Initial Postbuckling Behavior Research Articles

Nonlinear stability and design of prestressed stayed columns in fire considering non-uniform temperature distribution

Prestressed stayed columns are vulnerable to fire due to material weakening. The fire behaviour of the stayed columns is investigated with non-uniform temperature distributions considered using steady state analysis within ABAQUS. This study aims to reveal the load-carrying capacity reduction of stayed columns in fire and propose a design method. Results show that ultimate loads decrease with temperature increasing, and the stayed columns approximate failure at the temperatures higher than 600°C. The failure deflected mode maintains similar to the dominant linear buckling mode for the uniform temperature distribution, and the sensitivity of ultimate loads to prestressing levels becomes less significant at higher temperatures. Moreover, compared to buckling and ultimate behaviour, initial post-buckling behaviour shows more significant sensitivity to temperature distributions. The non-uniform distribution can lead to asymmetric deflected mode due to stiffness re-distribution from the material weakening at elevated temperatures, and the post-buckling behaviour of the cases with rotational symmetric mode dominance is sensitive to prestressing levels. Finally, a fire design method is proposed based on the ultimate load reduction in fire.

Postbuckling behavior and imperfection sensitivity of elastic–plastic periodic plate-lattice materials

Advances in additive manufacturing have enabled a new generation of materials with advantageous properties inherent to their architecture. Recently, architected materials with periodic arrangements of plates, called plate-lattice materials, have been developed to reach theoretical least upper bounds for stiffness and strength of isotropic materials. This work investigates the buckling behavior of several plate-lattice architectures with relative densities between 0.5% to 25% when subjected to uniaxial compression. Finite element unit cell models with shell elements and periodic boundary conditions are used to simulate the buckling and post-buckling behavior of plate-lattices made from an elastic-perfectly plastic parent material. Five plate-lattice architectures with cubic symmetry are investigated — three anisotropic architectures (simple cubic, body-centered cubic, face-centered cubic) and two isotropic architectures (simple cubic and body-centered cubic combination, simple cubic and face-centered cubic combination). Geometric imperfections of varying amplitude are applied to determine the imperfection sensitivity of these plate-lattice materials, and to calculate their buckling knockdown factors. Consistent with the behavior of the elastic–plastic plate building blocks that comprise the lattice, this investigation reveals that plate-lattice materials are generally imperfection insensitive when their constituent plates become thinner and first bifurcation occurs in the elastic range away from the yield stress. When this occurs, the initial post-buckling behavior is stable and the ultimate strength can be many times higher than the initial buckling stress.

Thermal buckling of thin injection-molded FRP plates

The occurrence of buckling during cooling of an injection-molded fiber-reinforced plastic (FRP) object is a complex phenomenon which can result in unacceptably large deformations. The present work focuses on the thermal buckling of injection-molded thin disks, as they are representative for the local behavior in parts of a more complex object. An analytical approach based on the Föppl-von Kármán theory for thin elastic plates is presented and used to analyze its buckling stability and initial post-buckling behavior. The system of equations is formulated in a coordinate-free approach for a thin anisotropic linear thermo-elastic body with residual thermal stresses. Large out-of-plane displacements are assumed. The particular case of a free disk with cylindrical orthotropy and uniform thermo-elastic properties through its thickness is considered. A general technique to study the bifurcation in the solutions (i.e., buckling) based on a perturbation approach and the Fourier-Galerkin method to determine post-buckling deflections is proposed. Experimental and numerical results available in the literature are used as validation cases. The results of the present model show qualitative agreement with the experimental data. It is found that the periodicity of the first buckling mode for orthotropic disks is fully determined by the ratio between the elastic moduli in the tangential and radial directions. The predictions are also in excellent quantitative agreement with the numerical results of FEM models for the same cases. The effect of the disk thickness on the buckling temperature and warpage magnitude is correctly captured.

Displacement-based formulation of Koiter's method: Application to multi-modal post-buckling finite element analysis of plates

Koiter's asymptotic method enables the calculation and deep understanding of the initial post-buckling behaviour of thin-walled structures. For the single-mode asymptotic analysis, Budiansky (1974) presented a clear and general formulation for Koiter's method, based on the expansion of the total potential energy function. The formulation from Budiansky is herein revisited and expanded for the multi-modal asymptotic analysis, of primordial importance in structures with clustered bifurcation modes. Given the admittedly difficult implementation of Koiter's method, especially for multi-modal analysis and during the evaluation of the third– and fourth–order tensors involved in Koiter's analysis; the presented study proposes a formulation and notation with close correspondence with the implemented algorithms. The implementation is based on state-of-the-art collaborative tools: Python, NumPy and Cython. The kinematic relations are specialized using von Kármán shell kinematics, and the displacement field variables are approximated using an enhanced Bogner-Fox-Schmit (BFS) finite element, modified to reach third-order interpolation also for the in-plane displacements, using only 4 nodes per element and 10 degrees-of-freedom per node, aiming an accurate representation of the second-order fields. The formulation and implementation are verified by comparing results for isotropic and composite plates against established literature. Finally, results for multi-modal displacement fields with up to 5 modes and corresponding post-buckling factors are reported for future reference.

