Incremental-iterative Procedure Research Articles

Analytical model for axially compressed circular concrete-filled double skin steel tubes (CFDSTs): Insights from concrete non-uniformly confined states

Concrete-filled double skin steel tube (CFDST) is becoming a competitive structural concept in deep-sea and deep-ground engineering structures. However, the passive non-uniform confinement of the sandwiched concrete needs to be further clarified and quantitatively described to achieve better design efficiency of CFDSTs in sophisticated application scenarios. This investigation suggests a development approach for the analytical model of the axially compressed circular CFDSTs considering the non-uniform confinement effect. The strength and deformation characteristics of the concrete subjected to constant lateral stresses have been first systematically analyzed through the extensively collected experimental database, and the corresponding constitutive models have been formulated accordingly. The stress distributions of the sandwiched concrete and the influence of the cross-sectional geometric properties have been discussed, and then the calculation method for the non-uniform confining factor has been developed. An efficient analytical model has been established on the basis of an incremental-iterative procedure, with the steel tube-concrete interaction and bonding behavior appropriately considered. The proposed analytical model has been verified through the widely collected experimental data of axially compressed CFDSTs, and the results show that it could effectively simulate the full-range axial load–strain curves and improve the prediction accuracy of ultimate loads by about 5% to 16% compared with the existing design formulas.

Analytical GNI Analysis for Lateral-torsional Behavior of Thin-walled Beams with Doubly-symmetric I-Sections

In this paper elastic lateral-torsional behavior of simple beams is discussed. The motivation of the presented research is the observation that classic analytical prediction and finite element prediction are, typically, considerably different, when the second-order nonlinear behavior of beams with initial imperfections is analyzed. In order to understand and explain the observed differences, a novel analytical solution is presented for the geometrically nonlinear analysis of beams with initial geometric imperfection. The presented analytical solution is derived for doubly-symmetric cross-sections, but with the novelty that it takes into consideration the changing geometry as the load is increasing. The most important steps of the derivations are summarized, and the resulted formulae are briefly discussed. Numerical studies are performed, too: the results of the new analytical formulae are compared to those from shell finite element analysis. The results suggest that the new formulae are able to capture the most important elements of the behavior. By the analytical and numerical results, it is proved that classic analytical solutions for the geometrically nonlinear analysis of beams with geometric imperfections are necessarily different from the numerical results obtained by incremental-iterative procedures.

A Low-Cost Algorithm for Uncertainty Quantification Simulations of Steady-State Flows: Application to Ocular Hemodynamics

An algorithm for the calculation of steady-state flowing under uncertain conditions is introduced in this work in order to obtain a probabilistic distribution of uncertain problem parameters. This is particularly important for problems with increased uncertainty, as typical deterministic methods are not able to fully describe all possible flow states of the problem. Standard methods, such as polynomial expansions and Monte Carlo simulations, are used for the formation of the generalized problem described by the incompressible Navier-Stokes equations. Since every realization of the uncertainty parameter space is coupled with non-linear terms, an incremental iterative procedure was adopted for the calculation. This algorithm adopts a Jacobi-like iteration methodology to decouple the equations and solve them one by one until there is overall convergence. The algorithm was tested in a typical artery geometry, including a bifurcation with an aneurysm, which consists of a well-documented biological flow test case. Additionally, its dependence on the uncertainty parameter space, i.e., the inlet velocity distribution, the Reynolds number variation, and parameters of the procedure, i.e., the number of polynomial expansions, was studied. Symmetry exists in probabilistic theories, similar to the one adopted by the present work. The results of the simulations conducted with the present algorithm are compared against the same but unsteady flow with a time-dependent inlet velocity profile, which represents a typical cardiac cycle. It was found that the present algorithm is able to correctly describe the flow field, as well as capture the upper and lower limits of the velocity field, which was made periodic. The comparison between the present algorithm and the typical unsteady one presented a maximum error of ≈2% in the common carotid area, while the error increased significantly inside the bifurcation area. Moreover, “sensitive” areas of the geometry with increased parameter uncertainty were identified, a result that is not possible to be obtained while using deterministic algorithms. Finally, the ability of the algorithm to tune the parameter limits was successfully tested.

