Shell Mechanics: Theory and Applications

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preamble
General principles of notation
Chapter 1 Introduction to the behaviour of shells
1.1 Some general ideas
1.1.1 Definitions
1.1.1.1 Shell theory
1.1.1.2 Middle surface and thickness
1.1.1.3 Particular case of plates
1.1.2 Importance of shape
1.1.2.1 Spherical tank under pressure
1.1.2.2 Cylindrical tank under pressure
1.1.2.3 Conclusion
1.1.3 Behaviour of the shell
1.1.4 Methods used
1.2 Nature of internal forces
1.2.1 Generalised actions applied to the shell
1.2.2 Link with three-dimensional stresses
1.2.2.1 Approximate values of stresses
1.2.2.2 Comparison with the three-dimensional solution
1.2.2.3 Resultant of three-dimensional stresses
1.2.3 Validity of the membrane theory
1.2.4 Contribution of global balances
1.2.4.1 Infinite plane in cylindrical bending
1.2.4.2 Spherical dome
Exercises
Chapter 2 Elements of tensorial algebra and analysis
2.1 Tensors on a vector space
2.1.1 Vector space and dual space
2.1.2 Bilinear forms on E
2.1.3 Endomorphisms on E
2.1.4 Tensor product space on E
2.1.5 Change of basis
2.1.6 Contraction
2.1.7 Symmetry and antisymmetry
2.1.8 Euclidean tensors
2.1.9 Levi-Civita tensor and cross product
2.2 Differentiation
2.2.1 Local basis in an affine space
2.2.2 Tensor fields
2.2.3 Differentiation of the vectors of the basis
2.2.4 Differentiation of tensor fields
2.2.5 Expression of Christoffel coefficients versus metric
2.2.6 Differential operators
2.2.7 Riemann–Christoffel tensor associated with a differentiation
Exercises
Chapter 3 Deformation of surfaces
3.1 Fundamental forms on a surface
3.1.1 Introduction to surface geometry
3.1.2 Definitions, local basis
3.1.2.1 Parameterisation of the surface
3.1.2.2 Tangent plane
3.1.2.3 Normal vector to the surface
3.1.3 Metric
3.1.3.1 Metric tensor
3.1.3.2 Fundamental antisymmetric tensor
3.1.4 Curvature
3.1.4.1 Definition
3.1.4.2 Geometric interpretation
3.1.4.3 Examples
3.1.4.4 Meusnier theorem
3.1.4.5 Principal curvatures
3.1.4.6 Invariants of the curvature tensor
3.1.4.7 Third fundamental form
3.1.5 Normal coordinates in the neighbourhood of the surface
3.1.5.1 Naturel coordinate system in the neighbourhood of the surface
3.1.5.2 Principal curvature coordinate system
3.1.5.3 Inverse endomorphism of m
3.1.5.4 Surface parallel to the middle surface
3.2 Differentiation on the surface
3.2.1 Levi-Civita differentiation
3.2.1.1 Definitions
3.2.1.2 Differentiation formulas
3.2.1.3 Expression of the connection coefficients in
3.2.1.4 Rules of differentiation of any tensor field
3.2.1.5 Differentiation of fundamental forms
3.2.1.6 Example: cylindrical coordinates on a cylinder
3.2.1.7 Example: spherical coordinates on a sphere
3.2.1.8 Example: surfaces defined in Cartesian coordinates
3.2.2 Mainardi–Codazzi and Gauss relations
3.2.2.1 Riemann–Christoffel surface tensor field associated with the Levi-Civita differentiation
3.2.2.2 Relations between metric and curvature
3.2.2.3 Consequences of the Gauss theorem
3.3 Deformation of surfaces
3.3.1 Characterisation of surface deformation
3.3.2 Tensors of deformation of a surface
3.3.3 Expression of deformation variables as a function of displacement
3.3.3.1 General case
3.3.3.2 Small displacement cases
3.3.4 Compatibility equations
