Limit Analysis of Structures with Stochastic Strength Variations∗

On the basis of a probabilistic fomulation of the fundamental theorems of “limit analysis,” a procedure is developed which allows, with a very limited amount of computing work, the determination of a domain containing the probability distribution curve of the collapse load factor of any structure that satisfies the usual conditions for validity of the limit analysis, but has randomly distributed limit strengths.

Further improvements of the bounds thus obtained can be achieved by the equivalent of either the equilibrium or the kinematic methods of limit analysis.     

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.     

Damage Diagnosis in Frame Structures with a Dynamic Response Method∗

ABSTRACT

The dynamic response of planar frame structures composed of damped Bernoulli-Euler beams is computed, with and without a crack present in the structure. The inertance changes due to the crack are investigated in relation to the crack location, with the aim of developing a diagnosis method. The optimum excitation location and frequency and the optimum locations for response measurement are determined for best diagnosis results. The effects of crack location and severity and of damping are investigated. Damping is accounted for by the complex Young's modulus. Frames are analyzed with the electrical analogy method. A crack is modeled as a torsional spring, which is represented with an electrical resistor in the analogy. The electrical analogy method is used only as an analysis tool in this study, with the resulting equations being solved on a digital computer.     

The Hamiltonian Structures of 3D ODE with Time-Independent Invariants

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Generalized Inverses in Elastic-Plastic Analysis of Structures∗

This paper deals with elastic-plastic analysis of skeletal structures that are subjected to proportionally increasing loading. It is assumed that no local unloading occurs (holonomic behavior) and that yield conditions are piecewise linearized. A quadratic programming problem, which arises from the application of the minimum complementary energy principle for such structures, is shown to have an explicit form of solution. The matrix expression of this solution involves certain modifications of the Bott-Duffin generalized inverse. This inverse can be effectively calculated for a given structure and allows one to obtain a unique stress distribution under agiven load. Moreover, if the load is prescribed up to an unknown scalar factor, the ultimate value of this multiplier (collapse load) and the elastoplastic stress state at collapse can be found by the same method     

The Minimal Thickness of a Semicircular Arch with a Tension-Free Crown∗

ABSTRACT

When a semicircular elastic arch is loaded with a single point force at the vertex, the stress at interior points depends on the thickness of the arch and on the way the abutments react. Applying the theory of plane elasticity, three typical load-bearing abutment constraints are evaluated to find the minimal thickness at which no tensile stress occurs at a given interior point. Such thickness is found to be quite large.     

Simulation of Initially Slackened and Stiffened Structures by Virtual Distortions∗

ABSTRACT

Variational principles and computational methods for analysis of initially slackened and stiffened structures are discussed. The simulation of clearances or internal dry fraction in structural elements by virtual (eigen) distortions is applied. Considerations presented are used in the problem of nonstandard design of structural settings, with clearances or friction in the structural joints, for load capacity maximization.     

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.     

Formability of Linearly Elastic Structures with Volume‐Type Actuation

Formation theory concerns the modification of the geometric configuration of an elastic structure by means of attached and/or embedded actuators. In this paper we consider "volume" type actuation, which involves application of an isotropic expansive/contractive stress to the elastic medium. The question of "formability", i.e., whether or not a given modified geometric configuration for the elastic body can be achieved with actuation of this type, is considered at length, in both the two- and three-dimensional contexts, along with related questions of optimal formability, expressed in terms of the L² norm of the volume controller employed. In two dimensions, with the aid of the Airy "stress" function, we establish connections between optimal formation, in the "L"² norm sense, and the standard theory of conformal mapping of simply-connected regions in the complex plane. Further results are presented for multiply-connected domains, including a complete discussion of the case of an annulus.     

The Elastic-Plastic Torsion Problem: A Direct Boundary Approach∗

A new mechanics theorem pertaining to the elastic-plastic torsion problem is shown; i.e., the complementary energy evaluated on some admissible set of functions is minimized with respect to the transient (free) boundary between the elastic and plastic zones. The elastic-plastic torsion problem is then formulated as an unconstrained shape optimal design problem which treats the free boundary directly as a unknown. Unlike other formulations there is no inequality constraint involved in this formulation. The derivation of shape design sensitivity and numerical results for circular cross section and Sokolovsky's oval are presented.     

