Showing papers on "Tangent stiffness matrix published in 2022"

An updated Lagrangian Bézier finite element formulation for the analysis of slender beams

Abstract: A G1-conforming finite element formulation based on the Kirchhoff beam model suitable for the analysis of structures composed by coupling of slender beams is presented. A new set of kinematic parameters is introduced in order to account for the continuity required by the rod model. This new set of kinematic parameters defines the G1-map that guarantees continuity of the rotations at the ends of the beam. The tangent stiffness matrix for the proposed Kirchhoff beam model is derived in a consistent way. It is shown that an additional geometric term, specific for the G1-conforming formulation, appears in the tangent stiffness matrix. In order to avoid the singularities arising with the introduction of the G1-map, an updated Lagrangian formulation is adopted. In this way, a G1-conforming Bézier finite element based on the Kirchhoff beam model able to model large deformations of space rod systems is obtained. Several numerical examples show the high accuracy and the robustness of the proposed conforming formulation.

A refined hyperbolic shear deformation theory for nonlinear bending and vibration isogeometric analysis of laminated composite plates

Abstract: In this paper, a refined hyperbolic tangent higher order shear deformation theory is developed. Assuming that the shear function is parameter dependent, the parameters are determined by the inverse method. The refined theory is assessed by using the NURBS based isogeometric analysis for the geometric linear and nonlinear bending and free vibration problems of laminated composite plates. The nonlinearity of the plates is based on the von-Karman strain assumptions. Numerical examples show that the refined hyperbolic tangent shear deformation theory combined with IGA has high accuracy in both linear and geometric nonlinear analysis of laminated composite plates. The effects of plate length–thickness ratio, modulus ratio and stacking sequence on the static bending and free vibration behaviors of laminated plate are also discussed.

Geometrically nonlinear formulation of Generalized Beam Theory for the analysis of thin-walled circular pipes

Abstract: In this paper, the formulation of Generalized Beam Theory (GBT) for the geometrically nonlinear analysis of thin-walled circular pipes is presented. GBT is a computationally efficient numerical method which is especially formulated for thin-walled members with a capacity of determining the cross-sectional deformation through a combination of a set of pre-determined cross-sectional deformation modes. In this study, the current GBT analysis of circular pipes which is limited to buckling analysis is enhanced to a full geometrically nonlinear analysis. This new formulation considers the nonlinear membrane kinematic description based on the Green–Lagrange strain definition. The nonlinear tangent stiffness matrices and the internal force vectors are derived from the variation of the internal energy which results in third and fourth-order GBT deformation mode coupling tensors. These tensors can predetermine the type of GBT deformation modes needed for the nonlinear analysis based on the applied loading conditions. In addition to the classical GBT deformation modes, the non-conventional GBT deformations modes have a vital role since without these modes the coupling tensors and the nonlinear stiffness matrix related to the transverse and the shear membrane energy will be lost. Here, to illustrate the application and capabilities of the developed GBT formulation, two numerical examples involving transverse and longitudinal bending are presented to show the nonlinear relationship between bending, cross-sectional ovalization, and higher local deformation modes. For the purpose of validation, these examples are compared with refined shell finite element models in both displacement and stress fields.

Intelligent stiffness computation for plate and beam structures by neural network enhanced finite element analysis

Abstract: In the present study, new methods are proposed to replace the constitutive law and the entire tangent stiffness matrix in finite element analysis by artificial neural networks (ANNs). By combining the FEM with ANN, so‐called intelligent elements are developed. First, as an extension to recent trends in model‐based material law replacement, we introduce an additional loss term corresponding to the material stiffness. This training procedure is referred to as Sobolev training and ensures that the ANN learns both the function approximating the stress behavior and its first derivative (material stiffness). In a following step, we introduce three methods to replace the entire local stiffness matrix of an element by approximating its generalized force‐displacement relations. These methods also adopt ANNs with Sobolev training procedure to predict the mentioned quantities. Since neural networks (NN) are universal function approximators, they are used to extract the stiffness information for elements undergoing plastic deformation. The focus of this work is to establish a neural network‐based FEM framework (independent of NN topology) to introduce an enhanced‐material law and in a consequent step also approximate stiffness information of truss, beam, and plate elements taking physical non‐linear behavior into account.

