Theory of Plates and Shells, Article 30, Levy’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 Navier’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:

DisplacementNumber of terms to include in the series expansions:

numM = 10;

The expression for the displacement is:

αm =m π b

2 a;

factor = 1 -αm Tanh[αm] + 2

2 Cosh[αm]Cosh

2 αm y

b +

αm

2 Cosh[αm]

2 y

bSinh

2 αm y

b;

w =4 q0 a4

π5 $Sum

1

m5factor Sin

m π x

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

The maximum displacement in mm appears at the middle of the plate:

1000 w /. x →a

2, y → 0 // N

1.28375which yields:

Timoshenko also provides this expression for the maximum displacement, here multiplied by 1000 to obtain an answer in mm:

10004 q0 a4

π5 $Sum

(-1)m-12

m51 -

αm Tanh[αm] + 2

2 Cosh[αm], {m, 1, (2 numM - 1), 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:

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

Examples Updated February 9, 2018 Page 2

Plot of the displacement:

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

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

Examples Updated February 9, 2018 Page 3

The maximum value appears at the middle of the plate:

Mxx /. x →a

2, y → 0

7.81984which 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

3.65772which yields:

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

Examples Updated February 9, 2018 Page 4

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 AbsMxy /. x → 0, y →b

2

9.1533which yields:

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

Examples Updated February 9, 2018 Page 5