Plates and Shells: Theory and Computation

- 4D9 -

Dr Fehmi Cirak (fc286@)

Office: Inglis building mezzanine level (INO 31)

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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