Finite Stiffness Research Articles

Rapid dynamic load estimation procedure for lifting propellers in forward flight

AbstractLifting propellers operate at oblique inflow and thus encounter severe dynamic loads during forward flight, impacting structural integrity, fatigue, and vibration. Numerical optimization approaches consider aerodynamic, structural mechanical, and aeroacoustic aspects within preliminary design. To also account for dynamic loads during forward flight, a novel procedure allows their rapid estimation. Based on steady-state simulations combining strip theory and finite beam elements, aerodynamic excitation, damping, and stiffness are defined in the frequency domain. Loads are derived through a linear inflow model and quasi-steady aerodynamics. Damping and stiffness loads are linearized and transferred into matrix form to calculate the frequency response. The computationally expensive need for simulations in the time domain is thus avoided. Applicability extends to both fixed and variable pitch lifting propellers utilized in large multicopters for cargo or passenger transportation. Comparisons to time-marching simulations show good agreement with deviations of approximately 10 %. The analytical derivation yields physical insights to understand and reduce dynamic loads and their magnification due to resonance.

Image-based finite element model stiffness and vBMD by single and dual energy CT reconstruction kernel

Single-energy quantitative computed tomography (SEQCT) provides volumetric bone mineral density (vBMD) measures for bone analysis and input to image-based finite element models (FEMs). Dual-energy CT (DECT) improves vBMD by accounting for voxel-specific material variations utilizing scans at multiple x-ray energies. vBMD is also altered by reconstruction kernel that cannot be accounted for using calibration phantoms. This study compared vBMD and FEM stiffness derived from SEQCT and DECT images reconstructed with two common kernels. SEQCT and DECT images of cadaveric shoulders (n = 10) were collected using standard (STD) and boneplus (BONE) kernels. Hounsfield Units were converted to vBMD using specimen-specific calibrations. DECT STD and BONE images were generated using an established material decomposition method with 40 and 90 keV simulated monochromatic images. A proximal humerus bone section below the anatomic neck was used for vBMD analysis and FEM generation. FEMs were loaded to 1 % apparent strain for stiffness measurements.Between STD and BONE kernel images, average vBMD differed 0.9 mgK2HPO4/cc and 4.1 mg K2HPO4/cc, in SEQCT and DECT images, respectively. Significant differences occurred in DECT images (p = 0.001). BONE reconstructed images produced higher vBMD measures across both SEQCT and DECT images. The difference between STD and BONE in both SEQCT- and DECT-based FEMs persisted, with larger estimated stiffness in BONE models. For six of the models DECT-based had higher stiffness than SEQCT-based models using the same kernel, although these models differed between STD and BONE kernels. Differences in stiffness between STD and BONE derived models were similar across image types (DECT: 17.5 kN/mm; SEQCT: 19.0 kN/mm). Stiffness values were significantly different within SECT kernels and between SEQCT BONE and DECT STD models. This study shows important differences in vBMD and FEM stiffness that occur due to CT-based imaging parameters alone. These results indicate that consistent imaging parameters should be used for vBMD analysis and FEM input to avoid systematic measurement errors.

A simple structure with two different linear regimes

We perform an asymptotic static analysis on a simple two degrees of freedom structure, made up of a highly stiff bar and two springs, which shows two different linear behaviours and different finite stiffness depending on the magnitude of an applied static force. Both regimes as well as the non-linear transition between them occur in the framework of small forces and displacements. The problem gives physical insight to the relations between linearity and small displacement hypothesis and it is suitable for a first course in asymptotic analysis to students with basic skills in structural or rational mechanics.