Two-step incremental procedure associated with the normal flow technique applied to trusses

To achieve the nonlinear structural behavior, there is a need to trace of their equilibrium path in the space of load-displacement. Truss systems are commonly implemented in several structural systems, including high-span bridges and bracing of the supporting structure of tall buildings. Our study adapts a two-step method with cubic convergence into an incremental-iterative procedure to analyze the geometric nonlinear behavior of trusses. The solution method is combined with the Linear Arc-Length path-following technique. To find the approximate root of nonlinear equation system in the two-step method, two formulas are used. Structures are discretized using the Positional Finite Element Method and all truss bars are assumed to remain linear elastic. The correction of the nodal coordinates subincrement vector is performed using the Normal Flow technique. A computational algorithm was implemented using the free program Scilab. Our numerical results show that, when compared to the standard and modified Newton-Raphson algorithms, the new algorithm decreases the number of iterations and the computing time in the nonlinear analysis of trusses. Equilibrium paths with force and/or displacement limit points are obtained with good precision.

Space-fractional small-strain plasticity model for microbeams including grain size effect

Elastic–plastic bending of microstructure-dependent beams is analyzed, i.e., beams whose external dimensions and characteristic internal length are close. The formulations presented concern space-fractional Euler–Bernoulli (s-FEBB) and Timoshenko (s-FTB) beams, enriched by including the plastic properties of the material. The Huber–Mises–Hencky plasticity criterion and isotropic hardening are applied in the developed models. The incremental iterative numerical procedure is elaborated to solve the elastic–plastic bending problem. The analysis covers not only the scale effect but also the grain size-dependent strengthening described by the general Hall–Petch relation. Based on the experimental results of WC-Co micropillars compression available in the literature, model parameters are identified for specific microstructures. Significant results were found — it was revealed that the length scale could be directly correlated with the grain size. Finally, a numerical study of elastic–plastic bending of a fractional beam demonstrated how the microstructure affects the propagation of plastic hinges.

BLOOD FLOW IN ARTERIAL BIFURCATION CALCULATED BY TURBULENT FINITE ELEMENT MODEL

In this paper, turbulent fluid flow is analyzed using a two-equation turbulent finite element model that can calculate values in the viscous sublayer. Implicit integration of the equations is used for determining the fluid velocity, fluid pressure, turbulence, kinetic energy, and dissipation of turbulent kinetic energy. These values are calculated in the finite element nodes for each step of the incremental-iterative procedure. Developed turbulent finite element model, with the customized generation of finite element meshes, is used for calculating complex blood flow problems. Analysis of results shows that a cardiologist can use the proposed tools and methods for investigating the hemodynamic conditions inside the bifurcation of arteries.

Turbulent finite element model applied for blood flow calculation in arterial bifurcation

