Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak Office: Inglis building mezzanine level (INO 31)

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1 Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak Office: Inglis building mezzanine level (INO 31)

2 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 Page 2

3 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 Page 3

4 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 Page 4

5 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 Page 5

6 Examples of Shell Structures -1! Civil engineering Masonry shell structure (Eladio Dieste)! Mechanical engineering and aeronautics Ship hull (sheet metal and frame) Page 6 Concrete roof structure (Pier Luigi Nervi) Fuselage (sheet metal and frame)

7 Examples of Shell Structures -2! Consumer products! Nature Crusteceans Page 7 Ficus elastica leaf Red blood cells

8 Representative Finite Element Computations Wrinkling of an inflated party balloon Virtual crash test (BMW) Sheet metal stamping (Abaqus) buckling of carbon nanotubes Page 8

9 Shell-Fluid Coupled Airbag Inflation m 0.49 m 0.74 m m 0.86 m m Shell mesh: elements Fluid mesh: 48x48x62 cells Page 9

10 Shell-Fluid Coupled Airbag Inflation -2- Page 10

11 Detonation Driven Fracture -1Fractured tubes (Al 6061-T6)! Modeling and simulation challenges!! Page 11 Ductile mixed mode fracture Fluid-shell interaction

12 Detonation Driven Fracture -2- Page 12

13 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 Page 13

14 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 infinitesimal volume element)! Axial strains Page 14

15 Elasticity Theory -2-! Shear components! Stresses! Normal stress components! Shear stress component! Shear stresses are symmetric Page 15

16 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) Page 16

17 Elasticity Theory -4-! Hooke s law (linear elastic material equations)! With the material constants Young s modulus and Poisson s ratio Page 17

18 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 Page 18

19 Index Notation -2-! Kronecker delta! Examples: Page 19

20 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 Page 20

21 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 v i which are zero at fixed boundaries but otherwise arbitrary Page 21

22 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 Page 22

23 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 Page 23

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