introduction to the theory of shellsmechanics.tamu.edu/.../03/lecture-09-linear_shells.pdf ·...
Post on 23-Jan-2021
1 Views
Preview:
TRANSCRIPT
JN Reddy
1
INTRODUCTION TO THE THEORY OF SHELLS
• Geometry of shells• Kinematics of shells• Displacements and strains• Strain-displacement equations• Stress resultants• Equations of motion• Shell constitutive relations• Specialization to cylindrical shells• Examples
CONTENTS
JN Reddy
GEOMETRY OF SHELLS
h
ζ
1ξ2ξMiddle surface
R2R1
1ξ
2ξ
= constant curve( curve)
1ξ
= constant curve( curve)
2ξ
2dξ
1dξ2g
1g
n
1x2x
3x
1i
rd+r r
dr
ds
Middle surface
2ξ
1ξ
2i
3i
JN Reddy
GEOMETRY OF SHELLS
1 11 2 22 1 2 1 2
1 2, ( , )
, , , cos
d d d
g a g a g a a
r rr g g
g g g g
R2
n
••
rRh
ζ2g
1x2x
3x
3i
2ξ
1i2i
JN Reddy Shells: 4
KINEMATICS OF SHELLS
1 2
1 2
2
2 2 2 21 1 2 2 1 2 1 22
ˆsin
( )
( ) ( ) cos
a a
ds d d g d d
a d a d a a d d
g g
r r
n
1
ˆ
ˆ
ˆ (no sum on )
ii
d d d d d d
R
R r
R R RR G
R rG g
n
n
n
R2
n
••
rRh
ζ2g
1x2x
3x
3i
2ξ
1i2i
JN Reddy Shells: 5
1 11 2 22 3 1
1
, , ,
(no sum on )
G A G A G A
A aR
G G
2 2 2 2 2 2 2 21 1 2 2 3
2 1 1 2
1 1 1 2 2 2 1 2
2 21 1 2 2 2 1
2 2 2 1 1 1
1 1 2
2 2 2 2 1 1 1
1 1
1 1
1 1 1 1
( ) ( ) ( ) ( ) ( )
,
,
dS d d G d d d A d A d A d
A A a aA A R R
A A A A A AA A
A a AA a A a
R R
2
1
a
KINEMATICS OF SHELLS
JN Reddy
DISPLACEMENT FIELD AND STRAINS
1 1 1 1 11
2 2 2 2 22
1
1
dS d A d d a d dR
dS d A d d a d dR
1 1 1 1 11
1 ,dS a d A dR
2 2 2 2 2
2
1dS a d A dR
ζ
12σ 21σ
13 5σ σ=23 4σ σ=
22 2σ σ=11 1σ σ=
1dξ2dξ
1dS2dS
1ξ 2ξ
JN Reddy Shells: 7
AREAS AND VOLUME OF A SHELL ELEMENT
0 1 2 1 2 1 2 1 21 2
1 2 1 2 1 2 1 21 2
dA d d d d a a d d
dA d d d d A A d d
r rr r n n
R RR R n n
1 2 1 2 1 2
1 2 1 21 2
1 1
dV d d d dA d A A d d d
a a d d dR R
R R n
Middle surface
dr1 dr2n
dA0
x1
x2
x3r2r1
rg2g1
ζ
1ξ 2ξ
0 1 2 1 2dA a a d d 1e 3e
2e
11N
1Q
12N21N
2Q
22N
12M11M
21M22M
11 12 21 22 11 12 21 22
1 2
Membrane forces Flexural forces, , , , , ,
,N N N N M M M M
Q Q
ξ1 ξ2
ζ Surface at +ζ
dR1 dR2n
dAζ
x1
x2
x3 R2
R1R G2G1
ζ
1 2 1 2dA A A d d
1e3e
2e
JN Reddy Shells: 8
STRAIN-DISPLACEMENT RELATIONS2
3 3
1 1
23
21
1 1 12
1 1 2 32 ,
( , , )
i k i i k ii
k ki i i k k i i i k k
k i i
k k ii i k k
j ji iij
j j i i i j
u u A u u AA A A A A A
u u A iA A
A uA uA A A A
3
3
1
1
1 1
1
,
j jk i i k
k i k j i j i k k j k k
j ji i k i
kj j i i i i i k k
j ji i
i i j j j
u Au u A uA A A A
u Au u u AA A A A A
u uu AA A
3
1
3 1 1 2 2 3 31 2
1
1 1 1, , ,
jk
kj j k k
AuA A A
A a A a A aR R
JN Reddy Shells: 9
2
1 2 1 1 1 2 1 11 11 3 32
1 1 2 2 1 1 1 2 2 1
2 2
32 1 1 11
1 2 2 1 1
2 1 2 22 22 3
2 2 1 1 2
1 12
1
u u a a u u a au uA a R A a R
uu u a a ua R
u u a a uA a R
2
2 1 2 232
