Equilibrium Displacement and Stress Distribution in a Two-Dimensional, Axially Moving Web Under Transverse Loading

1995 ◽  
Vol 62 (3) ◽  
pp. 772-779 ◽  
Author(s):  
C. C. Lin ◽  
C. D. Mote
Keyword(s):  
Stress Distribution ◽  
Plate Theory ◽  
Transverse Load ◽  
Transverse Displacement ◽  
Transverse Loading ◽  
Closed Form Solutions ◽  
Coupled Equations ◽  
Axially Moving ◽  
Axially Moving Web ◽  
The Web

Von Karman nonlinear plate equations are modified to describe the motion of a wide, axially moving web with small flexural stiffness under transverse loading. The model can represent a paper web or plastic sheet under some conditions. Closed-form solutions to two nonlinear, coupled equations governing the transverse displacement and stress function probably do not exist. The transverse forces arising from the bending stiffness are much smaller than those arising from the applied axial tension except near the edges of the web. This opens the possibility that boundary layer and singular perturbation theories can be used to model the bending forces near the edges of the web when determining the equilibrium solution and stress distribution. The present analysis is applied to two examples: (I) a web deflecting under its own uniformly distributed weight; (II) a web deflecting under a transverse load whose distribution is described by the product of sine functions in the axial and width directions. Membrane theory and linear plate theory solutions are used to characterize the importance of the web deformation solutions.

The Elastic Field of an Elliptic Cylindrical Inclusion in a Laminate With Multiple Isotropic Layers

1999 ◽  
Vol 66 (1) ◽  
pp. 165-171 ◽  
Author(s):  
H. G. Beom ◽  
Y. Y. Earmme
Keyword(s):  
Plate Theory ◽  
Elastic Field ◽  
Cylindrical Inclusion ◽  
Transverse Displacement ◽  
Laminated Plate ◽  
Displacement Fields ◽  
Elastic Fields ◽  
Closed Form Solutions ◽  
Main Plane ◽  
Interior Points

An elliptic cylindrical inclusion with an eigenstrain in an infinite laminate composed of multiple isotropic layers is analyzed. The problem is formulated by using the classical laminated plate theory in which displacement fields in the laminated plate are expressed in terms of in-plane displacements on the main plane and transverse displacement. Employing a method based on influence functions, an integral type solution to the equilibrium equation is expressed in terms of the eigenstrain. Closed-Form solutions for the elastic fields are obtained by evaluating the integrals explicitly for interior points and exterior points of the ellipse. The elastic fields caused by an elliptic cylindrical inhomogeneity with an eigenstrain in the infinite laminate are determined by the equivalent eigenstrain method. Solutions for a finite laminate with an eigenstrain in a circular cylindrical inhomogeneity are also obtained in terms of material and geometric parameters for each layer composing the laminate.

On the Frequency Response of an Axisymmetric Non-Flat Spinning Disk

2010 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Frequency Response ◽  
Stress Function ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Governing Equations ◽  
Coupled Equations ◽  
Stationary Disk ◽  
Spinning Disk

For rotating disks, the effect of axisymmetric runout is of interest. This study examines the frequency characteristics of thin rotating discs subjected to axisymmetric non-flatness. The equations of motion used are based on Von Karman’s plate theory. First, the eigenfunctions of the stationary disk problem corresponding to the stress function and transverse displacement are found. These eigenfunctions produce an equation that can be used in the Gelrkin’s method. The initial nonflatness is assumed to be a linear combination of the eigenfunctions of the transverse displacement of the stationary disk problem. Since the initial non-flatness is assumed to be axisymmetric, only eigenfunctions with no nodal diameters are considered to approximate the initial runout. It is supposed that the disk bending deflection is small compared to disk thickness, so we can ignore the second-order terms in the governing equations corresponding to transverse displacement and stress function. After simplifying and discretizing the governing equations of motion, we can obtain a set of coupled equations of motion which takes the effect of initial axisymmetric runout into account. These equations are then used to study the effect of initial runout on the frequency response of the stationary disk. It is found that the initial runout increases the frequencies of the oscillations of a stationary disk. In the next step, we study the effect of initial non-flatness on the critical speed behavior of a spinning disk.

