Structural Equilibrium Equations Research Articles

System reduction-based approximate reanalysis method for statically indeterminate structures with high-rank modification

Efficient structural reanalysis for high-rank modification plays an important role in engineering computations which require repeated evaluations of structural responses, such as structural optimization and probabilistic analysis. To improve the efficiency of engineering computations, a novel approximate static reanalysis method based on system reduction and iterative solution is proposed for statically indeterminate structures with high-rank modification. In this approach, a statically indeterminate structure is divided into the basis system and the additional components. Subsequently, the structural equilibrium equations are rewritten as an equation system with the stiffness matrix of the basis system and the pseudo forces derived from the additional elements. With the introduction of the spectral decomposition, a reduced equation system with the element forces of the additional elements as unknowns is established. The approximate solutions of the modified structure can then be obtained by solving the reduced equation system through a pre-conditioned iterative solution algorithm. The computational costs of the proposed method are compared with those of two other reanalysis methods, and numerical examples including static reanalysis and static nonlinear analysis are presented. The results demonstrate that the proposed method has excellent computational performance for both structures with homogeneous material and structures composed of functionally graded beams. Furthermore, the superiority of the proposed method suggests that the combination of system reduction and pre-conditioned iterative solution technology is an effective approach to develop high-performance reanalysis methods.

Algebraic multigrid methods for saddle point systems arising from mortar contact formulations

AbstractIn this article, a fully aggregation‐based algebraic multigrid strategy is developed for nonlinear contact problems of saddle point type using a mortar finite element approach. While the idea of extending multigrid methods to saddle point systems can already be found, for example, in the context of Stokes and Oseen equations in literature, the main contributions of this work are (i) the development and open‐source implementation of an interface aggregation strategy specifically suited for generating Lagrange multiplier aggregates that are required for coupling structural equilibrium equations with contact constraints and (ii) a review of saddle point smoothers in the context of constrained interface problems. The new interface aggregation strategy perfectly fits into an aggregation‐based multigrid framework and can easily be combined with segregated transfer operators, which allow to preserve the saddle point structure on the coarse levels. Further analysis provides insight into saddle point smoothers applied to contact problems, while numerical experiments illustrate the robustness of the new method. We have implemented the proposed algorithm within the MueLu package of the open‐source Trilinos project. Numerical examples demonstrate the robustness of the proposed method in complex dynamic contact problems as well as its scalability up to 23.9 million unknowns on 480 MPI ranks.

Numerical deflection and stress prediction of cutout borne damaged composite flat/curved panel structure

This article presents the deflection and stress responses of the concentric cutout impregnated composite curved/flat panel structure, including the variable cutout profiles (square, circular, and elliptical). The model can compute the structural configuration due to the curvature parameter change, i.e., plate, spherical, cylindrical, elliptical, and hyperboloid. The damaged composite structural model is derived using a cubic-order of displacement field kinematics to maintain the stress–strain continuity. The isoparametric type C0 continuous finite element (FE) modeling steps are adopted to solve the structural equilibrium equation of the cutout abided layered composite panels subjected to uniform mechanical loading via an in-house derived MATLAB code. The model accuracy has been established by comparing the deflections and the stresses with the previously published results. Finally, the layered composite shell panel's critical behavior under the static loading has been examined. Different types of numerical problems are solved by varying the associated design constraint parameter, including the cutout relevant factors (shape and size), to show the proposed model applicability.

Designing Self Supported SLM Structures via Topology Optimization

The potential of Additive Manufacturing (AM) is high, with a whole new set of manufactured parts with unseen complexity being offered. However, the process has limitations, and for the sake of economic competitiveness, these should also be considered. Therefore, a computational methodology, capable of including the referenced limitations and providing initial solid designs for Selective Laser Melting (SLM) is the subject of the present work. The combination of Topology Optimization (TO) with the simplified fabrication model is the selected methodology. Its formulation, implementation, and integration on the classic TO algorithm is briefly discussed, being capable of addressing the minimum feature size and the overhang constraint limitations. Moreover, the performance and numerical stability of the methodology is evaluated, and numerical variables, such as the accuracy of structural equilibrium equations and the material interpolation model, are considered. A comparative study between these variables is presented. The paper then proposes an enhanced version of the selected methodology, with a better convergence towards a discrete solution.

