Showing papers on "Tangent stiffness matrix published in 2018"

A method for stiffness analysis of overconstrained parallel robotic mechanisms with Scara motion

Abstract: This paper presents a general method for analyzing stiffness of overconstrained parallel robotic mechanisms with Scara motion. In the method, the stiffness model of a limb is derived by applying Castigliano's second theorem to strain energy of the limb with a structural decomposition strategy, and the stiffness matrix of a parallel mechanism is established based on stiffness models of limbs and the static equilibrium equation of the moving platform. Comparisons show that the stiffness model obtained from the proposed method is very close to the counterpart obtained from finite element analysis (FEA). In addition, a new index is proposed to evaluate stiffness performance of a parallel mechanism in a given configuration based on strain energy under external unit forces and moments. With this index, the dimensions of a parallel mechanism can be optimized and the path of a given task can be planned to obtain high stiffness.

An Improved Analytical Algorithm on Main Cable System of Suspension Bridge

Abstract: This paper develops an improved analytical algorithm on the main cable system of suspension bridge. A catenary cable element is presented for the nonlinear analysis on main cable system that is subjected to static loadings. The tangent stiffness matrix and internal force vector of the element are derived explicitly based on the exact analytical expressions of elastic catenary. Self-weight of the cables can be directly considered without any approximations. The effect of pre-tension of cable is also included in the element formulation. A search algorithm with the penalty factor is introduced to identify the initial components for convergence with high precision and fast speed. Numerical examples are presented and discussed to illustrate the accuracy and efficiency of the proposed analytical algorithm.

A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

Abstract: Due to the nonlocal property of the fractional derivative, the finite element analysis of fractional diffusion equation often leads to a dense and non-symmetric stiffness matrix, in contrast to the conventional finite element formulation with a particularly desirable symmetric and banded stiffness matrix structure for the typical diffusion equation. This work first proposes a finite element formulation that preserves the symmetry and banded stiffness matrix characteristics for the fractional diffusion equation. The key point of the proposed formulation is the symmetric weak form construction through introducing a fractional weight function. It turns out that the stiffness part of the present formulation is identical to its counterpart of the finite element method for the conventional diffusion equation and thus the stiffness matrix formulation becomes trivial. Meanwhile, the fractional derivative effect in the discrete formulation is completely transferred to the force vector, which is obviously much easier and efficient to compute than the dense fractional derivative stiffness matrix. Subsequently, it is further shown that for the general fractional advection---diffusion---reaction equation, the symmetric and banded structure can also be maintained for the diffusion stiffness matrix, although the total stiffness matrix is not symmetric in this case. More importantly, it is demonstrated that under certain conditions this symmetric diffusion stiffness matrix formulation is capable of producing very favorable numerical solutions in comparison with the conventional non-symmetric diffusion stiffness matrix finite element formulation. The effectiveness of the proposed methodology is illustrated through a series of numerical examples.

Meshless meso-modeling of masonry in the computational homogenization framework

Abstract: 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.

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

Abstract: 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 investig...

Buckling load optimization for 2D continuum models, with alternative formulation for buckling load estimation

Abstract: Buckling load estimation of continua modeled by finite element (FE) should be based on non-linear equilibrium. When such equilibrium is obtained by incremental solutions and when sensitivity analysis as well as iterative redesigns are included, the computational demands are large especially due to optimization. Therefore, examples presented in the literature relate to few design variables and/or few degrees of freedom. In the present paper a non-incremental analysis is suggested, and a simple sensitivity analysis as well as recursive redesign is proposed. The implicit geometrical non-linear analysis, based on Green-Lagrange strains, apply the secant stiffness matrix as well as the tangent stiffness matrix, both determined for the equilibrium corresponding to a given reference load, obtained by the Newton-Raphson method. For the formulated eigenvalue problem, which solution gives the estimated buckling load, the tangent stiffness matrix is of major importance. In contrast to formulations based on incremental solutions, the tangent stiffness matrix is here divided into two matrices, the stress stiffness matrix that is linear depending on stresses and the remaining part of the tangent stiffness matrix. Examples verify the effectiveness of the proposed procedure.

Free vibration behavior of a thermally post-buckled FG Timoshenko beam under large deflection using a tangent stiffness-based method

Abstract: For thermally postbuckled configurations, the free vibration behavior of functionally graded (FG) Timoshenko beams are investigated. The postbuckling configurations are obtained through a geometrically nonlinear static problem. The free vibration problem around the postbuckled configuration is formulated using its tangent stiffness. The energy based governing equations are solved following the Ritz method. The elements of the tangent stiffness matrix are obtained using the Ritz coefficients. The results are shown to exhibit the effects of FG material, material profile parameter, and length-thickness ratio. The comparative results are presented for both the cases of the physical neutral surface and the geometrical neutral surface.

