Tangent Stiffness Research Articles

Contact Stiffness Identification Method for Joint Surface and the Effect of Contact Stiffness on the Dynamic Response of the Cable Bracket

The cable bracket is one of the crucial components of high-speed Electric Multiple Units (EMUs), which is fixed to the axle box by bolts to support the cable. As an important parameter of the joint surface, contact stiffness significantly affects the dynamic response and service life of the cable bracket. However, it is difficult to directly measure the contact stiffness of the joint surface. As a result, an identification method is proposed in this investigation to identify the contact stiffness. A finite element model incorporating both normal and tangential stiffness parameters is developed. According to the finite element model and modal testing results, an improved particle swarm optimization algorithm is employed to identify the contact stiffness of the joint surface. The effect of the contact stiffness on the vibrations of the cable bracket is investigated. The findings indicate that pre-tightening torque significantly influences the contact stiffness of the cable bracket. The contact stiffness has a great effect on the vibrations of the cable bracket. Optimal contact stiffness notably reduces vibrations of the cable bracket, thereby extending its service life.

Nonlinear Model for LTB Prediction of RHS Beams Under Combined Axial and Bending Loads With Distortional Deformation Allowed

ABSTRACTThe aim of the present study is to develop an efficient and accurate analytical model for the lateral torsional buckling (LTB) prediction of RHS beams. To this end, a coherent nonlinear geometric theory is formulated assuming large torsion angles. This theory is based on a new kinematic model that incorporates the distortion deformations in addition to the bending and torsion ones. Ritz's method is employed to discretize the governing equilibrium equations basing on trigonometric shape functions, and then the eigenvalue problem is defined according to the fundamental state. The Buckling loads are obtained by imposing the singularity condition of the tangential stiffness matrix. The numerical investigation is carried out to prove the accuracy of the proposed model in LTB load evaluation, when compared with the nonlinear shell finite element simulation using Abaqus software. The numerical results reveal that the classical linear models associated with moderate torsion angles, such as those provided by Generalized Beam Theory (GBT) and the classical eigenvalue shell finite element method, tend to underestimate the correct value of LTB loads of simply supported RHS beams. In accordance with linear and non‐linear models, the numerical study is focused on the impact of load height, intensity of introduced compressive load and section ratio b/h in the LTB loads prediction.

Jacobian vs. disturbance method for UMATs in ABAQUS: An application to isotropic damage mechanics

This paper compares the computational efficiency of different methodologies to estimate the tangent stiffness applied to ABAQUS standard UMAT user-subroutine, using implicit formulation with predictor corrector methodology. It uses two new methodologies to estimate the stiffness matrix for the predictor-corrector iterative process, named Disturbance Method and Implicit to Explicit Method. With the Disturbance Method, it is enough to build UMATs using only a secant matrix, since an approximate tangential material matrix is automatically computed, promoting a faster implementation of new material constitutive relations in the ABAQUS standard. It is also studied the possibility of the use of a new algorithm that transforms the UMAT implicit formulation to an explicit formulation using a new secant predictor corrector algorithm. In this study, a new concrete damage model that considers fracture energy regularization for both tensile and compressive behaviour with Mode I Fracture is utilized. Classical and known benchmarks to test damage models in the scientific community are used to compare both methods for the calculation of the Jacobian matrix in terms of time and number of iterations.

Shaking table test and control effect analysis of structure with negative stiffness damped outrigger

A shaking table test was conducted on a negative stiffness damped outrigger (NSDO) structure, designed based on a real engineering project, to gain insights into the practical performance of the NSDO under earthquakes when equipped with nonlinear negative stiffness devices (NSD) and dampers. A low-friction NSD was designed, built, and verified by experiments to mitigate the potential issue adversely affecting its performance. The shaking table tests showed that NSD effectively increases the maximum deformation of the damper under moderate earthquakes. Nevertheless, the NSD had no influence on the initial and dynamic stiffness of the NSDO structure when the NSD was not activated due to low displacement. With the bell-shaped damping model to match modal damping ratios, a simplified fishbone numerical model was developed to simulate the response of the NSDO structure and to study the control effect of the NSDO through parametric analysis. The results revealed that the modal damping ratios were not uniform from mode to mode, and an inaccurate higher-mode damping ratio would significantly affect the response predictions during NSDO design. The initial tangential negative stiffness of the nonlinear NSD could be used to analyze the control effect under moderate earthquakes, while the secant stiffness was necessary for severe earthquakes. In addition, a combined seismic reduction curve was proposed for a two-level seismic mitigation analysis.

