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Plates on elastic foundationCircular elastic plate, axial-symmetric load, Winkler soil

(after Timoshenko & Woinowsky-Krieger (1959) - Chapter 8)

Prepared by Enzo Martinelli Draft version (3 April 2016)

IntroductionWinkler model: modulus of subgrade

N/mm3

kgf/cm3

0.0271

2.76

0.0407

4.15

0.0543

5.53

0.0679

6.92

0.136

13.83

𝛔𝐬 = 𝐤𝟎𝐰

0

Plates on elastic foundation: Circular elastic plate, axial-symmetric load, Winkler soil (Timoshenko & Woinowsky-Krieger, 1959)

IntroductionCircular elastic (thin) plates

The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments.

The following kinematic assumptions that are made in this theory:− straight lines normal to the mid-surface remain straight after deformation− straight lines normal to the mid-surface remain normal to the mid-surface after

deformation− the thickness of the plate does not change during a deformation.

(after Wikipedia)

Germain-Lagrange equation (cartesian coordinates)

4 4 4

4 2 2 4

w w w q2

x x y y D



IntroductionCircular elastic (thin) plates

Germain-Lagrange equation (cartesian coordinates)

2

qw w

D

Conversion to polar (cylindrical) coordinates

w w 1 wcos sin

x r r

w w 1 wsin cos

y r r

2 2

2 2x y

x r cos

y r sin

x

y

r

2 2

2 2 2

1 1

r r r r



IntroductionCircular elastic (thin) plates under axial-symmetric load

Germain-Lagrange equation (polar coordinates)

2

qw w

D

2 2

2 2 2

1 1

r r r r

2 2 2 2

2 2 2 2 2 2 2

1 1 w 1 w 1 w qw

r r r r r r r r D

2 2

2 2

d 1 d d w 1 dw q

dr r dr dr r dr D



IntroductionCircular elastic (thin) plates on Winkler soil under axial-symmetric load

2 20

2 2

q k wd 1 d d w 1 dw

dr r dr dr r dr D

dy

dx

q

k0w

4 3 20

4 3 3 2 2 3

q k wd w 2 d w 1 d w 1 dw

dr r dr r dr r dr D

Since an uniformly distributed load q applied on an elastic plates(or elastic beams) on Winkler soils with uniform k0 does notdetermine any stress within the plate, the sole effect of a pointload P applied in the centre of the plate is considered hereafter.

P

R

q=0



Approximate solutionConversion to a dimensionless form

2 2

2 2 4

d 1 d d w 1 dw w0

dr r dr dr r dr l

04

k 1

D l

3

4

[F ][L] 1

[F ][L] [L]

- Characteristic length:

wz

l

- Dimensionless quantities:

Dimensionless displacement

rx

lDimensionless radius

dw l dz dz

dr l dx dx

Dimensionless derivatives:

2 2

2 2

d w d dw d dz 1 d z

dr dr dr l dx dx l dx

i i

i i 1 i

d w d dw 1 d w

dr dr dr l dx

2 2

2 2

d 1 d d z 1 dzz 0

dx x dx dx x dx

4 3 2

4 3 3 2 2 3

d z 2 d z 1 d z 1 dzz 0

dx x dx x dx x dx

2 2

3 2 2 3

1 d 1 d d z 1 dz z0

l dx x dx dx x dx l



Approximate solutionGeneral solution

4 3 2

4 3 3 2 2 3

d z 2 d z 1 d z 1 dzz 0

dx x dx x dx x dx

The proposed mathematical transformations led to the following homogeneous 4th-order linear differential equation with variable coefficients:

Therefore, in principle the general solution of this differential equation might be written as follows:

1 1 2 2 3 3 4 4z A X x A X x A X x A X x

where Xi are four independent solutions of the differential equation under consideration and Ai are four integration constants depending on the actual boundary conditions.

Note:

22i i

i2 2

d X dXd 1 d 1X 0

dx x dx dx x dx

i 1..4



Approximate solutionApproximation by power series

T

T

T

NNn 1 2 n

i n 0 1 2 n Nn 0

X a x a a x a x ... a x ... a x

Since Xi are unknown (as the differential equation under consideration includes variable coefficients), an approximation based on power series may be firstly assumed. Hence, the general expression of Xi can be taken as follows:

where NT is the highest monomial in the series.

