Theory of Plates and Shells, Article 41, Plate with edge...

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