Appendix A

THE GREEN'S IDENTITIES

A.l The two-dimensional Green's identity

Given two continuous and twice differentiable functions P(x,y) and

Q(x,y) defined in terms of a cartesian coordinates sytem over

domain Q and its boundary r, with continuous first and second

derivatives, the two-dimensional Green's theorem can be written as

follows:

Ip(X,Y)dX

r

+ Q(x,y)dy I (_a_Q_:_: '-y-)-

Q

ap(x,y) ) --dQ

ay

A.2 The Generalised Rayleigh-Green identity for plates

Given a plate with domain Q and boundary r defined in terms of a

cartesian coordinates system, the strain energy density can be

written as follows:

E(w,u)= D a2w a2u

2----+ axay axay

where w is the plate deflection, u a weighting function, D the

plate rigidity and ~ the Poisson ratio of the plates's material.

Integrating the above expression by parts twice, using equations

that express the equivalent shear Vn , bending moment Mn and

twisting moment M in terms of w or u, and accounting for corner ns points of the plate we obtain:

186

N

J uV4w d!i2 - J (UVn(W) a ] c

[Mns(W)

+

E(w,u)= D - a: ttn(w) dr - LU !i2 !i2 i=l

N

J wv4u J (wVn(U) a ] c

(Mns(U)

+

E(u,w)= D d!i2 - - a: ttn(u) dr - LW

!i2 !i2 i=1

where N is the number of corners, n the direction normal to c boundary and s the boundary tangential direction. The term

+

(M is the corner jump of the twisting moment. Substracting the ~ ns

the last two equations from each other, we obtain the generalised

Rayleigh-Green identity for plates:

o J [uv"v - vv"u ]dO = L h (v) - :: Hn (v) - vVn (u) + :: Hn (u) ]dr !i2

Appendix H

FUNCTIONS OF THE FUNDAMENTAL SOLUTIONS

In what follows:

r is the magnitude of vector PO where P is the source point and 0

the field point.

a is the counter-clockwise angle from vector PO to the normal to

the boundary at P.

e is the counter-clockwise angle from the x-axis to PO.

~ is the counter-clockwise angle from vector PO to the normal to

the boundary at O.

~ is the counter clockwise angle from the x-axis to the normal to

the boundary at O.

(u2 is only used when source point P is on the boundary).

H.1 Fundamental Solution u1

r2 ln r

8nD

aU1 r --- (2ln r + 1) cos~ an 8nD

aU1 r --= (21n r + 1) sin~

as 8nD

188

1 1-v Vn (u1 )= - [2+ (l-v)cos2~1 cos~ + -- cos2~

4nr 4nr

l+v I-v Mn (u1 )= - (In r + 1) - -- cos2/3

4n 8n

I-v M (u1 )= sin2/3 ns

an

iU1

1 (1 + In r)

2nD

an 2nDr

B.2 Fundamental Solution u2

--= an

r - -- (21n r + l)cos 0:

8nD

1 1 - -- (21n r + 1)cos(j3-0:) - -- cosl3 coso:

8nD 4nD

1

as anD (2ln r + l)sin(j3-o:) + -- coso: sinl3

4nD

1 1-v I-v V n (u2 )=---2 [2+( 1-v)cos2131 cos (13+0:)- -----zsin2/3sino:+--2sino:cos/3sin213

4~ 2~ 2~

l+v I-v Mn (u2)= -- coso: + -- sin213 sino:

4nr 4nr

1-'\) Mns (u2)= -- sinet cos2/3

4nr

v'lu2 COSet

2nDr

av'lu2 cos( /3+et)

an 2nD/

B.3 Fundamental Solution Uxx

au 1

[cos xx /3 - sin

an 4nDr

v'lu cos 29

xx 2nDr2

av'luxx cos (29-/3)

an nDr3

B.4 Fundamental Solution u xy

sin 29

anD

sin /3 cos 29

4nDr

189

2 cos 9

4nD

/3 sin 29)

190

v'luxy sin 28

2nDr2

av'lu x;l

sin (2EHn

an nDr3

8.5 Fundamental Solution uyy

1 ( ) sin2e 8 nD 2ln r + 1 + -4-nD--

1 (cos 13 + sin 13 sin 29) 4nDr

cos 28

2nDr2

cos (29-13)

nDr3

Appendix C

TRIGONOMETRIC DEFLECTION FUNCTION5

In what follows:

s is the coordinate axis (either x or y)

L is the dimension of plate in direction s (either a or b)

k=2m-l

m-l z=(-l) (z=-l for even m, z=1 for odd m)

1. 55-55 :

Boundary Conditions: f(O)=O.,f"(O)=O.,f(L)=O.,f"(L):O.

mns f(s)= sin (m=1,2,3, ... )

L

2. 55-CL :

Boundary Conditions: f(O)=O.,f"(O)=O.,f(L)=O.,f'(L)=O.

ns mns f(s)= cos -- sin (m=1,2,3, ... )

2L L

3. 55-FR :

Boundary Conditions: f(O)=O.,f"(O)=O.,f"(L)=O.,f"'(L)=O.

