The diffusion of load from a stiffener into an infinite elastic sheet

1957 ◽  
Vol 239 (1218) ◽  
pp. 296-310 ◽  
Keyword(s):  
Mixed Boundary ◽  
Functional Differential Equations ◽  
Mixed Boundary Conditions ◽  
Limited Range ◽  
The Other ◽  
Shear Stresses ◽  
Variable Theory ◽  
Elastic Sheet ◽  
Complex Variable Theory ◽  
In Series

The use of complex variable theory to express problems in generalized plane stress is well known, but methods of finding particular solutions are available for only a limited range of problems. This paper and its sequel will develop a new technique, reducing certain problems with mixed boundary conditions to second order functional differential equations, whose solutions can be found in series form. Exact solutions are given to three fundamental problems of the diffusion of load in an infinite two-dimensional elastic sheet to which a semi-infinite elastic stiffener is continuously attached throughout its length. The first problem has a load applied to the end of the stiffener, with its line of action along the stiffener and its reactions at infinity. In the other two problems the stiffener end is unloaded but a uniform tension is applied to the sheet at infinity, in one case parallel to the stiffener, in the other perpendicular to it. Expressions for the load in the stiffener and for the direct and shear stresses in the sheet are found and plotted in non-dimensional form.

The Application of Complex Variable Theory to Reinforcing Ring Design

1983 ◽  
Vol 105 (2) ◽  
pp. 213-217
Author(s):  
M. Holland ◽  
M. A. Keavey
Keyword(s):  
Experimental Study ◽  
Complex Variable ◽  
Strain Energy ◽  
Elliptic Cylinder ◽  
The Other ◽  
Constant Depth ◽  
Variable Theory ◽  
Complex Variable Theory ◽  
Bending Strain ◽  
Energy Theory

The complex variable method is applied to the analysis of a traditional constant-depth reinforcing ring as used on a pressurized single discontinuous bend (Fig. 1). It is shown that there is a “cross-over” point where complex variable theory and bending strain energy theory take over one from the other. This condition occurs when ring depth divided by the cylinders’ mean radius is approximately equal to 0.3. This criterion is of special interest since it falls within the range of factors used in industrial designs, i.e., the Mackenzie and Beattie bend [1] has a ratio of 0.29. Limitations of the complex variable theory are investigated with a detailed theoretical and experimental study of an internally pressurized prismatic elliptic cylinder having a circular bore (Fig. 2). Three bore sizes are investigated to gain knowledge of the convergent/divergent characteristics of the theory. For the largest bore size, numerical results show a definite divergence.

Shear Buckling of Rectangular Plates with Mixed Boundary Conditions

1963 ◽  
Vol 14 (4) ◽  
pp. 349-356 ◽  
Author(s):  
I. T. Cook ◽  
K. C. Rockey
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
The Other ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Shear Buckling

SummaryThe paper presents a solution for the buckling under shear of a rectangular plate which is clamped along one edge and simply-supported along the other edges. The authors have also re-examined the case of one pair of opposite edges clamped and the other pair simply-supported.

Bending and Vibration of Plates of Variable Thickness

1976 ◽  
Vol 98 (1) ◽  
pp. 166-170 ◽  
Author(s):  
S. S. H. Chen
Keyword(s):  
Boundary Conditions ◽  
Variable Thickness ◽  
Natural Frequencies ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
The Other ◽  
Polar Coordinates ◽  
Rectangular Plates ◽  
Circular Plates ◽  
Plates Of Variable Thickness

The problem of bending and vibration of plates of variable thickness and arbitrary shapes and with mixed boundary conditions was solved by a modified energy method of the Rayleigh-Ritz type. General trial functions of deflection were obtained, one in Cartesian coordinates for rectangular plates and the other in polar coordinates for other shapes. The forced boundary conditions were satisfied approximately by introducing fixity factors which depended upon the prescribed conditions. Central deflections for circular plates subjected to static bending were within 0.2 percent of published results while they were within 1 percent for rectangular plates. The differences of natural frequencies of various rectangular plates were from 0.05 percent for simple, 2.9 percent for clamp, and up to 4.3 percent for free-free plates based on the published values.

