Department of Continuum Mechanics and Structural...

Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero

Department of Continuum Mechanics and Structural Engineering

Aerospace Structures

Chapter 4. Plates and ShellsShells of revolution


1. Introduction2. Membrane theory3. Basic geometrical relations4. Equilibrium equations for shells of revolution5. Displacement equations for shells of revolution6. References

Index

Chapter 4. Plates and Shells

Shells of revolution


Introduction





Introduction

Differences between plates and shells?

Plate Shell

Beam Arch

Differences between beams and arches?


A structural membrane or shell is a curved surface structure. Usually, it iscapable of transmitting loads in more than two directions to supports. It ishighly efficient structurally when it is so shaped, proportioned, andsupported that it transmits the loads without bending or twisting.

A membrane or a shell is defined by its middle surface, halfway between itsextrados, or outer surface and intrados, or inner surface. Thus, dependingon the geometry of the middle surface, it might be a type of dome, barrelarch, cone, or hyperbolic paraboloid. Its thickness is the distance, normalto the middle surface, between extrados and intrados.

Introduction

General definition


1. Efficiency of load-carrying behaviour

2. High degree of reserved strength and structural integrity

3. High specific strength (strength/ weight ratio)

4. Very high stiffness

5. Containment of space

Introduction

Advantages of shells structures


A thin shell is a shell with a thickness relatively small compared with itsother dimensions. But it should not be so thin that deformations would belarge compared with the thickness.

Calculation of the stresses in a thin shell generally is carried out in twomajor steps, both usually involving the solution of differential equations. Inthe first, bending and torsion are neglected (membrane theory). In thesecond step, corrections are made to the previous solution bysuperimposing the bending and shear stresses that are necessary to satisfyboundary conditions (bending theory)

Introduction

Thin shells


Introduction

Examples of thin shells

• Civil and architectural engineering Large span-roofs Liquid-retaining structures and water tanks Concrete arch domes

• Mechanical engineering Piping systems Turbine disks Pressure vessels

• Biomechanics

AEROSPACE STRUCTURES


Membrane theory





Kirchhoff-Love hypothesis

• Thickness is small comparing to the curvature radius of the mid-surface

• Small displacements and strains (equilibrium is verified in the undeformed shape)

• Straight lines normal to the mid-surface remain straight and normal after deformation

• Stresses in perpendicular direction to the mid-surface are neglected

Membrane theory


Geometrical relations





f: meridian angle

q: circumferential angle

r1: first main radius. It is the curvature radius of the meridian

r2: second main radius. It is the distance to the revolution axis in normal direction to the shell surface

Meridian

Axis of revolution

Meridian plane

Parallel

𝐴

𝑟2

𝜃

𝜙

𝑟1

𝐴

𝑟1

𝑟2

𝑅𝜃

𝜙

Meridian plane

Geometrical relations


Equilibrium equations





• Bending and torsion moments are neglected

(Membrane theory)

• Shear forces are neglected in mid-surface

(Axial-symmetry)

Internal forces:• Nf: per unit length in meridian direction• Nq: per unit length in circumferential direction (independent on q)

External loads per unit surface• pn: in normal direction• pf: in meridian direction• pq: does not exist (axial-symmetrical load)

𝑁𝜙

𝑁𝜙 + 𝑑𝑁𝜙

𝑁𝜃

𝑁𝜃𝑝𝜙

𝑝𝑛

𝜃

𝑑𝜃

𝑅

𝜙

𝑑𝜙𝑟1

𝑟2



Equilibrium in meridian direction

1 1 122

dF N rd sen N rd dq q

qf f q

Tangential component

1 1cos cosF N r d dqf f f q

2F N dRd RdN df fq q

2F d RN df q

Tangential component of the external load:

1p rRd df f q

1 1cos 0d RN N r d p rRdf q ff f f

𝑑𝜃

2

𝑁𝜃

𝑁𝜃𝑑𝜃

2

𝐹1R

𝑑𝜃

N

T

𝐹1

𝜙

𝑟1𝑟2

𝜙

𝑁𝜙

𝑁𝜙+d𝑁𝜙

d𝜙𝑟1

𝑟2𝜙

𝑑𝜙

2

𝑑𝜙

2

𝐹3

𝐹2



Equilibrium in normal direction

1 1F N rd dq f q

Normal component:

