Primary Equilibrium Path Research Articles

In-plane nonlinear postbuckling analysis of circular arches using absolute nodal coordinate formulation with arc-length method

In this study, Absolute Nodal Coordinate Formulation (ANCF) in conjunction with Crisfield’s arc-length method is utilized in order to predict the nonlinear postbuckling behaviour of circular arches. The whole primary equilibrium path in load-displacement space of circular arches under central concentrated load is obtained. Three ANCF based approaches, i.e., the conventional two-dimensional fully parameterized shear deformable ANCF beam element based on the General Continuum Mechanics (GCM) approach, the same element modified by the Strain Split Method (SSM) approach and the ANCF planar Higher Order Beam Element (HOBE) with GCM approach are used. Circular arches with various geometric configurations and boundary conditions such as clamped-clamped, hinged-hinged, clamped-hinged and three-hinged arches are studied which exhibit nonlinear response in the form of snap-through, snap-back and looping phenomenon. The obtained results are compared with the analytical solutions, experimental result (where available in the literature) and numerical approximations (by using the commercially available FEM package). In this paper, the recently proposed ANCF based approaches are successfully implemented which validate and verify the utility of ANCF in nonlinear postbuckling analysis. The characteristics of the three approaches with regard to the adoptability of arc-length method are compared and discussed.

Non-linear in-plane stability analysis of FGM circular shallow arches under central concentrated force

A comprehensive study on the non-linear in-plane stability behavior of shallow arches made of functionally graded materials (FGMs) is presented in this work. Simply supported–simply supported (S–S) and clamped–clamped (C–C) boundary conditions are considered as two types of well-known symmetric boundary conditions for this analysis. The arch is subjected to a central concentrated force and material dispersion is according to the power law distribution. For this aim, the classical single layer theory is adjusted to approximate the displacement field through the arch. Kinematical relations are reduced to suitable ones for shallow arches. Static version of the virtual displacement principle is used to obtaining the governing equations and the complete set of boundary conditions. In the presence of the highly non-linear behavior of shallow arches under central concentrated force, buckling analysis is preformed in the presence of pre-buckling deformations. Existence of secondary equilibrium paths for shallow arches is studied and stability behavior of FGM shallow arches is classified into non-linear bending, full snap-through, bifurcation from post-snap path, and bifurcation. Also, multiple snap-to-state condition is investigated for FGM shallow arches. Results are presented as primary equilibrium paths and effect of material dispersion, geometrical characteristics, and boundary conditions on the stability behavior of shallow arches under central concentrated force is studied.

Non-linear thermo-elastic and buckling analysis of FGM shallow arches

The paper presents a theoretical investigation on the non-linear thermo-elastic behaviour of pin-ended functionally graded material (FGM) circular shallow arches. Temperature dependency of constituents is taken into account and the arch is subjected to a uniform temperature field. Classical arches theory along with the non-linear shallow shell theory of Donnell are postulated as the basic assumptions. The virtual displacement principle and calculus of variation techniques are implemented to obtain the equilibrium equations. An analytical solution is presented to trace the primary equilibrium path of a pin-ended arch under in-plane thermal loading. Afterwards, closed form expressions are yielded to obtain the radial and axial displacement, stress, strain and bending moment of the arch as a function of temperature parameter. Adjacent equilibrium criterion is used to extract the stability equation associated to the non-linear primary equilibrium path. A closed form solution is presented to estimate the fundamental thermal bifurcation points of the arch. Illustrative results examine the role of the various involved parameters such as power law index, opening angle and length to thickness ratio. Numerical results reveal that, in most cases, critical buckling temperature difference of the FGM shallow arches are too high, even for thin class of arches.

Equilibria and stability boundaries of shallow arches under static loading in a thermal environment

The structural behaviour of shallow arches is complex and can be influenced by many parameters. In this paper, the response of a half-sine shallow arch under static loading in a thermal environment is investigated. The arch has pinned supports and the material behaviour is assumed elastic. The exact displacement field, load-bearing capacity and the locus of critical points are obtained. Boundaries of domains with different stability behaviours (e.g., different number of limit and bifurcation points) are also determined. Three types of loading (concentrated, uniform and asymmetrical uniform) are examined. The primary equilibrium paths are verified against results obtained from finite element simulations. The proposed method is robust and accurate.

