Theory of Plates and Shells, Article 30, Levys solution for uniform...

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