Showing papers on "Virtual work published in 1983"

Variational Principles in Mechanics

Abstract: The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. The belief that “something” should be minimized was in fact a long-standing conviction of natural philosophers who felt that God had constructed the universe to operate in the most efficient manner—but how that efficiency was to be assessed was subject to interpretation. However, Fermat (1657) had already invoked such a principle successfully in declaring that light travels through a medium along the path of least time of transit. Indeed, it was by recognizing that the brachistochrone should give the least time of transit for light in an appropriate medium that Johann Bernoulli “proved” that it should be a cycloid in 1697. (See Problem 1.1.) And it was Johann Bernoulli who in 1717 suggested that static equilibrium might be characterized through requiring that the work done by the external forces during a small displacement from equilibrium should vanish. This “principle of virtual work” marked a departure from other minimizing principles in that it incorporated stationarity—even local stationarity—(tacitly) in its formulation. Efforts were made by Leibniz, by Euler, and most notably, by Lagrange to define a principle of least action (kinetic energy), but it was not until the last century that a truly satisfactory principle emerged, namely, Hamilton’s principle of stationary action (c. 1835) which was foreshadowed by Poisson (1809) and polished by Jacobi (1848) and his successors into an enduring landmark of human intellect, one, moreover, which has survived transition to both relativity and quantum mechanics. (See [L], [Fu] and Problems 8.11 8.12.)

A simple contact–friction interface element with applications to buried culverts

Abstract: A simple friction–contact interface element is introduced which simulates frictional slippage, separation, and re-bonding of two bodies initially mating at a common interface and subsequently deforming with an arbitrary static loading schedule. Constraint equations between initially mating node pairs and the general principle of virtual work are used to formulate the interface element for a finite element solution procedure. Some advantages of the interface element include; easy implementation intostandard finite element programs, direct determination of interface forces without round-off problems, and the generality afforded by the virtual work formulation to include other non-conservative models in the system. The application of the interface element to an idealized buried culvert problem illustrates that the model behaves properly. A second application, for a long-span culvert installation with incremental soil loading, demonstrates the importance of modelling slippage at the culvert–soil interface in order to conform with experimental observations.

Ultimate Strength Analysis of Curved I‐Beams

Abstract: A method of analyzing the elastic and inelastic large‐displacement behavior of horizontally curved I‐beams based on the transfer matrix method is presented. The field transfer matrix is derived under the assumption that the elastic core in each segment is constant and the resultant force in the yielded portions of the cross section is replaced by the external force. The point transfer matrix is derived using the continuity of the displacements and the principal of virtual work taking into account the discontinuity of centroidal lines at the nodes. A computer program based on this theory has been developed and the theoretical values for the deformation and the ultimate strengths are shown to be in good agreement with experimental results. Some numerical examples are also presented. The effects of cross‐sectional dimensions, loading conditions, and residual stresses on the ultimate strength are analyzed for specified central angles of curved beams.

Dynamic elasto‐plastic response of shells in an acoustic medium—EPSA code

Abstract: A nonlinear, large deflection, elasto-plastic finite element code (EPSA) has been developed for the analysis of shells in an acoustic medium subjected to dynamic loadings. The nonlinear equations of shells are discretized with the aid of a finite difference/finite element method based upon the principle of virtual work. The resulting system of equations contains the nodal displacements as the generalized co-ordinates of the problem. The integration in time of the equations of motion is done explicitly via a central difference scheme. Shell strain-displacement relations are established by a two-dimensional finite difference scheme. The shell constitutive equations are formulated in terms of the shell stress resultants and the shell strains and curvatures. The fluid-structure interaction is accounted for by means of the doubly asymptotic approximation (DAA) expressed in terms of orthogonal fluid expansion functions. The analytically produced results satisfactorily reproduce available experimental data for dynamically loaded shells.

