Tangent Stiffness Research Articles

Bifurcations and Stability Analysis of Elastic Slender Structures Using Static Discrete Elastic Rods Method

Abstract Discrete elastic rods (DER) method provides a computationally efficient means of simulating the nonlinear dynamics of one-dimensional slender structures. However, this dynamic-based framework can only provide first-order stable equilibrium configuration when combined with the dynamic relaxation method, while the unstable equilibria and potential critical points (i.e., bifurcation and fold point) cannot be obtained, which are important for understanding the bifurcation and stability landscape of slender bodies. Our approach modifies the existing DER technique from dynamic simulation to a static framework and computes eigenvalues and eigenvectors of the tangential stiffness matrix after each load incremental step for bifurcation and stability analysis. This treatment can capture both stable and unstable equilibrium modes, critical points, and trace solution curves. Three representative types of structures—beams, strips, and gridshells—are used as demonstrations to show the effectiveness of the modified numerical framework, which provides a robust tool for unveiling the bifurcation and multistable behaviors of slender structures.

Intrinsic incremental evolution of hypoelastic discrete mechanical systems

AbstractThis paper provides an explicit geometric and coordinate‐free formulation of incremental discrete mechanics in the framework of potentially non integrable hypoelasticity. First, the general framework is developed in order to tackle hypoelasticity as an Ehresmann connection on the cotangent bundle . Two types of incremental evolutions may be distinguished, the weak or integrable incremental evolutions and the strong or non integrable incremental evolutions, according to the nature of the hypoelastic constitutive law. The geometric structure of the double tangent bundle is fully used in order to get the geometric counterpart κ of the so‐called tangent stiffness matrix. Subject to specific conditions in , the incremental evolution is then a well‐founded question. An hypoelastic four‐grains granular system illustrates in detail these general results.

Stiffness of three-bar tensegrity structure

PurposeIn this paper, the main factors influencing the structure stiffness will be analyzed by studying the tangent stiffness matrix based on different requirement in engineering practice. The authors can obtain the deformation of three-bar tensegrity basic unit in different load, and gain the primary factor by comparing the deformation, which will provide reference to concrete structure design in the engineering.Design/methodology/approachThe mathematical model of tensegrity structure was built by establishing generalized node coordinates and connective matrix. Three main factors that affect the structure deformation can be obtained by analyzing the stiffness matrix, which is preload, Young's modulus, and cross-sectional area, the thinking of deformation also be sorted out. The deformation analysis of the concrete structure is carried out, and it is concluded that increasing the cross-sectional area can quickly improve the stiffness of the structure, which provides a reference for the structural variable stiffness design in practical engineering.Findings(1) When the axial external force is applied to the structure, the torsion-angle deformation of the structure is the largest, and the radial deformation of the structure is the smallest. (2) The structure stiffness can be rapidly enhanced by increasing the cross-sectional area. But the cross-sectional area can't be increased indefinitely. Because the mass will be increased once increasing the cross-sectional area, which will destroy the structure of the advantages of light weight in engineering practice.Originality/valueThe deformation analysis of the concrete structure is carried out, and it is concluded that increasing the cross-sectional area can quickly improve the stiffness of the structure, which provides a reference for the structural variable stiffness design in practical engineering.

Second-order two-cycle analysis of frames based on interpolation functions from the solution of the beam-column differential equation

In geometrically nonlinear problems solved using the Finite Element Method (FEM), the structure response is directly influenced by the level of discretization and the nonlinear solution algorithm used. To reduce the discretization dependence, exact solutions are developed based on the deformed infinitesimal element equilibrium. To deal with the nonlinear solution problem, the two-cycle method can be used, since it is not dependent on load or displacement steps. The two-cycle method developed by Chen & Lui (1991) uses the classical geometric matrix and is not accurate for high axial loads. This happens because the geometric matrix is obtained using Hermitian polynomials which are approximate solutions. To circumvent this issue, the frame element’s tangent matrix is obtained using interpolation functions that match the homogeneous solution of the differential equation of the beam-column problem. The main objective of this study is to carry out a second order analysis of the frames and obtain equilibrium paths using the two-cycle method and the tangent stiffness matrix based on solutions of the differential equations obtained from the element’s deformed configuration. The results in terms of displacements and rotations for the examples studied are identical to the analytical solutions, showing that the combination of the two-cycle method with the exact element formulation is promising and can diminish the need for discretization.