3.3.4.1 Case where the reference surface is a plane
3.3.4.2 Case where the reference surface is developable
3.3.4.3 Case of shallow surfaces or almost developable
3.4 Curves drawn on a surface
3.4.1 Coordinate systems associated with a curve
3.4.1.1 Frénet–Serret coordinate system
3.4.1.2 Darboux coordinate system
3.4.2 Link with the curvature tensor of the surface
3.4.3 Special lines on a surface
Exercises
Chapter 4 The shell as a surfacic solid
4.1 Kinematics of surfaces
4.1.1 Velocity and acceleration
4.1.1.1 Parametrisation
4.1.1.2 Velocity
4.1.1.3 Strain and variation of curvature rates
4.1.1.4 Spin Tensor
4.1.1.5 Acceleration
4.1.2 Conservation laws
4.1.2.1 General form
4.1.2.2 Mass continuity equation
4.1.3 Virtual Movements
4.1.3.1 Characterisation of a virtual movement
4.1.3.2 Virtual rotation velocity
4.1.3.3 Virtual rates of deformation and curvature variation
4.1.3.4 Virtual transformation of a curve drawn on a surface
4.2 Equations of motion
4.2.1 Application of general momentum theorems
4.2.1.1 Introduction
4.2.1.2 Nature of actions applied to a shell part
4.2.1.3 Expression of internal actions
4.2.1.4 Expression of local equilibrium
4.2.1.5 Boundary conditions
4.2.2 Application of the principle of virtual powers
4.2.2.1 Virtual power of deformation
4.2.2.2 Virtual power of external actions
4.2.2.3 Virtual power of inertial actions
4.2.2.4 Application of the Principle of Virtual Powers
4.2.2.5 Local equations
4.2.2.6 Boundary conditions
4.2.2.7 Singular points of the edges
4.2.3 Special cases
4.2.3.1 Membranes
4.2.3.2 Plates
4.2.3.3 Shallow shells
4.2.4 Double surface concept
4.3 Problem of movement and constitutive laws
4.3.1 Need to take into account the behaviour of materials
4.3.2 Elastic global constitutive laws
4.3.2.1 Expression of elastic constitutive laws from the thermodynamic potential
4.3.2.2 Linearly elastic shells
4.4 Comparison of the equations obtained by the equilibrium equations and by the PVP
4.4.1 Comparison of local equations
4.4.2 Example: sphere under internal pressure
4.4.3 Boundary conditions
4.5 Mechanics of Cosserat-oriented surfaces
4.5.1 Kinematics
4.5.1.1 Principle
4.5.1.2 Parameterisation of the Cosserat surface
4.5.1.3 Kinematics of the director
4.5.1.4 Deformation and rate of deformation
4.5.1.5 Expression of deformation variables by displacement and director
4.5.1.6 Virtual movements
4.5.2 Equations of movement
4.5.2.1 Virtual power of deformation
4.5.2.2 Virtual power of external actions
4.5.2.3 Virtual power of inertial actions
4.5.2.4 Local equations
4.5.2.5 Boundary conditions
4.5.2.6 Conclusion
Exercises
Chapter 5 Shell as a three-dimensional solid
5.1 Stresses and equilibrium Equations
5.1.1 Parameter setting and metric
5.1.1.1 Normal coordinates
5.1.1.2 Metric in normal coordinates
5.1.1.3 Volumic element
5.1.2 Kinematics
5.1.3 Screw resultant of local stresses
5.1.4 Obtaining equilibrium equations from three-dimensional equations
5.1.5 Case of thin shells
5.1.5.1 Hypothesis of thin shells
5.1.5.2 Consequences on the resultant of stresses in the thickness
5.2 Theory of Kirchhoff–Love shells
5.2.1 Kirchhoff kinematic assumptions
5.2.2 Consequences of the kinematics of the transformation
5.2.2.1 Velocity and acceleration
5.2.2.2 Deformations
5.2.2.3 Virtual deformation