Equilibrium Paths of a Fractional Order Viscoelastic Two-Member Truss

The fractional dynamics equation of a viscoelastic two-member truss system, in which fractional derivative model introduced to simulate the materials’ characteristics, is proposed. The equilibrium paths under vertical loads, horizontal loads and combined loads are discussed respectively. The results show that: there are primary and secondary equilibrium paths under vertical and horizontal loads. Bifurcations will occur under combined loads and the ultimate bearing capacity of the system will reduce. The equilibrium paths became complex because of the horizontal disturbance, and the bigger of the horizontal disturbance the smaller of the system ultimate bearing capacity.

Nonlinear dynamic buckling of shallow circular arches under a sudden uniform radial load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly applied load is sufficiently large, the oscillation of the arch may reach a position at its primary unstable equilibrium path or secondary bifurcation unstable equilibrium path, leading the arch to buckle dynamically. This paper presents an analytical study of the nonlinear dynamic in-plane buckling of a shallow circular arch under a uniform radial load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. It is shown that under a suddenly applied uniform radial load, a shallow pinned–fixed arch has a unique possible dynamic buckling load, while shallow pinned–pinned and fixed–fixed arches may have two possible dynamic buckling loads: a lower dynamic buckling load and an upper dynamic buckling load. The dynamic buckling loads of a shallow arch under a suddenly applied uniform radial load with infinite duration are found to be lower than their static counterparts, and to increase with an increase of the arch included angle and slenderness. The effect of static preloading on the dynamic buckling of an arch is also investigated. It is found that the pre-applied static load decreases the dynamic buckling load of the arch, but increases the sum of the pre-applied load and the dynamic buckling load.

On the interdependency of primary and initial secondary equilibrium paths in sensitivity analysis of elastic structures

The scientific motivation for this paper is lack of clarity about the interdependency of primary and initial secondary equilibrium paths in the frame of sensitivity analysis of elastic structures. The investigation of this interdependency comprises of the following four cases: (1) nonlinear primary path, nonlinear stability problem, (2) linear primary path, nonlinear stability problem, (3) nonlinear primary path, linear stability problem, and (4) linear primary path, linear stability problem. The consistently linearized eigenproblem is used for differentiation of two classes of nonlinear stability problems with markedly different characteristics of both the prebuckling and the postbuckling behavior. For one of them, e.g. zero-stiffness postbuckling is impossible. For the other one, which is restricted to a prebuckling regime with axial deformations only, sensitivity analysis of the initial postbuckling behavior either exhibits its continuous improvement or its continuous deterioration, depending on whether the bifurcation point diverges from or converges to the snap-through point. In other words, a monotonic variation of the design parameter cannot result in a non-monotonic change of the initial postbuckling behavior. The practical motivation for this work is to explore the mechanical reasons for qualitatively different modes of transition from imperfection sensitivity to insensitivity in the course of sensitivity analysis for the purpose of improving the postbuckling behavior of structures by means of minor design changes. Results from a numerical investigation corroborate the theoretical findings.

ENERGY APPROACH FOR DYNAMIC BUCKLING OF AN UNDAMPED ARCH MODEL UNDER STEP LOADING WITH INFINITE DURATION

Performing a dynamic buckling analysis of structures is more difficult than carrying out its static buckling analysis counterpart. Some structures have a nonlinear primary equilibrium path including limit points and an unstable equilibrium path. They may also have bifurcation points at which equilibrium bifurcates from the primary equilibrium path to an unstable secondary equilibrium path. When such a structure is subjected to a load that is applied suddenly, the oscillation of the structure may reach the unstable primary or secondary equilibrium path and the structure experiences an escaping-motion type of buckling. For these structures, complete solutions of the equations of motion are usually not needed for a dynamic buckling analysis, and what is really sought are the critical states for buckling. Nonlinear dynamic buckling of an undamped two degree-of-freedom arch model is investigated herein using an energy approach. The conditions for the upper and lower dynamic buckling loads are presented. The merit of the energy approach for dynamic buckling is that it allows the dynamic buckling load to be determined without the need to solve the equations of motion. The solutions are compared with those obtained by an equation of motion approach.