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 noninvasive measurement technique for the initial stiffness of the radial artery

Arterial stiffness carries significant implications for cardiovascular disease. Monitoring changes in arterial stiffness is integral to proactive health management, however, current noninvasive methods of quantifying stiffness in vivo rely primarily on linear tangent stiffness, making the measurements vulnerable to the variability of blood pressure and thereby affecting the accuracy in portraying the health status of the arteries. This study proposed a novel methodology for evaluating arterial stiffness that is unaffected by changes in blood pressure. Ultrasound detection techniques are applied to accurately chronicle changes in radial artery diameters across varied blood pressures. Incorporating blood pressure measurements, the initial diameter at cuff blockade, and vessel diameters at systolic and diastolic pressures enables inverse determination of the unstressed initial radial artery stiffness. This method accurately mirrors the results of in vitro experiments employing porcine blood vessels at physiological pressures. The results underscore the technique's ability to quantify arterial mechanical properties precisely. This study offers a groundbreaking strategy for fostering the early detection of atherosclerosis, and aiding artery health regulation.

Stability Conditions and Stiffness Variability of General Tensegrity Systems with Kinematic Joints

Abstract Tensegrity systems represent promising candidate mechanisms with in-situ stiffness variability through changing the cables' prestress levels. However, prestress-based stiffness behaviors of tensegrity systems with arbitrary kinematic joints have not been analyzed systematically. This paper adopts the natural absolute coordinates for static modeling of tensegrity systems consisting of rigid members and tension elements. Then, a generic stiffness analysis method is developed to formulate the reduced-basis tangent stiffness matrix, which is found to include three parts: positive semi-definite material and geometric stiffness matrices, and an indefinite constraint stiffness matrix. Based on these findings, a systematic stability-checking procedure is derived to determine prestress and super stability, which are qualitative indicators of the softening and stiffening effects in different tensegrity systems. Then, we proceed to quantify the range of prestress-based stiffness variability by formulating semidefinite programming problems that numerically pinpoint the maximum and zero stiffness points. Furthermore, this paper reveals the composable nature of multiple self-stress states, enabling the composability of stiffness properties in mechanism designs. Several numerical examples verify the efficacy and versatility of the proposed method and demonstrate interesting stiffness behaviors of tensegrity systems with kinematic joints.

Determination of the plastic J integral of ductile material using the XFEM with only Heaviside function and variable-node elements

In this paper, the extended finite element method (XFEM) with only Heaviside function is proposed for the elastic–plastic fracture mechanics (EPFM) modeling. The proposed method removes tip enrichment functions depending on the polar coordinates at the crack front, and a step function only determined by the level set function is utilized to track the crack front. To alleviate the volumetric locking phenomena caused by the plastic incompressibility, the B-bar method is incorporated into the three dimensional (3D) XFEM program. Therefore, the fully integration scheme can be chosen to ensure the accuracy when addressing large plastic deformation in EPFM analysis. Additionally, the material tangent stiffness matrix of Ramberg-Osgood constitutive is given, and the local refinement technique using variable-node elements is adopted to reduce the number of elements and nodes for efficient analysis. A Newton-Raphson iterative algorithm is developed to solve the nonlinear algebraic equations caused by material nonlinearity. Several numerical examples including the determination of crack opening displacement, and the fully plastic J integral in the ductile materials are presented to test the performance of the proposed method. Comparisons with the results from the existing methodologies show that the new enrichment scheme can save computational cost and obtain sufficient accuracy even in the case of 3D curved crack.

Discrete element parameter calibration during cottonseed crushing process

In order to address the current deficit in research on discrete elements within the context of the cotton seed pressing process, a combination of physical tests and simulation experiments were conducted on Xinluzao 84 cottonseed. The Plackett-Burman experimental design was employed to identify the parameters that had a significant impact on the accumulation angle. Subsequently, the Box-Behnken design was employed to establish a second-order regression model for the accumulation angle and the significant parameters. This resulted in the optimal parameter combination for the maximum friction coefficient, minimum sliding friction coefficient and critical shear stress being 0.207, 0.388, and 5×10⁶ Pa, respectively. In order to calibrate the cottonseed bonding parameters, a Box-Behnken test was designed based on the results of the cottonseed particle crushing test. The optimal parameter combinations of normal stiffness, tangential stiffness, and bond radius were obtained as 1.25×10⁹ N/m², 7.57×10⁸ N/m², and 0.727 mm, respectively. The obtained parameters were verified by the simulation of stacking angle and the crushing test simulation. The relative error was 0.84% and 0.46%, respectively. This serves to validate the cottonseed bonding model and simulation parameters, thereby providing a foundation for optimising the core structural parameters of the cottonseed shelling machine.

