Prebuckling Displacements Research Articles

Nonlinear instability problem for geometrically exact beam under conservative and non-conservative loads

In this work we propose a numerical solution procedure for nonlinear instability problems for geometrically exact beams (representing arbitrary large displacements and rotations) under either conservative or non-conservative load. Both static and dynamics frameworks are developed, with the latter needed for non-conservative case. The proposed model can also successfully deliver the analytic solution to linearized instability problems (with small pre-buckling displacements) for critical loads of Euler (no shear) and Timoshenko (with shear) cantilever beam, which is used for validation for both conservative and non-conservative loads. We also validated the proposed geometrically exact beam models against the analytic solutions to nonlinear problem (with finite rotations) and pointed out their advantage in providing a more sound interpretation of instability for both conservative and non-conservative loads. Finally, the proposed models performance is illustrated on more complex problems with corresponding numerical solutions obtained by using both the geometrically exact finite elements for Reissner beam (with shear) and for the Kirchhoff beam (no shear).

Elastic stability of U-shaped members in bending considering pre-buckling displacements

The present paper describes a theoretical study on the effect of pre-buckling displacements on the elastic instability of U-shaped members. Expressions for the elastic critical bending moments under mono-axial bending and the elastic critical load amplification factor for the case of bi-axial bending are derived and validated through elastic geometrically non-linear analysis including a geometric imperfection. It will be shown that pre-buckling displacements have a non-negligible influence on the behaviour of U-shaped members, especially when subjected to minor-axis and bi-axial bending. It will be shown that, contrariwise to the natural feeling, U-shaped members under minor-axis bending may, in some cases, be sensitive to elastic instability (lateral-torsional buckling). Moreover, the validated analytical expressions show that shorter members are more sensitive to elastic instability under minor-axis bending than their longer counterparts!

Study of geometric non-linear instability of 2D frame structures

AbstractIn this work, we deal with the geometric instability problem of the two-dimensional (2D) elastic frame structures undergoing large overall motion. The geometrically exact beam model with total Lagrangian formulation is used to obtain the solution to non-linear instability problems with large pre-buckling displacements. We propose, in particular, a study of dynamic analysis that can deal with instability problems of this kind with no need for any load decrease. The dynamics approach provides a more realistic post-buckling behaviour for the case of snap-through or snap-back. The material damping is necessary when a classical time integration scheme like Newmark is used. The principal novelty in this work is to consider non-linear damping to avoid the vibration around the equilibrium point when a classical scheme as Newark is used. The efficiency of the damping model and methodology analysis are illustrated by a number of numerical simulations.

Decaying/conserving implicit scheme and non-linear instability analysis of 2D frame structures

In this work we deal with the geometric instability problem of the two-dimensional elastic frame structures undergoing large overall motion. The geometrically exact beam model with total Lagrangian formulation is used to obtain the solution to non-linear instability problems with large pre-buckling displacements. We have proposed, in particular, the use of the conserving and decaying scheme for dynamic analysis that can deal with instability problems of this kind with no need for structural damping, whereas it is necessary when a classical time integration scheme like Newmark is used. This scheme is an extension of time-integration energy conserving scheme presented in Ibrahimbegovic and Mamouri [1], redesigned to introduce desirable properties of controllable energy decay in higher modes, which improve its stability and accuracy. The efficiency of the proposed scheme in non-linear instability is illustrated by a number of numerical simulations.

Lateral buckling analysis of beams of arbitrary cross section by BEM

In this paper, the lateral buckling analysis of beams of arbitrary cross section is presented taking into account moderate large displacements and employing nonlinear relationships between bending moments and curvatures. The beam is supported by the most general boundary conditions including elastic support or restraint. Starting from a displacement field without any simplifying assumptions about the angle of twist amplitude and based on the total potential energy principle, the stability criterion is formulated taking into account the warping effects, the prebuckling displacements and the Wagner’s coefficients due to the asymmetric character of the cross section. The stability criterion is based on the positive definiteness of the second variation of the total potential energy and is established using the Analog Equation Method (AEM), a BEM based method. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross section’s torsional rigidity is evaluated exactly without using the so-called Saint–Venant’s torsional constant. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The effects of both the load height and prebuckling deflections as well as the discrepancy of the Eurocode 3 standard solutions are discussed.

Non-linear model for stability of thin-walled composite beams with shear deformation

A geometrically non-linear theory for thin-walled composite beams is developed for both open and closed cross-sections and taking into account shear flexibility (bending and warping shear). This non-linear formulation is used for analyzing the static stability of beams made of composite materials subjected to concentrated end moments, concentrated forces, or uniformly distributed loads. Composite is assumed to be made of symmetric balanced laminates or especially orthotropic laminates. In order to solve the non-linear differential system, Ritz's method is first applied. Then, the resulting algebraic equilibrium equations are solved by means of an incremental Newton–Rapshon method. This paper investigates numerically the flexural–torsional and lateral buckling and post-buckling behavior of simply supported beams, pointing out the influence of shear–deformation for different laminate stacking sequence and the pre-buckling deflections effect on buckling loads. The numerical results show that the classical predictions of lateral buckling are conservative when the pre-buckling displacements are not negligible, and a non-linear buckling analysis may be required for reliable solutions.

