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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
Piet Schreurs (TU/e) 3 / 38
Negative stiffness
0 5 10 15 20x
-40
-20
0
20
40
60
80
100
120
y, z
, s
Piet Schreurs (TU/e) 4 / 38
BUCKLING
Piet Schreurs (TU/e) 5 / 38
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
Piet Schreurs (TU/e) 6 / 38
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
Piet Schreurs (TU/e) 7 / 38
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
Piet Schreurs (TU/e) 8 / 38
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
Piet Schreurs (TU/e) 9 / 38
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
Piet Schreurs (TU/e) 10 / 38
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
Piet Schreurs (TU/e) 11 / 38
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
Piet Schreurs (TU/e) 12 / 38
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
Piet Schreurs (TU/e) 13 / 38
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
Piet Schreurs (TU/e) 14 / 38
POST BUCKLING
Piet Schreurs (TU/e) 15 / 38
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
]
Piet Schreurs (TU/e) 16 / 38
NONLINEAR DEFORMATION
ITERATIVE SOLUTION PROCEDURE
Piet Schreurs (TU/e) 17 / 38
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
Piet Schreurs (TU/e) 18 / 38
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
Piet Schreurs (TU/e) 19 / 38
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
Piet Schreurs (TU/e) 20 / 38
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∗)
Piet Schreurs (TU/e) 21 / 38
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∗
Piet Schreurs (TU/e) 22 / 38
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
Piet Schreurs (TU/e) 23 / 38
Convergence control
δu
u∗
fi (u)
r∗∗
u
fe
f∗∗
i
u∗∗
residual force |r∗∗| ≤ cr → stop iteration
iterative displacement |δu| ≤ cu → stop iteration
Piet Schreurs (TU/e) 24 / 38
Convergence
u
fi (u)
fe
Piet Schreurs (TU/e) 25 / 38
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
Piet Schreurs (TU/e) 26 / 38
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)
Piet Schreurs (TU/e) 27 / 38
SNAP-THROUGH
Piet Schreurs (TU/e) 28 / 38
Snap-through
0 2 4 6 8 10displacement
-1
-0.5
0
0.5
1
1.5
2
2.5
forc
e
Piet Schreurs (TU/e) 29 / 38
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 ]
Piet Schreurs (TU/e) 30 / 38
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 ]
Piet Schreurs (TU/e) 31 / 38
PRELOAD AND LOAD
Piet Schreurs (TU/e) 32 / 38
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 ]
Piet Schreurs (TU/e) 33 / 38
Stiffness of beam and support
-3 -2 -1 0 1 2 3displacement [mm]
-10
-5
0
5
10
reac
tion
forc
e [N
]
leftdisp = [ 0.01 0.015 0.02 0.025 0.03 ]
-3 -2 -1 0 1 2 3displacement [mm]
-15
-10
-5
0
5
10
reac
tion
forc
e [N
]
Piet Schreurs (TU/e) 34 / 38
Preload: edge displacement; load: beam center
-3 -2 -1 0 1 2 3displacement [mm]
-25
-20
-15
-10
-5
0
5
reac
tion
forc
e [N
]
leftdisp = [ 0.01 0.015 0.02 0.025 0.03 ]
Piet Schreurs (TU/e) 35 / 38
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 ]
Piet Schreurs (TU/e) 36 / 38
VARIABLES IN MENTAT
PROCEDURE FILES
MENTAT CALL FROM MATLAB
VARIATION OF VARIABLES
Piet Schreurs (TU/e) 37 / 38
Beam model
L
F
L = 100 mm ; h = 1 mm ; b = 10 mm ; E = 200 GPa ; ν = 0.3 ; F = 1 N
Piet Schreurs (TU/e) 38 / 38
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