Element Nodal Forces Research Articles

An Efficient and Robust Nonlinear Dynamic Analysis Method for Framed Structures Using the Rigid Body Rule

In this paper, by implanting the rigid body rule (RBR)-based strategy for static nonlinear problems into the implicit direct integration procedure, an efficient and robustness nonlinear dynamic analysis method for the response of framed structures with large deflections and rotations is proposed. The implicit integration method proposed by Newmark is improved by inserting an intermediate time into the time step and by adding the 3-point backward difference in the second substep so as to preserve the momentum conservation and to maintain the stability of the direct integration method. To solve the equivalent incremental equations of motion, the RBR is built in to deal with the rigid rotations and the resulting additional nodal forces of element. During the increment-iterative procedure, the use of RBR-qualified geometric stiffness in the predictor reduces the numbers of iterations, while the elastic stiffness alone in the corrector to update the element nodal forces makes the computation efficiency and convergence with no virtual forces caused by the ill geometric stiffness. The proposed algorithm is advanced in the applications of several framed structures with highly nonlinear behavior in the dynamic response by its simplicity, efficient and robustness.

Response of shear critical reinforced concrete frames and walls under monotonic loading

This paper focuses on a novel finite element formulation which can predict the bending moment-shear force-axial force interaction of reinforced concrete frames and walls, and validate it against 170 experiments available in literature. This distributed plasticity element is established on force-based finite element method, where the relationship between element nodal forces and section forces are exactly known. Hence, element discretization is nonessential when modelling frames using this formulation, reducing the number of degrees of freedom in the numerical model compared to displacement-based formulations. The computations are carried out at four hierarchical levels, namely structure, element, section and fibre. There are two nested iterative procedures at the structure level and the section level. In the existing formulation, these iterative procedures are computationally demanding due to use of initial stiffness matrices. Furthermore, it uses Modified Compression Field Theory at the fibre level, which has inherent drawbacks compared to its more evolved version, the Disturbed stress Field Model. The current study refines the iterative procedures at structure and section levels to fully operate with tangent stiffness matrices to improve the speed of convergence. In addition, the Modified Compression Field Theory is replaced with the Disturbed stress Field Model at the fibre level to compute fibre resisting force for a given fibre deformation, accounting for both averaged behaviour and local crack slip. The novel element is validated by comparing the predicted results with experimental results of 170 tests found in the literature. It is shown that the novel element predicts the load carrying capacity well with an average experimental-to-predicted load carrying capacity ratio of 0.99 and a coefficient of variation of 12.8%. Furthermore, the element can be used to discuss the different failure mechanisms of reinforced concrete frame elements.

Geometrically exact beam element with rational shear stress distribution for nonlinear analysis of FG curved beams

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.

Volumetric discretization element with invariant displacements approximation in the SSS numerical analysis problems for the building structures

The article describes the sought quantities invariant approximation method in relation to the volumetric discretization element in the form of a parallelepiped with the main nodes located at its vertices. The displacement vector components and their partial derivatives of the first order were selected as the sought-for unknowns of the volume discretization element. An approximating expression when implementing the invariant approximation technique is written for the displacement vector of an element inner zone point and not for components of this vector isolated from each other. The approximating expressions obtained on the sought quantities’ invariant approximation basis contain the curvilinear coordinates system parameters used. The discretization volume element compatibility is improved due to the use of Lagrange multipliers in additional nodes located in the middle of the parallelepiped bases’ edges. The Lagrange correction factors ensure the displacement vectors normal components derivatives’ equality along the normals to the element base edges’ midpoints. The expanded stiffness matrix and the column of the volumetric discretization element nodal forces are formed by minimizing the conditional Lagrange functional.

A three-dimensional dual-scale coupled approach and its application in geotechnics

To study the microscopic and macroscopic behaviour of geotechnical engineering materials, attention is increasingly being paid to the dual-scale coupled computation of continuum and discrete models. However, two-dimensional dual-scale coupling cannot reveal the actual behaviour of materials. Therefore, a three-dimensional dual-scale coupled approach to accomplish the information transmission between discrete-element model and finite-difference model was deduced, and a corresponding coupled program was developed. The point of application of the resultant force on the interface in the discrete-element model was sought by introducing the principle of least squares, and the element nodal forces at the interface in the continuum model were obtained based on the formulation of load transfer from the literature. The velocities acting on the interface in the discrete-element model were calculated by bilinear interpolation according to the nodal velocities of the interface in the continuum model, which achieved the velocity transmission from the continuous zone to the discrete-element zone. The coupling was accomplished through the velocity and force transmissions at the interface between discrete-element model and finite-difference model. Finally, the algorithm and program were validated through an example of dynamic compaction.

