Many shell structures consist of free form surfaces and/or have a complex ...
Computational methods are the only tool for designing such shell structures. □.
Plates and Shells: Theory and Computation - 4D9 Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31)
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 ! !
Page 2
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
F. Cirak
Outline -2!
There will be opportunities to gain hands-on experience with the implementation of finite elements using MATLAB ! !
Page 3
One hour lab session on implementation of beam finite elements (will be not marked) Coursework on implementation of plate finite elements and dynamics
F. Cirak
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 !
!
!
Page 4
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
F. Cirak
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
F. Cirak
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) F. Cirak
Examples of Shell Structures -2!
Consumer products
!
Nature
Crusteceans Page 7
Ficus elastica leaf
Red blood cells F. Cirak
Representative Finite Element Computations
Wrinkling of an inflated party balloon
Virtual crash test (BMW)
buckling of carbon nanotubes Sheet metal stamping (Abaqus) Page 8
F. Cirak
Shell-Fluid Coupled Airbag Inflation -1-
0.49 m
0.86 m
6m 0.8
0.025 m
0.74 m
0.123 m Shell mesh: 10176 elements
Page 9
Fluid mesh: 48x48x62 cells
F. Cirak
Shell-Fluid Coupled Airbag Inflation -2-
Page 10
F. Cirak
Detonation Driven Fracture -1Fractured tubes (Al 6061-T6)
!
Modeling and simulation challenges ! !
Page 11
Ductile mixed mode fracture Fluid-shell interaction
F. Cirak
Detonation Driven Fracture -2-
Page 12
F. Cirak
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 ! !
!
In the derivation of the weak form for beams, plates and shells the following approach will be pursued 1) 2) 3)
Page 13
The weak form A mesh and the corresponding shape functions
Assume how a beam, plate or shell deforms across its thickness Introduce the assumed deformations into the weak form of three-dimensional elasticity Integrate the resulting three-dimensional elasticity equations along the thickness direction analytically
F. Cirak
Elasticity Theory -1!
Consider a body in its undeformed (reference) configuration
!
!
Kinematic equations; defined based on displacements of an infinitesimal volume element) !
Page 14
The body deforms due to loading and the material points move by a displacement
Axial strains
F. Cirak
Elasticity Theory -2!
!
Page 15
Shear components
Stresses !
Normal stress components
!
Shear stress component
!
Shear stresses are symmetric
F. Cirak
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
!
Page 16
are the components of the external loading vector (e.g., gravity)
F. Cirak
Elasticity Theory -4!
Hooke’s law (linear elastic material equations)
!
Page 17
With the material constants Young’s modulus and Poisson’s ratio
F. Cirak
Index Notation -1!
!
Page 18
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 F. Cirak
Index Notation -2!
!
Page 19
Kronecker delta
Examples:
F. Cirak
Elasticity Theory in Index Notation -1!
Kinematic equations
!
!
Equilibrium equations !
!
Page 20
Note that these are six 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
F. Cirak
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 !
!
!
Page 21
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
F. Cirak
Weak Form of Equilibrium Equations -1!
!
!
Aside: divergence theorem !
Consider a vector field
!
The divergence theorem states
and its divergence
Using the divergence theorem equation (1) reduces to
!
Page 22
Further make use of integration by parts
which leads to the principle of virtual displacements F. Cirak
Weak Form of Equilibrium Equations -2!
!
!
!
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
!
Page 23
The integral on the left hand side is the internal virtual work performed by the internal stresses due to virtual displacements
Try to prove F. Cirak
