Showing papers on "Plate theory published in 1970"

Exact solutions for rectangular bidirectional composites and sandwich plates

Abstract: In a continuing study, three-dimensional elasticity solutions are constructed for rectangular laminates with pinned edges. The lamination geometry treated consists of arbitrary numbers of layers which can be isotropic or orthotropic with material symmetry axes parallel to the plate axes. Several specific example problems are solved, including a sandwich plate, and compared to the analogous results in classical laminated plate theory.

Shear Deformation in Heterogeneous Anisotropic Plates

Abstract: : A bending theory for anisotropic laminated plates developed by Yang, Norris,and Stavsky is investigated. The theory includes shear deformation and rotary inertia in the same manner as Mindlin's theory for isotropic homogeneous plates. The governing equations reveal that unsymmetrically laminated plates display the same bending-extensional coupling phenomenon found in classical laminated plate theory based on the Kirchhoff assumptions. Solutions are presented for bending under transverse load and for flexural vibration frequencies of symmetrical and nonsymmetrical laminates. Good agreement is observed in numerical results for plate bending as compared to exact solutions obtained from classical elasticity theory. For certain fiber reinforced composite materials, radical departure from classical laminated plate theory is indicated. (Author-PL)

Influence of Shear Coupling in Cylindrical. Bending of Anisotropic Laminates

Abstract: Investigation of the success of classical lamination theory in predicting the response of composite laminates under static bending is extended by consideration of the influence of shear coupling. Specifically, we treat the exact solution of the problem of a pinned-end laminate composed of N layers, each of which possesses only a single plane of elastic symmetry, under cylindrical bending. Several example problems, involving unidirectional and angle-ply composites, are solved and the detailed results compared to corresponding approximate solutions. Some observations are offered in regard to the general range of validity of classical laminated plate theory.

Comparison of theory and experiment for nonlinear flutter of loaded plates

Abstract: Flutter analysis of plates with inplane boundary support flexibility exposed to transverse pressure loading or buckled by uniform thermal expansion

Vibrations of short beams

Abstract: The paper presents a new formulation for transverse oscillations of uniform beams. The governing equations are two SUnultaneous partial integro-dilferential equations. From these equations, simpler governing equations to various orders of approximation are deduced. Well-known beam equations correspond to some special cases in the present formulation.Introduction of refined shear coefficient in the Timosbenko's theory seems to increase the discrepancy between theory and experiment, whereas the present formulation reduces this discrepancy. Second-order approximation equations are believed to be adequate for most engineering applications; for more accurate determination of the natural frequency higher-order approximations can be used.

Finite Element Bending Analysis of Reissner Plates

Abstract: A finite element plate bending analysis which includes the effects of transverse shear is described. A complete displacement formulation based on the governing equations of the Reissner theory is developed for application to the bending of rectangular plates. The finite element employed is a rectangle with 20 degrees of freedom which include both bending and shear deformation states and their interaction. Of primary importance is the ability of the finite element method to overcome certain of the anomalies found in classical plate theory. The presence of “Kirchoff shear forces” and concentrated corner reactions can be eliminated by the specification of more realistic boundary conditions. Results of several numerical examples are in excellent agreement with those of the Reissner theory for maximum displacements and for the distribution of shear stress resultants along plate edges. Also, the added transverse shear degrees of freedom allow shear deformations along an edge parallel to the edge. Thus, the vanishing edge twisting moment condition can be satisfied approximately. The formulation described offers a convenient method for approximating the Reissner theory with a discrete element approach.

Stress Distribution in a Bimaterial Plate under Uniform External Loadings

Abstract: Elastic fields in bimaterial plate under uniform compressive and anti-plane shear loadings, finding stress distribution and induced interfacial shear stresses

Exact Natural Frequencies for Cross-Ply Laminates

Abstract: Vibration of a simply supported cross-ply laminated plate infinite in length is analyzed. Existence of plane strain conditions allow an exact solution of the field equations. Curves of variation of exact natural frequencies versus wavelength for several cases of a two-ply laminated plate are calculated. The first two mode shapes are plotted for various wavelengths and compared to classical plate theory. Flex ural vibration appears to be well approximated by classical theory up to moderate wavelengths. Extensional vibration is approximated mar ginally only for very long wavelengths.

