finite element models of the fsdt shear and membrane...

JN Reddy

CONTENTS

FINITE ELEMENT MODELS OF NONLINEAR PLATE BENDING

• Finite element models of the FSDT• Shear and Membrane Locking• Numerical Examples

Nonlinear Plate Bending: 1

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Kinematics of the Classical and Shear Deformation Plate Theories

x

Undeformed Edge

First-Order (Mindlin)

Plate Theory (FST)

xz

z

u

xφ

xφ


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GOVERNING EQUATIONS OF THE FSDT with the von Karman Nonlinearity

2 231 1 1

2 2

2 232 1 1

2 2

3 31 2

,

,

xxx xz x

yyy yz y

yxxy

uu u w wzx x x x x x

uu v w wzy y y y y y

u uu u u v w w zy x x y y x x y y

x

Nonlinear strains

12

ji m mij

j i i j

uu u ux x x x

2 2 231 1 1 2

1 1 1 1 1 1 1

1 1 1 12 2 2 2

m mxx

u u uu u u ux x x x x x x

Von Karman Nonlinear strains


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EQUATIONS OF MOTION OF FSDT

22

0 12 2

22

0 12 2

xyxx xx

xy yy yy

NN uf I I

x y t t

N N vf I I

x y t t

yxxx xy

xy yy

QQ w wN N

x y x x y

w w wN N q I

y x y t

2

0 2

2 22 12 2

xyxx xx

MM uQ I I

x y t t

2 22 12 2

xy yy yy

M M vQ I I

x y t t


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FINITE ELEMENT MODELS OF (FSDT)

Weak Forms (from the principle of virtual displacements)

2

0 2

2

0 2

0

0

e e

e e

xx xy x n nn

xy yy y s ns

u u uN N u f I u dxdy u N ds

x y t

v v vN N v f I v dxdy u N ds

x y t


2

0 2

2

2 2

2

2 2

0

0

0

e e

e e

e

x y n

x x xxx xy x x x n nn

y y yxy yy y y y s

w w wQ Q w N I w dxdy wQ ds

x y t

M M Q I dxdy M dsx y t

M M Q I dxdyx y t

e

nsM ds

Qn = Qxnx+Qyny; Mnn = Mxxnx+Mxyny; Mns = Mxynx+Myyny

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THE FIRST-ORDER SHEAR DEFORMATION THEORYStress Resultants (Nonlinear)

22

11 12 16

11 12 16

2

12

1 12 2

12

xx

y y Tx xxx

yy

u w v w u v w wN A A Ax x y y y x x y

B B B Nx y y x

u wN Ax x

2

22 26

12 22 26

22

16 26

12

1 12 2

y y Tx xyy

xy

v w u v w wA Ay y y x x y

B B B Nx y y x

u w v wN A Ax x y y

66

16 26 66y y Tx x

xy

u v w wAy x x y

B B B Nx y y x

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THE FIRST-ORDER SHEAR DEFORMATION THEORYStress Resultants (Nonlinear)

22

11 12 16

2 2 2

11 12 162 2

2

12 22

1 12 2

2

1 12 2

xx

Txx

yy

u w v w u vM B B Bx x y y y x

w w wD D D Mx yx y

u w v wM B Bx x y y

2

26

2 2 2

12 22 262 2

22

16 26 66

2 2

16 262 2

2

1 12 2

2

Tyy

xy

u vBy x

w w wD D D Mx yx y

u w v w u vM B B Bx x y y y x

w wD D Dx y

2

66

55 45 45 44

Txy

x s x s y y s x s y

w Mx y

w w w wQ K A K A Q K A K Ax y x y

;

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FINITE ELEMENT MODELS OF (FSDT) (continued)

w(x; y; t) =mX

j=1

wj(t)Ã(1)j (x; y)

Áx(x; y; t) =nX

j=1

Sxj(t)Ã(2)j (x; y); Áy(x; y; t) =

nX

j=1

Syj(t)Ã(2)j (x; y)

Finite element approximation


(0) (0)1 1

( ) ( , ), ( ) ( , )m m

j j j jj j

u u t x y v v t x y

Although, in general, different degree of interpolation can be used for various field variables, the same degree of interpolation is used for all variables:

(0) (1) (2)j j j

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FINITE ELEMENT MODELS OF FSDT (continued)

M11ij = I0Mij ; M22ij = M

33ij = I2Mij; Mij =

Z

eÃiÃj dx dy

K11ij =

Z

e

µA55

@Ãi@x

@Ãj@x

+ A44@Ãi@y

@Ãj@y

¶dx dy

K12ij =

Z

eA55

@Ãi@x

Ãj dx dy

K13ij =

Z

eA44

@Ãi@y

Ãj dx dy

Semidiscrete finite element model

Use reduced integration

to avoid shear locking


M 0 0 M 0 K K K K Kuv0 M 0 0 M K K K K Kw0 0 M 0 0 K K K K KSM 0 0 M 0 K K K K KS0 M 0 0 M K K K K K

11 14 11 12 13 14 15

22 25 21 22 23 24 25

33 31 32 33 34 35

14 44 41 42 43 44 45

25 55 51 52 53 54 55

x

y

FuFv

w FS FS F

1

2

3

4

5

x

y

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FINITE ELEMENT MODELS OF FSDT (continued)

