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THE LO THEORY FOR SHELLS
Journal of Sound and
LETTERS TO THE EDITOR THE LO THEORY FOR SHELLS H. HASSIS Ecole Nationale d1ingeH nieurs de ¹unis-¸aboratoire ModeH lisation et Calcul de Structures, B.P. 37 ¸e BelveH de` re 1002 ¹unis, ¹unisia (Received 2 June 1999)
1. INTRODUCTION
Lo et al. [1] presented a high order theory for plates. It is appropriated to the following displacement forms: ; (x, x, x)"u (x, x)#xb (x, x)#(x) f (x, x)#(x) (x, x), ; (x, x, x)"u (x, x)#xb (x, x)#(x) f (x, x)#(x) (x, x), ; (x, x, x)"u (x, x)#xb (x, x)#(x) f (x, x),
(1)
where (x, x) are the surface co-ordinates and x is the normal co-ordinate to the surface. The Lo theory, proposed for plates, is here extended to shell structures.
2. DEFORMATION TENSOR}STRESS TENSOR}STRESS RESULTANT
2.1.
DEFORMATION TENSOR
Using the development presented in reference [2], and for small displacement, the deformation tensor associated with equation (1) is written as e "c (u )#xc (b)!xo (u )#(x) c (f)#(x) c ( ), ?@ ?@ S ?@ ?@ S ?@ ?@ 2e "(b #u CH#u )#(x b #(x) f ) ? ? H ? ? ? ? #(2x f #3(x) #(x) f CH#(x) CH), ? ? H ? H ? e "b #2x f ,
(2)
where
u u b f
u " u , u " u , b" b , f" f , "
S S u 0 0 0 0 0022-460X/00/400895#06 $35.00/0
2000 Academic Press
896
LETTERS TO THE EDITOR
with
c
(V)" ( a# b)!< C ?@ ?@ (3) o (V)" ( a CH# b CH )!< CH C , @ ? ? H@ ?@ where C@ is the coe$cients of the curvature tensor and ;a > b is the covariant derivate. ? 2.2.
STRESS RESULTANT
The stress resultants are de"ned by
[N]"
N N M M P P M M MM , [M]" , [P]" , [M M ]" , N N M M P P M M M M
Q"
Q S , S" , Q S
R"
R ¸ , L" R ¸
(4)
with
Q F 1 " [p p p p pp] dx , R x F S F (x ) " [p p p pp] dx . ¸ (x ) F The stress resultants in terms of displacements are given by N N N N Q X M M M M R X P P P S MM M M M M ¸
h N"h[2k c(u )#jTr+c(u ), ) 1]# [2k c(f)#jTr+c(f), ) 1]#jhb 1, S S 12
(5)
(6)
h M" [2k (c(b!o(u ))#jTr (c(b)!o(u )) ) 1] S S 12 h h # [2k(c( ))#jTr (c( ) ) 1]#2j f 1, 80 12
(7)
h h h P" [2k c(u )#jTr+c(u ), ) 1]# [2k c(f)#jTr+c(f), ) 1]#j b 1, S S 80 12 12 (8) h MM " [2k (c(b!o(u ))#jTr (c(b)!o(u )) ) 1] S S 80 h h # [2k(c( ))#jTr (c( )) ) 1]#2j f 1, 448 80 h Q"kh [b#u #C ) u ]#k [ f #3 #C) f], S 12 h h R"k [ b #2f]#k [C ) ], 12 80 S"k
h h [b#u #C ) u ]#k [ f #3 #C ) f], S 12 80
(9) (10) (11) (12)
LETTERS TO THE EDITOR
L"k
h h [b #2f]#k [C ) ], 80 448
897 (13)
h N "h [(2k#j) b #jTr(c(u ))]#j Tr(c (f)), X S 12
(14)
h h M " [2(2k#j)f #jTr(c(b)!o(u ))]#j Tr(c ( )), X 12 S 80
(15)
where 1 is the unit tensor and (k, j) are the LameH coe$cients, and Tr are, respectively, the gradient and the trace operators.
