An algorithm for the nonlinear analysis of compound bifurcation

1981 ◽  
Vol 300 (1455) ◽  
pp. 443-471 ◽  
Keyword(s):  
General Scheme ◽  
Full Range ◽  
Three Dimensional ◽  
Good Effect ◽  
Imperfection Sensitivity ◽  
Secondary Bifurcation ◽  
Plates And Shells ◽  
Loading Parameter ◽  
Overall Buckling ◽  
Interactive Buckling

The paper presents a procedure for the localized analysis of compound bifurcations. A full range of phenomena are embraced, including loci of equilibria, secondary bifurcation, and imperfection sensitivity, the scheme showing how to generate the appropriate lowest-order non-trivial equations of interest. A number of aids to solution are presented, including the concepts of generalized imperfection and generalized loading parameter. The scheme is developed by using a general non-diagonalized format suitable for numerical analysis, but the special diagonalized form can also be used to good effect. This is illustrated in the application to the interactive buckling of stiffened plates and shells, when local and overall buckling occur simultaneously or nearly so. The modelling relies heavily on the elimination-of-passive-coordinates routine of the general scheme. The study shows that the parabolic umbilic catastrophe is the key phenomenon for most such problems. Finally, the branching analysis is fully illustrated for semi-symmetric branching, where one of the contributing bifurcations is symmetric and the other is asymmetric. In all, ten different loci are treated, including the full imperfection sensitivity at complete and near coincidence plotted in three-dimensional form; these relate to an earlier stiffened-plate formulation. The general scheme is thus made directly accessible for any problem that exhibits a bifurcational manifestation of either the elliptic or hyperbolic umbilic catastrophe.

Interactive Buckling of Sandwich Plates Under Compression

2001 ◽  
Author(s):  
Srinivasan Sridharan ◽  
Kim Sunjung ◽  
Sami I. El-Sayed
Keyword(s):  
Maximum Load ◽  
Core Material ◽  
Local Mode ◽  
Modal Interaction ◽  
Compressive Behavior ◽  
Imperfection Sensitivity ◽  
The Core ◽  
Post Buckling ◽  
Overall Buckling ◽  
Interactive Buckling

Abstract Compressive behavior of two classes of “sandwich” structures is investigated. These structures have for their principal load bearing components two relatively stiff parallel horizontal sheets which are interconnected in one of the following ways: (i) by a highly compliant core material such as foam, or (ii) a set of discrete stiffeners connecting the parallel (top and bottom) sheets. In case (i), the structure can buckle in either a local mode in which the core and the facing bend together or a wrinkling mode in which the facing sheet undergoes severe bending with the core subjected to deformation in the transverse plane. It is found that these plates have neither post buckling stiffness nor do they exhibit any imperfection-sensitivity. In case (ii) the point of principal interest is the interaction of local and overall buckling. For the case of coincident local and overall buckling, it is found that 30% reduction in the maximum load can occur for modest levels of imperfections as a result of modal interaction.

On the Secondary Bifurcation of Three Dimensional Standing Waves.

1985 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Three Dimensional ◽  
Standing Waves ◽  
Secondary Bifurcation

Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky
Keyword(s):  
Mathematical Theory ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Theory Of Elasticity ◽  
Elastic Plates ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Complex Boundary ◽  
Plates And Shells ◽  
Basic Equations

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

Fracture Mechanics of Thin Plates and Shells Under Combined Membrane, Bending, and Twisting Loads

2005 ◽  
Vol 58 (1) ◽  
pp. 37-48 ◽  
Author(s):  
Alan T. Zehnder ◽  
Mark J. Viz
Keyword(s):  
Fracture Mechanics ◽  
Plate Theory ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Crack Tip ◽  
Thin Plates ◽  
Element Analysis ◽  
Crack Tip Fields ◽  
Plates And Shells ◽  
Membrane Bending

The fracture mechanics of plates and shells under membrane, bending, twisting, and shearing loads are reviewed, starting with the crack tip fields for plane stress, Kirchhoff, and Reissner theories. The energy release rate for each of these theories is calculated and is used to determine the relation between the Kirchhoff and Reissner theories for thin plates. For thicker plates, this relationship is explored using three-dimensional finite element analysis. The validity of the application of two-dimensional (plate theory) solutions to actual three-dimensional objects is analyzed and discussed. Crack tip fields in plates undergoing large deflection are analyzed using von Ka´rma´n theory. Solutions for cracked shells are discussed as well. A number of computational methods for determining stress intensity factors in plates and shells are discussed. Applications of these computational approaches to aircraft structures are examined. The relatively few experimental studies of fracture in plates under bending and twisting loads are also reviewed. There are 101 references cited in this article.