The Finite Element Desired Quantities Invariant Approximation Method in the Thin Shells Calculation Based on the Timoshenko Hypothesis

The displacement vector components vector (invariant) approximation implementation and the initial inclination angles by the hypothesis of S. P. Tymoshenko in obtaining the thin shell quadrangular finite element nodal forces stiffness matrix and the column is shown.

A buckling and postbuckling analysis of axially loaded thin-walled beams with point-symmetric open section using corotational finite element formulation

The axially loaded thin-walled beams with point symmetric open section are studied using a corotational finite element formulation. The kinematics of the beam element is described in the current element coordinate system. The element nodal forces are derived using the virtual work principle, and consistent second order linearization of the fully geometrically non-linear beam theory. Different axial load systems with the same centric resultant axial force but different resultant bimoments are considered. Numerical examples studied show that the effects of bimoments on the buckling and postbuckling behavior of point symmetric open section beams are remarked.

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.

Damage identification of truss structures based on force method and free vibration analysis

The improved structural analysis is presented to determine the sites and extents of multiple damages of truss structures. The algorithm uses the mathematical programming of the finite element technique based on the force method and the free vibration analysis. The equilibrium matrix is generated through the finite element analysis, while the compatibility matrix is obtained directly using the force–displacement relations and the singular value decomposition technique. The governing equations with the element nodal forces as variables are presented by combining the equilibrium and compatibility matrices with the force–displacement relations. Next, the genetic algorithm is used to properly identify the sites and extents of multiple damage cases in truss structures. In order to verify the accuracy and superiority of the proposed damage detection technique, numerical solutions are presented for the planar and space truss models. The numerical results indicate that the combination of the force method and genetic algorithm can provide a reliable tool to accurately and efficiently identify the multiple damages of truss structures.

A Simple Co-Rotational Finite Planar Beam Element of Geometrical and Material Nonlinearity

A simple geometrical and material nonlinear co-rotational planar beam element of field consistency is proposed. Herein the element which produces a local stiffness matrix of 3 by 3 other than 6 by 6 is developed. Material nonlinearity is taken into account on the base of yield function of element internal forces. By applying static equilibrium relationship of classic beam theory for the transferring of local element nodal force to global element nodal force, a new transformation matrix different from the nodal displacement transformation matrix is established. Although this results in an asymmetric global tangential stiffness matrix, the new transformation is simpler, and gives rise to field consistency and makes it possible to compute very large beam deflection without remeshing of the deformed structure. Computations of numerical example indicates that formulations for the nonlinear beam element are of validation and high efficiency

Nonlinear analysis of cable-supported structures with a spatial catenary cable element

This paper presents a spatial catenary cable element for the nonlinear analysis of cablesupported structures. An incremental-iterative solution based on the Newton-Raphson method is adopted for solving the equilibrium equation. As a result, the element stiffness matrix and nodal forces are determined, wherein the effect of self-weight and pretension are taken into account. In the case of the initial cable tension is given, an algorithm for form-finding of cable-supported structures is proposed to determine precisely the unstressed length of the cables. Several classical numerical examples are solved and compared with the other available numerical methods or experiment tests showing the accuracy and efficiency of the present elements.

Investigation on steady state deformation and free vibration of a rotating inclined Euler beam

The steady state deformation and infinitesimal free vibration around the steady state deformation of a rotating inclined Euler beam at constant angular velocity are investigated by the corotational finite element method combined with floating frame method. The element nodal forces are derived using the consistent second order linearization of the nonlinear beam theory, the d'Alembert principle and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The rotating element coordinates rotate about the hub axis at the angular speed of the hub. The equations of motion of the system are defined in terms of an inertia global coordinate system, which is coincident with a rotating global coordinate system rigidly tied to the rotating hub. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady state deformation and natural frequency of the rotating inclined beam.

Wave propagating across atomic–continuum interface

Motivated by eliminating the spurious wave reflection at the interface between atomic and continuum domains in the multiscale modeling and simulation, we present a novel numerical approach for the atomistic field theory without any special treatment at the interfaces. This novel approach abandons the method to calculate the finite element nodal forces in the continuum domain based on the strain–stress relation. Instead, it employs the interatomic forces as in molecular dynamics simulation. This nonlocal procedure overcomes the force mismatch between atomic and continuum domains and enables a seamless scale transition between fully atomistic resolution and continuum description. Meanwhile, to balance the trade-off between computational efficiency and numerical accuracy, we adopt a cluster-based sampling scheme which allows a group of unit cells around every nodal point to contribute to the calculation of nodal forces. Numerical results show that this novel numerical procedure drastically reduces the wave reflection at the atomic–continuum interface.

The use of nodal point forces to improve element stresses

We present in this paper a novel approach to stress calculations in finite element analysis. Rather than using the stress assumption employed in establishing the stiffness matrix, the element nodal point forces are used, in a simple way, to enhance the finite element stress predictions at a low computational cost. While this paper focuses on the improvement of the stress accuracy, the proposed procedure can also be used as a basis for error estimation. Moreover, the procedure is quite general, and has the potential for many applications in finite element analysis.