Transverse Vibrations and Stability of Systems with Gyroscopic Forces∗

Abstract

Rotating shafts and pipes conveying fluid are examples of systems involving gyroscopic forces. The vibration and stability properties of such systems are often of practical interest to structural engineers. In this paper attention is focused on the characteristic curves of gyroscopic conservative systems in an appropriately chosen loading-frequency space. An upper bound to the fundamental frequency is obtained via the concept of a “corresponding nongyroscopic system.” The choice of the parameters and the resulting

characteristic curves shed light on the stabilizing effect of gyroscopic forces. Special emphasis is placed on flutter instability. Three well-defined types of systems are discussed and several examples are analyzed. It is shown that various sequences of stable, divergence, and flutter regions may be exhibited as the loading parameter is increased, and that flutter instability may take place in an otherwise stable region.     

Upper Bounds on Deformations of Elastic-Workhardening Structures in the Presence of Dynamic and Second-Order Geometric Effects∗

Abstract

Discrete models of elastoplastic structures are considered, Piecewise linear yield conditions and hardening rules are assumed. On this basis, a deformation bounding method resting on the use of fictitious loads as proposed first by Ponter [6, 7], is developed for situations in which: (a) the geometry changes affect the equilibrium equations but their effects may be expressed by bilinear terms in the pre-existing stresses and additional displacements (“second-order geometric effects”); (b) inertia and viscous damping forces play a significant role. Comparisons are made with different bounding methods previously established by the author [3,4], for the same classes of structures and mechanical situations.     

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 Continuous Coagulation-Fragmentation¶Equations with Diffusion

Existence of global weak solutions to the continuous coagulation-fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition. Our approach relies on weak and strong compactness methods in L ¹ in the spirit of the DiPerna-Lions theory for the Boltzmann equation. Under the detailed balance condition the large-time behaviour is also studied.     

The Deflection of an Elastic Web Wrapped Around a Surface of Revolution∗

ABSTRACT

A computational method is described for predicting the deflection of an elastic web wrapped under tension around a (possibly compliant) surface of revolution. The authors validate the algorithm presented by comparison to several other solutions and to experimental data. Particular attention is paid to the interfacial loads between the web and roll, and to the redistribution of stress within the midsurface of the web.     

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.     

Longitudinal and Bending Stresses in a Beam with Saw Cut Edge Cracks∗

Abstract

The Griffith-Irwin theory of brittle fracture of elastic solids predicts the propagation of cracks on the basis of the energy release rate. This depends upon the stress intensity factors for a given crack configuration. The present paper provides these informations for the problem of an infinite number of periodic, non-coplanar, parallel edge cracks in a strip. Two types of crack configurations, namely, periodic cracks of equal length starting from one edge and a set of two coplanar symmetrical edge cracks of equal length are solved for constant and linearly varying pressure distributions. These problems arise naturally in structural mechanics while investigating stresses in extension and bending of cracked strips. Final results are obtained from the numerical solution of certain Fredholm integral equations of the second kind derived from a dual series of Papkovich-Fadle eigenfunctions     

Optimal Plastic Design of a Cross Section under Torsion with Small Bending∗

A doubly connected cross section of a bar under torsion with bending is optimized in two variants; either the internal contour is given and the external one is subject to optimization, or only the inside area is given and both contours are optimized. Minimal cross-sectional area is the design objective. The material is assumed to be perfectly plastic, incompressible, and subject to the Huber-Mises-Hencky yield condition. A specially adopted perturbation method is used and only the range of small bending is analyzed. The resulting optimal bars are regarded as thick-walled, hence no wall stability constraints are applied.     

Vibration of Shallow Shells with Two Adjacent Edges Clamped and the Others Free∗

ABSTRACT

Considerable information is available in the published literature on the free vibration frequencies and mode shapes of rectangular flat plates having two adjacent edges clamped and the other two free. However, no results appear to have been published previously for shallow shells having such edge conditions. The present work uses the Ritz method with displacement components in the form of algebraic polynomials to obtain accurate frequencies. Frequencies are determined for the first eight modes of shallow shells having spherical, cylindrical, and hyperbolic paraboloi-dal curvatures and square planforms. Beginning with the plate, the curvatures are incrementally increased in each case to the limits of shallow     

On a Class of Deformations of a Material with Nonconvex Stored Energy Function∗

This paper considers the problem of finding the deformation of a nonlinear elastic layer contained between two infinite parallel rigid plates, each of which undergoes the same finite rotation, but about noncoincident axes. Each plane of the layer is assumed to rotate about a point whose position depends on its distance from the rigid bounding plates. The locus of these centers of rotation satisfies a differential equation which depends on the strain energy density of the material. In the case of a Mooney material, the locus is a straight line connecting the centers of rotation of the bounding plates, as is to be expected. It is shown that for a certain class of strain energy density functions, the solution is nonunique and consists of piecewise linear segments.