Membrane structures have been used since the earliest of times. Until recently, their analysis has relied chiefly on trial and error; however, modern methods of analysis are evolving. The deformations are nearly always of the large rotation and/or strain type and are thus inherently nonlinear. Static analysis can be considered as a special case of the dynamic analysis. This paper is concerned then with reviewing methods of nonlinear dynamic analysis of membrane structures. Two problems of analysis are associated with membrane structures: (i) shape (or form) finding; (ii) response (deformation and/or stress) analysis. Shape finding (ie, determination of the surface geometry given an initial prestress, generation of cutting patterns, etc) is nontrivial but well documented in the literature and is not considered in this paper. In this review attention is instead focused on formulation of field equations, wrinkling analysis, fluid/structure interactions, material nonlinearities, and computational methods.

Nonlinear Dynamic Response of Membranes: State of the Art

Simple and economical procedures for large-deformation elasto-plastic analysis of frames, whose members can be characterized as beams, are presented. An assumed stress approach is employed to derive the tangent stiffness of the beam, subjected in general to non-conservative type distributed loading. The beam is assumed to undergo arbitrarily large rigid rotations but small axial stretch and relative (non-rigid) point-wise rotations. It is shown that if a plastic-hinge method (with allowance being made for the formation of the hinge at an arbitrary location or locations along the beam) is employed, the tangent stiffness matrix may be derived in an explicit fashion, without numerical integration. Several examples are given to illustrate the relative economy and efficiency of the method in solving large-deformation elasto-plastic problems. The method is of considerable utility in analysing off-shore structures and large structures that are likely to be deployed in outerspace.

Large-deformation, elasto-plastic analysis of frames under nonconservative loading, using explicitly derived tangent stiffnesses based on assumed stresses

A critical review of the heavy elastica

Many design codes do not give methods for designing steel arches against in-plane failure. The few that do provide methods that are essentially based on a linear interaction equation for the in-plane strengths of an equivalent beam-column, which uses the maximum elastic bending moment and axial compression in the arch. However, the linear interaction equation for a beam-column may not be suitable for an arch because it does not consider the strength characteristics of steel arches. This paper studies the in-plane buckling of arches in uniform compression and uses a nonlinear inelastic finite-element model to develop a method for designing steel arches against uniform compression, and also to develop an interaction equation for the design of steel arches against nonuniform in-plane compression and bending. Analytical solutions for the buckling loads of shallow arches in uniform compression are obtained. It is found that the design equation for steel columns cannot be used directly for steel arches in uniform compression, nor can the design interaction equations for steel beam-columns be used directly for steel arches under nonuniform compression and bending. The proposed design equations provide close predictions for the in-plane buckling strengths of both shallow and nonshallow steel arches in uniform compression. The modified interaction equation proposed provides good lower bounds for the in-plane strengths of both shallow and nonshallow steel arches in bending and compression because it considers the nonuniform distributions of the bending moment and axial compression around the arch, the behavior of shallow arches, and the favorable moment redistribution after the first hinge forms.

In-Plane Buckling and Design of Steel Arches

This paper presents a new perturbation method of analysis applicable to a class of geometrically non-linear problems of shells, plates, and membranes with translationally restrained edges. The perturbation parameter is a linear function of Poisson's ratio. The convergence of successive perturbations (i.e., approximations) is independent of the magnitudes of deflections. The method also offers a rational explanation of the efficacy of Berger's approximate equations, thus placing Berger's method on a firmer foundation while at the same time weakening his hypothesis of vanishing second membrane strain invariant in the strain energy integral. Several solutions and results are obtained for the purposes of illustration and discussion. Whenever possible, calculated values are compared with results obtained by other means.

A new approach to the analysis of shells, plates and membranes with finite deflections

Unter Vernachlassigung der Schubverzerrung und der Dehnbarkeit der elastischen Linie werden die kritischen Werte der auf einen schlanken Zweigelenkkreisbogentrager niederwarts wirkenden Gipfeleinzellast auf der Basis der nichtlinearen Eulerschen Theorie berechnet. Der Zustand des Bogentragers im Augenblick der seitlichen Schwenkung sowie dessen Verhalten vor und nach der Knickung, werden fur vier verschiedene Massverhaltnisse des Bogens untersucht. Die bisherigen Naherungslosungen werden mit den berechneten Ergebnissen verglichen. A bar on top of a letter indicates that the entity pertains to the right half of the arch.

Nonlinear analysis of buckling and postbuckling behavior of circular arches

Auf Grund der Hypothesen von Ebenbleiben und Normalitat der Querschnitte werden die Differentialgleichungen der nichtlinearen Theorie der Bogentrager abgeleitet und im Falle des schlanken, durch Einzellasten belasteten Kreisbogentragers mit undehnbarer Mittellinie auf die Form der Pendelgleichung gebracht. Diese Gleichung wird dann benutzt, um die grossen Durchbiegungen und die Spannungsresultierenden eines Zweigelenkkreisbogens, der durch eine lotrechte exzentrische Einzellast belastet wird, zu berechnen. In der Nahe der kritischen Last bewirken kleine Exzentrizitaten bedeutende Grossenanderungen der Spannungsresultierenden und der Durchbiegungen.

https://link.springer.com/content/pdf/10.1007/BF01594857.pdf

Large deflections of eccentrically loaded arches

Large deflections of the heavy horizontal cantilever of uniform cross section have been investigated by Hummel and Morton [1], Bickley [2], and Rohde [3]. No exact analysis of the postbuckling deflections of the heavy column could be found in technical and mathematical publications. The purpose of this note is to extend and improve Rohde's results for the horizontal cantilever as well as to present an exact analysis of large postbuckling deflections of the heavy column. For notations refer to Fig. 1. According to the Bernoulli-Euler hypothesis for slender beams and columns, the bending couple M is proportional to the curvature dip/ds of the deflected inextensional elastica. Hence, m where EI = const, is the flexural stiffness of the column and dM/ds = ws sin by the conditions of equilibrium. Thus, d2\\p ws . . ,0. -5 + -sin* = 0, (2) in which w is the weight per unit length of the column. Let

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Large deflections of heavy cantilever beams and columns

Numerical method for determining response of elasto-plastic structure of finite number of degrees of freedom to dynamic loads by high speed digital computer; damping, either linear or nonlinear, may exist; method consists of numerical integration combined with matrix solution for elasto-plastic constraints; epecial case of multi-story elasto-plastic structure responding to lateral dynamic forces.

/pdf/dynamic-analysis-of-elasto-plastic-structures-1od0ej39sa.pdf

Donald A. DaDeppo

Papers

A new approach to the analysis of shells, plates and membranes with finite deflections

Nonlinear analysis of buckling and postbuckling behavior of circular arches

Large deflections of eccentrically loaded arches

Large deflections of heavy cantilever beams and columns

Dynamic Analysis of Elasto-Plastic Structures