Stability of a Sandwich Cylindrical Shell with Core Subject to External Pressure and Pressure in the Inner Cylinder*

A generalized design model is proposed to study the stability and initial post-buckling equilibrium trajectory of sandwich cylindrical shells with elastic core that resists only transverse tension–compression. The model includes nonlinear equilibrium mixed-form equations, asymptotic equations obtained by the Koiter–Budiansky method, and the analytical solution of a homogeneous eigenvalue problem and inhomogeneous problem to find the values of the unknown functions at the critical point. The numerical results obtained indicate that the internal pressure strongly influences the critical load and initial post-buckling behavior of the shells.

Closed solution for initial post-buckling behavior analysis of a composite beam with shear deformation effect

Closed solution for initial post-buckling behavior analysis of a composite beam with shear deformation effect

P.347 Alterations in key elements of the immune system in suicide victims

The paper concerns flexural buckling and initial post-buckling of axially compressed columns made of aluminium alloy described by the Ramberg-Osgood relationship. The non-linear differential equation of the problem is derived using the stationary total energy principle and the assumptions of classical beam theory within a finite range. The approximate analytical solution of the equation leading to the buckling loads and initial post-buckling equilibrium path is determined by means of the perturbation approach. Numerical examples dealing with simply supported and clamped I-columns are presented, the effect of the material non-linearity on the critical loads and initial post-buckling behaviour in comparison to linear one is discussed too. The analytical results are compared with the FEM solutions to present a good agreement.

Stability and Initial Post-Buckling Behavior of Orthotropic Cylindrical Sandwich Shells with Unidirectional Elastic Filler

The problem of the stability and initial post-buckling behavior of sandwich shells with transversely elastic light-weight filler is solved. The governing system of nonlinear equations is derived with allowance for structural discreteness that allows independent deformation of load-bearing layers. The system is solved with the asymptotic Koiter–Budiansky method. Structures composed of two cylindrical carbon-fiber shells interacting through a spring-type filler are analyzed. It is shown that the critical load under axial compression with like forces falls between the critical loads for the inner and outer shells. In the post-buckling state, the bending modes may be different. It is rational to increase the filler stiffness to a certain level.

Localization in spherical shell buckling

This paper addresses localization of the deformation due to buckling that occurs immediately following the onset of bifurcation in the axisymmetric buckling of a perfect spherical elastic shell subject to external pressure. The localization process is so abrupt that the buckling mode of the classical eigenvalue analysis, which undulates over the entire shell, becomes modified immediately after bifurcation transitioning to an isolated dimple surrounded by an unbuckled expanse of the shell. The paper begins by revisiting earlier attempts to analyze the initial post-buckling behavior of the spherical shell, illustrating their severely limited range of validity. The unsuccessful attempts are followed by an approximate Rayleigh-Ritz solution which captures the essence of the localization process. The approximate solution reveals the pathway that begins at bifurcation from the classical mode shape to the localized dimple buckle. The second part of the paper presents an exact asymptotic expansion of the initial post-buckling behavior which accounts for localization and which further exposes the analytic details of the abruptness of the transition.

Correction to: Stability and initial postbuckling behavior of anisotropic cylindrical shells under external pressure

The second author’s affiliation should read: National University of Water and Environmental Engineering, Rivne, Ukraine.

