Summary
Axisymmetric stability and thermal-buckling equations are established for circular plates composed of polarorthotropic layers subjected to mechanical loads depending only onr and thermal fieldsT=T(r, z). Alternative fourth order systems of two coupled differential equations are suggested in terms of the slope β andeither the radial displacementu or a stress function Ψ.
The eigenvalue problem is formulated for quasi-heterogeneous composite plates and a closed-type solution is given for certain thermal-buckling problems in the form of Bessel functions of first kind and fractional order. Possibility of an analogy between a thermal and mechanical stability problem is shown and variation of eigenvalues with anisotropy parameters is noted. Numerous examples are presented indicating that suitable lamination of composite circular plates may transcend the buckling loads of individual constituents.
Zusammenfassung
Gleichungen für axialsymmetrische Stabilität und thermisches Beulen runder Platten aus polar-orthotropen Schichten und unter nur vonr abhängiger Belastung und TemperaturfeldernT=T(r, z) werden aufgestellt. Als Alternative werden Systeme 4. Ordnung von zwei gekoppelten Differentialgleichungen in der Neigung β undentweder der Radialverschiebungu oder einer Spannungsfunktion Ψ vorgeschlagen.
Das Eigenwertproblem wird für quasiheterogene Verbundplatten formuliert und eine geschlossene Lösung durch Bessel-Funktionen erster Art für bestimmte Probleme thermischen Beulens angegeben. Die Möglichkeit einer Analogie zwischen einem thermischen und mechanischen Stabilitätsproblem wird aufgezeigt und die Veränderung des Eigenwertes in Abhängigkeit von Parametern der Anisotropie festgestellt. Zahlreiche Beispiele, die zeigen, daß durch geeignete Schichtung die Beullast von runden Verbundplatten die der einzelnen Bestandteile übersteigt, werden angegeben.
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Abbreviations
- a :
-
radius of plate
- A i :
-
thermal coefficient defined by Eq. (2.4)
- A ij :
-
elastic area
- A * ij :
-
extensional rigidity
- B i :
-
constant
- B ij :
-
elastic statical moment
- B * ij :
-
matrix element defined by Eq. (2.34.2)
- C i :
-
constant
- C * ij :
-
matrix element defined by Eq. (2.34.3)
- d :
-
orthotropy parameter in bending
- D ij , D *ij , D:
-
elastic moment of inertia
- e ij :
-
elastic compliance
- E ij :
-
elastic stiffness
- h :
-
plate thickness
- H :
-
horizontal stress resultant
- J v :
-
Bessel function of first kind and orderv
- k :
-
thermal parameter defined by Eq. (3.2.1)
- K i :
-
coefficient of thermal conductivity ini-th direction
- L i ,L ij :
-
functional operator
- M i :
-
bending moment
- M iT :
-
thermal quantity defined by Eq. (2.10.2)
- N i :
-
in-plane force
- N iT :
-
thermal quantity defined by Eq. (2.10.1)
- N 0c ,N 0s :
-
critical radial compressions for clamped and simply-supported plate, respectively
- p i :
-
component of load intensity
- p :
-
parameter defined by Eqs. (3.26), (3.55)
- P i :
-
constant
- Q :
-
transverse shear resultant
- r :
-
radial coordinate
- s :
-
orthotropy parameter in stretching
- t :
-
parameter defined by Eq. (3.53.1)
- T :
-
temperature change
- u, w :
-
radial and transverse displacements, respectively
- V :
-
vertical component ofN r andQ
- x :
-
non-dimensional variable
- z :
-
vertical coordinate
- α i , α:
-
coefficient of thermal expansion
- β:
-
slope in radial direction
- γ, δ:
-
parameters in Bessel-type equation
- ε i0 :
-
strain component atz=0
- θ:
-
circumferential coordinate
- ϰ i :
-
curvature
- λ:
-
eigenvalue parameter defined by Eq. (3.17)
- v :
-
order of Bessel function, Poisson's ratio
- ∂:
-
non-dimensional radial coordinate
- τ ij :
-
stress component
- Φ:
-
sloping angle of tangent to radial curve
- Ψ:
-
stress resultant function
References
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Dedicated to the memory of my father, Mr. Israel Hacohen Stavsky.
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Stavsky, Y. Thermoelastic stability of laminated orthotropic circular plates. Acta Mechanica 22, 31–51 (1975). https://doi.org/10.1007/BF01170618
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DOI: https://doi.org/10.1007/BF01170618