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