Tangent Stiffness Matrix Research Articles

Meshless meso-modeling of masonry in the computational homogenization framework

In the present study a multi-scale computational strategy for the analysis of structures made-up of masonry material is presented. The structural macroscopic behavior is obtained making use of the Computational Homogenization (CH) technique based on the solution of the Boundary Value Problem (BVP) of a detailed Unit Cell (UC) chosen at the mesoscale and representative of the heterogeneous material. The attention is focused on those materials that can be regarded as an assembly of units interfaced by adhesive/cohesive joints. Therefore, the smallest UC is composed by the aggregate and the surrounding joints, the former assumed to behave elastically while the latter show an elastoplastic softening response. The governing equations at the macroscopic level are formulated in the framework of Finite Element Method (FEM) while the Meshless Method (MM) is adopted to solve the BVP at the mesoscopic level. The material tangent stiffness matrix is evaluated at both the mesoscale and macroscale levels for any quadrature point. Macroscopic localization of plastic bands is obtained performing a spectral analysis of the tangent stiffness matrix. Localized plastic bands are embedded into the quadrature points area of the macroscopic finite elements. In order to validate the proposed CH strategy, numerical examples relative to running bond masonry specimens are developed.

Improved Craig-Bampton method for transient analysis of structures with large-scale plastic deformation

This paper reports on the improvement of Craig-Bampton (CB) method for transient analysis of structures with large-scale plastic deformation. As is known, the CB method is effective and accurate in reduced order modeling for linear system. In contrast to this, an improved CB method using tangent modes for nonlinear dynamic analysis has been developed. To do this, the incremental governing equations of nonlinear system are linearized in each time step by using tangent stiffness matrix, and the corresponding tangent modes are proved to be orthogonal with respect to mass matrix as well as with respect to tangent stiffness matrix by incorporating the elastic-plastic material behavior. Thus, the tangent modes can be used to assemble the transformation matrix of CB method in nonlinear dynamic analysis. Using the proposed method, two elastic-plastic beams loaded impulsively are examined. Simulation results show that the improved CB method is valid and accurate for transient analysis of structures with large-scale plastic deformation and has lower computational cost compared with full order model.

Elasto-plastic analysis of functionally graded metal-ceramic beams under mechanical loading

The elasto-plastic analysis of functionally graded (FG) metal-ceramic beams under mechanical loading by using the finite element method is presented. A bilinear stress-strain relation with isotropic hardening is assumed for elasto-plastic behaviour of metal, and the effective elasto-plastic properties of the functionally graded material are evaluated by using Tamura-Tomota-Ozawa (TTO) model. A nonlinear beam element based on the classical beam theory is formulated and employed in the analysis. The element employed nonlinear von K\'am\'an strain-displacement relationship is derived by using the neutral surface as reference plane. The layer beam approach, in which the plastic rate equation is solved at Gauss points, is adopted in updating the stress and evaluating the element nodal force vector and tangent stiffness matrix. Numerical examples are given to show the accuracy of the derived formulation and to illustrate the effect of the material distribution and plastic deformation on the behavior of the beams. The formation and propagation of plastic zone during the loading process is also examined and highlighted.

EFESTS: Educational finite element software for truss structure – Part 3: Geometrically nonlinear static analysis

This article covers the modeling, formulation, solution, and software development of the geometrically nonlinear static finite element method of truss structure. Firstly, we summarize the total Lagrange bar elment formulation, which includes the tangent stiffness matrix and the internal force vector. Secondly, static class diagrams and dynamic sequence diagrams assist students in designing software architecture. Thirdly, the analytical example of the 2-bar truss structure and the numerical example of the 10-bar truss structure are presented to promote students’ understanding of geometrically nonlinear finite element theory and application. Finally, the developed software is free for educational research and can be downloaded from the website: http://mach.jlu.edu.cn/hb_images/xygk/xssz_sz_js.php?id=395 .

A second-order flexibility-based model for steel frames of tapered members

The paper presents a new computer method for nonlinear inelastic analysis of steel frames consisting with members with non-uniform cross-sections. A novel second-order flexibility-based element has been developed. The behaviour model accounts for material inelasticity due to combined bending and axial force, element geometrical nonlinear effects in conjunction with initial geometric imperfections using only one element per structural member. The proposed element formulation combines the power series approach to obtain the general solution of the second-order bending moments with the Maxwell-Mohr method to compute the force-displacement relationship of the general continuously non-prismatic Timoshenko-Euler beam-column element. The method ensures also that the plastic strength interaction requirements are always satisfied in the plastic hinges developed at the ends of the member or within the member length. The second-order elasto-plastic tangent stiffness matrix and equivalent nodal loads vector of non-uniform 2D steel members with semi-rigid connections is developed and the proposed nonlinear analysis formulation has been implemented in a computer program. In order to verify the efficiency and accuracy of the proposed approach, several benchmark problems have been studied and the results prove the performance of the proposed method.