Energy approach for dynamic buckling of shallow fixed arches under step loading with infinite duration

Shallow fixed arches have a nonlinear primary equilibrium path with limit points and an unstable postbuckling equilibrium path, and they may also have bifurcation points at which equilibrium bifurcates from the nonlinear primary path to an unstable secondary equilibrium path. When a shallow fixed arch is subjected to a central step load, the load imparts kinetic energy to the arch and causes the arch to oscillate. When the load is sufficiently large, the oscillation of the arch may reach its unstable equilibrium path and the arch experiences an escaping-motion type of dynamic buckling. Nonlinear dynamic buckling of a two degree-of-freedom arch model is used to establish energy criteria for dynamic buckling of the conservative systems that have unstable primary and/or secondary equilibrium paths and then the energy criteria are applied to the dynamic buckling analysis of shallow fixed arches. The energy approach allows the dynamic buckling load to be determined without needing to solve the equations of motion.

Is hilltop buckling imperfection sensitive or insensitive?

AbstractHilltop buckling is the coincidence of a snap‐through point with a bifurcation point on the primary equilibrium path. An originally imperfection sensitive structure cannot be modified such that it shows both hilltop buckling and imperfection insensitivity. This is a rule for designing structures for imperfection insensitivity and guarantees validity of Koiter's postbuckling analysis, which would fail for imperfection insensitive hilltop buckling. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Effects of thickness and transverse shear modulus nonlinearity on the post-“yield” and post-localization response of an externally pressurized imperfect cross-ply ring

Abstract The present investigation is concerned with the prediction of localization (onset of deformation softening) and post-localization and post-“yield” equilibrium paths for moderately thick and thick cross-ply [90/0/90] imperfect plane strain rings. These paths are often unstable in the presence of modal imperfections and material nonlinearity, and are considered to “bifurcate” from the primary equilibrium paths, representing periodic buckling patterns pertaining to global or structural level stability. The fully nonlinear finite element analysis, based on the total Lagrangian formulation, employs a three-dimensional theory, known as layer-wise linear displacement distribution theory (LLDT), to capture the three-dimensional interlaminar (especially, shear) deformation behavior, associated with the localized interlaminar shear-crippling failure. The combined effects of modal imperfections, interlaminar shear/normal deformation and nonlinear (hypo-elastic) material property for the transverse shear modulus, G TT , on the localization and delocalization (deformation hardening) phenomena are thoroughly investigated, and physically meaningful conclusions are drawn from these numerical results.

Sensitivity of the post-localization response of a thick cross-ply imperfect ring to transverse Young’s modulus nonlinearity

Abstract A fully nonlinear finite elements analysis for prediction of localization (onset of deformation softening) and post-localization response of a thick cross-ply [90/0/90] imperfect plane strain ring (infinitely long cylindrical shell) under applied hydrostatic pressure is presented. The present investigation is concerned with the prediction of post-“yield” and post-localization equilibrium paths for moderately thick and thick cross-ply rings, which are often unstable in the presence of modal imperfections and material nonlinearity, and which are considered to “bifurcate” from the primary equilibrium paths, representing periodic buckling patterns pertaining to global or structural level stability. The present nonlinear finite element solution methodology, based on the total Lagrangian formulation, employs a quasi-three-dimensional hypothesis, known as layer-wise linear displacement distribution theory (LLDT) to capture the three-dimensional interlaminar (especially, shear) deformation behavior, associated with the localized interlaminar shear-crippling failure. The combined effects of modal imperfections, interlaminar shear/normal deformation and nonlinear (hypoelastic) material property for the transverse Young’s modulus, ETT, on the localization phenomenon are thoroughly investigated, and physically meaningful conclusions are drawn from these numerical results.