Section Properties and Buckling Behavior of Pultruded FRP Profiles

Experimental determination of the flexural and torsional properties of pultruded FRP profiles, based on full section and coupon tests, is described. Results for a range of I-profiles indicates that transverse shear moduli, determined from full section three point bending tests, are influenced significantly by localised deformation at the supports. Closed form solutions for the influence of shear deformation on global flexural, torsional and lateral buckling of pultruded FRP profiles have been developed. Parametric studies for a range of I-profiles indicate that shear deformation can reduce flexural and torsional buckling loads by up to 10%. For wide flange I-profiles pre-buckling displacements can increase lateral buckling moments by more than 20% while the influence of shear deformation is relatively minor.

Non-linear inplane deformation and buckling of rings and high arches

A non-linear theory is presented for stretching and inplane-bending of isotropic beams which have constant initial curvature and lie in their plane of symmetry. For the kinematics, the geometrically exact one-dimensional (1-D) measures of deformation are specialized for small strain. The 1-D constitutive law is developed in terms of these measures via an asymptotically correct dimensional reduction of the geometrically non-linear 3-D elasticity under the assumptions of comparable magnitudes of initial radius of curvature and wavelength of deformation, small strain, and small ratio of cross-sectional diameter to initial radius of curvature ( h/R). The 1-D constitutive law contains an asymptotically correct refinement of O(h/R) beyond the usual stretching and bending strain energies which, for doubly symmetric cross sections, reduces to a stretch–bending elastic coupling term that depends on the initial radius of curvature and Poisson’s ratio. As illustrations, the theory is applied to inplane deformation and buckling of rings and high arches. In spite of a very simple final expression for the second variation of the total potential, it is shown that the only restriction on the validity of the buckling analysis is that the prebuckling strain remains small. Although the term added in the refined theory does not affect the buckling loads, it is shown that non-trivial prebuckling displacements, curvature, and bending moment of high arches are impossible to calculate accurately without this term.

A consistent finite element formulation for linear buckling analysis of spatial beams

A consistent co-rotational finite element formulation and numerical procedure for the linear buckling analysis of three-dimensional elastic Euler beam is presented. A mechanism for generating configuration dependent conservative moment is proposed and the corresponding load stiffness matrix is derived. It is assumed that the prebuckling displacements and rotations of the structure and the corresponding deformations of the elements are linearly proportional to the external loading. The prebuckling rotations of the structure are fixed axis rotations or small rotations, and the effect of the prebuckling displacement on transformation matrix for the coordinates transformation can be ignored. All coupling among bending, twisting and stretching deformations for beam element is considered by consistent linearization of the fully geometrically nonlinear beam theory. An inverse power method for the solution of the generalized eigenvalue problem is used to find the buckling load and buckling mode. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Prebuckling Deflections and Lateral Buckling. I: Theory

When the ratio of the minor axis flexural stiffness to the major axis flexural stiffness is large, classic analysis may lead to inaccurate predictions of the lateral buckling loads of beams and beam‐columns because the effects of prebuckling deformations are not considered. The energy equation for the elastic lateral buckling of monosymmetric beam‐columns that includes the effects of prebuckling deformations is derived, and the buckling differential equilibrium equations are obtained. A finite element algorithm for the prediction of the lateral buckling loads of monosymmetric beam‐columns that includes the effects of prebuckling deformations is developed from the energy equation. This includes the effects of second‐order moments due to the prebuckling displacements and the axial loads. An iterative procedure for determining the lateral buckling loads is recommended. The finite element results, given in a companion paper, show that the classic predictions of the lateral buckling loads of beams and beam‐columns are generally conservative, but that the predictions by a linearized procedure are overestimated. The predictions by the recommended nonlinear iteration procedure agree well with the experimental results.

Buckling of cylindrical shells under the action of nonuniform external pressure

Over the last few decades, storage tanks have become bigger and thinner. Because of this, the buckling capacity of these cylindrical shells may well be the determining factor of shell thickness. In this paper, the critical buckling load of isotropic and orthotropic cylinders subjected to different types of wind load distributions is investigated. The prebuckling displacements are obtained by using the membrane theory of shell analysis. The principle of minimum potential energy in conjunction with Ritz's approach is used to obtain the stability matrix. The size of the stability matrix in this analysis is (81 × 81). By solving the stability matrix as an eigenvalue problem, the critical pressures are obtained as eigenvalues and the deflection shapes as eigenvectors. In the present study cylindrical shells of various dimensions, which are fixed at the base and free at the top, are investigated. The buckling load curves for isotropic and orthotropic cylinders of various dimensions are given for practical use.