3. EQUATIONS OF MOTION
The principle of the virtual work is used to derive the governing equations of motion. For Shellls subjected to the volume and surface densities f and f , it is found that (for T Q more details see Appendix A): h fG ?, [N?@ > !(MB@ C? )>d!QBC?#F?]"oh uK ?#o B @ @ 12 h h M?@ bG ?#o
G ?, > !Q?#m?"o @ 12 80 h fG , N?@ C !(M?@ CB C )#Q?> #F"oh uK #o ?@ ? B@ ? 12 h h R?> !N #F"o uK #o fG , ? X 12 80 h h P?@ uK ?#o fG ?, > !2R?!SB C?#F? "o @ B 12 80 h S?> !2M #F"o b , ? X 12 h h M M ?@ bG ?#o
G ? > !3S?!¸BC?#F? "o @ B 80 448 or h div N!div (C ) M)!C )Q#F "oh n(uK )#o fG , S S 12
(16.a)
898
LETTERS TO THE EDITOR
oh h div M!Q#m " bG #o
G , S 12 80 h N : C!(C ) M): C#div (Q)#F"oh uK #o fG , 12 h h div R!N #F"o uK #o fG , X 12 80 h h div P!2R!C ) S#F "o n (uK )#o fG , S 12 80 h div S!2M #F"o b , X 12 h h div MM !3S!C ) L#F "o bG #o
G , 80 448
(16.b)
where o is the mass density, h is the thickness of the shell, n(uK ) is the projection on surface S of the surface's acceleration uK , F is the middle surface forces vector, m is the middle S S S surface moments vector. F , F , F and F are de"ned by
F? "
F \F
(x) f ? dx, F? " T
F
\F
(x) f ? dx, F" T
F
\F
x f dx, T
F
(x) f dx. (17) T \F For a shell loaded by a surface density forces applied on *u x ]!h/2, h/2[ (*u is the boundary of the surface u), the boundary conditions are
F"
!N?@ l #(MB@ C?) l #F?"0, @ @ B Q !M?@ l #m?"0, @ Q !Q? l #F"0, ? Q !R? l #(F) "0, ? Q !P?@ l #(F? ) "0, @ Q !S? l #(F) "0, ? Q !M M ?@ l #(F? ) "0 @ Q or !N ) m#C ) M ) m#(F ) "0, SQ !M ) m#(m ) "0, SQ !Q ) m#F"0, 1 !R ) m#(F) "0, Q
(18.a)
899
LETTERS TO THE EDITOR
!P ) m#(F ) "0, Q !S ) m#(F) "0, Q !M M ) m#(F ) "0, Q
(18.b)
where (F ) is the in-surface boundary forces vector and (m ) is the in-surface boundary SQ SQ moments vector. (F ) , (F ) , (F) and (F) are de"ned by Q Q Q Q
F
F
(F? ) " Q (F) " Q
\F
\F
(x) f ? dx, (F? ) " Q Q
x f dx, (F) " Q Q
F
\F
F
\F
(x) f ? dx, Q
(x) f dx. Q
(19)
REFERENCES 1. K. H. LO, R. M. CHRISTENSEN and E. M. WU 1997 Journal of Applied Mechanics 44, 663}676. A high order theory of plate deformation. Part 1: Homogeneous plates. 2. H. HASSIS 1999 Journal of Sound and
APPENDIX A
Using an integration over the normal co-ordinate, the external virtual work is (for the densities f and f ): T Q
=" C
[F?u*#m?b*#F? f*#F? *#Fu*#F b*#F f*] du ? ? ? ?
