Nonlinear Equilibrium Paths Research Articles

Finite element analysis of the static properties and stability of a 800 m Kiewitt type mega-latticed structure

A comparison was made of the static behavior of a double-layer reticulated shell, a local triple-layered reticulated shell, a ribbed type mega-latticed structure and a Kiewitt type mega-latticed structure with the same geometrical parameters and loads. The results show that the Kiewitt type mega-latticed structure has the best structural performance. Then a complex structural configuration was analyzed in detail, using CAD 3D modeling, and a command-flow was developed to create the new structure automatically by ANSYS Parameter Design Language as an innovative structural-programming method. The static analysis and comprehensive stability analysis were conducted. The reasonable values of parameters for the overall buckling mode were summarized and the ultimate bearing capacity was accurately obtained by considering geometrical nonlinear, material nonlinear and initial curvature of the members. The nonlinear equilibrium paths were traced using the Arc-length method. Results indicate that the 800m Kiewitt type mega-latticed structure has a very low sensitivity to geometrical nonlinear and a low sensitivity to initial curvature of members, but a high sensitivity to material nonlinear. Therefore, the Kiewitt type mega-latticed structure could be regarded as a reasonable super-large span structure with several advantages, including reasonable stresses, clear force transmission lines, economic steel consumption, high stiffness and good bearing capacity.

A robust Riks-like path following method for strain-actuated snap-through phenomena in soft solids

In this paper, a practical and robust Riks-like path following method is developed to automatically trace the nonlinear equilibrium path for initial-finite-strain-induced snap-through phenomena in soft solids. The deformation gradient is decomposed into an elastic part and an initial-strain part, where the initial strain is characterized through a prescribed pattern tensor and an unknown strain factor describing the magnitude. In the non-linear finite element (FE) solution scheme, an equivalent body force proportional to the increment of the strain factor is derived, which varies in each iteration step due to its dependence on deformed configuration and stress. Both the displacement and the strain factor increments are introduced as unknowns in the linearized equation, which permits to predict the critical equilibrium states before and after the snap-through. Several examples with various configurations and strain histories are constructed, including multilayer beams and film–substrate systems, which demonstrate the effectiveness and robustness of the algorithm to capture the strain-induced snap-through phenomena.

CFS lipped channel columns affected by L-D-G interaction. Part I: Experimental investigation

This work deals with the structural behaviour and ultimate strength of fixed-ended cold-formed steel (CFS) lipped channel columns experiencing various levels of local-distortional-global buckling mode interaction. After briefly addressing the column specimen geometry selection (carefully chosen to ensure more or less closeness between the critical buckling stresses associated with these three buckling mode types), the paper reports the results of an experimental investigation involving a set of 17 columns with several cross-section dimensions, lengths and yield stresses, which is aimed at (i) providing experimental evidence of the occurrence of the triple mode interaction under consideration and (ii) quantifying its effect on the column deformed configuration evolution and failure load erosion (drop). Various aspects concerning the test set-up and procedure are described in some detail before presenting and discussing the obtained column experimental results, which basically consist of (i) measured cross-section dimensions, lengths and initial displacements (geometrical imperfections), (ii) stress-strain curves and yield stresses determined from tensile coupon tests, (iii) recorded non-linear equilibrium paths (applied load vs. various relevant displacements) and ultimate strength values, and (iv) observed deformed configurations and collapse mechanisms. The test results are used, in a companion paper (Part II), (i) to calibrate and validate numerical simulations and (ii) to support the proposal and to assess the merits of design approaches based on the Direct Strength Method (DSM) and intended to handle L-D-G interactive failures.

Effects of shear deformation on Mechanical and thermo-mechanical nonlinear stability of FGM shallow spherical shells subjected to uniform external pressure

In the present paper, the asymmetrical nonlinear response of a clamped functionally graded shallow spherical shell is subjected to uniform external pressure. It considers the effects of thermal stresses by both of the theories, Classical Laminate Theory, CLT and First-Order Shear Deformation Theory FSDT. Material properties are graded in the thickness direction according to the power-law distribution in terms of the volume fraction of the constituents. Mechanical and thermo-mechanical properties are assumed to be temperature-independent and linear elastic. All of the governing equations are derived by aid of first-order transverse shear deformation theory considering geometrical nonlinearity. The nonlinear differential equation system is solved employing Galerkin method. Buckling and post-buckling analysis have been done according to one-term deformation mode by the closed form relation of load-deflection that shows the equilibrium path. Parametric studies are conducted to bring out the effects of shear deformation on the equilibrium path in different geometries and boundary conditions. Numerical results are presented in graphical arrangement, showing that the geometrical nonlinear equilibrium paths. The effects of shear deformation on the equilibrium path are considered by comparing the results from FSDT and CLT and was verified by nonlinear finite-element method.

