Buckling and Post-Bucklingpiet/edu/ogo1920/doc/COC20191129sheets.pdfMARC/Mentat ADD NODES / ADD ELEMENTS GEOMETRIC / MATERIAL PROPERTIES BOUNDARY CONDITIONS : ﬁxed displacements

Buckling and Post-Buckling

Piet Schreurs

Department of Mechanical EngineeringEindhoven University of Technologyhttp://www.mate.tue.nl/∼piet

November 29, 2019

NEGATIVE STIFFNESS

Piet Schreurs (TU/e) 2 / 38

Mass support

F


Negative stiffness

0 5 10 15 20x

-40

-20

0

20

40

60

80

100

120

y, z

, s


BUCKLING


Buckling : clamped-clamped

Pb =π2

EI

( 12L)2

; v(x) = α

{

1 − cos

(

2πx

L

)}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

cc : Le = 1/2 L


Buckling

K u˜

= f˜

e→ u

˜= K

−1f˜

e

K : structural stiffness matrixu˜

: nodal displacementsf˜

e: external nodal forces

u˜

→ ε → σ

σ → Kσ(σ)

Kσ

: geometric or stress stiffness matrix: proportional to f

˜e

[ K + λKσ(σ) ] α

˜= 0

˜→ λi → α

˜i

λi : buckle factor (load multiplication factor)αi : buckle mode


MARC/Mentat

ADD NODES / ADD ELEMENTS

GEOMETRIC / MATERIAL PROPERTIES

BOUNDARY CONDITIONS : fixed displacements + point load

LOAD CASE : Buckle

JOB : choose loadcase ’Buckle’

RUN → Post Processing: deformed scaled automatically


Buckling: clamped-clamped

Pb =π2

EI

( 12L)2

; v(x) = α

{

1 − cos

(

2πx

L

)}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

cc : Le = 1/2 L

[ b h L ] = [ 25 1 400 ] mm , E = 200 GPa

[ Fc Pb ] = [ 102.8 102.81 ] N


Buckling: pinned-clamped

Pb =π2

EI

(0.7L)2; v(x) =

α

nLsin(n(L − x)) − α cos(n(L − x)) + α

{

1 −(L − x)

L

}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pc : Le = 0.7 L

[ b h L ] = [ 25 1 400 ] mm , E = 200 GPa

[ Fc Pb ] = [ 52.58 52.45 ] N


Buckling: free-clamped

Pb =π2

EI

(2L)2; v(x) = α

{

1 − cos

(

π

2

(L − x)

L

)}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

fc : Le = 2 L

[ b h L ] = [ 25 1 400 ] mm , E = 200 GPa

[ Fc Pb ] = [ 6.426 6.43 ] N


Buckling: pinned-pinned

Pb =π2

EI

L2; v(x) = α sin

(πx

L

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pp : Le = L

[ b h L ] = [ 25 1 400 ] mm , E = 200 GPa

[ Fc Pb ] = [ 25.7 25.70 ] N


Buckling: supported beam, clamped-clamped

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

cc : Le = 1/4 L

[ b h L] = [ 25 1 400] mm , E = 200 GPa , k = 4 N/mm

[ Fc Pb ] = [ 411.2 411.23 ] N

This is the 2nd mode of the unsupported case: Pb2 =16π2

EI

L2


Buckling: supported beam, pinned-pinned

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pp : Le = 1/2 L

[ b h L] = [ 25 1 400] mm , E = 200 GPa , k = 4 N/mm

[ Fc Pb ] = [ 102.8 102.81 ] N

This is the 2nd mode of the unsupported case: Pb2 =4π2

EI

L2


POST BUCKLING


Post-buckling of a beam

Load

Inc: 100Time: 1.000e+02

X

Y

Z

1

Load

Inc: 100Time: 1.000e+02

X

Y

Z

1

0 50 100 150 200 250displacement [mm]

0

50

100

150

forc

e [N

]


