Theory of Plates and Shells, Article 28, Naviers solution...

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