Dynamic Relaxation Method Research Articles

Non-Linear Analysis of Structures Utilizing Load-Discretization of Stiffness Matrix Method with Coordinate Update

This paper proposes a stiffness method based structural analysis algorithm for geometrically non-linear structures. In this study, the applied load on the joints has been discretized to a sequence of a few loadings applied. Each loading step produces incremental external nodal displacements, which are added to the corresponding coordinates to get a new geometrical shape of the structure. This process is iteratively repeated until the sum of the loading of all iterations matches the total initial applied loading. The size of the increments affects the technique’s accuracy, subsequently affecting the number of iterations. The configuration of non-linear geometrical structures is vital in the work; a slight change of the coordinates makes a considerable variation of nodal displacements. In this paper, three pin-jointed assemblies and a cantilever beam have been examined using the proposed technique; significantly reasonable outcomes emerged, compared to the non-linear approaches, such as Dynamic Relaxation Method (DRM) and Non-linear approach by Kwan. In a numerical sense, the dissimilarity between the results of the conventional Stiffness Matrix (SM) method and the non-linear method is about 228%, while the maximum discrepancy between the proposed approach and the non-linear methods is just above 15%.

A PD-FEM approach for fast solving static failure problems and its engineering application

A method of coupling of peridynamics (PD) and finite element method (FEM) for fast solving static failure problems is proposed. It is based on the force coupling method and adopts interface elements to couple the PD and FE subregions. By assembling the global matrix of the coupling model and using the adaptive dynamic relaxation (ADR) method to solve it, the static solution of elasticity and crack growth problems can be obtained quickly. By introducing short-range repulsive force and re-defining the coupling force, it is not only suitable for simulating tensile failure problems, but also suitable for compression failure problems. Through comparative analysis, the accuracy and effectiveness of this approach are verified. And then the method is successfully applied to simulate the excavation process of an underground cavern group, and the simulation of damage and failure process of engineering rock mass under complex tension–compression conditions has been realized. The stability characteristics of the surrounding rock are revealed by analyzing the damage and deformation evolution laws. It provides an effective numerical simulation method for predicting the excavation damage zone (EDZ) distribution characteristics and deformation law of the surrounding rock during the excavation of an underground cavern group. At the same time, compared with pure PD, the calculation efficiency is greatly improved, which is of great significance for the realization of engineering-scale simulation.

A Quasi-Brittle damage model in the framework of Bond-based Peridynamics with Adaptive Dynamic Relaxation method

Peridynamics (PD) is a new tool, based on the non-local theory for modelling fracture mechanics, where particles connected through physical interaction used to represent a domain. By using the PD theory, damage or crack in a material domain can be shown in much practical representation. This study compares between Prototype Microelastic Brittle (PMB) damage model and a new Quasi-Brittle (QBR) damage model in the framework of the Bond-based Peridynamics (BBPD) in terms of the damage plot. An in-house code using Matlab was developed for BBPD with inclusion of both damage models, and tested for a quasi-static problem with the implementation of Adaptive Dynamic Relaxation (ADR) method in the theory in order to get a faster steady state solutions. This paper is the first attempt to include ADR method in the framework of BBPD for QBR damage model. This paper analysed a numerical problem with the absence of failure and compared the displacement with literature result that used Finite Element Method (FEM). The obtained numerical results are in good agreement with the result from FEM. The same problem was used with the allowance of the failure to happen for both of the damage models; PMB and QBR, to observe the damage pattern between these two damage models. PMB damage model produced damage value of roughly twice compared to the damage value from QBR damage model. It is found that the QBR damage model with ADR under quasi-static loading significantly improves the prediction of the progressive failure process, and managed to model a more realistic damage model with respect to the PMB damage model.

