The Effect of a Rigid Circular Inclusion on the Bending of a Thick Elastic Plate

1952 ◽  
Vol 19 (1) ◽  
pp. 28-32
Author(s):  
R. A. Hirsch
Keyword(s):  
Plane Strain ◽  
Elastic Plate ◽  
Plate Theory ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Inclusion Size ◽  
Circular Inclusion ◽  
Stress Concentrations ◽  
Classical Plate Theory ◽  
Rigid Circular Inclusion

Abstract The three-dimensional problem of the effect of a rigid circular inclusion on the bending of a thick elastic plate is solved approximately by the method of E. Reissner (1, 2). Comparison is made for the limiting cases of vanishing inclusion size, (plane strain), and vanishing thickness (Poisson-Kirchoff plate theory), with the work of J. N. Goodier (3) and M. Goland (4). Graphs showing the transition from the plane-strain solution to the Poisson-Kirchoff solution are given. Stress concentrations are calculated and plotted versus the inclusion diameter-plate thickness ratio. The stress concentrations are found to be less than predicted by the classical plate theory when the inclusion diameter approaches the same order of magnitude as the plate thickness.

The Effect of a Rigid Elliptic Inclusion on the Bending of a Thick Elastic Plate

1961 ◽  
Vol 28 (3) ◽  
pp. 379-382
Author(s):  
Fu Chow
Keyword(s):  
Stress Concentration ◽  
Elastic Plate ◽  
Plate Theory ◽  
Circular Inclusion ◽  
Elliptic Inclusion ◽  
Focal Distance ◽  
Classical Plate Theory ◽  
Concentration Factors ◽  
Stress Concentration Factors ◽  
Rigid Circular Inclusion

The effect of a rigid elliptic inclusion on both plain bending and pure twist of a thick elastic plate is investigated on the basis of Reissner’s plate theory [1, 2]. Comparison is made for the limiting cases of vanishing focal distance of the elliptic inclusion (a rigid circular inclusion), and vanishing thickness (Poisson-Kirchhoff plate theory), with the solutions of C. Pai [3], R. A. Hirsch [4], and M. Goland [5]. The stress-concentration factors are lower than those predicted by the classical plate theory.

Refined hyperbolic shear deformation plate theory

2014 ◽  
Vol 229 (15) ◽  
pp. 2675-2686 ◽  
Author(s):  
K Nareen ◽  
RP Shimpi
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Governing Equations ◽  
Dimensional Theory ◽  
Classical Plate Theory ◽  
Shear Deformation Plate Theory ◽  
Elasticity Solutions

The paper presents a novel shear deformation plate theory involving only two variables. Taking a cue from exact three-dimensional theory of elasticity solutions for a plate, hyperbolic functions are used for describing displacement variation across plate thickness. The theory involves only two governing equations, which are uncoupled for statics and are only inertially coupled for dynamics. The shear stress free surface conditions are satisfied. No shear correction factor is required. The theory is variationally consistent, has a strong similarity with classical plate theory, and is simple, yet accurate. Illustrative examples for free vibration and for static flexure demonstrate the effectiveness of the theory.

Generalized Bending of a Large, Shear Deformable Isotropic Plate Containing a Circular Hole or Rigid Inclusion

2000 ◽  
Vol 68 (2) ◽  
pp. 230-233 ◽  
Author(s):  
C. W. Bert ◽  
H. Zeng
Keyword(s):  
Circular Hole ◽  
Isotropic Plate ◽  
Plate Theory ◽  
Three Dimensional ◽  
Bending Moment ◽  
Circular Inclusion ◽  
Experimental Result ◽  
Classical Plate Theory ◽  
Geometric Ratio ◽  
Three Dimensional Elasticity

The problem of a large isotropic plate with a circular hole or rigid circular inclusion is considered here. The plate experiences transverse shear deformation and is subjected to an arbitrary bending field. By using Reissner’s plate theory, a general solution, in terms of Poisson’s ratio ν, a geometric ratio, and bending moment ratio B, is obtained to satisfy both the boundary conditions along the edge and at great distances from the edge. The stress couple concentration factors are calculated and compared with classical plate theory, three-dimensional elasticity theory, higher-order plate theory, and an experimental result.