Optimal lamination angles via exact and efficient differentiation of the geometrically nonlinear finite element solution

Laminates allow tailoring the fiber orientations in the layers to obtain the desired mechanical response. The optimal layup design is a challenging task in the case of finite deformations and buckling. For an assigned design, an incremental-iterative finite element analysis is needed to compute the structural response. Gradient-based methods are very often the most efficient optimization tools. Their bottleneck is the gradient evaluation, generally possible only approximately by finite differences. This article shows how to compute the exact gradient of the geometrically nonlinear finite element solution with respect to the stacking sequence. The strategy relies on the implicit differentiation of the nonlinear discrete equations of a control equilibrium point corresponding to an assigned displacement. This provides the load factor gradient by a single fast-solution linear system with the already factorized tangent stiffness matrix, regardless of the number of design variables, and scalar products involving the partial derivatives of the discrete internal force vector, calculated in an exact and efficient way. Several applications demonstrate the efficiency and robustness of the approach.

Nonlinear analysis of spatial trusses with different strain measures and compressible solid

This paper investigates the nonlinear behavior of spatial truss elements under finite deformations, focusing on the impact of various strain measures in compressible materials. We examine both Total Lagrangian (using engineering and Green–Lagrange strains) and Eulerian formulations (using natural, Biot, and Almansi strains). The analysis assumes a linear spatial hyperelastic material where Cauchy stress is proportional to axial natural strain via Young’s modulus. For infinitesimal strains, Young’s modulus remains consistent across different stress/strain pairs. In the finite strain regime, we derive a nonlinear secant modulus based on Young’s modulus. Internal force vectors and tangent stiffness matrices are computed using the direction cosines of the truss element in its deformed state. The paper demonstrates that for infinitesimal deformations, adjusting the modulus of elasticity when using different stress/strain pairs is unnecessary. However, for finite deformations, it is essential to adjust the modulus of elasticity. Numerical simulations validate the performance of the proposed 3D truss element against established formulations. This research offers critical insights into the nonlinear response of spatial trusses, guiding the selection of appropriate strain measures for enhanced accuracy in engineering applications. These findings contribute to more reliable and efficient structural designs, especially in scenarios involving finite deformations and compressible materials.

A novel cohesive interlayer model considering friction

To understand the influence of friction on the shear-slip behavior of heterogeneous brittle composites, a novel cohesive interlayer model that can effectively capture the friction effect was proposed based on the classical Park-Paulino-Roesler model. Meanwhile, the unified potential energy function governing the interface tangential and normal behaviors was introduced to realize the mechanical interaction between Mode I fracture and Mode II fracture, and a smooth friction growth function was added in the elastic deformation stage for calculating the accurate contact pressure and friction force. Furthermore, the capability of the proposed model in addressing unloading and reloading was improved, and the fracture energy can vary accordingly during cyclic loading. To verify the effectiveness of the proposed model, it was examined by modelling the shear behavior of a masonry wallette. The results show that the relative error of the proposed model is 14.92% which is much lower than those of the other three pre-existing models when calculating the displacement corresponding to peak shear stress. Meanwhile, in terms of peak shear stress and initial displacement at residual stage, the relative errors of the proposed model are only 1.82% and 5.04%, respectively, indicating the high accuracy. Besides, the tangent stiffness determined by the second-order integration of the potential energy function is also continuous and smooth, which ensures the effective convergence of the proposed cohesive model.

Cyclic behavior of cold-formed stainless steel sandwich tubular brace: An experimental investigation