Background and objectiveDue to the relatively low fluid velocities in major arteries and veins, blood flow is by default laminar, however, turbulence can occur as a result of stenosis or other obstacles. Hemodynamic parameters like Wall Shear Stress or Oscillatory Shear Index can be used for plaque formation prediction, and these parameters are depended on the nature of the flow. Implementation of the k-ω turbulent flow in the Finite Element solver aims to improve numerical analysis of cardio-vascular condition development and progression. Calculation of turbulent fluid flow in this paper is performed using a two-equation turbulent finite element model that can calculate values in the viscous sublayer. MethodsImplicit integration of the equations is used for determining the fluid velocity, turbulent kinetic energy and dissipation of turbulent kinetic energy. These values are calculated in the finite element nodes for each step of the incremental-iterative procedure. Developed turbulent finite element model with the customized generation of finite element meshes is used for calculating complex blood flow problems. ResultsTurbulent model is verified on an example of fluid flow in the backward-facing step channel and analysis results correspond well with the experimental ones from the literature. Further, a turbulent model is applied for the simulation of blood flow through artery bifurcation. Verification of numerical examples obtained using different commercial software packages (Ansys, COMSOL Multiphysics) ensuring usage and accuracy of PAK in-house solver. ConclusionsAnalysis results show that turbulence cannot be neglected in the modelling of cardio-vascular conditions and that cardiologists can use the proposed tools and methods for investigating the hemodynamic conditions inside the bifurcation of arteries. Appropriate agreement between experimental results, and results obtained using commercial solutions and the k-ω turbulent flow in the Finite Element solver PAK, validate methodology presented in this paper. However, small deviations between the results underline the importance of the proper boundary condition prescription and mesh size and node distribution, which is also discussed in this paper. Due to the implicit integration implemented in PAK solver, time step size has an insignificant influence on the analysis results, assuming the initial time increments are sufficiently small to ensure proper discretization of velocity and pressure pulsatile functions.

Obtaining interaction diagram of concrete section exposed to fire by mathematical optimization

This paper proposes a novel method for obtaining the axial load–moment (P–M) interaction diagrams of reinforced concrete sections exposed to fire. The problem of obtaining the strength of a reinforced concrete section subjected to an eccentric load was converted into determining an optimum solution of a constrained nonlinear optimization problem. The proposed method does not limit the strains of both concrete and reinforcing bars. A sequential quadratic programming algorithm was used to solve the constrained nonlinear optimization. Discontinuity in the arctangent function was avoided by rotating the atan2 function. A bending moment offset from the tilted axis was introduced to solve the asymmetric section and uneven heating condition. The results of the examples confirm that the proposed method can produce the entire P–M interaction surface of an arbitrarily shaped reinforced composite section exposed to fire. The proposed method is much faster and more versatile than the incremental-iterative procedure. The proposed method can calculate equally spaced points on the Mx–My interaction curve which the incremental-iterative procedure cannot obtain. The proposed method can be applied to any combination of cross-sectional shape, material, and temperature history.

Optimal dynamic control for a maglev vehicle moving on multi-span guideway girders

Abstract An optimal control algorithm using a virtual tuned mass damper called virtual TMD to control the levitation force of a magnetic system is developed for resonance suppression of a maglev vehicle moving on multi-span guideway girders. Since the optimal dynamic parameters of a TMD in vibration control are well developed, the optimal tuning gains required to control the magnetic oscillations of the maglev bogie can be directly used and fed back to the maglev control system. To address the dynamic interaction analysis from the maglev vehicle to the guideway girders and vice versa, the entire coupling system is decomposed into two subsystems, one is the moving vehicle subsystem and another the stationary guideway subsystem. Then, an incremental–iterative procedure associated with the Newmark method is presented to solve the two sets of subsystem equations. Finally, the control effectiveness and parametric studies of the optimal virtual TMD scheme on resonance reduction of the moving maglev vehicle are demonstrated.

Nonlinear static finite element analysis of sinking a floating pipeline to the bottom in calm water

ABSTRACT This paper describes a nonlinear static finite element analysis for simulating a process of submerging a floating pipeline to the bottom in calm water. Soil–pipe interaction elements, fully submerged pipe elements, partially submerged pipe elements, and floating pipe elements were used. Also, a load stiffness matrix associated with hydrostatic pressure was illustrated. Then, an incremental iterative procedure was developed to analyze large deformation of the pipeline during the sinking process and to take into account pipe-seabed interaction. The paper concentrates on the modeling of filled-in water loads and boundary conditions related to an offshore tugboat. Some numerical examples are provided to show the capabilities of the proposed analysis to model the continuous float-and-sink procedure of the pipeline.

