INTRODUCTION TO THE THEORY OF SHELLSmechanics.tamu.edu/.../03/Lecture-09-Linear_Shells.pdf ·...

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


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 =


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 =


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 =


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


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


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


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


JN Reddy Shells: 34

Finally, using the second equation, we can write


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


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


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)


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.)



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


2 2, , ,xx xx

pR pR pRN N pRh h


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


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


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.

INTRODUCTION TO THE THEORY OF SHELLSmechanics.tamu.edu/.../03/Lecture-09-Linear_Shells.pdf ·...

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Transcript of INTRODUCTION TO THE THEORY OF SHELLSmechanics.tamu.edu/.../03/Lecture-09-Linear_Shells.pdf ·...