Nonlinear Solution Scheme Research Articles

Initial load factor adjustment through genetic algorithm for the generalized displacement control method: Implementation on non-rigid origami analysis

The nonlinear analysis of structural systems relies on effective nonlinear solution schemes, known as nonlinear solvers or path-following methods. This paper explores the challenge of determining the initial load factor in such analyses, with a particular focus on complex origami structures. This study presents a novel approach that leverages genetic algorithms (GA) to efficiently determine the initial load factor for the generalized displacement method. Building upon the concept of stiffness parameters, our method offers a systematic solution. In this work, GA serves as a tool to discover the appropriate initial load factor based on changes in the stiffness parameter, providing a more rational and user-friendly approach. To demonstrate the practical applicability of our approach, we apply it to a range of complex origami problems, including the folding of Miura-ori, folding of an egg-box origami, bending of an egg-box origami, and folding of an arc Miura-ori. These examples, along with a truss example, encompass highly nonlinear behaviors, including limit points, extreme stiffness fluctuations, snap-through, and snap-back. Our results showcase the effectiveness of our method in accurately tuning the initial load factor to capture and track intricate nonlinear behaviors. Furthermore, it offers the flexibility to fine-tune the distribution of points along the path and manage the resolution of the depicted equilibrium path. This versatility underscores its potential applications in fields such as deployable structures and complex large deformation structural analysis.

A time-adaptive FE2-approach within the method of vertical lines

FE2-approaches are chosen to incorporate micro-mechanical information into macroscopical computations. Although the approach is applied for several years, there are some open theoretical aspects in the algorithmic description. In this contribution the entire concept is brought into the method of vertical lines to solve initial boundary-value problems based on the homogenization concept under consideration. This requires a clear description of known and unknown quantities in the space and time discretization procedure. Dependent on the constitutive model, systems of algebraic or differential-algebraic equations (DAE) arise. Here, we concentrate on models of evolutionary type. Frequently, the resulting DAE-systems are solved using the Backward-Euler method, which are embedded in the more general class of diagonally, implicit Runge-Kutta methods. After the time discretization the resulting systems of non-linear equations have to be solved. This can be done using various schemes to obtain more efficient computations. Here, the three-level Newton algorithm, the Newton-Raphson-Schur method, numerical differentiation and a Chord-version are derived in the context of FE2 and discussed. Finally, step-size-controlled FE2-problems are solved, highlighting the importance of time-adaptive computations, particularly, in combination with the non-linear solution scheme, where the Multilevel-Newton-Chord method shows high efficiency. This contributes to an additional reduction of the overall computational times.

Computational modelling of built-up cold-formed steel columns and parametric studies

The paper develops a detailed finite-element model for the reliable nonlinear collapse analysis of built-up cold-formed steel columns with discrete fastener connections. First, the main components of the computational model are discussed, including their available options and comparison between their results, followed by a rational justification for the options chosen in this study. Specifically, a detailed theoretical background is provided with conclusive comparisons of the options for enforcing end support conditions, applying contact conditions between the constituent plates, selecting connection elements for representing intermediate screw fasteners, and finally, the incremental load stepping and nonlinear solution schemes. The accuracy and performance of the proposed FE modelling strategy are verified against the experimental data, contributing to the calibration of the influencing parameters. The predictions of the proposed methodology are shown to be in excellent agreement with the test data in terms of the failure mode and the inelastic capacity curve up to ultimate capacity and beyond. Upon its successful calibration, the FE model is utilised to perform extensive parametric studies into the effects of various design parameters on the ultimate strength of built-up columns. This includes practical variations of cross-section dimensions, sectional slenderness, built-up section geometry, and fastener spacing. The parametric study results suggest a meaningful dependency of the ultimate capacity of built-up sections on the fastener spacing ratio, the critical buckling half-wavelength of the built-up section and the relative number of restrained components in the built-up configuration undergoing sectional buckling.