2 [kns kns kns f(s)=- 3 + cos-- + O.5z sin-- - 4 cos-- + 4z sin

17 L L 2L ~]

2L

(m=1,2,3, •.• ) (k=1,3,5, •.• )

192

4. CL-SS :

Boundary Conditions: f(O)=O.,f'(O)=O.,f(L)=O.,f"(L)=O.

ns mns f(s)= sin -- sin

L L (m=1,2,3, ... )

5. CL-CL :

Boundary Conditions: f(O)=O.,f'(O)=O.,f(L)=O.,f'(L)=O.

f(s)= 1

2

6. CL-FR :

(m-1)ns

L

(m+1)ns 1 -cos-

L (m=1,2,3, ... )

Boundary Conditions: f(O)=O.,f'(O)=O.,f"(L)=O.,f"'(L)=O.

2 [17 1 kns z kns kns kns 1 f(s)=- - - -cos-- + --sin-- - 4cos-- - z sin--

17 4 4 L 2 L 2L 2L

(m=1,2,3, ... ) (k=1,2,3, ... )

7. FR-SS :

Boundary Conditions: f"(O)=O.,f"'(O)=O.,f(L)=O.,f"(L)=O.

f(S)=-=-[3 _ cos kns + ~in kns + 4cos 17 L 2 L

kns

2L kns 1 4zsin --2L

(m=1,2,3, ... ) (k=1,3,S, ..• )

193

8. FR-CL :

Boundary Conditions: f"(O)=O.,f"'(O)=O.,f(L)=O.,f'(L)=O.

1 f(s)=

2

9. FR-FR :

[ 1 _ z (13 _ cos 14

kns

L

_kns ] ) sin

2L (m=1,2,3, ... ) (k=1, 3,5, ... )

Boundary Conditions: f"(O)=O.,f"'(O)=O.,f"(L)=O.,f"'(L)=O.

For m=O

f(s)=1.

For m=1,3,5, ... i.e. k=1,5,9, ...

1 [1 kns kns kns 1 kns knS] f(s)=- -(2cos- - sin-) + sin- - -(4 + cos-)cos-

3 8 L L 2L 9 L 2L

For m=2,4,6, ... i.e. k=3,7,11, ...

1 [1 kns kns kns 1 kns knS] f(s)=- -(2cos- - sin-) + sin- - -(9 - cos-)cos-

3 8 L L 2L 4 L 2L

Alfutov,

Solving

N.A.

Plate

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SUBJECT INDEX

Automatic discretization 47-48,72-73,76,86, 116,180,182

Bifurcation 3,9-11,13,26,118,133-134,139-141 Buckling load 1,3,5,12-14,23-24,26,35,54, 59,73,77,81,83-84,151,171-172,179- 181

Cell boundary nodes 63,72 Circular plates 24,47,72,75,81-82,84-85 Concentrated forces 47,50-51,55,182 Corners 9,42,56,59,60-61,83-84,145,162,181

Dirac delta function 40 Dirichlet conditions 94 Discontinuous elements 43-44,59-60,62-63, 66-67,70,74,76-83,156,162,181

Eccentric loads 36,172,175-177 Effective width 2,26,36 Elastic foundations 7

Field point 39-40,55,112 Finite difference method 6,35-36,158 Finite element method 6,11,35-36,65,79,86-88,171,174,180,813 Flexural rigidity 2,4,22,42,55,153 Free boundary 6,9,29-31,36,50-51,55,67-70,73-75,84-85,87-88,99-104,114,117,122,129-132,136-140,142,145,149,152,164,172,176-177,181-182 Fundamental solution 6-8,38-42,52,55-56,59, 89,116,129,133,151,154,162-163,182- 184