Analytical and differential reformulations of the Vaiana–Rosati model for complex rate-independent mechanical hysteresis phenomena

A recent analytical model of hysteresis, developed by the authors to simulate a great variety of complex uniaxial rate-independent mechanical hysteresis phenomena, is reformulated in a twofold manner. First, it is proposed an analytical reformulation that, compared to the current version of the model, does not require the evaluation of any internal variable. Second, the closed form expressions provided by the analytical reformulation are expressed in rate form to foster its use especially in the field of nonlinear dynamics. To verify their accuracy, both formulations are first employed to reproduce four different complex hysteresis phenomena and the related results are compared in terms of generalized force–displacement hysteresis loops, tangent stiffness functions, and work-displacement relations. Subsequently, nonlinear time history analyses are carried out on four single degree of freedom mechanical systems for three different types of external generalized forces. To this end, two solution strategies are adopted. The former combines the model analytical reformulation with an explicit structure-dependent time integration method used to directly integrate the second-order ordinary differential equation of motion. The latter employs the model differential reformulation in conjunction with the Runge–Kutta method implemented to solve the equivalent system of three first-order ordinary differential equations. The comparison of the numerical results obtained by the two solution procedures shows an excellent agreement.

Experimental characterization and parameter identification of bolted joints under vibratory loading

The paper experimentally investigates the frictional behaviour of bolted joints with varying preload, inclined bolt head interface, and varying contact area between plates subjected to tangential loading. Experimental result shows a reduction macroslip force and energy dissipation with decrease in bolt preload. Similarly, increase in inclination of bolt bearing surface reduces macroslip force and energy dissipation. The reduction in contact area does not affect the tangential stiffness or frictional energy dissipation when contact pressure profile on plate interface is fully developed. For truncated pressure profile, the reduction in contact area reduces both the tangential stiffness and the energy dissipation. The experiments are conducted to determine the energy dissipation during pinning reveals differential pinning for lower displacements. At higher displacements, the crack initiates into the bolt shank which reduces the energy dissipation.

Regulating the fretting behavior of diamond-like carbon films by changing the composition and structure

Reducing or eliminating the slip regime and mix regime is an effective way to reduce the fretting wear, while the diamond-like carbon (DLC) film, as a self-lubricating solid lubricant, can reduce friction coefficient while improving the surface strength. The most valuable thing is that DLC film can change the film's composition, structure, and physical and chemical properties by controlling its deposition process. This regularability can realize the reduction or elimination of slip regime and mix regime under different fretting conditions, so it is important to explore the changing regulation law of the fretting regime by changing the sp2/sp3 hybrid bond content ratio and microstructure of DLC film. Here, DLC films with different sp2/sp3 hybrid carbon content ratios were obtained by annealing; the fretting performance was studied under different normal loads and amplitudes. The following rules are obtained by drawing the running conditions fretting map and analyzing the tangential stiffness, friction coefficient and dissipation energy: with increasing the sp2/sp3 hybrid carbon content of DLC film, the fretting regime moves towards the slip regime, while the friction coefficient and dissipated energy will increase. In addition, the short-range ordered microstructure, such as onion-like carbon structure, will facilitate the fretting regime towards the partial slip regime. This work demonstrates the controllability and potential of DLC films in fretting.