Accelerated nonlinear domain decomposition solver for multi-phase flow and transport in porous media

Development of robust and efficient nonlinear solution strategies for multi-phase flow and transport within natural porous media is challenging. Fully Implicit Method (FIM) is widely used in reservoir simulation for solving nonlinear algebraic systems. Previous studies showed that nonlinearities of flow and transport problems could exhibit large degrees of locality across timesteps and nonlinear iterations. Nonlinear domain decomposition (NDD) is an attractive class of methods for achieving localization and resolving spatially unbalanced nonlinearities. However, NDD is not robust as iterative solvers, because the outer iterations are essentially fixed-point type.We design a solution framework to accelerate the outer-loop convergence of NDD. We first apply a nonlinear acceleration (NA) technique to the fixed-point iterations, for correcting the boundary conditions of subdomains. Because flow problems are parabolic and exhibit global behaviors, long-range information is essential to capture main features of the flow field. We incorporate a preconditioning step based on CPR-AMG before each timestep, to provide better initial solutions. Additionally, smoother iterations can be taken to improve the boundary conditions during the NDD process.We evaluate the NDD methods using heterogeneous multi-phase models. The nonlinearities associated with the flow, transport and the coupling are locally resolved through the NDD iterations. Results demonstrate that the basic NDD solver is afflicted with severe nonlinear difficulties. By comparison, the accelerated NDD solver greatly improves the outer convergence performance. Moreover, the new solver can effectively exploit locality on the Newton level, for both the flow and transport problems.

Nonlinear static analysis of extreme structural behavior: Overcoming convergence issues via an unconditionally stable explicit dynamic approach

Simulating extreme structural behavior is of vital importance as it is a major means to study the damage and failure mechanism of engineering structures. However, convergence issues frequently occur when simulating structural response under extreme loading using traditional nonlinear solution strategies, e.g., the Newton–Raphson family of methods. Though commercial finite element software provides explicit dynamic methods to overcome this issue, they are only conditionally stable and require a small time step size. In this paper, a new explicit dynamic approach to simulate static problems involving extreme structural behavior is presented. The multi-support excitation pattern is employed to apply a nodal displacement history of the problem in a dynamic way, then the unconditionally stable explicit KR-α method is used to eliminate the convergence issues. Three examples are presented to illustrate the effectiveness of the proposed dynamic approach, and a parametric study is conducted to investigate the influence of the algorithmic parameters. The results indicate that the proposed method provides the same accuracy as the traditional static method without any convergence issues.

Computational bifurcation analysis for hyperelastic residually stressed tubes under combined inflation and extension and aneurysms in arterial tissue

In this investigation, the mechanical modelling of a hyperelastic residually stressed thick-walled circular cylindrical tube under inflation and extension is performed from a numerical standpoint. The constitutive relation is derived for residually stressed solids using an invariant-based free energy approach. A three-dimensional residual stress field is introduced in the argument of the free-energy function. This formulation is incorporated into a numerical procedure using the Finite Element Method (FEM), with the use of a nonlinear solution scheme via the modified Riks method in order to capture bifurcation and post-bifurcation of the tube. The FE implementation of the proposed formulation is carried out in the general purpose code ABAQUS by means of user-defined capabilities. A comprehensive analysis of the structural performance of the tube is conducted with special focus on the effect of the residual stresses on the bifurcation behaviour. Results are analysed mainly in the context of aneurysms formation and propagation in arterial wall tissues, although it has other applications such as in the context of venous tortuosity. Bulging and bending modes are obtained, and post-bifurcation is also captured. Results suggest that the onset and location of the bulge along the axis of the tube depend mainly on the axial stretch and on the residual stress field. Computations also show that for sufficiently large values of axial stretch the onset of bifurcation is bulging while for small values of axial stretch the onset of bifurcation is bending. Post-bifurcation behaviour of this latter case gives rise to bulges on one side of the tube, which is an irregular shape that appears in the advancement of abdominal aortic aneurysms (AAA). • Bifurcation and postbifurcation of an inflated and extended residually-stressed circular cylindrical tube are analysed. • Bending and bulging motion are captured depending on the axial stretch of the structure. • The onset of bifurcation and the motion of the bifurcation are exploited in the context of aneurysm formation and propagation. • The combination of bending and bulging is consistent with the development of abdominal aortic aneurysms (AAA).