2 2 1 1 2
2 2
31 2 2 22
2 1 1 2 2
2 2 2
3 31 23 33
12
12
u u a a uA a R
uu u a a ua R
u uu u
STRAIN-DISPLACEMENT RELATIONS(simplified relations)
JN Reddy
STRAIN-DISPLACEMENT RELATIONS(simplified relations - continued)
3 2 2 2 1 2 24 23 2 3
2 2 2 2 2 1 1 2
3 31 1 2 2 22
2 1 1 2 2
3 15 13 1
1 1 1
1 12
1 12
u u u u u a aA uA A A a R
u uu u u a a ua R
u uAA A A
1 1 2 1 1
31 1 2 2 1
3 32 2 1 1 11
1 2 2 1 1
2 2 1 1 1 2 26 12
1 1 2 2 2 1 1 2 2 1 1
12
u u u a a ua R
u uu u u a a ua R
A u A u u u aA A A A A A a
1 1 1
2 31 2 2 1
3 32 1 1 2 2 2 1 21 3 1 2
1 2 2 2 1 1 2 1 1 2 2
1
1
u a au ua R
u uu u a u a a a au u u ua a R R R
JN Reddy Shells: 11
/2 /211 2 2 11 2 11 2 2/2 /2
2
/2 /211 2 2 11 2 11 2 2/2 /2
2
11 11/2
12 12 21/22
1
2
21
2
3
1
1
1 ,
h h
h h
h h
h h
h
h
s
dS d a d d N a dR
dS d a d d M a dR
N NN d N
RQ K Q
− −
− −
−
= + ≡
= + ≡
= +
∫ ∫
∫ ∫
∫/2
21/2
23
/2 /211 11 22/2 /2
12 12
22
1
2
21 2 12
2
1
1
1 , 1
h
h
s
h h
h h
dR
K
M MM
d dR M R
−
− −
= +
= + = +
∫
∫ ∫
STRESS RESULTANTS
JN Reddy
ASSUMPTIONS 1. The transverse normal is inextensible (i.e., ) and the
transverse normal stress is small compared with the other normal stress components and may be neglected.
2. Normals to the undeformed middle surface of the shell before deformation remain straight, but not necessarily normal after deformation.
3. The deflections and strains are sufficiently small so that the quantities of second- and higher-order magnitude, except for second-order rotations about the transverse normals, may be neglected in comparison with the first-order terms.
4. The rotations about the and axes are moderate so that we retain second-order terms (i.e., terms that are products and squares of the terms ) in the strain-
displacement relations (the von K'arm'an nonlinearity).
3 0
1 2
3u a uR
JN Reddy 2-D Problems: 13
DISPLACEMENTS AND STRAINS0
1 1 2 1 1 2 1 1 2
02 1 2 2 1 2 2 1 2
03 1 2 3 1 2
( , , , ) ( , , ) ( , , )( , , , ) ( , , ) ( , , )( , , , ) ( , , )
u t u t tu t u t tu t u t
20 00 0 0
1 3 31 2 1 1 1 1 2 11
1 1 2 2 1 1 1 1 1 2 2
20 00 0 02 3 32 1 2 2 2 2 1 2
22 2 1 1 2 2 2 2 2 1 1
1 12
1 12
a u uu u a a u aA a R A R a
a u uu u a a u aA a R A R a
0 00 03 32 2 1 1
3 4 2 5 12 2 2 2 1 1 1 1
0 00 00 03 32 2 1 1 1 2
6 1 21 1 2 2 2 1 1 2 1 1 2 2
1 10
1
, ,u ua u a uA a R A a R
u uA u A u a au uA A A A A A R R
2 2 1 1
1 1 2 2 2 1
A AA A A A
JN Reddy Shells: 14
EQUATIONS OF MOTION(obtained through the use of virtual work principle)
1 2 12 11 1 21 12 22 1
2 1 1 2 2 1 1
0 00 0 2 0 23 311 1 1 12 2 2 1 1
0 12 21 1 1 1 2 1 2 2
21 22 2 12 21
1 2 2 1 1
1
1
ˆ
( ) ( )
( ) ( )
a a Qa N a N N N fa a R
u uN a u N a u uI Ia R R a R R t t
aa N a N Na a