Linear and geometrically nonlinear analysis of non-uniform shallow arches under a central concentrated force

In this paper an integral equation solution to the linear and geometrically nonlinear problem of non-uniform in-plane shallow arches under a central concentrated force is presented. Arches exhibit advantageous behavior over straight beams due to their curvature which increases the overall stiffness of the structure. They can span large areas by resolving forces into mainly compressive stresses and, in turn confining tensile stresses to acceptable limits. Most arches are designed to operate linearly under service loads. However, their slenderness nature makes them susceptible to large deformations especially when the external loads increase beyond the service point. Loss of stability may occur, known also as snap-through buckling, with catastrophic consequences for the structure. Linear analysis cannot predict this type of instability and a geometrically nonlinear analysis is needed to describe efficiently the response of the arch. The aim of this work is to cope with the linear and geometrically nonlinear problem of non-uniform shallow arches under a central concentrated force. The governing equations of the problem are comprised of two nonlinear coupled partial differential equations in terms of the axial (tangential) and transverse (normal) displacements. Moreover, as the cross-sectional properties of the arch vary along its axis, the resulting coupled differential equations have variable coefficients and are solved using a robust integral equation numerical method in conjunction with the arc-length method. The latter method allows following the nonlinear equilibrium path and overcoming bifurcation and limit (turning) points, which usually appear in the nonlinear response of curved structures like shallow arches and shells. Several arches are analyzed not only to validate our proposed model, but also to investigate the nonlinear response of in-plane thin shallow arches.

Numerical study for cohesive zone model in delamination analysis based on higher-order B-spline functions

A numerical aspect of the implementation cohesive zone model for delamination analysis is presented in this work by employing interface elements and higher-order B-spline functions. The stress oscillation is addressed, and Newton–Cotes integration scheme is considered as a good candidate to overcome this phenomenon. In order to track the nonlinear equilibrium path, a general arc-length constraint named dissipation-based arc-length method is applied. A numerical example is presented to perform the ability of the formulation in predicting the delamination behavior of studied structures.

Stability Analysis of a Shallow Shell

Elastic very shallow shell subjected to the external pressure is analysed in the paper numerically by FEM. Nonlinear equilibrium paths are calculated for the different boundary conditions. ANSYS system was used for nonlinear analysis, arc-length method was chosen to obtain fundamental load-displacement path of solution. Effects of geometrical imperfections on equilibrium paths are investigated. Presented results confirm the fact that the nonlinear approach is necessary. The difference between the critical load of buckling analysis and the load value at the limit point of load-displacement path of nonlinear solution is in the tens of percent (depending on the boundary conditions).

Snap-Through of the Very Shallow Shell with Initial Imperfection

Abstract Elastic shallow shell of translation subjected to the external pressure is analysed in the paper numerically by FEM. Nonlinear equilibrium paths are calculated for the different boundary conditions. Special attention is paid to the influence of initial imperfection on the limit load level of fundamental load-displacement path of nonlinear analysis. ANSYS system was used for analysis, arclength method was chosen for obtain fundamental load-displacement path of solution.

Nonlinear bending, postbuckling and snap-through of circular size-dependent functionally graded piezoelectric plates

In the present work, the nonlinear thermoelectrical stability of perfect/imperfect circular size-dependent functionally graded piezoelectric plates is studied according to modified couple stress theory. The second, concurrent aim is to address snap-through phenomenon in the thermally preloaded plates due to concentrated/uniform lateral loads. Ritz finite element method is implemented into virtual displacement principle to construct the matrix representation of nonlinear governing equations. Under certain circumstances, bifurcational instability may occur in which case a direct displacement control scheme is utilized. In other cases, the response is unique and stable to which any standard load control strategy seems appropriate and thus Newton-Raphson method is selected. Standard load control strategies, however, fail to trace nonlinear equilibrium paths through limit points and path following methods must be employed in snap-through problems. Being more popular among the existing path following solution methods, cylindrical arc-length method is adopted. Two types of thermal loading as well as two cases of boundary conditions are considered. Moreover, various parametric studies are conducted to assess the influence of involved parameters.

Koiter–Newton analysis of thick and thin laminated composite plates using a robust shell element

The Koiter–Newton method is a novel reduced-order modeling technique to efficiently trace the geometrically nonlinear equilibrium path of the structure in the presence of buckling. In this paper, the existing method is extended for nonlinear buckling analysis of thick and thin laminated composite plates. A 4-node quadrilateral element S4AT is developed as a geometric linear element in the co-rotational formulation of the Koiter–Newton method. The developed element S4AT is a combination of the membrane part and the plate-bending part. The quadrilateral area coordinate method is applied to make the element insensitive to the mesh distortion. The Reissner–Mindlin laminated composite plate theory is used to consider the transverse shear deformation effect, and the shear-locking phenomenon is eliminated using the Timoshenko laminated composite beam theory. The performance of the method in terms of reliability, accuracy and computational effort is demonstrated with several examples.

