Large deflection analysis of planar solids based on the ...paulino.ce.gatech.edu/conferences/presentations/09yu_largedef... · Large deflection analysis of planar solids based on

Large deflection analysis of planar solids

based on the Finite Particle Method

Presenter: Ying Yu

Advisors: Prof. Glaucio H. Paulino

Prof. Yaozhi Luo

Date: July 17th

1 Department of Civil and Environmental Engineering, University

of Illinois at Urbana-Champaign, U.S. & NSF

2 Space Structures Research Center, Zhejiang University, China

10th US National Congress on Computational Mechanics

[email protected]

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Motivation of the Finite Particle Method (FPM)

Fundamentals of the FPM

Numerical Examples

Concluding Remarks

Future Work

Outline

[email protected] 10th US National Congress on Computational Mechanics

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Structural nonlinear problems:

— large deformation, large rotation

Discontinuous problems:

— fracture, collapse, fragmentation

Motivation of the FPM


Methods:

Finite Element Method (FEM):

— general method

Mesh-free Methods: (DEM, SPH, and others)

— suited for particulate material, such as sand and concrete

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Finite Particle Method (FPM)

Based on Vector Mechanics

The procedure of this method is simple and unified. Without

special considerations, strong nonlinear and discontinuous

problems can be solved.

Ting, E.C., Shi, C., Wang, Y.K.: Fundamentals of a vector form intrinsic finite

element: Part I. Basic procedure and a planar frame element. J. Mech. 20, 113–

122 (2004)

Motivation of the FPM


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Assumptions:

Particle

Cell (―element‖ like)

The relationship between particle and element

The particle motion undergoes a time history.


Fundamentals of the FPM1) Discretization of structure.

cell

particle

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Assumptions:

The reference configuration for stress analysis of a continuous body is its configuration at time t1.

The deformation of the continuous body is infinitesimal (small deformation) .


Fundamentals of the FPM2) Discrete Path

Path unit (interval)

The effect due to geometrical changes within the time interval t1-t2 can be neglected

The use of infinitesimal strain and engineering stress for evaluating stresses and computing virtual work is feasible.

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Fundamentals of the FPM3) Particle Motion Equation

1 1

n next ext int

i i

f f

M d F&&

ext

yF

ext

xF

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a) Evaluate the rigid body motion

b) Use reverse motion to remove

rigid translation and rotation

c) Use deformed coordinate to

reduce element degree of freedomd) Calculate the internal force

at the deformed coordinate

system

g) Transform the internal force

back to global coordinate

f) Use forward motion to get

the element back to the original

position

Fundamentals of the FPM4)Evaluation of deformations and internal forces

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Fundamentals of the FPM5) Time Integration

To avoid iteration, explicit time integration is used..

If a second order, explicit, central difference time integrator is

adopted:

1 1

1( ),

2n n n

t

d d d

1 12

1( 2 ).n n n n

t

d d d d

2

1

1 1

1

2( )

2

4 2.

2 2

n next ext int

n

i i

n n

tf f

t m

t

t t

Fd

d d

Velocity:

Acceleration:

Displacement:

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Fundamentals of the FPM6) Flow Chart

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Goal: a) verify the accuracy of internal force evaluation

b) effect of the Young’s modulus

Numerical examples 1. A square plane subjected to an initial angular velocity


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• AnimationThe Young’s modulus and initial angular velocity are the same in

these two cases.

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Displacement of point 1 in x direction



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Numerical examples 2. A square frame under tension and compression


Goal: test the capability of FPM in simulating large deformation

of planar solids

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Numerical examples 2. A square frame under tension


Modeled with 228 particles

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Numerical examples 2. A square frame under compression


Modeled with 228 particles

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Numerical examples 2. A square frame under tension and compression


under tension under compression

Compare with analytical solution of Euler beam

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Remarks on the Finite Particle Method


1. FPM is based on the combination of the vector mechanicsand numerical calculations. It enforces equilibrium on each particle.

2. No iterations are used to follow nonlinear laws, and no matrices are formed. The procedures are quite simple and robust.

3. The examples demonstrate performance and applicability of the proposed method on large deflection analysis of planar solids.

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1. Expand the present work into 3D;

2. Use FPM in failure and collapse simulation.

Future work


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References

Ying Yu, Yaozhi Luo. Motion analysis of deployable

structures based on the rod hinge element by the finite particle

method. Proc. IMechE Part G: J. Aerospace Engineering.

2009, 223: 1-10.

Ying Yu, Yaozhi Luo. Finite particle method for kinematically

indeterminate bar assemblies. J Zhejiang Univ Sci A. 2009, 10

(5): 667-676.

Ying Yu, Glaucio Paulino, Yaozhi Luo. Finite particle method

for progressive failure simulation of framed structures .

(finished and will be submitted for publication )

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Thank you!!

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a) Evaluate the rigid body motion

3

1

1

3i

i

translation rotation

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b) Using reverse motion to remove rigid translation

and rotation

1 1( )( ), ( 1,2,3)d d

u u u

R I x x

d r

i i i

i i i

cos( ) sin( )

sin( ) cos( )R

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c) Using deformation coordinate to reduce element degree

of freedom

1 2

1

1 22

1ˆ

d

x

ddy

l u

m uu

e

1

2

1

ˆm

l

e

1ˆˆ ( )Q x xx

1 11

1 12

ˆˆ

ˆ

T

T

l m

m l

eQ

e

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d) Calculate the internal force at the deformation

coordinate

1 1 2 2 3 3

1 1 2 2 3 3

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

u N u N u N u

v N v N v N v

2 2 3 3

3 3

ˆ ˆ ˆ

ˆ ˆ

u N u N u

v N v

1 1

ˆ ˆ 0x y

1 1 2ˆ ˆ ˆ 0u v v

1 1 2 2 3 3

T

x y x y x yf f f f f f

2 2 2 2 2

3 3 3 3 3

3 3 3 3 3

ˆ

ˆ[ ]Β EB

x xa x xa x

T

x xa x xa xA

y ya y ya y

f f f f u

f f f f t dA u

f f f f v

Principle of virtual work:

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g) Transform the internal force back to global

coordinate

1 11

1 12

ˆˆ

ˆ

T

T

l m

m l

eQ

e

ˆ, 1,2,3

ˆQ

ix ixT

iy iy

f fi

f f

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f) Using forward motion to get the element back to the

original position

Transform the internal force back to the original

direction

, 1,2,3Rix ix

iy iy

f fi

f f

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Large deflection analysis of planar solids based on the ...paulino.ce.gatech.edu/conferences/presentations/09yu_largedef... · Large deflection analysis of planar solids based on