Lecture on Plates and Shells

7. Plates and Shells

MEEM4405 Introduction to Finite Element Analysis

7.1 Plate Formulation  Plates Plates may be considere considered d similar to beams, however: – Plates can bend in two directions and twist – Plates must be flat (or else they are shells)

• For thin plate on z = 0 plane, with thickness t , and neglecting shear strain: w = w( x , y ) ∂w u = − z ∂ x γ yz = γ zx = 0 ∂w v = − z ∂ y MEEM4405 Introduction to Finite Element Analysis

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Thin Plate Formulation • A differential slice from plate:


Thin Plate Formulation • For the thin plate, we assume σz = 0. Therefore:


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Thin Plate Formulation • These stresses give rise to moments:

• Maximum stresses are therefore given by: 6 M x  2 z  σ x = σ x = σ x  , since  2 t  t  6 M y

This is similar to the beam formula, but since the plate is very wide we t have a situation similar to plain 6 M xy τ xy = strain. Flexural rigidity D=EI = Et 3 /12 2 t with EI = Et 3 /12, but since strain it is MEEM4405 Introduction to Finite Element Analysis very wide (like σ y

=

2

,

Thin Plate Formulation • This is similar to the beam formula, but since the plate is very wide we have a situation similar to plain strain. • For a unit width beam, flexural rigidity D= EI = Et 3 /12.

• For a unit width plate, flexural rigidity /(1-ν 2)= Et 3 /[12(1-ν 2)]. D= EI

• This thin plate theory is also called the “Kirchhoff” plate theory. MEEM4405 Introduction to Finite Element Analysis

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Mindlin Plate Theory • Mindlin Plate Theory assumes that transverse shear deformation occurs.


Mindlin Plate Theory • The deformations and strains are therefore given by:


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Large Displacements and Membrane Forces • A beam with fixed supports will exhibit “string action” axial forces as shown.

• If we consider both string action and bending stresses, a beam can carry a distributed load of:


Large Displacements and Membrane Forces • A similar situation arises with plates, however linear plate elements are not set up to handle “membrane” forces. • If w / t is large (e.g. greater than 0.1), a nonlinear analysis must be performed using elements that handle membrane forces. • In general, however, tensile membrane forces will have a stiffening effect and compressive membrane forces will decrease stiffness. MEEM4405 Introduction to Finite Element Analysis

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7.2 Plate Finite Elements • Plate elements must be able to show constant σ x , σ y and τ xy at each z level to pass a patch test. They must pass the test for constant M x , M y and M xy. • Kirchhoff elements can be implemented with 12 dof elements.

• However, they are awkward to use because of the question of how to handle the twist dof. MEEM4405 Introduction to Finite Element Analysis

Plate Finite Elements • Mindlin plate elements are more common. • The displacement interpolation is given by:

• N i can be the same shape functions as for

Q4 and Q8 quadrilateral elements. MEEM4405 Introduction to Finite Element Analysis

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Plate Finite Elements • “Discrete Kirchhoff Elements ” are also available as triangular elements.


Support Conditions • Support Conditions are similar to those for beams:

θn, M n – rotation and moment normal to edge θs, M s– rotation and moment perpendicular to edge

For Mindlin plates, do not restrain θn, to avoid accuracy problems. MEEM4405 Introduction to Finite Element Analysis

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Test Cases • For plate elements, patch tests and single element tests include the cases shown

• Many element formulations perform poorly for these tests. MEEM4405 Introduction to Finite Element Analysis

7.4 Shells and Shell Theory • Shell elements are different from plate elements in that: – They can be curved – They carry membrane and bending forces

• A thin shell structure can carry high loads if membrane stresses predominate. • However, localized bending stresses will appear near load concentrations or geometric discontinuities. MEEM4405 Introduction to Finite Element Analysis

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Shells and Shell Theory • Localized bending stresses


Shells and Shell Theory • For a cylindrical shell of radius R and thickness t , the localized bending dies out after a distance λ :

• Membrane stresses do not die out.


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7.5 Shell Finite Elements • The most simple shell elements combine a membrane element and bending element. E.g. combining plane stress and plate elements. • These elements are flat. • When flat elements, it is important that elements are not all coplanar where they meet at a node.


Shell Finite Elements • Curved shell elements can be derived from shell theory • Isoparametric shell elements can also be obtained by starting with a solid element and reducing degrees of freedom. • Thin shell behavior varies widely between formulations and should be tested before use. MEEM4405 Introduction to Finite Element Analysis

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Shells of Revolution • In axisymmetric problems, shells resemble beam elements. • Conical elements have problems similar to flat shell elements.


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Lecture on Plates and Shells

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