An efficient multiscale computational formulation for geometric nonlinear analysis of heterogeneous piezoelectric composite

In this contribution, an efficient multiscale computational formulation is developed for the geometric nonlinear analysis of heterogeneous piezoelectric materials. In this formulation, the relations between the microscopic heterogeneous properties and macroscopic behaviors of the piezoelectric composite are established by the numerically constructed displacement and electric potential base functions. The heterogeneous and complex microstructure can be equivalent to a simple macroscopic piezoelectric coarse element through those constructed multiscale base functions. Then the equivalent tangent stiffness matrix and internal force vector of the macroscopic coarse element are deduced based on the co-rotational approach, which can describe the motion of macroscopic coarse element clearly and efficiently. Thus the original electro-mechanical coupling geometric nonlinear problem could be solved iteratively on the macroscopic scale, which will save a tremendous amount of computing time and cost. After all the macroscopic calculations, the microscopic mechanical and electrical responses could be retrieved and calculated from the macroscopic solutions by using the above-mentioned multiscale shape functions. To verify the validation and high-efficiency of the proposed multiscale computation formulation, several typical numerical examples are carried out. All the computation results indicate that the developed multiscale formulation not only could provide high precision solutions but also has high efficiency.

Kink-antikink asymmetry and impurity interactions in topological mechanical chains.

We study the dynamical response of a diatomic periodic chain of rotors coupled by springs, whose unit cell breaks spatial inversion symmetry. In the continuum description, we derive a nonlinear field theory which admits topological kinks and antikinks as nonlinear excitations but where a topological boundary term breaks the symmetry between the two and energetically favors the kink configuration. Using a cobweb plot, we develop a fixed-point analysis for the kink motion and demonstrate that kinks propagate without the Peierls-Nabarro potential energy barrier typically associated with lattice models. Using continuum elasticity theory, we trace the absence of the Peierls-Nabarro barrier for the kink motion to the topological boundary term which ensures that only the kink configuration, and not the antikink, costs zero potential energy. Further, we study the eigenmodes around the kink and antikink configurations using a tangent stiffness matrix approach appropriate for prestressed structures to explicitly show how the usual energy degeneracy between the two no longer holds. We show how the kink-antikink asymmetry also manifests in the way these nonlinear excitations interact with impurities introduced in the chain as disorder in the spring stiffness. Finally, we discuss the effect of impurities in the (bond) spring length and build prototypes based on simple linkages that verify our predictions.

Dynamic Response Analysis of Elastic-plastic Structures with Component Mode Synthesis

This paper evaluates the validity of component mode synthesis (CMS) technique for dynamic response of elastic-plastic structures. A method incorporating elastic-plastic material behaviour and the fixed interface mode synthesis technology is presented. To do this, the incremental governing equations of elastic-plastic transient problem derived from nonlinear finite element theory are linearized in each time step, the tangent stiffness matrix is used to generate the basis vector of transformation matrix. Using the proposed method, the response of a simply supported beam loaded impulsively is examined. The Numerical results indicate that the proposed method is accurate, convergent and more efficient compared with the full order finite element model.

Non-linear finite element analysis of an elastic structure loaded by hydrostatic follower forces

This study deals with the non-linear finite element computation of thin flexible structures loaded by hydrostatic forces due to the presence of an internal liquid at rest. In aerospace application, the dynamic behaviour of structures containing an inviscid incompressible fluid (i.e. launcher with liquid propellant, tank of satellite, etc.) is generally computed considering an equilibrium state resulting from a linear fluid-structure interaction problem. It is important to note that in this case, the gravity plays an important role in vibrations and the so called elastogravity operator should be taken into account [1,2]. In the present work, we consider the large deformation behavior of a structure loaded by hydrostatic follower forces in order to obtain an accurate static equilibrium state. The solution is computed using a Newton-Raphson algorithm considering the geometrical and material tangent stiffness matrices as well as the contribution of the follower forces [3]. Those linearized operators of the iterative algorithm are used to compute the dependancy between the incremental horizontal level of the free surface, under the fluid volume conservation constraint [4]. Special attention is paid to the finite element discretization of the wetted surface, considering a level-set method [6,7]. Some numerical examples are analyzed to show the efficiency of the proposed approach.