Semi-analytical Stiffness Model of Bolted Joints in Machine Tools Considering the Coupling Effect

This study proposes an improved semi-analytical approach for contact stiffness modeling of bolted joints in a machine tool system. First, nonlinear contact stress distribution within a single-bolted joint is obtained from the simulation results of finite element analysis software. Second, employing the Hertz contact theory and fractal theory, the contact stiffness model of a single asperity is formulated, affording analytical expressions for normal and tangential contact stiffnesses of a single-bolted joint by integrating multi-asperities in the contact area. Subsequently, considering two test specimens as illustrations, the mode shapes and natural frequencies of the proposed model and modal analysis tests are compared, and the influence of coupling effects between two adjacent bolts is illustrated. The maximum error in the natural frequencies of the proposed approach is < 2.73% relative to the experimental results. Finally, the measurements of frequency response functions on a box-in-box precision horizontal machine tool are conducted to demonstrate the accuracy and efficiency of the proposed model. The proposed model is highly efficient in revealing the influence of microcontact factors on the contact stiffness of bolted joints and in guiding the optimal functional design of bolt arrangements under the framework of virtual machine tools.

Uncertainty in identifying contact stiffness in a dovetail attachment for turbine blades

Designing bladed disks in turbo engines requires accurate dynamic models to correctly estimate resonance frequencies and related stresses. The contact parameters – stiffness and damping in the blade attachments – are currently among the most significant uncertainties of such models. Dry friction and alternating relative motions between the contact interfaces determine the transition between stick and slip that causes the nonlinear behavior of attachments. Commercial and in-house finite element software make use of specific contact elements to simulate dry friction and the resulting nonlinear behavior. These elements require the friction coefficient as input while the normal and tangential contact stiffness can be directly evaluated by the software or set up by the user. The main objective of the present research is to discuss the uncertainty associated with identifying with experimental data the normal and tangential stiffness in a dovetail coupling. In addition, the reliability of available theoretical contact models will be addressed. The response of the blade/attachment system was measured as a function of the axial load (simulating the centrifugal force) and of the blade vibration amplitude. An identification procedure using a finite element model was set up to identify the normal and tangential contact stiffness of the attachment. These stiffnesses were compared with the values predicted by a theoretical model. The comparison highlights a significant difference between measured and predicted stiffness. The final discussion focuses on the significance of theoretical contact stiffness and its use in finite element models.

Inovativna metoda za krivulje inkrementalne dinamičke analize djelomično izoliranih mostova

This paper proposes an innovative method for determining the approximate incremental dynamic analysis curves of semi-isolated bridges, including the derivation of statistical mathematical formulas. Ultimately, an innovative method is devised based on the primary and limit state tangential stiffness values of semi-isolated bridges. The bridges selected for this study are two reinforced concrete highway bridges in the United States, which have been optimally re-designed using lead rubber bearing isolators. The established procedure can be used in future studies to extract the probabilistic responses of semi-isolated bridges via a simplified and efficient method.

轮胎与土壤相互作用的离散元仿真参数标定

To carry out simulation research on tire-soil interaction, EDEM software was used to calibrate the test soil and the contact parameters between the tire and soil. The soil contact model was the Edinburgh Elasto-Plastic Adhesion (EEPA) model. Using the soil repose angle as the repose value, the contact plasticity ratio, the soil-soil rolling friction coefficient, and the tensile exponential (Tensile exp) were respectively calculated using the Plackett-Burman test, the steepest climbing test, and the Box-Behnken test, and the optimal combination of parameters was found to be E = 0.08, B = 0.1, and F = 4.8. The values of the remaining parameters were as follows: a soil-soil static friction coefficient of 0.45, a restitution coefficient of 0.5, a surface energy of 4, and a tangential stiffness multiplier of 0.35. Based on the slope sliding method, the coefficient of static friction between soil and rubber was calculated as 0.88. On this basis, a central combination test was designed to calibrate the rubber-soil rolling friction coefficient and coefficient of restitution, the optimal combination of which was found to be H = 0.18 and I = 0.55. A soil tank model was created using the optimal parameters, and the correctness of the established soil discrete element model and rubber-soil contact parameters was validated by comparing the simulation results and the results of an experiment of the tire driving process.

