Large Deflexions Research Articles

Prestressed Morphing Bistable and Neutrally Stable Shells

This study deals with prestressed shells, which are capable of “morphing” under large deflexions between very different load-free configurations. Prestressing involves plastically curving a flat, thin shell in orthogonal directions either in the opposite or same sense, resulting in two unique types of behavior for isotropic shells. Opposite-sense prestressing produces a bistable, cylindrically curved shell provided the prestress levels are large enough and similar in size: This effect forms the basis of a child’s “flick” bracelet and is well known. On the other hand, same-sense prestressing results in a novel, neutrally stable shell provided the levels are also sufficiently large but identical: The shell has to be made precisely, otherwise, it is monostable and is demonstrated here by means of a thin, helically curved strip. The equilibrium states associated with both effects are quantified theoretically and new expressions are determined for the requisite prestress levels. Furthermore, each stability response is revealed in closed form where it is shown that the neutrally stable case occurs only for isotropic materials, otherwise, bistability follows for orthotropic materials, specifically, those, which have a shear modulus different from the isotropic value. Finally, prestressing and initial shape are considered together and, promisingly, it is predicted that some shells can be neutrally stable and bistable simultaneously.

Large deflexions of a uniformly loaded, elastically restrained composite strip

An analysis is given of the large-deflexion behaviour under uniform normal load of a multi-layered composite strip that exhibits coupling between moments and planar strains and whose planar boundary displacements are elastically restrained. The strip forms part of a uniformly loaded infinite plate supported on a grid of stringers that is itself supported on an orthogonal grid of ribs.

High Reynolds number flow past an equatorial island

A survey of the density field in the immediate vicinity of Jarvis Island was made in April 1971. The island is isolated and is situated in the Pacific Equatorial Undercurrent, thus presenting the opportunity to study a high Reynolds number (∼ 109) stratified shear flow. Large deflexions of the isopycnals were observed near the island. Solutions derived from the perfect-fluid equations of Drazin (1961) axe in reasonable agreement with the observed mass field in the upstream region. Downstream, a wake region was apparent, but the dynamics of the wake are obscure. A coefficient of pressure deficit in the wake is in the same range as values computed from homogeneous laboratory flows, but baroclinic effects are observed.

Large deflexions of thin elastic bars by the electronic analogue computer

The paper presents a general method, using an electronic analogue computer, of determining large deflexions of thin, piecewise prismatic elastic bars which may be initially straight or bent to any shape, carrying arbitrary discrete loads and moments and subjected to arbitrary boundary conditions. Numerical examples are given to illustrate the method.

Large deflexions of cantilevers by the electronic analogue computer

The paper presents a method of finding large deflexions of initially straight and initially curved cantilevers and circular rings subjected to point loads, using an electronic analogue computer. The method is illustrated with numerical examples and the results are compared with those obtained from the exact and numerical methods.

A grid analogy for the non-linear behaviour of elastic flat plates

Deflexions and stresses for a plate with small deflexion can be obtained from an equivalent grid. In this paper, a similar grid is introduced for problems with large deflexions. The technique used is to separate the problem into two sections, one dealing with out-of-plane forces (stress resultants), the other dealing with in-plane forces and to link these and introduce the effect of large deflexions through an equilibrium routine. The non-linear effect is obtained by an iteration process between the two parts. A computer programme has been developed for the solution of large deflexion problems of plates using this method. The analysis compares well in the case of a worked solution and can deal with problems which cannot be solved by direct mathematical analysis.

Tensile membrane behaviour of uniformly loaded rectangular reinforced concrete slabs with fully restrained edges

Summary A theory is developed for the load carried at large deflexions by the tensile membrane action of uniformly loaded rectangular slabs with edges restrained against all translation. It is assumed that the reinforcement acts as a plastic membrane. On comparison with test results the theory is shown to be conservative. The load carried by tensile membrane action is found to be considerable and for heavily reinforced slabs it can exceed the ultimate flexural strength.

Thin heavy elastic strips at large deflexions

The deflected forms taken up by an elastic strip of uniform thickness and constant width hanging under its own weight as (i) a cantilever, (ii) a heart-shaped loop and (iii) with position-fixed ends in the same horizontal plane, are presented, the results having been obtained using an analogue computer. Some comparisons with small-deflexion theory are drawn. Results showing deflected form are also given for a uniformly thick strip whose width tapers uniformly (triangular strip) and for a long, thin, heavy wire held at its top end and deflected by a uniform horizontal wind pressure.

THE LARGE DEFLEXION OF THIN CANTILEVERED PLATES

A scheme of solution, especially suited for digital computers, is presented for the case of the bending of a cantilevered plate of small constant thickness, under the action of prescribed normal loads which produce moderately large deflexions. It takes the form of an asymptotic expansion of the solution of the von Kármán equations in terms of a large non-dimensional loading parameter. The leading term of the expansion is a combination of the solution obtained using the inextensional theory of bending and a suitable boundary layer solution. The form of the higher order terms is deduced and it is shown how they may be obtained in terms of quadratures and solutions of ordinary linear differential equations.

