Thermal stresses in rectangular plates

Summary A method of determining the thermal stresses in a flat rectangular isotropic plate of constant thickness with arbitrary temperature distribution in the plane of the plate and with no variation in temperature through the thickness is presented. The thermal stress have been obtained in terms of Fourier series and integrals that satisfy the differential equation and the boundary conditions. Several examples have been presented to show the application of the method.Nomenclature x, y rectangular coordinates - _x, _y direct stresses - _xy shear stress - ø Airy's stress function - E Young's modulus of elasticity - coefficient of thermal expansion - T temperature - ² Laplace operator: - ⁴ biharmonic operator - 2a length of the plate - 2b width of the plate - a/b aspect ratio - a _mr, b_ms, c_nr, d_ns Fourier coefficients defined in equation (6) - _m=m/a m=1, 2, 3, ... _n=n/2a n=1, 3, 5, ... - _r=r/b r=1, 2, 3, ... _s=s/2b s=1, 3, 5, ... - A _m, B_m, C_n, D_n, E_r, F_r, G_s, H_s Fourier coefficients - K _rand L _s Fourier coefficients defined in equation (20) - direct stress at infinity - T ₁(x, y) temperature distribution symmetrical in x and y - T ₂(x, y) temperature distribution symmetrical in x and antisymmetrical in y - T ₃(x, y) temperature distribution antisymmetrical in x and symmetrical in y - T ₄(x, y) temperature distribution antisymmetrical in x and y     

Shell-Core Method for the Analysis of a Long Circular Cylindrical Sandwich Shell Subjected to Axisymmetric Loading

The analysis of a sandwich shell having a thick core and thin outer and inner layers (facings) and subjected to axisymmetric loads is considered. The problem is solved by applying the three-dimensional theory of elasticity to the core and the classical thin shell theory for the outer and inner facings. The displacement and stress continuity conditions are satisfied along the junctions of the facings and core. The results obtained from this solution have been compared with the results obtained from the sandwich shell theory of Fulton.     

Static Response of Composite Circular Cylindrical Shells Studied by Different Theories

A detailed study, on the static response of cross-ply laminated composite circular cylindrical shell of revolution and shell panel with various support conditions, has been made using Levy type of solution and the classical shell theories of FlSanders, Love and Donnell in an unified form. It has been shown that while developing a Levy type of solution using aforementioned theories, certain difficulty is encountered for determining the particular integral in respect of Fl Sanders and Love theories. This difficulty has been overcome by making use of the membrane solution as a particular integral. A comparative study has been carried out using the above shell theories for different geometrical parameters, lamination schemes and support conditions. It has been shown that Donnell theory predicts inaccurate results for certain lamination schemes, support conditions and geometrical parameters of the shell. It is suggested that, for developing shear deformation shell theories, it would be better to use a more accurate shell theory like Flügge Sommario. La risposta statica di gusci cilindrici circolari di materiale composito laminato, a strati incrociati, e di pannelli con varie condizioni di supporto viene analizzata utilizzando in una forma unificata soluzioni tipo Levy e le classiche teorie dei gusci di Flügge Sanders, Love e Donnell. Si mostra che nello sviluppare una soluzione di tipo Levy si incontra una certa difficoltà nel determinare l'integrale particolare rispetto alle teorie di Flügge, Sanders e Love. Tale difficoltà viene superata usando la soluzione di membrana come integrale particolare. Viene sviluppato uno studio comparativo facendo uso delle suddette teorie dei gusci per differenti parametri geometrici, schemi di laminazione e condizioni di vincolo. Si mostra che la teoria di Donnell fornisce risultati non accurati per certi schemi di laminazione, condizioni di supporto e parametri geometrici del guscio. Si suggerisce che per sviluppare teorie dei gusci che tengano conto delle deformazioni di scorrimento sarebbe più opportuno l'uso di una teoria dei gusci più accurata come ad esempio quella di Flügge.     

Stress analysis of a thick plate having a circular hole under axisymmetric radial load

Summary In this paper a solution for the determination of stresses and displacements in a thick plate having a cylindrical hole subjected to localised hydrostatic loading has been given. Detail numerical results have been presented and compared with the results of an infinite hole subjected to localised hydrostatic load and a semiinfinite hole subjected to localised end load. It has been shown that for certain ratio of thickness of the pate to the radius of the hole and loading, the results could be obtained by using the solution of infinite or semiinfinite hole subjected to the same hydrostatic loading.

