Summary
An approximate analytical solution of the large deflection axisymmetric response of polar orthotropic thin truncated conical and spherical shallow caps is presented. Donnell type equations are employed. The deflection is approximated by a one term mode shape satisfying the boundary conditions. The Galerkin's method is used to get the governing equation for the deflection at the hole. Nonlinear free vibration response and the response under uniformly distributed static and step function loads are obtained. The effect of various parameters is investigated.
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Abbreviations
- A, A * :
-
Inward and outward amplitudes
- a, b, h :
-
Base radius, inner radius and thickness of the cap
- D :
-
M θ h 3/[12(β−v 2 θ )]
- E τ,E θ :
-
Young's moduli
- H *,H :
-
Apex height, dimensionless apex heght:H */h
- N τ,θ :
-
Stress resultants
- p :
-
α 1/2
- q :
-
Uniformly distributed load
- Q,Q0 :
-
Dimensionless load:\(\frac{{qa^4 }}{{E_\tau h^4 }}\), dimensionless step load
- Q, Q 0 :
-
Dimensionless load:\(\frac{1}{8}\left[ {3\left( {1 - v_\theta ^2 } \right)} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \frac{{qa^4 }}{{E_\tau h^4 H^2 }}\), step load
- t, τ:
-
Time, dimensionless time:\(\left[ {\frac{D}{{\gamma ha^4 }}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} t\) t
- T A :
-
Ratio of nonlinear periodT for inward amplitudeA and the linear periodT L
- w * :
-
Normal displacement at middle surface
- w :
-
Dimensionless displacement:w */h
- α1 :
-
Linear parameter of static response
- β:
-
Orthotropic Parameter:E θ/E τ
- γ:
-
Mass density
- ɛ2,ɛ3 :
-
Quadratic and cubic nonlinearity parameters
- η:
-
b/a
- v θ,v τ :
-
Poisson's ratios
- ϖ:
-
Dimensionless radius:r/a
- ψ*, ψ:
-
Stress function, dimensionless stress function:\(\frac{a}{D}\psi ^* \)
- ω *0 ,ω0 :
-
Linnear frequency, dimensionless frequency:\(\omega _0^* \left( {{{\gamma ha^4 } \mathord{\left/ {\vphantom {{\gamma ha^4 } D}} \right. \kern-\nulldelimiterspace} D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \)
References
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Hübner, W.: Deformationen und Spannungen bei Tellerfedern. Konstruktion34, 387–392 (1982).
Alwar, R. S., Reddy, B. C.: Dynamic buckling of isotropic and orthotropic shallow spherical cap with central hole. Int. J. Mech. Sci.21, 681–688 (1979).
Dumir, P. C., Gandhi, M. L., Nath, Y.: Axisymmetric static and dynamic buckling of orthotropic shallow spherical cap with circular hole. Comput. Struct.19, 725–736 (1984).
Dumir, P. C., Khatri, K. N.: Axisymmetric static and dynamic buckling of orthotropic truncated shallow conical caps. Comput. Struct. (To appear) (1985).
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Dumir, P.C. Nonlinear axisymmetric response of orthotropic thin truncated conical and spherical caps. Acta Mechanica 60, 121–132 (1986). https://doi.org/10.1007/BF01302946
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DOI: https://doi.org/10.1007/BF01302946