Application of the Generalized Airy Stress Function to Problems on Elastic Vibrations of Hollow Cylinders

1973 ◽  
Vol 40 (4) ◽  
pp. 1140-1141
Author(s):  
R. A. Bourgois
Keyword(s):  
Stress Function ◽  
Laplace Transformation ◽  
Stress Waves ◽  
Elastic Response ◽  
Impact Loads ◽  
Airy Stress Function ◽  
Hollow Cylinders ◽  
Thick Cylinder ◽  
Basic Equations ◽  
Dynamic Plane

This Note concerns the use of the generalized Airy stress function, introduced by Radok [1], for dynamic plane-stress and plane-strain problems with axial symmetry. The solution of the basic equations for stress waves may be accomplished by Laplace transformation. The case of the dynamic elastic response of a thick cylinder to internal axial symmetric impact loads will be studied.

Dynamic Elastic Response of a Ring to Transient Pressure Loading

1966 ◽  
Vol 33 (2) ◽  
pp. 261-266 ◽  
Author(s):  
Shin-Ichi Suzuki
Keyword(s):  
Stress Analysis ◽  
Laplace Transformation ◽  
Static Load ◽  
Transformation Method ◽  
Elastic Response ◽  
Impact Loads ◽  
Pressure Loading ◽  
Transient Pressure ◽  
Laplace Transformation Method

Stress analysis is carried out for distributed impact loads applied along the inner and outer edges of a ring. These loads are assumed to be of the form q(θ)e−αt. The relationships between the stresses at both edges, the dimensions, and time are obtained. The results show that the ratio μ of the stress under impact to that under static load is different from μ = 2. The problem is analyzed by using the Laplace transformation method.

scholarly journals Form-Finding of Shells Containing Both Tension and Compression Using the Airy Stress Function

2020 ◽  
Author(s):  
Masaaki Miki ◽  
Emil Adiels ◽  
William Baker ◽  
Toby Mitchell ◽  
Alexander Sehlstrom ◽  
...  
Keyword(s):  
Stress Function ◽  
Mixed Type ◽  
Airy Stress Function ◽  
Form Finding ◽  
Graphic Statics ◽  
Tension And Compression ◽  
Stress Functions ◽  
Complete Set ◽  
Pure Compression ◽  
Asymptotic Lines

Pure-compression shells have been the central topic in the form-finding of shells. This paper studies tension-compression mixed type shells by utilizing a NURBS-based isogeometric form-finding approach that analyzes Airy stress functions to expand the possible plan geometry. A complete set of smooth version graphic statics tools is provided to support the analyses. The method is validated using examples with known solutions, and a further example demonstrates the possible forms of shells that the proposed method permits. Additionally, a guideline to configure a proper set of boundary conditions is presented through the lens of asymptotic lines of the stress functions.

Alternative Derivation of Marguerre’s Displacement Solution in Plane Isotropic Elasticity

1999 ◽  
Vol 67 (2) ◽  
pp. 419-421 ◽  
Author(s):  
X.-L. Gao
Keyword(s):  
Stress Function ◽  
Present Approach ◽  
Function Approach ◽  
Airy Stress Function ◽  
Green's Theorem ◽  
Alternative Derivation ◽  
Green’S Theorem ◽  
Isotropic Elasticity

An alternative derivation of Marguerre’s solution for displacements in plane isotropic elasticity is provided. It is shown that the present approach, which is based on Green’s theorem and parallel to the Airy stress function approach, is straightforward. Also, the current derivation establishes the completeness of the Marguerre solution. [S0021-8936(00)00302-0]

scholarly journals A Multi-Parameter Perturbation Solution for Functionally Graded Piezoelectric Cantilever Beams under Combined Loads

Materials ◽  
2018 ◽  
Vol 11 (7) ◽  
pp. 1222 ◽  
Author(s):  
Yongsheng Lian ◽  
Xiaoting He ◽  
Sijie Shi ◽  
Xue Li ◽  
Zhixin Yang ◽  
...  
Keyword(s):  
Cantilever Beam ◽  
Functionally Graded ◽  
Airy Stress Function ◽  
Parameter Perturbation ◽  
Combined Loads ◽  
Piezoelectric Cantilever ◽  
Basic Equations ◽  
Perturbation Parameters ◽  
Functionally Graded Piezoelectric ◽  
Parameter Perturbation Method

In this study, we use a multi-parameter perturbation method to solve the problem of a functionally graded piezoelectric cantilever beam under combined loads, in which three piezoelectric coefficients are selected as the perturbation parameters. First, we derive the two basic equations concerning the Airy stress function and electric potential function. By expanding the unknown Airy stress function and electric potential function with respect to three perturbation parameters, the two basic equations were decoupled, thus obtaining the corresponding multi-parameter perturbation solution under boundary conditions. From the solution obtained, we can see clearly how the piezoelectric effects influence the behavior of the functionally graded piezoelectric cantilever beam. Based on a numerical example, the variations of the elastic stresses and displacements as well as the electric displacements of the cantilever beam under different gradient exponents were shown. The results indicate that if the pure functionally graded cantilever beam without a piezoelectric effect is regarded as an unperturbed system, the functionally graded piezoelectric cantilever beam can be looked upon as a perturbed system, thus opening the possibilities for perturbation solving. Besides, the proposed multi-parameter perturbation method provides a new idea for solving similar nonlinear differential equations.

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