Response of a Plate Supported by a Fluid Half Space to a Moving Pressure

1970 ◽  
Vol 37 (4) ◽  
pp. 1050-1054 ◽  
Author(s):  
D. H. Y. Yen ◽  
C. C. Chou
Keyword(s):  
Half Space ◽  
Elastic Plate ◽  
Plate Theory ◽  
Fluid Pressure ◽  
Integral Transforms ◽  
Classical Plate Theory ◽  
Plate Deflection

The response of an elastic plate supported by a fluid half space to a steadily moving pressure is studied. The Timoshenko plate theory is used in the study. By the method of integral transforms, solutions for both the plate deflection and the interaction fluid pressure are obtained. The results are then compared in detail with those obtained previously using the classical plate theory.

Three-Dimensional Elastic Stress Fields of Finite Thickness Plates with Elliptical Hole

2007 ◽  
Vol 353-358 ◽  
pp. 74-77
Author(s):  
Zheng Yang ◽  
Chong Du Cho ◽  
Ting Ya Su ◽  
Chang Boo Kim ◽  
Hyeon Gyu Beom
Keyword(s):  
Stress Concentration ◽  
Plane Strain ◽  
Plane Stress ◽  
Stress Concentration Factor ◽  
Concentration Factor ◽  
Maximum Stress ◽  
Elastic Stress ◽  
Finite Thickness ◽  
Plate Thickness ◽  
Three Dimensional

Based on detailed three-dimensional finite element analyses, elastic stress and strain field of ellipse major axis end in plates with different thickness and ellipse configurations subjected to uniaxial tension have been investigated. The plate thickness and ellipse configuration have obvious effects on the stress concentration factor, which is higher in finite thickness plates than in plane stress and plane strain cases. The out-of-plane stress constraint factor tends the maximum on the mid-plane and approaches to zero on the free plane. Stress concentration factors distribute ununiformly through the plate thickness, the value and location of maximum stress concentration factor depend on the plate thickness and the ellipse configurations. Both stress concentration factor in the middle plane and the maximum stress concentration factor are greater than that under plane stress or plane strain states, so it is unsafe to suppose a tensioned plate with finite thickness as one undergone plane stress or plane strain. For the sharper notch, the influence of three-dimensional stress state on the SCF must be considered.

scholarly journals Breakage of Ice Sheets by Underwater Explosions

1977 ◽  
Vol 19 (81) ◽  
pp. 661-663
Author(s):  
E. Vittoratos ◽  
M. E. Charles
Keyword(s):  
Elastic Plate ◽  
High Speed ◽  
Plate Theory ◽  
Ice Sheets ◽  
Theoretical Development ◽  
Small Scale ◽  
Theory Approach ◽  
Classical Plate Theory ◽  
Theoretical Understanding ◽  
Deformation And Failure

AbstractWe have attempted to develop a theoretical understanding of the field experience on the destruction of floating ice sheets by explosives, by performing small-scale laboratory explosions and developing a theory based on the elastic plate. Glass spheres 4.5 × 10¯2 m in diameter and c. 4 × 10¯4 m thick were immersed in water underneath a floating ice sheet c. 2 × 10¯2 m in thickness. The air pressure was raised till the sphere burst at pressures around 15 atmospheres. A high-speed camera (up to several thousand frames per second) recorded the details of the explosion: the growth of the gas bubble and the corresponding deformation and failure of the ice. We observed radial and circumferential cracks develop within 2 ms of the bursting of the sphere.As a first step in the theoretical development, we have considered the response of an infinite elastic plate to impulsive pressure loading due to an underwater explosion. We have assumed potential, incompressible flow which is a valid approximation for the case of the above experiments and the analogous compressed-gas blasting in the field (Mellor and Kovacs, 1972). However the effects caused by the intense shock wave that is radiated by the detonation of a high explosive are thus not considered. We relate the maxima in the tensile stress with the crack pattern and the eventual damage, and have achieved qualitative agreement with the laboratory observations. The model does reproduce and clarify some aspects of the field data, in particular the role of the thickness (Mellor, unpublished); but it fails to relate the crater diameter to the weight of the explosive. It appears that at optimum blasting conditions with high explosives an incompressible-fluid, classical-plate-theory approach is inadequate.

