Showing papers on "Bending of plates published in 1973"

Method of Laurent series expansion for internal crack problems

Abstract: This paper presents a general method of analysis of internal cracks in isotropic homogeneous elastic media. The proposed method is based on the Laurent series expansions of the complex potentials which are consistent with the singlevaluedness of displacements as well as stresses and strains. It applies to the problems of longitudinal shear, plane stress or plane strain and classical plate bending in which the stress state is completely characterized by one or two complex potentials.

Rigid circular plate resting on a nonhomogeneous elastic halfspace

Abstract: A FINITE ELEMENT COMPUTER PROGRAM IS USED TO SOLVE THE SETTLEMENT AND STRESSES INDUCED BY A RIGID CIRCULAR PLATE RESTING ON A NON-HOMOGENEOUS ELASTIC HALF-SPACE. THE SOLUTIONS ARE COMPARED WITH CLOSED FORM SOLUTIONS AND THE COMPUTER SOLUTIONS AGREE WITH THE THEORETICAL SOLUTIONS. THE COMPUTER SOLUTIONS ALLOW AN ENGINEER TO CALCULATE THE SETTLEMENT OF A PLATE FOR ANY COMBINATION OF YOUNG'S MODULUS OF ELASTICITY, SLOPE OF THE CURVE EXPRESSING THE LINEAR INCREASE OF YOUNG'S MODULUS WITH DEPTH Z, POISSION'S RATIO, V, G (THE UNIT LOAD ON THE PLATE) AND D (THE PLATE DIAMETER). THIS PAPER SHOWS HOW THE SOLUTIONS MAY BE USED TO IMPROVE THE PREDICTION OF SETTLEMENTS AND STRESSES BENEATH A FOOTING RESTING ON A CLAY OR SAND. FOR MOST PRACTICAL PROBLEMS, THE SOLUTION FOR A ROUGH PLATE IS THE SAME AS THE SOLUTION FOR A SMOOTH PLATE. /TRRL/

Triangular, nine-degrees-of-freedom, ⁰ plate bending element of quadratic accuracy

Abstract: Introduction. The triangular plate bending element with the three nodal values w, wx and wv at the vertices has a particular appeal—its simplicity. But a C1 polynomial interpolation scheme defined over the whole element does not exist. To overcome this Clough and Tocher [1] resorted to subdividing the element into three subelements with the transverse displacement w interpolated individually [2] over each subtriangle so as to maintain a C1 continuity both in the interior of this complex element and on its boundaries. Bazeley et al. [3] interpolated w by some rational functions to obtain the desired variation of w and its derivatives along the sides of the element for assuring a C1 continuity of displacements. They considered also the use of elements which violate the continuity requirements (non-conforming) and for which the variational principle on minimum total potential energy does not hold any more. These elements may, nevertheless, sometimes produce a valid, stable difference scheme and converge to a useful solution. But their excessive flexibility and precarious convergence did not endear them to engineers. Severn, Taylor and Dungar [4, 5] and Allman [6] used a mixed variational principle [7] for generating a compact hybrid [8] finite element. Stricklin et al. [9] (and also others) made a more radical approach to the generation of plate bending elements in general and the nine-degrees-of-freedom triangular element in particular. They started from the basic equations of elasticity rather than from a ready plate theory and did away with the Kirchhoff assumption (the shear) except at the vertices and along the sides of the element (this is the \"discrete Kirchhoff assumption\" [10]). Since the plate is now considered a three-dimensional solid, the in-plane displacements are introduced independently of the transverse displacements and the continuity requirement for them is only C°, as in three-dimensional elasticity. Assuming that the shear energy is at any rate negligible in thin plates, Stricklin neglected it altogether. Removing the Krichhoff assumption from the finite-element analysis of plates and starting with the basic three-dimensional elasticity would seem the most natural approach to the generation of bending elements, particularily since the thin plate is obtained as a limiting case from a three-dimensional solid. However, without the a priori assumptions of Kirchhoff, and with the shear energy retained, the global stiffness matrix becomes violently ill-conditioned [11] as the thickness t of the structure is reduced. The difficulties, then, in constructing thin-plate bending elements directly from three-dimensional elasticity are of a numerical or computational nature. The decline in the conditioning of the matrix may cause, in the computational stage of the solution, grave numerical (round-off) errors, or for a computer with insufficient significant digits the matrix may even be numerically singular.

Dynamic buckling of a rectangular orthotropic plate

Abstract: The elastic buckling of a simply-supported rectangular orthotropic plate, with initial imperfections, under a rapidly applied compressive load is analyzed. The large deflection plate equations are used to study inertial effects in the postbulkling phase. Numerical results for a controlled rate of load application show that the critical load is increased over the corresponding static case. The load-deflection relation is oscillatory in the postbuckling phase, thereby increasing the stresses in the plate over those in the static case. Initial imperfections decrease the critical load as well as increase the frequency and decrease the amplitude of the oscillations.