1 1F sen N rsen d dqf f f q

32 2

d dF N Rd sen N dN R dr d senf f f

f fq q

3F RN d df q f

Normal component of the external load:

1np rRd df q1 2

n

N Np

r r

f q

𝑑𝜃

2

𝑁𝜃

𝑁𝜃𝑑𝜃

2

𝐹1R

𝑑𝜃

N

T

𝐹1

𝜙

𝑟1𝑟2

𝜙

𝑁𝜙

𝑁𝜙+d𝑁𝜙

d𝜙𝑟1

𝑟2𝜙

𝑑𝜙

2

𝑑𝜙

2

𝐹3

𝐹2



1 2

n

N Np

r r

f q 1 1cos 0d RN N r d p rRdf q ff f f

21 2 1

1

cosn

rd RN r r p N d r Rp d

rf f ff f f

Multiplying by send

ff

and considering 2R r senf

2

2 2

1 2 1 2osn

d r sen Nrr p c sen rr sen p d

d

f

f

ff f f f

f

1 2

2

2

cosnrr sen p p sen d kN

r sen

f

f

f f f f

f

Integrating over f

k is obtained from boundary conditions



Displacement equations





Displacements: v, tangent to meridian; w, in normal direction

1 1

1A B AB dv wd dvw

AB rd r df

f

f f

2 2

2

R dR R dR

R Rq

cosdR v wsenf f

2

cos 1cot

v wsenv w

R rq

f f f

with 2R r senf

Strains

𝑑𝜙

𝑤 + 𝑑𝑤𝑟1

A´

B´

A

B

𝑣 + 𝑑𝑣

𝑑𝜙

A

B

𝑤𝑑𝜙

𝑤

𝑑𝜙

A

B𝑣 + 𝑑𝑣

𝑣

𝑟2

A𝑣

𝑣

𝑤

𝜙

𝑅

𝑣

𝑑𝑅



1

2cot

dvw r

d

v w r

f

q

f

f

Strains

1 2cotdv

v r rd

f qf f

Hook laws

1 1

1 1

N NE Eh

N NE Eh

f f q f q

f q f q f

1 2 2 1

1cot

dvv r r N r r N

d Ehf qf

f

1 2 2 1

1 1v sen r r N r r N d k

Eh senf qf f

f

22 cot cot

rw r v N N v

Ehq q f f f

k is obtained from boundary conditions

d vsen

d senf

f f



Meridian rotation

g1: rotation motivated by the displacement v of point A

g2: rotation motivated by the difference between displacements

w of points A and B

g : meridian rotation

1 2

1 1

v dw

r rdg g g

f

d : horizontal displacement

2

1R r sen N N

Ehq q fd f

𝑑𝜙

A

B𝑟1

𝑤d𝑤

𝑤𝛾2

A

𝑟1

𝑣𝛾1𝛾1



Meridian rotation

1 2

1 1

v dw

r rdg g g

f

22cot cot

dd dv vv w r

d d sen d

qf f

f f f f

2

1 2 2 12

cos cotcot cot cot

dv dvv v r r N r r N

d d sen Ehf q

f ff f f

f f f

On the other hand

Subtracting the above equations and dividing by r1

1 2 2 1

1

1 cot 1dr r N r r N N N

r Eh d Ehf q q f

fg

f



References

Chapter 4. Plates and ShellsShells of revolution



References

1. J. A. Jurado Albarracín-Martinón y S. Hernández Ibáñez, “Análisis

estructural de placas y láminas”. Tercera Edición. Andavira editora

2014

Cap.7

2. Timoshenko, Stephen “Theory of plates and shells “

McGraw-Hill, 1959

Cap.14,15,16

3. Ugural, A.C. “Stresses in beams, plates and shells”. CRC.

Taylor & Francis, 2009

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