Computational Wind Engineering of a Mast-supported Tensile Structure

The Computational Fluid Dynamics (CFD) analysis of wind actions on a cube and on the hyperbolic surface of a mast-supported tensile structure is presented. As standard in Computational Wind Engineering (CWE) the models were analyzed in incompressible fluid flow. For the CWE study, the Reynolds-Averaged Navier-Stokes (RANS) equations were solved with two of the most widely used turbulence models, Realizable k−ε and SST k−ω. The validation of the CWE models included the comparison of the numerically determined pressure coefficient fields with existing Wind Tunnel test (WT) results. The membrane forces and the displacements of the membrane roof were calculated by the Dynamic Relaxation Method (DRM), corresponding to the different (CWE and WT) pressure coefficient fields of the tensile roof. In addition to the pressure coefficients, the membrane force and the displacement fields and their significant values were also compared. It is presented that both turbulence models provide a suitable solution; the pressure coefficients and the membrane forces are also acceptable approaches to the experimental results.

Development of an adaptive explicit algorithm for static simulation using the vector form intrinsic finite element method

Abstract The vector form intrinsic finite element (VFIFE) method is a solution technique for nonlinear structural problems, which describes a continuous body using a set of particles instead of a mathematical function. Thus, a dynamic particle equation can be established by Newton's law of motion, and a viscous or kinetic damping can be introduced to obtain the steady state of the structure. This paper focuses mainly on the development of a stability condition regarding the explicit central difference method used in VFIFE to guarantee the system's convergence. The process is established and evaluated in combination with a dynamic relaxation method with kinetic damping and discrete control theory. Four numerical examples of structure nonlinear problems are used to verify the accuracy, stability and efficiency of the method.

Simulation of Deformations of Pre-Stressed Steel Trusses in Emergency Situations

Purpose of research is to make methodology and algorithm for finite element modeling in a single computational scheme of deformation of flat steel trusses, previously stressed using high-strength ropes, in accordance with the chronology of impacts on the object in the form of prestresses, normative loads and emergency destruction of one of the bearing elements.Methods. The solution of the problem is carried out in geometrically nonlinear staging using numerical integration based on Newmark approach with the construction of equilibrium equations of the finite element model of the structure in a deformed state at each integration step. Structural nonlinearity related to structural restructuring and consideration of ropes operation for tension only is described. The application of gravity forces of the carrier system, sequential introduction of tightening and their prestress, the application of payload and emergency impact in the form of instantaneous local destruction are traced. Before failure occurs, static loading condition is simulated using dynamic relaxation method. Methodology of accounting within numerical integration of emergency impact is formulated by application of dummy forces, values of which are calculated in excluded structural element before its destruction.Results. Performance of presented computational procedure is illustrated by the example of a flat steel truss calculation with a span (54 m), including two ropes. Object behavior is considered considering the break of one of the ropes subjected to preliminary stress. It was revealed that the investigated emergency does not lead to the destruction of the second rope and the occurrence of plastic deformations in the truss rods.Conclusion: Completed developments can be used to ensure the survivability of pre-stressed steel trusses under beyond design basis effects, leading to the destruction of individual structural elements.

High surface accuracy and pretension design for mesh antennas based on dynamic relaxation method

Cable truss structures are widely used in large satellite antennas with ultrahigh electromagnetic performance. The electromagnetic performance is directly determined by the mesh reflector's surface accuracy, which triggers research on the pretension design. In this paper, a novel pretension design method for both high surface accuracy and uniform tension distribution is proposed based on the dynamic relaxation (DR) method. In this approach, members of the cable network are regarded as string elements with 3 degrees of freedom to resist the axial tensile force, and members of the supporting truss are regarded as Euler-Bernoulli beams with 6 degrees of freedom to resist the axial and transverse forces and the bending and torsion moments. The DR method is adopted to find the static equilibrium of the whole mesh antenna, including both the cable mesh and the supporting truss. Thus, the deformation of the cables and rods and their interactive effects are fully captured. Then, to find an expected high-precision mesh surface with a uniform tension distribution, the inverse iteration algorithm (IIA) is adopted to adjust the original cable lengths of the cable network in each DR calculation cycle until the convergence criterion is met. Finally, this approach is effectively applied to the pretension design of symmetric and asymmetric mesh antennas, and the results indicate that the method is effective in obtaining both high surface accuracy and uniform tension distribution with high computational efficiency.