Refined Beam Theory for Geometrically Nonlinear Pre-Twisted Structures

Abstract: This paper proposes a novel fully nonlinear refined beam element for pre-twisted structures undergoing large deformation and finite untwisting. The present model is constructed in the twisted basis to account for the effects of geometrical nonlinearity and initial twist. Cross-sectional deformation is allowed by introducing Lagrange polynomials in the framework of a Carrera unified formulation. The principle of virtual work is applied to obtain the Green–Lagrange strain tensor and second Piola–Kirchhoff stress tensor. In the nonlinear governing formulation, expressions are given for secant and tangent matrices with linear, nonlinear, and geometrically stiffening contributions. The developed beam model could detect the coupled axial, torsional, and flexure deformations, as well as the local deformations around the point of application of the force. The maximum difference between the present deformation results and those of shell/solid finite element simulations is 6%. Compared to traditional beam theories and finite element models, the proposed method significantly reduces the computational complexity and cost by implementing constant beam elements in the twisted basis.

Quasistatic Fracture using Nonlinear-Nonlocal Elastostatics with Explicit Tangent Stiffness Matrix

Abstract: We apply a nonlinear-nonlocal field theory for numerical calculation of quasistatic fracture. The model is given by a regularized nonlinear pairwise (RNP) potential in a peridynamic formulation. The potential function is given by an explicit formula with and explicit first and second derivatives. This fact allows us to write the entries of the tangent stiffness matrix explicitly thereby saving computational costs during the assembly of the tangent stiffness matrix. We validate our approach against classical continuum mechanics for the linear elastic material behavior. In addition, we compare our approach to a state-based peridynamic model that uses standard numerical derivations to assemble the tangent stiffness matrix. The numerical experiments show that for elastic material behavior our approach agrees with both classical continuum mechanics and the state-based model.The fracture model is applied to produce a fracture simulation for a ASTM E8 like tension test. We conclude with an example of crack growth in a pre-cracked square plate. For the pre-cracked plate, we investigated {\it soft loading} (load in force) and {\it hard loading} (load in displacement). Our approach is novel in that only bond softening is used as opposed to bond breaking. For the fracture simulation we have shown that our approach works with and without initial damage for two common test problems.

An Effective Tangent Stiffness of Train–Track–Bridge Systems Based on Artificial Neural Network

Abstract: Dynamic response analysis of a train–track–bridge (TTB) system is a challenging task for researchers and engineers, partially due to the complicated nature of the wheel–rail interaction (WRI). When Newton’s method is used to solve implicit nonlinear finite element equations of a TTB system, consistent tangent stiffness (CTS) is essential to guarantee the quadratic convergence rate. However, the derivation and software implementation of CTS for the WRI element require significant efforts. Artificial neural network (ANN) can directly obtain a potentially good tangent stiffness by a trained relationship between input nodal displacement/velocity and output tangent stiffness. In this paper, the backpropagation neural-network-based tangent stiffness (BPTS) of the WRI element is derived and implemented into a general finite element software, OpenSees, and verified by dynamic response analysis of a high-speed train running on a seven span simply supported beam bridge. The accuracy and efficiency are compared between the use of BPTS and CTS. The results demonstrate that BPTS can not only save the significant efforts of deriving and software implementing CTS but also improve computational efficiency while ensuring good accuracy.

Stiffness analysis of a 3-DOF parallel mechanism for engineering special machining

Abstract: Abstract. There are considerably rigorous requirements for accuracy and stability of the mechanism to accomplish large-scale and complex surface machining tasks in the aerospace field. In order to improve the stiffness performance of the parallel mechanism, this paper proposes a novel three degrees of freedom (DOF) redundantly actuated 2RPU-2SPR (where R, P, U and S stand for revolute, prismatic, universal and spherical joints, respectively) parallel mechanism. Firstly, the kinematics position inverse solution is derived and a dimensionless generalized Jacobian matrix is established through the driving Jacobian matrix and constraint Jacobian matrix. Secondly, the stiffness model of the parallel mechanism is deduced and the accuracy of the stiffness model is verified through finite-element analysis. Using eigenscrew decomposition to illustrate the physical interpretation of the stiffness matrix, the stiffness matrix is equivalent to six simple screw springs. Finally, the simulation experiment results demonstrate that redundantly actuated parallel mechanism has better stiffness performance compared to the traditional 2RPU-SPR parallel mechanism.