NONLINEAR DEFORMATION

ITERATIVE SOLUTION PROCEDURE


Equilibrium

u

fil , A

P

Pfi

l , A

l0, A0

fe

fe

external force fe

internal force fi = fi (u)

equilibrium of point P fi (u) = fe


Linear deformation

u

P

Pl ≈ l0, A ≈ A0

l ≈ l0, A ≈ A0

fi

l0, A0

fe

fefi

external force fe

internal force fi = σnA0 = EεA0 =EA0

l0u = Ku

equilibrium fi = fe → Ku = fe → u = us =fe

K=

l0

EA0fe


Equilibrium : nonlinear

u

fi (u)

fe

uexact

external force fe

internal force fi = σA = fi (u)

equilibrium of point P fi (u) = fe

fi (u) non-linear iterative solution process needed


Iterative solution procedure

uu∗

f∗

i

fi (u)

r∗

fe

uexact

analytic solution fi (uexact ) = fe → fe − fi (uexact) = 0

approximation u∗

fe − fi (u∗) = r(u∗) 6= 0

residual r∗ = r(u∗)


Newton-Raphson iteration procedure

fi (uexact) = fe

uexact = u∗ + δu

}

→ fi (u∗ + δu) = fe

fi (u∗) +

dfi

du

∣

∣

∣

∣

u∗

δu = fe → f∗

i+ K

∗δu = fe

K∗ δu = fe − f

∗

i= r

∗ → δu =1

K ∗r∗

u

δu

u∗

K∗

f∗

i

fi (u)

fe

r∗


New approximate solution

uexact

δu

u∗

K∗

f∗

i

fi (u)

fe

r∗

uu∗∗

new approximation u∗∗ = u

∗ + δu

error uexact − u∗∗

error smaller → convergence


Convergence control

δu

u∗

fi (u)

r∗∗

u

fe

f∗∗

i

u∗∗

residual force |r∗∗| ≤ cr → stop iteration

iterative displacement |δu| ≤ cu → stop iteration


Convergence

u

fi (u)

fe


MARC/Mentat

ADD NODES / ADD ELEMENTS

GEOMETRIC / MATERIAL PROPERTIES

BOUNDARY CONDITIONS : fixed displacements and/or point loads

TABLES : time tables for time-dependent loading

LOAD CASE : BC’s, nr. steps, convergence criterion

JOB : choose subsequent loadcasesNO initial loadsJob parameters : large strain

RUN → Post Processing: deformed scaled manual

HISTORY PLOT


Tables

0 2 4 6 8 10 12time t

0

0.5

1

1.5

2

2.5

F(t

)

0 2 4 6 8 10 12 14time t

-1.5

-1

-0.5

0

0.5

1

1.5

F(t

)

F (t) = set data points F (t) = sin(t) + 12 cos(t)


SNAP-THROUGH


Snap-through

0 2 4 6 8 10displacement

-1

-0.5

0

0.5

1

1.5

2

2.5

forc

e


Beam with horizontal spring

-5 0 5displacement [mm]

-30

-20

-10

0

10

20

30re

actio

n fo

rce

[N]

Kv = [ 40 400 4000 40000 ]


Snap-through for supported beam

-5 0 5displacement [mm]

-30

-20

-10

0

10

20

30

reac

tion

forc

e [N

]

Kv = [ 0.04 0.4 2 4 ]


PRELOAD AND LOAD


Preload: edge displacement; load: beam center

-3 -2 -1 0 1 2 3displacement [mm]

-12

-10

-8

-6

-4

-2

0

2

reac

tion

forc

e [N

]

leftdisp = [ 0.01 0.015 0.02 0.025 0.03 ]


Stiffness of beam and support


-10

-5

0

5

10

reac

tion

forc

e [N

]

leftdisp = [ 0.01 0.015 0.02 0.025 0.03 ]


-15

-10

-5

0

5

10

reac

tion

forc

e [N

]


Preload: edge displacement; load: beam center


-25

-20

-15

-10

-5

0

5

reac

tion

forc

e [N

]

leftdisp = [ 0.01 0.015 0.02 0.025 0.03 ]


Preload: edge displacement; load: spring support

-1 -0.5 0 0.5 1displacement [mm]

-3

-2

-1

0

1

2

3

reac

tion

forc

e [N

]

leftdisp = [ 0.001 0.005 0.01 0.015 ]


VARIABLES IN MENTAT

PROCEDURE FILES

MENTAT CALL FROM MATLAB

VARIATION OF VARIABLES


Beam model

L

F

L = 100 mm ; h = 1 mm ; b = 10 mm ; E = 200 GPa ; ν = 0.3 ; F = 1 N


Documents

Buckling and Post-Bucklingpiet/edu/ogo1920/doc/COC20191129sheets.pdfMARC/Mentat ADD NODES / ADD ELEMENTS GEOMETRIC / MATERIAL PROPERTIES BOUNDARY CONDITIONS : ﬁxed displacements