Theory of Plates and Shells, Article 41, Plate with Edge Moments

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. The coordinate system from Levy’s solution used:

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;M0 = 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;

γm = αm Tanh[αm] Coshm π y

a -

m π y

aSinh

m π y

a;

w =2 M0 a2

π3 $Sum

1

m3 Cosh[αm]γm Sin

m π x

a , {m, 1, (2 numM - 1), 2};

The displacement at the middle of the plate is, in mm:

1000 w /. x →a

2, y → 0 // N

0.398809which yields:

Plot of the displacement:

Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca

Examples Updated February 9, 2018 Page 2

Bending moment about the x-axisMxx = -$ (D[w, {x, 2}] + ν D[w, {y, 2}]);

The maximum value appears at the middle of the plate:

Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca

Examples Updated February 9, 2018 Page 3

Mxx /. x →a

2, y → 0

2.27313which 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

-0.0597135which yields:

The comparable value for a simply supported beam with that span is:

Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca

Examples Updated February 9, 2018 Page 4

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

22.183which yields:

Professor Terje Haukaas The University of British Columbia, Vancouver terje.civil.ubc.ca

Examples Updated February 9, 2018 Page 5