Postbuckling optimisation of a variable angle tow composite wingbox using a multi-modal Koiter approach

The stiffness-tailoring capability of Variable Angle Tow (VAT) laminates gives enhanced freedom to design thin-walled structures. One key advantage of tow steering is the ability to redistribute stresses improving buckling performance, leading to reduction in material weight and costs. The aim of this work is to optimise the initial postbuckling behaviour of a recently proposed VAT composite wingbox. The optimisation process is based on a fibre path parameterisation. It involves seeking the stacking sequence that minimises the displacements occurring in the postbuckling regime. This problem is solved by coupling the multi-modal Koiter asymptotic approach implemented with a solid-shell Finite Element environment through stochastic optimisation strategies. Results obtained regarding different optimisation scenarios show a much improved performance for the buckling and postbuckling response of the wingbox with respect to the initial VAT design. Additionally, manufacturing constraints are readily included in the optimisation program. The possibility of performing an efficient and robust optimisation process of a complex structure with a multi-modal Koiter asymptotic approach is demonstrated, showing its viability as a design tool for buckling dominated structures. A parametric study regarding the influence of steering radii shows that overcoming the current manufacturing constraint on minimum radius is worthy of investigation.

Buckling and initial post-local buckling behaviour of cold-formed channel member flange

The initial post-buckling behaviour of a cold-formed channel member flange after its local buckling is investigated. An axially compressed column or beam subjected to pure bending is considered. The member material is assumed to follow a linear stress-strain relationship. The governing non-linear differential equation of the problem is derived using the minimum total potential energy principle. An approximate solution for the equation is found by means of the perturbation approach, which allows obtaining the critical buckling stress and the initial post-buckling equilibrium path. The bifurcation point is shown to be symmetric and stable. The proposed analytical solution is compared with finite element method (FEM) and finite strip method (FSM) results to check the validity and range of applicability.

Flexural buckling and post-buckling of columns made of aluminium alloy

The paper concerns flexural buckling and initial post-buckling of axially compressed columns made of aluminium alloy described by the Ramberg-Osgood relationship. The non-linear differential equation of the problem is derived using the stationary total energy principle and the assumptions of classical beam theory within a finite range. The approximate analytical solution of the equation leading to the buckling loads and initial post-buckling equilibrium path is determined by means of the perturbation approach. Numerical examples dealing with simply supported and clamped I-columns are presented, the effect of the material non-linearity on the critical loads and initial post-buckling behaviour in comparison to linear one is discussed too. The analytical results are compared with the FEM solutions to present a good agreement.

Torsional buckling and post-buckling of columns made of aluminium alloy

• The closed-form analytical solution of torsional buckling load is derived. • The initial post-buckling behaviour is presented. • The non-linear differential equation of torsional buckling is derived and solved. • The effect of non-linear elastic material with comparison to linear one is discussed. • Comparison of analytical results to FEM analysis is presented. The paper concerns torsional buckling and the initial post-buckling of axially compressed thin-walled aluminium alloy columns with bisymmetrical cross-section. It is assumed that the column material behaviour is described by the Ramberg–Osgood constitutive equation in non-linear elastic range. The stationary total energy principle is used to derive the governing non-linear differential equation. An approximate solution of the equation determined by means of the perturbation approach allows to determine the buckling loads and the initial post-buckling behaviour. Numerical examples dealing with simply supported I-column are presented and the effect of material elastic non-linearity on the critical loads and initial post-buckling behaviour are compared to the linear solution.

Length effects on interactive buckling in thin-walled rectangular hollow section struts

A variational model formulated using analytical techniques describing the nonlinear coupling between local and global buckling modes within an elastic thin-walled rectangular hollow section strut is presented. A system of nonlinear differential and integral equations subject to boundary conditions is derived and solved using numerical continuation techniques. The nonlinear behaviour of four representative lengths is investigated, which are characterized by the post-buckling equilibrium paths. The numerical results from the variational model are validated using a nonlinear finite element model and largely show excellent comparisons, particularly for the practically important ultimate load and the initial post-buckling behaviour. Boundaries for the four distinct length-dependent zones are identified and the most unstable zone is demonstrated to have a considerably narrower length range than previously determined for practical corner boundary conditions within the cross-section.

Evolution and verification of a kinematic hypothesis for splitting of the strain energy

Splitting of the strain energy into its “non-membrane” and membrane percentage provides insight into the load-carrying mechanism of structures, subjected to proportional loading. It may be useful, for example, for sensitivity analysis of the initial postbuckling behavior of beams, arches, plates, and shells, and assemblies of such structures. The task of this work is to determine this percentage without computing insignificant numbers such as the values of the strain energy and its membrane part. It is hypothesized that this percentage is proportional to the acceleration of a fictitious particle, moving along a curve on the unit sphere. The curve is described by the vertex of the normalized “fundamental eigenvector” of the so-called “consistently linearized eigenvalue problem”. The proportionality factor is obtained from the initial condition for the “non-membrane” percentage of the strain energy, hypothesized as twice the initial velocity of the particle. The lower bound of this factor signals the constancy of this percentage with increasing load, whereas the upper bound indicates a monotonic increase or decrease up to its ab initio predictable value at a stability limit or to an unphysical asymptotic limiting value. The proof of the universal validity of the two hypotheses begins with their verification for the special cases of a membrane stress state and pure bending. The assertion that this is a sufficient condition for the universal validity of these hypotheses is subsequently verified for an example with a monotonically increasing “non-membrane” percentage of the strain energy. It is finally confirmed by an indirect proof of their validity for a non-monotonic course of this percentage. A by-product of this work are conditions for extreme values of the stiffness of structures, subjected to proportional loading.