Large scale simulation of pressure induced phase-field fracture propagation using Utopia

Non-linear phase field models are increasingly used for the simulation of fracture propagation problems. The numerical simulation of fracture networks of realistic size requires the efficient parallel solution of large coupled non-linear systems. Although in principle efficient iterative multi-level methods for these types of problems are available, they are not widely used in practice due to the complexity of their parallel implementation. Here, we present Utopia, which is an open-source C++ library for parallel non-linear multilevel solution strategies. Utopia provides the advantages of high-level programming interfaces while at the same time a framework to access low-level data-structures without breaking code encapsulation. Complex numerical procedures can be expressed with few lines of code, and evaluated by different implementations, libraries, or computing hardware. In this paper, we investigate the parallel performance of our implementation of the recursive multilevel trust-region (RMTR) method based on the Utopia library. RMTR is a globally convergent multilevel solution strategy designed to solve non-convex constrained minimization problems. In particular, we solve pressure-induced phase-field fracture propagation in large and complex fracture networks. Solving such problems is deemed challenging even for a few fractures, however, here we are considering networks of realistic size with up to 1000 fractures.

A unified approach to the Timoshenko 3D beam-column element tangent stiffness matrix considering higher-order terms in the strain tensor and large rotations

A structural geometric nonlinear analysis using the finite element method (FEM) depends on the consideration of five aspects: the interpolation (shape) functions, the bending theory, the kinematic description, the strain–displacement relations, and the nonlinear solution scheme. As the FEM provides a numerical solution, the structure discretization has a great influence on the analysis response. However, when applying interpolation functions calculated from the homogenous solution of the differential equation of the problem, a numerical solution closer to the analytical response of the structure is obtained, and the level of discretization could be reduced, as in the case of linear analysis. Thus, to reduce this influence and allow a minimal discretization of the structure for a geometric nonlinearity problem, this work uses interpolation functions obtained directly from the solution of the equilibrium differential equation of a deformed infinitesimal element, which includes the influence of axial forces. These shape functions are used to develop a complete tangent stiffness matrix in an updated Lagrangian formulation, which also integrates the Timoshenko beam theory, to consider shear deformation and higher-order terms in the strain tensor. This formulation was implemented, and its results for minimal discretization were compared with those from conventional formulations, analytical solutions, and Mastan2 v3.5 software. The results clearly show the efficiency of the developed formulation to predict the critical load of plane and spatial structures using a minimum discretization.

An algebraic flux correction scheme facilitating the use of Newton-like solution strategies

Building on recent advances in the analysis and design of algebraic flux correction (AFC) schemes, new differentiable limiter functions are constructed for efficient nonlinear solution strategies. The proposed scaling parameters are used to limit artificial diffusion operators incorporated into the residual of a high order target scheme to produce accurate and bound-preserving finite element approximations to hyperbolic problems. Due to this stabilization procedure, the occurring system becomes highly nonlinear and the efficient computation of corresponding solutions is a challenging task. The presented regularization approach makes the AFC residual twice continuously differentiable so that Newton’s method converges quadratically for sufficiently good initial guesses. Furthermore, the performance of each nonlinear iteration is improved by expressing the Jacobian as the sum and product of matrices having the same sparsity pattern as the Galerkin system matrix. Eventually, the AFC methodology constructed is validated numerically by applying it to several numerical benchmarks.