Distribution of Wheel Loads on Highway Bridges

Abstract: PROPOSED REVISIONS ARE PRESENTED TO THOSE SECTIONS OF THE AASHO SPECIFICATIONS FOR HIGHWAY BRIDGES HAVING TO DO WITH THE DISTRIBUTION OF WHEEL LOADS ON BRIDGE DECKS. THE STATIC DISTRIBUTION OF MOVABLE WHEEL LOADS WAS STUDIED IN A BROAD RANGE OF BRIDGE TYPES. HOWEVER, THE STUDY WAS LIMITED TO SHORT-AND MEDIUM-SPAN BRIDGES; BRIDGES WITH SPANS UP TO ABOUT 120 FEET. THE BRIDGE TYPES IN THIS SPAN RANGE CAN BE CLASSIFIED: BEAM AND SLAB, MULTI-BEAM, AND CAST-IN-PLACE CONCRETE BOX GIRDER. THE BEHAVIOR OF THESE BRIDGES CAN BE CHARACTERIZED BY THE FOLLOWING MAJOR VARIBLES: ASPECT RATIO (BRIDGE WIDTH/BRIDGE SPAN), RELATIVE STIFFNESS OF BEAMS AND FLOOR, RELATIVE DIAPHRAGM STIFFNESS, AND EXTENT OF BRIDGE CONTINUITY. FROM AN EXTENSIVE LITERATURE SEARCH AND A STUDY OF AVAILABLE METHODS OF ANALYSIS, IT WAS FOUND THAT THE DISTRIBUTION OF REAL LOADS IN THSES BRIDGE TYPES COULD BE DETERMINED USING THE FOLLOWING THEORIES: (1) ORTHOTROPIC PLATE THEORY, (2) ARTICULATED PLATE THEORY, AND (3) PRISMATIC FOLDED-PLATE THEORY. THE MAJOR VARIABLES WHICH AFFECT THE LOAD DISTRIBUTION IN EACH OF THE MAJOR BRIDGE TYPES ARE: RELATIVE FLEXURAL STIFFNESS IN LONGITUDINAL AND TRANSVERSE DIRECTIONS, RELATIVE TORSIONAL STIFFNESS IN THE SAME DIRECTIONS, BRIDGE WIDTH, AND EFFECTIVE BRIDGE SPAN. IT IS FELT THAT WITH THE NEW DESIGN CRITERIA PRESENTED, PREDICTION OF WHEEL LOAD DISTRIBUTION WILL BE MORE ACCURATE, AND WILL MORE TRULY INDICATE THE BEHAVIOR OF THE BRIDGE TYPES STUDIED.

Theoretical elastic deflections of a thick horizontal circular mirror on a ring support.

Abstract: This paper shows how exact, closed-form expressions can be determined for the deflections under its own weight of a thick, horizontally oriented, circular mirror on a ring support. A theory developed by Reissner for thick plates that includes shear deformations is used, and the results are reduced to those of classical plate theory. An optimum support radius is obtained, for which the center deflection is equal to the deflection of the outer edge. For mirrors having thickness-to-diameter ratios greater than approximately one-tenth, shearing deformations can contribute significantly to the total deflection and hence should not be neglected.

Calculations of plate frequencies from complementary energy formulation

Abstract: The complementary variational principle has been used to derive the differential equations and the associated boundary conditions of the vibrating plate in terms of bending moments. It is shown that in this formulation, the plate possesses an infinite number of zero frequency modes in which the plate remains in a state of constant strain under a set of self-equilibrating bending moments. In applying the Rayleigh Ritz procedure for the non-zero frequency modes of the plate, it is shown that it the assumed functions are orthogonal to only a finite number of zero frequency modes, then one may obtain frequencies which are lower than the true frequencies of the plate. An iliustrative example is given in the paper.

Design of steel bearing plates

Abstract: BASE PLATES FOR COLUMNS AND BEARING PLATES FOR BEAMS RESTING ON MASONRY ARE DETAILS ASSOCIATED WITH THE DESIGN OF ALL STEEL STRUCTURES. GIVEN THE LOAD AND ALLOWABLE BEARING PRESSURE, THE AREA OF THE PLATE IS EASILY COMPUTED. FROM THIS, THE OVERHANGING CANTILEVER SPAN CAN BE DETERMINED. IF THE YEILD POINT OF THE STEEL IS KNOWN, THE THICKNESS OF THE PLATE REQUIRED CAN BE DETERMINED BY PROCEDURES OUTLINED IN THE AMERICAN INSTITUTE OF STEEL CONSTRUCTION (AISC) MANUAL. HOWEVER, DESIGN PROCEDURES PRESENTED IN THE AISC MANUAL ARE SILENT ON TWO QUESTIONS: NO LIMIT IS PLACED ON PLATE DEFLECTIONS, AND COLUMN BASE PLATES NEARLY THE SAME SIZE AS THE COLUMN CAN NOT BE PROPERLY DESIGNED FOR THICKNESS. A MATHEMATICAL ANALYSIS IS PRESENTED OF PLATE DEFLECTIONS AND MINIMUM COLUMN BASE PLATE THICKNESS. AN EQUATION IS DERIVED FOR THE MAXIMUM DEFLECTION AT THE MIDDLE OF THE FREE EDGE OF A PLATE WHICH IS FIXED ON THE OPPOSITE EDGE AND SUPPORTED ON THE OTHER TWO EDGES. AN ASSUMPTION MADE IN THE DERIVATION OF THIS EQUATION IS THAT THE PLATE IS NOT STRESSED BEYOND THE ELASTIC LIMIT. THE MINIMUM THICKNESSES COMPUTED ARE BASED ON ASSUMED UNIFORM BEARING PRESSURE.