Effect of Substrate Roughness and Contact Scale on the Tribological Performance of MoS2 Coatings

This present study aimed to clarify the effect of contact scale and surface topography of substrates with different roughnesses on the actual contact area, tangential stiffness, and tangential deformation of the substrate at micro- and macro-scales via finite element method (FEM) simulations, as well as the final tribological performances of MoS2 coatings by experiments. The MoS2 coatings were deposited on stainless steel (SS) substrates with different roughnesses, and the settings in the simulation models were based on the roughness of the SS substrates. The predicted tribological behavior of the simulation results was confirmed by the morphological and compositional analysis of the wear track using scanning electron microscopy (SEM), energy dispersive X-ray spectroscopy (EDS), 3D profilometer, and Raman spectroscopy. The results showed that the substrate with a surface roughness of Ra 600 nm (R600), coated by MoS2 nanosheets, exhibited excellent tribological properties at both micro- and macro-scales. At the micro-scale, the lubrication lifetime of R600 was as long as 930 cycles, while the substrates with surface roughnesses of Ra 60 nm (R60) and Ra 6 nm (R6) had a lubrication lifetime of 290 cycles and 47 cycles, respectively. At the macro-scale, the lifetime of the substrate R600 was 9509 cycles, which was nearly six times longer than the 1616 cycles of substrate R60. For the rough surface of substrate, the surface grooves could not only effectively preserve the lubricant but also continuously release them, ensuring that the lubricants with low shear strength were always present in the contact interface. It was further verified that the high surface roughness of the substrate reduced friction and wear by reducing the actual contact area and enhancing the tangential stiffness of asperities, thereby prolonging the lubrication lifetime. The wear mechanisms were discussed in terms of the morphology and chemical composition of the wear tracks.

Coupled hydromechanical analysis of sparsely fractured crystalline rocks

This paper addresses the issue of evolution and coalescence of localized damage zones in sparsely fractured crystalline rocks. The approach incorporates a constitutive law with embedded discontinuity, which is phrased in terms of both the hydraulic and mechanical response. The formulation takes into account the hydromechanical interaction in regions intercepted by discontinuities. An internal length scale parameter is employed in the definition of equivalent hydraulic conductivity and the tangential stiffness operators, and the onset of newly developed macrocracks is detected by the bifurcation analysis. An enhanced mixed u-p finite element formulation is derived which considers the effect of progressive evolution of the fracture aperture in the weak statements of balance equations. Fully implicit temporal discretization is employed, and the finite element formulation is stabilized by invoking the Polynomial-Pressure-Projection (PPP) technique. A coupled FE analysis is conducted examining the response of Luc du Bonnet granite, with pre-existing fracture network, subjected to plane strain compression that triggers the crack propagation and coalescence. The approach is first verified on some benchmark problems that involve the presence of a dominant fracture. The results of simulations are compared with those obtained using a very fine mesh incorporating interface elements. Later, a series of coupled analyses are carried out examining the hydromechanical response in the presence of multiple fractures.

An approach of micro-slip modeling based on asymmetric contact pressure distribution and its application in nonlinear boundary beam system

A novel micro-slip friction modeling approach based on the Iwan model is presented here. The normal load and tangential stiffness are reassigned reasonably to obtain the tangential force–displacement expression, and the relationship between the asymmetry of contact pressure distribution and the Iwan density function is established. The micro-slip model depends on the distribution of contact pressure, and has the ability to degenerate into a macro-slip model. Based on the model, a simplified non-linear dynamics model of a rotating blade with a dovetail joint is developed, and its effectiveness is verified by comparing it with a relevant reference. The first-order bending vibration of the blade under distributed excitation is analyzed. The results indicate that the pressure distribution of the contact interface has a significant effect on the amplitude–frequency response of the blade and the contact behavior on the joint interface. Compared with the proposed micro-slip model, the single point contact model overestimates the response amplitude. The amplitude–frequency curve obtained by this model is obviously different from the macro-slip model. It is proved that the micro-slip model can more accurately simulate the nonlinear vibration characteristics of tenoned blades, which provides new insights for the study of the vibration characteristics of tenoned blades.