A conservative sequential fully implicit method for compositional reservoir simulation

A conservative sequential fully implicit method is derived for compositional reservoir simulation. Multi-phase flow in porous media comprises coupled complex processes: i.e. elliptic flow equation, hyperbolic transport equation and highly nonlinear phase equilibrium equation. These processes contain very different mathematical characteristics that cannot be efficiently solved by one numerical method. As a result, the fully implicit method may become numerically complex and inefficient because the Jacobian includes the derivatives w.r.t. the variables from all of the different processes involved.Jenny et al. (2004) [12] and Lee et al. (2015) [20] demonstrated that flow (pressure) and transport (saturation) for multi-phase flow without compositional effect can be efficiently solved by a sequential fully implicit method. However, the characteristics of the phase equilibrium equations are very different from those of the transport equations. This paper proposes an iterative method that solves the flow, transport and phase equilibrium equations in a sequential manner. The transport of hydrocarbons through porous media is governed by the multi-phase Darcy's equation, which is used to compute the phase velocities. The hydrocarbon components belonging to the same phase are transported with the same phase velocity. Upon arrival in the destination grid cell, these components are redistributed via a phase equilibrium calculation. This observation leads to simplification of the governing equations by reducing primary variables to four (i.e., pressure and three phase saturations). The nonlinear solution scheme composed of the stages outlined above is proven to preserve mass conservation, while a new degree of freedom, “thermodynamic flux”, is introduced to ensure volume conservation. The sequential algorithm is solved iteratively until pressure, saturation, and phase composition are fully converged.It is well-known that sequential solution schemes may require many iterations or fail to converge if the phase equilibrium calculation involves phase transition with a large volume change. This indicates that the current governing equations may not adequately describe fluid flux during rapid phase transition. With numerical examples we demonstrate that such numerical difficulties are successfully resolved via the thermodynamic flux term.

Localized nonlinear solution strategies for efficient simulation of unconventional reservoirs

Accurate and efficient numerical simulation of unconventional reservoirs is challenging. Long periods of transient flow and steep potential gradients occur due to the extreme conductivity contrast between matrix and fracture. Detailed near-well/near-fracture models are necessary to provide sufficient resolution, but they are computationally impractical for field cases with multiple hydraulic-fracture stages.Previous works in the literature of unconventional simulations mainly focus on the gridding level that adapts to wells and fractures. Limited research has been conducted on nonlinear strategies that exploit locality across timesteps and nonlinear iterations. It was reported that an individual Newton update is typically sparse and nonlinear convergence is constrained by a small portion of the model. To perform localized computations, an a-prioristrategy is essential to first determine the active subset of simulation cells for the subsequent iteration. The active set flags the cells that will be updated, and then the corresponding localized linear system is solved.The objective of this work is to develop localization methods that are readily applicable to complex fracture networks and flow physics in unconventional reservoirs. By utilizing the diffusive nature of pressure updates, an adaptive algorithm is proposed to make adequate estimates for the active domains. In addition, we further develop a localized solver based on nonlinear domain decomposition (DD). Compared to a standard DD method, domain partitions are dynamically constructed. The new solver provides effective partitioning that adapts to flow dynamics and Newton updates.We evaluate the developed methods using several complex problems with discrete fracture networks. The problems consider multi-phase and compositional fluid systems with phase changes. The results show that large degrees of solution locality present across timesteps and iterations. Compared to a standard Newton solver, the new solvers enable superior computational performance. Moreover, Newton convergence behavior is preserved, without any impact on solution accuracy.