Stainless steel shows promise for application in seismic-active regions. Previous research showed that stainless steel dampers could exhibit good energy dissipation capacity and high post-yield stiffness under cyclic loading. In light of that, a novel type of cold-formed stainless steel sandwich tubular brace (STB) is presented in this study, featuring one stainless steel square hollow section (SHS) tube as the load-carrying component and two additional stainless steel SHS tubes providing a buckling-restraining effect. Six specimens were tested under cyclic loading, varying parameters including the cross-section slenderness of the core tube, restraining ratio and loading histories. Loading protocols encompassed standardized ones following the Chinese specification (CN) and American provision (US), as well as a protocol representing the near-fault earthquakes (NF). Failure modes, hysteretic response and seismic performance indexes were obtained. Results showed that the hysteretic curves could maintain stable. Under the CN loading protocol, the specimen with a slender core cross-section exhibited uniform local buckling. The strain hardening coefficient ω stayed within the range of 1.46–1.58. The specimens exhibited a slow decay rate of tangent stiffness, highlighting their relatively high post-yield stiffness. The cumulative energy dissipation factor could reach 658, signifying good energy dissipation capacity. Additionally, under NF loading, specimens exhibited enhanced cumulative ductility compared to CN or US loading. Furthermore, an evaluation was conducted to assess the suitability of existing design methods for preventing overall buckling and local bulging in stainless steel STBs. It indicated that the current design approaches tended to be overly conservative for both overall and local stability design of stainless steel STBs, primarily due to the lack of consideration for SHS core stiffness and the interactions between flanges and webs, respectively.

Modeling static friction behavior of elastic–plastic spherical adhesive microcontact in full-stick condition

The static friction behavior of an elastic–plastic spherical adhesive microcontact between a rigid flat and a deformable sphere under combined normal and tangential loading is studied by the finite element method (FEM). The contact between the sphere and the rigid flat is assumed to be full-stick, and the sliding inception is related to a loss of tangential stiffness. The intermolecular force between the rigid flat and the sphere is assessed by the Lennard–Jones (LJ) potential, which is applied to the sphere and the rigid flat by a user subroutine. The evolution of the adhesive force with tangential displacement in the full-stick condition is revealed. The results indicate that the increasing effect of adhesive energy on the static friction coefficient gradually diminishes with an increase in the adhesive energy and the external normal load. Finally, based on an extensive parametric study, an empirical dimensionless expression is obtained to predict the static friction coefficient of the spherical adhesive microcontact considering the intermolecular force.

A novel model for detecting prestress by external anchor head section of prestressed anchor cables

Prestressed anchor cables are widely used in ocean engineering, the accurate and efficient prestress measurement is crucial for ensuring their safety and stability. In this study, a new theoretical model based on the Hertz contact theory and fractal theory was proposed to establish the relationship between tangential contact stiffness (TCS) at the interface of the anchorage and the anchor bedding plate in the external anchor head section. This model was validated through indoor static load tests, and its feasibility for practical application was explored by indoor dynamic load tests. The results demonstrate that both tangential load and tensile load are the primary factors determining the value of TCS. TCS decreases with increasing tangential load, but increases with growth in tensile force. The calculated theoretical values based on the proposed model align closely with test data from indoor static load tests, indicating the efficiency of the proposed model in prestress evaluation. The correlation trends between TCS and tensile force observed in dynamic load tests exhibit good consistency with those found in static load tests, which also align with calculated values based on the proposed model, demonstrating its feasibility in dynamic load tests. The proposed model is modified by considering the interactions between the micro-convex bodies, and other parameter such as fractal dimension D need to be considered when applying the proposed model. These findings can provide valuable insights for the application of prestressed anchor cables in ocean engineering.

Nonlinear Analysis of Plane Frames Using a Co-Rotating Timoshenko Beam Element

The present work describes a co-rotating shear flexible beam element without shear locking and integrating Euler-Bernoulli's and Timoshenko's beam theories. The co-rotational kinematics is based on the separation of the motion in deformational and rigid body components. The deformation of the beam element is composed by three natural modes of deformation: the extension mode, the symmetric bending mode, and the anti-symmetric bending mode. The respective generalized stresses from these natural modes are self-balanced, allowing the achievement of a consistent tangent stiffness matrix. In this paper, it is detailed and deduced all the algebraic steps for the deduction of the elastic stiffness matrix, the geometric stiffness matrix, and the co-rotation stiffness matrix. Some examples are presented and the numerical results demonstrate that the beam element here presented is able to handle large rotations.