Study of hull girder ultimate strength at elevated temperatures

ABSTRACT Fire poses a significant threat in the shipping industry not only for humans (passenger, seafarers) but also for the ship structure. It is of great concern how elevated temperatures from a fire can influence the bending moment capacity of a hull girder due to the degradation of steel’s mechanical properties. The aim of the paper is to evaluate the ultimate bending moment capacity of a chemical oil/product carrier at ambient and elevated temperature conditions. The study defines the parameters and modelling that need to be considered in the finite element analysis (FEA) and proposes a modified incremental–iterative procedure for the bending moment capacity calculation at elevated temperatures. Smith Method was extended in order to incorporate neutral axis rotation and steel’s mechanical properties at elevated temperatures. The proposed modified Smith Method’s results present good agreement with the equivalent FEA results. It is concluded that with the proposed modified Smith Method, it is possible to determine the loading that can be carried by a hull girder with a specified temperature profile. The study also determines an analysis technique for the stiffened panel ultimate strength analysis at elevated temperatures.

A co-rotational triangular finite element for large deformation analysis of smooth, folded and multi-shells

A six-node co-rotational curved triangular shell finite element with a novel rotation treatment for folded and multi-shell structures is presented. Different from other co-rotational triangular element formulations, rotations are not represented by axial (pseudo) vectors, but by components of polar (proper) vectors, of which additivity and commutativity lead to symmetry of the tangent stiffness matrices in both local and global coordinate systems. In the co-rotational local coordinate system, the two smallest components of the shell director are defined as the nodal rotational variables. Similarly, the two smallest components of each director in the global coordinate system are adopted as the global rotational variables for nodes located either on smooth shells or away from non-smooth shell intersections. At intersections of folded and multi-shells, global rotational variables are defined as three selected components of an orthogonal triad initially oriented along the global coordinate system axes. As such, the vectorial rotational variables enable simple additive update of all nodal variables in an incremental-iterative procedure, resulting in significant enhancement in computational efficiency for large deformation analysis. To alleviate membrane and shear locking phenomena, an assumed strain method is employed in obtaining the element tangent stiffness matrices and the internal force vector. The effectiveness of the presented co-rotational triangular shell element formulation is verified by analyzing several benchmark problems of smooth, folded and multi-shell structures undergoing large displacements and large rotations.

Post-buckling behaviour in variable stiffness cylindrical panels under compression loading with modal interaction effects

Variable Angle Tow (VAT) composite panels, whereby fibre angle distributions vary continuously in-plane, have more scope for tuning structural properties than traditional straight fibre laminated composite materials. Currently, the geometrically non-linear response of such structures is typically modelled using path-following methods implemented in commercial finite element analysis. However the high computational cost given from both the incremental-iterative procedure and fine mesh required for accurate modelling of the variable trajectory of fibre direction, can be excessively time consuming. Driven by these shortcomings, we use a fast multi-modal Koiter asymptotic method for investigating the nonlinear buckling behaviour of VAT cylindrical panels in compression. In doing so, this paper provides new insight into the multi-modal description and nonlinear buckling mode interactions which are responsible for the highly nonlinear behaviour of cylindrical panels in compression with a particular emphasis on the nonlinear components in the mathematical asymptotic description. As such, through an extensive parametric virtual testing programme we construct, for each panel under investigation, an asymptotic solution projecting the equilibrium equations in the subspace of its first 30 buckling modes. Then, at several points of the resulting equilibrium path, the percentage contribution of each of these modes to the solution is measured. On the basis of the percentage obtained, those modes which give the largest contribution to the solution are identified. The main novelty of this work is that these results are used as a priori information for re-projecting the equilibrium equations into a new modal subspace to which only the buckling modes with the largest participation belong. Moreover, for the first time, we observe that in a multi-modal Koiter-inspired description the modal shapes giving the largest contribution to the asymptotic solution have the same degree of symmetry. In this way, for a generic panel under consideration, we show that it is possible to obtain a more computationally-efficient asymptotic description than current approaches. Finally, the new computed equilibrium paths are compared to benchmark results using the commercial finite element software ABAQUS and the computational advantages given by using a priori information in the asymptotic expansion are commented upon.