1 211 2
2 2
0 00 0 2 0 23 322 2 2 12 1 1 2 2
0 12 22 2 2 2 1 2 1 1
0 00 03 32 1 1 2 2
11 121 2 1 1 1 1 2 2
122
2 2
1
ˆ
ˆ
ˆ
( ) ( )
a QN fR
u uN a u N a u uI Ia R R a R R t t
u ua a u a uN Na a a R R
a Na
0 00 03 32 2 1 1
12 2 1 1 22 2 1 1 1 2
2 0311 22
3 0 21 2
( ) ( )u ua u a uN a Q a QR R
uN N f IR R t
JN Reddy 2-D Problems: 15
1 22 11 1 21 12 22 1
1 2 1 2 2 1
2 0 21 1
1 22 2
2 12 12 1 22 21 11 2
2 1 1 2 1 2
2 0 22 2
1 22 2
1
1
( ) ( )
( ) ( )
a aa M a M M M Qa a
uI It t
a aa M a M M M Qa a
uI It t
EQUATIONS OF MOTION
JN Reddy Shells: 16
Stress-Strain Relations
11 1255
1255
66
1 12 2 211 12 22
12 21 12 21 1
22
2 21
00
00
0 0
1 1 1
,
, , ,
yz
xx xxxz xz
yy yyyz
xy xy
Q QQ
Q QQ
Q
E E EQ Q Q
66 12 4 23 55 134, , ,Q G Q G Q G
CONSTITUTIVE RELATIONS
011 11 12 1 11 1
1
11 12 1
022 12 22 2 22 12 22 2
012 6 12
166 66 6
0 00 0
0 0 0 0 ,
MN A AN
D DM D DM
AD
AN A
0442 4
0551 5
00s
AQK
AQ
Resultant-Strain Relations
322 2
2 2 12,
h h
h hij ij ij ij ij ijhA Q d Q h D Q d Q
JN Reddy Shells: 17
STRAIN-DISPLACEMENT RELATIONS00 0
1 31 2 1
1 1 2 2 1
00 02 32 1 20
12 2 1 1 20
0 00 3 2
20 2 2 2
0 00 3 1
11 1 1
0 02 1 1
1 1 2
2
4
5
6
2
1
1
1
1
1
a uu u aa a R
a uu u aa a R
u ua R
u ua R
u u aa a
0 01 2 2
2 2 1 1
1 u u aa a
1 2 1
1 1 2 2111 2 1 2
2 2 1 11
2 1 1 1 2 2
1 1 2 2 2 2 1 1
2
6
1
1
1 1
aa a
aa a
a aa a a a
JN Reddy
2 0 211 1 1
21 0 12 1 0 12 21 2
2 0 222 2 2 2
12 0 12 2 0 12 21 2
2 031 2 22
3 0 21 2
2 0 211 21 1 1
1 1 22 21 2
2 012 22 2
2 1 21 2
( )
( )
N uN C M f I Ix x t t
N Q uN C M f I Ix x R t t
uQ Q N f Ix x R t
M M uQ I Ix x t t
M M uQ Ix x t
22
2 2I
t
1 2 1 1 1 2 2 2 1 21 0 1( / ) , , , , ,R R R a x a x a a R
SPECIALIZATION TO CYLINDRICAL SHELLS
R1 xξ =
y2ξ θ=
+
zζ =
yfxf
zf
Nθθ
Nθθ
xN θ
xxxN
xN θ
zζ =xxN
xN θ
xN θ
2 yξ =
JN Reddy 2-D Problems: 19
THIN CYLINDRICAL SHELLS
1 1 1 2 2 2 0 11 12 22
11 12 22 1 2
0 0 01 0 2 0 3 0 1 2
1 2, , / , , , ,
, , , , ,
, , , ,
xx x
xx x x
x
a x x a x R C R N N N N N N
M M M M M M Q Q Q Q
u u u v u w
R1 xξ =
y2ξ θ=
+
zζ =
•
θ
1 xξ =
zζ =2 yξ =
yfxf
zf
Nθθ
Nθθ
xN θ
xxxN
xN θ
zζ =xxN
xN θ
xN θ
2 yξ =
JN Reddy Cylindrical shells: 20
THIN CYLINDRICAL SHELLS
1 1 0 12
1 1 0 22
1 0 3
1 0 4
1 0 5
( )
( )
( )
( )
( )
xxx x x
x x
xz
xx xx
x
N N M fx R R
N QN M fx R R R
Q Q N fx R R
M M Qx R
M M Qx R
R1 xξ =
y2ξ θ=
+
zζ =
yfxf
zf
Nθθ
Nθθ
xN θ
xxxN
xN θ
zζ =xxN
xN θ
xN θ
2 yξ =
JN Reddy Cylindrical shells: 21
CONSTITUTIVE EQUATIONS FOR CYLINDRICAL SHELLS
0
0 02
0 0
1 011 0
11 10 0
2
( )
xx
x
uxN
v wEhNR R
N v ux R
0 03
20
11 011 0
12 11 10 0
2
( ),
x
x
xx
s
xxx
x w vMQEh R RM K GhQR w
Mx
x R
JN Reddy Pure membrane state: 22
Internal pressure
Rapid change of curvature causes bending deformation under any load
