FEM analysis of plates and shells

Jerzy Pamin

e-mail: JPamin@L5.pk.edu.pl

With thanks to:

M. Radwańska, A. WosatkoANSYS, Inc. http://www.ansys.com

Comp.Meth.Civ.Eng., II cycle

Lecture contents

Classification of models and finite elements

Finite elements for plate bending

Finite elements for shells

Theory of moderately large deflections

[1] T. Kolendowicz Mechanika budowli dla architektów. Arkady, Warszawa,1996.[2] G. Rakowski, Z. Kacprzyk. Metoda elementow skończonych w mechanicekostrukcji. Oficyna Wyd. PW, Warszawa, 2005.

[3] M. Radwańska. Ustroje powierzchniowe, podstawy teoretyczne orazrozwiązania analityczne i numeryczne. Wydawnictwo PK, Kraków, 2009.

Comp.Meth.Civ.Eng., II cycle

Classification of models and finite elements

Reduction of model dimensions:I bar structures (one-dimensional geometry)I surface structures (small third dimension)I volume structures (three-dimensional)

Finite elements for mechanics:I 1D - truss (kratowy)I 1.5D - beam (belkowy), frame (ramowy)I 2D - plane stress - panel (PSN), plane strain (PSO), axial symmetry

(symetria osiowa)I 2.5D - plate/slab (płytowy), shell (powłokowy)I 3D - volume (bryłowy)

Comp.Meth.Civ.Eng., II cycle

Plate/slab - 2.5D structure [1]

Bending Transverse shear Torsion

Figures from [1]

Comp.Meth.Civ.Eng., II cycle

Shell - 2.5D structure

Figures from [1]

ANSYS simulation:Shell deflectionunder constant load

Comp.Meth.Civ.Eng., II cycle

Plate in bending

Fundamental unknown:deflection w(x , y)

Figures taken from [2]Generalized stresses

Comp.Meth.Civ.Eng., II cycle

Bending - generalized strains and stressesThin plate theory of Kirchhoff-Love

Curvatures and twist (spaczenie)em = {κx , κy , κxy}

Bending and torsional momentsm = {mx ,my ,mxy}

Figures taken from [2]

Comp.Meth.Civ.Eng., II cycle

Rectangular element for plate bending

Nodal degrees of freedom and forces

Hermite shape functions

Figures from [2]

Be careful with imposing boundary conditions (kinematic and/or static)

Comp.Meth.Civ.Eng., II cycle

Geometry of shell

Figures taken from [2]

Comp.Meth.Civ.Eng., II cycle

Shell - generalized stresses

Stresses in shell cross-sectionFigures taken from [2]

Membrane and bending forces(transverse shear neglected)

Comp.Meth.Civ.Eng., II cycle

Finite elements for plates and shellsReissner-Mindlin theory of moderately thick shells

Rotation anglesapproximated independently

Ahmad finite element- degenerated continuum

Transverse sheartaken into account

Figures from [2]

Comp.Meth.Civ.Eng., II cycle

Geometrical nonlinearity

Equilibrium of discretized system [3]:

K ∆d = ft+∆text − ftint

where tangent stiffness matrix:

K = K0 + Ku + Kσ

K0 - linear stiffness matrix

Ku - initial displacement matrix(discrete kinematic relations matrix B dependent on displacements)

Kσ - initial stress matrix (dependent on generalized stresses)

Comp.Meth.Civ.Eng., II cycle

Karman theory of moderately large deflectionsDeflection of the order of thickness admitted [3]

Medium plane deflectionsεx = ε

Lx + ε

Nx =

∂u∂x +

12

(∂w∂x

)2εy = ε

Ly + ε

Ny =

∂v∂y +

12

(∂w∂y

)2γxy = γ

Lxy + γ

Nxy =

∂u∂y +

∂v∂x +

∂w∂x

∂w∂y

Curvatures and twist as in linear theoryκx = κ

Lx = −∂

2w∂x2 , κy = κ

Ly = −∂

2w∂y2 , χxy = χ

Lxy = −2 ∂

2w∂x∂y

Two equations of Karman theory for moderately large plate deflections

∇2∇2F (x , y) + Eh2 L(w ,w) = 0Dm∇2∇2w(x , y)− L(w ,F )− p̂z = 0where:F (x , y) - stress function (nx = F,yy , ny = F,xx , nxy = −F,xy )L(a, b) = a,xxb,yy − 2a,xyb,xy + a,yyb,xx

Comp.Meth.Civ.Eng., II cycle

Theory of moderately large deflectionsFEM approximation [3]

Total potential energy (additional membrane state energy)

Π̃m = Um + Ũn −Wm

Discretization

un ={u(x , y)v(x , y)

}=

[Nu 0Nv 0

]{dn

dm

}

wm = [w(x , y)] =[

0 Nw]{ dn

dm

}Nonlinear kinematic equations

B(6×LSSE) = BL(6×LSSE) + BN(6×LSSE)

BL(6×LSSE) =[

Bn 00 Bm

], BN(6×LSSE) =

[0 Bnw0 0

]

Comp.Meth.Civ.Eng., II cycle

Theory of moderately large deflectionsFEM approximation [3]

Matrices of discrete kinematic relations

Bn =

Nu,xNv ,yNu,y + Nv ,x

, Bm = −Nw ,xx−Nw ,yy−2Nw ,xy

Bnw =

w,xNw ,xw,yNw ,yw,xNw ,y + w,yNw ,x

Deflection gradients

g ={w,xw,y

}=

[Nw ,xNw ,y

]dm = Gmdm

Comp.Meth.Civ.Eng., II cycle

Theory of moderately large deflectionsFEM approximation [3]

Element tangent stiffness

keT = ke0 + k

eu + k

eσ

ke0 =∫ ∫

Ae

[BnTDnBn 0

0 BmTDmBm

]dA

keu =∫ ∫

Ae

[0 BnTDnBnw

BnTw DnBn 0

]dA

keσ =∫ ∫

Ae

[0 00 GmTSnGm

]dA, Sn =

[nx nxynxy ny

]Vector of nodal forces representing stresses

feint =∫ ∫

Ae

[BnT 0BnTw B

mT

] [Sn

Sm

]dA

Comp.Meth.Civ.Eng., II cycle

Square plate [3]Moderately large rotations

p̂z

x

z ,w

y

a

a

C

Dane:L = Lx = Ly = 2a = 1.0 mh = 0.002 mE = 200000 MPa, ν = 0.25

Relation deflection-loading:

0.001 0.003

0.30

0.15

0.002

wC [m]

p̂z [kPa]

FEM(geometrical nonlinearity)

wC = 0.00406p̂zL4

D

linear solution:

Comp.Meth.Civ.Eng., II cycle

Classification of models and finite elementsFinite elements for plate bendingFinite elements for shellsTheory of moderately large deflections