On Newton-Raphson formulation and algorithm for displacement based structural dynamics problem with quadratic damping nonlinearity

Quadratic damping nonlinearity is challenging for displacement based structural dynamics problem as the problem is nonlinear in time derivative of the primitive variable. For such nonlinearity, the formulation of tangent stiffness matrix is not lucid in the literature. Consequently, ambiguity related to kinematics update arises when implementing the time integration-iterative algorithm. In present work, an Euler-Bernoulli beam vibration problem with quadratic damping nonlinearity is addressed as the main source of quadratic damping nonlinearity arises from drag force estimation, which is generally valid only for slender structures. Employing Newton-Raphson formulation, tangent stiffness components associated with quadratic damping nonlinearity requires velocity input for evaluation purpose. For this reason, two mathematically equivalent algorithm structures with different kinematics arrangement are tested. Both algorithm structures result in the same accuracy and convergence characteristic of solution.

Geometrical nonlinear elasto-plastic analysis of tensegrity systems via the co-rotational method

An efficient finite element formulation is presented for geometrical nonlinear elasto-plastic analyses of tensegrity systems based on the co-rotational method. Large displacement of a space rod element is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. A new form of tangent stiffness matrix, including elastic and elasto-plastic stages is derived based on the proposed approach. An incremental-iterative solution strategy in conjunction with the Newton-Raphson method is employed to obtain the geometrical nonlinear elasto-plastic behavior of tensegrities. Several numerical examples are given to illustrate the validity and efficiency of the proposed algorithm for geometrical nonlinear elasto-plastic analyses of tensegrity structures.

Practical nonlinear inelastic analysis method of composite steel-concrete beams with partial composite action

This paper presents an efficient computer method for nonlinear inelastic analysis of composite steel-concrete beams with partial composite action. The proposed formulation is intended to model the combined effects of partial composite action and distributed plasticity using only one 2-noded beam-column element per structural member. Based on elasto-plastic cross-sectional analyses the behaviour model is able to take into account the effects of partial composite action between the concrete slab and the steel beam. Gradual yielding throughout the cross-section is described through basic equilibrium, compatibility, material and shear connection nonlinear constitutive equations. Tangent flexural and axial rigidity of the cross-section are derived and then using the flexibility approach the elasto-plastic tangent stiffness matrix and equivalent nodal loads vector of the beam-column element including the shear deformability of the partially connected composite beam has been developed. The proposed nonlinear analysis formulation has been implemented in a general nonlinear static purpose computer program, NEFCAD. Advanced finite element simulations have been conducted by using the specialized software for nonlinear analysis of structures, ABAQUS. Several computational examples are given to validate the accuracy and efficiency of the proposed method by comparing the results predicted by NEFCAD with those given by the ABAQUS software and other results retrieved from the open literature.

Effect of intermediate principal stress and loading-path on failure of cementitious materials using granular micromechanics

The failure behavior of rock-like cemented granular materials shows dependency upon both the intermediate principal stress as well as the loading path. Here, we apply a thermo-mechanics based granular micromechanics constitutive relationships of cementitious materials to predict such failure phenomena and investigate the macro- and the micro-scale mechanisms that govern the predicted behavior. True triaxial tests with different levels of minor and intermediate principal stress are simulated. The simulations clearly show that for a given loading path, the maximum principal stress at failure depends upon both the minor and the intermediate principal stress components in a manner that conform closely with recent experimental observations. The path dependent nature of the material response to loading is also demonstrated in the failure behavior by simulations performed for two different types of true triaxial loading paths. The macro-scale failure mechanisms are investigated using the eigenvalues of the tangent stiffness matrix which indicate the corresponding Kelvin mode at failure. From microscopic viewpoint, the evolution of normal and tangential components of inter-granular force vectors is analyzed to reveal the inter-granular mechanisms associated with the load bearing and failure. With respect to the localized failure modes, the eigenvalue analysis of the localization tensor is performed and the direction of failure plane is studied. The direction of failure plane is seen to depend on all component of the principal stress. Further, the results show a change in the localized macro-scale failure mechanism from shear dominated at low confinement to a mixture of shear and compaction at higher confinement, which agrees with the decreasing trend observed in fault angle by increasing confinement. This result is in agreement with experimental results of true triaxial tests on rock samples which report similar transition from shear to compaction bands.