p
Stiffening ring
Local bending
pDeformed centerline of the shell
MEMBRANE AND BENDING STATES
JN Reddy Pure membrane state: 23
Stiffening ring
Local bending
Deformed centerline of the shell Temperature
change, T∆
○○○
Allows pure membranestate of stress
○○○
Membrane stateof stress is onlyapproximate
Bending state ofstress exists
MEMBRANE AND BENDING STATES
JN Reddy Membrane theory of shells: 24
1 0
1 0
0
xx xx
x
z
N N fx R
N N fx R
N fR
MEMBRANE THEORY OF CYLINDRICAL SHELLS
The equilibrium equations governing the membrane state of deformation and stress, called the membrane theory, are obtained by setting bending moments and transverse shear forces to zero:
R1 xξ =
y2ξ θ=
+
zζ =
yfxf
zf
Nθθ
Nθθ
xN θ
xxxN
xN θ
zζ =xxN
xN θ
xN θ
2 yξ =
JN Reddy Membrane theory of shells: 25
1 0 1
1 0 2
0 3
( )
( )
( )
xx xx
x
z
N N fx R
N N fx R
N fR
ANALYTICAL SOLUTIONSof the membrane theory of shells
EXAMPLE 1: Consider a circular cylindrical shell of radius R and thickness h, filled with liquid, and simply supported at itsends. Determine assuming that there are no axial forces at the ends of the shell and the bending deformation is negligible.
, , andx xxN N N
R
+
θz
L
zx
+
θcosR
JN Reddy Membrane theory of shells: 26
The components of load for this case are is the pressure at the axis of the tube and is the specific weight of the liquid. We have
0
0
0 and cos ,where is the pressure at the axis of the tube and
is the specific weight of the liquid.
x zf f f p Rp
2
00 cosz
N f N p R RR
From the Eq. (3) we obtain
EXAMPLE 1: CYLINDRICAL SHELL FILLED WITH LIQUID
JN Reddy Exact solution of membrane shells: 27
2
1
12
sin sin ( )
( )cos cos ( ) ( )
xx
xx xxx
N N R N x R Ax R
N N A x xx N A Bx R R R
From the Eqs. (1) and (2) we obtain
EXAMPLE 1 CONTINUED
2 20 0 5
2( ) ( )cos , . sin , cosx xxN p R R N R L x C N xL x
Using the end conditions
Thus, we have
0 0 0 0 0 01 102 2
( , ) , ( , ) , ( , ) : ( )
( , ) : ( ) cos ( ) sin
xx xx xx
xx
N N L N B
N L A RL A RL C
In the absence of any torsional moment, we have C=0
JN Reddy Exact solutions of shells: 28
EXAMPLE 2: CYLINDRICAL PANEL UNDER ITS OWN WEIGHT
θθNxxN
θxN
x0.5L
+R
xθ
A
BC
D
z ζ=
coszf p θ= −
sinf pθ θ= p
0.5L
Consider a cylindrical shell of semicircular cross section supporting its own weight, which is assumed to be distributed uniformly over the surface of the shell. The shell is assumed to be supported at the four corners A, B, C, and D, but the edges AB and CD are free, as shown in the figure. Using the membrane theory of shells and assuming that there are no axial forces at the ends of the shell, , determine the forces, and displacements .
