Theory of Plates and Shells, Article 28, Navier’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 Levy’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 q 0 : a = 3; b = 5; q0 = 10; Plate thickness, Young’s modulus, and Poisson’s ratio: h = 0.1; Ε= 63000000; 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
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Theory of Plates and Shells, Article 28, Navier’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 Levy’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:
LoadNumber of terms to include in the series expansions:
numM = 10;numN = numM;
Series expansion of the load, summing over odd indices only:
f = SumSum16 q0
π2 m nSin
m π x
a Sin
n π y
b, {m, 1, (2 numM - 1), 2},
{n, 1, (2 numN - 1), 2};
Plot of the load:
DisplacementThe expression for the displacement is:
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 2
w =16 q0
$ π6SumSum
1
m n m2
a2+ n2
b22Sin
m π x
a Sin
n π y
b,
{m, 1, (2 numM - 1), 2}, {n, 1, (2 numN - 1), 2};
The maximum displacement in mm is:
1000 w /. x →a
2, y →
b
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:
Plot of the displacement:
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 3
Bending moment about the x-axisMxx = -$ (D[w, {x, 2}] + ν D[w, {y, 2}]);