A new formulation for kinetic dynamic relaxation method based on the Lagrangian interpolation

This research presents a new algorithm for the Dynamic Relaxation (DR) method with kinetic damping by utilizing Lagrangian interpolation functions to derive new iterative kinetic DR equations. This procedure leads to an algorithm that, unlike the ubiquitous DR methods, does not require the calculation of nodal velocities, thereby marching forward only through successive nodal displacement. Also, the power iteration method is used to determine the optimal time step ratio. By utilizing this time step, the restarting analysis phase, which is one of the drawbacks of the kinetic DR strategy, is eliminated. In order to evaluate the performance and efficiency of the proposed method, some truss and frame structures with geometrically nonlinear behavior are analyzed. Results prove that the number of convergence iterations and analysis duration is reduced in the new kinetic DR scheme compared to other conventional DR methods.

Ratcheting Analysis of Steel Plate under Cycling Loading using Dynamic Relaxation Method Experimentally Validated

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.

Shape Generation of Bending-Active Braced Arches Based on Elastica Curves

The active bending concept provides a new perspective for a well-established structural type which has been used at various scales: the beam-string, consisting of a beam with an attached lower tie in tension and bracing struts balancing the forces between them. The idea goes back to the gutter beams of the Crystal Palace and has been widely used to the present for large-scale structures. When a slender beam is used, the tension in the tie induces curvature in the beam and increases the structural depth of the system; this opens new formal possibilities and results in lightweight structures at the expense of increasing their overall flexibility. Systems of this kind fall within the realm of active bending. We name them bending-active braced arches. The target shape of the system follows the tensioning process and needs to be pre-determined by means of a specific analysis, typically involving dynamic relaxation or optimization-based methods. In this paper, we propose an analytical method to generate shapes for bending-active braced arches. It assumes that each segment of the activated rod between deviators behaves as a segment of elastica; this enables the use of closed-form expressions to evaluate the shape and induced stress level in the active member. Taking advantage of this idea, it is possible to devise a procedure to carry out the shaping process in a sequential way by adequately choosing the design parameters. When alternative choices for the parameters are selected, the problem becomes non-linear and can be solved using suitable techniques. Some examples with different design constraints have been reproduced to illustrate the possibilities of the method.

Improving the convergence rate of kinetic dynamic relaxation method with new R-point

Improving the convergence rate of kinetic dynamic relaxation method with new R-point

Bending analysis of functionally graded nanoplates based on a higher-order shear deformation theory using dynamic relaxation method

In this paper, bending analysis of rectangular functionally graded nanoplates under a uniform transverse load has been considered based on the modified couple stress theory. Using Hamilton’s principle, governing equations are derived based on a higher-order shear deformation theory. The set of coupled equations are solved using the dynamic relaxation method combined with finite difference discretization technique for clamped and simply supported boundary conditions. Finally, the effects of aspect ratio, thickness-to-length ratio, boundary conditions, transverse load, and length scale parameter are studied in detail. The results showed that by rising the length scale-to-thickness ratio, the influence of the grading index on the deflection decreased.

Buckling analysis of CNTRC plates in the thermal environment based on combination of the incremental load technique and dynamic relaxation method

In this article, the buckling of composite plates reinforced with functionally graded distributions of single-walled carbon nanotubes along the thickness direction are studied in the thermal surroundings. The properties of carbon-nanotubes reinforced composite (CNTRC) plate were determined through the modified rule of mixture. All governing equations are obtained incrementally based on first-order shear deformation theory and von Karman nonlinear strain. The nonlinear equilibrium equations are solved through a combination of dynamic relaxation and incremental load methods. A detailed parametric study is carried out to investigate the buckling load of nanocomposite plates with different boundary conditions.