A family of structures for the double cusp catastrophe

Different types of loss of stability of elastic structures are usually illustrated by simple models. This paper presents a family of structures which can demonstrate four cases of the double cusp catastrophe by the variation of a parameter. The path defined by this parameter is calculated in the diagram of the classes of the double cusp catastrophe. In one of the four cases the primary equilibrium path of the perfect structure intersects a secondary equilibrium surface at the critical point. In the other cases there are two secondary equilibrium paths, one of which belongs to critical state in all points. In all four cases the equilibrium paths of the imperfect structure are determined as well as the critical load in terms of the value of the simplest imperfections.

Large rotations in first-order shear deformation FE analysis of laminated shells

The paper deals with the geometrically non-linear analysis of laminated composite beams, plates and shells in the framework of the first-order transverse shear deformation (FOSD) theory. A central point of the present paper is the discussion of the relevance of five- and six-parameter variants, respectively, of the FOSD hypothesis for large rotation plate and shell problems. In particular, it is shown that the assumption of constant through-thickness distribution of the transverse normal displacements is acceptable only for small and moderate rotation problems. Implications inherent in this assumption that are incompatible with large rotations are discussed from the point of view of the transverse normal strain–displacement relations as well as in the light of an enhanced, accurate large rotation formulation based on the use of Euler angles. The latter one is implemented as an updating process within a Total Lagrangian formulation of the six-parameter FOSD large rotation plate and shell theory. Numerical solutions are obtained by using isoparametric eight-node Serendipity-type shell finite elements with reduced integration. The Riks–Wempner–Ramm arc-length control method is used to trace primary and secondary equilibrium paths in the pre- and post-buckling range of deformation. A number of sample problems of non-linear, large rotation response of composite laminated plate and shell structures are presented including symmetric and asymmetric snap-through and snap-back problems.

Dynamic buckling of a 2-DOF imperfect system with symmetric imperfections

Non-linear dynamic buckling of a two-degree-of freedom (2-DOF) imperfect planar system with symmetric imperfections under a step load of infinite duration (autonomous system) is thoroughly discussed using energy and geometric considerations. This system under the same load applied statically exhibits (prior to limit point) an unstable symmetric bifurcation lying on the non-linear primary equilibrium path. With the aid of the total energy-balance equation of the system and the particular geometry (due to symmetric imperfections) of the plane curve corresponding to the zero level total potential energy “surface” exact dynamic buckling loads are obtained without solving the non-linear initial-value problem. The efficiency and the reliability of the technique proposed herein is demonstrated with the aid of various dynamic buckling analyses which are compared with numerical simulation using the Verner–Runge–Kutta scheme, the accuracy of which is checked via the energy-balance equation.

Localization and shear-crippling (kinkband) instability in a thick imperfect laminated composite ring under hydrostatic pressure

A fully nonlinear finite elements analysis for prediction of localization representing shear-crippling (kinkband) instability in a thick laminated composite (plane strain) ring (infinitely long cylindrical shell) under applied hydrostatic pressure is presented. The primary accomplishment of the present investigation is prediction of meso(lamina)-structure-related equilibrium paths, which are often unstable in the presence of local imperfections and/or material nonlinearity, and which are considered to “bifurcate” from the primary equilibrium paths, representing periodic buckling patterns pertaining to global or structural level stability of the thick cross-ply ring with modal or harmonic imperfection. The present nonlinear finite elements solution methodology, based on the total Lagrangian formulation, employs a quasi-three-dimensional hypothesis, known as layerwise linear displacement distribution theory (LLDT) to capture the three-dimensional interlaminar (especially, shear) deformation behavior, associated with the localized interlaminar shear-crippling failure. A thick laminated composite [90/0/90] imperfect (plane strain) ring is investigated with the objective of analytically studying its premature compressive failure behavior. Numerical results suggest that interlaminar shear/normal deformation (especially, the former) is primarily responsible for the appearance of a limit (maximum pressure) point on the post-buckling equilibrium path associated with a periodic (modal or harmonic) buckling pattern, for which a modal imperfection serves as a perturbation. Localization of the buckling pattern results from “bifurcation” at or near this limit point, and can be viewed as a symmetry breaking phenomenon. In order to investigate a localization of the buckling pattern, a local or dimple shaped imperfection superimposed on a fixed modal one is selected. With the increase of local imperfection amplitude, the limit load (hydrostatic pressure) decreases, and also the limit point appears at an increased normalized deflection. Additionally, the load–deflection curves tend to flatten (near-zero slope) to an undetermined lowest pressure level, signaling the onset of “phase transition” in the localized region, and coexistence of two “phases”, i.e., a highly localized band of shear crippled (kinked) phase and its unshear-crippled (unkinked) counterpart along the circumference of the ring. Interlaminar shear-crippling triggered by the combined effect of imperfection, material nonlinearity and interlaminar shear/normal deformation appears to be the dominant compressive failure mode. A three-dimensional or quasi-three-dimensional theory, such as the afore-mentioned LLDT is essential in order to capture the meso-structure-related instability failure such as localization of the interlaminar shear crippling, triggered by the combined presence of local imperfection and material nonlinearity.