The parametric HFGMC micromechanical model for soft UHMWPE laminated composites

The 3D parametric High-Fidelity-Generalized-Method-of-Cells (HFGMC) micromechanical model was used to predict the mechanical tensile behavior of ultra-high-molecular-weight-polyethylene (UHMWPE) cross-ply laminated composite (Dyneema®). The composite microstructure was obtained from optical microscope images and was digitized using CAD software. Two configurations of 3D representative-volume-elements (RVE)s were constructed and analyzed to demonstrate the capability of the HFGMC in predicting the effective behavior of UHMWPE laminated composites. The elastic properties of both constituents (fiber and matrix) were obtained from published experimental data and from in-house experiments. Linear and nonlinear constitutive laws were applied for the constituents for comparison. The parametric HFGMC predictions for the overall mechanical properties were compared with extensive tensile experiments. The nonlinear behavior under in-plane shear was also examined in order to evaluate the effect of the proposed RVE configurations for the laminated composite. The parametric HFGMC (PHFGMC) is formulated using the virtual work for both the global and local fields. This allows the development of a new incremental-iterative nonlinear solution scheme. Special periodicity constraints are proposed for the special, yet prevalent case, of non-periodic RVE mesh. The parametric HFGMC micromechanical model is shown to be capable to simulate the nonlinear behavior of soft composites. This model can be generalized for rate dependent constituents.

Direct Numerical Simulation of Flow on Pore-Scale Images Using the Phase-Field Method

SummaryAdvances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow on pore structures feasible. In this paper, we describe a two-phase-flow simulator that solves mass- and momentum-balance equations valid at the pore scale (i.e., at scales where the Darcy velocity homogenization starts to break down). The simulator is one of the key components of a molecule-to-reservoir truly multiscale modeling work flow.A Helmholtz free-energy-driven, thermodynamically based diffuse-interface/phase-field method is used for the effective simulation of numerous advecting interfaces, while honoring the interfacial tension (IFT). The advective Cahn-Hilliard (CH) (mass-balance, energy dissipation) and Navier-Stokes (NS) (momentum-balance, incompressibility) equations are coupled to each other within the phase-field framework. Wettability on rock/fluid interfaces is accounted for by means of an energy-penalty-based wetting (contact-angle) boundary condition. Individual balance equations are discretized by use of a flexible discontinuous Galerkin (DG) method. The discretization of the mass-balance equation is semi-implicit in time using a convex/concave splitting of the energy term. The momentum-balance equation is split from the incompressibility constraint by a projection method and linearized with a Picard splitting. Mass- and momentum-balance equations are coupled to each other by means of operator splitting, and are solved sequentially.We discuss the mathematical model and its DG discretization, and briefly introduce nonlinear and linear solution strategies. Numerical-validation tests show optimal convergence rates for the DG discretization, indicating the correctness of the numerical scheme and its implementation. Physical-validation tests demonstrate the consistency of the phase distribution and velocity fields simulated within our framework. Finally, two-phase-flow simulations on two real pore-scale images demonstrate the usefulness of the pore-scale simulator. The direct pore-scale numerical-simulation methodology rigorously considers the flow physics by directly acting on pore-scale images of rocks without remeshing. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.

Sequential implicit nonlinear solver for geothermal simulation

Modeling of thermal multiphase fluid flow in subsurface porous media poses significant difficulties for nonlinear solvers. These difficulties are a result of the strong coupling between the mass and energy conservation equations through the thermodynamic relationships. In particular, for geothermal problems, this tight coupling results in an apparent “negative compressibility” for cells (control volumes) that have two fluid phases (liquid and vapor) during the time interval of interest. This behavior is associated with the flow of cold water into a cell that is at saturated conditions. As steam condensation occurs, the reduction in fluid volume due to the phase change is larger than the expansion of the liquid and vapor due to injection. As a result, the cell pressure decreases as condensation proceeds. This pressure decrease leads to an increase in the inflow of cold water until the condensation is complete. During the condensation period, the overall compressibility of the system is negative, and this poses serious convergence problems even when a fully implicit (i.e., backward Euler) time discretization is used. In order to deal with this problem, nonlinear solvers must be modified. In this work, we present a nonlinear solution strategy that overcomes the negative compressibility associated with the condensation front. We first consider a single-cell problem that demonstrates the negative compressibility phenomenon. Then, we describe a solution strategy that overcomes the nonlinear convergence difficulties. We then derive a criterion that can be used as a nonlinear preconditioning strategy to solve multicell problems. The effectiveness of the proposed nonlinear solution scheme is then demonstrated for more complex models. It was found that for one-dimensional problems, this nonlinear solution strategy converges for all timesteps sizes, while the fully coupled strategy only converges for a limited sized timestep. Furthermore, for a two-dimensional heterogeneous problem, the largest Courant–Friedrichs–Lewy (CFL) number for this method is at least an order of magnitude larger than the CFL number that can be used with the fully coupled approach.