Dual-support smoothed particle hydrodynamics in solid: variational principle and implicit formulation

Abstract: In this paper, we derive the dual-support smoothed particle hydrodynamics (DS-SPH) in solid in the framework of variational principle. The tangent stiffness matrix of SPH is obtained with ease, which can be served as the basis for implicit SPH. We propose a hourglass energy functional, which allows the direct derivation of hourglass force and hourglass tangent stiffness matrix. The dual-support is identified in all derivation based on variational principles and is automatically satisfied in the assembling of stiffness matrix. The implementation of stiffness matrix comprises with two steps, the nodal assembly based on deformation gradient and global assembly. Several numerical examples are presented to validate the method.

Geometrically Nonlinear Analysis of Beam Structures via Hierarchical One-Dimensional Finite Elements

Abstract: The formulation of a family of advanced one-dimensional finite elements for the geometrically nonlinear static analysis of beam-like structures is presented in this paper. The kinematic field is axiomatically assumed along the thickness direction via a Unified Formulation (UF). The approximation order of the displacement field along the thickness is a free parameter that leads to several higher-order beam elements accounting for shear deformation and local cross-sectional warping. The number of nodes per element is also a free parameter. The tangent stiffness matrix of the elements is obtained via the Principle of Virtual Displacements. A total Lagrangian approach is used and Newton-Raphson method is employed in order to solve the nonlinear governing equations. Locking phenomena are tackled by means of a Mixed Interpolation of Tensorial Components (MITC), which can also significantly enhance the convergence performance of the proposed elements. Numerical investigations for large displacements, large rotations, and small strains analysis of beam-like structures for different boundary conditions and slenderness ratios are carried out, showing that UF-based higher-order beam theories can lead to a more efficient prediction of the displacement and stress fields, when compared to two-dimensional finite element solutions.

An Investigation on Optimal Initial Self-Stress Design of Tensegrity Grid Structures

Abstract: To achieve the optimal feasible force density vector of a given geometry configuration tensegrity grid structure, an efficient procedure is presented for optimal initial self-stress design of tensegrity grid structures by consecutively solving two linear homogeneous systems in conjunction with a minimization problem. The nonlinear constrained optimization algorithm, known as the Interior-Point Method (I-PM), is utilized to obtain the minimal solution, leading to a set of force densities which guarantee the non-degeneracy condition of the force density matrix. The evaluation of the eigenvalues of tangent stiffness matrix is also introduced to check the geometric stability of the tensegrity grid structures. Finally, three numerical examples have been investigated comprehensively to prove the capability of the proposed method in optimal initial self-stress design of tensegrities. Furthermore, division of number of member group has been discussed in detail for the purpose of demonstrating the efficiency of the proposed method in seeking initial force densities of tensegrity grid structures.

Improved Prediction Model for Dynamic Resilient Modulus of Subgrade Silty Clay in Eastern Hunan and Its Relevant Finite Element Method Implementation

Abstract: With the enhancement of transportation speed and axle load, dynamic response of subgrade increases significantly. In order to improve the calculation accuracy of subgrade response under complex stress state, it is necessary to use dynamic indicators instead of static indicators in calculative process. For the sake of investigating the influence factor of dynamic resilient modulus of subgrade silty clay in Eastern Hunan, resilient modulus tests were carried out by conducting repeated load tri-axial tests. Based on available model, an improved resilient modulus prediction model considering four parameters was proposed by introducing k4. Corresponding accurate consistent tangent stiffness matrix was derived. Afterward, the improved model was implemented into finite element method software and verification work was put forward both on single element and pavement-subgrade structure. Finally, calculated results were compared with in-site measured results. Study achievements demonstrate that the improved model exhibits a higher precision and efficiency on single element because k4 can better adjust the affecting proportion of octahedral shear stress. When applied to analysis on pavement-subgrade structure, the improved model can reflect subgrade resilient modulus distribution and evolution more factually. In addition, numerical calculated result nearly coincides with measured results, which shows the application value of the improved model.

Element Stiffness Matrix

Abstract: First, the element stiffness matrix [k] for a 2-node bar is generated using three approaches: direct, variational, and weighted residuals. For the weighted residuals method, emphasis is placed on the use of the Galerkin's method. We conclude from this exercise that the element stiffness matrix can be directly obtained from integration of the element shape functions. Second, we calculate numerical entries of element stiffness matrices by using the Gauss numerical integration scheme for the simplest and most commonly used element types. Also introduced are the full-integration, selective-reduced-integration, and reduced-integration schemes. When the reduced-integration scheme is used, concerns regarding the potential hourglass mode are discussed. Third, for a new element type, which can be decomposed into two or more basic element types, we discuss the principle of superposition for creating the stiffness matrix of this new element type. Finally, the rotational matrix is presented to transfer vectors and stiffness matrices from locally derived stiffness matrices, which are based on the natural coordinate system, to the global coordinate system.