Parallel FEM for structural analysis of reinforced systems

Calculations in software systems based on the finite element method (FEM), performed with account for genetic nonlinearity, i.e., assembling, are not feasible for a number of construction problems. In particular, in buildings made of factory-made and cast-in-place concrete, the latter is gradually involved in deformation and external load processes. In reconstructed buildings, during the increase in the element cross-section (e.g., reinforced concrete or metal casing), the reinforcing and reinforced materials are also involved in deformation and external load processes at different time. As a result, there are initial stresses and strains in the "old" part of the element by the time its "new" part installation and strength gain. At the same time, in the reinforcing material or cast-in-place element, the stress-strain state is zero until the additional load is applied.Thus, during the structural analysis, a change in the finite element stiffness is required. But FEM software systems have no finite element capable of changing (increasing) the stiffness. This determines the relevance of the developed algorithm for the structural analysis of factory-made and cast-in-place concrete buildings using the parallel FEM.Purpose: The aim of this work is to improve the parallel FEM and develop a calculation algorithm based on it.Research findings: Based on the parallel FEM, the calculation algorithm is proposed for factory-made and cast-in-place concrete buildings. The proposed algorithm accounts for the structural properties of such buildings, including the gradual involvement of different-aged materials in the deformation process. Using the proposed algorithm, the structural analysis is conducted for the reconstructed system, in which the finite element is replaced by the parallel element, and the hinge coupling of elements is simultaneously replaced by a rigid one.

Multi-criteria parametric optimization of the displacement and weight of a shell of minimal surface on a circular contour consisting of two inclined ellipses under thermal and power loading with consideration of geometric nonlinearity

Thin shells are well suited to optimal design problems, and the finite element method and gradient descent method make it possible to solve inverse problems in structural and applied mechanics. The calculation process takes into account geometric nonlinearity. In modern numerical studies, a formulation taking into account geometric nonlinearity in the finite element method is used, namely, a stepwise loading procedure. It becomes necessary to take into account the relations between displacement vectors and their derivatives and strain increments. These relations help to determine the stiffness matrix of the finite element at each loading step, which leads to a qualitative study of real displacements in the structure. The geometrically nonlinear calculation of a shell of minimum surface on a circular contour consisting of two inclined ellipses in a multi-criteria parametric optimization is performed by the finite element method. The finite element method is a universal variational method that is focused on solving the most complex problems of elasticity theory and applied and structural mechanics using separate calculation complexes. Stiffness matrix - for the whole body is formed on the basis of the finite element stiffness matrix. The system of solving equations of the finite element method is formed using the Lagrange's variational principle, according to which the total potential energy P of a finite element model of a body in a state of stability and equilibrium has a minimum value. As part of the sensitivity analysis, gradients of the design variables of the structure, displacements in the form of partial derivatives of these characteristics along the design variables, and shell thickness are calculated. The sensitivity information serves as the basis for building an optimal design algorithm using the gradient descent method of the objective function. Standard multi-criteria parametric optimization allows for an average of 10% steel savings, with geometric nonlinearity increasing to 20%. At this study site, 24.4% of sheet steel was saved, which is a significant relative saving. This methodology of the authors shows high results for innovative design and production of steel thin shells in Ukraine and around the world, and also makes it possible to apply several types of optimization to one research object.

Чисельне дослідження впливу довжини стрижня на його критичні сили

The effect of changing the length of a longitudinally compressed rod on its critical forces is numerically investigated. The research is carried out on the example of a two-span rectilinear rod of bending stiffness constant along length, which is compressed by a constant lengthwise longitudinal force and hinged on one of the ends on an absolutely rigid support, and inside - on a support of finite stiffness. The change in the length of the rod occurs due to the movement of the end hinge support with the corresponding increase or decrease of the adjacent section of the rod without changing the position and characteristics of the remaining constraints. The dependence of the critical forces of the rod on the position of this support and, accordingly, on the length of the adjacent compressed section of the rod is investigated. Calculations are performed on the basis of the use of known exact analytical expressions of the influence functions of a rod of constant cross-section compressed by a longitudinal force constant by length. In the considered examples, qualitative signs of increase, decrease, and extremum of simple critical forces when changing the length of the rod, related to the qualitative features of the corresponding buckling forms, established earlier theoretically, were fully confirmed. In particular, exact calculations have shown that the increase or decrease of the simple critical force when the length of the fragment of the rod adjacent to the movable support is changed is determined by the type of the corresponding buckling form in the neighborhood of this support. Different possible configurations of buckling forms are considered, and the behavior of critical forces when changing the length of the rod are considered for each of the configurations. In order to verify the previously established theoretical results, which relate to the study of the behavior of not only the main critical forces, but also higher simple critical forces, which have an arbitrary number in the spectrum, the calculations are carried out in the article for the second critical forces of the rods considered in the given examples. The results of the calculations are shown in the form of graphs, which represent configurations of buckling forms of various possible types in connection with the corresponding changes in critical forces. Graphs of the dependence of the second critical force of the studied rods on their length are also given. It has been demonstrated that under certain conditions, reducing the length of the rod can lead to a reduction in its critical force.