2 0 2 0( / ) , ( / )xx xxN L N L )( , ,x xxN N N
0 0 0, , u v w
JN Reddy
EXAMPLE 2: CYLINDRICAL PANEL UNDER ITS OWN WEIGHT
The body force components arewhere p is the weight per unit area. From Eq. (3), we obtain
0, sin , cosx zf f p f p
cosN pR
21 1 22 'sin ( ), cos ( ) ( )x xx
p xN xp C N x C CR R
Equations (2) and (1), respectively, yield
2 2( / , ) ( / , )x xN L N L Since (by symmetry), we must have. . Also, the boundary conditions give1 0( )C 2 0( / )xxN L
2
2 4( ) cospLC
R
2 22 44
cos , sin , cos( )x xxpN pR N xp N L xR
Thus, the complete solution is
JN Reddy Shells: 30
EXAMPLE 2: Cylindrical panel under its own weight (continued)
Plots of the variations of these forces with $\theta$, for a fixed $x$, are shown in figure below.
Exact solutions of shells: 30
C
θθθN θxN xxNθf
zf
g p
pR2 24
4( )p L x
R
2px
JN Reddy Cylindrical shells: 31
The displacements can be determined using the constitutive relations
0
0 02
0 0
1 011 0
11 10 0
2
( )
xx
x
uxN
v wEhNR R
N v ux R
0
0 0
0 0
1 01 1 1 0
0 0 2 11 ( )
xx
x
ux N
v w NEh R R
Nu vR x
Inverting the relations
EXAMPLE 2: Cylindrical panel under its own weight (continued)
JN Reddy Shells: 32
EXAMPLE 2: Cylindrical shell filled with liquid(continued)
From the first equation, we have
2
2 20 14
cosxxu p LN N x Rx Eh ERh
Integrating with respect to x gives2 2
20 33 4( , ) cos ( )px x Lu x R C
ERh
Using the boundary condition (by symmetry)
0 30 0 0( , ) , ( )u C
The third equation can now be expressed as2 2
20 02
1 2 1 4 33 4
( ) ( ) sinxv u px x LN Rx R Eh ER h
JN Reddy Shells: 33
Upon integration, we have2 2 2
20 42 4 3
2 6 4( , ) ( ) sin ( )px x Lv x R C
ER h
The boundary condition allows us to calculateas
2 22
4 2
5 4 38 24
( ) ( ) sinpL LC RER h
We have2 2 2 2 2
0 2 4 5 4 24 4 3192
( , ) ( ) sin( ) ( )pv x L x L x REhR
0 2 0( / , )v L
4( )C
EXAMPLE 2: Cylindrical panel under its own weight (continued)
JN Reddy Shells: 34
Finally, using the second equation, we can write
EXAMPLE 2: Cylindrical panel under its own weight (continued)
00
2 2 2 2 2 42 4 5 4 24 4 192
192( ) cos{( ) ( ) }
xxv Rw N N
Ehp L x L x R R
EhR
Thus, the displacement field is given by
2 2 20
2 2 2 2 20 2
2 2 2 2 2 40 2
4 3 1212
4 5 4 24 4 3192
4 5 4 24 4 192192
( , ) cos
( , ) ( ) sin
( , ) ( ) cos
( ) ( ){( ) ( ) }
pxu x x L RERh
pv x L x L x REhR
pw x L x L x R REhR
We note that implying that there is some bending state of stress at x = L/2.
0 2 0( / , ) ,w L
JN Reddy Flexure of shells: 35
Flexural Theory for Axisymmetric Loads
0 0, ,x xxxx z x
dQ dMNN f Qdx R dx
If the cylindrical shell is axisymmetrically loaded, i.e., the shell is subjected to only forces normal to the surface, the deformation is independent of , , are zero, and
are constant. Then the second and fifth equations of equilibrium of cylindrical shells (slide 20) are trivially satisfied, and the remaining three equations take the form
( ), ,x xN M Q 0 0(i .e., )v
( , )N M
There are two equations in three unknowns, requiring us to use kinematic relations. We consider here isotropic material:
211 22 12
3 211 22 12
112 1/ ( ),/ ( ),
A A A Eh A AD D D Eh D D
JN Reddy Flexure of axismmetric cylindrical shells: 36
Flexural Theory of Thin Shells for Axisymmetric Loads
0 0 0 02
0 0 02
2301
2 2
2 30 0
2 3
01
1
12 1( )
,
xx
xx
xxxx x
du w du wEhNdx R dx Rdu w EhwEhNdx R R
d wdEhM Ddx dx
d w dM d wM M D Q Ddx dx dx
2 220 0
2 2 2 2xx
z zd M d w EhwN df D f
dx R dx dx R
Stress resultant-displacement relations
Equilibrium equation in terms of transverse deflection
JN Reddy Flexure of shells: 37
EXAMPLE 3: Flexure of Thin Shells for Axisymmetric Loads
Consider a long circular cylindrical shell of radius R, subjected to uniform bending moment and shearing force at the end x = 0. Determine the deflection .