Impacts of the mechanical connectors on seismic performance of the restored masonry Plaka Bridge using experimental and numerical studies

The use of clamp and dowel materials in the arch of the masonry bridges holds pivotal significance in enhancing the seismic resilience of these structures, as these elements contribute crucially to the structural integrity and dynamic response of these venerable structures. For this reason, the seismic performance of the restored historic Plaka Bridge, originally constructed in 1866 in Arta, Greece, and later restored in 2015 after flooding, was investigated considering the dowels and clamps in the arch section of this study. A combination of experimental and numerical analyses was conducted and verified to understand the in-situ interface behavior of the arch stones. In the first phase of the study, the mechanical properties of the dowels and clamps were characterized by the experimental tests. Compression and tension tests were conducted on the stone samples containing dowel bars and clamps respectively to determine the mechanical properties, The experimental outcomes were subsequently validated through Three-Dimensional (3D) Finite Difference Models (FDM). It was obtained that the dowels and clamps represent the bi-linear material model at the interfaces. Finally, the idealized interface behavior was constructed with normal and tangential spring stiffness properties in 3D-FD Models. These springs successfully simulate the interface delamination failure of the stones in all directions. In the final phase of the study, a 3D-FD model of the Plaka Bridge was constructed employing the previously validated normal and tangential stiffness elements in the arch of the bridge. A free-field non-reflecting boundary condition was imposed on the base and lateral boundaries of the bridge model. Seismic analyses were conducted utilizing the significant seismic events including 2023 Kahramanmaraş earthquakes. The results revealed significant alterations in the structural behavior of the bridge under considered earthquakes. Furthermore, the displacements, principal stresses and tensile-compressive damages in the main arch significantly reduced with the adopted stiffness values. This study proposes the undeniably realistic and efficient implementation of interface material modeling for the mechanical connections as opposed to the traditionally assumed mortar properties in literature. Furthermore, the combined behavior of the dowel, clamp, and Khorasan mortar proposes non-linear interface material model to the existing literature by invalidating commonly assumed linear interface behavior in masonry structures.

A Foundational Study on Rational Optimization of Damping Ratio for Accurate Dynamic Simulation with Ultra Large Displacement

The integration of dynamic simulation analysis has become widespread in general-purpose software, providing enhanced capabilities. However, accurately tracking deformations based on complete equilibrium solutions remains a significant challenge in problems characterized by strong geometric nonlinearity. This study examines the accuracy of the combined Newmark β method and Tangent stiffness method in dynamic analysis with ultra large displacements and evaluates the utility of Rayleigh proportional damping in numerical simulations compared to experimental models. An experimental model of a slender steel plate undergoing free vibrations after being released from a deformed state was created. Video footage capturing the deformation histories was compared to computational simulations to verify accuracy. The study also examines the appropriate values of the damping ratio (ζ) and the Newmark β value in the simulations. The results indicate that adjusting various damping ratio and a β value of 1/2 yield more realistic simulations with longer conservation of mechanical energy. The findings suggest that incorporating numerical damping into actual damping settings can achieve a more realistic simulation of dynamic behaviour with ultra-large displacements. Furthermore, experiments and analyses were performed to correct natural frequency by changing Young’s modulus, which is a key factor influencing natural frequencies, with observed correlations to plate thickness, setting the stage for further research under varied conditions to develop a more rational methodology for structural analysis.

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.

Nonlinear dynamics of a dual-rotor system with active elastic support/dry friction dampers based on complex nonlinear modes

In this study, the nonlinear dynamics of a dual-rotor system with active elastic support/dry friction dampers (ESDFDs) are investigated based on complex nonlinear modes (CNMs). The finite element method (FEM) combined with a full-3D friction model is introduced to construct the governing equation for the system. Additionally, the Craig–Bampton technique is applied to downscale the finite element model of the system. Based on the reduced order model (ROM), the nonlinear modal damping ratio of the target mode is employed to measure the dry friction damping performance of active ESDFD. The effects of the active ESDFD position, normal force, and tangential contact stiffness on the nonlinear modal damping ratio and modal frequency are analysed. Moreover, the softening characteristics of the active ESDFD are revealed, and the critical speed intervals of the active ESDFD/dual-rotor system are determined. Furthermore, by using the harmonic balance–alternating frequency/time domain (HB–AFT) method, the steady-state response of the system under unbalanced excitation is calculated. The accuracy and effectiveness of nonlinear modal analysis are validated based on the relationships between nonlinear modes and steady-state unbalanced responses. Conversely, the vibration mitigation effects of active ESDFD are determined by the unbalanced response amplitude. Additionally, the controllable region and optimal normal force for effective vibration control in the target mode are defined. Depending on the controllable region, a control strategy for turning on/off the optimal normal force is developed. The findings demonstrate that the developed control strategy enables the active ESDFD to significantly reduce the response amplitude of the dual-rotor system across various excitation levels, showing substantial potential for engineering applications.