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Showing papers on "Tangent stiffness matrix published in 2021"


Journal ArticleDOI
TL;DR: The SPD-NN weakly imposes convexity on the strain energy function, satisfies time consistency for path-dependent materials, and therefore improves numerical stability, especially when theSPD-NN is used in finite element simulations.

97 citations


Journal ArticleDOI
TL;DR: In this article, a three-dimensional constitutive model for shape memory alloys considering the TRsformation-Induced Plasticity (TRIP) as well as the Two-Way Shape Memory Effect (TWSME) through a large deformation framework is presented.

47 citations


Journal ArticleDOI
TL;DR: In this paper, a rank-constrained linear matrix inequality (LMI) based approach was proposed to solve the tense-grity structure topology problem, where the rank constraint on the rank of the force density matrix was considered.

14 citations


Journal ArticleDOI
TL;DR: In this paper, a tensorial coordinate free form formulation of the Kirchhoff model for curved rods and rod assemblies is presented, where the tangent stiffness matrix is derived in tensor-free form and the Levi-Civita connection of the configurations manifold of the rod is needed.

12 citations


Journal ArticleDOI
TL;DR: A simplified nonlinear equations-model is presented that is intended to design shape memory alloy (SMA) dampers in three-dimensional structures of unsymmetrical-plan and in yielding shear-frames and takes no more than two iterations.

10 citations


Journal ArticleDOI
TL;DR: In this paper, a first-order shear deformable curved beam element is developed for geometrically nonlinear analysis of functionally graded (FG) curved beams, and the stress equilibrium condition is introduced into the element formulation by using the mixed finite element method with displacements and internal forces as independent fields.
Abstract: Based on the geometrically exact beam theory, a first-order shear deformable curved beam element is developed for geometrically nonlinear analysis of functionally graded (FG) curved beams. In order to accurately predict the distribution of transverse shear stress, the stress equilibrium condition is introduced into the element formulation by using the mixed finite element method with displacements and internal forces as independent fields. The element nodal force vector and consistent tangent stiffness matrix for nonlinear iterative solution are obtained by going through a consistent linearization procedure. Numerical examples are presented to validate the present formulation. As indicated by the numerical results, the proposed element demonstrates a high level of solution accuracy and good applicability. Furthermore, using the proposed element, an investigation is conducted into the nonlinear stability of FG structures with different material distribution parameters, and the accuracy loss caused by unreasonable distribution of shear stress is discussed.

9 citations


Journal ArticleDOI
TL;DR: A shifting technique is used to enforce compatibility in the direction perpendicular to the crack path, while a blending technique has been adopted to remove the effect of the discontinuity in the extension of the cohesive crack.

9 citations


Journal ArticleDOI
TL;DR: In this article, a three-dimensional corotational total Lagrangian beam element is formulated and implemented in the OpenSees framework to account for these coupling effects by invoking Green-Lagrange strains referenced to a basic system.

9 citations


Journal ArticleDOI
TL;DR: In this paper, a structural geometric nonlinear analysis using the finite element method (FEM) depends on the consideration of five aspects: the interpolation (shape) functions, the bending theory, the kinematic description, the strain-displacement relations, and the nonlinear solution scheme.

8 citations


Journal ArticleDOI
TL;DR: A new shell superelement is presented to study linear/nonlinear static and free vibration analysis of spherical structures with partial or full spherical geometries that exist in many industrial applications and compared with available results in the literature and conventional shell elements in a commercial software.
Abstract: Finite element analysis of huge and/or complicated structures often requires long times and large computational expenses. Superelements are huge elements that exploit the deformation theory of a specific problem to provide the capability of discretizing the problem with minimum number of elements. They are employed to reduce the computational cost while retaining the accuracy of results in FEM analysis of engineering problems. In this research, a new shell superelement is presented to study linear/nonlinear static and free vibration analysis of spherical structures with partial or full spherical geometries that exist in many industrial applications. Furthermore, this study investigates the effects of changing the superelement size and its number of nodes on solution accuracy. The governing equations of composite spherical shells are derived based on the first-order shear deformation theory and considering large deformations. For solving the nonlinear governing equations, the tangent stiffness matrix has been extracted and the Newton–Raphson method is employed. The capability of the presented shell superelement is investigated in several problems under linear/nonlinear static and free vibration analysis. The results acquired by the presented shell superelements are compared with available results in the literature and conventional shell elements in a commercial software. Results comparisons reveal high accuracy at a reduced computational cost in the superelement model.

6 citations


Journal ArticleDOI
TL;DR: In this article, a tangent plasticity matrix approach was proposed to solve the unit cell boundary value problem. And the tangent FVDAM theory was implemented using a quasi-Newton-Raphson strategy that quantifies errors in the evaluation of surface-averaged stresses due to the linearization, hence allowing large load increments.