0Q0M0w
0 1 2 3 4( ) cos sin cos sinx xw x e K x K x e K x K x
The displacement field is given by
424
EhR D
where .
Since the applied loads and are expected to produce local bending and shear and their influence on the solution is expected to die out rapidly with x increasing (St. Venant's principle), the constants and must be zero, giving the solution
0M 0Q0w
1K 2K
JN Reddy Flexure of shells: 38
EXAMPLE 3: Flexure of Thin Shells for Axisymmetric Loads (cont.)
2 30 0
0 0 0 02 30 0( ) , ( )( ) ( )xx x x xd w d wM D M Q D Qdx dx
We obtain
0 3 4( ) cos sinxw x e K x K x
The remaining constants, and , are determined using theboundary conditions at x = 0:
3K 4K
0 0 03 43 22 2
,Q M MK KD D
0 0 03
12
( ) cos sin cos( )xw x e M x x Q xD
The solution becomes
(1)
JN Reddy Flexure of shells: 39
z
x+0M
0Q
L
R+θ
0Q0M
z
xxM
xQxxN
x
θ
z
θθN
00 0 2 0 3 0 3 0 13 2
0 1 0 4 0 1 0 4
0 2 0 33
1 2
3 4
1 1 22 2
1
2
( ) ( ) , ( ) ( )
( ) ( ) , ( ) ( )
( ) ( )
( ) cos sin ( ) cos sin( ) cos ,
, ,
xx
x x
x
dww M f x Q f x M f x Q f xD dx D
M M f x Q f x M M f x Q f x
EhN M f x Q f xR D
f x e x x f x e x xf x e x f
( ) sinxx e x
EXAMPLE 3: Flexure of Thin Shells for Axisymmetric Loads (cont.)
JN Reddy Flexure of shells: 40
EXAMPLE 4: Flexure of Thin Shells for Axisymmetric Loads
Consider a long isotropic circular cylindrical shell of radius R and thickness h, subjected to uniform internal pressure of intensity p. Determine the deflection and bending moment
when the edges are built-in.0w
xxM
L
2R
x0p
pp 2pR
h θσ
h θσ
pRhθ
σ =
L
2R 0p
JN Reddy Flexure of shells: 41
2 2, , ,xx xx
pR pR pRN N pRh h
EXAMPLE 4: Flexure of Thin Shells for Axisymmetric Loads
If the shell is free of any geometric constraints, the shellexperiences membrane forces and hoop and circumferential stresses of
where h is the thickness of the cylinder. The cylinder experiences an increase in the radius of the cylinder by the amount
2
0 22xx
R pRRE Eh
Since the ends of the cylinder are restrained from moving out, the shell develops local bending stresses at the edges. If the shell is sufficiently long, we can use the solution (1) of Example 3 to
JN Reddy Flexure of shells: 42
EXAMPLE 4: Flexure of Thin Shells for Axisymmetric Loads (continued)
determine the bending moment and shear force developed at the ends that produce zero deflection and slope there. Thus, the deflection for the present problem is the sum of the deflection in Eq. (1) and
0M 0Q
0
0 0 0 03
12
( ) cos sin cos( )xw x e M x x Q xD
The boundary conditions yield00 0 0 0, atdww x
dx
2 30 0 0 022 4
2,p pM D Q D
JN Reddy Flexure of shells: 43
2
0
2
1
2
( ) cos sin ,
( ) sin cos , ( ) cos
( )x
x xxx x
pRw x e x xEh
p pM x e x x Q x e x
EXAMPLE 4: Flexure of Thin Shells for Axisymmetric Loads (continued)
The solution becomes
The maximum deflection occurs for large values of x and it is equal to ; the maximum bending and shear forces occur at x = 0 and they are
0
0 22max max max, ,p pw M Q
More examples can be found in the author’s book on plates and shells.
top related