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Plates and Shells: Theory and Computation
Dr. Mostafa Ranjbar
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Outline -1-
! This part of the module consists of seven lectures and will focus on finite
elements for beams, plates and shells. More specifically, we will consider
! Review of elasticity equations in strong and weak form
! Beam models and their finite element discretisation
! Euler-Bernoulli beam
! Timoshenko beam
! Plate models and their finite element discretisation
! Shells as an assembly of plate and membrane finite elements
! Introduction to geometrically exact shell finite elements
! Dynamics
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Outline -2-
! There will be opportunities to gain hands-on experience with the
implementation of finite elements using MATLAB
! One hour lab session on implementation of beam finite elements (will be not marked)
! Coursework on implementation of plate finite elements and dynamics
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Why Learn Plate and Shell FEs?
! Beam, plate and shell FE are available in almost all finite element software
packages
! The intelligent use of this software and correct interpretation of output requires basic
understanding of the underlying theories
! FEM is able to solve problems on geometrically complicated domains
! Analytic methods introduced in the first part of the module are only suitable for computing plates
and shells with regular geometries, like disks, cylinders, spheres etc.
! Many shell structures consist of free form surfaces and/or have a complex topology
! Computational methods are the only tool for designing such shell structures
! FEM is able to solve problems involving large deformations, non-linear
material models and/or dynamics
! FEM is very cost effective and fast compared to experimentation
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Literature
! JN Reddy, An introduction to the finite element method, McGraw-Hill (2006)
! TJR Hughes, The finite element method, linear static and dynamic finite element
analysis, Prentice-Hall (1987)
! K-J Bathe, Finite element procedures, Prentice Hall (1996)
! J Fish, T Belytschko, A first course on finite elements, John Wiley & Sons (2007)
! 3D7 - Finite element methods - handouts
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Examples of Shell Structures -1-
! Civil engineering
! Mechanical engineering and aeronautics
Masonry shell structure (Eladio Dieste) Concrete roof structure (Pier Luigi Nervi)
Fuselage (sheet metal and frame)Ship hull (sheet metal and frame)
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Examples of Shell Structures -2-
! Consumer products
! Nature
Red blood cellsFicus elastica leafCrusteceans
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Representative Finite Element Computations
Virtual crash test (BMW)
Sheet metal stamping (Abaqus)
Wrinkling of an inflated party balloon
buckling of carbon nanotubes
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0.74 m
0.02
5 m
Shell-Fluid Coupled Airbag Inflation -1-
Shell mesh: 10176 elements
0.86 m
0.49
m
0.86 m
0.123 m
Fluid mesh: 48x48x62 cells
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Shell-Fluid Coupled Airbag Inflation -2-
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Detonation Driven Fracture -1-
! Modeling and simulation challenges
! Ductile mixed mode fracture
! Fluid-shell interaction
Fractured tubes (Al 6061-T6)
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Detonation Driven Fracture -2-
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Roadmap for the Derivation of FEM
! As introduced in 3D7, there are two distinct ingredients that are combined
to arrive at the discrete system of FE equations
! The weak form
! A mesh and the corresponding shape functions
! In the derivation of the weak form for beams, plates and shells the
following approach will be pursued
1) Assume how a beam, plate or shell deforms across its thickness
2) Introduce the assumed deformations into the weak form of three-dimensional elasticity
3) Integrate the resulting three-dimensional elasticity equations along the thickness direction
analytically
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Elasticity Theory -1-
! Consider a body in its undeformed (reference) configuration
! The body deforms due to loading and the material points move by a displacement
! Kinematic equations; defined based on displacements of an infinitesimalvolume element)
! Axial strains
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Elasticity Theory -2-
! Shear components
! Stresses
! Normal stress components
! Shear stress component
! Shear stresses are symmetric
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Elasticity Theory -3-
! Equilibrium equations (determined from equilibrium of an infinitesimal
volume element)
! Equilibrium in x-direction
! Equilibrium in y-direction
! Equilibrium in z-direction
! are the components of the external loading vector (e.g., gravity)
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Elasticity Theory -4-
! Hooke’s law (linear elastic material equations)
! With the material constants Young’s modulus and Poisson’s ratio
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Index Notation -1-
! The notation used on the previous slides is rather clumsy and leads to very
long expressions
! Matrices and vectors can also be expressed in index notation, e.g.
! Summation convention: a repeated index implies summation over 1,2,3, e.g.
! A comma denotes differentiation
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Index Notation -2-
! Kronecker delta
! Examples:
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Elasticity Theory in Index Notation -1-
! Kinematic equations
! Note that these are six equations
! Equilibrium equations
! Note that these are three equations
! Linear elastic material equations
! Inverse relationship
! Instead of the Young’s modulus and Poisson’s ratio the Lame constants can be used
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Weak Form of Equilibrium Equations -1-
! The equilibrium, kinematic and material equations can be combined into
three coupled second order partial differential equations
! Next the equilibrium equations in weak form are considered in preparation
for finite elements
! In structural analysis the weak form is also known as the principle of virtual displacements
! To simplify the derivations we assume that the boundaries of the domain are fixed (built-in, zero
displacements)
! The weak form is constructed by multiplying the equilibrium equations with test functions vi which
are zero at fixed boundaries but otherwise arbitrary
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Weak Form of Equilibrium Equations -1-
! Further make use of integration by parts
! Aside: divergence theorem
! Consider a vector field and its divergence
! The divergence theorem states
! Using the divergence theorem equation (1) reduces to
! which leads to the principle of virtual displacements
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Weak Form of Equilibrium Equations -2-
! The integral on the left hand side is the internal virtual work performed by the internal stresses due to virtual
displacements
! The integral on the right hand side is the external virtual work performed by the external forces due to virtual
displacements
! Note that the material equations have not been used in the preceding derivation.
Hence, the principle of virtual work is independent of material (valid for elastic, plastic,
…)
! The internal virtual work can also be written with virtual strains so that the principle of
virtual work reads
! Try to prove
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