Material nonlinear topology optimization using the ground structure method with a discrete filtering scheme

Topology optimization of truss lattices, using the ground structure method, is a practical engineering tool that allows for improved structural designs. However, in general, the final topology consists of a large number of undesirable thin bars that may add artificial stiffness and degenerate the condition of the system of equations, sometimes even leading to an invalid structural system. Moreover, most work in this field has been restricted to linear material behavior, yet real materials generally display nonlinear behavior. To address these issues, we present an efficient filtering scheme, with reduced-order modeling, and demonstrate its application to two- and three-dimensional topology optimization of truss networks considering multiple load cases and nonlinear constitutive behavior. The proposed scheme accounts for proper load levels during the optimization process, yielding the displacement field without artificial stiffness by simply using the truss members that actually exist in the structure (spurious members are removed), and improving convergence performance. The nonlinear solution scheme is based on a Newton-Raphson approach with line search, which is essential for convergence. In addition, the use of reduced-order information significantly reduces the size of the structural and optimization problems within a few iterations, leading to drastically improved computational performance. For instance, the application of our method to a problem with approximately 1 million design variables shows that the proposed filter algorithm, while offering almost the same optimized structure, is more than 40 times faster than the standard ground structure method.

A unified approach for beam-to-beam contact

Existing beam contact formulations can be categorized in point-to-point contact models that consider a discrete contact force at the closest point of the beams, and line-to-line contact models that assume distributed contact forces. In this work, it will be shown that line contact formulations applied to slender beams provide accurate and robust mechanical models in the range of small contact angles, whereas the computational efficiency considerably decreases with increasing contact angles. On the other hand, point contact formulations serve as sufficiently accurate and very efficient models in the regime of large contact angles, while they are not applicable for small contact angles as a consequence of non-unique closest point projections. In order to combine the advantages of these basic formulations, a novel all-angle beam contact (ABC) formulation is developed that applies a point contact formulation in the range of large contact angles and a recently developed line contact formulation in the range of small contact angles, the two being smoothly connected by means of a variationally consistent model transition. Based on a stringent analysis, two different transition laws are investigated, optimal algorithmic parameters are suggested and conservation of linear momentum, angular momentum and total energy is shown. All configuration-dependent quantities within the point-, the line- and the transition-contact regime are consistently linearized, thus allowing for their application within implicit time integration schemes. Furthermore, a step size control of the nonlinear solution scheme is proposed that allows for displacement increments per time step that exceed the order of magnitude of the beam cross-section radius. For many standard beam-to-beam contact algorithms, this is the typical limitation concerning possible time step sizes, especially when considering high beam slenderness ratios. Finally, an efficient two-stage contact search based on dynamically adapted search segments is proposed. This algorithm yields in a tight set of potential contact pairs and enables a subdivision into potential point and potential line contact pairs, which is essential in order to fully exploit the efficiency potential of the proposed contact formulation. A series of numerical test cases is analyzed in order to verify the accuracy and consistency of the proposed contact model transition regarding contact force distributions and conservation properties, but also for quantifying the efficiency gains as compared to standard beam contact formulations.