An Approximate Hybrid Type of Virtual Work Principle for Two Elastoimpact Contact Bodies

Abstract: This paper describes the typical finite element methods and the variational principles for two elastic contact bodies. The following results are obtained in this paper : (1) The two-dimensional behavior for the longitudinal impact of two uniform rods is calculated, using the tinite element method for elastoimpact contact structures. An impact contact force is decomposed into a stationary component due to contact displacement and a variable component due to contact acceleration. The variable component is within the range of a maximum ±10.4 %. Therefore, the impact force can be approximately expressed by the stationary component. (2) The stationary component of the impact force corresponds to a Lagrange multiplier in the variational principle. An approximate hybrid type of virtual work principle is formulated in various contact and separate states of two bodies. If the components of velocities and accelerations are eliminated in this principle, a precise virtual work principle for static contact structures can be derived. Using the concept of relative movement, the matrices degree of the finite element method based on this principle remains constant in various contact and separate states.

Instability Analysis of Pressure-Loaded Thin Arches of Arbitrary Shape

Abstract: The governing differential equations and the virtual work expressions for the large displacement analysis of thin arches of arbitrary shape, subjected to pressure loads, are derived. The virtual work expressions are employed as the basis for formulation of finite element stiffness equations. Classical solutions are obtained, from the differential equations, for the buckling of circular rings under uniform “follower” (hydrostatic) and “dead” (constant direction) pressure loadings. Finite element solutions are calculated for elliptical rings for a wide range of axis ratios.

Cooling Towers on Flexible Foundations

Abstract: When large cooling towers are placed directly on moderately soft soils or on long piles, the foundation flexibility has a marked influence on tower behavior. Early studies showed that axisymmetric vertical flexibility of the foundations would reduce the capacity of the tower to carry wind loads by the meridional membrane stresses so that the shell would then develop substantial circumferential bending moments over the entire height of the tower. The present paper extends these studies to include non-axisymmetric foundation stiffness which can arise frequently in practical cases. The method of analysis given here uses a set of soil-structure interactive equations derived from the principle of virtual work. Displacements and forces are developed by harmonic analysis and the results give the influence of a nonuniform foundation stiffness on the tower behavior. Numerical studies of two typical tower configurations show that, for gravity load it is the variation of foundation stiffness that influences design, while for wind load it is the average uniform foundation stiffness that governs. The method can be used for any axisymmetrical structure.

Studies of nonlinear theories for thin shells

Abstract: In order to formulate the equations for the study here, the Fourier expansions upon the system of orthonormal polynomials are used. It may be considerably convenient to obtain the expressions of displacements as well as stresses directly from the solutions. Based on the principle of virtual work the equilibrium equations of various orders are formulated. In particular, the system of thirdorder is given in detail, thus providing the reference for accuracy analysis of lower-order equations. A theorem about the differentiation of Legendre series term by term is proved as the basis of mathematical analysis. Therefore the functions used are specified and the analysis rendered is no longer a formal one. The analysis will show that the Kirchhoff-Love's theory is merely of the first-order and the theory which includes the transverse deformation but keeps the normal straight is essentially of the first order, too.

Virtual Work Principles for Two Elastoimpact Contact Bodies

Abstract: The virtual work principles for two elastoimpact bodies are required to formulate the finite element method for elastoimpact contact structures with translational and rotational motions. At first, the virtual work principles and their subsidiary contact conditions for the two bodies are two-dimensionally formulated in the basic contact and separate states of the two bodies, based on the conventional principle for a body. In the separate state of the two bodies, the virtual work principle and its subsidiary conditions are obtained by introducing a relative movement corresponding to clearance and regarding the bodies as a connective body. The relative movement in a slip direction is applied to the subsidiary conditions of a slip state in the same way as to the subsidiary conditions of the separate state Finally, the virtual work principles and their subsidiary conditions for the two bodies are three-dimensionally established on the basis of the two-dimensional principles and their subsidiary conditions.

Orthotropic hole element

Abstract: A finite element was developed to adequately represent the state of stress in the region around a circular hole in orthotropic material experiencing reasonably general loading. This has been achieved through a complementary virtual work formulation of the stiffness and stress matrices for a square element with a center circular hole. The element has been incorporated into COSMIC/NASTRAN as a dummy element. Sample problems have been solved and these results are presented.