S
[F? u*#m? b*#(F? ) f*#(F? ) *#F u*#(F) b*#(F) f*] dC, Q ? Q ? Q ? Q ? Q Q Q
#
*
S
where
FG"
F \F
f G dx, FG" T Q
F
\F
f G dx, i"1, 2 or 3. Q
Using an integration over the normal co-ordinate, the inertial virtual work
= "! H
S
h oh (uK ? u*#uK u*)#o (bG ? b*#uK ? f*#fG ? u*#uK f*#bG b* ? ? ? ? 12
#fG u*) du!
o
S
h h G (bG ? *# G ? b*#fG ? f*#fG f* )#o ( ? * ) du. ? ? ? ? 80 448
900
LETTERS TO THE EDITOR
Using an integration over the normal co-ordinate and using the stress resultants de"ned by equation (5), the internal virtual work is
= "! G
[N?@ c (u*)#M?@ c (b*)!M?@ o (u*)#P?@ c (f*) ?@ S ?@ ?@ S ?@
S #M M ?@ c ( *)] du! ?@
[Q? (b*#CH u*#u* ")#(2R? f*#3S? * ? ? H ? ? ? S #S?CH f*#¸?CH *)] du ? H ? H
!
[N b*#2M f*] du! X X
[R? b* #S? f* ] du. ? ?
S S Using relation (2) and after integration by parts, it becomes
=" G
S
[N?@ > !(MB@C? )>d!QB C? ] u* du# @ @ B ?
#
S
#
S
#
S
!
*
S
*
S
!
[P?@ > !2R? !SBC?)] f* du# @ B ?
S
S
[M?@ >b!Q?] b* du ?
[M M ?@ >b!3S?!¸BC?] * du B ?
[N?@ C !(M?@ CB C )#Q?>a ] u* du# ?@ ? B@
[S?> !2M ] f* du# ? X
M?@b* l dC! ? @
Q? l u* dC! ?
*
S
*
S
*
S
S
[R?!N ] b* du ? X
[!N?@l #(MB@C? ) l ] u* dC @ @ B ?
P?@ l f* dC! @ ?
*
S
M M ?@ * l dC ? @
[R?l b*#S? l f*] dC. ? ?
The application of the virtual work leads to equations (16) and (18).
LETTERS TO THE EDITOR THE LO THEORY FOR SHELLS H. HASSIS Ecole Nationale d1ingeH nieurs de ¹unis-¸aboratoire ModeH lisation et Calcul de Structures, B.P. 37 ¸e BelveH de` re 1002 ¹unis, ¹unisia (Received 2 June 1999)
1. INTRODUCTION
Lo et al. [1] presented a high order theory for plates. It is appropriated to the following displacement forms: ; (x, x, x)"u (x, x)#xb (x, x)#(x) f (x, x)#(x) (x, x), ; (x, x, x)"u (x, x)#xb (x, x)#(x) f (x, x)#(x) (x, x), ; (x, x, x)"u (x, x)#xb (x, x)#(x) f (x, x),
(1)
where (x, x) are the surface co-ordinates and x is the normal co-ordinate to the surface. The Lo theory, proposed for plates, is here extended to shell structures.
2. DEFORMATION TENSOR}STRESS TENSOR}STRESS RESULTANT
2.1.