SOFTENING BRANCHES OF A TWO-DEGREE-OF-FREEDOM SYSTEM INDUCED BY SPATIAL BUCKLING

2006 ◽  
Vol 06 (04) ◽  
pp. 493-512 ◽  
Author(s):  
NOËL CHALLAMEL
Keyword(s):  
Structural Model ◽  
Structural Parameters ◽  
Degree Of Freedom ◽  
Lateral Loading ◽  
Load Parameter ◽  
Imperfection Sensitivity ◽  
Softening Effect ◽  
Out Of Plane ◽  
Secondary Bifurcation ◽  
Two Degree Of Freedom

The aim of this paper is to show how geometrical non-linearity may induce equivalent softening in a simple two-degree-of-freedom spatial elastic system. The generic structural model studied is a generalization of Augusti's spatial model, incorporating lateral loading. This model could be used as a teaching model to understand the softening effect induced by out-of-plane buckling. The lateral loading in the plane of maximal stiffness is considered as the varying load parameter, whereas the vertical load is perceived as a constant parameter. It is shown that a bifurcation occurs at the critical horizontal load. The fundamental path becomes unstable, beyond this critical value. However, two symmetrical bifurcate solutions appear, whose stability depend on the structural parameters value. No secondary bifurcation is observed for this system. The presented system possesses imperfection sensitivity, and imperfection insensitivity, depending on the values of the structural parameters. In any case, for sufficiently large rotations, collapse occurs with unstable softening branches induced by spatial buckling.

Geometrical characterisation of fault arrays in three dimensions

2021 ◽  
Author(s):  
Vincent Roche ◽  
Giovanni Camanni ◽  
Conrad Childs ◽  
Tom Manzocchi ◽  
John Walsh ◽  
...  
Keyword(s):  
Full Range ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Normal Faults ◽  
Fault Geometry ◽  
Surface Geometry ◽  
Fault Surface ◽  
Mechanical Stratigraphy ◽  
Quantitative Basis ◽  
Geometry Analysis

<p>Normal faults are often complex three-dimensional structures comprising multiple sub-parallel segments separated by intact or breached relay zones. In this study we outline geometrical characterisations capturing this 3D complexity and providing a semi-quantitative basis for the comparison of faults and for defining the factors controlling their geometrical evolution. Relay zones are classified according to whether they step in the strike or dip direction and whether the relay zone-bounding fault segments are unconnected in 3D or bifurcate from a single surface. Complex fault surface geometry is then described in terms of the relative numbers of different types of relay zones to allow comparison of fault geometry between different faults and different geological settings. A large database of 87 fault arrays compiled primarily from mapping 3D seismic reflection surveys and classified according to this scheme, reveals the diversity of 3D fault geometry. Analysis demonstrates that mapped fault geometries depend on geological controls, primarily the heterogeneity of the faulted sequence and the presence of a pre-existing structure. For example, relay zones with an upward bifurcating geometry are prevalent in faults that reactivate deeper structures, whereas the formation of laterally bifurcating relays is promoted by heterogeneous mechanical stratigraphy. In addition, mapped segmentation depends on resolution limits and biases in fault mapping from seismic data. In particular, the results suggest that the proportion of bifurcating relay zones increases as data resolution increases. Overall, where a significant number of relay zones are mapped on a single fault, a wide variety of relay zone geometries occurs, demonstrating that individual faults can comprise segments that are both bifurcating and unconnected in three dimensions. Models for the geometrical evolution of fault arrays must therefore account for the full range of relay zone geometries that appears to be a characteristic of all faults.</p>

scholarly journals Overall buckling analysis on different formed circular steel tubular columns under axial compression

2019 ◽  
Vol 277 ◽  
pp. 03017
Author(s):  
Xi Feng Yan
Keyword(s):  
Numerical Analysis ◽  
Axial Compression ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Element Model ◽  
Welded Steel ◽  
Eurocode 3 ◽  
Column Design ◽  
Hot Rolled ◽  
Overall Buckling

This paper presents a numerical finite element model (FEM) investigation on the overall buckling behaviour of hot-rolled (HR), submerged arc welded (SAW) and high-frequency welded (HFW) steel circular columns under axial compression. Three dimensional FEM of circular hollow sections were developed using shell elements considering material nonlinearities, geometric imperfections and residual stress. The established FEM was used to simulate experimental studies conducted by past researchers. Good agreement has been found between numerical analysis and past researchers results, which has validated the reasonability of the FEM to carry out further investigation. Based on the validated FEM, numerical analysis incorporating 180 numerical generated HR, SAW and HFW steel circular columns with various section sizes and slenderness were carried out. The numerical analysis results were compared with the existing column design curves in Chinese, European and American codes. The numerical results showed that the design resistances for hot-rolled and welded steel circular columns calculated based on design curve a in both GB 50017-2003 and Eurocode 3 and the design formula in ANSI/AISC 360-2016 should be accepted. In addition, to further improve the design efficiency, new column design curves for hot-rolled and welded steel circular columns were recommended based on the expressions in GB 50017-2003 and Eurocode 3.

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