Quasistatic fracture using nonlinear‐nonlocal elastostatics with explicit tangent stiffness matrix

Abstract: We apply a nonlinear-nonlocal field theory for numerical calculation of quasistatic fracture. The model is given by a regularized nonlinear pairwise potential in a peridynamic formulation. The potential function is given by an explicit formula with an explicit first and second derivatives. This fact allows us to write the entries of the tangent stiffness matrix explicitly thereby saving computational costs during the assembly of the tangent stiffness matrix. We validate our approach against classical continuum mechanics for the linear elastic material behavior. In addition, we compare our approach to a state-based peridynamic model that uses standard numerical derivations to assemble the tangent stiffness matrix. The numerical experiments show that for elastic material behavior our approach agrees with both classical continuum mechanics and the state-based model. The fracture model is applied to produce a fracture simulation for a ASTM E8 like tension test. We conclude with an example of crack growth in a pre-cracked square plate. For the pre-cracked plate, we investigated load in force (soft loading) and load in displacement (hard loading). Our approach is novel in that only bond softening is used as opposed to bond breaking. For the fracture simulation we have shown that our approach works with and without initial damage for two common test problems.

Finite element formulation of Timoshenko tapered beam-column element for large displacement analysis based on the exact shape functions

Abstract: ABSTRACT A numerical formulation was carried out in this paper to produce the tangent stiffness matrix for two-nodal tapered Timoshenko beam-column elements for geometrically nonlinear analysis. The proposed solution is based on the exact shape functions and their derivatives describing the non-uniformity of the element properties. The section properties were presented as exponential functions with tapering indices to illustrate the variations in section properties along the tapered element length. The model is applicable for elements with different solid and hollow cross-sections. The proposed formulation is embedded into a Visual Basic code to carry out the analysis accompanied by many examples for validating its accuracy and efficiency. The model results are compared with those of commercial software and cited references that showed high accurate results with a small number of elements.

Crystal plasticity algorithm based on the quasi‐extremal energy principle

Abstract: The direct incremental energy minimization in rate‐independent plasticity does not account for the skew‐symmetric part of the tangent stiffness matrix. In crystal plasticity, this corresponds to neglecting the asymmetry of the matrix of interaction moduli for active slip‐systems. This limitation has been overcome in the recently proposed quasi‐extremal energy principle (QEP) applicable to nonpotential problems. In the present article it is shown how to extend QEP to finite increments in the backward‐Euler computational scheme. A related constitutive algorithm is proposed which enables automatic selection of active slip systems using an energetic criterion, along any path of large deformation of a rate‐independent single crystal with a nonsymmetric slip‐system interaction matrix. Numerical examples have been calculated for a fcc single crystal subjected to simple shear or uniaxial tension. The slip system activity predicted by using the QEP algorithm has been found to be more reliable in describing the actual plastic response of metal crystals than conventional rate‐dependent modeling in cases where the selection of active slip‐systems is essential.

A symmetric tangent stiffness approach to cohesive mechanical interfaces in large displacements

Abstract: The present article proposes a formulation for a cohesive interface element in large displacement conditions. Theoretical and computational aspects, useful for an effective and efficient finite element implementation, are examined in details. A six-node (or higher) isoparametric interface element for two dimensional cohesive fracture propagation problems is developed. The element operators are consistently derived by a variational approach enforced in the current configuration, where a current frame is defined with axes tangential and normal to the middle line of the interface opening displacement gap. Under the constitutive assumption of small value of the modulus of the vector product between the Cauchy traction and the displacement jump vector, explicit expression of the interface nodal force vector and of the consistent tangent stiffness matrix are derived in a closed form. At difference with most of the available formulations, the proposed approach allows to naturally achieve a symmetric tangent stiffness matrix, provided that all the involved terms are maintained in the development, namely without neglecting second-order partial derivatives involved in the exact linearization procedure.

