Plates and Shells This course explores the following topics: derivation of elastic and plastic stress-strain relations for plate and shell elements; the bending and buckling of rectangular plates; nonlinear geometric effects; post-buckling and ultimate strength of cold formed sections and typical stiffened panels used in naval architecture; the general theory of elastic shells and axisymmetric shells; buckling, crushing and bending strength of cylindrical shells with application to offshore structures; and the application to crashworthiness of vehicles and explosive and impact loading of structures. The class is taught during the first half of term.
7 Plates and Shells MEEM4405 Introduction to Finite Element Analysis 7.1 Plate Formulation • Plates may be considered 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 = plane, with thickness t, and neglecting shear strain: w = w( x, y ) ∂w u = −z ∂x ∂w v = −z ∂y γyz = γzx = MEEM4405 Introduction to Finite Element Analysis Thin Plate Formulation • A differential slice from plate: MEEM4405 Introduction to Finite Element Analysis Thin Plate Formulation • For the thin plate, we assume σz = Therefore: MEEM4405 Introduction to Finite Element Analysis Thin Plate Formulation • These stresses give rise to moments: • Maximum stresses are therefore given by: 6M x 2z since σ x = σ x , t t 6M This is similar to the beam formula, σy = y , but since the plate is very wide we t have a situation similar to plain 6M τ xy = xy strain Flexural rigidity D=EI=Et3/12 t with EI=Et3/12, but since strain it is σx = MEEM4405 Introduction to Finite Element Analysis very wide (like 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=Et3/12 • For a unit width plate, flexural rigidity D=EI/(1-ν2)=Et3/[12(1-ν2)] • This thin plate theory is also called the “Kirchhoff” plate theory MEEM4405 Introduction to Finite Element Analysis Mindlin Plate Theory • Mindlin Plate Theory assumes that transverse shear deformation occurs MEEM4405 Introduction to Finite Element Analysis Mindlin Plate Theory • The deformations and strains are therefore given by: MEEM4405 Introduction to Finite Element Analysis 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: MEEM4405 Introduction to Finite Element Analysis 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 7.2 Plate Finite Elements • Plate elements must be able to show constant σx, σy and τxy at each z level to pass a patch test They must pass the test for constant Mx, My and Mxy • 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: • Ni can be the same shape functions as for Q4 and Q8 quadrilateral elements MEEM4405 Introduction to Finite Element Analysis Plate Finite Elements • “Discrete Kirchhoff Elements” are also available as triangular elements MEEM4405 Introduction to Finite Element Analysis Support Conditions • Support Conditions are similar to those for beams: θn, Mn – rotation and moment normal to edge θs, Ms– rotation and moment perpendicular to edge For Mindlin plates, not restrain θn, to avoid accuracy problems MEEM4405 Introduction to Finite Element Analysis 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 Shells and Shell Theory • Localized bending stresses MEEM4405 Introduction to Finite Element Analysis 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 not die out MEEM4405 Introduction to Finite Element Analysis 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 MEEM4405 Introduction to Finite Element Analysis 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 10 Shells of Revolution • In axisymmetric problems, shells resemble beam elements • Conical elements have problems similar to flat shell elements MEEM4405 Introduction to Finite Element Analysis 11 ... Element Analysis Shells and Shell Theory • Localized bending stresses MEEM4405 Introduction to Finite Element Analysis Shells and Shell Theory • For a cylindrical shell of radius R and thickness... Conditions are similar to those for beams: θn, Mn – rotation and moment normal to edge θs, Ms– rotation and moment perpendicular to edge For Mindlin plates, not restrain θn, to avoid accuracy problems... 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