Lateral instability of monosymmetric beams with initial curvature

A theoretical study of the flexural-torsional, or lateral, instability of monosymmetric beams is presented. The analysis is based on beam theory and energy considerations and incorporates the influence of pre-buckling displacements and initial curvature in the plane of major axis bending. Closed form solutions are obtained for simply supported beams subjected to uniform moment, central concentrated and uniformly distributed loads, which are valid for a wide range of section properties. Where the closed form solutions become inaccurate due to the complexity and variability of the buckled shape, numerical results are presented.

Theoretical elastoplastic buckling of stringer stiffened cylindrical shells

A linear analysis method is offered to predict the theoretical elastoplastic buckling of stringer stiffened cylindrical shells subjected to longitudinal loading. Welding residual stresses are taken into account in the calculation, but effects of geometrical imperfections and pre-buckling displacements are ignored. The examples analysed show a good correlation between the analytical results and those obtained experimentally with stocky models of moderate geometrical imperfections.

Simplified theory for the behaviour of buried flexible cylinders under the influence of uniform hoop compression

The elastic behaviour of a long flexible cylinder buried in an infinite elastic medium and acted on by uniform hoop compressions is examined. A simplified solution is obtained, which can be used to accurately and efficiently determine the cylinder response. The simplified theory is used to obtain a parametric solution to the cylinder buckling problem. The effects of interface condition, elastic ground parameters, load behaviour, and finite thickness on the critical hoop compression are considered. A straightforward method is developed for predicting the prebuckling displacements and bending moments which result from the influence of initial geometrical imperfections and nonhydrostatic field stresses.

Instability of monosymmetric I-beams and cantilevers

The elastic flexural-torsional or lateral instability of monosymmetric I-beams and cantilevers is investigated using a general energy approach based on the vanishing of the second variation of the total potential energy in which the buckling displacements are represented by trigonometric series. For simply supported beams, closed-form solutions can be obtained which include the influence of pre-buckling displacements and which are valid for a wide range of section properties. Where the closed-form solutions become inaccurate, and for cantilevers, numerical results are presented and compared with existing numerical results based on the governing differential equations. The convergence of the series solution and variability of the eigenvectors defining the buckled shape are also discussed.

Instability of Monosymmetric I‐Beams

The elastic flexural-torsional instability of monosymmetric I-beams and cantilevers is investigated using a general energy approach based on the vanishing of the second variation of the total potential energy. For simply supported beams, closed form solutions are presented which are valid for a wide range of section properties and which include the influence of pre-buckling displacements. Where the closed form solutions become inaccurate, and for cantilevers, numerical results are presented.

Instability of short stiffened composite cylindrical shells under bending with prebuckling displacements

This paper presents results for buckling of a stiffened cylindrical shell with cutouts and both isotropic and composite shells without cutouts acting under end bending moments. The STAGS-C program has been used in the analysis.

Influence of pre-buckling displacements on the elastic critical loads of thin walled bars of open cross section

Nonlinear expressions for the strains occurring in thin walled bars of open cross section, when subjected to axial, flexural and torsional displacements, are incorporated in a general instability analysis based on the vanishing of the second variation of the total potential energy. It is shown that the influence of the pre-buckling displacements is automatically included in the analysis. A closed form solution for the lateral buckling of a simply supported beam subjected to uniform bending agrees exactly with a solution based on the governing differential equations. Solutions obtained using numerical methods are also presented. The significance of the second order axial strains induced by rotation about the shear centre, is investigated by considering the instability of an inverted T- beam subjected to uniform bending.

Destabilizing effect of additional restraint on elastic bar structures

This paper revises the well-known theorem; “The addition of restraint to a structure cannot decrease its elastic critical load”. It is pointed out by theoretical arguments and simple examples that this theorem is valid only if the displacement components in the buckling direction are additionally restrained. But additional restraints on pre-buckling displacements, which are orthogonal to the direction of buckling, can decrease the elastic critical load factor of the structure. In the case of additional restraints on prebuckling displacements, by the Dunkerley theorem, a bracketing formula is presented which gives lower and upper bounds to the minimum positive critical load factor of the modified structure with the help of minimum positive critical load factors of different loading patterns on the original structure.

Post-Buckling Analysis of Cylindrical Shells

A method is presented for the calculation of the critical load and post-buckling displacements of elastic structures, and is demonstrated on circular cylindrical shells. The critical load is found by successive extrapolations of the load with the aid of the tangent stiffness matrix determinant. The buckling mode shape is calculated at the instability point and the prebuckling displacements are perturbed a small amount proportional to this mode shape in order to determine an equilibrium state in the post-buckling range. A modified Newton-Raphson iteration technique is utilized to trace the post-buckling load-displacement curve.