Asymmetric Bifurcation Point Research Articles

Dynamic buckling of partially-sway frames with varying stiffness using catastrophe theory

This work deals with the static and dynamic stability analysis of imperfect partially-sway frames with non-uniform columns. The examined two-bar frames are elastically supported and subjected to an eccentrically vertical load at their joint. Through a linear stability analysis, the effect of the taper ratio of the column cross-section on the buckling capacity of the partially-sway frame is thoroughly discussed. Using a non-linear method an accurate formula has been established for determining the exact asymmetric bifurcation point associated with the maximum load carrying capacity. These findings have been re-derived more readily using Catastrophe Theory (CT) and considering the frame as a one degree-of-freedom (1-DOF) system through an efficient technique. A local analysis allows us to classify, after reduction, the total potential energy (TPE) function of the system to one of the seven elementary Thom׳s catastrophes (with known properties) and to obtain static and dynamic singularity as well as bifurcational sets. It has been found that geometrical and loading imperfections, which are always present in structural engineering problems, have a significant effect on the dynamic buckling loads. The efficiency of the present approach is illustrated via several examples, while results from finite element analyses are in good agreement with the analytical solution presented herein.

Imperfection sensitivity of degenerate hilltop branching points

Imperfection sensitivity is investigated for a degenerate hilltop branching point, where a degenerate bifurcation point exists at a limit point. A degenerate hilltop branching point is important as it is a byproduct of optimization of shallow shell structures under non-linear buckling constraints. A systematic procedure is presented for asymptotic sensitivity analysis based on enumeration of vertices of a convex region defined by linear inequality constraints on the orders of the variables. The effectiveness of the proposed method is demonstrated by sensitivity analysis of degenerate hilltop branching points, considering minor and major imperfections, corresponding to an unstable-symmetric or asymmetric bifurcation point at the limit point. It is found that a hilltop branching point can be imperfection sensitive.

Maximum load factors corresponding to a slightly asymmetric bifurcation point

Explicit formulations are derived for quantitative evaluation of imperfection sensitivity of critical load factors corresponding to a slightly asymmetric bifurcation point; i.e. the perfect system itself has slight asymmetry. The load factor corresponding to local minimum or maximum along the fundamental equilibrium path and the complementary path are simultaneously obtained as functions of the imperfection parameter. The upper bound of the imperfection parameter is also derived for existence of critical points along the fundamental equilibrium path. This way, interpolation formulas are presented between symmetric and asymmetric systems. The accuracy of the proposed formulas is verified by the examples of rods supported by translational and rotational springs.

A simplified nonlinear stability analysis of an imperfect rectangular two-bar frame

A simplified nonlinear stability analysis for moderately large rotations and small strains is performed on a rectangular two-bar frame subjected to a concentrated eccentrically applied joint load. Such a simplification consisting of adopting linear kinematic relations leads to very reliable results for the initial postbuckling path in the vicinity of the critical point of the above imperfect frame. The existence of an asymmetric bifurcation point is thoroughly discussed and a direct evaluation of the bifurcational load is readily obtained. Using this technique the effect of imperfection sensitivity is also addressed. A qualitative analysis associated with the physical phenomenon yields a substantial reduction of the computational work. The efficiency and reliability of this approximate nonlinear stability analysis proposed herein is illustrated by means of several examples for which a lot of numerical results based on a more accurate nonlinear analysis are available.

Sensitivity analysis of an optimized bar-spring model with hill-top branching

Characteristics of optimal solutions under nonlinear buckling constraints are investigated by using a bar-spring model. It is demonstrated that optimization under buckling constraints of a symmetric system often leads to a structure with hill-top branching, where a limit point and bifurcation points coincide. A general formulation is derived for imperfection sensitivity of the critical load factor corresponding to a hill-top branching point. It is shown that the critical load is not imperfection-sensitive even for the case where an asymmetric bifurcation point exists at the limit point.