How Fast are Elastic Domino Waves?

The paper is concerned with the problem of toppling propagation velocity in elastic, domino-like mechanical systems. We build on the work of Efthimiou and Johnson, who developed the theory of perfectly elastic collisions of thin rigid dominoes on a frictional foundation. This theory has been criticised for the lack of correspondence with the experimental observations, in particular, prediction of infinite propagation velocity for zero spacing between dominoes, as well as the inability to represent the collective nature of collisions in real domino systems. In our work we consider a more realistic scenario of dominoes of finite stiffness and obtain a theory of fast elastic domino waves, taking into account a limit velocity of the perturbation propagation in the system of dominoes. Moreover, finite collision time allows to extract dynamic quantities of collisions and establish upper and lower borders for domino separations where the theory could still be applied. Our discrete element simulations support our theoretical findings and shed light on the nature of collective interactions in the nearly-elastic domino chains.Graphical abstract

Calculation and manufacture of multilayer composite structures

A brief overview of the work done to solve the problems of bending of composite plates and shells is presented. A mathematical model has been developed that takes into account the physically nonlinear properties of the layer’s material. The seams between the layers have finite longitudinal stiffness. Seams can be continuous or discrete (anchors). The proposed mathematical model makes it possible to take into account the creep properties of the aging material. The reliability of calculations is substantiated by comparison with experimental data and known calculations. Taking into account the temperature mode made it possible to develop a technology for manufacturing multilayer structures with predicted properties of the material of the layers. Based on the results presented, methods for calculating multilayer structures and technologies for their manufacture have been developed. It is proposed to create products from cement-based composites with unique planned characteristics, oriented to the intended purpose of the manufactured structures. The intended purpose of products can be achieved by using a material with special characteristics in each layer. To protect against rodents and microorganisms, light antiseptics can be used in the process of making the composite. The solution of these issues in thin-walled structures is associated with the development of technologies for creating multilayer composite systems. The results of the creation of composites are presented.

CALCULATION OF SHELLS BASED ON THE MIXED FEM VARIANT WITH TENSOR-VECTOR APPROXIMATION OF THE DESIRED VALUES

An algorithm for calculating thin shells based on the mixed finite element method (FEM) in a two-dimensional formulation is proposed. To correctly take into account possible displacements of the shell as a rigid body, the algorithm implements the developed tensor-vector form of the interpolation procedure of the desired values, which were chosen as the tensors of deformations and curvature at the point of the middle surface and the displacement vectors of this point. The discretization element was a quadrangular fragment of the middle surface with unknown nodes in the form of location and their first derivatives, as well as components of strain and curvature tensors among the inner surface at the nodes of the finite element. After minimizing the modified mixed functional, a 36?36 finite element stiffness matrix is formed. In order to verify the developed algorithm, a number of test problems were solved for calculating a fragment of an elliptical cylinder with an analytical solution, as well as for calculating a shell with spring supports that allow the ellipsoidal shell to move as an absolutely rigid body. Analysis of the obtained results showed that when calculating a fragment of an elliptical cylinder, the calculation scheme of which makes it possible to obtain an analytical solution, the numerical values of bending moments and normal stresses completely coincide with the values calculated from the condition of static equilibrium. An analysis of the calculation results for an ellipsoidal shell with spring supports led to the conclusion that the developed tensor-vector interpolation procedure in the mixed FEM version allows one to correctly take into account the displacements of shells as solids and obtain an adequate assessment of their stress-strain state.

The linear stability of plane Couette flow with a compliant boundary

The linear stability of plane Couette flow subject to one rigid boundary and one flexible boundary is considered at both finite and asymptotically large Reynolds number. The wall flexibility is modelled using a very simple Hooke-type law involving a spring constant K and is incorporated into a boundary condition on the appropriate Orr–Sommerfeld eigenvalue problem. This problem is analyzed at large Reynolds number by the method of matched asymptotic expansions and eigenrelations are derived that demonstrate the existence of neutral modes at finite spring stiffness, propagating with speeds close to that of the rigid wall and possessing wavelengths comparable to the channel width. A large critical value of K is identified at which a new short wavelength asymptotic structure comes into play that describes the entirety of the linear neutral curve. The asymptotic theories compare well with finite Reynolds number Orr–Sommerfeld calculations and demonstrate that only the tiniest amount of wall flexibility is required to destabilize the flow, with the linear neutral curve for the instability emerging as a bifurcation from infinity.

