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Theory of Plates and Shells, Article 30, Levy’s Solution for Uniform Load
This example is found in the book Theory of Plates and Shells by S. P. Timoshenko & S. Woinowsky-Krieger, published in 1959 by McGraw-Hill. When reading the solution then remember the coordinate system is slightly different from Navier’s solution:
x
y
a
b/2
b/2
x
y
a
b
Coordinate system for Navier’s
solution
Coordinate system for Levy’s solution
Origin Origin
Input values (kN, m)The length of the plate is a in the x-direction and b in the y-direction. The uniformly distributed load has intensity q0:
a = 3;b = 5;q0 = 10;
Plate thickness, Young’s modulus, and Poisson’s ratio:
h = 0.1;Ε = 63 000 000;ν = 0.2;
The resulting “plate stiffness” is:
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 1
$ =Ε h3
12 1 - ν2
5468.75which yields:
DisplacementNumber of terms to include in the series expansions:
numM = 10;
The expression for the displacement is:
αm =m π b
2 a;
factor = 1 -αm Tanh[αm] + 2
2 Cosh[αm]Cosh
2 αm y
b +
αm
2 Cosh[αm]
2 y
bSinh
2 αm y
b;
w =4 q0 a4
π5 $Sum
1
m5factor Sin
m π x
a , {m, 1, (2 numM - 1), 2};
The maximum displacement in mm appears at the middle of the plate:
1000 w /. x →a
2, y → 0 // N
1.28375which yields:
Timoshenko also provides this expression for the maximum displacement, here multiplied by 1000 to obtain an answer in mm:
10004 q0 a4
π5 $Sum
(-1)m-12
m51 -
αm Tanh[αm] + 2
2 Cosh[αm], {m, 1, (2 numM - 1), 2}
1.28375which yields:
The comparable displacement, also in mm, of a simply supported beam of unit width and length the shortest of a and b is:
5 q0 Min[a, b]4
384 Ε h3
12
1000
2.00893which yields:
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 2
Plot of the displacement:
Bending moment about the x-axisMxx = -$ (D[w, {x, 2}] + ν D[w, {y, 2}]);
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 3
The maximum value appears at the middle of the plate:
Mxx /. x →a
2, y → 0
7.81984which yields:
The comparable value for a simply supported beam with that span is:
q0 b2
8// N
31.25which yields:
Bending moment about the y-axisMyy = -$ (D[w, {y, 2}] + ν D[w, {x, 2}]);
The maximum value appears at the middle of the plate:
Myy /. x →a
2, y → 0
3.65772which yields:
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 4
The comparable value for a simply supported beam with that span is:
q0 a2
8// N
11.25which yields:
Twisting moment & Kirchhoff uplift shearMxy = -$ (1 - ν) D[w, x, y];
The uplift force at the corners is twice the twisting moment at those locations:
2 AbsMxy /. x → 0, y →b
2
9.1533which yields:
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 5