Finite element solution of stress and flexural strength of functionally graded doubly curved sandwich shell panel

The finite solutions of deflection and the corresponding in-plane stress values of the graded sandwich shallow shell structure are computed in this research article via a higher-order polynomial shear deformation kinematics. The shell structural equilibrium equation is derived using the variational principle in association with a nine noded isoprametric element (nine degrees of freedom per node). The deflection values are computed via an own customized MATLAB code including the current formulation. The stability of the current finite element solutions including their accuracies have been demonstrated by solving different kind of numerical examples. Additionally, a few numerical experimentations have been conducted to show the influence of different design input parameters (geometrical and material) on the flexural strength of the graded sandwich shell panel including the geometrical configurations.

Theoretical deflection analysis of multi-walled carbon nanotube reinforced sandwich panel and experimental verification

In this article, the influence of the multi-walled carbon nanotube reinforcement on the stiffness of sandwich curved panel is examined theoretically via deflection analysis and compared with own experimental data for the verification of accuracy. The nanotube-reinforced sandwich structural panel model is derived theoretically using the higher-order polynomial functions and displacement finite element steps adopted for the numerical solution purpose. The structural stiffness values are measured from the deflection resistance of the theoretical structural model by computing the structural equilibrium equation with the help of an own customized MATLAB code. Firstly, the numerical solution accuracy and the corresponding reliability of the present solutions are cross-checked through the element sensitivity including the comparison test. Further, the multi-walled carbon nanotube reinforced sandwich plate is fabricated for the required experimentation including the mechanical as well as the material characterization. Finally, the validity of theoretically predicted deflection data of sandwich structure demonstrated by comparing with the own experimental results. In addition, the effect of various design parameters on the stiffness behavior of the own fabricated sandwich construction is computed using the proposed theoretical model and discussed in detail.

Design of a Bi-stable Airfoil with Tailored Snap-through Response Using Topology Optimization

This study aims to harness the geometric non-linearity of structures to design a novel camber morphing mechanism for a bi-stable airfoil using topology optimization. The goal is to use snap-through instabilities to actuate and maintain the shape of the morphing airfoil. Topology optimization has been used to distribute material over the design domain and to tailor the nonlinear response of the baseline structure to achieve the desired bi-stable behavior. The large scale deformation undergone by the structure is modeled using a hyperelastic material model. The non-linear structural equilibrium equations are solved using arc-length and displacement-controlled Newton–Raphson (NR) analysis. Isoparamteric finite element evaluation is used for analyzing kinematic and deformation characteristics of the structure. The optimization problem is solved using a computationally efficient nonlinear optimization algorithm, the Method of Moving Asymptotes (MMA), with a Solid Isotropic Material Penalization (SIMP) scheme. The gradient information required for the optimization has been evaluated using an adjoint sensitivity formulation. Two different design domains, one with a structured quadrilateral mesh and other with an unstructured triangular mesh, are investigated and compared. The effect of different optimization parameters on the final optimized structure and its behavior has also been analyzed. The final result is a novel camber morphing mechanism without the disadvantages of increased weight and higher maintenance costs associated with conventional actuation mechanisms.

Frequency and Deflection Responses of Shear Deformable Skew Sandwich Curved Shell Panel: A Finite Element Approach

The eigenfrequency and transverse deflection values of the sandwich shell panel structure including the skew angle effect are examined numerically in this article. The sandwich shell panel is modelled via the higher-order displacement polynomial functions in the framework of the equivalent single-layer theory including the thickness stretching term effect. The numerical solutions are obtained via an own finite element code (MATLAB platform) in association with the derived mathematical model. The variational technique has been adopted to solve the sandwich structural equilibrium equation and the eigenvalue parameter under the influence of mechanical loading. The solution stability including the validity of the current numerical solutions has been verified via solving the adequate number of examples as same as the available published data. Finally, the current model is extended further to explore the probable effect of one or more parameters (geometrical, material and end constraint) on the final structural performances (frequency, deflection and stresses) including the fibre skew angle.

Nonlinear finite element solutions of thermoelastic deflection and stress responses of internally damaged curved panel structure

• A novel numerical model is developed to study the deflection and stress values of the pre-damaged composite structure. • The internal damage has been modeled via sub-laminate approach including two kinematic theories. • The nonlinear FEM solutions are obtained under the combined thermomechanical loading. • The numerical solutions (deflection and stress) are evaluated for different panel geometries via the derived models. Thermoelastic deflection and corresponding stresses of the pre-damaged layered panel structure are investigated numerically in this article including the large deformation kinematics under the linearly varying temperature field. The composite structural deformation kinematics is derived using two different polynomial type of kinematic theories including the through-thickness stretching effect. The inter-laminar separation between the adjacent layers is incurred via the sub-laminate approach and Green–Lagrange strain to count the total structural deformation. Also, the intermittent displacement continuity conditions are imposed in the current mathematical model to establish the displacement continuity between the separated layers. The variational principle is adopted for the evaluation of the nonlinear structural equilibrium equations and solved via total Lagrangian approach . The convergence and the corresponding validity of the currently derived nonlinear finite element solutions are checked by solving different sets of numerical examples. Additionally, the comprehensive inferences are drawn from various numerical examples for the well-defined important input parameter including the size, position, and location of delamination .

