Summary
The stress problem of a thin cylindrical shell supported by an elastic core of a different material and subjected to arbitrary loading on its curved surface is considered. The problem is solved by applying the three-dimensional theory of elasticity to the core and using membrane or bending solutions for the shell. Equilibrium and compatibility equations are satisfied at the junction of the shell and the core. It is pointed out that the procedure can easily be extended to the case of a hollow core with or without another shell of another material in it. Numerical results are presented to illustrate the effectiveness of even a weak core in reducing the shell stresses.
Zusammenfassung
Gegenstand der Untersuchung ist eine dünne Kreiszylinderschale, die durch einen elastischen Kern aus einem anderen Werkstoff gestützt ist und eine beliebige Belastung trägt. Die Lösung verbindet die strenge, dreidimensionale Theorie des zylindrischen Kerns mit der Membran- oder Biegetheorie der Schale. An der Grenze zwischen beiden Teilen müssen die Verschiebungen und gewisse Spannungskomponenten stetig übergehen. Es wird darauf hingewiesen, daß die Lösung leicht auf den Fall ausgedehnt werden kann, daß der Kern ein Hohlzylinder ist, der möglicherweise auf der Innenseite mit einer zweiten Zylinderschale verbunden ist. Zahlenergebnisse zeigen, daß selbst ein verhältnismäsig nachgiebiger Kern einen großen (und günstigen) Einfluß auf die Spannungen in der Schale ausübt.
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Abbreviations
- a :
-
Radius of the middle surface of the shell
- t :
-
Thickness of the shell
- ι:
-
=1−t/2a
- u c,v c,w c :
-
Displacements respectively in the axial, circumferential and radial directions of a point in the core
- X(x), Φ(φ), Δ(λr/a):
-
3×3 square matrices
- λ,m :
-
Parameters
- l :
-
Length of the cylinder
- c:
-
A vector containing constantsc 1,c 2 andc 3
- ϱ:
-
=λr/a
- γ:
-
=m+4(1−v e)
- E c,v e :
-
Elastic constants for the core material
- \(\left. {\begin{array}{*{20}c} {\sigma _x ,\sigma _\varphi ,\tau _{x\varphi } } \\ {\tau _{rx} ,\tau _{r\varphi } ,\sigma _r } \\ \end{array} } \right\}\) :
-
Stresses at a point in the core
- D c :
-
\(\frac{{E_c \alpha }}{{1 + v_e }}\)
- τ:
-
A vector containing τ rx ,τ rϕ and σ r
- Ω(λr/a):
-
A 3×3 matrix
- \(\bar u,\bar v,\bar w\) :
-
Displacements at the surfacer=aι of the core
- \(\bar u\) :
-
A vector containing\(\bar u,\bar v,\bar w\)
- \(\bar u_{mn} ,\bar v_{mn} ,\bar w_{mn} \) :
-
Amplitudes of displacements\(\bar u,\bar v,\bar w\)
- \(\bar u_{mn} \) :
-
A vector containing\(\bar u_{mn} ,\bar v_{mn} ,\bar w_{mn} \)
- \(\bar \tau \) :
-
=τ(x, φ,aσ)
- α ij :
-
Constants
- A :
-
A square matrix containing constants α ij
- \(\left. {\begin{array}{*{20}c} {N_x ,N_\varphi ,N_{x\varphi } } \\ {N_{\varphi x} ,M_x ,M_\varphi } \\ {M_{x\varphi } ,M_{\varphi x} ,Q_x ,Q_\varphi } \\ \end{array} } \right\}\) :
-
Stress resultants in the shell as defined in reference [3]
- p x,p φ,P r :
-
Components of applied loading per unit area of shell's middle surface
- ()′:
-
\( = \frac{{\alpha \partial ()}}{{\partial x}}\)
- ()·:
-
\( = \frac{{\partial ()}}{{\partial \varphi }}\)
- u, v, w :
-
Displacements of a point on the middle surface of the shell
- E s,v s :
-
Elastic constants for the shell material
- D s :
-
\( = \frac{{E_s t}}{{1 - v_s^2 }}\)
- K :
-
\( = \frac{{E_s t^3 }}{{12(1 - v_s^2 )}}\)
- k :
-
\( = \frac{K}{{D_s \alpha ^2 }}\)
- p xmn,p φmn,p rmn :
-
Amplitudes of loadsp x,p φ, pr
- u mn, vmn,w mn :
-
Amplitudes of displacementsu, v, ω
References
Kelkar, V. S.: “The Problem of an Elastic Circular Cylinder With or Without a Thin Skin of a Different Material,” Ph D. Dissertation, Stanford University, Stanford, California, 1966.
Flügge, W. andV. S. Kelkar: “The Problem of an Elastic Circular Cylinder,” Int. J. of Solids and Structures,4, (1968), 397–420.
Flügge, W.: Stresses in Shells, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960.
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Kelkar, V.S., Flügge, W. Stresses in a cylindrical shell supported by a core of a different material. Acta Mechanica 6, 165–179 (1968). https://doi.org/10.1007/BF01170381
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DOI: https://doi.org/10.1007/BF01170381