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