DEFORMATION TENSOR
Using the development presented in reference [2], and for small displacement, the deformation tensor associated with equation (1) is written as e "c (u )#xc (b)!xo (u )#(x) c (f)#(x) c ( ), ?@ ?@ S ?@ ?@ S ?@ ?@ 2e "(b #u CH#u )#(x b #(x) f ) ? ? H ? ? ? ? #(2x f #3(x) #(x) f CH#(x) CH), ? ? H ? H ? e "b #2x f ,
(2)
where
u u b f
u " u , u " u , b" b , f" f , "
S S u 0 0 0 0 0022-460X/00/400895#06 $35.00/0
2000 Academic Press
896
LETTERS TO THE EDITOR
with
c
(V)" ( a# b)!< C ?@ ?@ (3) o (V)" (
STRESS RESULTANT
The stress resultants are de"ned by
[N]"
N N M M P P M M MM , [M]" , [P]" , [M M ]" , N N M M P P M M M M
Q"
Q S , S" , Q S
R"
R ¸ , L" R ¸
(4)
with
Q F 1 " [p p p p pp] dx , R x F S F (x ) " [p p p pp] dx . ¸ (x ) F The stress resultants in terms of displacements are given by N N N N Q X M M M M R X P P P S MM M M M M ¸
h N"h[2k c(u )#jTr+c(u ), ) 1]# [2k c(f)#jTr+c(f), ) 1]#jhb 1, S S 12
(5)
(6)
h M" [2k (c(b!o(u ))#jTr (c(b)!o(u )) ) 1] S S 12 h h # [2k(c( ))#jTr (c( ) ) 1]#2j f 1, 80 12
(7)
h h h P" [2k c(u )#jTr+c(u ), ) 1]# [2k c(f)#jTr+c(f), ) 1]#j b 1, S S 80 12 12 (8) h MM " [2k (c(b!o(u ))#jTr (c(b)!o(u )) ) 1] S S 80 h h # [2k(c( ))#jTr (c( )) ) 1]#2j f 1, 448 80 h Q"kh [b#u #C ) u ]#k [ f #3 #C) f], S 12 h h R"k [ b #2f]#k [C ) ], 12 80 S"k
h h [b#u #C ) u ]#k [ f #3 #C ) f], S 12 80
(9) (10) (11) (12)
LETTERS TO THE EDITOR
L"k
h h [b #2f]#k [C ) ], 80 448
897 (13)
h N "h [(2k#j) b #jTr(c(u ))]#j Tr(c (f)), X S 12
(14)
h h M " [2(2k#j)f #jTr(c(b)!o(u ))]#j Tr(c ( )), X 12 S 80
(15)
where 1 is the unit tensor and (k, j) are the LameH coe$cients, and Tr are, respectively, the gradient and the trace operators.
3. EQUATIONS OF MOTION
The principle of the virtual work is used to derive the governing equations of motion. For Shellls subjected to the volume and surface densities f and f , it is found that (for T Q more details see Appendix A): h fG ?, [N?@ > !(MB@ C? )>d!QBC?#F?]"oh uK ?#o B @ @ 12 h h M?@ bG ?#o
G ?, > !Q?#m?"o @ 12 80 h fG , N?@ C !(M?@ CB C )#Q?> #F"oh uK #o ?@ ? B@ ? 12 h h R?> !N #F"o uK #o fG , ? X 12 80 h h P?@ uK ?#o fG ?, > !2R?!SB C?#F? "o @ B 12 80 h S?> !2M #F"o b , ? X 12 h h M M ?@ bG ?#o
G ? > !3S?!¸BC?#F? "o @ B 80 448 or h div N!div (C ) M)!C )Q#F "oh n(uK )#o fG , S S 12
(16.a)
898
LETTERS TO THE EDITOR
oh h div M!Q#m " bG #o
G , S 12 80 h N : C!(C ) M): C#div (Q)#F"oh uK #o fG , 12 h h div R!N #F"o uK #o fG , X 12 80 h h div P!2R!C ) S#F "o n (uK )#o fG , S 12 80 h div S!2M #F"o b , X 12 h h div MM !3S!C ) L#F "o bG #o
G , 80 448
(16.b)
