Department of Continuum Mechanics and Structural...
Transcript of 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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Introduction
Chapter 4. Plates and Shells
Shells of revolution
1. Introduction2. Membrane theory3. Basic geometrical relations4. Equilibrium equations for shells of revolution5. Displacement equations for shells of revolution6. References
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Introduction
Differences between plates and shells?
Plate Shell
Beam Arch
Differences between beams and arches?
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Membrane theory
Chapter 4. Plates and Shells
Shells of revolution
1. Introduction2. Membrane theory3. Basic geometrical relations4. Equilibrium equations for shells of revolution5. Displacement equations for shells of revolution6. References
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Geometrical relations
Chapter 4. Plates and Shells
Shells of revolution
1. Introduction2. Membrane theory3. Basic geometrical relations4. Equilibrium equations for shells of revolution5. Displacement equations for shells of revolution6. References
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Equilibrium equations
Chapter 4. Plates and Shells
Shells of revolution
1. Introduction2. Membrane theory3. Basic geometrical relations4. Equilibrium equations for shells of revolution5. Displacement equations for shells of revolution6. References
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
• 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 equations
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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 equations
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Equilibrium equations
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Equilibrium equations
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Displacement equations
Chapter 4. Plates and Shells
Shells of revolution
1. Introduction2. Membrane theory3. Basic geometrical relations4. Equilibrium equations for shells of revolution5. Displacement equations for shells of revolution6. References
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Displacement equations
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Displacement equations
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Displacement equations
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Displacement equations
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
References
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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