Partially conforming plate bending elements for static and dynamic analyses

Abstract: Two partially conforming flat-plate bending elements have been developed for static and dynamic analyses through the principle of virtual work.One of these elements is triangular and the other quadrilateral and both have corner nodes only, with three degrees of freedom per node. By use of these elements in the matrix displacement method, the natural frequencies of vibration of a square cantilevered plate have been calculated; these calculated results have been compared with experimental results. Agreement has been found to be good, and in general, convergence appears to take place.

Dynamic moiré methods for the bending of plates

Abstract: Two approaches are proposed to extend the Ligtenberg’s reflective moire method to dynamic-bending problems of plates. Partial slope contours of two orthogonal directions can be obtained simultaneously with the proposed schemes. Vibrating modes of plates can also be obtained by the methods.

Vibration of Circular Plates with Edge-Beams

Abstract: Extensive research in the general area of circular plate vibration has been conducted by numerous researchers, and a compilation of results in the general subject of the vibration of plates has been published by Leissa. The majority of the research has considered the idealized problem in which the plate is characterized by simply supported or clamped boundary conditions. In actuality, plate structures rarely posses simply supported or clamped edges but instead are subjected to conditions somewhat between these two extreme cases. When conditions on the boundary of a plate are changed from those approximating a simple support to a clamped one, small deflections and a more favorable distribution of stress usually result. It is very difficult to actually clamp the edges to take advantage of this fact in design, but a similar stiffening effect can be achieved by providing the plate with an edge-beam, as is shown in this note. By analyzing the coupled problem of the vibrating plate and edge-beam structure simultaneously, an exact solution to the problem is obtained.

A plastic theory for collapse of plate girders under combined shearing force and bending moment

Abstract: The end panel of a plate girder is analysed for plastic collapse under high shearing load by means of the upper bound theorem. Two possibilities are considered for the behaviour of the web: either (a) it remains unbuckled or (b) it develops a full plastic tension field. Curves are presented showing how the collapse load depends on the dimensions and on a parameter signifying the full plastic bending strength of the flanges in relation to the web strength. In general, the numerous experimental results available in the literature lie between the predictions based on (a) and (b) respectively. The analysis is developed to account for collapse under the action of shearing force and bending moment simultaneously. There is a section on collapse by a mechanism which involves plastic bending in the vertical 'end post', which includes a design method for suppression of this mode. Finally there is a worked example.(a)

Poisson's ratio and the deflection of a viscoelastic plate

Abstract: A new method for the measurement of the Poisson's Ratio function of the more rigid plastics is described. The deflection of a freely-supported circular plate under uniformly distributed load is combined with modulus data obtained in an independent experiment. The apparatus is described, the validity of the relevant elastic stress-strain formulae are discussed and the results obtained on PMMA and PVC are compared with other published values. It is concluded that this method is at least as accurate as the alternatives, all of which have particular deficiencies, and that it is possibly superior in some respects.

The Vibrations of a Circular Plate with Uniformly Distributed Load around the Outer Periphery

Abstract: The vibrations of a circular plate with uniformly distributed load around the outer periphery are studied. The circular plate has a hole at the center and the profile is of an exponential curve. The frequencies of circular plates with various boundary conditions are also obtained and the deflections, the inclinations, the bending moments and the shearing forces are shown in graphs.

Anisotropic parametric plate discrete elements

Abstract: A limitation of most plate and shell discrete elements now in use is the shape of their undeformed geometry Typically, the plan form of these elements is a straight-sided triangle or quadrilateral that linearly approximates the undeformed geometry while often using higher-order polynomials to approximate the deformed geometry This modelling difference leads to inefficiencies that can be eliminated, as demonstrated by a new parametric discrete element based entirely on bicubic Hermite polynomials This representation of element geometry corresponds to the bicubic Coon's surface patch widely used in design, which allows a common mathematical model for design and analysis Consideration is given to automating the generation of these patches Solutions are presented for several plate bending and plate stretching problems The solutions are in good agreement with closed-form solutions and photoelastic results in the case of a stress-concentration problem These data demonstrate that the new parametric discrete element maintains solution accuracy for plates with curved boundaries

Sector elements for matrix displacement analysis

Abstract: Three sector elements, two for plane stress problems (PSL and PSN) and one for plate bending problems (BSN), have been developed. Of the three elements, two are based on the natural mode technique. The elements have been applied to typical problems to test their performance.

Driving mechanisms for plate tectonics

Abstract: It is generally accepted that some form of thermal convection drives plate tectonics. The driving force can act within the plate or can be transmitted to the plate from below. Forces acting within the plate are: 1) the negative buoyancy and elevated phase boundary in the descending plate and 2) gravitational sliding of the plate off the ocean ridge. Both of these forces can be calculated with reasonable accuracy. Forces transmitted to the plate by shear stresses acting on its base are much more difficult to evaluate. The form of the convective flow in the mantle is not known and estimates of viscosity may be in error by orders of magnitude. It is only possible to speculate whether such forces contribute significantly to driving plate tectonics.

