Thermoelastic Displacement Function

2014 ◽  
pp. 5684-5684
Keyword(s):  
Displacement Function ◽  
Thermoelastic Displacement

Buckling of Circular Cylindrical Shells Using a Displacement Function

1974 ◽  
Vol 96 (4) ◽  
pp. 1322-1327
Author(s):  
Shun Cheng ◽  
C. K. Chang
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Cylindrical Shells ◽  
External Pressure ◽  
Displacement Function ◽  
Combined Loading ◽  
Trial Solution ◽  
Governing Differential Equation ◽  
Circular Cylindrical Shells ◽  
The Stability

The buckling problem of circular cylindrical shells under axial compression, external pressure, and torsion is investigated using a displacement function φ. A governing differential equation for the stability of thin cylindrical shells under combined loading of axial compression, external pressure, and torsion is derived. A method for the solutions of this equation is also presented. The advantage in using the present equation over the customary three differential equations for displacements is that only one trial solution is needed in solving the buckling problems as shown in the paper. Four possible combinations of boundary conditions for a simply supported edge are treated. The case of a cylinder under axial compression is carried out in detail. For two types of simple supported boundary conditions, SS1 and SS2, the minimum critical axial buckling stress is found to be 43.5 percent of the well-known classical value Eh/R3(1−ν2) against the 50 percent of the classical value presently known.

The Effect of Two Rigid Spherical Inclusions on the Stresses in an Infinite Elastic Solid

1966 ◽  
Vol 33 (1) ◽  
pp. 68-74 ◽  
Author(s):  
Joseph F. Shelley ◽  
Yi-Yuan Yu
Keyword(s):  
Stress Field ◽  
Uniaxial Tension ◽  
Elastic Solid ◽  
Displacement Function ◽  
Hydrostatic Tension ◽  
Series Form ◽  
Spherical Inclusions ◽  
Spherical Cavities ◽  
In Series ◽  
The Common

Presented in this paper is a solution in series form for the stresses in an infinite elastic solid which contains two rigid spherical inclusions of the same size. The stress field at infinity is assumed to be either hydrostatic tension or uniaxial tension in the direction of the common axis of the inclusions. The solution is based upon the Papkovich-Boussinesq displacement-function approach and makes use of the spherical dipolar harmonics developed by Sternberg and Sadowsky. The problem is closely related to, but turns out to be much more involved than, the corresponding problem of two spherical cavities solved by these authors.

Free Vibration of Orthotropic Skew Plates

1999 ◽  
Vol 122 (3) ◽  
pp. 313-317 ◽  
Author(s):  
A. M. Farag ◽  
A. S. Ashour
Keyword(s):  
Boundary Conditions ◽  
Transition Matrix ◽  
Longitudinal Direction ◽  
Displacement Function ◽  
Characteristic Matrix ◽  
Skew Angle ◽  
Kantorovich Method ◽  
Vibration Effect ◽  
Finite Strip Method ◽  
Semianalytical Method

The main purpose of this paper is to develop a fast converging semianalytical method for assessing the vibration effect on thin orthotropic skew (or parallelogram/oblique) plates. Since the geometry of the skew plate is not helpful in the mathematical treatments, the analysis is often performed by more complicated and laborious methods. A successive conjunction of the Kantorovich method and the transition matrix is exploited herein to develop a new modification of the finite strip method to reduce the complexity of the problem. The displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction. Using the new transition matrix, after necessary simplification and the satisfaction of the boundary conditions, yields a set of simultaneous equations that leads to the characteristic matrix of vibration. The influence of the skew angle, the aspect ratio, the properties of orthotropy, and the prescribed boundary conditions are investigated. Convergence of the solution is investigated and the accuracy of the results is compared with that available from other numerical methods. The numerical results show that the convergence is rapidly deduced and the comparisons agree very well with known results. [S0739-3717(00)00202-6]

scholarly journals Analytical solution to bending of stiffened and continuous antisymmetric laminates

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Liecheng Sun ◽  
Issam E. Harik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Displacement Function ◽  
The Other ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Coupling Problem ◽  
Simply Supported ◽  
General Response

AbstractAnalytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

An Improved Fourier–Ritz Method for Analyzing In-Plane Free Vibration of Sectorial Plates

2017 ◽  
Vol 84 (9) ◽  
Author(s):  
Siyuan Bao ◽  
Shuodao Wang ◽  
Bo Wang
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Displacement Function ◽  
Accurate Solution ◽  
Previous Method ◽  
Fourier Series Expansion ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Radial Variable

