DYNAMIC RELAXATION METHOD is an iterative procedure for solving the simultaneous system of equations. This technique is used in the static and DYNAMIC nonlinear structural analysis. One of the most important parameters in this approach is fictitious damping factor. If this factor is selected more accurately, convergence rate will rise. In this paper, inverse vector iteration METHOD is utilized to find the damping factor in the DYNAMIC RELAXATION iterations, and a new formulation is proposed. The geometric nonlinear analysis of several plane and space trusses and also structural frames are performed using the suggested METHOD. The numerical results indicate that the convergence rate improves compared with the conventional DYNAMIC RELAXATION procedure so that the number of iterations and the analysis time decrease significantly. Consequently, the authors' technique makes the solving process faster and more capable.

DYNAMIC RELAXATION is an iterative technique, which is used as an equation solver. In this paper, a new time step will be formulated for DYNAMIC RELAXATION METHOD. The suggested technique is based on the minimization of residual force in each iteration. Mathematical theories and numerical examples are used to verify efficiency of the formulation. By using optimum time step, mathematical convergence rank of DR algorithm will be infinite and two in linear and nonlinear analyses, respectively. To investigate the capability of the proposed formulation, isotropic plates and frame structure by linear and geometrical nonlinear behaviors are analyzed. This study shows that optimum time step reduces number of convergence iterations.Therefore, the cost and the computational time will be reduced. As a result, the suggested formulation for optimum time step has higher mathematical and numerical efficiency than other common METHODs, such as constant time step. Therefore, the convergence rate of DR iterations will considerably improve.

The common DYNAMIC RELAXATION algorithm (DR) does not have the ability to trace the static path. In these techniques, the jumps occur at the limit points. A variable load factor is used to fix this defect. Here, a new procedure is suggested to calculate the load factor. The authors’ relationship is achieved by minimizing external work and residual energy, simultaneously. It should be stated that the proposed load factor depends only on the DR artificial parameters. To show the ability of the new formulation, several truss and shell structures with nonlinear geometrically behavior are analyzed. All used METHODs are ranked by the number of iterations, numbers of convergence points and total duration analysis. Numerical solutions show the high efficiency of the new METHOD. In other words, the authors' technique, in addition to good accuracy, has higher convergence rate, in comparison to the other strategies. On the other hand, the time duration of the proposed METHOD to trace the static paths has been reduced appropriately compared to other techniques.

In this paper a new time step will be formulated for DYNAMIC RELAXATION METHOD. For this purpose, residual force is minimized in each iteration. Calculation of the convergence rate shows the advantage of the recommended technique. For illustrating usability of this METHOD, some plane and space trusses have been analyzed. In this paper, geometrical non-linearity is considered. Moreover, proposed METHOD is widely examined by author's computer program and some examples will be given.

In this paper, new algorithm is proposed for fictitious mass of DYNAMIC RELAXATION (DR) METHOD with viscous damping. First, the incremental equations are derived for DR procedure. By using the transformed Gershgö rin theory, new boundaries are achieved for fictitious mass. This formulation leads to a new algorithm for viscous DR METHOD. For evaluating the efficiency of the proposed METHOD, some 2D and 3D truss and frame structures are analyzed by elastic linear and geometrically nonlinear behaviors. Results show that the proposed scheme for fictitious mass improves the convergence rate of DR METHOD so that the suggested algorithm presents the structural response with less iteration in comparison with other common DR techniques.