Free vibrations of plates and beams of pyrolytic graphite type materials

Abstract: A set of general governing equations is derived for the study of free vibrations of rectangular plates composed of a transversely isotropic material, including the effects of transverse shear deformation and rotatory inertia. Such a formulation is necessary for even geometrically thin plates, when the ratio of in-plane modulus of elasticity to transverse shear modulus is large (i.e., 20-50) which occurs in vapor deposited materials such as pyrolytic graphite, and in many fiber reinforced composite materials. The corresponding equations for beams are also derived. The case of a simply supported plate is treated in detail. Numerical results show that significant differences occur in predicting natural frequencies for various modes when the present theory is used compared to the use of classical methods for plates of these material systems. Classical methods can predict frequencies that are erroneous by a factor of nearly three.

Effects Of Lamination And Spacing On Finite Thickness Plate Perforation

Abstract: In determining ballistic limits and residual projectile characteristics for very thick targets (situations where the ratio of target thickness T to projectile diameter D exceeds 10), resort must frequently be made to constructing the target from a number of layers whose thickness is less than that of the monoblock target. This holds true for determining penetration depths in semiinfinite plates as well. This paper presents results of numerical simulations comparing projectile residual characteristics, primarily residual mass, for monoblock and equivalent thickness layered targets for a number of situations involving T/D 10. It is found that for thick plates, results obtained from layered target perforation compare favorably with those from monoblock targets provided that the layering is not excessive and care is taken to insure that the individual layers have the same material properties as the monoblock target. For thin targets, the correlation ranges from poor to non-existent. Introduction Impact and impulsive loading onto layered media targets consisting of different materials is a problem of long standing. It occurs naturally when dealing with impact effects into geological media, where different strata have different material properties. It can occur in the design of protective structures where materials of different density, strength and cross-sectional area are employed to reduce the intensity of the impact stress. This aspect of the impact problem is well understood and is covered in modern textbooks and reference books dealing with transient phenomena. Another aspect of layering involves the impact of projectiles onto targets consisting of multiple layers of plates of the same density] In impact testing, this often occurs when very thick targets need to be constructed yet the target material in question is not manufactured in the required thickness. Take, for example, the requirement to construct an effective "semi-infinite" target, one where the rear of the plate does not influence the penetration process. It is Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 104 Structures Under Shock And Impact desired to build such a target of rolled, homogeneous armor (RHA) steel. The maximum thickness of RHA commercially available is 20.32 cm. At this thickness, uniformity of material properties is a problem, as is cost. Hence, targets for deep penetration studies are often constructed by using a number of plates of smaller thickness, stacking them until the desired thickness is reached. This target stack is then contained in some fashion (e.g. trapped, welded at the periphery) and the test conducted. In the course of testing the restraints are broken and the front and rear plates are observed to move considerable distances, even for tests involving target plates weighing several terns. Each layer acts as a momentum trap and the outermost layers dissipate the residual energy through rigid body motion. Several questions must now be answered before the test results may be accepted as valid: (a) does the penetration event occur on the same time scale as the rigid body motion of the target plates? In other words, is a layered target an effective simulant of a monoblock target? (b) what is the effect of layering as the number of plates required to simulate the monoblock thickness increases? (c) if target plates do separate before completion of the penetration/perforation process, what is the effect on the penetration depth (or, if a perforation, on the projectile residual mass and velocity) These questions must be answered for three classes of targets: (a) thin targets (T/D 10) Thin and Intermediate Thickness Targets For thin and intermediate thickness targets the answers may be readily inferred from the existing literature. In their study of containment structures, Zaid, ElKalay and Travis (1973) point out that for very thin plates (thicknesses < 2 mm), lamination greatly reduces the resistance of the target plate to ballistic impact. Netherwood (1979), conducting in situ pressure measurements of impacted plates found the laminated target to be much weaker than a solid one of the same thickness so that the mechanism of penetration of a laminated target was different than that for a solid target. Nixdorff (1984) examined analytically the effect on lamination on the ballistic limit for up to five layers and found considerable differences as the number of layers increased. Similar conclusions were reached by Segletes and Zukas (1989) in a numerical analysis Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 Structures Under Shock And Impact 105 of laminated plates. Other studies could be cited but these suffice to show that for thin targets, lamination can alter the response mechanism under impact loading and fail to correlate with the behavior of a solid target, especially if the number of layers is large. The problem for intermediate thickness targets can be seen from the results of the following calculations. The ZeuS code [Segletes and Zukas (1987), Janzon et al (1992), Zukas (1993)], a two-dimensional explicit finite element code for fast, transient analysis on personal computers, was used to calculate the impact of a 64.5 gram S-7 tool steel projectile with length-todiameter (L/D) ratio of 5 into a single RHA plate with a thickness of 3.18 cm. The projectile had a diameter of 1.3 cm and a striking velocity of 1164 m/s. Experimental data was taken from the report by Lambert (1978). The experimentally determined values of projectile residual mass and residual velocity were 22.9 grams and 223 m/s, respectively. ZeuS calculations indicated a residual mass of 25.5 grams and a residual velocity of 233 m/s. These were deemed acceptably close. Next, a series of calculations was undertaken where the solid target above was assumed to consist of 2, 4 and 6 layers, each with properties identical to those of the solid target. Penetration of the four-layer target at various times is shown in Figure 1. The variation of projectile normalized residual mass ( mjm^ and normalized residual velocity (V,IV^ V striking velocity) can be seen in Figures 2 and 3. With the 4-layer laminated target, the difference between Lambert's data for the solid target and the computed residual masses is 43% while for the residual velocity it is 143%. The differences continue to increase with increasing lamination. Even though the plates making up the laminated target have the same density and material properties as the solid target, the differences noted above could be anticipated. The plates in the laminated target are not restrained and are allowed to slip freely over each other. As they separate after the passage of the projectile, a free surface is created. The inability of a free interface to support rarefaction waves changes the stress wave propagation characteristics of multiplate penetration events at early times. As these stress differences are integrated in time, the difference between the simulations becomes more visible, with the multiplate case demonstrating more bending than the equivalent solid plate case (Figure 4). This can also be inferred from plate theory which gives for the bending stiffness of the plate E7 /12(1 -v*), where E is the elastic modulus, T the plate thickness and v Poisson's ratio. Since bending stiffness follows plate thickness to the third power, simply cutting a monoblock plate in half reduces its bending stiffness by a factor of 8. Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509

