Post on 06-Mar-2018
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