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Stability of clamped skew plates

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Abstract

The stability problems of clamped skew plates are considered with the inplane stresses represented in terms of oblique components. Deflection is expressed in terms of a double series of beam characteristic functions of clamped-clamped beam. Energy method is used to obtain buckling coefficients under individual loadings and for a few cases of combined loading. Convergence is examined in a few representative cases. For buckling in shear, two critical values exist the magnitude of negative shear being much larger than that of positive shear.

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Abbreviations

a, b :

dimensions of the plate

C rs :

coefficient in the series expansion of deflection

D :

flexural rigidity of the plate (Eh 3/12(1−v 2))

E :

Young's modulus of the material of the plate

G, H (1),H (2),H (3) :

matrices defined in (15)

G 1 :

matrix defined in (16)

h :

plate thickness

I pqmr , J pqns :

integrals defined in (11)

K :

order of the matrix

M, N :

maximum number of terms inx- andy-directions respectively

N x,N y,N xy :

midplane forceshσ x,hσ y,hσ xy respectively

\(\bar R_x^* ,\bar R_y^* ,\bar R_{xy}^* \) :

nondimensional midplane force parameters,\(\frac{{h\sigma _x a^2 \cos ^3 \psi }}{D},\frac{{h\sigma _y a^2 \cos ^3 \psi }}{D},\frac{{h\sigma _{xy} a^2 \cos ^3 \psi }}{D}\) respectively

\(\bar R_x ,\bar R_y ,\bar R_{xy} \) :

buckling coefficients,\(\frac{{\sigma _x hb^2 }}{{\pi ^2 D}},\frac{{\sigma _y hb^2 }}{{\pi ^2 D}},\frac{{\sigma _{xy} hb^2 }}{{\pi ^2 D}}\) respectively

U :

strain energy of the plate

V :

potential energy of the middle surface forces

W(ξ, η):

deflection of the plate

X m(ξ),Y n(η):

beam characteristic functions

x, y, z :

oblique coordinate system defined in Fig. 1

x 1,y 1,z :

orthogonal coordinate system defined in Fig. 1

ξ, η :

nondimensional coordinates,x/a andy/b respectively

ν :

ratio

σ x,σ y,σ xy :

oblique stress components defined in Fig. 1

λ :

side ratio (a/b)

ψ :

skew angle

▽ 2 :

skew differential operator\(\left( {\sec ^2 \psi \left( {\frac{{\partial ^2 }}{{\partial x^2 }} - 2\sin \psi \frac{{\partial ^2 }}{{\partial x^2 \partial y}} + \frac{{\partial ^2 }}{{\partial y^2 }}} \right)} \right)\)

▽ 21 :

skew differential operator in nondimensional coordinates\(\left( {\sec ^2 \psi \left( {\frac{{\partial ^2 }}{{\partial \xi ^2 }} - 2\lambda \sin \psi \frac{{\partial ^2 }}{{\partial \xi ^2 \partial \eta }} + \lambda ^2 \frac{{\partial ^2 }}{{\partial \eta ^2 }}} \right)} \right)\)

α, β :

ratios ofσ y/σ x andσ xy/σ x respectively

ε m,ε n :

beam eigenvalues, (7) and (8)

α m,α n :

parameters in beam characteristic functions, (7) and (8)

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Authors and Affiliations

  1. Department of Aeronautical Engineering, Indian Institute of Science, Bangalore, India

    M. S. S. Prabhu & S. Durvasula

Authors
  1. M. S. S. Prabhu
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  2. S. Durvasula
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Prabhu, M.S.S., Durvasula, S. Stability of clamped skew plates. Appl. Sci. Res. 26, 255–271 (1972). https://doi.org/10.1007/BF01897854

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  • DOI: https://doi.org/10.1007/BF01897854

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