Confined collapse of unbonded multi-layer pipe subjected to external pressure

If a leakage exists on the outer sheath of an unbonded multi-layer pipe, external hydrostatic pressure will be applied on the surface of the internal sheath inducing the possibility of a confined collapse. Also, when the pipe bore is depressurized at a fast rate, the pressure remaining in the pipe wall can cause the similar collapse of the internal sheath. The present paper uses a flexible pipe with multi-layer structure to examine the mechanical response of the internal cylinder confined by outer cylinders under external pressure. The previous related researches mainly focus on the rigid or almost rigid confinement conditions. However, this study emphasizes on structural elastic stability in terms of the different confinement levels and D/t ratios of the internal cylinder. The initial investigation is numerical and employs two-dimensional models to obtain the nonlinear pressure-deformation equilibrium paths using a Riks algorithm. The results are fitted as uniform formulas for the prediction to the critical pressure and corresponding collapse modes. The simplified analytical model followed is developed through the principle of virtual work to compare with the fitting formulas. Experimental tests of an unbonded flexible pipe with simple construction are also conducted to validate and agree with the theoretical estimations.

Thermal Buckling and Postbuckling Analysis of Functionally Graded Carbon Nanotube-Reinforced Composite Beams

Thermal buckling and postbuckling of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams are investigated in this paper based on Timoshenko beam theory within the framework of von Kármán geometric nonlinearity. The material properties of FG-CNTRCs are assumed to be temperature-dependent and vary in the beam thickness direction. The governing equations are derived by employing Hamilton’s principle then discretized by using differential quadrature (DQ) method. An iterative scheme is used to obtain the critical buckling temperature and nonlinear thermal postbuckling equilibrium path of the FG-CNTRC beam. Numerical results are presented for FG-CNTRC beams hinged or clamped at both ends, with particular focuses on the effects of the volume fraction of carbon nanotubes (CNTs), slenderness ratio, and end supports on the thermal buckling and postbuckling characteristics.

Analytical solution for thermal buckling analysis of rectangular plates with functionally graded coatings

In this paper, an analytical solution is developed to investigate the thermal buckling behavior of the imperfect rectangular plates with functionally graded (FG) coatings under uniform temperature rise. Material properties of the FG coatings are assumed to be temperature-dependent and to continuously vary in the thickness direction using the power law distribution of volume fraction of metal and ceramic. Theoretical formulations are based on the classical plate theory with von-Karman nonlinear kinematic relations. The initial geometrical imperfections of plates are also accounted. By using the Galerkin method and the Airy stress function, the resulting equations are solved to obtain the closed form expressions for nonlinear equilibrium paths. Effects of power law index, imperfection, geometric parameters, and temperature distributions on the response of rectangular plates with FG coatings are discussed in details.

Stability Analysis of an Imperfect Slender Web Subjected to the Shear Load

The stability analysis of an imperfect slender web subjected to the shearing load is presented, a specialized code based on FEM has been created. The nonlinear finite element method equations are derived from the variational principle of minimum of total potential energy. To obtain the nonlinear equilibrium paths, the Newton-Raphson iteration algorithm is used. Corresponding levels of the total potential energy are defined. The peculiarities of the effects of the initial imperfections are investigated. Special attention is paid to the influence of imperfections on the post-critical buckling mode. Obtained results are compared with those gained using ANSYS system.

A robust macroscopic model for normal–shear coupling, asymmetric and anisotropic behaviors of polycrystalline SMAs

The aim of this article is to develop a robust macroscopic bi-axial model to capture self-accommodation, martensitic transformation/orientation/reorientation, normal–shear deformation coupling and asymmetric/anisotropic strain generation in polycrystalline shape memory alloys. By considering the volume fraction of martensite and its preferred direction as scalar and directional internal variables, constitutive relations are derived to describe basic mechanisms of accommodation, transformation and orientation/reorientation of martensite variants. A new definition is introduced for maximum recoverable strain, which allows the model to capture the effects of tension–compression asymmetry and transformation anisotropy. Furthermore, the coupling effects between normal and shear deformation modes are considered by merging inelastic strain components together. By introducing a calibration approach, material and kinetic parameters of the model are recast in terms of common quantities that characterize a uniaxial phase kinetic diagram. The solution algorithm of the model is presented based on an elastic-predictor inelastic-corrector return mapping process. In order to explore and demonstrate capabilities of the proposed model, theoretical predictions are first compared with existing experimental results on uniaxial tension, compression, torsion and combined tension–torsion tests. Afterwards, experimental results of uniaxial tension, compression, pure bending and buckling tests on rods and tubes are replicated by implementing a finite element method along with the Newton–Raphson and Riks techniques to trace non-linear equilibrium path. A good qualitative and quantitative correlation is observed between numerical and experimental results, which verifies the accuracy of the model and the solution procedure.

