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The bending of flat plate structures with rectangular symmetry

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Summary

A collocation technique is used in conjunction with complex variable methods and conformal transformation to determine the elastic bending moments and shear forces in a uniformly loaded infinite flat plate structure, supported at each node of a regular rectangular lattice by rigid rectangular columns of finite dimensions.

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Abbreviations

A n :

coefficients in the series solution of the deflection function

a, b :

lengths of slab panel sides

C :

edge of column capital

c 1, c 2 :

column side dimensions

D :

plate rigidity

f 1, f 2 :

functions defining the boundary conditions of the problem

k x , k y , kα :

numerical factors for bending moments

k′ :

value characterizing the aspect ratio of the column sides

k n :

parameters associated with complex potentials

m, n :

coefficients defining the mapping function

M x , M y :

bending moments in x and y directions

M ρ, M α :

radial and tangential bending moments

Q x , Q y :

shear forces

q :

uniformly distributed load acting on plate surface

R :

constant of the mapping function

r, φ :

polar coordinate system

S :

plate region in the (x, y) plane

w :

deflection function in the plate region

α n , β n :

parameters associated with the deflection functions

γ :

unit circle

ζ :

complex mapping plane

ρ, θ :

curvilinear coordinate system

μ :

Poisson's ratio of the slab material

φ(ζ), x (ζ), ψ(ζ), Φ(ζ), Ψ(ζ) :

complex potentials defining the deflection functions

σ :

value of ζ on the unit circle

ω(ζ):

mapping function

References

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    Woinowsky-Krieger, S., J. Appl. Mech. 21 (1954) 263.

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    Girkman, K., Flächentragwerke, p. 284, Springer, Vienna 1963.

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    Tolke, F., Ingenieur Archiv 5 (1934) 187.

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    Savin, G. N., Stress Concentration around Holes, Pergamon Press, Oxford 1961.

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    Muskhelishvili, N. I., Some basic problems of the mathematical theory of elasticity, Noordhoff, Groningen (Holland) 1963.

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    Sokolnikoff, I. S., Mathematical theory of elasticity, McGraw-Hill, New York 1956.

  7. [7]

    Gupta, K. K., Distribution of elastic moments in flat slabs with particular reference to Lift slab structures. Doctoral thesis, London University, London 1965.

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    Gupta, K. K. and R. C. Vaughan, J. Strain Anal. 2 (1967) 109.

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    Timoshenko, S. P. and S. Woinowsky-Krieger, Theory of Plates and Shells, Second Edition, p. 249, McGraw-Hill, New York 1959.

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Gupta, K.K. The bending of flat plate structures with rectangular symmetry. Appl. Sci. Res. 20, 115–130 (1969). https://doi.org/10.1007/BF00382387

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Keywords

  • Shear Force
  • Variable Method
  • Flat Plate
  • Conformal Transformation
  • Finite Dimension