CONTINUUM SHELL ELEMENT AND POSTBUCKLING ANALYSIS

Description of continuum shell element(principle of virtual displacements)

Post-buckling analysis of composite panels and comparison with experimental results

CONTINUUM SHELL ELEMENT

J N Reddy

1 1,X x

2 2,X x

3 3,X x

e3ˆ e1ˆE1ˆE3

ˆ

e2ˆE2ˆ

00Ω Ω

1Ω

10S

0d V

12Ω 2

rΩ2Ω1

2rΩ

0d S

10E

0

Equilibrium iterations

0S

20S20E

Continuum Shell Element 2

CONTINUUM SHELL ELEMENT

Z

0V

t+¢t0 Sij ±(t+¢t

0 "ij)0dV = t+¢tR

t+¢tR =

Z

0A

t+¢t0 tk ±uk

0dA +

Z

0V

0½ t+¢t0 fk ±uk

0dV

Z

t+¢tV

t+¢t¾ij ±t+¢teijt+¢tdV = t+¢tR

Z

t+¢tV

t+¢ttij ±(t+¢teij)t+¢tdV =

Z

0V

t+¢t0 Sij ±(t+¢t

0 "ij)0dV

Principle of virtual displacement applied to the configuration to be determined

Use energy conjugacy to write all integrals on the reference configuration

Continuum Shell Element 3J N Reddy

CONTINUUM SHELL ELEMENT

±(t+¢t0 "ij) = ±(t

0"ij) + ±(0"ij) = ±(0"ij)

0Sij = 0Cijrs 0"rs; 0Sij = 0Cijrs 0`rs

Z

0V0Cijrs 0`rs ±(0"ij)

0dV +

Z

0V

t0Sij ±(0´ij)

0dV

= t+¢tR ¡Z

0V

t0Sij ±(0`ij)

0dV

Final virtual work statement

0`ij =1

2(0ui;j + 0uj;i + t

0uk;i 0uk;j + 0uk;it0uk;j)

Decompositionst+¢t0 Sij = t

0Sij + 0Sij

t+¢t0 "ij = t

0"ij + 0"ij 0´ij =1

20uk;i 0uk;j

0"ij = 0`ij + o´ij

Continuum Shell Element 4J N Reddy

Assumed Kinematics of the Shell Element

CONTINUUM SHELL ELEMENT(continued)

Continuum Shell Element 5J N Reddy

txi =nX

k=1

Ãk(»; ´)[txki +

³

2hk

tek3i]

tui = txi ¡ 0x1 =nX

k=1

Ãk(»; ´)[tukk +

³

2hk(tek

3i ¡ 0ek3i)]

ui = t+¢tui ¡ tui =nX

k=1

Ãk(»; ´)[uki +

³

2hk(t+¢tek

3i ¡ tek3i)]

Continuum Shell Element 6

E = 3103 Ν/mm2, ν = 0.3R = 2540 mm, L = 254 mm

(b)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 5 10 15 20Center deflection, u3 (mm)

Load

, P(k

N)

•

-

-

-

L

Rx1, u1

x2, u2

x3, u3

h = 6.35 mm

Free edge

•••••

••••••••••

••••••••••

Simply supported:u1 = u2 = u3 = φ2 = 0

Symmetry line: u1 = φ1 = 0

Simply supported:u1 = u2 = u3 = φ2 = 0

0.1LSymmetry line:

u2 = φ2 = 0

P

A NUMERICAL EXAMPLE

J N Reddy

POSTBUCKLING AND FAILURE ANALYSIS - AN EXAMPLE

• Nonlinear FE analysis • Comparison with experimental results of

Starnes and Rouse• Progressive failure analysis

a

P

x

y

b

0=w0

u0 constant=w0 0=

∂x∂w0

0=∂x

∂w0

=u0 0=w0

u0

a = 50.8 cm (20 in.), b = 17.8 cm (7 in.), hk = 0.14 mm (0.0055 in.)

24-ply laminate: [45/-45/02 /45/-45/02 /45/-45/0/90]s

E1 = 131 Gpa (19,000 ksi), E2 = 13 Gpa (1,890 ksi),

G12 = 6.4 Gpa (930 ksi), ν12 = 0.38 (graphite-epoxy)

J N Reddy Continuum Shell Element 7

(a) Typical panelwith test fixture

(b) A transverse shear failure mode

Experimental Setup and Failure Region(Starnes & Rouse, NASA Langley)

J N Reddy Continuum Shell Element 8

POST-BUCKLING OF A COMPOSITE PANELUNDER IN-PLANE LOADING

20.0

in

7.0 in

Finite Element Models

• Mesh has six elements per bucklingmode half wave in each direction

• Elements used:(1) Four-node C1 -based flat shell

element (STAGS)(2) Nine-node continuum shell

element (Chao & Reddy)(3) For-node and nine-node ANS

shell elements (Park & Stanley)

• All meshes have 72 elements (91 nodes for the meshes with four-node elements and 325 nodesfor meshes with nine-node elements).

Continuum Shell Element 9J N Reddy

Comparison of the Experimental (Moire)and Analytical Out-of-Plane Deflection

Patterns

J N Reddy

ExperimentalTheoretical (FEM)

(b) Photograph of Moiréfringe pattern

(a) Contour plot of theanalytical results

Continuum Shell Element 10

Comparison of the Experimental and Analytical Solutions for End Shortening

J N Reddy Continuum Shell Element 11

Postbuckling of a Composite Panel

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0End shortening, u0/ucr

0.0

0.5

1.0

1.5

2.0

2.5

p,

c

AnalysisTest

0.0 0.5 1.0 1.5 2.0 2.5Maximum deflection, w0/h

0.0

0.5

1.0

1.5

2.0

2.5

p,

AnalysisTest

J N Reddy Continuum Shell Element 12

Postbuckling of a Composite Panel(Stress Distributions)

P

x

y

b

a 2

a 2

0.0 0.2 0.4 0.6 0.8 1.0y/b

-80-70-60-50-40-30-20-10

01020

,

xx (

)

-500

-400

-300

-200

-100

0

100(MPa)

P/Pcr = 2.1

P/Pcr = 1.5

P/Pcr = 1.0

Axi

al S

tres

s, s x

x(k

si)

J N Reddy Continuum Shell Element 13

0.0 0.2 0.4 0.6 0.8 1.0y/b

-9.0-8.0-7.0-6.0-5.0-4.0-3.0-2.0-1.00.01.0

,

xz (

)

-60

-50

-40

-30

-20

-10

0

(MPa)

P/Pcr = 1.0

P/Pcr = 1.5

P/Pcr = 2.1

from constitutive relationsfrom equilibrium relations

Shear Stress, σxz (ksi)

Postbuckling of a Composite Panel(Stress Distributions)

J N Reddy Continuum Shell Element 14

Description of continuum shell element(principle of virtual displacementsconjugate pairs, decompositions, approximation of the geometry and displacement field)

Post-buckling analysis of composite panels and comparison with experimental results

(problem description, boundary conditions, fringe patters of the displacement field, post-buckling path; failure analysis)

J N Reddy Continuum Shell Element 15

SUMMARY