paid to the differences or interactions between damage factors. In order to meet the actual engineering needs of high altitude and cold areas needs to expand the temperature range of freeze–thaw tests and consider the dynamic loads impact on concrete damage, the establishment of constitutive model of concrete under the actual freeze–thaw damage is the focus of frost-resistant durability research of hydraulic concrete in cold regions.