scholarly journals Breakage of Ice Sheets by Underwater Explosions

1977 ◽  
Vol 19 (81) ◽  
pp. 661-663
Author(s):  
E. Vittoratos ◽  
M. E. Charles
Keyword(s):  
Elastic Plate ◽  
High Speed ◽  
Plate Theory ◽  
Ice Sheets ◽  
Theoretical Development ◽  
Small Scale ◽  
Theory Approach ◽  
Classical Plate Theory ◽  
Theoretical Understanding ◽  
Deformation And Failure

Abstract We have attempted to develop a theoretical understanding of the field experience on the destruction of floating ice sheets by explosives, by performing small-scale laboratory explosions and developing a theory based on the elastic plate. Glass spheres 4.5 × 10¯2 m in diameter and c. 4 × 10¯4 m thick were immersed in water underneath a floating ice sheet c. 2 × 10¯2 m in thickness. The air pressure was raised till the sphere burst at pressures around 15 atmospheres. A high-speed camera (up to several thousand frames per second) recorded the details of the explosion: the growth of the gas bubble and the corresponding deformation and failure of the ice. We observed radial and circumferential cracks develop within 2 ms of the bursting of the sphere. As a first step in the theoretical development, we have considered the response of an infinite elastic plate to impulsive pressure loading due to an underwater explosion. We have assumed potential, incompressible flow which is a valid approximation for the case of the above experiments and the analogous compressed-gas blasting in the field (Mellor and Kovacs, 1972). However the effects caused by the intense shock wave that is radiated by the detonation of a high explosive are thus not considered. We relate the maxima in the tensile stress with the crack pattern and the eventual damage, and have achieved qualitative agreement with the laboratory observations. The model does reproduce and clarify some aspects of the field data, in particular the role of the thickness (Mellor, unpublished); but it fails to relate the crater diameter to the weight of the explosive. It appears that at optimum blasting conditions with high explosives an incompressible-fluid, classical-plate-theory approach is inadequate.

Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

2021 ◽  
pp. 095440622110197
Author(s):  
Bhupesh K Chandrakar ◽  
NK Jain ◽  
Ankur Gupta
Keyword(s):  
Governing Equation ◽  
Orthotropic Plate ◽  
Multiple Scales ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Plate Thickness ◽  
Solution Technique ◽  
Classical Plate Theory ◽  
Non Linear ◽  
The Impact

The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.

scholarly journals Buckling and Vibration of Non-Homogeneous Rectangular Plates Subjected to Linearly Varying In-Plane Force

2013 ◽  
Vol 20 (5) ◽  
pp. 879-894 ◽  
Author(s):  
Roshan Lal ◽  
Renu Saini
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Material ◽  
Classical Plate Theory ◽  
Simply Supported

The present work analyses the buckling and vibration behaviour of non-homogeneous rectangular plates of uniform thickness on the basis of classical plate theory when the two opposite edges are simply supported and are subjected to linearly varying in-plane force. For non-homogeneity of the plate material it is assumed that young's modulus and density of the plate material vary exponentially along axial direction. The governing partial differential equation of motion of such plates has been reduced to an ordinary differential equation using the sine function for mode shapes between the simply supported edges. This resulting equation has been solved numerically employing differential quadrature method for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. The effect of various parameters has been studied on the natural frequencies for the first three modes of vibration. Critical buckling loads have been computed. Three dimensional mode shapes have been presented. Comparison has been made with the known results.

EXACT SOLUTION FOR BENDING OF AN ELASTIC PLATE CONTAINING A CRACK AND SUBJECTED TO A CONCENTRATED MOMENT

2007 ◽  
Vol 04 (02) ◽  
pp. 265-281
Author(s):  
LALITHA CHATTOPADHYAY ◽  
S. SRIDHARA MURTHY ◽  
S. VISWANATH
Keyword(s):  
Stress Intensity Factor ◽  
Elastic Plate ◽  
Integral Transform ◽  
Plate Theory ◽  
Infinite Plate ◽  
Bending Moment ◽  
Single Line ◽  
Bending Stress ◽  
Single Crack ◽  
Classical Plate Theory

The problem of estimating the bending stress distribution in the vicinity of cracks located on a single line in an elastic plate subjected to concentrated moment is examined. Using classical plate theory and integral transform techniques, the general formulae for the bending moment and twisting moment in an elastic plate containing cracks located on a single line are derived. The solution is obtained in detail for the case in which there is a single crack in an infinite plate, and the bending stress intensity factor is determined in a closed form. Two examples are considered to illustrate the present approach.

Sign in / Sign up

Export Citation Format

Share Document