Green's theorems 40-41,58 Grillage analogy method 100

Hook's law 18-19

In-plane boundary conditions 9,26,31-36, 38,42,153,171-172,177,184

Mesh nodes 35-36,44,48,59,63,72

Navier's equation 2-4,23,54

Parallelogram plates 83-85 Plate bending 1,2,4,7-8,17,22,24,31,55,151 Post-buckling 1,3,10-11 ,13-14,26,35, 151, 172,178

Rayleigh method 5 Rayleigh-Green equation 9-10,38,40-41, 56,90,92,109,112,124,179-180

Secondary buckling 36,172,175,178 Second-order terms 10,152,167 Shear deformations 4,16 Snap load 13-14 Source point 39-43,45-46,55,67-70,73,84, 104,107,109,112,114,117,125,149,162- 163 Stress function 2,9,21,25,31-35,38-39,48, 52,152-153,179-180 Stretching strains 4,9,16-17,22,24,26

Top-of-the-knee approach 26 Traction boundary conditions 9,31-35,46,153 Triangular plates 74-75,81-83,85,144-145

Von Karman's equations 2-5,9-11,25-26,38,152

SYMBOLS

a

h

n

s

s1' s2' s3

u1,u2'Uxx'Uxy'Uyy

ux,uy,uz

uxs,uys

uxb,uyb

w

aw

as

LIST OF SYHBOLS

depth of plate

width of plate

width of load strip

discontinuity jump of a fUnction f at corners

thickness of plate

direction normal to the boundary

incremental membrane stress resultants

distance between source and field points

direction tangential to the boundary

local coordinates of a domain cell

fundamental solutions

displacements in the x, y and z directions

stretching components of Ux and uy

bending components of Ux and uy

Uz displacement of the mid-plane of the plate

initial w pr~or to loading

gaussian weight

total w

derivative of w with respect to a then ~

normal slope of deflection

tangential slope of deflection

x,y,z

BEM

D

E

F

FO' FI FI n' nm

FEM

IJ I

KO' KI KI n' nm

M xx,Mxy,Myy

M n

M ns

N c

Nbe

Nbu

Nde

Ndn

Nd , Ndu

Neb

Nfl' Nf2

Nfe

N g

Ngb

Ngd

N ne

Nz

203

cartesian coordinates system

Boundary Element Method

plate flexural rigidity

Young's modulus

Airy's stress function

Fourier trigonometric functions

Finite Element Method

jacobian

coefficients of Fourier transforms

bending moments

normal bending moment

transverse bending moment

number of corners

number of boundary elements

number of boundary unknowns

number of domain cells

number of domain nodes

number of domain unknowns

number of boundary equations for critical loads

order of Fourier analysis in the x and y directions respectively

number of elements on free boundaries

number of Gaussian points

number of Gaussian points for boundary integrals

number of Gaussian points for domain integrals

number of nodes per element

number of fictitious domain nodes

P

Q

Ot

t. 1

"

C1XX ' C1yy ' C1ZZ

'xy' 'XZ ' 'YZ

r

INTEGRALS

204

source point

field point

shear force

radius of circular plate

normal and tangential boundary tractions

equivalent shear

angle between vector r and normal n to boundary

initial rotation of a rigid bar

direct strains in the x, y and Z directions

corner angle

ith interpolation function

shear strains

load factor

critical load factor

Poisson ratio

domain of plate

extended Fourier domain

critical stress resultant

direct stresses in the x, y and z directions

shear stresses

boundary of plate

Poisson operator

biharmonic operator

boundary integral of equation for critical loads

boundary integral of bending equation for large deflections

I~ J

ARRAYS

{b f }

{bw}

lew]

{Fw}

{n}

{Nail}

{T f }

{Ub}

{Ud}

{w}

{w Q} ,a ....

205

boundary integral of stretching equation for large deflections

domain integral of equation for critical loads, with curvatures terms

domain integral of bending equation for large deflections

domain integral of stretching equation for large deflections

domain integral of equation for critical loads, with deflections

traction boundary integral of equation for cri tical loads

traction boundary integral of bending equation for large deflections

corner jump term of transverse moment in equation for critical loads

corner jump term of transverse moment in bending equation for large deflections

vector of stress function unknowns

vector of bending boundary unknowns

matrix relating curvatures to deflections

vector relating deflections to domain unknowns

vector of incremental membrane stresses

vector of membrane stresses

vector containing Fourier functions

vector of boundary unknowns in dual-reciprocity

vector of domain unknowns in dual-reciprocity

vector of domain deflections

vector of curvatures

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