Post-critical analysis of structures with a nonlinear pre-buckling state in the presence of imperfections

In the context of the geometrically nonlinear analysis of structures by continuation methods, a technique for the determination of the imperfection sensitivity is carried out. The technique is based on Galerkin's reduction of equilibrium equations in the subspace of extrapolation branches emanating from the bifurcation points. The pre- and post-critical behaviour in the neighbourhood of bifurcation points, however, is well approximated provided that extrapolations have a large convergence domain. To this end an efficient asymptotic representation of the primary and secondary equilibrium paths and procedures that overcome the ill-conditioning of the systems defined in the critical points is used here.

A Dynamical Systems Approach to Bifurcation and Instability of Constitutive Relations for Brittle Damage Materials

A methodology based on stability theory of dynamical systems is proposed to study the stability of the constitutive relations for brittle damage materials. The bifurcation conditions and the subsequent stability issues of the equilibrium paths are examined in the framework of bifurcation and stability theories of nonlinear dynamical systems. This framework also demonstrates that stability is not only a constitutive property, it also depends on the imposed traction and displacement boundary conditions. Moreover, the loss of uniqueness and stability of the primary equilibrium path does not necessarily imply failure in the dynamical sense since there may exist a stable secondary equilibrium path. Stability criteria for both stress-controlled and displacement-controlled conditions are formulated and an equivalent stability condition for the corresponding statical system is also presented.

Bifurcation of Elastoplastic Pressure-Sensitive Hollow Cylinders

A linear bifurcation analysis is presented for pressure sensitive elastoplastic hollow cylinders under radial surface loads. Material response is modeled by flow and deformation theories of the Drucker-Prager solid accounting for arbitrary hardening. Sample calculations are given for cylinders that deform in axially symmetric patterns under uniform radial pressure applied at the boundaries. No bifurcation points were found with flow theory in the realistic range of stress though the primary equilibrium path is nearly identical for both theories. For thick-walled cylinders the dominant bifurcation mode predicted by deformation theory appears to be a circumferential surface instability. Deformation theory results for bifurcations are apparently not sensitive to deviations from associativity.

A Geometric Approach for Establishing Dynamic Buckling Loads of Autonomous Potential Two-Degree-of-Freedom Systems

Nonlinear dynamic buckling of autonomous potential two-degree-of-freedom nondissipative systems with static unstable critical points lying on nonlinear primary equilibrium paths is studied via a geometric approach. This is based on certain salient properties of the zero level total potential energy “surface” which in conjunction with the total energy-balance equation allow establishment of new dynamic buckling criteria for planar systems. These criteria yield readily obtained “exact” dynamic buckling loads without solving the highly nonlinear initial-value problem. The simplicity, reliability, and efficiency of the proposed technique is illustrated with the aid of various dynamic buckling analyses of two two-degree-of-freedom models which are also compared with those obtained by the Verner-Runge-Kutta scheme.