where o is the mass density, h is the thickness of the shell, n(uK ) is the projection on surface S of the surface's acceleration uK , F is the middle surface forces vector, m is the middle S S S surface moments vector. F , F , F and F are de"ned by
F? "
F \F
(x) f ? dx, F? " T
F
\F
(x) f ? dx, F" T
F
\F
x f dx, T
F
(x) f dx. (17) T \F For a shell loaded by a surface density forces applied on *u x ]!h/2, h/2[ (*u is the boundary of the surface u), the boundary conditions are
F"
!N?@ l #(MB@ C?) l #F?"0, @ @ B Q !M?@ l #m?"0, @ Q !Q? l #F"0, ? Q !R? l #(F) "0, ? Q !P?@ l #(F? ) "0, @ Q !S? l #(F) "0, ? Q !M M ?@ l #(F? ) "0 @ Q or !N ) m#C ) M ) m#(F ) "0, SQ !M ) m#(m ) "0, SQ !Q ) m#F"0, 1 !R ) m#(F) "0, Q
(18.a)
899
LETTERS TO THE EDITOR
!P ) m#(F ) "0, Q !S ) m#(F) "0, Q !M M ) m#(F ) "0, Q
(18.b)
where (F ) is the in-surface boundary forces vector and (m ) is the in-surface boundary SQ SQ moments vector. (F ) , (F ) , (F) and (F) are de"ned by Q Q Q Q
F
F
(F? ) " Q (F) " Q
\F
\F
(x) f ? dx, (F? ) " Q Q
x f dx, (F) " Q Q
F
\F
F
\F
(x) f ? dx, Q
(x) f dx. Q
(19)
REFERENCES 1. K. H. LO, R. M. CHRISTENSEN and E. M. WU 1997 Journal of Applied Mechanics 44, 663}676. A high order theory of plate deformation. Part 1: Homogeneous plates. 2. H. HASSIS 1999 Journal of Sound and
APPENDIX A
Using an integration over the normal co-ordinate, the external virtual work is (for the densities f and f ): T Q
=" C
[F?u*#m?b*#F? f*#F? *#Fu*#F b*#F f*] du ? ? ? ?
S
[F? u*#m? b*#(F? ) f*#(F? ) *#F u*#(F) b*#(F) f*] dC, Q ? Q ? Q ? Q ? Q Q Q
#
*
S
where
FG"
F \F
f G dx, FG" T Q
F
\F
f G dx, i"1, 2 or 3. Q
Using an integration over the normal co-ordinate, the inertial virtual work
= "! H
S
h oh (uK ? u*#uK u*)#o (bG ? b*#uK ? f*#fG ? u*#uK f*#bG b* ? ? ? ? 12
#fG u*) du!
o
S
h h G (bG ? *# G ? b*#fG ? f*#fG f* )#o ( ? * ) du. ? ? ? ? 80 448
900
LETTERS TO THE EDITOR
Using an integration over the normal co-ordinate and using the stress resultants de"ned by equation (5), the internal virtual work is
= "! G
[N?@ c (u*)#M?@ c (b*)!M?@ o (u*)#P?@ c (f*) ?@ S ?@ ?@ S ?@
S #M M ?@ c ( *)] du! ?@
[Q? (b*#CH u*#u* ")#(2R? f*#3S? * ? ? H ? ? ? S #S?CH f*#¸?CH *)] du ? H ? H
!
[N b*#2M f*] du! X X
[R? b* #S? f* ] du. ? ?
S S Using relation (2) and after integration by parts, it becomes
=" G
S
[N?@ > !(MB@C? )>d!QB C? ] u* du# @ @ B ?
#
S
#
S
#
S
!
*
S
*
S
!
[P?@ > !2R? !SBC?)] f* du# @ B ?
S
S
[M?@ >b!Q?] b* du ?
[M M ?@ >b!3S?!¸BC?] * du B ?
[N?@ C !(M?@ CB C )#Q?>a ] u* du# ?@ ? B@
[S?> !2M ] f* du# ? X
M?@b* l dC! ? @
Q? l u* dC! ?
*
S
*
S
*
S
S
[R?!N ] b* du ? X
[!N?@l #(MB@C? ) l ] u* dC @ @ B ?
P?@ l f* dC! @ ?
*
S
M M ?@ * l dC ? @
[R?l b*#S? l f*] dC. ? ?
The application of the virtual work leads to equations (16) and (18).