Experimental and Numerical Investigations of RC Frame Stability Failure under a Corner Column Removal Scenario

In recent decades, interest in the resistance of buildings and structures to progressive collapse has been increasingly sparked in research communities. Although several experimental, numerical, and analytical research projects on the robustness of building frames under a column removal scenario have been implemented, some aspects of this problem remain understudied. These aspects encompass failure mechanisms of reinforced concrete frames with slender columns, as well as criteria used to evaluate such failures. This paper focuses on experimental and numerical investigations of the structural behavior and failure of a scale reinforced concrete frame with slender columns under a sudden corner column removal scenario. In addition, we analyze the stability failure mechanism of a reinforced concrete frame with slender columns and the tangent stiffness criterion, which allow for evaluation of the ultimate state of a structure subjected to an accidental impact. A scale physical model of a reinforced concrete frame of a multistory building was designed and tested using the theory of functional similarity. For numerical study purposes, a finite element model was made that exactly the same as the test frame. We validated the findings by comparing simulation results and experimental data. The studies on the behavior of a reinforced concrete frame subjected to quasistatic loading with unequal concentrated loads identified the load transfer between columns through beams. Although these effects were minor in the frame under consideration, they can become more significant in cases of long-term loading. Numerical simulation and physical modeling of an accidental impact allowed for identification of the mechanism of load capacity exhaustion triggered by stability failure. Such failure was fragile. The moment of stability failure of the column of the experimental frame corresponded to the extremum on the force–displacement curve, indicating that zero tangent stiffness was reached. Hence, a criterion of tangent stiffness can be proposed for evaluation of the ultimate state of a structure subjected to an accidental impact.

Material‐tangent‐stiffness‐proportional viscous damping for nonlinear geometric inelastic frame analysis

AbstractThis work presents a stiffness‐proportional viscous damping model based on the material component of the total tangent stiffness matrix. It is developed for nonlinear geometric inelastic frame analysis and implemented with distributed plasticity formulations. The use of a corotational approach allows isolating the nonlinear effects arising from material and geometric sources, hence allowing to preclude the contribution of the latter to simulate non‐modelled sources of energy dissipation. Following the presentation of the theoretical framework and a discussion on the physical meaningfulness of the proposed approach, several illustrative case studies are presented. The first is based on the damped free vibration response of a slender cantilever with elastic response, whereas the second addresses the seismic behavior of an inelastic reinforced concrete cantilever column. Comparisons with the total‐tangent‐stiffness‐proportional approach showcase the advantages of the current proposal.

Efficient Method for Derivatives of Nonlinear Stiffness Matrix

Structural design often includes geometrically nonlinear analysis to reduce structural weight and increase energy efficiency. The full-order finite element model can perform the geometrically nonlinear analysis, but its computational cost is expensive. Therefore, nonlinear reduced-order models (NLROMs) have been developed to reduce costs. The non-intrusive NLROM has a lower cost than the other due to the approximation of the nonlinear internal force by a polynomial of reduced coordinates based on the Taylor expansion. The constants in the polynomial, named reduced stiffnesses, are derived from the derivative of the structure’s tangential stiffness matrix with respect to the reduced coordinates. The precision of the derivative of the tangential stiffness affects the reduced stiffness, which in turn significantly influences the accuracy of the NLROM. Therefore, this study evaluates the accuracy of the derivative of the tangential stiffness calculated by the methods: finite difference, complex step, and hyper-dual step. Analytical derivatives of the nonlinear stiffness are developed to provide references for evaluating the accuracy of the numerical methods. We propose using the central difference method to calculate the stiffness coefficients of NLROM due to its advantages, such as accuracy, low computational cost, and compatibility with commercial finite element software.