In this paper, a new algorithm is presented for DYNAMIC RELAXATION (DR) METHOD with kinetic damping. In the kinetic DYNAMIC RELAXATION algorithm, some successive points with maximum kinetic energy are traced in the course of numerical fictitious time integration. In the absence of damping forces, the points with maximum kinetic energy are close to the static equilibrium position of structure. This paper deals with a new formulation for kinetic DR METHOD. For this purpose, Lagrangian interpolation functions were utilized to derive iterative DYNAMIC RELAXATION equations. In the Lagrangian interpolation functions, new estimation of structural displacement vector was obtained based on previous estimations of displacement vector. Therefore, this procedure leads to adopting a trial and error METHOD. On the other hand, this procedure leads to a new formulation that, unlike the ubiquitous DR METHODs, does not require the calculation of nodal velocities, thereby marching forward only through successive nodal displacement. Elimination the nodal velocities from DYNAMIC RELAXATION process increases the simplicity of DR algorithm. Moreover, the requirement analysis memory is reduced using the suggested technique so that velocity vectors would not be stored in the program memory. Also, the power iteration METHOD was used to determine the optimal time step ratio. By utilizing this time step, the restarting analysis phase, considered as one of the drawbacks of the common kinetic DR strategies, is eliminated. To evaluate the performance and efficiency of the proposed METHOD, several truss and frame structures were analyzed. These structures had geometrically nonlinear behavior (Large Deflection). Results of these analyses were also compared with those of other conventional DYNAMIC RELAXATION METHODs. Numerical results showed that the convergence rate of the proposed kinetic DR technique was higher than that of common DR algorithms. In other words, the number of the required DR iterations for convergence was reduced using the proposed DR algorithm in comparison with other DR schemes. Moreover, the analysis time of the proposed METHOD was shorter than that of other common techniques.

The present study aimed to introduce a numerical METHOD to study ratcheting strains of rectangular plates. A new numerical analysis was conducted by development of DYNAMIC RELAXATION METHOD combined with MATLAB software to evaluate the ratcheting behavior of the thin steel plate under mentioned loading condition. In order to verify the results, experimental tests were performed under stress-controlled conditions by a zwick/roell amsler HB100 machine and bending ratcheting of CK45 steel plate at room temperature was studied. Under stress-controlled conditions with non-zero mean stress, ratcheting behavior occurred on thin plate. Moreover, a finite element analysis was carried out by Abaqus using nonlinear isotropic/kinematic (combined) hardening model. The results showed that the rate of ratcheting strain decreased with an increase in cycle number. It was found that the hysteresis loops were wider in experimental METHOD than those of other METHODs because of more energy dissipation. The numerical results are in a good agreement with the simulation and experimental data. Comparison of errors between these METHODs obviously demonstrate high accuracy of the new introduced METHOD.

In this paper, nonlinear static analysis of moderately thick plate made of functionally graded materials subjected to mechanical transverse loading is carried out using DYNAMIC RELAXATION METHOD. Mindlin first order shear deformation theory is employed to consider thick plate. Discretized equations are extracted for geometrically nonlinear behavior analysis. Loading Conditions and boundary conditions of the plate are uniformly distributed transverse load and simply supported at the four edges of the thick plate, respectively. In order to generalize the obtained results, the equations are solved by applying DYNAMIC RELAXATION METHOD based on central finite deference discretization in the non-dimensional form. The effects of problem parameters such as gradient constant of the functionally graded material and the side to thickness ratio of plate on the results are investigated. According to the obtained results, the need of including elastic large deflection and applying the theory which considers the effects of plate thickness on the plate bending response and also finally the need of employing DYNAMIC RELAXATION solution METHOD despite the non-linear terms resulted from large deflection of the functionally graded thick plate are discussed.

The main purpose of this study is to investigate nonlinear buckling analysis of Functionally Graded (FG) cylindrical shells under uniform axial compressive loads by DYNAMIC RELAXATION (DR) METHOD. The mechanical properties of shell vary continuously throughout the thickness direction according to the power-law, exponential function and the Mori-Tanaka distribution. The poisson’s ratio of the FG cylindrical shell is constant for power-law and exponential function. But in the Mori-Tanaka distribution variations of poisson's ratio is determined as a function of the thickness direction. The incremental form of nonlinear formulations are based on first order shear deformation theory (FSDT) and large deflection von Karman equations. The DR METHOD combined with the finite difference discretization technique is employed to solve the equilibrium equations. Some comparison study is carried out to compare the current solution with the results reported in the literature and the ones obtained by the Abaqus finite element software for the isotropic cylindrical shells. Finally, numerical results are presented for critical buckling load with various boundary conditions, grading indices, radius -to- thickness ratio, length -to- radius ratio and variation of poisson’s ratio.