Imperfection sensitivity of optimal symmetric braced frames against buckling

Imperfection sensitivity characteristics of the non-linear buckling load factors of non-optimal and optimal symmetric frames are investigated. The frames are subjected to symmetric proportional vertical loads. The optimization problem is formulated under constraints on linear buckling load factors. Although the buckling load factors corresponding to sway and non-sway modes coincide at the optimum design, the non-sway-type asymmetric bifurcation point disappears as a result of geometrically non-linear analysis. Therefore, the maximum allowable load factors of perfect and imperfect systems should be determined by assigning displacement constraints. It is shown that quantitative evaluation is possible for imperfection sensitivity and mode interaction based on the higher order differential coefficients of the total potential energy even for frames of which the critical points should be numerically obtained. Numerical examples are presented to show that the properties of the non-sway bifurcation point are similar to those of a symmetric bifurcation point, and the interaction between sway and non-sway modes does not always lead to enhancement of imperfection sensitivity.

Axisymmetric postbuckling and secondary bifurcation buckling of circular plates

The exact axisymmetric-postbuckling equilibrium path has been obtained by the use of the power series method in solving the non-linear differential equations for the circular plates subjected to uniform radial compression. The problem of the asymmetric-bifurcation buckling from the axisymmetric-postbuckling deformation state has been investigated according to the adjacent equilibrium criterion (Brush and Almroth, Buckling of Bars, Plates and Chells, McGraw-Hill, New York, 1975; Ziegler, Principles of Structural Stability, Blaisdell, 1968), and the critical loads corresponding to the asymmetric-bifurcation point have been calculated for both simply supported and clamped plates. The von Karman non-linear equations in the incremental form have been solved by the use of the power-series expansion and the Fourier series expansion, and the postbuckling behaviour of the circular plate beyond the asymmetric-bifurcation point has been investigated. The given results show that in advanced axisymmetric-postbuckling stage, the high circumferential compressive stress can induce the circular plate to buckle with an asymmetric mode, and the equilibrium of the circular plate at the asymmetric-bifurcation point is unstable.

Energy-based dynamic buckling estimates for autonomous dissipative systems

The dynamic buckling global response of a nonlinear, 3-degree-of-freedom dissipative model under a step loading of infinite duration is thoroughly discussed. Geometrically imperfect models with symmetric or antisymmetric imperfections losing their static stability through a limit point and an asymmetric bifurcation point, respectively, are considered. Emphasis is given to the combined effect of nonlinearities (geometric and/or material) and damping. Exact, approximate, and lower/upper bound estimates based on energy criteria for establishing the dynamic buckling response of such autonomous models without solving the highly nonlinear initial-value problem are assessed. The reliability and efficiency of the proposed readily obtained estimates is illustrated via numerical simulation, the accuracy of which is checked using energy balance considerations. Certain interesting byproducts associated with a postlimit point bifurcation and breakdown of the symmetry of deformation are also revealed.

The effect of interaction of loadings and geometry imperfection on the buckling state of frames

In this paper the effect of the interaction of two loadings, and of the deviation from right angle on the buckling mechanism of a rectangular two-bar frame is discussed. Using a nonlinear stability analysis it is found that there is a critical value of the ratio of the two loadings, as well as of the deviation from right angle, for which the physical equilibrium path and the physically unacceptable complementary path meet each other at an asymmetric bifurcation point. At this point, the response of the frame changes from an elastic limit point instability to a stable equilibrium path associated with inelastic buckling. The analysis is supplemented by a variety of numerical results obtained by an efficient and reliable simplified nonlinear stability analysis applied for the first time to a non-rectangular frame.

The effects of material nonlinearity and compressibility in the buckling of nonlinear elastic systems

The stability and post-buckling response of simple perfect and imperfect systems made of nonlinear elastic material are thoroughly discussed. The perfect systems correspond to the three types of bifurcation points, i.e. stable symmetric, unstable symmetric and asymmetric. Material-dependent stability conditions are properly established. It is found that the mechanism of buckling of perfect systems associated with a stable symmetric bifurcation point may become unstable depending on the value of the material nonlinearity parameter. Moreover, it is established that the effect of compressibility on the buckling load may be considerable in case of imperfect systems which in their ideal state are associated with an asymmetric bifurcation point.

The effect of interaction of loadings on the change of buckling response of framed structures

In this investigation the effect of two loadings interaction and right angle deviation on the response of a rectangular two-bar frame is thoroughly discussed. It is found that there is a critical value of the two loadings ratio as well as of the right angle deviation for which the physical equilibrium path and the physically unacceptable complementary path meet each other at an asymmetric bifurcation point. This state constitutes the boundary between stability and instability, since the response of the frame changes from snapthrough buckling to a stable equilibrium path.