Research on nonlinear, postbuckling and elasto-plastic analyses of framed structures and curved beams

Nonlinear analysis of structures is an effective way for predicting the bearing capacity or ultimate strength of a structure. This paper will review researches on the nonlinear, buckling, postbuckling and elasto-plastic analyses of three-dimensional (3D) framed structures. Also considered are curved beams since they share the problem of joint equilibrium in simulation. The following are the key issues to be covered in this paper: (a) analytical or semi-analytical approaches; (b) formulation for incremental-iterative nonlinear analysis; (c) joint equilibrium of 3D frames at the deformed configuration; (d) rigid body test for nonlinear finite elements; (e) rigid-body-qualified geometric stiffnesses and derivation; (f) simulation of buckling of curved beams; (g) elasto-plastic analysis of steel frames; (h) mechanism of iterations in incremental analysis; (i) recovery of element forces; and (j) path-tracing scheme for the postbuckling response. In this paper, the theories and methods for solving the postbuckling responses of structures involving multi-loop curves are considered. In fact, the incremental-iterative procedure for the geometric and/or inelastic nonlinear analysis can be greatly simplified if it is well rooted in physics. Namely, if the rigid body rule is fully obeyed at each stage of analysis, then the entire incremental-iterative procedure can be made simpler and more efficient. Recently, it was demonstrated that satisfactory results can be obtained for the postbuckling response of a number of structures using the elastic stiffness alone. A total of 154 references are cited in this paper.

Geometrical Nonlinear Analysis of Slender Micro Arch-Beams Pulled by Magnetic Loadings

In this study, a nonlinear model of micro arch-beams under the action of electrostatic loadings with an initial air gap is presented. Based on the well-developed 3D beam theory proposed by Yang and Kuo [4], the arch beam is modelled as a series of uniform straight beam elements and the pull-in magnetic force is simulated as an electric-induced force by an electrode lying. Then the nonlinear structural analysis of snap-through and pull-in stability for a simply supported shallow arch subjected to a concentrated electrostatic loading will be investigated. To tackle the path-dependent features of electric forces due to presence of air gaps and nonlinear deformations, this study regards the electric forces as pseudo-forces acting at the arch-beam and solves the structure equations using an incremental-iterative procedure. From the numerical results, the present approach demonstrates that the micro-arch beam has the capacity to withstand further electrostatic loading after the snapthrough jump.

Vibration reduction for interaction response of a maglev vehicle running on guideway girders

As a vehicle moves on multiple equal-span beams at constant speed, the running vehicle would be subjected to repetitive excitations from the beam vibrations under it. Once the exciting frequency caused by the vibrating beams coincides with any of the vehicle's frequencies, resonance would take place on the vehicle. A similar resonance phenomenon occurs on a beam subject to sequential moving loads with identical axle-intervals. To reduce both resonant phenomena of a vehicle moving on guideway girders, this study proposed an additional feedback controller based the condensed virtual dynamic absorber (C-VDA) scheme. This condensation scheme has the following advantages: (1) the feedback tuning gains required to adapt the control currents or voltages are directly obtained from the tuning forces of the VDA; (2) the condensed VDA scheme does not need additional DoFs of the absorber to control the vibration of the maglev-vehicle/guideway system. By decomposing the maglev vehicle-guideway coupling system into two sub-systems (the moving vehicle and the supporting girders), an incremental-iterative procedure associated with the Newmark method is presented to solve the two sets of sub-system equations. From the present studies, the proposed C-VDA scheme is a feasible approach to suppress the interaction response for a maglev vehicle in resonance moving on a series of guideway girders.