A numerical study on the symmetrization of tangent stiffness matrix in non-linear analysis using corotational approach

In the corotational kinematics (EIRC), the movement is decomposed in deformational and rigid body components using projection operators. Large rigid body translations and rotations, and infinitesimal deformation tensors are adopted. In this way, a non-symmetric stiffness matrix is obtained for the finite element. Literature points that this matrix may be symmetrized in the usual way and the original non-symmetric matrix tends to be in a symmetric form as equilibrium is approached. This paper performed symmetrization studies using Frobenius norm and Absolute Maximum Coefficient of the anti-symmetric part of stiffness matrix in several numerical analyses with high non-linearity solved with an incremental-iterative scheme. An element independent corotational (EICR) kinematics is used to analyze non-linear spatial frames with 3D Euler–Bernoulli beam elements. The results show that this symmetrization not always occurs and depends significantly on the magnitude of displacements and finite-element mesh refinement. The authors demonstrated that, in some cases, the symmetrization would only take place with very refined meshes when the discretized structure approaches the continuum. In this way, the current literature and general mathematical proof of that symmetrization have to be restated.

A [formula omitted]-integral-based arc-length solver for brittle and ductile crack propagation in finite deformation-finite strain hyperelastic solids with an application to graphene kirigami

Arc-length methods based on Newton–Raphson (NR) iterative numerical solvers are indispensable in non-linear mechanics to trace the equilibrium curves of systems with snap-back or snap-through behaviours. An example is a crack propagating in a rubber-like solid, which can generate extremely sharp snap-backs when the elastic energy is released.A standard arc-length solver satisfies the arc-length constraint which only limits the increment in displacements and in load level to a fixed amount. Thus, it cannot guarantee the satisfaction of the energy balance in the rate form. This work proposes an arc-length based on the J-integral, and assumes the Griffith balance as a constraint equation.Firstly, a fracture criterion (critical load and direction of crack increment) is formulated, consistent with the thermodynamic principle of maximum dissipation: a crack propagates in the direction of the maximum strain energy release rate. For this purpose, this paper provides an explicit and simple expression of the J-integral for varying angles. This fracture criterion does not require the computation of mixed mode Stress Intensity Factors, or asymptotic solutions or derivatives of the tangent stiffness matrix. By doing so, it is shown that the proposed method easily reconciles with the theory of Configurational Forces.Secondly, because of the explicit expression of J provided in this paper, it is easy to linearize the discretized equations of motions consistently.The method proves able to handle very sudden snap-backs occurring under large strains, and both brittle and ductile crack propagation. A particular example of such behaviour is kirigami (paper-cuts) structures made of graphene. The proposed arc-length can capture, for these structures, the transitions from brittle to ductile crack propagation for the same crack patterns, but different pre-crack lengths.

A 4-Node Co-Rotational Quadrilateral Composite Shell Element

A 4-node co-rotational quadrilateral composite shell element is presented. The local coordinate system of the element is a co-rotational framework defined by the two bisectors of the diagonal vectors generated from the four corner nodes and their cross product. Thus, the element rigid-body rotations are excluded in calculating the local nodal variables from the global nodal variables. Compared with other existing co-rotational finite-element formulations, the present element has two features: (i) The two smallest components of the mid-surface normal vector at each node are defined as the rotational variables, leading to the desired additive property for all nodal variables in a nonlinear incremental solution procedure; (ii) both element tangent stiffness matrices in the local and global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of strain energy with respect to the local nodal variables and, through chain differentiation with respect to the global nodal variables. In the modeling of composite structures, the first-order shear deformable laminated plate theory is adopted in the local element formulation, where both the thickness deformation and the normal stress in the direction of the shell thickness are ignored, and an assumed strain method is employed to alleviate the membrane and shear locking phenomena. Several examples involving composite plates and shells with large displacements and large rotations are presented to testify to the reliability and convergence of the present formulation.