On the unity of the constant strain/constant moment finite element methods

Abstract: The paper is an attempt to unify the finite element methods of ‘first’ order in plate stretching and plate bending, i.e. those assuming in each element constant strains in stretching or constant moments in bending. Reference is made to the stretching–bending analogy between the method of Turner, Clough, Martin and Topp and the method of Elias. By another analogy property a plate stretching method which is the dual to the constant moment method of Herrmann (and Hellan) is derived. The transformation of variables which carries the last pair of dual methods identically into the first pair is finally demonstrated.

Bending of Sectorial Plates with Clamped Edges

Abstract: A solution for the deflection of a uniformly loaded sectorial plate with all edges clamped is presented. Numerical results for both 60 and 90° sectors are given, but the method of solution applies to plates of other sectorial angles. The governing equation for the small deflection theory of thin plates is satisfied. The boundary conditions on the radial edges of the plate are satisfied exactly, while those on the circumferential edge are satisfied to vanishingly small residues. A homogeneous solution is derived to satisfy the boundary conditions along the radial boundaries and then superposed on the known solution for a clamped, uniformly loaded semi-infinite plate to correct for the slope and deflection along the circumferential edge. Carrier and Shaw presented a solution for the 60° sectorial plate using an infinite series of eigenfunctions and an averaging technique on the circumferential edge. This study employs the method of collocations on the same boundary. Carrier and Shaw outlined a solution but presented no numerical results. Numerical results which can be compared with those of this study were generated by Conway and Huang, who used the method of superposition in conjunction with a series of line loads along the radii of the plate defining a particular sector angle. The line loads were adjusted to correct for the boundary conditions along the radial edges. Their results, which do not include results for the 60° sector, compare very closely to the results of this investigation for both the magnitude and location of the maximum deflection for the 90° plate.

Plate bending machines

Abstract: A plate bending machine particularly for heavy duty applications including a vertically disposed roll member rotatably disposed in a bipartite frame, one part of the frame being hinged to be moved away from the roll member. A pair of hydraulic power cylinders forming part of a movable bolster mounted in the other part of the frame, the bolster being reciprocable relative to the roll member and including plate contacting bolster abutments positioned one on each side of the roll member centre to bend a plate about the roll member. Sensing arrangements are included to ensure alignment of the bolster with the roll member. The bolster abutments are adjustable relative to the roll member centre to lessen the amount of "end flat" obtained on a bent workpiece. A hydraulic intensifier is also disclosed.

Model studies of curved box girder bridges

Abstract: THIS REPORT PRESENTS THE RESULTS OF LABORATORY TESTS ON A SERIES OF 1/29 SCALE ALUMINUM MODELS OF CURVED BOX-GIRDER BRIDGES THE TESTS WERE CONDUCTED PRIMARILY TO PROVIDE EXPERIMENTAL RESPONSE DATA TO CHECK THE ACCURACY OF PROPOSED ANALYTICAL SOLUTIONS FOR THE ELASTIC BEHAVIOR OF BRIDGES OF THIS TYPE THE PROTOTYPE DESIGN WAS A FOUR CELL BOX SECTION WITH A TOTAL WIDTH OF 33 FT 10 IN, THE RADIUS OF CURVATURE WAS 282 FT IN ALL CASES, AND THE SPAN VARIED FROM 1450 FT TO 725 FT ALL TESTS WERE CONDUCTED ON A SINGLE MACHINED ALUMINUM MODEL STARTING WITH THE LONGEST SPAN OF 60 IN, THE MODEL WAS CUT BACK TO 45 IN AND 30 IN SPANS RESPECTIVELY ALL MODELS WERE TESTED BOTH WITH AND WITHOUT A MIDSPAN RADIAL DIAPHRAGM FOR 10 DIFFERENT LOCATIONS OF A SINGLE VERTICAL POINT LOAD REACTIONS WERE MEASURED BY FOUR LOAD CELLS, DEFLECTIONS AT 9 LOCATIONS BY LOW VOLTAGE DIFFERENTIAL TRANSFORMERS, AND STRAINS BY 68 ROSETTE GAGES AND FOUR SINGLE ELEMENT GAGES LOCATED ON A RADIAL SECTION CLOSE TO MIDSPAN ALL LOAD AND STRAIN READINGS WERE RECORDED AND REDUCED BY COMPUTER, AND THIS DATA WAS CHECKED FOR SELF-CONSISTENCY BY THREE INDEPENDENT STATIC CHECKS THE DATA PRESENTED IN THIS REPORT INCLUDE EXTERNAL REACTIONS, DEFLECTIONS AT SELECTED POINTS ALONG THE WEBS, TANGENTIAL PLATE FORCES AND RADIAL PLATE BENDING MOMENTS ON A RADIAL SECTION CLOSE TO MIDSPAN ALSO DEDUCED IN EACH TEST IS THE DISTRIBUTION OF TOTAL MOMENT AT MIDSPAN BETWEEN THE INDIVIDUAL I-BEAMS THAT COMPRISE THE BOX SECTION THE REPORT CONCLUDES WITH SOME OBSERVATIONS ON THE GENERAL BEHAVIOR OF THE CURVED BOX-GIRDER BRIDGES TESTED, AND A BRIEF COMPARISON OF TEST AND THEORETICAL RESULTS /AUTHOR/