Theory of Plates and Shells, Article 29, Naviers solution...

Theory of Plates and Shells, Article 29, Navier’s Solution for Point 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. Another example posted on this webpage employs Navier’s solution to a plate with uniformly distributed load. The same solution approach is here used for a point load.

Input values (kN, m)Dimensions of the plate:

a = 3;b = 5;

Position of the point load relative to the origin of the x-y coordinate system:

ξ =a

2;

η =b

2;

Load value:

P = 15;

Plate thickness, Young’s modulus, and Poisson’s ratio:

h = 0.1;Ε = 63 000 000;ν = 0.2;

The resulting “plate stiffness” is:

& =Ε h3

12 1 - ν2

5468.75which yields:

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

Examples Updated February 9, 2018 Page 1

LoadNumber of terms to include in the series expansions:

numM = 20;numN = numM;

Series expansion of the load, summing over odd indices only:

qmn =4 P

a bSin

m π ξ

a Sin

n π η

b;

f = m=1

numMn=1

numNqmn Sin

m π x

a Sin

n π y

b;

Plot of the load:

DisplacementThe expression for the displacement is:

w =4 P

& π4 a bm=1

numMn=1

numN Sin m π ξa

Sin n π ηb

m2

a2+ n2

b22

Sinm π x

a Sin

n π y

b;

The displacement in the middle of the eplate is, in mm:

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

Examples Updated February 9, 2018 Page 2

1000 w /. x →a

2, y →

b

2

0.391732which yields:

The displacement of a simply supported beam of unit width and length the shortest of a and b, with a point load at midspan, is in mm:

P Min[a, b]3

48 Ε h3

12

1000

1.60714which yields:

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 →

b

2

5.34316which yields:

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

P b

4// N

18.75which yields:

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

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

Examples Updated February 9, 2018 Page 4

The maximum value appears at the middle of the plate:

Myy /. x →a

2, y →

b

2

4.30078which yields:

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

P a

4// N

11.25which yields:

Twisting moment & Kirchhoff uplift shearMxy = -& (1 - ν) D[w, x, y];

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

Examples Updated February 9, 2018 Page 5

The uplift force at the corners is twice the twisting moment at those locations:

2 Abs[Mxy /. {x → 0, y → 0}]

1.42935which yields:

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

Examples Updated February 9, 2018 Page 6