Tunable topological phase transition in soft Rayleigh beam system with imperfect interfaces

Acoustic metamaterials, especially the topological insulators, have garnered substantial research attention due to their unique wave properties. Recent research emphasizes the crucial role of imperfect interfaces in accurately simulating and understanding wave propagation within these metamaterials. This paper studies the mechanical properties of soft imperfect interfaces and derives the effects of finite deformations on their stiffness. We present a comprehensive model of a soft Rayleigh beam system that incorporates these interfaces, allowing for the observation of the topological phase transition by altering the distance between them. Our investigation into the impacts of finite deformations and interface stiffness adjustment reveals that longitudinal waves' phase transition points are mainly influenced by interface stiffness. In contrast, transverse waves are predominantly affected by finite deformations. The decisive parameters driving these phenomena are also identified theoretically. The transmission studies of supercells further confirm the presence of topologically protected interface modes and their extensive tunability. Overall, this study offers a novel methodology and framework for managing topological phase transition in composite and soft topological insulator systems.

РАСЧЕТ ОСАДОК ПРОТЯЖЕННОГО ПЛИТНОГО ФУНДАМЕНТА КОНЕЧНОЙ ЖЕСТКОСТИ НА ОСНОВЕ ДАННЫХ КОМПЬЮТЕРНОГО МОДЕЛИРОВАНИЯ

The paper presents a database for estimation the magnitude of the settlement of an extended slab foundation of finite stiffness in the initial phase of projecting. The offered solution uses only interpolation methods without involving powerful computing devices. The computer program FEA, which formalises the finite element method, was used to obtain the result. The maximum horizontal and vertical dimensions of the computational schemes were 25H3•46H3. The computational scheme consisted of 57 500 finite elements in the form of right isosceles triangles conjugated at 29 106 nodes. The width of the stiffness matrix of the linear equation system was 466. This made it possible to eliminate the influence of trivial boundary conditions on the results. 560 computational operations were performed. It corresponded to the number of possible combinations of numerical values of the variables calculated parameters adopted in computer modelling of process of the slab foundation settlement. As an upshot of the computations, the authors made a table of coefficients of approximating expressions for the curves constructed from the results of the calculated dependences of slab foundation settlement on the variable design parameters. The article provides a calculation of settlements of two extended slab foundations of different widths, using a constructed database and interpolation methods. The control calculation in the FEA showed the difference between the values of the control settlements and the values obtained using the proposed tables and the linear interpolation method for 13.45 and 22.08 %. Summing up the results, the offered database can be used for preliminary (estimation) calculations of extended slab foundations settlements.

Thermal vibration analysis of cracked nanobeams submerged in elastic foundations by nonlocal continuum mechanics

Understanding the mechanical behavior of nanoscale structures is crucial in the development of advanced nanotechnologies. In this study, a novel approach to investigate the thermal lateral vibration of cracked nanobeams immersed in an elastic matrix is investigated. For this purpose, Reddy’s third-order shear deformation theory (TSDT) is considered. In contrast to Timoshenko beam theory (First-order Shear Deformation Theory, FSDT), TSDT does not depend on a shear correction coefficient. The nano-scale effect is modeled using Eringen’s nonlocal continuum mechanics theory. The nonlocal form of the governing equation is obtained through the application of Hamilton’s principle. The weak form of the finite element global mass and stiffness matrices are obtained using Lagrange linear and Hermitian cubic interpolation. To model the crack in bending vibration, two rotational springs are used for TSDT, unlike the use of a single rotational spring in traditional Bernoulli–Euler (Classical Beam Theory, CBT) and FSDT. The stiffness of the springs is adjusted based on the severity of the crack. The influences of the nonlocal parameter, beam slenderness, position of crack, crack severity, Pasternak and Winkler foundation parameters, thermal effects and boundary conditions on the natural frequencies are investigated. The model’s outcomes are compared with findings from prior publications, demonstrating a strong level of agreement. This study contributes to the growing research on nanostructures by presenting a novel approach to understanding the dynamics of cracked nanobeams using Reddy beam analysis-based solutions.

Study of the stress-deformation state of the soil base and foundations for equipment under significant horizontal loads

The study of the possibility of calculating foundations for stability against displacement along the sole and overturning under significant horizontal loads was carried out by the LIRA-SAPR computer software complex.
 The study was carried out by analyzing the amount of horizontal (sliding) and vertical (overturning) displacement of the foundation under the shunting winch under two variants of the engineering and geological conditions of the construction site.
 Through "manual" calculation, it was found that the stability of the foundation is ensured in the first variant of geological conditions, but not in the second one. Therefore, such geological conditions are illustrative for the study of calculation possibilities using the software complex.
 The foundation with width b=1.8 m, length l=4.2 m and height h=2.35 m is modeled by volume finite elements. The bottom layer, which is in contact with the soil base, is modeled by quadrilateral finite elements of the slab (slab on an elastic base).
 Friction under the footing is modeled by finite stiffness links (single-node finite elements) that are added at each node of the slab that models the bottom layer of the foundation.
 The stiffness of such finite elements in the direction of the global axis X and axis Y is determined automatically in the "GRUNT" system according to the methodology of SNiP 2.02.05-87.
 Summarizing the conducted research, we can say that the advanced capabilities of the LIRA-SAPR PC, in particular, the implementation of the algorithm for determining the stiffness coefficients of the base according to the SNiP 2.02.05-87 method, allow calculating foundations for resistance against displacement along the sole and overturning by numerical methods. This allows, firstly, to completely automate the calculation of the foundations, and secondly, during the automated calculation, the negative influence of the neighboring foundations is additionally taken into account, which is relevant, since the foundations for machines and mechanisms are mainly subjected to significant horizontal loads.
 In addition, the conducted research revealed the absence of a kinematic criterion for assessing the stability of the foundation. In manual calculation, the criterion for ensuring the stability of the foundation against displacement is the condition of static equilibrium. When calculating by the method of finite elements, implemented in PC "LIRA" and similar, displacements are decisive, so it is considered promising to study the issue of the criterion of stability of the foundation by displacements.

Model order reduction of time-domain vibro-acoustic finite element simulations with non-locally reacting absorbers

Time-domain vibro-acoustic finite element simulation gained considerable interest in recent years because of the rise of some applications such as auralization and virtual sensing. Efficiency and stability are of great importance for this topic. The number of elements needed per wavelength to reach acceptable accuracy often results in a large model size, which requires lots of computational resources and cannot run efficiently. Model order reduction can significantly alleviate this problem by reducing the size of these models, while maintaining the high-fidelity property. However, many model order reduction techniques fail to preserve the stability for non-locally reacting absorbers, which are often accounted for by using the Helmholtz equation with frequency-dependent density and bulk modulus, known as the equivalent fluid method. When discretized into a finite element model, frequency-dependent mass and stiffness matrices are introduced, which hinder the preservation of stability in the context of model order reduction. This paper presents a method that enables the creation of a stable reduced order model of a vibro-acoustic system with non-locally absorbers. This method reforms the Helmholtz equation and considers it as the interconnection of several passive systems. Stacking the states of these subsystems gives the final descriptor representation of the non-locally reacting absorbers. Furthermore, a possible second-order representation of the descriptor model is chosen such that it satisfies the stability-preserving condition under model order reduction. The resulting second-order model is coupled with a vibro-acoustic system in a velocity potential-displacement formulation, leading to a second-order model satisfying the aforementioned stability-preserving condition. The proposed method is successfully verified by several numerical simulations.

Factors Influencing the Expansion of Arch-Shaped Electromagnetic Railguns with Pre-Stressed Composite Barrels.

Rail expansion significantly impacts the launch precision of a railgun system. Higher precision can be achieved when the extent of expansion is low. This paper investigates three main factors that influence the extent of rail expansion using the finite element method, including pre-stress, electromagnetic load, and stiffness of the insulators. The mean squared error between experiment results and simulating results is less than 0.06, validating the finite element model. The simulated results reveal that the extent of rail expansion increases with a decrease in pre-stress and an increase in electromagnetic pressure and the stiffness of the insulator is the most significant influencing factor, as the use of a stiff insulator not only results in a small extent of rail expansion but also delays the separation between the rails and insulators. The mechanism of how pre-stress influences the railgun system has been proposed. It has been expressed that the pre-stress maintains the integrity of the railgun system by hindering the process of a decrease in the contact surface area between rails and insulators during launch. The study provides a platform to improve the design of the railgun system.