Theoretical elastic deformations of a thick horizontal circular mirror on a double-ring support

Abstract: This paper gives exact, closed-form expressions for the deflection, under its own weight, moments, and shears, in a thick, horizontally oriented, circular mirror on a double-ring support. A theory developed by Reissner for thick plates that includes shear deformations is used, and the results are reduced to those of classical plate theory. It is found that for mirrors having thickness-to-diameter ratios greater than approximately one-tenth, shearing deformations can contribute significantly to the total deflection, and hence should not be neglected. Numerical results are presented and interpreted in detail.

Transverse shear deformation in multicell box beam bridges

Abstract: A THEORETICAL ANALYSIS IS PRESENTED OF MULTICELL, RECTANGULAR BOX BEAM BRIDGES WITHOUT INTERMEDIATE DIAPHRAGMS, WHICH CONSIDERS THE EFFECT OF VIERENDEEL DISTORTION OF THE DECK. THE SOLUTION IS EXPRESSED IN FOURIER HALF-RANGE SINE SERIES AND, AS SUCH, IT IS LIMITED TO SIMPLY SUPPORTED BRIDGE DECKS. THE EDGES MAY BE FREE OR ELASTICALLY RESTRAINED BY EDGE BEAMS OF KNOWN ELASTIC RIGIDITIES IN FLEXURE AND TORSION. THE EQUATIONS FOR DEFLECTION, MOMENTS, AND SHEARS ARE DERIVED AND AN ILLUSTRATIVE EXAMPLE IS INCLUDED TO SHOW THE EFFECT OF VIERENDEEL DISTORTION ON THE LOAD DISTRIBUTION CHARACTERISTICS OF THE DECK. COMPARISON IS MADE WITH CONVENTIONAL ORTHOTROPIC PLATE THEORY. /AUTHOR/

Dynamic stability of circular plates under stochastic excitations

Abstract: This paper deals with a Lyapunov-type analysis of a linear structural dynamic system \vhich is excited by a stochastic parametric load. The technique of solution is developed on the basis of the linearity of the elastic system but with modification in developing the Lyapunov function, the technique can be used to consider nonlinear systems. Stochastic convergence and a theory for mean square global stability are discussed. A radially loaded circular plate is considered, whose dynamic motion can be described by a Hill-type equation, with one parameter being a random variable. A Lyapunov function, satisfying the stochastic stability theorem, is developed and sufficient conditions are established to insure mean square global stability. These conditions are applicable for any general continuous random process, and are expressed in terms of the excitation, physical parameters of the system and a measure of the stochastic dependency of the output on the input.