K22ij =

Z

e

µD11

@Ãi@x

@Ãj@x

+ D66@Ãi@y

@Ãj@y

+ A55ÃiÃj

¶dxdy

K23ij =

Z

e

µD12

@Ãi@x

@Ãj@y

+ D66@Ãi@y

@Ãj@x

¶dx dy

K33ij =

Z

e

µD66

@Ãi@x

@Ãj@x

+ D22@Ãi@y

@Ãj@y

+ A44ÃiÃj

¶dx dy

F 1i =

Z

eqÃi dxdy +

I

¡eQnÃi ds

F 2i =

I

¡eM̂nnÃi ds; F

3i =

I

¡eM̂nsÃi ds

Finite Elements

a

b

z

x

y4

32

1

a

b

z

x

y4

32

1

5

6

7

8

9

( , , , , )x y

u v w ( , , , , )x y

u v w

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Shear and Membrane Locking (Revisit)


Shear Locking

Membrane Locking

Use reduced integration to evaluate all shear stiffnesses(i.e., all Kij that contain transverse shear terms)

Use reduced integration to evaluate all membrane stiffnesses(i.e., all Kij that contain von Kármán nonlinear terms)

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Simply Supported Plate (SS2)

NUMERICAL EXAMPLES

x

y

b2

a 2

= w= u φx= 0 = w= v φy= 0

φx = 0= u φy= 0= v

¹w = w0(0; 0)E2h

3

a4q0; ¹¾xx = ¾xx(0; 0;

h

2)

h2

b2q0

¹¾yy = ¾yy(0; 0;h

4)

h2

b2q0; ¹¾xy = ¾xy(

a

2;b

2;¡h

2)

h2

b2q0

¹¾xz = ¾xz(a

2; 0;¡h

2)

h

bq0; ¹¾yz = ¾yz(0;

b

2;h

2)

h

bq0

¾xx(A;A;h

2); ¾xy(B;B;¡

h

2); ¾xz(B;A;¡

h

2)

Table: The Gauss point locations at which the stresses are computedin the ¯nite element analysis of simply supported plates.

Point 2L 4L 8L 1Q9 2Q9 4Q9

A 0:125a 0:0625a 0:0312a 0:1056a 0:0528a 0:0264aB 0:375a 0:4375a 0:4687a 0:3943a 0:4472a 0:4736a

Plate bending: 12

xy z

xy

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Effect of Quadrature Rules and Shear Deformation on Deflection and Stresses

a=h Mesh ¹w £ 102 ¹¾xx ¹¾xy ¹¾xz

Finite Element Solutionsy10 2L{F 2.4742 0.1185 0.0727 0.2627

2L{S 4.7120 0.2350 0.1446 0.27502L{R 4.8887 0.2441 0.1504 0.27501Q{F 4.5304 0.2294 0.1610 0.28131Q{S 4.9426 0.2630 0.1639 0.28471Q{R 4.9711 0.2645 0.1652 0.28864L{F 3.8835 0.2160 0.1483 0.33664L{S 4.7728 0.2661 0.1850 0.33564L{R 4.8137 0.2684 0.1869 0.33562Q{F 4.7707 0.2699 0.1930 0.34372Q{S 4.7989 0.2715 0.1939 0.34242Q{R 4.8005 0.2716 0.1943 0.34258L{F 4.5268 0.2590 0.1891 0.37008L{S 4.7966 0.2743 0.2743 0.20148L{R 4.7866 0.2737 0.2737 0.20084Q{F 4.7897 0.2749 0.2044 0.37374Q{S 4.7916 0.2750 0.2043 0.37354Q{R 4.7917 0.2750 0.2044 0.3735

Analytical Solutions

[15] 4.7914 0.2762 0.2085 0.3927

Square plate under UDLF – full integration

S – Selective integrationR- Reduced integration

L – Linear elementQ- Qudratic element

Plate bending: 13

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a=h Mesh ¹w £ 102 ¹¾xx ¹¾xy ¹¾xz

Finite Element Solutions

100 2L{F 0.0469 0.0024 0.0014 0.26352L{S 4.4645 0.2350 0.1446 0.27502L{R 4.6412 0.2441 0.1504 0.27501Q{F 4.0028 0.2040 0.1591 0.27331Q{S 4.7196 0.2629 0.1643 0.28371Q{R 4.7483 0.2645 0.1652 0.28864L{F 0.1819 0.0108 0.0071 0.34624L{S 4.5481 0.2661 0.1850 0.33564L{R 4.5890 0.2684 0.1869 0.3356

