Complete Equilibrium Path Research Articles

Nonlinear in-plane buckling of fixed FGP-GPLRC arches subjected to a half-span distributed radial load

In this work, nonlinear in-plane buckling analysis of fixed functionally graded porous graphene nanoplatelet reinforced composite (FGP-GPLRC) shallow arches under a half-span distributed radial load is presented. Material properties of the arch with different porosity distribution modes are determined via a modified Halpin–Tsai micromechanical model. Neutral plane-based nonlinear governing equations of motion are established based on the principle of virtual work from which analytical solutions for the critical buckling load are derived and the complete equilibrium path is traced. Possible buckling modes of fixed FGP-GPLRC arches are determined by employing a perturbation technique. Key parameters controlling the buckling configuration switching are also proposed. FE analysis is then carried out to verify the accuracy of the presented solutions. Effects of porosity distributions, GPL weight fraction, and porosity coefficient on the buckling behaviors are comprehensively examined. It was found that under the action of a half-span distributed radial load, the fixed FGP-GPLRC shallow arches buckle in a limit point configuration only, and the equilibrium path especially the per-buckling branch is significantly affected by porosity coefficient.

Geometrically nonlinear behavior of lattice domes coupled with local Eulerian instability

Abstract Structural analysis is an intricate subject when nonlinearities occur. They make the structural behavior complex and may have important consequences in the design choice as well. Especially for lattice domes, as snap-through phenomena and local Eulerian instabilities generally affect the structural response, linear analysis is not enough. In this paper, a semi-analytical formulation is used in order to study the geometrically nonlinear behavior of lattice domes subject to vertical loads. The formulation is derived from the equilibrium equations written in the deformed configuration, considering large displacements and taking also into account local buckling conditions. The resulted system of equations, being strongly nonlinear, has been solved by means of a numerical procedure, based on a mixed load-displacement control scheme, leading to the evaluation of the complete equilibrium path. The influence of geometrical parameters on the critical load multiplier and shape of the load-displacement curve is also discussed. In particular, a complex equilibrium path for a sixteen-member five-node lattice structure is analyzed, which is characterized by several branches which can generate ‘snapping’ conditions.

An efficient mixed variational reduced‐order model formulation for nonlinear analyses of elastic shells

SummaryThe Koiter‐Newton method had recently demonstrated a superior performance for nonlinear analyses of structures, compared to traditional path‐following strategies. The method follows a predictor‐corrector scheme to trace the entire equilibrium path. During a predictor step, a reduced‐order model is constructed based on Koiter's asymptotic postbuckling theory that is followed by a Newton iteration in the corrector phase to regain the equilibrium of forces. In this manuscript, we introduce a robust mixed solid‐shell formulation to further enhance the efficiency of stability analyses in various aspects. We show that a Hellinger‐Reissner variational formulation facilitates the reduced‐order model construction omitting an expensive evaluation of the inherent fourth‐order derivatives of the strain energy. We demonstrate that extremely large step sizes with a reasonable out‐of‐balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path. More importantly, the numerical effort of the corrector phase involving a Newton iteration of the full‐order model is drastically reduced thus revealing the true strength of the proposed formulation. We study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems.

Survey on the interface behaviour in reinforced concrete beams strengthened with externally bonded FRP reinforcement

Abstract The use of FRP materials to strengthen reinforced concrete elements has gained more and more popularity in the last decades, for their relevant properties such as the high strength to weight ratio and the excellent corrosion resistance. However, debonding of FRP lamina from the substrate may prevent the achievement of the full load bearing capacity of the reinforced element. In the present study, a novel numerical method recently proposed by the authors is adopted in order to model 25 experimental tests available in literature. The numerical approach, able to follow the complete equilibrium path of a reinforced concrete beam externally strengthened by means of FRP sheets, let shedding further light on the behaviour of the interface between concrete and FRP lamina, during the different stages in the loading process from the service behaviour to the ultimate limit state. A comparison with the approach proposed by the codes shows that the error committed when adopting the hypothesis of perfect bond is still acceptable.

Elastoplastic analysis of pressure‐sensitive materials by an effective three‐dimensional mixed finite element

This work is focused on the analysis of three‐dimensional bodies whose mechanical behavior can be modeled by an elastoplastic pressure‐sensitive material description. To this end the yield condition is assessed with respect to a general quadratic function capable to represent both standard surfaces, such as Drucker–Prager surface, or more generic surfaces. For the sake of the efficiency and robustness of the numerical analysis of general‐shape solids, an effective mixed tetrahedral finite element is used to perform a step‐by‐step analysis obtaining the complete equilibrium path and the collapse load. Some numerical results regarding technical problems, such as masonry walls and footing problems, show the effectiveness of the mechanical model and of the numerical strategy.

