Abstract
Dynamic stability of a thin spherical shell is investigated analytically under a uniform normal pressure.
The purpose of this paper is to present a dynamic stability criterion which together with the energy method result and the numerical integration of the asymptotic nonlinear shell equations permit to find a closed form analytic expression for the lower critical pressure.
The dynamic stability criterion states that the change in kinetic energy is equal to the work of all the forces between the initial and the buckled position after the dynamic stage of buckling.
The solution of this nonlinear problem can be interpreted as the trajectory of a material point moving in a nonconservative force field.
The resulting lower critical pressure curve lies along all the lowest known experimental data. It determines the boundary for the absolute dynamic stability and can be very useful for the practical shell design to prevent buckling.
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Abbreviations
- A :
-
constant value 2.2
- dA s :
-
elemental area of the surface of the shell
- a, b :
-
constants of integration
- C :
-
elasticity modulus of elastic foundation
- C 1, C 2, C 3 :
-
constants of integration
- D :
-
Eh 3/12(1−ν 2), stiffness of the shell
- e:
-
base of natural logarithms
- E :
-
modulus of elasticity
- F :
-
“force” vector
- F x , F y :
-
components of the force vector in the x, y directions
- h :
-
thickness of the shell
- H :
-
height of a segment of a shell
- In :
-
ξ*3 A
- i:
-
\(\sqrt { - 1}\) imaginary number
- I :
-
moment of inertia of a beam
- K s , K θ :
-
changes of curvature in the s, and θ directions
- M :
-
bending moment in a beam
- M s ,M θ :
-
shell moment resultants
- N s ,N θ ,N sθ :
-
shell membrane resultants
- N 0 s :
-
initial value of the membrane force in the meridional direction
- P :
-
external uniform normal pressure
- p cl :
-
“classical” value of critical pressure (obtained by Von Kármán from the linear analysis of the shell)
- Q :
-
shell transverse shear resultant
- R :
-
initial radius of curvature of the middle surface
- r :
-
distance from the point on the shell to the axis of symmetry of the shell
- S 40 =DR 2/Eh :
-
“characteristic” length of the shell
- s :
-
distance measured along the meridian of the shell
- t=ξ − ξ* :
-
distance measured from the transition zone
- \(\left. \begin{gathered} u_x = x - 1 \hfill \\ u_y = y \hfill \\ \end{gathered} \right\}\) :
-
new variables to study the behavior of the shell in the vicinity of x=1
- V :
-
“velocity” vector
- Vol :
-
total change of volume of the shell
- \(x = \frac{\psi }{\varphi }\) :
-
relative change of the curvature in the θ direction
- \(y = \frac{{S_0^2 N_s }}{D}\) :
-
non-dimensional membrane force
- Z=u x +iu y :
-
complex variable
- w :
-
deflection of the beam
- dW b :
-
bending energy per unit surface of the shell
- W b :
-
change of bending energy of the shell
- W c :
-
energy of initial uniform compression of the shell
- dW m :
-
membrane energy per unit surface of the shell
- W m :
-
change of membrane energy
- W p :
-
total work done by the external pressure
- W T :
-
total work of the buckled shell
- ε s , ε θ :
-
membrane deformations
- ϕ :
-
initial angle of the shell
- ψ :
-
angle of the deformed shell
- ν :
-
Poisson's ratio
References
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Part of this research was carried out at Princeton University.
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Zagustin, A.I., Zagustin, E.A. Lower critical pressure for dynamic stability of a spherical shell. Appl. Sci. Res. 23, 337–367 (1971). https://doi.org/10.1007/BF00413210
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DOI: https://doi.org/10.1007/BF00413210