Nonlinear buckling analysis of variable stiffness composite plates based on the reduced order model

Abstract The variable-stiffness fiber composite plates which have an enhanced design flexibility, largely rely on laminate optimizations to maximize the buckling performance. The corresponding computational efficiency becomes a key issue, in particular when the nonlinear structural behavior is considered. The finite element method based on a full nonlinear analysis is a standard technique for nonlinear structural analysis, however the high computational complexities generated from both the incremental-iterative procedure and the very refined mesh needed for the discrete modeling of curved fibers, are still a decisive cost factor on modern computers. In this work, the Koiter-Newton method is further extended to nonlinear buckling analysis, including the pre and post buckling stage, of variable stiffness composite plates. A four-node quadrilateral element based on the classical laminated plate theory is developed in framework of the von Karman kinematics, for the finite element implementation of the proposed asymptotic method . The reduced order model , with or without imperfections, is constructed using the improved Koiter’s asymptotic expansion , for both the symmetrical and unsymmetrical laminates. The nonlinear response curve of loaded structure can be traced automatically, using the nonlinear predictor and corrections both generated from the reduced order model. This leads to a fairly large step size in the path-tracing process, compared to that for the classical Newton method . The reduced order model largely reduces the computational burden produced by the high-density FE mesh for the varied fiber path. Numerical results indicate the overall high quality and efficiency of the proposed method.

Nonlinear bending of elastoplastic functionally graded ceramic-metal beams subjected to nonuniform distributed loads

Nonlinear bending of elastoplastic functionally graded (FG) ceramic-metal beams subjected to nonuniform distributed loads is investigated by the finite element method. A bilinear stress-strain relation with isotropic hardening is assumed for elastoplastic behavior of metal, and the elastoplastic properties of the FG ceramic-metal material are evaluated by using Tamura–Tomota–Ozawa (TTO) model. Based on Euler–Bernoulli beam theory, a nonlinear finite element formulation, taking the effect of plastic deformation into account, is derived and used in the investigation. The formulation employing nonlinear von Kámán strain-displacement relationship is derived by using the physical neutral surface as reference plane. An incremental-iterative procedure based on Newton–Raphson method, in which the plastic equation is solved at Gauss points for updating the stress and evaluating the element formulation, is employed to solve nonlinear equilibrium equations. The elastoplastic behavior is illustrated for a FG beam composed of TiB and Ti. The numerical results show that yielding in the FG beam occurs at the layer near the ceramic surface earlier than it does at the layer near the metal surface. The effect of the material distribution, plastic deformation on the nonlinear behavior of the beam with various end conditions is investigated in detail. The formation and propagation of plastic zone inside the beam during the loading process is also examined and highlighted.

A New Procedure for Nonlinear Dynamic Analysis of Structures Under Seismic Loading Based on Equivalent Nodal Secant Stiffness

This paper presents a dynamic analysis procedure for predicting the responses of large, highly nonlinear, discontinuous structural systems subjected to seismic loading. The concept of equivalent nodal secant stiffness is adopted to diagonalize the conventional stiffness matrix of the structure. With the lumped-mass idealization, the decoupled equilibrium equations of the structure are then solved by the implicit Newmark integration method. Additionally, an incremental-iterative procedure is performed to ensure that the equilibrium conditions are satisfied at the end of each time step. The proposed analysis procedure has the advantages of both the conventional explicit and implicit integration procedures, but with their disadvantages removed. Through extensive applications, the results demonstrate that the proposed procedure is simple and robust for analyzing practical structural systems in terms of computational efficiency and stability.

A new implicit dynamic finite element analysis procedure with damping included

Abstract The concept of equivalent nodal secant damping coefficients is proposed to diagonalize the stiffness-proportional damping matrix. Additionally, a novel method is proposed to rapidly evaluate stiffness-proportional damping nodal forces for nonlinear elements on the element level. With the assumption of lumped-mass idealization an incremental-iterative procedure is performed to solve the decoupled equilibrium equations for damped systems using the implicit Newmark integration method. The proposed analysis procedure has the advantages of both traditional explicit and implicit integration methods without their disadvantages. Through extensive practical implementations, the results demonstrate that the proposed procedure is simple and robust for analyzing practical damped systems with excellent computational efficiency and stability.