Nonlinear analysis of cable structures using the dynamic relaxation method

The analysis of cable structures is one of the most challenging problems for civil and mechanical engineers. Because they have highly nonlinear behavior, it is difficult to find solutions to these problems. Thus far, different assumptions and methods have been proposed to solve such structures. The dynamic relaxation method (DRM) is an explicit procedure for analyzing these types of structures. To utilize this scheme, investigators have suggested various stiffness matrices for a cable element. In this study, the efficiency and suitability of six well-known proposed matrices are assessed using the DRM. To achieve this goal, 16 numerical examples and two criteria, namely, the number of iterations and the analysis time, are employed. Based on a comprehensive comparison, the methods are ranked according to the two criteria. The numerical findings clearly reveal the best techniques. Moreover, a variety of benchmark problems are suggested by the authors for future studies of cable structures.

Lagrangian interpolation for kinetic dynamic relaxation method with the variable load factor

Common dynamic relaxation (DR) methods use central finite differences to derive fundamental equations. In this study, Lagrangian interpolation functions are utilized to formulate iterative DR relationships with kinetic damping. This procedure leads to an algorithm that, unlike the ubiquitous DR methods, does not require the calculation of nodal velocities thereby marching forward only through successive nodal displacement. Also, the power iteration method is used to determine the optimal time-step ratio. By utilizing this time-step, the restarting analysis phase which is one of the drawbacks of the kinetic DR strategy is eliminated. Furthermore, a new variable load factor is developed based on the concept of minimization of displacement increment function. To evaluate the performance and efficiency of the proposed method, some truss and frame structures with geometrically nonlinear behavior are analyzed. Results prove that the number of convergence iterations and analysis time are reduced in the new kinetic DR scheme in comparison with other conventional DR methods.

An axisymmetric ordinary state-based peridynamic model for thermal cracking of linear elastic solids

This paper proposes an axisymmetric ordinary state-based peridynamic model considering the thermal expansion effect for temperature-induced cracking of linear elastic solids. Both the force state and bond failure criterion of the model are built in axisymmetric domain. The direct calculation for peridynamic bond energy density of the axisymmetric model considering thermal terms is given. The adaptive dynamic relaxation method is adopted for the quasi-static peridynamic simulations. Validity and reliability of the proposed axisymmetric model are illustrated by numerical examples of cold-induced fracture in a pipe with an inner crack. It is shown that the displacement predicted by the proposed peridynamic model is in good agreement with the corresponding finite element method. The critical bond energy density criterion for thermal cracking is verified via comparisons with the critical fracture temperature loads computed from the virtual crack closure technique and released energy based on Griffith’s theory. The sensitivity of the present model on the peridynamic discretization is investigated and discussed in both the continuous deformation and cracking cases. The performance of the peridynamic model is further demonstrated for cold-induced fracture of double-layered pipes with inner cracks, and the cracking behaviors influenced by the interfacial toughness and number of inner cracks are captured, respectively. The present model is capable of solving axisymmetric thermal cracking problems of linear elastic solids effectively, and it can be applied in more advanced axisymmetric structures which involve more complicated cracking issues.

Numerical study on the rate-dependent behavior of geogrid reinforced sand retaining walls

Time effect on the deformation and strength characteristics of geogrid reinforced sand retaining wall has become an important issue in geotechnical and transportation engineering. Three physical model tests on geogrid reinforced sand retaining walls performed under various loading conditions were simulated to study their rate-dependent behaviors, using the presented nonlinear finite element method (FEM) analysis procedure. This FEM was based on the dynamic relaxation method and return mapping scheme, in which the combined effects of the rate-dependent behaviors of both the backfill soil and the geosynthetic reinforcement have been included. The rate-dependent behaviors of sands and geogrids should be attributed to the viscous property of materials, which can be described by the unified three-component elasto-viscoplastic constitutive model. By comparing the FEM simulations and the test results, it can be found that the present FEM was able to be successfully extended to the boundary value problems of geosynthetic reinforced soil retaining walls. The deformation and strength characteristics of the geogrid reinforced sand retaining walls can be well reproduced. Loading rate effect, the trends of jump in footing pressure upon the step-changes in the loading rate, occurred not only on sands and geogrids but also on geogrid reinforced sands retaining walls. The lateral earth pressure distributions against the back of retaining wall, the local tensile force in the geogrid arranged in the retaining wall and the local stresses beneath the footing under various loading conditions can also be predicted well in the FEM simulations.

