FEM analysis of plates and shells

FEM analysis of plates and shells

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

With thanks to:

M. Radwańska, A. Wosatko

ANSYS, Inc. http://www.ansys.com

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 mechanice kostrukcji. Oficyna Wyd. PW, Warszawa, 2005.

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

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)

Plate/slab - 2.5D structure [1]

Bending Transverse shear Torsion

Figures from [1]

Shell - 2.5D structure

Figures from [1]

ANSYS simulation:

Shell deflection under constant load

Plate in bending

Fundamental unknown:

deflection w (x , y )

Figures taken from [2] Generalized stresses

Bending - generalized strains and stresses

Thin plate theory of Kirchhoff-Love

Curvatures and twist (spaczenie) e^m= {κx, κy, κxy}

Bending and torsional moments m = {mx, my, mxy}

Figures taken from [2]

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)

Geometry of shell

Figures taken from [2]

Shell - generalized stresses

Stresses in shell cross-section Figures taken from [2]

Membrane and bending forces (transverse shear neglected)

Finite elements for plates and shells

Reissner-Mindlin theory of moderately thick shells

Rotation angles

approximated independently

Ahmad finite element - degenerated continuum

Transverse shear taken into account Figures from [2]

Geometrical nonlinearity

Equilibrium of discretized system [3]:

K ∆d = f_ext^t+∆t− f_int^t

where tangent stiffness matrix:

K = K0+Ku+Kσ

K0- linear stiffness matrix Ku - initial displacement matrix

(discrete kinematic relations matrix B dependent on displacements) K_σ - initial stress matrix (dependent on generalized stresses)

Karman theory of moderately large deflections

Deflection of the order of thickness admitted [3]

Medium plane deflections εx = ε^L_x+ ε^N_x =^∂u_∂x +¹₂ ^∂w_∂x
² ε_y = ε^L_y + ε^N_y = ^∂v_∂y +¹₂

∂w

∂y

2

γ_xy = γ_xy^L + γ_xy^N = ^∂u_∂y +^∂v_∂x +^∂w_∂x ^∂w_∂y Curvatures and twist as in linear theory

κx = κ^L_x = −^∂_∂x^2w2, κy = κ^L_y = −^∂_∂y^2w2, χxy = χ^L_xy = −2_{∂x ∂y}^∂2w

Two equations of Karman theory for moderately large plate deflections

∇²∇²F (x , y ) + ^Eh₂L(w , w ) = 0 D^m∇²∇²w (x , y ) − L(w , F ) − ˆpz= 0 where:

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

Theory of moderately large deflections

FEM approximation [3]

Total potential energy (additional membrane state energy) Π˜^m= U^m+ ˜Uⁿ− W^m

Discretization

uⁿ=

 u(x , y ) v (x , y )



=

 N_u 0 N_v 0

  dⁿ d^m



w^m= [w (x , y )] =

0 Nw



 dⁿ d^m



Nonlinear kinematic equations B_{(6×LSSE )}= B^L_{(6×LSSE )}+ B^N_{(6×LSSE )}

B^L_{(6×LSSE )}=

 Bⁿ 0 0 B^m



, B^N_{(6×LSSE )}=

 0 Bⁿ_w 0 0



Theory of moderately large deflections

FEM approximation [3]

Matrices of discrete kinematic relations

Bⁿ=





N_u,x N_{v ,y} N_u,y+ N_{v ,x}



, B^m=





−Nw ,xx

−N_{w ,yy}

−2N_{w ,xy}





Bⁿ_w =





w,xNw ,x

w,yNw ,y

w_,xN_{w ,y} + w_,yN_{w ,x}





Deflection gradients

g =

 w,x

w,y



=

 Nw ,x

Nw ,y



d^m= G^md^m

Theory of moderately large deflections

FEM approximation [3]

Element tangent stiffness k^e_T = k^e₀+ k^e_u+ k^e_σ

k^e₀= Z Z

A^e

 B^nTDⁿBⁿ 0 0 B^mTD^mB^m

 dA

k^e_u= Z Z

A^e

 0 B^nTDⁿBⁿ_w B^nT_w DⁿBⁿ 0

 dA

k^e_σ= Z Z

A^e

 0 0

0 G^mTSⁿG^m



dA, Sⁿ=

 n_x n_xy n_xy n_y



Vector of nodal forces representing stresses

f_int^e = Z Z

A^e

 B^nT 0 B^nT_w B^mT

  Sⁿ S^m

 dA

Square plate [3]

Moderately large rotations

ˆ pz

x

z, w y

a

a C

Dane:

L = Lx = Ly = 2a = 1.0 m h = 0.002 m

E = 200000 MPa, ν = 0.25

Relation deflection-loading:

0.001 0.003

0.30

0.15

0.002

wC [m]

ˆ pz [kPa]

FEM

(geometrical nonlinearity)

wC= 0.00406^ˆpz_D^L4 linear solution: