Equivalent Stiffness Matrix Research Articles

Numerical Analysis of Footing Settlement Problem by Subloading Surface Model

This article addresses the numerical analysis of the load-settlement relation of footing on a soil ground by the finite difference method incorporating the subloading surface model. The subloading surface model is the elastoplastic constitutive model describing most pertinently the plastic strain rate due to the rate of stress inside the yield surface so as to fulfill the continuity and the smoothness conditions. For the finite difference program FLAC3D is used, which adopts the explicit dynamic relaxation method without solving the total stiffness matrix equation and thus has the capability for the remarkable reduction of calculation time. The relevance of the present approach to the description of the load-settlement curve exhibiting the ultimate load is demonstrated by the numerical experiments. Further, the applicability to the prediction of real behavior of footing-settlement is verified by comparison with the test data for sand grounds with a high friction property.

Application of rigid(visco)–plastic element free Galerkin method to simulation of plane strain forming

A rigid–plastic/viscoplastic element free Galerkin (EFG) method is introduced based on the flow formulation for rigid–plastic/viscoplastic materials to analyse the metal forming processes. The penalty function method is employed to introduce the volume constancy condition, and the stiffness matrix equation is derived according to incomplete generalised variational principle. The transformation method is adopted to impose the essential boundary condition in the local coordinate system. In order to implement the frictional boundary conditions, the transform matrix from the global coordinate system to the local coordinate system is given, and the arctangent frictional model is used. The meshless analysis techniques for metal forming processes with arbitrary shaped die are developed. The plane strain forming process is analysed numerically by using the rigid(visco)–plastic EFG method. The simulation results of the EFG method are in good agreement with those obtained by using the rigid(visco)–plastic FEM, and the effectiveness of the model and key techniques is verified.

Large free vibration of thin plates: Hierarchic finite Element Method and asymptotic linearization

The aim of this work is to study the free dynamic response of thin plates characterized by geometrical nonlinearities. To achieve this task, the equation of motion of the plate is first carried out through modeling by hierarchical finite element method whose interpolating shape functions are sinusoidal. Then, the study of the nonlinear vibrations was carried out by the development of asymptotic linearization and equivalent linearization methods in modal space. The nonlinear angular frequencies are successively deduced by exciting the corresponding vibrating mode of the structure. The confrontation of these results to those obtained by the iterative method in the physical space and to those found in the literature, showed a very good agreement between the various methods. From the elementary nonlinear frequencies we showed that there exists an equivalent linear dynamical system characterized by only one equivalent linear stiffness matrix. Numerical experiments were carried out on beams and thin plates of various dimensions ratios and boundary conditions. These numerical test simulations, whether in time space or frequency space, have showed that the nonlinear elastic energy is restored by the equivalent linear dynamical system. Nevertheless, we have to say that the dynamic effects of modes above the excited one are neglected.

Frame Flexibility Effects on Engine Mount Optimization for Vibration Isolation in Motorcycles

This paper examines the influence of frame flexibility on the optimization of an engine mounting system for enhanced vibration isolation in motorcycles. A theoretical model is developed to represent the structural dynamics of an engine mount system in motorcycles. The model consists of the power-train assembly, modeled as a six-degree-of-freedom (DOF) rigid body; the swing arm assembly, connected to the power-train through a coupler shaft assembly; and the frame, connected to the power-train by elastomeric mounts and to the swing-arm through the rear suspension. Two models of the flexible frame are developed for analysis. The first model uses an equivalent stiffness matrix of the frame, derived from its finite element model, in terms of the nodes connecting the frame to the other subsystems. The second model is based on a dynamic model of the frame as well as the swing arm derived from their respective finite element models. The optimization procedure minimizes the load transmitted to the frame while constraining the engine displacement due to imposed loads within prescribed limits. The mount stiffnesses, locations and orientations are used as design variables. Examples are presented to demonstrate the influence of frame flexibility on the force transmitted to the frame.

Massive metal forming process simulation based on rigid/visco-plastic element-free Galerkin method

Abstract The advantages of meshless method are analyzed. Applications of meshelss method in numerical simulations of metal forming processes are summarized. The element-free Galerkin method based on the hypothesis of the rigid/visco-plastic material is introduced. Using the incomplete generalized variational principle and employing penalty function method to introduce volume constancy, the stiffness matrix equation is established. The meshless analysis techniques for metal forming processes with arbitrarily shaped die, such as the description of dies or workpieces with arbitrarily shaped boundary, iteration convergence criterions and detachment and contact criterion, are developed. The phenomenon of volumetric locking and pressure oscillation is released. The rigid/visco-plastic element-free Galerkin method nalysis of the non-steady metal forming processes with arbitrarily shaped dies is realized. The simulation results of the meshless method are in agreement with those obtained by using the rigid/visco-plastic FEM, and the effectiveness of the model and key techniques is verified.

Seismic Response of R‐C Frame Structure Controlled by Ved

Based on the character of R‐C frame, the calculation formula of VED is derived from the sub‐structure plan of lateral resistant force, the definition of equivalent stiffness matrix and equivalent additional viscoelastic force vector of VED is given and the seismic dynamic equation of R‐C frame controlled by VED is set up, the viscoelastic additional force vector of VED should be considered as outside load of structure and be calculated through the pseudo‐loading method, the programs of calculating the dynamic characteristic, calculating the elastic seismic response and calculating the elastic‐plastic seismic response of R‐C frame controlled by VED are compiled. Through the calculation of a inclined crossed R‐C frame, the results show that VED is effective in controlling the seismic response of frame and the pseudo‐loading method is reliable in the calculation.

Finite-volume direct averaging micromechanics of heterogeneous materials with elastic–plastic phases

In this communication, we extend the recently re-constructed micromechanics model called high-fidelity generalized method of cells (Bansal, Y., Pindera, M.-J., 2005. A second look at the higher-order theory for periodic multiphase materials. J. Appl. Mech. 72 (2), 177–195.) by incorporating inelastic response capability for the individual phases. The re-construction, based on the local/global stiffness matrix approach, has simplified the model’s theoretical framework and substantially increased its computational efficiency as well as implementability, enabling analysis of unit cells with realistic multiphase microstructures previously unattainable in the original formulation developed by Aboudi et al. (Aboudi, J., Pindera, M-J., Arnold, S.M., 2001. Linear thermoelastic higher-order theory for periodic multiphase materials. J. Appl. Mech. 68 (5), 697–707; Aboudi, J., Pindera, M-J, Arnold, S.M., 2003. Higher-order theory for periodic multiphase materials with inelastic phases. Int. J. Plasticity 19 (6), 805–847.) with an accuracy approaching finite-element solutions. Just as importantly, the re-construction has revealed the model to be based on a finite-volume, direct averaging approach with clearly discernible similarities to, and differences with, the finite-element method and the finite-volume technique used in computational fluid mechanics. Herein, easily programmable closed-form expressions have been derived for the thermo-inelastic contributions to the local stiffness matrix equations that facilitate incorporation of different inelastic constitutive theories for the phase response. The re-constructed model is then employed to investigate orientational and architectural effects in unidirectional metal matrix composites characterized by multi-inclusion unit cells. Classical incremental plasticity theory with isotropic hardening is employed for the matrix response for consistency and comparison with previously reported results by Aboudi et al. (2003). Unit cells representative of a square array of fibers rotated by an angle about the fiber axis, which lack planes of material symmetry in the rotated coordinate system in which the micromechanical analysis is performed, belong in the first category. New results are presented for such rotated unidirectional porous composites which suggest guidelines for optimizing stiffness and ductility of this class of light-weight materials relative to dominant loading directions. Strengthening effects due to fiber clustering, which require highly discretized multi-inclusion unit cells, fall in the second category. It is demonstrated that the previously documented results for particulate composites, explained by the clustering-induced alteration of stress invariants which govern plastic strain evolution, are recovered for unidirectional composites as well.

Finite element simulation of the squaring circular tube

The squaring tube process is examined by an incremental elasto-plastic finite-element method based on an updated Lagrangian formulation in which a sliding-sticking friction mode is specifically considered. The high nonlinearity of the process due to the geometric changes, the inelastic constitutive behavior, and the deformation-dependent boundary conditions are taken into account in an incremental manner. A static explicit approach to the solution is applied, the tangent stiffness matrix equation is solved without iteration and a weighting factor rmin is employed to limit the step size to linear relation. The simulated geometries of squaring clearly demonstrate the processes of square tube until unloading. The formation of squaring defects both collapse and asymmetry are reported in a theoretical manner. Accordingly, the effects of various parameters of the process, such as geometric ratio R/t, strain hardening exponent n, and the friction coefficient μ, on the occurrence of collapse (collapse ratio C/t) and on the extent of asymmetry (deviation ratio C1/C2) for the squaring process are discussed and interpreted in simulation. Mainly it is expected that formation of a square tube for industrial use that does not collapse will be found during the design stage, before trials begin. The present work may be expected to improve the understanding of the formation of the square tube .

Application of parallel processing in three-dimensional bulk forming finite element analysis

The design of bulk metal forming parts with improved mechanical properties and higher production rates can greatly benefit from accurate three-dimensional analyses. However, efforts are still needed to make such analyses more attractive in terms of less computation time and memory requirements. For this purpose, an investigation of parallel processing of the three-dimensional rigid-viscoplastic finite element analysis has been made in this study. The domain decomposition method was used to divide physical domain of the problem into sub-domains and obtain a reduced form of the global stiffness matrix equation. Methods to solve the reduced system of equations using iterative and direct solvers were developed and applied to determine the numerical benefit of using them through simple block upsetting and swash-plate forging simulations. It was found that direct solver was more beneficial for the practical forming example of swash-plate forging. Also, advantage of using domain decomposition was examined in terms of contact treatment and stiffness matrix formulation of the problem solved.

Frequency-shifting analysis of electrostatically tunable micro-mechanical actuator

A numerical approach for eigenvalue analysis of electrostatically tunable micro-mechanical actuators is proposed. An effi- cient algorithm for calculating the natural frequwncy shifting due to applied DC turning voltage is proposed. In the calculation of the coupled field problem, 3-D FEM and BEM are employed. The numerical examples are presented and the numerical analysis results are compared with experimental data. Previous works on the analysis of tunable electrostatic actuators used very simplified models in which micro- mechanical actuators are assumed as single mass-spring systems (3-5). However, actuator systems must be modeled to continuum structures because the actuators assumed as rigid body in previous works are generally deformable. Moreover, in real situations, the electrostatic field and the elastic deformation field of the system are 3-dimensionally coupled. An advanced numerical approach, which can treat continuous fields, is used only in the calculation of the electrostatic force applied on the device (6). In order to accurately predict the behavior of the real system, the analysis of the coupled fields in 3-D continuous system and an algorithm for evaluating the effects of electrostatic field on natural frequency are necessary. In this work, a numerical scheme for eigenvalue analysis of electrostatically tunable micro-mechanical actuators is proposed. We have used the boundary element method (BEM) and the finite element method (FEM) for the 3-D coupled analysis of electrostatic field and deformation of a device. In order to consider the effects of electrostatic field on natural frequency, an equivalent stiffness matrix for electrostatic tuning voltage is introduced. We can perturb the equilibrium structure using a concerned eigenvector of operating mode and then linearize the corresponding electrostatic force variation in order to determine the equivalent stiffness. For examples, a simple beam-shaped structure and a tunable micromirror are analyzed and the numerical analysis results are compared with experimental data.

Simulation model of impact on reinforced concrete

This paper proposes a simple but more effective way to perform finite element analyses of impact on reinforced concrete. The equivalent inclusion method is applied and considers the reinforced concrete as a homogeneous material, simplifying the finite element meshes and greatly reducing the computational cost of analyses. Using Mori–Tanaka's average strain theory, the equivalent stiffness matrix of the homogenized material and the associated equivalent material moduli are derived for finite element analyses. The residual velocity at which a projectile penetrates into an equivalent reinforced concrete slab is studied based on the strength of the equivalent material. Two examples are presented to demonstrate the proposed method. One involves the impact of an ogive-nose projectile on a reinforced concrete slab. The FEM computational results obtained using this method are very close to the test data, implying that the proposed method will be promising in future studies of impact analyses of reinforced concrete structures. Another example of 50° oblique impact is then presented to demonstrate the dependence of projectile ricochet upon the impact velocity.

A two-dimensional beam–column finite element with embedded rotational discontinuities

This paper presents a new two-dimensional two-noded beam–column finite element for the inelastic analysis of plane frames. The proposed element is capable of having multiple internal inelastic hinges whose positions are not limited to the ends of the element. Consequently, it is possible to use one element for one member of a frame structure even when distributed loads are present. At each inelastic hinge, the inelastic rotation is considered while the inelastic axial deformation is neglected. The proposed element is derived by the use of new shape functions. In the development of the shape functions, each inelastic hinge is treated as a location where a rotational discontinuity occurs. By embedding these rotational discontinuities into the expressions of the shape functions, the effect of the inelastic hinges can be included. The obtained stiffness matrix equation of the proposed element is simple and, therefore, practical. Finally, the validity and efficiency of the proposed element are verified by means of solving several numerical examples.

ランニングにおける地面反力に対する等価接地剛性行列の算出(身体・運動解析)

The purpose of this study was to calculate an equivalent stiffness matrix with respect to ground reaction force during running motion. Eight Subjects' 2-D images for the motion analysis and ground reaction forces were recorded by using a high speed camera and two forceplatforms. The angular velocities which contribute to vertical direction movement were extracted from joint angular velocities using Jacobian Matrix. Joint stiffness of each joint was calculated from a joint torque and the angular velocity. An equivalent stiffness matrix on center of pressure was transformed from the joint stiffnesses. At comparison of heel striker with forefoot striker, differences were observed at the equivalent stiffness matrix. The results of this study suggest that the equivalent stiffness matrix has a potential to quantify the characteristics of runners and running forms.

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.

Variational formulations for TFEM of piezoelectricity

The paper presents a family of variational formulations of Trefftz finite elements wherein the assumed internal displacement and electric potential fields a priori fulfil the governing differential equations of the problem over the element sub-domain, while the inter-element continuity and the boundary conditions are enforced using a modified variational principle together with an independent frame field defined on each element boundary. It is based on four free energy densities, each with two kinds of independent variables as basic independent variables, i.e. ( σ, D ) , ( ε, E ) , ( ε, D ) , and ( σ, E ) . Based on the assumed intra-element and frame fields, an element stiffness matrix equation is obtained which is implemented into computer programs for numerical analysis. Some numerical examples are considered to show the application of the proposed formulation.

FRACTAL TWO-LEVEL FINITE ELEMENT MESH FOR ACOUSTIC SCATTERING PROBLEMS

The problems of acoustic waves scattered by scatterer immersed in unbounded domain is an essential ingredient in the study of acoustic-structure interaction. In this paper the problems of acoustic scattering in an infinite exterior region are investigated by using a fractal two-level finite element mesh with self-similar layers in the media which encloses the conventional finite element mesh for the cavity. The similarity ratio is bigger than one so that the fractal mesh extends to infinity. Because of the self-similarity, the equivalent stiffness (mass) matrix of one layer is proportional to the others. By means of the Hankel functions automatically satisfying Sommerfeld's radiation conditions at infinity, the different unknown nodal pressures on different layers are transformed to some common unknowns of the Hankel coefficients. The set of infinite number of unknowns of nodal pressure is reduced to the set of finite number of Hankel's coefficients. All layers have the same matrix dimension after the transformation and the respective matrices of each layer are summed. Due to the proportionality, the infinite number of layers can be summed in closed form as the entries of each matrix are in geometric series. That is, processing one layer is enough to virtually represent a set of infinite number of layers covering an infinity domain. No new elements are created. Numerical examples show that this method is efficient and accurate in solving unbounded acoustic problems.

Interlocked Composite Grids Design and Manufacturing

Composite grid structures made from pultruded unidirectional glass or carbon ribs promise to provide unmatched performance/cost combination of any composite panels. A new manufacturing method for a square grid using slotted joint and adhesive bonding (Interlocked Composite Grid or ICG) with pultruded ribs has been developed. Compared to previously proposed grids, it is mass producible at low cost. It has large potential for many applications, particularly for large civil structures. Equivalent stiffness models for the proposed ICG, in equivalent plate stiffness matrix form and equivalent engineering constant form, have been formulated and verified for the flexural behavior of ICGs. The effects of the slots have been considered and integrated in the model. Design guidelines with equivalent engineering constants, and approximate cost estimation rules have been established. An example problem has been solved to demonstrate the simplicity of the design rules. ICG beams and panels have been built and tested under static and dynamic flexural loading. Valuable test data are provided and superior mechanical properties, such as high damage tolerance, resilience, and durability have been demonstrated.

Investigation of random response of rotational shell when considering geometric nonlinear behaviour

An iteration method of statistic linearization (IMSL) is presented. By this method, an equivalent linear term was formed in geometric relation and then an equivalent stiffness matrix for nonlinear term in vibration equation was established. Using the method to solve the statistic linear vibration equations, the effect of geometric nonlinearity on the random response of rotational shell is obtained.

Finite-element simulation of the lateral compression of aluminium tube between rigid plates

The lateral compression of aluminium and clad tubes owing to a large deformation is examined by an incremental elasto-plastic finite-element method based on an updated Lagrangian formulation in which a sliding-sticking friction mode is specially considered. It is mainly expected to predict the buckling process and load–deflection curves for energy dissipation capacity during the design stage, before trials. The high non-linearity of the process due to geometric changes, the inelastic constitutive behavior, and the deformation-dependent boundary conditions are taken into account in an incremental manner. A static explicit approach to the solution is applied, tangent stiffness matrix equation is solved without iteration and the r min technique is employed to limit the step size to linear relation. The simulated load–deflection curve agrees with a published experimental result. The predicted geometries of the compressed tube clearly demonstrate the processes of the formation of buckling until unloading. The effects of various parameters of the process, such as elastic modulus, strain hardening exponent, tube thickness, friction coefficient and configurations of the clad tube, on the occurrence of buckling of tube are discussed and interpreted in simulation. The present work may be expected to improve the understanding of the buckling mechanism of lateral compression.

A study of an oblique cutting model

Abstract The finite deformation theory and the updated Lagrangian formulation (ULF) were used to describe the oblique cutting process in an attempt to develop an oblique cutting elastic–plastic finite-element analytical model. In this paper, an elastic–plastic stiffness matrix equation was first derived. The geometrical non-linear quality of the material was also taken into consideration to infer the appropriate geometrical stiffness matrix. Under the condition of balanced variation rate of the surface force before and after deformation, the load corrective matrix was derived. At the same time, the friction force on the chip–tool interface was transformed into a friction corrective matrix based on the Coulomb friction law. The above matrices were combined into an elastic–plastic finite-element governing equation. With either the strain energy density constant or the geometrical condition as the chip-separation criterion, an equation of 3-D tool face geometrical limitation was established to examine and correct the chip node. Based on this approach, the tool advancement was achieved in displacement increments step-by-step from the initial tool contact with the workpiece until the formation of a steady cutting force. Mild steel was used as the workpiece in this study and P20 as the tool. The cutting simulation was conducted under a lower cutting speed of 274.8 mm s−1 and a constant temperature of 25°C to analyze the variation of the physical properties of workpiece. During the deformation process, the chip flow angle along the chip node flow direction approximately matched the tool inclination angle, which satisfied Stabler’s criterion and proved the accuracy of the chip geometrical shape of the present model. At the same time, the multiplication of cutting speed and cutting force obtained by the coordinate transformation of the force component on the chip was used to establish the specific cutting power on the basis of the unmachined workpiece cross-section. This simulated value also matched experimental findings, which further verified the feasibility of the present model.