Comparative analysis of three-dimensional frames by dynamic relaxation methods

ABSTRACTAnalysis of three-dimensional frames is a complex process. Each node of these structures has six degrees of freedom, which results in a large set of governing equations. Investigators have utilized the computer power to extend the application of numerical approaches. One of these methods is named dynamic relaxation technique. This strategy explicitly solves the simultaneous system of equations. In this scheme, the static structural equilibrium equations are converted into a dynamic one by adding fictitious mass and damping. To perform nonlinear geometric analysis of 3D frames, 12 classical DR methods are exploited. Previously, the abilities of these approaches in analyzing these structures have not been compared. In each technique, time step, mass, and damping matrices are obtained based on other researchers' works. The structural behavior is assessed with and without considering the shear deformations. For both cases, the load–displacement curves are depicted. In this article, 11 different frames are analyzed. Since a few nonlinear solutions of 3D frames are available, these structures can be used as a benchmark in the future studies. Finally, the used algorithms are graded based on their required number of iterations and analysis time. Findings prove that the Qiang, Rezaiee-Pajand and Sarafrzi techniques perform more successfully in comparison to the other schemes.

Three-dimensional structural topology optimization of aerial vehicles under aerodynamic loads

A previously developed density distribution-based structural topology optimization algorithm coupled with a Computational Fluid Dynamics (CFD) solver for aerodynamic force predictions is extended to solve large-scale problems to reveal inner structural details of a wing wholly rather than some specific regions. Resorting to an iterative conjugate gradient algorithm for the solution of the structural equilibrium equations needed at each step of the topology optimizations allowed the solution of larger size problems, which could not be handled previously with a direct equation solver. Both the topology optimization and CFD codes are parallelized to obtain faster solutions. Because of the complexity of the computed aerodynamic loads, a case study involving optimization of the inner structure of the wing of an unmanned aerial vehicle (UAV) led to topologies, which could not be obtained by intuition alone. Post-processing features specifically tailored for visualizing computed topologies proved to be good design tools in the hands of designers for identifying complex structural components.

Aerodynamic unstable critical wind velocity for three-dimensional open cable-membrane structures

The aerodynamic unstable critical wind velocity for three-dimensional open cable-membrane structures is investigated. The geometric nonlinearity is introduced into the dynamic equilibrium equations of structures. The disturbances on the structural surface caused by the air flow are simulated by a vortex layer with infinite thickness in the structures. The unsteady Bernoulli equation and the circulation theorem are applied in order to express the aerodynamic pressure as the function of the vortex density. The vortex density is then obtained with the vortex lattice method considering the coupling boundary condition. From the analytical expressions of the unstable critical wind velocities, numerical results and some useful conclusions are obtained. It is found that the initial curvature of open cable-membrane structures has clear influence on the critical wind velocities of the structures.

A Hybrid State Vector Approach to Aeroelastic Analysis

A computational technique has been developed for performing preliminary design aeroelastic analyses of large-aspect-ratio lifting surfaces. This technique, applicable to both fixed and rotating wing configurations, is based upon a formulation of the structural equilibrium equations in terms of a hybrid state vector containing generalized force and displacement variables. An integrating matrix is employed to solve these equations for divergence and flutter eigenvalues and steady aeroelastic deformation. Results are presented for simple examples that verify the technique and demonstrate how it can be applied to analyze lifting surfaces, including those constructed from composite materials.

The solution of structural equilibrium equations by the conjugate gradient method with particular reference to plane stress analysis

AbstractA computer program is described for the analysis of plates in extension using the equivalent plane‐framework idealization. The conjugate gradient method is adopted to solve the resulting large number of simultaneous equations. Performance figures are quoted for the program when used on an ICL KDF9 machine with a restriction of 18K on the available core store. The method is generally shown to provide a satisfactory solution to a heavy computational task and thus permits the solution of very large structural systems on a small computer.

Design Principles for Underground Salt Cavities

Design principles for construction of salt cavities as containers for high level radioactive waste; studies on reduction of cavity volume, development of plastic zone, and stress redistribution around cavity as functions of cavity depth, strength of salt, and physicochemical effects of waste; structural equilibrium equations for designing cavity.