5.2.3 Continuity equation and conservation equations
5.2.3.1 Density and mass per unit area
5.2.3.2 Continuity equation
5.2.3.3 Conservation equations
5.2.4 Equilibrium equations
5.2.4.1 Expressions of deformation energy
5.2.4.2 Equilibrium equations
5.2.4.3 Relationships between generalised stresses and resultant stresses
5.2.5 Case of thin shells
5.2.5.1 Consequences for deformations
5.2.5.2 Consequences for generalised stresses
5.3 Theory of Reissner–Mindlin shells
5.3.1 Reissner–Mindlin hypotheses
5.3.2 Consequences for kinematics
5.3.2.1 Velocity and acceleration
5.3.2.2 Continuity equation
5.3.2.3 Deformations
5.3.2.4 Deformation in small displacements
5.3.2.5 Virtual deformations
5.3.3 Application of PVP
5.3.3.1 Virtual power of deformation
5.3.3.2 Virtual power of external actions
5.3.3.3 Equilibrium equations
5.4 Shell constitutive laws
5.4.1 Hypotheses of plane stresses
5.4.1.1 Stress T33
5.4.1.2 Case of Kirchhoff hypothesis
5.4.2 Linearly elastic shells
5.4.2.1 Expression of the local law in-plane stresses
5.4.2.2 Case of homogeneous and isotropic material
5.4.2.3 Obtaining the shell constitutive law
5.4.3 Elastic thin shells in Love theory
5.4.3.1 Hypotheses of Love theory
5.4.3.2 Linearly elastic solid
5.4.3.3 Homogeneous and isotropic solid
5.4.3.4 Shear stress due to shear force in a homogeneous and isotropic shell
5.4.4 Thin elastic shells in Reissner–Mindlin theory
5.4.4.1 Membrane stresses
5.4.4.2 Taking into account the distortion energy
5.4.4.3 Elastic potential of Reissner-Mindlin thin shells
5.5 Shell equilibrium equations obtained from local equilibrium equations
5.5.1 Expression of the local equilibrium equations
5.5.2 Integration method
5.5.3 Resultant of stresses
5.5.3.1 Normal component
5.5.3.2 Tangential components
5.5.3.3 Resultant moment
5.5.3.4 Symetries
5.6 Complements on shell constitutive laws
5.6.1 Thin elastic shells in Love theory
5.6.1.1 Materials with symmetry of revolution
5.6.1.2 Orthotropic material
5.6.2 Thick elastic shells
5.6.2.1 General method
5.6.2.2 Shell constitutive law in the main curvature coordinate system
5.6.2.3 Flügge–Lüre–Byrne method
5.6.3 Shells of composite materials
5.6.3.1 Multilayer materials
5.6.3.2 Sandwich shell
Exercises
Chapter 6 Equilibrium of membrane shells
6.1 The classical membrane theory
6.1.1 Assumptions and evolution equations
6.1.1.1 What is a membrane theory?
6.1.1.2 Equilibrium equations
6.1.1.3 Membrane forces / local stress relationships
6.1.1.4 Deformations
6.1.1.5 Homogeneous and isotropic linear elastic behaviour
6.1.1.6 General linearly elastic behaviour
6.1.2 Validity of the membrane theory
6.1.2.1 Preponderance of membrane energy
6.1.2.2 Connection of two shells with different curvatures
6.1.2.3 Conclusions
6.1.3 General methods of resolution
6.1.3.1 Direct methods
6.1.3.2 Case of Euclidean surfaces: stress function
6.2 Cylindrical Shells
6.2.1 Consequences of geometry
6.2.1.1 Parameterisation, natural basis, fundamental forms
6.2.1.2 Equilibrium equations
6.2.1.3 Deformation
6.2.1.4 Linearly elastic behaviour
6.2.1.5 Case of circular cylinders
6.2.2 Some classic examples of problems with cylindrical shells
6.2.2.1 Vault in half circular cylinder
6.2.2.2 Pipe under pressure
6.2.2.3 Non-axisymmetric loading on a cylindrical tank
6.2.2.4 Parabolic vault