4D inversion of borehole gravity data for monitoring fluid fronts

SUMMARY Monitoring fluid movement is an important component in enhanced oil recovery (EOR) and CO2 sequestration. The newly available slim-hole gravimeter operating at high temperature offers a new avenue for such monitoring efforts because of the direct sensitivity to the change in the density distribution. We present a time-lapse gravity inversion algorithm for recovering the front of injected fluid using borehole gravity measurements. We assume that the horizontal extent of the fluid can be represented by a polygon with known but variable thickness and density contrast due to fluid substitution. We represent the evolution of the front as a 4D function of the spatial position and time since the initiation of the injection. The inversion can be carried out either independently at discrete time points or as a single inversion simultaneously over all time points. We demonstrate that the latter approach is superior in that it is more stable and offers improved capability in detecting break-through events at later times. In this paper, we will describe the details of the two inversion approaches, including two different model objective functions in polar coordinates and the nonlinear solution strategies. We will illustrate the advantages and drawbacks of independent and simultaneous 4D inversions using numerical examples

Finite element implementation of a multiscale model of the human lens capsule.

An axisymmetric finite element implementation of a previously described structural constitutive model for the human lens capsule (Burd in Biomech Model Mechanobiol 8(3):217-231, 2009) is presented. This constitutive model is based on a hyperelastic approach in which the network of collagen IV within the capsule is represented by an irregular hexagonal planar network of hyperelastic bars, embedded in a hyperelastic matrix. The paper gives a detailed specification of the model and the periodic boundary conditions adopted for the network component. Momentum balance equations for the network are derived in variational form. These balance equations are used to develop a nonlinear solution scheme to enable the equilibrium configuration of the network to be computed. The constitutive model is implemented within a macroscopic finite element framework to give a multiscale model of the lens capsule. The possibility of capsule wrinkling is included in the formulation. To achieve this implementation, values of the first and second derivatives of the strain energy density with respect to the in-plane stretch ratios need to be computed at the local, constitutive model, level. Procedures to determine these strain energy derivatives at equilibrium configurations of the network are described. The multiscale model is calibrated against previously published experimental data on isolated inflation and uniaxial stretching of ex vivo human capsule samples. Two independent example lens capsule inflation analyses are presented.

Evaluation of wall deflections and ground surface settlements in deep excavations

A study to illustrate the ground surface and wall movements due to excavation is undertaken based on two-dimensional nonlinear finite element (FE) modeling and analysis. The FE model and nonlinear solution strategy are built in the open-source software platform, OpenSees, of the Pacific Earthquake Engineering Research Center. A drained analysis was carried out using Drucker–Prager multisurface kinematic plasticity model for sand. The proposed model is verified using a case history in an urban excavation. The ground surface settlement and wall deflection for the various conditions of excavation are calculated. The results indicate that the extent of the settlement zone depends on excavation depth and is predicted as 2.5–3 times excavation depth, and the upper limit of the maximum ground surface settlement is equal to the maximum lateral wall deflection.

Estabilidade e vibração não linear de pórticos em aço usados em galpões industriais: análise paramétrica

Esse trabalho apresenta um estudo referente à análise da estabilidade e vibração não linear de pórticos de aço planos do tipo galpão. Especial atenção é voltada à influência do pré-carregamento estático nas frequências naturais e nos modos de vibração dessas estruturas. Nesse sentido, foi utilizada uma formulação não linear de elementos finitos de viga-coluna que permite considerar as ligações entre os membros estruturais como semirrígidas. Para resolver o problema de equilíbrio estático não linear, um processo incremental e iterativo do tipo Newton acoplado ao método do comprimento de arco é adotado. Uma vez obtida convergência em cada passo de carga, emprega-se o método de Jacobi, para se determinarem as frequências naturais e os modos de vibração da estrutura pré-carregada. Todo o processo de solução não linear adotado é detalhado em um algoritmo. A influência de um parâmetro geométrico (altura da tesoura) e outro físico (rigidez dos apoios) no comportamento do pórtico tipo galpão é destacada nas análises. Objetiva-se, assim, obter uma melhor compreensão de alguns fenômenos de instabilidade e sensibilidade estrutural e mostrar a importância desses parâmetros e do pré-carregamento estático na flambagem e na vibração dessa estrutura de grande aplicação prática.