Improved Finite Element Method for Inflated Beams with Local Wrinkles

Abstract: An improved finite element method is proposed for inflatable beams with a global finite deformation and a local wrinkling behavior, based on the corotational approach and tension field theory. The finite deformation is decomposed into a rigid-body displacement in the global coordinate system and a small strain measured in the local coordinate system, by using the corotational approach. The tangent stiffness matrix of a three-node triangular membrane element is presented. With the modified bimodulus constitutive law, a wrinkling model is constructed for determination of the status of element (tension, wrinkled, or slack). To improve the rate of convergence, a consistent tangent material stiffness matrix is embedded into the local coordinate system. Numerical results are in good agreement with the existing experimental data. In some case, the proposed method can obtain more accurate deflection of inflatable beams than the postbuckling analysis of thin shells. A quadratic rate of convergence is observed for the proposed method, and thus the computational efficiency is improved greatly. The limitation of the method is also discussed.

A novel machine learning based tangent stiffness calculation method for 3D wheel-rail interaction element

Abstract: A machine learning (ML) based method is presented in this paper for obtaining tangent stiffness of a complicated three-dimensional wheel-rail interaction element that is able to practically and effectively simulate the complicated dynamic responses of vehicle-track problems. The element tangent stiffness, defined as differentiation of element insisting force to nodal displacement, is important in improving efficiency when Newton’s method is used to solve implicit nonlinear finite element equations. However, deriving and software implementing the tangent stiffness require significant efforts, and calculating the tangent stiffness in each iteration of the Newton method is usually time consuming. On the other hand, ML can directly obtain the implicit mapping between inputs and outputs of complex calculation process with limited programming effort and high computational efficiency, and is potentially a good alternative way to calculate the tangent stiffness of complicated element. In this paper, a feedforward artificial neural network is trained for obtaining the tangent stiffness, while inputs are the displacement and velocity of the element and outputs are the entries of the tangent stiffness matrix. The ML based tangent stiffness are implemented in an open source finite element software framework, OpenSees, and verified by application examples of a wheelset and a light rail vehicle running on straight rigid rail. The accuracy and efficiency are compared between the use of ML based tangent stiffness (MLTS) and the consistent tangent stiffness obtained at different speeds. The results demonstrate the MLTS can ensure the calculation accuracy and significantly improve the calculation efficiency.

Nonlinear Analysis of Rotational Springs to Model Semi-Rigid Frames

Abstract: This paper explains the mathematical foundations of a method for modelling semi-rigid unions. The unions are modelled using rotational rather than linear springs. A nonlinear second-order analysis is required, which includes both the effects of the flexibility of the connections as well as the geometrical nonlinearity of the elements. The first task in the implementation of a 2D Beam element with semi-rigid unions in a nonlinear finite element method (FEM) is to define the vector of internal forces and the tangent stiffness matrix. After defining the formula for this vector and matrix in the context of a semi-rigid steel frame, an iterative adjustment of the springs is proposed. This setting allows a moment–rotation relationship for some given load parameters, dimensions, and unions. Modelling semi-rigid connections is performed using Frye and Morris’ polynomial model. The polynomial model has been used for type-4 semi-rigid joints (end plates without column stiffeners), which are typically semi-rigid with moderate structural complexity and intermediate stiffness characteristics. For each step in a non-linear analysis required to adjust the matrix of tangent stiffness, an additional adjustment of the springs with their own iterative process subsumed in the overall process is required. Loops are used in the proposed computational technique. Other types of connections, dimensions, and other parameters can be used with this method. Several examples are shown in a correlated analysis to demonstrate the efficacy of the design process for semi-rigid joints, and this is the work’s application content. It is demonstrated that using the mathematical method presented in this paper, semi-rigid connections may be implemented in the designs while the stiffness of the connection is verified.