FUNDAMENTAL MODELS OF STRUCTURAL STABILITY

In this paper, basic structural stability phenomena are described. After some general comments about stability in the fi eld of civil engineering, four elementary sources of nonlinearity are mentioned: of equilibrium equations, strain (geometry) relations, material (stress-strain) law, force and displacement boundary conditions. Four fundamental stability models are analysed, both ideal (perfect) and with geometric imperfection. Besides geometrically exact theory, initial post-buckling behaviour and linearization are briefl y sketched. This paper is concluded with comments about the infl uence of plasticity.

Koiter asymptotic analysis of multilayered composite structures using mixed solid-shell finite elements

In this paper we propose a powerful tool for the evaluation of the initial post-buckling behavior of multi-layered composite shells and beams in both bifurcation and limit load cases, including mode interaction and imperfection sensitivity. This tool, based on the joint use of a specialized Koiter asymptotic method and a mixed solid-shell finite element model, is accurate, simple and characterized by a computational cost far lower than standard path-following approaches and many advantages with respect to asymptotic analysis performed with shell elements. The method is very simple and easy to include in existing FE codes because it is based on the same ingredients of a linearized buckling analysis, with very light formula due to the presence of displacement degrees of freedom only. Due to its efficiency it is suitable for layup design when geometrical nonlinearities have to be considered.

The buckling sphere: A symbiosis of mechanics and geometry

The basis of this work is a novel symbiosis of mechanics of solids and spherical geometry to quantify and illustrate the variation of the “non-membrane” percentage of the strain energy in the prebuckling region of linear elastic beams, arches, plates and shells, and structures assembled of such one-dimensional and two-dimensional members. The zenith angle of an arbitrary point of a specific curve on an octant of the unit sphere, called buckling sphere, is related to this energy percentage. For the limiting case of buckling from a membrane stress state this curve degenerates to a point, characterized by zero values of both spherical coordinates. For all other stress states the azimuth angle increases with the proportionally increasing load. Its magnitude at the stability limit correlates with a global quantity that depends on both the “non-membrane” deformations and the stiffness of the structure at incipient buckling. The azimuth angle is computed with the help of the so-called consistently linearized eigenproblem, which is solved by means of the Finite Element Method. This eigenvalue problem is the basis for a hypothesis for the “non-membrane” percentage of the strain energy. In the theoretical part of the paper, the concept of the buckling sphere is presented. The subsequent numerical investigation consists of four examples, referring to buckling from a membrane stress state, a pure bending stress state, and a general stress state. The practical motive for this research is the intention to investigate the influence of “non-membrane” action just before buckling on the initial postbuckling behavior of elastic structures.

Thermomechanical initial post-buckling of Timoshenko's beam on elastic foundations

ABSTRACTThe objective of this work is to propose an analytical solution for the thermomechanical initial post-buckling response of a thick beam resting on a linear elastic foundation and subject to a uniform temperature rise throughout its cross-section. The thermal strain is assumed to follow a linear law with the temperature rise and the material properties are considered temperature independent. The beam cross-section geometrical properties are constant along the beam length, and the formulation is consistent with the small strain assumption. The beam ends are assumed pinned and immovable, thermal expansion is not allowed and as a consequence compressive forces arise and the beam may buckle. If the temperature is increased further, the beam continues to deflect laterally, hence the problem is geometrically non-linear. In addition, the model is appropriate to describe the behavior of short beams as it takes into account transverse shear deformations. The governing equations are derived and made non-dimensional, and it is seen that three non-dimensional parameters control the thermomechanical initial post-buckling problem: The elastic foundation stiffness, the beam slenderness ratio and the beam cross-section shear coefficient. A classical perturbation method is applied to the non-linear set of differential governing equations, therefore the critical buckling temperatures (loads) and modes and the initial post-buckling behavior may be analytically determined. The change in length, reaction forces at the supports and geometric configurations are obtained as a function of temperature, the elastic foundation.