Since Xi should be a solution of the general differential equation, the 2nd order laplacian of each term anxn should find a similar monomial such that:

2 n n2n n n

n 2 2

d a x da xd 1 d 1a x

dx x dx dx x dx

2 n n

n 2 n 2 2 n 2n nn n n2

d a x da x1n n 1 a x na x n a x

dx x dx

2

22 n 2 2 n 4 2 n 4 2 n 4n n n n2

d 1 dn a x n n 2 n 3 a x n n 2 a x n n 2 a x

dx x dx

22 n 4 n 4

n n 4n n 2 a x a x 0

n 4

n 22

aa

n n 2

Recursive definition of an


0XX ii

Approximate solutionApproximation by power series: first independed solution X1

4 8 121

1 1 1X x 1 x x x ...

64 147456 2123366400

n 4

n 22

aa

n n 2

Based on the recursive definition of the an coefficients, a first independent solution X1 can be “built up” by assuming a0 as the first nonzero coefficient:

0a 1

0

4 2 2 22

a 1a

4 24 4 2

4

8 2 2 2 2 22

a 1a

8 6 4 28 8 2

n 0 :

n 4 :

n 8 :

n 12 :

812 2 2 2 2 2 2 22

a 1a

12 10 8 6 4 212 12 2

1 2 3a a a 0

1 1 2 2 3 3 4 4z A X x A X x A X x A X x



Approximate solutionApproximation by power series: second independed solution X2

2X x ....

n 4

n 22

aa

n n 2

Before proceeding with “constructing” the second independent solution X2 of z, it should be noted that the assumption of a1=1 is not admissible.In fact, if one assumes a1=1 (and, at the same time, a0=a2=a3=0) the resulting solution X2 of z would have a first term x:

which implies a nonzero first derivative of X2 (and, hence, of the general solution z) which cannot be accepted because of the fact that rotations (j=dX2/dx=1+…) should be zero at x=0.

this leads to an expression of X2 whose first nonzero monomial is of 2nd order and, hence, the second derivative of X2 is non zero (c=d2X2/dx2=1+…) ): this is acceptable because the curvature is nonzero at the plate centre (x=0).

22X x ....

Therefore, the second solution X2 should be built by assuming a2=1 and, at the same time, a0=a1=a3=0:



Approximate solutionApproximation by power series: second independed solution X2

2 6 10 142

1 1 1X x x x x x ...

576 3686400 104044953600

n 4

n 22

aa

n n 2

Based on the recursive definition of the an coefficients, a first independent solution X2 can be “built up” by assuming a2 as the first nonzero coefficient:

2a 2

2

6 2 2 22

a 1a

6 46 6 2

6

10 2 2 2 2 22

a 1a

10 8 6 410 10 2

n 2 :

n 6 :

n 10 :

n 14 :

1014 2 2 2 2 2 2 22

a 1a

14 12 10 8 6 414 14 2

0 1 3a a a 0



Approximate solutionApproximation by power series: third independed solution X3

n 4

n 22

aa

n n 2

33X x ....

The assumption a3=1 (and, at the same time, a0=a1=a2=0) is not even admissible, as it would lead to the a function having the following expression:

This is not admissible because it would lead to a finite value of shear for x=0:

23 3

2

d X dXd 1V 2 ....

dx dx x dx

whereas shear should be divergent (it should tend to infinity) for x0, as a finite force P should be in equilibrium with shear stresses on an infinitesimal circumference of radius e.

P

x=eV




Since,on the one hand, shear V for x0,andon the other one hand, both X1 and X2 would lead to a finite value of shear for x=0,the solutions X3 and X4 should be capable to reproduce the condition V for x0.

Therefore, a different solution, including the term log x (which diverges for x0 along with all its derivatives) ad an new unknown function F3, is preliminarily defined:

31

3 1 33

d X4X log x X F

x dx

The function X3 should be, in turn, a solution of the general differential equation:

i iX X 0 22

i ii2 2

d X dXd 1 d 1X 0

dx x dx dx x dx

First of all, the 2nd laplacian of X3 is determined

3 1 3X X log x F




Then:

31

3 3 1 3 1 33

d X4X X log x X F X log x F 0

x dx

3 3X X 0

The following condition has to be met:

and, after collecting the term log x,

3

11 1 3 33

d X4log x X X F F 0

x dx

the following relationship is derived for F3:

31

3 3 3

d X4F F

x dx

4 83 3

2 3 4 6 7 8 10 11 12F F 4 x x ...

64 147456 2123366400




4 83 3 2 2 2 2 2 2 2 2 2 2 2 2

2 3 4 6 7 8 10 11 12F F 4 x x ...

