Buckling of Columns with Movable Boundaries∗

The Euler buckling problem of a slender tubular column subject to its own weight, tension or compression exerted at its top, and internal and external variable static fluid pressure is studied. This problem finds many applications in drilling and production risers, mining risers, hydraulic columns, and legs of Tension Leg Platforms. The supports at the upper end of the column are considered movable to properly simulate drill ships or platforms that support such columns. The corresponding eigenvalue problem is comprised of a fourth-order differential equation with variable coefficients and four homogeneous boundary conditions. Two methods of solution derived in previous work are implemented numerically. The first solution is expressed in terms of Airy functions of the first and second kind and the second in terms of power series. The combined results of the two methods yield the critical buckling curves over the entire domain of practical interest. The critical curves are plotted in the plane of the two loading variables for the first six buckling modes for four different sets of boundary conditions. The results reveal the dependence of the asymptotic behavior of the critical curves for long columns on the boundary conditions.     

Engineering Models for Numerical Analysis of Composite Bending Members∗

Abstract

ABSTRACT The two-step numerical analysis of a composite beam structure is presented in this paper. The first step, based on the idea of dividing the cross section into laminas, leads to the estimation of the moment-curvature relation for different types of cross sections used in composite beams. The second step adopts this constitutive relation, which is expressed in the space of generalized stresses and strains, into finite element nonlinear code. Some numerical examples are given, to show the agreement of numerical calculations with results of the authors' experiments, when the shrinkage of a concrete encasement and stresses due to welding processes in steel beams are considered. In addition, the numerical concept presented here seems to reduce the sensitivity of the final results obtained to finite element discretization error.     

Constrained Optimal Design of Columns Against Buckling∗

The buckling loads of straight, simply supported columns, loaded by axial thrust are maximized, subject to the restriction that the available volume of structural material is specified. In addition, a constraint is placed upon the maximum allowable prebuckling stress. Analytic solutions are presented for a variety of different cross-sectional geometries.     

Nonlinear Interactive Buckling: Sensitivity and Optimality∗

ABSTRACT

The nonlinear buckling behavior of discrete systems in which two eigen-modes interact is studied. A variable parameter affecting the design is introduced to allow a change in the closeness of the two eigenvalues. The sensitivity of the critical equilibrium states to combined changes in the design and imperfection parameters is considered through the concept of “sensitivity surfaces.” The analysis is used to discuss the sensitivity of structural systems optimized on the basis of the Simultaneous Mode Design Principle.     

Buckling of an Axially Compressed Incompressible Half Space∗

Symmetric and nonsymmetric buckling solutions are obtained for an axially compressed half space. The half space is assumed to be made of a general in compressible isotropic elastic material. Solutions are made possible via the use of a direct approach that does not employ conventional Fourier-analysis calculations.     

Constrained Optimal Design of Circular Plates Against Buckling∗

Several problems are investigated in which the buckling loads of structures are maximized, subject to the restriction that the volume of structural material is specified. In addition, a constraint is placed upon either the maximum allowable prebuckling stress, or the minimum allowable thickness. The structures analyzed are flat, axisymmetric circular plates, loaded by uniform radial thrust, and assumed to buckle axisymmetrically. The plates are investigated for both sandwich and solid cross-sectional geometries.     

Elastic Critical Buckling of Welded Thin Steel Rectangular Plates∗

A theoretical approach for the study of the effect of residual stresses due to welding on the elastic critical buckling behavior of thin steel rectangular plates is described. A finite difference technique is utilized for the determination of in-plane residual stresses due to a weld, and the Rayleight-Ritz method is used for the critical buckling problem, with stresses due to external loads being superimposed on the residual stresses. A number of illustrative examples are included, showing the possible detrimental effect of residual stresses due to welding. An approximately linear relationship is shown to exist between the square of the natural frequency of lateral vibration and the drop in buckling strength, for certain loading conditions     

Dynamic Elastic Buckling of Complete Spherical Shells with Initial Imperfections∗

The dynamic elastic buckling behavior of a geometrically imperfect complete spherical shell that is subjected to a uniform external step pressure is examined using Sander's equilibrium and kinematic equations, appropriately modified to include the influence of inertia forces and initial stress-free imperfections in the radius. A finite-difference procedure with either the Houbolt or Park methods of time integration is used to predict the axisym-metric dynamic elastic buckling pressures and associated critical mode numbers. The dynamic buckling pressure is significantly smaller than the corresponding static value for small initial imperfections, but is less imperfection     

The Shallow Arch—General Buckling,Postbuckling, and Imperfection Analysis∗

A general discussion of the behavior of the shallow circular arch is presented. It is shown that, irrespective of specific loading or boundary conditions, it is possible to arrive at general conclusions regarding buckling, postbuckling, and imperfection sensitivity. General methods of analysis are established which lead to the determination of points of bifurcation and of postbuckling paths under symmetric loads. Modifications accounting for antisymmetric load components are introduced, with special emphasis on their asymptotic and limit load effect.

A typical numerical example is carried through in detail. The solution is “exact” in the sense of shallow arch theory. Its asymptotic behavior conforms to the asymptotic theory of Koiter.     