A Corotational Finite Element Method Combined with Floating Frame Method for Large Steady‐State Deformation and Free Vibration Analysis of a Rotating‐Inclined Beam

A corotational finite element method combined with floating frame method and a numerical procedure is proposed to investigate large steady‐state deformation and infinitesimal‐free vibrationaround the steady‐state deformation of a rotating‐inclined Euler beam at constant angular velocity. The element nodal forces are derived using the consistent second‐order linearization of the nonlinear beam theory, the d′Alembert principle, and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The governing equations for linear vibration are obtained by the first‐order Taylor series expansion of the equation of motion at the position of steady‐state deformation. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady‐state deformation and natural frequency of the rotating beam with different inclined angle, angular velocities, radius of the hub, and slenderness ratios.

Nonlinear static and dynamic analysis of cable structures

This paper presents a catenary cable element for the nonlinear analysis of cable structures subjected to static and dynamic loadings. The element stiffness matrix and element nodal forces, which account for self-weight and pretension effects, are derived based on exact analytical expressions of elastic catenary. Cables encountered in cable networks as well as cable-supported bridges can be modeled using the proposed element. An incremental-iterative solution based on the Newmark direct integration method and the Newton–Raphson method is adopted for solving the nonlinear equation of motion. The accuracy and reliability of the present element are verified by comparing the predictions with those generated by commercial finite element package SAP2000, and the results given by other authors using different analytical or numerical approaches.

The geometric stiffness of triangular composite-materials shell elements

This paper is concerned with the development of the geometric stiffness matrix for Newton type large rotation analysis of composite thin shell structures. The geometric stiffness matrix is derived from load perturbation of the discrete equilibrium equations of a given linear finite element formulation. The geometric stiffness matrix is extracted from the gradient, in global coordinates, of the element nodal force vector when stresses are kept fixed. In order to overcome the difficulties in taking derivatives of the rotation matrix with respect to the nodal coordinates, gradient evaluations are performed in the local coordinate system to result in an in-plane geometric stiffness matrix. An out-of-plane geometric stiffness matrix is then introduced to account for the effect of rigid body rotations on member forces. A unique procedure is used for the removal of rigid body displacements and rotations that enables stress recovery via linear, kinematic and constitutive, relationships. The geometric stiffness matrix derived was used to study several examples whose results compare well with the literature.

Co-rotational finite element formulation for thin-walled beams with generic open section

A consistent co-rotational total Lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of thin-walled beams with generic open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross-sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element. The element nodal forces are derived using the virtual work principle. The virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces. A procedure is proposed to determine the virtual rigid body motion. The way used to determine the element coordinate system and element nodal deformations corresponding to the virtual nodal displacements and that corresponding to the incremental nodal displacement are consistent. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. In the derivation of the element tangent stiffness matrix, the change of element nodal forces induced by the element rigid body rotations should be considered for the present method. Thus, a stability matrix is included in the element tangent stiffness matrix. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. The effect of the terms in the element nodal force and tangent stiffness matrix, which will converge to zero with the decrease of element size, on the convergence rate of solution and accuracy for the buckling load and nonlinear behavior of three dimensional beam structures are also investigated through numerical examples.

Object-oriented finite element-based design and progressive steel weight minimization

An object-oriented environment for reinforcement dimensioning of two- and three-dimensional concrete panel structures is suggested and a new finite element design (FED) object is introduced. In the framework of an object-oriented environment for finite element analysis, the FE object commonly providing the stiffness matrix for analysis is extended to the FED object used for design. The FED object provides the finite element yield conditions in terms of the element nodal forces as opposed to conventional yield conditions formulated for stresses at an infinitesimal point. It can be used for any kind of material provided the corresponding resistance values are used in the yield condition. Class definitions and methods for the FED object are defined. Other major design classes to be used in an object-oriented finite element-based design environment of concrete panel structures are presented. Further, a progressive linear programming approach is suggested in which the tableau is progressively expanded to include new optimization variable columns and constraint rows while most of the computation is performed outside of the tableau. This method is applied to the reinforcement minimization. Possibilities of parallelization of the method on a network of workstations are explained.

Co-rotational formulation for geometric nonlinear analysis of doubly symmetric thin-walled beams

A doubly symmetric thin-walled beam element with open section is derived using co-rotational (CR) total Lagrangian (TL) formulation. The effects of deformation-dependent third-order terms of element nodal forces on the buckling load and post-buckling behavior are investigated. All coupling among bending, twisting, and stretching deformations for beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, all third-order terms of nodal forces, which are relevant to the twist rate, rate of twist rate and curvature of the beam axis are also considered. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A parabolic interpolation method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the effect of third-order terms of element nodal forces on the buckling load and post-buckling behavior of doubly symmetric thin-walled beams.