Model Analysis of Steel Frame Structures Considering Interactions between Racks and the Frame

The steel racks on the floor are seen as live loads in the current design process, ignoring the interaction with the supporting frames. In this paper, multiple steel racks with different masses and stiffnesses are placed on the first floor of a two-story main structure to form different real structures (RS). The corresponding simplified structures (SS) are frames with the mass of steel racks concentrated on the first floor of the main structure. Modal analysis is performed to analyze the relationship between the periods of RS and SS in the cross-aisle direction. Firstly, the beams on the first floor are assumed to be infinitely rigid. The relationship between the periods of the rack TRk, the simplified structure TSS, and the real structure TRS under different mass ratios α is established, and an accurate equation relating TRS with TRk and TSS is proposed. Moreover, by considering the influence of finite beam stiffness, the interaction between racks and the main structure is studied by constructing different analysis models. The effect of the main structure on the racks is reflected by a combined system consisting of beams and racks. A modified model, distinguished from SS by considering the effect of no-mass racks, is constructed to study the strengthening effect of the racks on the first-floor beams. The effect of the top connecting bars is also analyzed.

Bond of steel reinforcing bars in self-prestressed hybrid steel fiber reinforced concrete

Reinforcing bar bond is of paramount importance for the structural integrity of reinforced concrete and can be enhanced by fiber inclusion and self-prestressing. Previous research has developed bio-inspired self-prestressed fiber reinforced concrete (SP-FRC) utilizing fibers and expansive agent, and multiscale hybrid steel fiber reinforced concrete (HSFRC). In this research, SP-FRC and HSFRC were combined to develop bio-inspired self-prestressed hybrid steel fiber reinforced concrete (SP-HSFRC) in which the bond of steel reinforcing bars was studied experimentally by pull-out tests. From the bond-slip curves, various bond parameters, namely, bond strength, slip at peak bond stress, initial bond stiffness, and secant bond stiffness, were determined and correlated to a hybrid fiber factor to establish analytical models for design calculations. Then, a new bond model that yields finite initial bond stiffness for more realistic bond-slip analysis than the Model Code 2010 was extended to SP-HSFRC, as evidenced by good agreement between the predicted and experimental bond-slip curves. Lastly, the beneficial effect of the improved bond in crack control of steel bar reinforced HSFRC was rigorously proved.

Bath-induced phase transition in a Luttinger liquid

We study an XXZ spin chain, where each spin is coupled to an independent ohmic bath of harmonic oscillators at zero temperature. Using bosonization and numerical techniques, we show the existence of two phases separated by an Kosterlitz-Thouless (KT) transition. At low coupling with the bath, the chain remains in a Luttinger liquid phase with a reduced but finite spin stiffness, while above a critical coupling the system is in a dissipative phase characterized by a vanishing spin stiffness. We argue that the transport properties are also inhibited: the Luttinger liquid is a perfect conductor while the dissipative phase displays finite resistivity. Our results show that the effect of the bath can be interpreted as annealed disorder inducing signatures of localization.

Classification system for inter- and intra-module joints in non-sway steel MiC structures

The lack of guidelines for determining the rigidity of inter- and intra-module joints in corner-supported steel modular integrated construction (MiC) buildings can lead to unsafe or over-conservative designs. To address this issue, this study proposes a classification system for non-sway steel MiC buildings that covers both inter- and intra-module joints. First, the semi-rigidity of the joint regions in steel MiC buildings was clarified and the necessary degrees of freedom for joint classification were determined. Then a minimum substructure including one joint region was extracted from a steel MiC building, and its carrying capacity was determined as a simplified analytical model that considering the stiffnesses of both the inter- and intra-module joints. The simplified model was adopted to derive explicit formulas for the boundary limits of rigid/non-rigid inter- and intra-module joints, based on the criterion that the carrying capacity of the substructure with finite joint stiffnesses is 95% of that with rigid properties. Finally, the feasibility of the proposed classification system was verified by applying it to prototype MiC structures.