Blown Flaps for Stol Operation

There has been some speculation anions STOL aircraft designers (hat a net increase in lift might be realized from a combination of blown flap and slip‐stream deflexion. A recent wind tunnel investigation confirms the speculation for large flap deflexions. At small flap deflexions, however, the lift gained by the combination is less than the sum of the separate contributions. The effect at medium flap deflexion is dependent on the strength of the slipstream. These results should be applicable to most configurations with a blown plain flap in the slipstream behind the propellers.

Some further results on the rubber membrane theory and Laplace's equation

Some further results and points are presented to underline the theory of constant horizontal tension of the rubber membrane as discussed in a previous paper by the author.(3) Small deflexion theory which assumes constant resultant tension fails, in the case of coaxial geometry, to account at large deflexions for (a) the linear force-deflexion law obtained and (b) the close correspondence of the membrane's contour to Laplace's equation. Both observations are successfully predicted by the revised theory which places no limitations on geometry or deflexion for the above two conditions to hold. Experiment also confirmed a linear force-deflexion law for an asymmetric geometry.

A characteristic type of instability in the large deflexions of elastic plates III. Curved rectangular plates in axial compression

The analysis of part I is extended to deal with the case of free-edged rectangular plates having an initial curvature about an axis parallel to one pair of opposite edges and loaded by distributed bending moments applied to the straight edges and compressive forces applied to the curved edges. In particular, the stability and post-buckling behaviour of such plates subjected to the compressive forces alone is studied. The axially symmetrical buckling of thin-walled circular tubes in axial compression is also considered. Experimental plates are found to buckle at loads rather lower than those predicted.

A characteristic type of instability in the large deflexions of elastic plates. I. Curved rectangular plates bent about one axis II. Flat square plates bent about all edges

Part I. From a general equation governing the bending of thin elastic plates into certain types of surfaces of revolution are derived expressions for the behaviour of rectangular plates with initial curvatures, subjected to pure bending about one axis. It is found that such plates exhibit the type of instability characteristic of thin-walled structures which depend for their stiffness on curvature. Curves are drawn showing the deformation suffered by such plates, and an expression for the critical bending moment at which instability occurs is obtained. Experimental results show satisfactory agreement.Part II. The analysis of part I is extended to deal with the case of flat square or rectangular plates loaded by distributed bending moments applied to all four edges. Curves are drawn to describe their behaviour, and they are found to exhibit the characteristic instability displayed by thin-walled curved structures. Experimental verification is satisfactory.

LXXXVI.The large deflexions of a thin circular ring

LXXXVI.<i>The large deflexions of a thin circular ring</i>

Bakerian Lecture: Nuclear constitution of atoms

Introduction .—The conception of the nuclear constitution of atoms arose initially from attempts to account for the scattering of α-particles through large angles in traversing thin sheets of matter. Taking into account the large mass and velocity of the α-particles, these large deflexions were very remarkable, and indicated that very intense electric or magnetic fields exist within the atom. To account for these results, it was found necessary to assume that the atom consists of a charged massive nucleus of dimensions very small compared with the ordinarily accepted magnitude of the diameter of the atom. This positively charged nucleus contains most of the mass of the atom, and is surrounded at a distance by a distribution of negative electrons equal in number to the resultant positive charge on the nucleus. Under these conditions, a very intense electric field exists close to the nucleus, and the large deflexion of the α-particle in an encounter with a single atom happens when the particle passes close to the nucleus. Assuming that the electric forces between the α-particle and the nucleus varied according to an inverse square law in the region close to the nucleus, the writer worked out the relations connecting the number of α-particles scattered through any angle with the charge on the nucleus and the energy of the α-particle. Under the central field of force, the α-particle describes a hyperbolic orbit round the nucleus, and the magnitude of the deflection depends on the closeness of approach to the nucleus. From the data of scattering of α-particles then available, it was deduced that the resultant charge on the nucleus was about ½ A e , where A is the atomic weight and e the fundamental unit of charge. Geiger and Marsden made an elaborate series of experiments to test the correctness of the theory, and confirmed the main conclusions. They found the nucleus charge was about ½ A e , but, from the nature of the experiments, it was difficult to fix the actual value within about 20 per cent. C. G. Darwin worked out completely the deflexion of the α-particle and of the nucleus, taking into account the mass of the latter, and showed that the scattering experiments of Geiger and Marsden could not be reconciled with any law of central force, except the inverse square. The nuclear constitution of the atom was thus very strongly supported by the experiments on scattering of α-rays. Since the atom is electrically neutral, the number of external electrons surrounding the nucleus must be equal to the number of units of resultant charge on the nucleus. It should be noted that, from the consideration of the scattering of X-rays by light elements, Barkla had shown, in 1911, that the number of electrons was equal to about half the atomic weight. This was deduced from the theory of scattering of Sir J. J. Thomson, in which it was assumed that each of the external electrons in an atom acted as an independent scattering unit.