Shallow Shell Equations at a Glance1,2

SHALLOW

SHELL

EQUATIONS

AT A GLANCE

1,2

1. Linearly elastic clamped shallow shell 2. Nonlinearly elastic clamped shallow shell 3. Marguerre-von Kgrmgn shallow shell 1.

LINEARLY

ELASTIC

CLAMPED

SHALLOW

SHELL

(Sect. 3.7) The shell is subjected to the boundary conditions g~ - 0 on the portion F0 - O~(% x [-~, c]) of its lateral face.

Minimization problem" ~ E V(a;) and j ~ ( ~ ) -

inf j~(r/), ncv(~)

where

V(w) -

{n-

J~(~7) - -~

(r]i) e H i ( w )

x Hi(w)

x

H2iw);Wi- ~u713 - 0

-~a:zo.Oo.qaO~zrl3 + ea:z~.e:.(rl)e:z(rl)

-

o n 70},

dw

q~0~r/a dw ) ,

1

1All notations and definitions used in this section are recalled in the section "Main notations and definitions" under the heading "Shallow shells". 2The shallow shell equations listed here are expressed in terms of Cartesian coordinates; those in terms of curvilinear coordinates are studied in Vol. III.

lvii

Shallow shell equations at a glance

1viii

Variational problem: ~ - (~:) E V(a~) a n d for all r~ - (~7~) v(~),

rl3dco + L ~cg~O~O~rladw + L n~O~rl~,d~ -

_

C3

a ~

e

Lp~

d~ -

/L

G ~ 3

dw,

n~,-ca~zo,eo.(r

Boundary value problem"

- O ~ m : z - 0 ~ ( ~ 0 ~ 0 ~) - P3 + O~q-~ in w,

-OZ~Z - / ~ ~: = o.r

in co

= 0 on'~o,

m~z~,~,Z - 0 on "71,

(O~m ~ + ~;~0~0~).~ + O.(m~z ~ u ~TZ) -- --q~v~ -~ on ')'1 , --E

n~L,Z - 0 on 71.

2. N O N L I N E A R L Y E L A S T I C C L A M P E D S H A L L O W SHELL (Sect. 4.14) T h e shell is s u b j e c t e d to the b o u n d a r y c o n d i t i o n s ~ - 0 on t h e p o r t i o n O~(-y0 x [ - ~ , e]) of its lateral face.

Minimization problem:

~

C V ( w ) a n d j~((2 ~) -

inf

j~(r/),

lix

Shallow shell equations at a glance

where"

V(w) - { r / - (r/i) e Hi(w) • Hi(w) • H2(w); r / i - 0.~3 - 0 on 70}.

j~(,7)

- .

~l J i { e 3- ~ a ~ o ~ . v ~ o ~ ~

+ ~ a ~ $ o : ( , 7 ) E~Z -o,~( r t )}d w ~

Pi rli d w

-

-

q~

Oa?73 d w

,

- o : (~) -- ~(0af]/3 1 Ec~/3 -~- 0/3T]c~ -~- Oqc~0 ~0/3?']3 -Jr- 0~0~0,~ ?']3 + O~r/aO~r/a)

Variational problem: (~ -

(~)

e V(w)

and for all r/ -

(r/i) e

V(~),

L

9~ o ~

L

d~ +

TI~, ~

Nt~O~(r + 0 ~ ) 0 ~ d~ +

L

N.~O~,7~d~

- Ji p:rl, dw - Ji q-:O~3 dw. s

N~ - ~a~~ -

e

e

K~O,e r

( ).

Boundary value problem: -O~zrn,~z - Oz{N~zO~(~ 3 + 0~)} - P3 + O~q~ in w, -OzN~z - p~ in w,

r

-

0.r

-

0 on

70,

ma~euauO -- 0 on 71, --

s

s

--~

Naeflyfl -- 0 on 71-

lx

Shallow shell equations at a glance

3. M A R G U E R R E - V O N 5.12)

K/i~RMAN SHALLOW

S H E L L (Sect.

The shell is subjected to the boundary conditions: / u ~ i ; d e p e n d e n t ~ 1 ^7 6 1

^

{ (~;~ + ~ 0 k / t ; )

o O ~} u~ dx;

h; o O ~ on "7.

The functions h~ - h~ o O ~ 97 ~ R satisfy the compatibility conditions

h i d ' y - ~ h~ d'y - ~ ( X l h : - x2h~l) d'y - O, and the "horizontal" components j~ 9~ - - IR and g~ ^~ 9F+U ^~ F~ - , R of the applied body and surface forces vanish. ~" vertical component of the displacement of the middle surface. r Airy stress function. The unknowns ~ and r satisfy the Marguerre yon Kdrrndn equa_

tions: 8" (A +

3(A~ + 2#~)

- 5[r

~a + 0~] + i0~ in w,

A2r ~ = -#~(3A~ + 2#~)[~, ~ + 20 ~] in w,

r - r --

C

and/).r

--

- r

g

on "~',

where [~, ~D] -- 011~(022~/2 -~- (~22~011~D-

r

(y)

h~d"/+y2f,

Jr (y)

(y)

2012X012r h~d'y+f, Jr(~)

(y)

(Xlh~-x2h~l)d"y, yE'y, 7.

Shallow shell equations at a glance

T h e f u n c t i o n s N ~Z ~ -

~ (~) ~~-~ % z o . E o~

are then given by

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