Journal ArticleDOI
TL;DR: In this paper, a cable net structure consisting of a wind-resistance main cable, windresistance secondary cable, and pulleys is proposed to improve the wind resistance stability of narrow suspension bridges.

Journal ArticleDOI
TL;DR: In this paper, a geometrically exact (GeX) hybrid-mixed four-node solid-shell element is developed using a sampling surfaces (SaS) method, based on the choice of N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns.
Abstract: In this paper, the nonlinear three-dimensional (3D) stress analysis of shell structures in buckling and snapping problems is presented. The exact geometry or geometrically exact (GeX) hybrid-mixed four-node solid-shell element is developed using a sampling surfaces (SaS) method. The SaS formulation is based on the choice of N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. The SaS are located at the Chebyshev polynomial nodes (roots of the Chebyshev polynomial of degree N), that is, the outer surfaces are not included into a set of SaS. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-the-thickness distributions of displacements, strains and stresses leads to an efficient higher-order shell formulation. The incremental equilibrium equations are solved by the Newton–Raphson method combined with the Crisfield arc-length algorithm. The tangent stiffness matrix is evaluated by effective 3D analytical integration. As a result, the proposed GeX hybrid-mixed solid-shell element exhibits superior performance for coarse meshes and allows much larger load increments than is possible with existing displacement-based solid-shell elements. This can be useful for the 3D stress analysis of thin and thick shells in different states such as pre-buckling, bifurcation and post-buckling.

Journal ArticleDOI
TL;DR: In this article, the Mises truss with out-of-plane lateral linear spring was analyzed as a three DOF system and the authors derived the tangent stiffness matrix and analyzed the stability of the equilibrium solutions.
Abstract: This paper deals with the equilibrium problem of the Mises truss, with out-of-plane lateral linear spring, analyzed as a three DOF system. It is shown that, as a consequence of the geometry of the structure, the system can undergo three buckling modes which are asymmetric in-plane buckling, symmetric out-of-plane buckling and asymmetric out-of-plane buckling. The analysis takes into account the influence of local buckling and yielding of bars on global instabilities. The Green–Lagrange strain is adopted as the strain measure and the theorem of the stationarity of the total potential energy is employed to derive the nonlinear equilibrium equations. The tangent stiffness matrix is derived and, through the solution of the eigenvalue problem, the stability of the equilibrium solutions is investigated. Analytical formulations for the instabilities of the truss are presented. For the numerical approach, a linear elastic constitutive model is assumed for the uniaxial stress–strain relationship of the truss bars. To take into account the yielding of bar elements, a perfect elastoplastic model is assumed. A computer program was developed in Fortran to perform comparisons with the results of the theoretical formulation. Finally, the numerical results obtained demonstrate the accuracy and effectiveness of the presented truss element. The main novelty of this paper is the introduction of an additional DOF in the Mises truss which allows to study a more complex scenario of equilibrium paths and instabilities.

Journal ArticleDOI
TL;DR: In this paper, a three-dimensional mixed beam element formulation for fully nonlinear distributed plasticity analysis of members composed of sections with no significant torsional warping such as steel angles and tees is presented using a corotational total Lagrangian approach.
Abstract: This paper presents a three-dimensional mixed beam element formulation for fully nonlinear distributed plasticity analysis of members composed of sections with no significant torsional warping such as steel angles and tees. This formulation is presented using a corotational total Lagrangian approach and implemented in the OpenSees corotational framework. In this context, a basic coordinate system is lined up with the element chord and translates and rotates as the element deforms. The element tangent stiffness matrix and resisting forces in the basic system are derived through linearization of the two-field Hellinger-Reissner variational principle. The displacement shape functions are cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation. The generalized stress resultant shape functions are linear for moments and constant for axial force and torque with the P - δ effect considered, which are developed from equilibrium equations. The fiber section method with uniaxial constitutive laws is adopted to account for material nonlinearity. Since the degrees-of-freedom in the basic system are defined with respect to different reference points, all element responses are transformed to acting about the shear center before conducting the corotational transformation. The mixed element is validated through a number of experimental and numerical examples.