Equilibrium and stiffness study of clustered tensegrity structures with the consideration of pulley sizes

Abstract: This paper presents the equilibrium and stiffness study of clustered tensegrity structures (CTS) considering pulley sizes. We first derive the geometric relationship between clustered strings and pulleys, where the nodal vector is chosen as the generalized coordinate. Then, the equilibrium equations of the clustered tensegrity structure with pulleys based on the Lagrangian method are given. Since the stiffness of a structure is usually weakened when using clustering strings, we formulate the tangent stiffness matrix equations for analysis. It is also shown that as pulley sizes go to zero, the governing equations of the clustered tensegrity system with pulleys yield to the classical clustered tensegrity structure without pulleys, which is consistent with the existing literature. Three examples are demonstrated to validate the given theory. The proposed method allows one to conduct equilibrium, stiffness, and robustness studies of cluster tensegrity structures with pulleys. Nevertheless, the approach developed in this paper is not limited to the tensegrity structures. It can also be applied to a wide range of applications with pulley-rope systems, such as drilling rigs, ocean platform anchors, and cargo cranes.

An Exact Consistent Tangent Stiffness Matrix for a Second Gradient Model for Porous Plastic Solids: Derivation and Assessment

Abstract: It is well known that the use of a consistent tangent stiffness matrix is critical to obtain quadratic convergence of the global Newton iterations in the finite element simulations of problems involving elasto-plastic deformation of metals, especially for large scale metallic structure problems. In this paper we derive an exact consistent stiffness matrix for a porous material model, the GLPD model developed by Gologanu, Leblond, Perrin, and Devaux for ductile fracture for porous metals based on generalized continuum mechanics assumptions. Full expressions for the derivatives of the Cauchy stress tensor and the generalized moments stress tensor the model involved are provided. The effectiveness and robustness of the proposed tangent stiffness moduli are assessed by applying the formulation in the finite element simulations of ductile fracture problems. Comparisons between the performance our stiffness matrix and the standard ones are also provided.

Size dependent large displacements of microbeams and microframes

Abstract: The size dependent large displacement behavior of planar microbeams and microframes is studied in this paper using a corotational beam element. To account for the size effect, the modified couple stress theory (MCST) is employed in conjunction with Euler-Bernoulli beam theory in deriving the internal force vector and the tangent stiffness matrix of the beam element. The Newton-Raphson based iterative procedure is used in combination with the arc-length method to solve the nonlinear equilibrium equation and to trace the equilibrium paths. Various microbeams and microframes are analyzed to show the influence of the size effect on the large deflection behavior of the microstructure. The obtained result reveals that the size effect plays an important role on the large deflection response, and the displacements of the structure are over estimated by ignoring the size effect. A parametric study is carried out to highlight the influence of the material length scale parameter on the large displacement behavior of the microbeams and microframes.

Method of analytical inverting of nonlinear constitutive relations of the combined model of phase and structural deformation of shape memory alloys

Abstract: The equations of the combined model of phase and structural deformation of shape memory alloys (SMA) express the increments of deformations in terms of the increments of stresses, martensite volume part parameter and temperature, stresses themselves, deformations, and the temperature. However, to solve the stability problems of long-or thin-walled elements from SMA, as well as to formulate the tangent stiffness matrix of the finite element method for SMA, it is necessary to have an explicit expression of stress increments in terms of stresses, deformations, strain increments, temperature increments, and temperature. The paper presents an analytical method for obtaining such inverting.

Upper and lower bounds to the overall incremental stiffness of hyperelastic composites

Abstract: Abstract Bounds to the overall stiffness of a composite are well-known within the classical theory of elasticity. They are based on the positive-definiteness of the local stiffness. A transfer to a prestressed state is not trivial. We may study the incremental stiffness that connects the nominal stress rate with the velocity gradient. But when there are mainly compressive stresses, then positive-definiteness can only be secured if this stiffness is replaced by a pseudo-stiffness. Its existence is equivalent to a strengthened form of uniform infinitesimal polyconvexity and is independent of the geometry. The same is the case with the crude Voigt and Reuss bounds. More refined kinematic or dynamic approximations do, of course, depend on the geometry. This is demonstrated with the unidirectional reinforcement of a matrix.

Efficient formulation of a two‐noded geometrically exact curved beam element

Abstract: The article extends the formulation of a 2D geometrically exact beam element proposed by Jirásek et al. (2021) to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first-order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The article develops the theoretical framework based on the Navier–Bernoulli hypothesis, with a possible extension to shear-flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral-shaped beam, and a spring-like beam with a zig-zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, that is, the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the Appendix.