An equivalent continuum multiscale formulation for 2D geometrical nonlinear analysis of lattice truss structure

In this work, an equivalent continuum multiscale formulation is presented for the geometrical nonlinear analysis of the structures with lattice truss materials. This formulation is established by combining the extended multiscale finite element method and the co-rotational approach. Firstly, the lattice truss unit cell is equivalent to a continuum coarse element by using a numerical constructed interpolation function in the local coordinate system. Then the tangent stiffness matrix of this coarse element is derived by employing the basic idea of the co-rotational approach in the global coordinate system. Thus, the global nonlinear equilibrium equations of the structure at the macroscopic level can be solved by using the general displacement control algorithm to capture the equilibrium path with multiple critical points. After performing all of the incremental steps and the iterative steps on the macroscopic scale, the microscopic information, such as the displacement, stress and strain, can be obtained easily by virtue of the afore-constructed numerical interpolation functions once again. In addition, several numerical examples are carried out to study the effects of the layout and size of unit cell, investigate the sensitivity of coarse-scale meshes and verify the validation and efficiency of the presented multiscale formulation.

Generalized Potential Energy Finite Elements for Modeling Molecular Nanostructures.

The potential energy of molecules and nanostructures is commonly calculated in the molecular mechanics formalism by superimposing bonded and nonbonded atomic energy terms, i.e. bonds between two atoms, bond angles involving three atoms, dihedral angles involving four atoms, nonbonded terms expressing the Coulomb and Lennard-Jones interactions, etc. In this work a new, generalized numerical simulation is presented for studying the mechanical behavior of three-dimensional nanostructures at the atomic scale. The energy gradient and Hessian matrix of such assemblies are usually computed numerically; a potential energy finite element model is proposed herein where these two components are expressed analytically. In particular, generalized finite elements are developed that express the interactions among atoms in a manner equivalent to that invoked in simulations performed based on the molecular dynamics method. Thus, the global tangent stiffness matrix for any nanostructure is formed as an assembly of the generalized finite elements and is directly equivalent to the Hessian matrix of the potential energy. The advantages of the proposed model are identified in terms of both accuracy and computational efficiency. In the case of popular force fields (e.g., CHARMM), the computation of the Hessian matrix by implementing the proposed method is of the same order as that of the gradient. This analysis can be used to minimize the potential energy of molecular systems under nodal loads in order to derive constitutive laws for molecular systems where the entropy and solvent effects are neglected and can be approximated as solids, such as double stranded DNA nanostructures. In this context, the sequence dependent stretch modulus for some typical base pairs step is calculated.

A new tangent stiffness-based formulation to study the free vibration behavior of a transversely loaded Timoshenko beam with geometric nonlinearity

A new formulation is introduced to study the free vibration behavior of a statically loaded beam with geometric nonlinearity. The tangent stiffness of the statically loaded beam is used to investigate the free vibration behavior of the beam about its loaded configuration. The problem is formulated for a linearly tapered beam, and a uniform beam is obtained as a special case. Energy principles based on the variational approach are used to derive the governing equations for the static and dynamic problems. The Ritz method of approximate displacement field is followed to solve the governing equations. The Ritz coefficients are used to derive the tangent stiffness of the loaded beam. Components of the tangent stiffness matrix are derived for a Timoshenko beam with von Kármán-type nonlinearity. Illustrative results are presented for four different classical boundary conditions having in-plane restraint. Results for the first two modes of transverse vibration are presented in the nondimensional deflection-frequency plane. Validation of the work is carried out using finite element software ANSYS. The formulation is new of its kind and can be used for any displacement-based problem following the Ritz method.

Unified formulation of geometrically nonlinear refined beam theories

ABSTRACTBy using the Carrera Unified Formulation (CUF) and a total Lagrangian approach, the unified theory of beams including geometrical nonlinearities is introduced in this article. According to CUF, kinematics of one-dimensional structures are formulated by employing an index notation and a generalized expansion of the primary variables by arbitrary cross-section functions. Namely, in this work, low- to higher-order beam models with only pure displacement variables are implemented by utilizing Lagrange polynomial expansions of the unknowns on the cross section. The principle of virtual work and a finite element approximation are used to formulate the governing equations, whereas a Newton-Raphson linearization scheme along with a path-following method based on the arc-length constraint is employed to solve the geometrically nonlinear problem. By using CUF and three-dimensional Green-Lagrange strain components, the explicit forms of the secant and tangent stiffness matrices of the unified beam element are provided in terms of fundamental nuclei, which are invariants of the theory approximation order. A symmetric form of the secant matrix is provided as well by exploiting the linearization of the geometric stiffness terms. Various numerical assessments are proposed, including large deflection analysis, buckling, and postbuckling of slender solid cross-section beams. Thin-walled structures are also analyzed in order to show the enhanced capabilities of the present formulation. Whenever possible, the results are compared to those from the literature and finite element commercial software tools.