Variational Principles for EIastoplastic Continua∗

ABSTRACT

A mixed variational principle is constrained by a homogeneous yield function using a Lagrange multiplier. The Lagrange factor corresponds to the scalar factor in Prager's normality rule for the plastic strain increments. Several reduced functionals and their associated constitutive equations are derived by eliminating some variables.     

On the Buckling of an Infinite Beam of Finite Width Embedded in an Isotropic Elastic Solid∗

The present paper considers the problem of buckling of a beam of finite width that is embedded in bonded contact with an isotropic elastic solid. Analysis of the buckling problem is restricted to the class of slender beams of narrow width that exhibit flexure only in the longitudinal direction. The governing integral equations are solved in an approximate fashion. Numerical results presented indicate the manner in which the buckling load is influenced by the relative flexibility of the beam-elastic medium system.     

Comparison Study of Constitutive Models for Overconsolidated Clays

Widely distributed in natural deposits,the overconsolidated(OC)clays have attracted extensive experimental investigations on their mechanical behaviors,especially in the 1960s and 1970s.Based on these results,numerous constitutive models have also been established.These models generally fall into two categories:one based on the classical plasticity theory and the other the bounding surface(BS)plasticity theory,with the latter being more popular and successful.The BS concept and the subloading surface(SS)concept are the two major BS plasticity theories.The features of these two concepts and the representative models based on them are introduced,respectively.The unified hardening(UH)model for OC clays is also based on the BS plasticity theory but distinguishes itself from other models by the integration of the reference yield surface,unified hardening parameter,potential failure stress ratio,arid transformed stress tensor.Modification is made to the Hvorslev envelop employed in the UH model to improve its capability of describing the behaviors of clays with extremely high overconsolidation ratio in this paper.The comparison among the BS model,SS model,and UH model is performed.Evidence shows that all these three models can characterize the fundamental behaviors of OC clays,such as the stress dilatancy,strain softening and attainment of the critical state.The UH model with the revised Hvorslev envelop has the fewest parameters which are identical to those of the modified Cam-Clay model.     

Higher Integrability for Minimizers of the Mumford–Shah Functional

We prove higher integrability for the gradient of local minimizers of the Mumford–Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55–62, 1991).     

Buckling and post-buckling analysis for magneto-elastic–plastic ferromagnetic beam-plates with unmovable simple supports

This paper presents an analysis of buckling/snapping, bending and post-buckling/snapping behaviors of magneto-elastic–plastic interaction and coupling for soft ferromagnetic beam-plates with geometrically nonlinear deformation and unmovable simple supports at the ends of the plates. Based on the expression of magnetic force from the variational principle of ferromagnetic plates, the theory of thin plates with the nonlinear deformation of van Karman’s type, and the Mises yield criterion and the increment theory for plastic deformation, here, we establish a numerical code to quantitatively simulate the behaviors of the nonlinearly multi-coupling problems by the finite element method. Along with that the phenomena of buckling, bending, and post-buckling/snapping, or the characteristic curve of deflection versus magnitude of applied magnetic fields are numerically displayed, the critical values of buckling/snapping and yield magnetic fields, and the expansibility of plastic region after the plates undergo plastic deformation with increasing of the applied magnetic fields, as well as the evolvement of deflection configuration of the plate are numerically obtained in a case study.     

An Improved Computational Approach for Multilevel Optimum Design∗

Recently there has been a growing interest in methods that decompose complex optimization problems and employ a multilevel or hierarchical approach. One of the most irksome problems with the hierarchical approach is the discontinuous behavior of derivatives that are transferred from the lower levels of the hierarchy to the upper levels. This paper proposes a hierarchical algorithm that is free of such difficulties. A penalty function method is employed, in combination with Newton's method with approximate second derivatives, to perform the optimization. The penalty function formulation is shown to be natural for multilevel problems. A simple three-bar truss and a simple frame problem are used for demonstration     

A KAM Theorem for Higher Dimensional Nonlinear Schrödinger Equations

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional nonlinear Schrödinger equation $$\begin{aligned} iu_t-\triangle u +M_\xi u+f(|u|^2)u=0, \quad t\in \mathbb{R }, x\in \mathbb{T }^d \end{aligned}$$ with periodic boundary conditions, where $M_\xi $ is a real Fourier multiplier and $f(|u|^2)$ is a real analytic function near $u=0$ with $f(0)=0$ . We obtain for the equation a Whitney smooth family of real-analytic small-amplitude linearly-stable quasi-periodic solutions with a nice linear normal form.     

Stability Conditions for Brittle-Plastic Structures with Propagating Damage Surfaces∗

The incremental problem for a brittle-plastic body is considered for which stable propagation of a damage interface occurs within the material. The stability conditions are derived and applied to several simple illustrative examples.     

Nonlinear Correction to the Euler Buckling Formula for Compressed Cylinders with Guided-Guided End Conditions

Euler’s celebrated buckling formula gives the critical load N for the buckling of a slender cylindrical column with radius B and length L as

A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrödinger equations with outer force

$$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$

where \(M_\xi \) is a real Fourier multiplier,\(f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)\) is real analytic and the forced frequencies \(\bar{\omega }\) are fixed Diophantine vectors.