The Effect of Axisymmetric Nonflatness on the Oscillation Frequencies of a Rotating Disk

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Stress Function ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Governing Equations ◽  
Coupled Equations ◽  
Stationary Disk ◽  
Frequency Behavior

This study examines the frequency characteristics of thin rotating disks subjected to axisymmetric nonflatness. The equations of motion used are based on Von Karman’s plate theory. First, the eigenfunctions of the stationary disk problem corresponding to the stress function and transverse displacement are found. These eigenfunctions produce an equation that can be used in Galerkin’s method. The initial nonflatness is assumed to be a linear combination of the eigenfunctions of the transverse displacement of the stationary disk problem. Since the initial nonflatness is assumed to be axisymmetric, only eigenfunctions with no nodal diameters are considered to approximate the initial runout. It is supposed that the disk bending deflection is small compared with disk thickness, so we can ignore the second-order terms in the governing equations corresponding to the transverse displacement and the stress function. After simplifying and discretizing the governing equations of motion, we can obtain a set of coupled equations of motion, which takes the effect of the initial axisymmetric runout into account. These equations are then used to study the effect of the initial runout on the frequency behavior of the stationary disk. It is found that the initial runout increases the frequencies of the oscillations of a stationary disk. In the next step, we study the effect of the initial nonflatness on the critical speed behavior of a spinning disk.

Vibration-Induced Loosening Performance of Preloaded Threaded Fasteners

2010 ◽  
Author(s):  
Xianjie Yang ◽  
Sayed Nassar
Keyword(s):  
Nonlinear Model ◽  
Excitation Amplitude ◽  
The Self ◽  
Transverse Displacement ◽  
Transverse Loading ◽  
Threaded Fasteners ◽  
Bolt Preload ◽  
Proposed Model ◽  
Number Of Cycles ◽  
Insight Into

In an effort to establish a theoretical outline of a criterion for preventing the vibration-induced loosening of preloaded threaded fasteners, this paper provides an experimental and analytical insight into the effect of the initial bolt preload and the excitation amplitude on the self loosening performance of cap screw fastener. A nonlinear model is used for predicting the clamp load loss caused by the vibration-induced loosening of cap screw fasteners under cyclic transverse loading. Experimental verification was conducted on the twisting torque variation and the effect of the preload level and transverse displacement amplitude. Comparison of the experimental and analytical results on the clamp load loss with the number of cycles verifies that the proposed model accurately predicts self-loosening performance.

Limit Loads for Laterally Loaded I-Section Beams With Consideration of Shear

2003 ◽  
Vol 47 (02) ◽  
pp. 83-91
Author(s):  
L. Belenkiy ◽  
Y. Raskin
Keyword(s):  
Physical Model ◽  
Limit Load ◽  
Nonlinear Finite Element ◽  
Shear Forces ◽  
Element Analysis ◽  
Limit Loads ◽  
Closed Form Solutions ◽  
Engineering Technique ◽  
Laterally Loaded ◽  
The Web

The paper examines an effect of shear forces on limit load for I-section beams carrying later alloads. The problem is solve don the basis of a physical model, which enables one to take into account the effect of a resistance of beam flanges to the plastic shears train in the web of the beam. The physical model for the evaluation of limit loads was veriŽed using nonlinear finite element analysis. An engineering technique for the calculation of limit loads for shiphull beams subjected to large shear forces was developed using this model. As illustrative examples, the paper shows the application of the proposed technique to obtain closed-form solutions for the prediction of limit loads.

Reinforced Circular Holes in Bending With Shear

1953 ◽  
Vol 20 (2) ◽  
pp. 279-285
Author(s):  
S. R. Heller
Keyword(s):  
Stress Distribution ◽  
Arbitrary Shape ◽  
Theoretical Solution ◽  
Bead Type ◽  
Circular Holes ◽  
Radial Thickness ◽  
Bending With Shear ◽  
The Web

Abstract The object of this paper is the determination of the effect of the reinforcement of circular holes on the stress distribution in the webs of beams subjected to bending with shear. A theoretical solution for a bead-type reinforcement, i.e., small radial thickness, is developed. The stress distribution in the web for arbitrary shape reinforcement is based on the work of Reissner and Morduchow (1). The theory developed is valid provided the diameter of the hole does not exceed one fourth of the depth of the beam.

