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}]);
Plot3D[Mxx, {x, 0, a}, {y, 0, b}, AxesLabel → {"x", "y", "Mxx"},PlotRange → All, ViewPoint → {Pi, Pi / 2, 2}]
The maximum value appears at mid-span:
Mxx /. x →a
2, y →
b
2
7.81945which 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 mid-span:
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 4
Myy /. x →a
2, y →
b
2
3.65574which yields:
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 Abs[Mxy /. {x → 0, y → 0}]
9.14041which yields:
Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca
Examples Updated February 9, 2018 Page 5