Journal ArticleDOI
TL;DR: In this paper, a total Lagrangian finite element formulation was developed for investigating large deflection bending of nanobeams based on the modified couple stress theory, and the tangent stiffness matrix was extracted based on this formulation.
Abstract: A total Lagrangian finite element formulation was developed for investigating large deflection bending of nanobeams based on the modified couple stress theory. Timoshenko beam theory accompanied by the axial displacement with least kinematic assumptions has been used to model nanobeams. For the first time, the tangent stiffness matrix for nanobeams is extracted based on the modified couple stress theory. The present finite element formulation provides the possibility of solving the problems of nanobeams with arbitrary loading, boundary conditions and cross-sectional variation. The obtained results have been validated with the results reported in the other works. The influence of material length scale parameter on the maximum amount of displacement and rotation of nanobeams is discussed for different ratios of the nanobeam length to the thickness. Besides, for the illustration of abilities of the present formulation, large deflection of a tapered nanobeam under the sinusoidal distributed load for two types of boundary conditions (clamped and simply supported) has been presented.

Journal ArticleDOI
TL;DR: In this article, a computational tool called SLUGFLEX has been developed, which is composed by two computational codes: the code that calculates the structural dynamic response of the flexible riser, and the other one that predicts the development of the internal slug flow.

Journal ArticleDOI
TL;DR: In this article, a non-classic double bifurcation of discrete static systems, occurring along a non trivial equilibrium path, is studied, where, by varying a parameter, a pair of eigenvalues of the tangent stiffness matrix first merge and then disappear.
Abstract: A non-classic double bifurcation of discrete static systems, occurring along a non trivial equilibrium path, is studied. It occurs when, by varying a parameter, a pair of eigenvalues of the tangent stiffness matrix first merge and then disappear. A paradigmatic 2 degrees of freedom system with some symmetry, consisting of an inverted extensible pendulum, so far studied in literature in the linear range only, is proposed. Exact nonlinear solutions are derived, showing the mechanism which leads two distinct bifurcated paths to locally touch each other at the double bifurcation point, and then to detach from the fundamental path. The detrimental effects of the interaction on stability are discussed. An asymptotic bifurcation analysis is also carried out, able to explain how a unique buckling mode existing at the double bifurcation is able to generate, in the perturbation process, two distinct deflections on close bifurcated paths.

Book ChapterDOI
01 Jan 2021
TL;DR: In this article, a methodology for discrete parametric optimization of space steel frames based on a metaheuristic job search inspired strategy has been developed, which considers minimizing the weight of frame structures made of bars with closed cross-sections, taking into account active constraints on overall stability.
Abstract: A methodology for discrete parametric optimization of space steel frames based on a metaheuristic job search inspired strategy has been developed. The paper considers minimizing the weight of frame structures made of bars with closed cross-sections, taking into account active constraints on overall stability, including the stability of individual bars, stresses and displacements. The search is carried out on the sets of admissible options for the cross-sections of the bars. Job search inspired strategy does not require the introduction of penalty functions when taking into account task constraints. Previously completed developments on the use of this strategy in the optimization of flat frames envisaged the introduction of small fictitious forces and the implementation of the iterative process of calculating the object from the deformed state to assess the degree of dissatisfaction with the overall stability condition. In the paper, this issue is solved for space frames based on LDLT decomposition of the tangent stiffness matrix of the finite element model of the structure. The auxiliary function of the goal uses the values of the elements of the diagonal matrix obtained from this decomposition. This approach provides an efficient stability test without directly considering the generalized eigenvalue problem for matrices. The performance of the proposed algorithm is illustrated using the example of optimization of a console made of round pipes.

Journal ArticleDOI
24 Nov 2021-MethodsX
TL;DR: In this article, the tangent stiffness matrix is calculated from the differential equation solution of deformed infinitesimal element equilibrium, considering the axial load and the shear deformation in this relation.

Journal ArticleDOI
17 Aug 2021
TL;DR: In this article, a numerical model based on the improved interpolating complex variable element free Galerkin (IICVEFG) method and the incremental tangent stiffness matrix method is proposed.
Abstract: A numerical model for the two-dimensional nonlinear elastic–plastic problem is proposed based on the improved interpolating complex variable element free Galerkin (IICVEFG) method and the incremental tangent stiffness matrix method. The viability of the proposed model is verified through three elastic–plastic examples. The numerical analyses show that the IICVEFG method has good convergence. The solutions using the IICVEFG method are consistent with the solutions obtained from the finite element method using the ABAQUS program. Moreover, the IICVEFG method shows greater computing precision and efficiency than the non-interpolating meshless methods.

Journal ArticleDOI
TL;DR: In this paper, an imperfect nonlinear beam-column element considering the hybrid effects of component stiffness is proposed to achieve accurate structural calculation of pultruded fiber reinforced polymer (PFRP) space frame structures by a single element per member.