2Q{F 4.4822 0.2644 0.1893 0.34852Q{S 4.5799 0.2715 0.1941 0.34142Q{R 4.5815 0.2716 0.1943 0.34258L{F 0.6497 0.0401 0.0275 0.38478L{S 4.5664 0.2737 0.2008 0.36918L{R 4.5764 0.2743 0.2014 0.36914Q{F 4.5530 0.2741 0.2020 0.37494Q{S 4.5728 0.2750 0.2044 0.37344Q{R 4.5729 0.2750 0.2044 0.3735

Analytical Solutions

[15] 4.5698 0.2762 0.2085 0.3927

Effect of Quadrature Rules and Shear Deformation on Deflection and Stresses

Plate bending: 14

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Bending of Simply Supported Plates


x

y

a 2

b2

b2

a 2

0xu w = = =0yv w = = =

0yv w = = = 0xu w = = =

SS-1

x

y

a 2

b2

b2

a 2

0u v w= = =

0u v w= = =

SS-3

0u v w= = =

0u v w= = =

0 50 100 150 200 250Load parameter,

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Defle

ction

, SS-3 (CPT)SS-1 (CPT)

SS-1 (FSDT)

SS-3 (FSDT)

0

4

4,q aww P

h Eh

wwh

0

4

4

q aPEh

Isotropic plate under uniformly distributed transverse load

610 1 7 8 10 0 3in, in, . psi, .a b h E

Displacements

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Bending of Simply Supported Plates


Isotropic plate under uniformly distributed transverse load

P0 50 100 150 200 250

Load parameter,

0

4

8

12

16

20

24

Stre

sses

,SS-3 (CPT)

SS-1 (CPT)

SS-1 (FSDT)

SS-3 (FSDT)

Membrane stresses

xx610 1

7 8 10 0 3in, in,

. psi, .a b hE

2

2,xx xx

aEh

0

4

4

q aPEh

Stresses

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0 20 40 60 80 100 120Load parameter,

0.0

1.0

2.0

3.0

4.0

Defle

ctio

n,

Mesh of 5-Q9 elements

w0/h

(q0a4/Eh4)

E = 106 psi, ν = 0.3a = 100 in., h = 10 in.

••

• •

•

•

••

•

•

• •

•

•

•

•

••

•••

•

• •

••

•

••

x

y

a = 100 in.

E = 106 psi, ν = 0.3

h = 10 in.

0at0,0

===

yyv φ

0at0,0

===

xxu φ

=====

edge clamped on the00

yxwuv

φφ

1 234

5 10 201525

26

272829

9

241914

Clamped Circular Plate under UDL


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0.0 0.4 0.8 1.2 1.6 2.0Pressure, q0 (psi)

0.00

0.10

0.20

0.30

0.40

0.50

,

0 (

)

Linear

Nonlinear

Experimental [8]CLPTFSDT

Simply Supported (SS2) Orthotropic* Plate

Geometry and Material Propertiesa = b = 12 in, h = 0.138 inE1 = 3×10

6 psi, E2 = 1.28×106 psi

G12 = G23 = G13 = 0.37×106 psi

ν12 = ν23 = ν13 = 0.32

[8] Zaghloul, S. A. and Kennedy, J. B., ``Nonlinear Behavior of Symmetrically Laminated Plates,” Journal of Applied Mechanics, 42, 234-236, 1975.


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DEFLECTION VS. LOAD PARAMETER FOR (0/90/90/0) LAMINATE UDL


0.0 0.4 0.8 1.2 1.6 2.0 2.4Intensity of the distributed load,

0.00

0.05

0.10

0.15

0.20

Defle

ction

,

Linear (FSDT; CC)

Nonlinear (FSDT; CC)

Square laminate (0/90/90/0) under UDL h = 0.096 in, a = b = 12 in.

Linear (FSDT; SS3)

Nonlinear (FSDT; SS3)

E1 = 1.8282 × 106 psi, E2 = 1.8315 × 106 psi,

G12 = G13 = G23 = 0.3125 × 106 psi,

ν12 = 0.2395

w

q0

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DEFLECTION VS. LOAD PARAMETER FOR TWO-AND SIX-LAYER CROSS-PLY LAMINATES UDL


E1/E2 = 40, G12 = G13 = 0.6E2 , G23 = 0.5E2

0 400 800 1200 1600Intensity of the distributed load, q0

0.00

0.10

0.20

0.30

0.40

0.50

0.60

Defle

ction

, w0

Linear (2 layers)

Nonlinear, CC[FSDT, (0/90)]

Clamped square laminates under UDL h = 0.3 in, a = b = 12 in (a/h = 40)

Nonlinear, CC [FSDT, (0/90/0/90/0/90)]

Linear (6 layers)

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SUMMARY

In this lecture we have covered the following topics:

• Governing equations of FSDT• Finite element models of FSDT• Shear and membrane locking• Numerical examples


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finite element models of the fsdt shear and membrane...

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