A modified Fourier–Ritz approach is developed in this study to analyze the free in-plane vibration of orthotropic annular sector plates with general boundary conditions. In this approach, two auxiliary sine functions are added to the standard Fourier cosine series to obtain a robust function set. The introduction of a logarithmic radial variable simplifies the expressions of total energy and the Lagrangian function. The improved Fourier expansion based on the new variable eliminates all the potential discontinuities of the original displacement function and its derivatives in the entire domain and effectively improves the convergence of the results. The radial and circumferential displacements are formulated with the modified Fourier series expansion, and the arbitrary boundary conditions are simulated by the artificial boundary spring technique. The number of terms in the truncated Fourier series and the appropriate value of the boundary spring retraining stiffness are discussed. The developed Ritz procedure is used to obtain accurate solution with adequately smooth displacement field in the entire solution domain. Numerical examples involving plates with various boundary conditions demonstrate the robustness, precision, and versatility of this method. The method developed here is found to be computationally economic compared with the previous method that does not adopt the logarithmic radial variable.

Torsional Vibration Analysis of Shafts With Sections of Tapered Diameter

1993 ◽  
Author(s):  
John R. Baker ◽  
Keith E. Rouch
Keyword(s):  
Torsional Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Displacement Function ◽  
The Other ◽  
Mode Shapes ◽  
Axial Distance ◽  
Rotational Degree ◽  
Rotor Systems ◽  
Constant Diameter

Abstract This paper presents the development of two tapered finite elements for use in torsional vibration analysis of rotor systems. These elements are particularly useful in analysis of systems that have shaft sections with linearly varying diameters. Both elements are defined by two end nodes, and inertia matrices are derived based on a consistent mass formulation. One element assumes a cubic displacement function and has two degrees of freedom at each node: rotation about the shaft’s axis and change in angle of rotation with respect to the axial distance along the shaft. The other element assumes a linear displacement function and has one rotational degree of freedom at each node. The elements are implemented in a computer program. Calculated natural frequencies and mode shapes are compared for both tapered shaft sections and constant diameter sections. These results are compared with results from an available constant diameter element. It is shown that the element derived assuming a cubic displacement function offers much better convergence characteristics in terms of calculated natural frequencies, both for tapered sections and constant diameter sections, than either of the other two elements. The finite element code that was developed for implementation of these elements is specifically designed for torsional vibration analysis of rotor systems. Lumped inertia, lumped stiffness, and gear connection elements necessary for rotor system analysis are also discussed, as well as calculation of natural frequencies, mode shapes, and amplitudes of response due to a harmonic torque input.

The displacement function and the return map

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Displacement Function ◽  
Semisimple Element ◽  
Fixed Point Set ◽  
Critical Set ◽  
Quantitative Estimates ◽  
Point Set ◽  
Toponogov’S Theorem

This chapter studies the displacement function dᵧ on X that is associated with a semisimple element γ‎ ∈ G. If φ‎″, t ∈ R denotes the geodesic flow on the total space X of the tangent bundle of X, the critical set X(γ‎) ⊂ X of dᵧ can be easily related to the fixed point set Fᵧ ⊂ X of the symplectic transformation γ‎⁻¹φ‎₁ of X. The chapter studies the nondegeneracy of γ‎⁻¹φ‎₁ − 1 along Fᵧ. More fundamentally, this chapter gives important quantitative estimates on how much φ‎ ½ differs from φ‎ ˗½γ‎ away from Fᵧ. These quantitative estimates are based on Toponogov's theorem.

Multiple-Points Constraints Based Deformation for Free-Form Surfaces

1999 ◽  
Author(s):  
J. M. Zheng ◽  
K. W. Chan ◽  
I. Gibson
Keyword(s):  
Surface Deformation ◽  
Single Point ◽  
Displacement Function ◽  
Surface Shape ◽  
Free Form ◽  
Surface Model ◽  
Deformation Method ◽  
Multiple Points ◽  
Free Form Surface ◽  
Free Form Surfaces

Abstract There is an increasing demand in the conceptual design for more intuitive methods for creating and modifying free-form curves and surfaces in CAD modeling systems. The methods should be based not only on the change of the mathematical parameters but also on the user’s specified constraints and shapes. This paper presents a new surface representation model for free-form surface deformation representation. The model is a combination of two functions: a displacement function and a function for representing an existing NURBS surface called parent surface. Based on the surface model, the authors develop two deformation methods which are named SingleDef (Single-point constraint based deformation method), and MultiDef (Multiple-points constraints based deformation method). The techniques for free-form surface deformation allow conceptual designer to modify a parent surface by directly applying point constraints to the parent surface. The deformation methods are implemented and taken in an experimental CAD system. The results show that the designer can easily and intuitively control the surface shape.

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