A NONCRITICAL RAMOND–NEVEU–SCHWARZ STRING WITH ONE END FIXED

2006 ◽  
Vol 21 (01) ◽  
pp. 185-204
Author(s):  
SAVAŞ ARAPOGLU ◽  
CIHAN SAÇLIOGLU
Keyword(s):  
Boundary Conditions ◽  
Heavy Quark ◽  
Ground State ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Casimir Energy ◽  
Space Time ◽  
The Other ◽  
Virasoro Constraints ◽  
Fermion Masses

We study a Ramond–Neveu–Schwarz string with one end fixed on a D0-brane and the other end free as a qualitative guide to the spectrum of hadrons containing one very heavy quark. The mixed boundary conditions break half of the worldsheet supersymmetry and allow only odd α and even d modes in the Ramond sector, while the Neveu–Schwarz oscillators b's become odd-integer moded. Boson-fermion masses can still be matched if space–time is nine-dimensional; thus SO(8) triality still plays a role in the spectrum, although full space–time supersymmetry does not survive. We quantize the system in a temporal-like gauge where X0 ~ τ. Although the gauge choice eliminates negative-norm states at the outset, there are still even-moded Virasoro and even (odd) moded super-Virasoro constraints to be imposed in the NS(R) sectors. The Casimir energy is now positive in both sectors; there are no tachyons. States for α′ M2 ≤ 13/4 are explicitly constructed and found to be organized into SO(8) irreps by (super)constraints, which include a novel "[Formula: see text]" operator in the Neveu–Schwarz and Γ0 ± I in the Ramond sectors. GSO projections are not allowed. The preconstraint states above the ground state have matching multiplicities, indicating space–time supersymmetry is broken by the (super)constraints. A distinctive physical signature of the system is a slope twice that of the open RNS string. When both ends are fixed, all leading and subleading trajectories are eliminated, resulting in a spectrum qualitatively similar to the J/ψ and ϒ particles.

On the Determination of Stresses, Displacements, and Stress-Intensity Factors in Edge-Cracked Sheets With Mixed Boundary Conditions

1979 ◽  
Vol 46 (3) ◽  
pp. 611-617 ◽  
Author(s):  
P. J. Torvik
Keyword(s):  
Stress Intensity ◽  
Boundary Conditions ◽  
Edge Crack ◽  
Variational Principles ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Single Edge ◽  
Aspect Ratios ◽  
In Series

The stresses and displacements in a finite, isotropic, elastic sheet containing a single edge crack are considered. Solutions are constructed through superposition of the solutions due to Williams. Variational principles are used to obtain solutions for mixed boundary conditions on the edge of a sheet containing a single, straight, unloaded edge crack. Trial solutions are constructed by expanding stresses and displacements in series of Williams’ solutions and a procedure for determining the coefficients for the best finite expansion is developed through the use of Reissner’s principle. The method is applicable to any combination of prescribed displacements and prescribed tractions on the boundary. The method is used to determine the stress-intensity factor for a rectangular edge-cracked sheet with two edges free of traction and a prescribed extension of the ends. Values of KI and compliance are given for various aspect ratios and crack lengths.

Integral Representations for the Solution of Dynamic Bending of a Plate with Displacement-Traction Boundary Data

2003 ◽  
Vol 10 (3) ◽  
pp. 467-480
Author(s):  
Igor Chudinovich ◽  
Christian Constanda
Keyword(s):  
Existence And Uniqueness ◽  
Boundary Data ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Boundary Integral ◽  
Integral Representations ◽  
Transverse Shear Deformation ◽  
Uniqueness Of Solutions ◽  
Nonstationary Boundary

Abstract The existence of distributional solutions is investigated for the time-dependent bending of a plate with transverse shear deformation under mixed boundary conditions. The problem is then reduced to nonstationary boundary integral equations and the existence and uniqueness of solutions to the latter are studied in appropriate Sobolev spaces.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

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