6.2.3 Quasi-cylindrical shell
6.2.3.1 Equilibrium equations of the quasi-cylindrical shell
6.2.3.2 Case of the barrel
6.2.3.3 Case of the diabolo
6.3 Shells of Revolution
6.3.1 Consequences of geometry
6.3.1.1 Parameterisation, natural basis
6.3.1.2 Equilibrium equations
6.3.1.3 Deformation
6.3.1.4 Problems with symmetry of revolution
6.3.1.5 Problems without symmetry of revolution
6.3.2 Some classic examples of problems involving shells of revolution
6.3.2.1 Spherical shells
6.3.2.2 Parabolic domes
6.3.2.3 Conical shells
6.3.2.4 Hyperboloid of revolution
6.3.3 Cylindrical tanks with bottom under pressure
6.3.3.1 General Relations
6.3.3.2 Spherical bottom
6.3.3.3 Ellipsoidal bottom
6.3.3.4 Cylindrical tank with torispherical bottom
6.4 Shells Defined by an Explicit Function
6.4.1 General characteristics of shells defined by an explicit function
6.4.1.1 Parametrisation; natural basis
6.4.1.2 Equilibrium equations
6.4.1.3 Stress function
6.4.2 Translation shells
6.4.2.1 Shells in the form of hyperbolic paraboloid
6.4.2.2 Conoid-shaped vaults
6.5 Helical Shells
Exercises
Chapter 7 Plates in flexion
7.1 Flexion of plates: classical theory
7.1.1 Equilibrium equations
7.1.1.1 Recalling the hypotheses
7.1.1.2 Equilibrium equations and boundary conditions
7.1.1.3 Link with local stresses
7.1.1.4 Extreme values of stresses
7.1.2 Deformations
7.1.2.1 Displacement
7.1.2.2 Deformations
7.1.3 Linear elastic plates
7.1.3.1 Homogeneous and isotropic plate
7.1.3.2 Anticlastic surface
7.1.3.3 Stresses due to a thermal gradient in a plate clamped on its edge
7.1.3.4 Orthotropic plate
7.2 Classical examples
7.2.1 Circular plates
7.2.1.1 General results
7.2.1.2 Case of axisymmetric bending
7.2.1.3 Free plate subject to a moment on the edge
7.2.1.4 Uniformly loaded clamped plate
7.2.1.5 Simply supported plate loaded in the centre
7.2.1.6 Plate simply supported, uniformly loaded
7.2.1.7 Bottom of a tank
7.2.1.8 Any loading
7.2.1.9 Vibrations of a circular plate
7.2.2 Rectangular plates
7.2.2.1 Isotropic plate supported on all four sides; Navier’s solution
7.2.2.2 Isotropic plate supported on two opposite edges; Lévy solution
7.2.2.3 Plates of other shapes
7.2.3 Orthotropic plate
7.3 Plate bending taking into account membrane forces
7.3.1 Equilibrium equations
7.3.2 Isotropic elastic plates
7.3.2.1 Equations of flexion with membrane forces
7.3.2.2 Rectangular plate supported on four sides subject to an action in its plane
7.3.2.3 Föppl-Von Kármán equations
7.3.2.4 Plate in axisymmetric bending
7.3.2.5 Plate in cylindrical bending
7.4 Reissner–Mindlin plates
7.4.1 Equilibrium equations
7.4.1.1 Scope
7.4.1.2 Equilibrium equations and boundary conditions
7.4.1.3 Local stresses
7.4.2 Deformations
7.4.2.1 Displacement
7.4.2.2 Deformations
7.4.3 Linear elastic plate
7.4.4 Equation of isotropic elastic plates
7.4.4.1 Equations of flexion
7.4.4.2 Looking for solutions
Exercises
Chapter 8 Shells in bending
8.1 Main theories and approximations
8.1.1 Kinematics of transformation
8.1.1.1 General
8.1.1.2 Indicators
8.1.1.3 Expression of deformations
8.1.1.4 Usual theories
8.1.1.5 Order of linearisation according to x
8.1.1.6 Hypothesis of small deformations and small rotations
8.1.1.7 Love approximation
8.1.1.8 Donnell approximation
8.1.1.9 Classical theory of shells in bending