Nonlinear thermal stability and snap-through behavior of circular microstructure-dependent FGM plates

Abstract The aim of the present work is twofold. The First is to present a study on nonlinear thermal stability of perfect/imperfect circular size-dependent functionally graded plates according to modified couple stress theory. The second, concurrent aim is to address snap-through phenomenon in the thermally preloaded plates due to concentrated/uniform lateral loads. Ritz finite element method is implemented into generalized form of Hamilton’s principle to construct the matrix representation of nonlinear governing equations. Under certain circumstances, bifurcation type instability may occur in which case a direct displacement control scheme is utilized. In other cases, the response is unique and stable to which any standard load control strategy seems appropriate and thus Newton-Raphson method is selected. Standard load control strategies, however, fail to trace nonlinear equilibrium paths through limit points and path following methods must be employed in snap-through problems. Being more popular among the existing path following solution methods, cylindrical arc-length method is adopted. Two types of thermal loading as well as two cases of boundary conditions are considered. Moreover, various parametric studies are conducted to assess the influence of involved parameters.

Closed-form expression for nonlinear analysis of imperfect sigmoid-FGM plates with variable thickness resting on elastic medium

With the aim of reducing the weight of structures, functionally graded materials (FGM) plates with variable thickness have been widely used in various engineering applications such as aeronautical, mechanical and ocean structures. However so far, the analytical approaches for analyzing the instability behaviors of FGM plates with variable thickness are still somehow limited in the literature. The paper hence presents an analytical approach to investigate the influences of variable thickness on buckling and postbuckling behavior of imperfect sigmoid FGM (S-FGM) plates resting on elastic medium subjected to compressive loading. The material properties of S-FGM plates are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Governing equations are based on the classical plate theory with von Karman-type geometric nonlinearity. The initial geometrical imperfections of plates are also accounted. By using the Galerkin procedure and the Airy stress function, the resulting equations are solved to obtain the closed form expressions for nonlinear equilibrium paths. The effects of power-law indices, coefficients of foundation, initial geometrical imperfections and geometrical parameters on nonlinear stability of plates are comprehensively investigated. The results reveal that the variable thickness of plate has a significant effect on the buckling behavior of S-FGM plates under compression loading.

Buckling And Postbuckling Of An Imperfect Plate Subjected To The Shear Load

Abstract The stability analysis of an imperfect plate subjected to the shear load is presented. To solve this problem, a specialized computer program based on FEM has been created. The nonlinear finite element method equations are derived from the variational principle of minimum of total potential energy. To obtain the nonlinear equilibrium paths, the Newton-Raphson iteration algorithm is used. Corresponding levels of the total potential energy are defined. Special attention is paid to the influence of imperfections on the post-critical buckling mode. Obtained results are compared with those gained using ANSYS system.

Effect of a Nonlinear Term in a Strain-Displacement Relationship on the Load-Displacement Path of Von Mises Truss

Von Misses truss is one of the best examples to explain different theoretical approaches, nature of non-linear solution, define the snap-through, illustrate interactive buckling, etc. The presented paper compares two nonlinear approaches to the problem. Effect of nonlinear terms in strain-displacement relationship on the load level in critical point of nonlinear solution is analyzed. To obtain the nonlinear equilibrium paths, the Newton-Raphson iteration algorithm is used. Corresponding levels of the total potential energy are defined. The peculiarities of the effects of the initial imperfections are investigated. Custom FEM computer program has been used for analysis. Full Newton-Raphson procedure, in which the stiffness matrix is updated at every equilibrium iteration, has been applied. Obtained results are compared with results of the nonlinear analysis using ANSYS system, element type BEAM3 is used. The arc-length method is chosen for analysis, the reference arc-length radius is calculated from the load increment. Only fundamental path of nonlinear solution has been presented.

Effect of a Bending Stiffness on the Load-Displacement Path of Von Mises Truss

Von Misses truss is one of the best examples to explain nature of non-linear solution and define the snap-through. Linear buckling analysis and nonlinear finite element approaches are compared in presented paper. At the present time theoretical models for the evaluation of the ultimate load assume a structure with imperfections. The peculiarities of the effect of the magnitude and mode of initial imperfections are investigated. Effect of member stiffness on the load level in critical point of nonlinear solution, as well as the relative position with respect to the critical load from buckling analysis are analyzed. To obtain the nonlinear equilibrium paths, Newton-Raphson iteration algorithm has been used. Obtained results are compared with those gained using ANSYS system.