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Übersicht Es wird eine Lösung zur Bestimmung der Spannungen und Verschiebungen für eine dicke Platte mit einem zylindrischen Loch gegeben, die durch örtlichen hydrostatischen Druck belastet ist. Numerische Ergebnisse werden mit entsprechenden Ergebnissen verglichen, die für ein unendlich sowie für ein halbunendliches Loch unter gleichen Belastungen erhalten wurden. Es wird gezeigt, daß für bestimmte Verhältnisse der Plattendicke zu den Radien von Loch und Belastung die für unendliches oder halbunendliches Loch erhaltenen Ergebnisse bei gleicher Belastung ausreichend genau sind.

     

Static analysis of a long and thick orthotropic circular cylindrical shell of revolution

Summary An elasticity solution has been obtained for a long, thick transversely isotropic circular cylindrical shell subjected to distributed pinch load using a set of three displacement functions. Numerical results are presented for different materials and thickness to mean radius ratios. The results obtained from this analysis have been compared with classical and first-order shear deformation shell theories of Flugge, Sanders, Love and Donnell.     

An Experimental-Numerical Hybrid Method for Elastic Contact Problems*

ABSTRACT

An experimental-numerical hybrid technique for determining the contact stress distribution between two elastic bodies having both frictionless as well as bonded contact is discussed in this paper. The hybrid method makes use of experimental data collected at a section far from the contact surface and the numerically generated influence coefficients, in terms of the applied unit normal and shear stresses. The experimental data, i.e., the differences in normal stresses and the shear stress, are obtained using photoelastic analysis for the examples illustrated in this paper. When substituted into equations corresponding to the unit normal and shear stress applied in the contact region, this results in a set of algebraic equations which, when solved, allow the contact stress distribution to be obtained. This method is illustrated with examples involving simple and complex geometries of the contacting bodies.     

Sound radiation from inflow turbulence in axial flow fans

The sound radiated when inflow turbulence is present in axial flow fans has been investigated. Theoretically, two noise radiating mechanisms can be identified: (i) interaction of turbulence with the rotor potential field results in a quadrupole-type volume source distribution, producing “flow-interaction” noise; (ii) impingement of turbulence on the blades results in a dipole-type (fluctuating force) surface source distribution, producing “fluctuating lift” noise. A theoretical expression for the flow interaction sound power in the upstream radiation field has been developed, in terms of parameters that can be experimentally determined by near field flow measurements involving spatial cross-correlations of the fluctuating axial velocity, with respect to both radial and circumferential position. Both these measurements and radiated sound pressure measurements have been made for eight- and ten-bladed rotors of relatively low tip Mach number (< 0·3). The sound pressure measurements revealed the occurrence of band-spreading of discrete tones at the blade passing frequency and its harmonics, as would be theoretically predicted for quadrupole-type sources here. The theoretical predictions and the measurements, respectively, of the sound power radiated upstream were compared. The results indicated that, for the fans tested, the “fluctuating lift” noise strongly predominated over the “flow-interaction” noise. The observed sound power levels were consistent with levels estimated from the theory.     

On the analysis of thick rectangular plates

Summary Thick rectangular plates are investigated using the method of initial functions proposed by Vlasov. The governing equations are derived from the three-dimensional elasticity equations using a MacLaurin series approach. As the governing equations can be obtained in the form of series, approximate theories of any desired order can be constructed easily by proper truncation. An exact solution is obtained for an allround simply supported thick plate using a Navier type solution. A Levy type solution for higher order theories is illustrated for the case of a thick plate with two opposite edges simply supported and other two edges clamped. Numerical results obtained are compared with those of classical, Reissner and Srinivas et al. solutions.

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Übersicht Mit Hilfe der Methode der Initial-Funktionen von Vlasov werden rechteckige Platten untersucht. Die zugehörigen Gleichungen werden aus den Gleichungen für das dreidimensionale Problem durch eine Entwicklung in MacLaurin-Reihen gewonnen. Durch Abbrechen dieser Reihen können Näherungen beliebiger Ordnung erhalten werden. Für den Fall einer allseitig einfach gelagerten dicken Platte wird eine exakte Lösung erhalten, bei der eine Lösung vom Navier-Typ verwendet wird. Eine Lösung vom Levy-Typ höherer Ordnung wird am Beispiel einer dicken Platte abgeleitet, von der zwei gegenüberliegende Ecken einfach gelagert, die anderen fest eingespannt sind. Die numerischen Ergebnisse werden mit den klassischen, von Reissner, Srinivas u. a. erhaltenen Resultaten verglichen.

     

Geometrically non-linear transient analysis of laminated,doubly curved shells

A dynamic, shear deformation theory of a doubly curved shell is used to develop a finite element for geometrically non-linear (in the von Karman sense) transient analysis of laminated composite shells. The element is employed to determine the transient response of spherical and cylindrical shells with various boundary conditions and loading. The effect of shear deformation and geometric non-linearity on the transient response is investigated. The numerical results presented here for transient analysis of laminated composite shells should serve as references for future investigations.