Journal ArticleDOI
TL;DR: In this article, the stability of non-uniform crane structures with stepped and tapered columns is analyzed using a hybrid of the transfer matrix method (TMM) and the finite element method (FEM).
Abstract: To analyze the stability of non-uniform crane structures with stepped and tapered columns, equivalent element stiffness derived from hybrid of the transfer matrix method (TMM) and the finite element method (FEM) is proposed. The second-order theory is applied to establish the differential equations of the stepped column with any number of sections, and the exact slope-deflection equations of the equivalent element are derived by TMM. Furthermore, the slope-deflection equations are formulated into the exact symmetric tangent stiffness matrix in FEM. Based on the FEM buckling criterion, the critical forces of stepped columns can be obtained precisely in Bernoulli–Euler sense. Buckling criterion equations of the stepped column with three widely used constraints in crane structures are given as the application of the proposed stiffness. Meanwhile, the proposed stiffness matrix of the stepped column is extended to analyze the buckling of the tapered column, which can be simulated by the four-section stepped column with the selective equivalent inertia moment. Finally, verification examples demonstrate that the proposed method can be used to accurately and efficiently evaluate the critical load of the non-uniform crane structures.

Journal ArticleDOI
TL;DR: In this article, a fully implicit backward Euler integration is proposed and validated in a commercial finite element code through a user-defined material subroutine, and the advantages of the proposed implicit formulation in terms of stability with respect to an explicit formulation were assessed.
Abstract: The viscoplastic model proposed by Vermeer and Neher in 1999 is still currently used in the oil and gas industry for subsidence modeling, to predict the deformation of the ground surface induced by hydrocarbon withdrawal from underground reservoirs. Even though several different implementations of this model have been proposed in the literature, also very recently, a consistent fully implicit implementation is still missing, probably due to the technical difficulties involved in the rigorous derivation of the analytical tangent matrix. To fill this gap and to provide an effective tool to the engineering community, a fully implicit backward Euler integration is proposed and validated in this work. The consistent expression of the tangent stiffness matrix is also derived analytically, and its validation strategy is described in detail. The model was implemented in a commercial finite element code through a user-defined material subroutine. The advantages of the proposed implicit formulation in terms of stability with respect to an explicit formulation were assessed and validated. The examples include studies at material point level and at field scale for a case study of subsidence in a synthetic reservoir.

Journal ArticleDOI
TL;DR: In this article, the authors investigated the numerical stability of the dynamic relaxation arc-length method for solving the snap-back problem and showed that the spectral radius of the amplification matrix is always greater than one, leading to unconditional instability.
Abstract: Incorporating the arc-length constraint, the dynamic relaxation strategy has been widely used to trace full equilibrium path in the post-buckling analysis of structures. This combined numerical scheme has been shown to be successful for solving snap-through problems, but its applicability to snap-back problems has been rarely investigated and remains unclear. This paper proposes a direct and more general finite-difference equation to investigate the numerical stability of this combined numerical scheme, which is dominated by the spectral radius of amplification matrix. And a key discovery of this paper is that a first minor of the tangent stiffness matrix is always negative once snap back occurs. Due to this negative minor stiffness, the spectral radius is invariably greater than one, resulting in unconditional instability, which demonstrates the invalidity of dynamic relaxation arc-length method for snap-back problems. These important conclusions are corroborated by the numerical results of three representative examples in one-, two- and three-dimensional spaces.

Journal ArticleDOI
TL;DR: In this article, the authors proposed a simple and unified formulation for different kinds of elements that can be cooperated in a co-rotational analysis, which is done by following the conventional element-independent CR formulation but dropping the terms involving projector matrix and therefore yields simple formulations of the internal force and the tangent stiffness matrix.

Journal ArticleDOI
TL;DR: Automatic differentiation (AD) is an ensemble of techniques that allows to evaluate accurate numerical derivatives of a mathematical function expressed in a computer programming language as discussed by the authors, and it has been used for stating and solving solid mechanics problems.
Abstract: Automatic differentiation (AD) is an ensemble of techniques that allows to evaluate accurate numerical derivatives of a mathematical function expressed in a computer programming language. In this paper we use AD for stating and solving solid mechanics problems. Given a finite element discretization of the domain, we evaluate the free energy of the solid as the integral of its strain energy density, and we make use of AD for directly obtaining the residual force vector and the tangent stiffness matrix of the problem, as the gradient and the Hessian of the free energy respectively. The result is a remarkable simplification in the statement and the solution of complex problems involving non trivial constraints systems and both geometrical and material non linearities. Together with the continuum mechanics theoretical basis, and with a description of the specific AD technique adopted, the paper illustrates the solution of a number of solid mechanics problems, with the aim of presenting a convenient numerical implementation approach, made easily available by recent programming languages, to the solid mechanics community.

Journal ArticleDOI
TL;DR: In this paper, the authors evaluated the accuracy of using the concentrated mass matrix and the Rayleigh damping matrix for the case of steel buildings and concluded that the consistent mass matrix should be used in the structural modelling of the structural system under consideration.