Parametric Optimization of Steel Frames Using the Job Search Inspired Strategy

Abstract: An algorithm for parametric optimization of steel plane frames is proposed, which is based on a meta-heuristic job search inspired strategy and genetic algorithms using the main and elite populations. A feature of this computational scheme is the ability to effectively solve an optimum problem without introducing penalty functions. This allows us to strictly consider the constraints in any algorithm run. Tension-compression deformations, bending and pure torsion of the rods are taken into account. The goal is to minimize the mass of the frame rods, taking into account active constraints on stresses, displacements and overall stability, including the stability of individual rods. The cross sections of rods vary on discrete sets of admissible options. Analyzing the considered structure’s deformations is performed using the finite-element method in the form of the displacement approach. The constraints on strength and stiffness are checked by iterative calculation of the stress-strain frame state, using a tangential stiffness matrix, which is formed considering the influence of normal forces on the bending of the rods. Information about the convergence of this computational process is required for the stability assessment of the structure. At the elite population formation stage, additional control is carried out on the stability of the frame’s options on the basis of checking the positive definiteness of the tangent stiffness matrix. The efficiency of the proposed approach to the optimal design of frame structures is illustrated by the example of a rod system made of round tubes.

Fiber Beam Analysis of Reinforced Concrete Members with Cyclic Constitutive and Material Laws

Abstract: This paper presents a non-linear Timoshenko beam element with axial, bending, and shear force interaction for nonlinear analysis of reinforced concrete structures. The structural material tangent stiffness matrix, which relates the increments of load to corresponding increments of displacement, is properly formulated. Appropriate simplified cyclic uniaxial constitutive laws are developed for cracked concrete in compression and tension. The model also includes the softening effect of the concrete due to lateral tensile strain. To establish the validity of the proposed model, correlation studies with experimentally-tested concrete specimens have been conducted.

Third-order elastic coefficients and logarithmic strain in constitutive and finite element modelling of anisotropic elasticity

Abstract: Third-order elastic coefficients (TOECs) have been measured experimentally and tabulated with pretty good accuracy since the middle of the previous century. In the classical acoustic measurement method the recalculation of instantaneous stiffness change onto TOECs is based on the use of Green strain. In recent calculations performed by means of atomistic and quantum methods many different strain measures are employed. In result, quite different sets of TOECs can be obtained for the same material. In this paper, it is shown how dramatically the coefficients obtained depend on the choice of strain measure. The known formulas for calculation of the second derivative of a tensor-valued function of tensor variable are corrected. The formulas are essential for the correct analytic calculation of the tangent stiffness matrix in finite element method, among others.

A nonlocal operator method for solving PDEs

Abstract: We propose a nonlocal operator method for solving PDEs. The nonlocal operator is derived from the Taylor series expansion of an unknown field, and can be regarded as the integral form `equivalent' to the differential form in the sense of nonlocal interaction. The variation of a nonlocal operator is similar to the derivative of shape function in meshless and finite element methods, thus circumvents difficulty in the calculation of shape function and its derivatives. {The nonlocal operator method is consistent with the variational principle and the weighted residual method, based on which the residual and the tangent stiffness matrix can be obtained with ease.} The nonlocal operator method is equipped with an hourglass energy functional to satisfy the linear consistent of the field. High-order nonlocal operators and high-order hourglass energy functional are generalized. The functional based on the nonlocal operator converts the construction of residual and stiffness matrix into a series of matrix multiplications on the nonlocal operator. The nonlocal strong forms of different functionals can be obtained easily with the aid of support and dual-support, two basic concepts introduced in the paper. Several numerical examples are presented to validate the method.

Internal Hinges—Modified Stiffness Matrix Method

Abstract: In Chapter 8, the internal releases are alternatively handled by modifying the element stiffness matrix.

A nonlocal operator method for solving partial differential equations

Abstract: We propose a nonlocal operator method for solving partial differential equations (PDEs). The nonlocal operator is derived from the Taylor series expansion of the unknown field, and can be regarded as the integral form "equivalent" to the differential form in the sense of nonlocal interaction. The variation of a nonlocal operator is similar to the derivative of shape function in meshless and finite element methods, thus circumvents difficulty in the calculation of shape function and its derivatives. {The nonlocal operator method is consistent with the variational principle and the weighted residual method, based on which the residual and the tangent stiffness matrix can be obtained with ease.} The nonlocal operator method is equipped with an hourglass energy functional to satisfy the linear consistency of the field. Higher order nonlocal operators and higher order hourglass energy functional are generalized. The functional based on the nonlocal operator converts the construction of residual and stiffness matrix into a series of matrix multiplications on the nonlocal operators. The nonlocal strong forms of different functionals can be obtained easily via support and dual-support, two basic concepts introduced in the paper. Several numerical examples are presented to validate the method.