A closed form solution for bending/stretching analysis of functionally graded circular plates under asymmetric loading using the Green function

2010 ◽  
Vol 224 (6) ◽  
pp. 1153-1163 ◽  
Author(s):  
A R Saidi ◽  
A Naderi ◽  
E Jomehzadeh
Keyword(s):  
Green Function ◽  
Closed Form ◽  
Volume Fraction ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Circular Plates ◽  
Equilibrium Equations

In this article, a closed-form solution for bending/stretching analysis of functionally graded (FG) circular plates under asymmetric loads is presented. It is assumed that the material properties of the FG plate are described by a power function of the thickness variable. The equilibrium equations are derived according to the classical plate theory using the principle of total potential energy. Two new functions are introduced to decouple the governing equilibrium equations. The three highly coupled partial differential equations are then converted into an independent equation in terms of transverse displacement. A closed-form solution for deflection of FG circular plates under arbitrary lateral eccentric concentrated force is obtained by defining a new coordinate system. This solution can be used as a Green function to obtain the closed-form solution of the FG plate under arbitrary loadings. Also, the solution is employed to solve some different asymmetric problems. Finally, the stress and displacement components are obtained exactly for each problem and the effect of volume fraction is also studied.

The Interface Boundary Conditions for Bolted Flanged Connections

1976 ◽  
Vol 98 (4) ◽  
pp. 277-282 ◽  
Author(s):  
J. C. Thompson ◽  
Y. Sze ◽  
D. G. Strevel ◽  
J. C. Jofriet
Keyword(s):  
Stress Distribution ◽  
Contact Stress ◽  
Plate Theory ◽  
Contact Region ◽  
Three Dimensional ◽  
Surface Region ◽  
Prestressing Force ◽  
Flange Thickness ◽  
Interface Boundary Conditions ◽  
Contact Stress Distribution

In most bolted connections, the unknown interface pressure distribution and the extent of the contact region are essential parameters in any stress analysis. Concerning these parameters, experimental and numerical studies of a model of an isolated single-bolt region show the following. The contact region between the flanges depends almost exclusively on the ratio of the flange thickness to the diameter of the surface region of each flange over which the bolt prestressing force is distributed; the contact zone is virtually independent of both the level of prestressing force and of the size of the bolt hole; and the contact stress distribution for a typical range of parameters is very closely approximated by a truncated conical distribution. The studies also delineate the regions of the flanges around each bolt where the stress state is strongly three-dimensional and regions where simple plate theory is applicable. The relationships established between the contact stress distribution and the various geometric parameters are presented in a form immediately applicable by designers.

Sharp Corner Functions for Mindlin Plates

2005 ◽  
Vol 72 (1) ◽  
pp. 1-9 ◽  
Author(s):  
O. G. McGee ◽  
J. W. Kim ◽  
A. W. Leissa
Keyword(s):  
Plate Theory ◽  
Thin Plates ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Transverse Displacement ◽  
Thick Plates ◽  
Thin Plate Theory ◽  
Mindlin Plate Theory ◽  
Moderately Thick Plates ◽  
Sharp Corners

Transverse displacement and rotation eigenfunctions for the bending of moderately thick plates are derived for the Mindlin plate theory so as to satisfy exactly the differential equations of equilibrium and the boundary conditions along two intersecting straight edges. These eigenfunctions are in some ways similar to those derived by Max Williams for thin plates a half century ago. The eigenfunctions are called “corner functions,” for they represent the state of stress currently in sharp corners, demonstrating the singularities that arise there for larger angles. The corner functions, together with others, may be used with energy approaches to obtain accurate results for global behavior of moderately thick plates, such as static deflections, free vibration frequencies, buckling loads, and mode shapes. Comparisons of Mindlin corner functions with those of thin-plate theory are made in this work, and remarkable differences are found.

Comparison of First-Order Shear and Plane Strain Assumptions in Warpage Prediction of Simply Supported Printed Wiring Boards

2000 ◽  
Vol 123 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Yarom Polsky ◽  
I. Charles Ume
Keyword(s):  
Plane Strain ◽  
Plate Theory ◽  
Shear Deformation Theory ◽  
Transverse Load ◽  
Transverse Shear Strain ◽  
Printed Wiring Board ◽  
Simply Supported ◽  
First Order ◽  
Support Conditions ◽  
Wiring Board

The influence of transverse shear strain in the lamination theory modeling of Printed Wiring Board (PWB) deflection due to support conditions was examined. The in-plane mechanical properties of the core materials of a commercial PWB were measured as a function of temperature. Classical laminated plate theory and first-order shear deformation theory solutions for the out-of-plane deflection of a bare board configuration with two opposite edges simply supported and the remaining edges free were obtained. The weight of the board was approximated as a distributed transverse load. The effect of material property decrease with temperature and FR-4 layer thickness were examined to compare first-order shear and plane strain assumptions for the predicted warpage.

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