8.1.2 Equilibrium equations in the main approximations
8.1.2.1 General
8.1.2.2 Small deformations and small rotations (first order)
8.1.2.3 Small deformations and small rotations (second order)
8.1.2.4 Linearised approximation
8.1.2.5 Love approximation
8.1.2.6 Donnell approximation
8.1.2.7 Classical approximation
8.1.3 Classical resolution methods
8.1.3.1 Displacement method
8.1.3.2 Stress method
8.2 Circular cylindrical thin shells
8.2.1 Deformations and equilibrium equations
8.2.1.1 Geometry and displacement
8.2.1.2 Calculation of deformations
8.2.1.3 Hypothesis of small deformations and small rotations
8.2.1.4 Love approximation
8.2.1.5 Donnell approximation
8.2.1.6 Classical approximation
8.2.1.7 Inextensional transformation
8.2.1.8 Equilibrium equations in first-order approximation
8.2.1.9 Constitutive laws
8.2.1.10 Form of solutions for closed cylinders
8.2.2 Resolution methods
8.2.2.1 Equilibrium equations
8.2.2.2 Airy function method
8.2.3 Some classical solutions
8.2.3.1 Cylinder in axisymmetric bending
8.2.3.2 Cylindrical vibrations
8.2.3.3 Cylindrical vault
8.2.4 Instability of cylindrical shells
8.2.4.1 Equilibrium in second-order approximation
8.2.4.2 Study of buckling of closed cylinders
8.2.4.3 Introduction to the stability of a cylindrical vault
8.3 Thin shells of rEvolution
8.3.1 Deformations
8.3.1.1 Geometry
8.3.1.2 Deformations in small axisymmetric transformation
8.3.2 Resolution methods
8.3.2.1 Equilibrium equations
8.3.2.2 Homogeneous and isotropic elastic shell
8.3.2.3 Resolution from equilibrium equations
8.3.2.4 Resolution by the energy approach
8.3.2.5 Cylindrical tank with a hemispherical bottom
8.3.3 Introduction to any loading
8.4 Example of application: flexion of a pipe elbow
Exercises
Chapter 9 Shell finite elements
9.1 The finite elements method
9.1.1 Purpose of the chapter
9.1.2 The principles of the finite element method
9.1.3 Formulation of a finite element
9.1.4 Expression of elementary energies
9.1.5 Expression of the energies of the complete system
9.1.6 Equations of motion (displacement-based method)
9.1.7 Finite element quality tests
9.2 Plate elements in bending
9.2.1 The constitution of the plate elements
9.2.1.1 Determination of interpolation functions
9.2.1.2 Strain energy of a plate in flexion
9.2.1.3 Kirchhoff plate elements
9.2.1.4 Reissner–Mindlin plate elements
9.2.2 An example of a Kirchhoff plate element construction: the rectangular four-node plate element
9.2.3 Some examples of plate elements
9.2.3.1 Kirchhoff plate elements
9.2.3.2 Reissner–Mindlin plate elements
9.3 Shell elements
9.3.1 General
9.3.2 Flat shell elements
9.3.2.1 Properties of flat elements
9.3.2.2 Examples of flat elements
9.3.3 Parametric curved elements
9.3.3.1 Curved geometry of the elements
9.3.3.2 Examples of parametric curved elements
9.3.4 Shell elements of revolution
9.3.4.1 Element of revolution in axisymmetric bending
9.3.4.2 An example of the construction of an axisymmetric membrane element: the isoparametric element with three nodes
9.3.4.3 Element of revolution in non-axisymmetric bending
9.3.4.4 Examples of elements of revolution
9.3.5 Curved isoparametric elements (degenerated 3D-element)
9.3.5.1 General principle of the formulation of these elements
9.3.5.2 Examples of isoparametric elements
Exercises
Index