Analytical Model of Hydraulic Fracturing for Low Permeability Hot Dry Rock Reservoirs and DEM Simulation Base on Fluid-Solid Coupling

The formation of a rich underground-seam network is the key problem in the development of low-permeability hot dry rock (HDR) resources. Considering the lack of macroscopic continuum theory to study hydraulic fracturing having preset fracture-interface element, the particle-flow method of micro-mechanical discrete-element theory is introduced to simulate the mechanical behavior of hydraulic fracturing for HDR low permeability reservoirs. The reservoir is simulated as a round particle; the fracturing fluid movement is described by the seepage field equation, and rock movement is described by the displacement field equation. Finally, the particle-flow numerical model of hydraulic fracturing for HDR low permeability reservoirs is established under the condition of fluid-solid coupling: the model contains two parts (rock and fracture). Based on the parallel-bond model, a definition of micro-fractures of hydraulic fracturing is given. The relation between the fracturing effect and influence parameters is discussed. The results show that the fracture-initiation pressure is proportional to the magnitude of minimum horizontal stress, particle normal-contact stiffness, and particle normal- and tangential-connection strengths; the pressure is also independent of maximal horizontal stress and tangential contact stiffness. At the same time, the formation temperature of dry hot rock will reduce the strength of the rock, so particle-flow numerical models of hydraulic fracturing in different temperatures are discussed. Results show that fracture length and width show a trend of increase before decrease with the increase of injection pressure, an inverse relationship with minimums horizontal principal stress, and a positive relationship with HDR reservoir permeability.

Free vibration response of bidirectional functionally graded rotating micro-disk under mechanical and thermal loading

The present work studies the free vibration response of functionally graded rotating micro-disks subjected to transverse pressure and thermal loading based on the modified couple stress theory. The disk material is considered to be functionally graded along the radial and thickness directions, and its properties are assumed to be temperature-dependent following the Touloukian model. The mathematical formulation is based on an energy functional involving the von Kármán type non-linearity, in which appropriate displacement derivatives and its conjugate stress measures are used to define the strain energy of the micro-disk. The minimum potential energy principle is employed to develop the governing equations for determining the deformed configuration of the micro-disk under combined centrifugal, pressure and thermal loading. Further, the governing equations for free vibratory motion of the micro-disk are derived following Hamilton’s principle and incorporating the tangent stiffness of the deformed micro-disk. The governing equations are discretized and solved employing the Ritz method. The mathematical model is successfully validated with different reduced problems available in the literature. The influence of rotational speed, transverse pressure, thermal loading, size-dependent thickness and volume fraction indices are investigated for a wide range of parametric values. Some illustrative mode shapes along with the contour have also been presented. The present study is first of its kind and the presented results would definitely serve as benchmarks for any further study in this field.

Self-equilibrium, stability, and accuracy degradation of imperfect prismatic tensegrities with additional cable nets

Prismatic tensegrity is an attractive structure due to the unique advantages of self-equilibrium, lightweight, small compression volume, specific super stability, ease of deployment, and functional extension. However, it is subject to geometric large deformation when suffering from imperfect factors like manufacturing errors, thermal deformation, slips of tensioning knots, and mechanical relaxation of members, which will eventually reduce the accuracy and reliability. We develop a kind of prismatic tensegrities with additional cable nets for space deployable constructions. For the specific requirements of high accuracy and high reliability, we fully consider the influence of imperfect factors, and explore their environment- and time-dependent characteristics from the view of self-equilibrium, stability, and accuracy. First, a preload optimization method is established for high-accuracy determination of the shape and uniform distribution of preload. Then, a high-fault-tolerance accuracy estimation method is proposed for accuracy degradation research, for which nonlinear force density equations and time-dependent tangential stiffness matrix are formulated to obtain deformation shapes and stability criteria beforehand. Eventually, the characteristics of the prismatic tensegrities with additional cable nets, under the influence of imperfect factors, are fully revealed and forecasted. The methods and some conclusions of this study can also be applied to other complex tensegrities or prestressed pin-jointed systems.