4 2 8 6 4 2 12 10 8 6 4 2

The function F3 may also be approximated by the following power series:

4 8 123 4 8 12F b x b x b x ...

whose 2nd laplacian can be expressed as follows:

2n 2 n 4

n nb x n n 2 b x

T nn1

n 443 3 2 2 2

n 4

n (n 1) (n 2)F F 4 ( 1) x

n (n 2) .. 2

n

122 n 4 n 4 n 44n n 4 2 2 2

n (n 1) (n 2)n n 2 b x b x ( 1) x

n (n 2) .. 2

n

n1

4n n 42 2 2 22

1 n (n 1) (n 2)b b ( 1) 4

n (n 2) .. 2n n 2




n1

4n n 42 2 2 22

1 n (n 1) (n 2)b b ( 1) 4

n (n 2) .. 2n n 2



Approximate solutionApproximation by power series: fourth independed solution X4

A similar procedure leads to defining the fourth independent solution:

4 2 4X X log x F

and, similar considerations, can be done in defining F4 for X4 is a solution of the general differential equation.

6 10 144 6 10 14F c x c x c x ...



Approximate solutionApproximation by power series: fourth independed solution X4



Approximate solutionBoundary conditions

1 1 2 2 3 3 4 4z A X x A X x A X x A X x

The general solution, whose expression was defined by 4th approximate solutions, is reported below:

Four boundary conditions should be written for determining the four constants A1…A4. P

R

r R

2

2

r R

d w dw0

dr r dr

The first two ones are imposed for r=R (namely, for x=R/l):

Dimensional expression Dimensionless expression

2

2

r R

d d w 1 dw0

dr dr r dr

2

2R

xl

d d z 1 dz0

dx dx x dx

2

2R

xl

d z dz0

dx x dx

x R l



Approximate solutionBoundary conditions

1 1 2 2 3 3 4 4z A X x A X x A X x A X x

The general solution, whose expression was defined by 4th approximate solutions, is reported below:

P

R

r 0

r R

dw0

dr

Other two boundary conditions are imposed for r=0 (namely, for x=0):

Dimensional expression Dimensionless expression

2

20r l

d d w 1 dwlim 2 l D P 0

dr dr r dree

e

x 0

dz0

dx

x R l

23

0 20x

d d z 1 dzlim 2 k l P 0

dx dx x dxee

e

P

elV



Approximate solutionBoundary conditions: expressions of constants

Since X1 and X2 are polynomials with even order power terms (i.e. x0, x4, x8 … and x2, x6, x10 …, respectively), their first derivative is zero for x=0.

Moreover, the first derivative of X4 is also zero for x=0 (or, better, in the limit of x→0):

2

42 4

0 0xx 0 x

dX d xlim X log x F lim 2x log x ... ... 0

dx dx xe ee e

Therefore, the first derivative of z in x=0 only takes a nonzero contribution from X3:

33 3 1 3 3

0 0x 0 x xx 0

dXdz d 1A A lim X log x F A lim ... 0

dx dx dx xe ee e

which is only satisfied for A3=0.

x 0

dz0

dx

First of all, let’s consider the third boundary condition:




and, once again, no contribution comes from X1 and X2, but the only nonzero terms is given by the third derivatives of X4:

22 24 4

4 4 2 42 2 2

x xx

2 22

4 2 4 42 2

x x

d X dXd d z 1 dz d 1 d d 1 dA A X log x F

dx dx x dx dx dx x dx dx dx x dx

d d 1 d d d 1 dA X log x F A x log x ...

dx dx x dx dx dx x dx

e ee

e e

4 4

x

2 2 4A A

x x e e

Then, the fourth condition can be considered:

23

0 20x

d d z 1 dzlim 2 k l P 0

dx dx x dxee

e

Therefore:

30 4

0

4lim 2 k l A P 0e

ee

30 48 k l A P 0 4 3

0

PA

8 k l




2

2R

xl

d d z 1 dz0

dx dx x dx

2

2R

xl

d z dz0

dx x dx

Finally, the first two equations can be considered for determining the two constants A1 and A2:

1 1 2 2 4 4z x A X x A X x A X x

r

w r l zl

2

r 2

d w dwM r D

dr r dr

2

2

1 dw d wM r D

r dr dr

2

r 2

d d w 1 dwV r D

dr dr r dr



Approximate solutionComparisons: proposed procedure vs FEM solution



1000 2000 3000 4000 5000

-0.04

-0.02

0.02

0.04

0.06

1000 2000 3000 4000 5000

-0.04

-0.02

0.02

0.04

0.06

NT=4 NT=12

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