Numerical stability analysis of arch-supported membrane roofs

Steel arches are widely used to support large span membrane roofs. In order to obtain a lightweight solution, these arches might be formed as a truss. Both the strength of the material and the critical load can limit the load-bearing capacity of the steel arch. Due to the out-of-plane slenderness of a large span arch, it is with utmost importance to utilize the membrane as a restraint against the buckling of the arch. Hence, the investigated structure is supported along a curve in an elastic way, but this support has a highly nonlinear response.The presented analysis of the roof is based on the Dynamic Relaxation Method (DRM). To model the membrane, an 8-node quadrilateral membrane element is used. The truss is modeled by straight bars. To obtain an appropriate model for the out-of-plane behavior of the truss, the flanges of the truss are modeled as sub-trusses: four flange 3D trusses are used to represent the proper stiffness of the real bending resistance of a flange. The truss model fits nicely into the highly nonlinear DRM algorithm of membrane structures.In the equilibrium state of the structure, the classical eigenvalue problem is solved to obtain the critical parameter of the actual load. The elastic and the geometric stiffness of the membrane and respectively the truss elements must be selected to withstand the destabilizing effects safely.

Analysis of wrinkled membrane structures using a Plane Stress projection procedure and the Dynamic Relaxation method

Deployable membrane structures such as inflatable stratospheric balloons are known to be sensitive to the occurrence of local instabilities such as wrinkles. The wrinkling phenomenon affects the working performances of the membrane and the occurrence of this phenomenon has to be controlled numerically in order to predict the best means of deployment during the inflation of aerospace balloons. To improve their performances and reliability during flight, the balloons also need to be sized appropriately without the stress field being disturbed by the wrinkles. These instabilities originate numerically from the membrane elements which have a negligible bending stiffness. Several wrinkling models have been presented in the literature in order to solve this problem. However, in most of these models an elastic law and the Green deformation approach have been used for this purpose.The new model called the PS-DPS model presented here for correcting the effects of wrinkles on membrane structures was implemented in the in-house finite element software Herezh++. A projection technique based on a Newton–Raphson method is used to control the stress plane and the in-plane contraction. Using the Almansi strain formulation, this model also accounts for the changes in membrane thickness liable to occur during simulations. The problems due to numerical instabilities are overcome by determining the equilibrium with the so-called Dynamic Relaxation method using kinetic damping procedures. Unlike other membrane models of literature, the PS-DPS model can be used with materials showing complex mechanical behaviour of all kinds. Several benchmark problems are analysed with the present wrinkling model and compared with results available in the literature, focusing first on an elastic law and then on a non-linear hyperelastic law. Lastly, the inflation of a square cushion test and that of a Zero Pressure Balloon are simulated with this non-linear law. The results obtained indicate that the PS-DPS model is valid and accurate to take into account the wrinkles in flexible structures with all these linear and non-linear behaviours.

Stress initialization methods for dynamic numerical simulation of rock mass with high in-situ stress

In the context of deep rock engineering, the in-situ stress state is of major importance as it plays an important role in rock dynamic response behavior. Thus, stress initialization becomes crucial and is the first step for the dynamic response simulation of rock mass in a high in-situ stress field. In this paper, stress initialization methods, including their principles and operating procedures for reproducing steady in-situ stress state in LS-DYNA, are first introduced. Then the most popular four methods, i.e., explicit dynamic relaxation (DR) method, implicit-explicit sequence method, Dynain file method and quasi-static method, are exemplified through a case analysis by using the RHT and plastic hardening rock material models to simulate rock blasting under in-situ stress condition. Based on the simulations, it is concluded that the stress initialization results obtained by implicit-explicit sequence method and dynain file method are closely related to the rock material model, and the explicit DR method has an obvious advantage in solution time when compared to other methods. Besides that, it is recommended to adopt two separate analyses for the whole numerical simulation of rock mass under the combined action of in-situ stress and dynamic disturbance.