Modified force method for the nonlinear analysis of FRP reinforced concrete beams

Among the different techniques adopted to retrofit structures that had become inadequate, the use of fiber reinforced composite materials (FRP) has gained more and more popularity for their relevant properties such as high strength to weight ratio and excellent corrosion resistance. However, debonding of FRP lamina from the substrate, made of either concrete or steel, may prevent the achievement of the full load bearing capacity of the reinforced element. In this paper a numerical method, able to follow the complete equilibrium path of a reinforced concrete beam externally strengthened by means of FRP sheets up to the complete detachment of the external reinforcement, is proposed. According to this method, the FRP sheets are separated from the reinforced concrete substructure by displaying the interface stresses that are taken into account by means of the corresponding interface slips (which are the primary unknowns). Compatibility equations are then imposed at the interface in order to restore the behavior of the externally reinforced beam. The method is first validated by benchmarking experimental results taken from literature and then exploited to perform a parametric analysis in order to point out the effect of the main design parameters on the response of the reinforced beam.

A crack length control scheme for solving nonlinear finite element equations in stable and unstable delamination propagation analysis

An automatic incremental solution algorithm for nonlinear static finite element analysis of delamination propagation problems is presented. This load–displacement–constraint method iterates in the load displacement space utilizing the Newton–Raphson method, allowing to trace out the complete equilibrium path. The procedure can calculate pre- and post-critical responses of bi-dimensional problems, ranging from stable delamination growth to structural collapse by unstable growth or delamination buckling with arbitrarily sharp snap-back instabilities. Benchmark solution examples of established composite materials structural problems illustrate the effectiveness of the method.

Complete equilibrium paths for Mises trusses

This paper presents determination of equilibrium paths for Mises trusses with different ratio of height to span. Unsymmetrical deformation modes are considered and the structure is treated as a two DOF system. First, a few special equilibrium configurations are resolved from considerations of free body diagrams. Complete equilibrium paths are determined by solving numerically the governing non-linear equilibrium equations. The stability of possible equilibrium configurations is checked using the second partial derivative test for the total potential energy. The positive definiteness of the appropriate Hessian matrices is checked numerically using the Sylvester criterion.

Ultimate Strength and Ductility Characteristics of Intermittently Welded Stiffened Plates Under In-Plane Axial Compression

The ultimate strength and ductility characteristics of the intermittently welded stiffened plates under the action of in-plane axial compression are investigated in this paper. A series of detailed numerical analyses of longitudinally stiffened steel plates subjected to in-plane compressive load is performed using the ADINA commercial finite element code. Complete equilibrium paths are traced up to collapse for a nonlinear elastic-plastic response of stiffened plates. Analyzed stiffened plates are imperfect and their aspect ratio, plate slenderness, and column slenderness are changed in a systematic manner. Different types of stiffener are chosen for stiffened plate models. Three different stiffener-to-plate welding procedures are considered: continuous, chain intermittent, and staggered intermittent fillet welding.

A comparative study on three different construction methods of stiffened plates-strength behaviour and ductility characteristics

Strength and ductility characteristics of non-continuously welded stiffened plates under in plane axial compression are the main focus of this research. A series of detailed numerical analyses of stiffened steel plates subjected to in plane compressive load is performed. Complete equilibrium paths are traced up to collapse for non-linear elastoplastic response of stiffened plates. Stiffened plates are selected from the deck structure of real sea-going ships and inland waterway vessels. Three different stiffener-to-plate welding procedures are considered: continuous, chain intermittent and staggered intermittent fillet welding. Special attention is paid to the finite element modeling of the fillet welds in either of welding practices. Some available tests are simulated applying finite element method.

Numerical study on the permissible gap of intermittent fillet welds of longitudinally stiffened plates under in plane axial compression

Intermittent fillet welding of stiffeners to plates and its influence on the collapse behaviour of stiffened plates is investigated applying the finite element method. Special attention is paid to the modelling of the fillet welds at the plate-to-stiffener junction. Some available experimental results are simulated in order to obtain reliable numerical results. A series of numerical analyses of stiffened steel plates subjected to an in plane axial compressive load has been performed. Stiffened plates are selected from the deck structure of real sea-going ships and inland waterway vessels. Complete equilibrium paths are traced up to collapse for the non-linear elastoplastic response of stiffened plates. Finally a proposal is presented for the permissible gap of welds in intermittent fillet welding of stiffened plates.

A finite element formulation for finite static axisymmetric deformation of hyperelastic membranes

A simple finite element formulation is presented in this paper for finite static axisymmetric deformation of isotropic incompressible hyperelastic membranes. This finite element procedure is derived from a recently published finite deformation membrane theory, in which Lagrangian type equilibrium equations, expressed in terms of the Biot stresses, are employed along with constitutive equations relating the principal components of the Biot stress tensor and of the principal stretches. Through consistent linearization of the equilibrium equations, a closed-form expression is obtained for the tangent stiffness matrix. In order to obtain the complete equilibrium path, the Newton-Raphson incremental-iterative method with displacement control is utilized. Numerical results are presented for inflation of a circular hyperelastic membrane. Various hyperelastic materials are considered.