theoretical details
Basic Concept – folding of “thin structures”
Folding mechanisms- Basic concept:
quasi-isometric transformations
Folding Modes – experimental program
Superfolding Element: Basic Folding Modes
Asymmetric folding mode Symmetric folding mode “Inverted” folding mode
SuperFolding Element: Energy dissipation mechanisms
Basic folding mechanisms in a deformed SuperFolding Element (SE):
1 Deformation of a floating toroidal surface.
2 Bending along stationary hinge lines.
3 Rolling deformations.
4 Opening of a conical surfaces.
5..Bending deformations along inclined, stationary, hinge lines following locking of the traveling hinge line 3.
SFE: Example of Elementary Calculations 1
(internal energy dissipation)
u N E
and M
E b o m o
4
2
/ t M
o
ot
N
o
o1. Rates of energy dissipation in bending and compression/tension
o
b
C M
E 8
2. Total energy dissipation
t M H H
N
dy yN E
o o
H
o m
2 2
0
4 4 2
SFE: Example of Elementary Calculations 2 (energy balance equation)
eff ext m
eff m m
ext
P E P
d P d
P E
eff eff
0 0
) (
Energy supplied to the system
H t
M c t
M H H H
P
o oeff m
2 4 ) 2
(
Mean crushing force - a function of the plastic folding wave 2H
t C t
H
t C M
P
o m
2 73 . 0
2 4
Result of minimization procedure
73 . 0 2
/ H
effSFE – mechanisms of internal energy dissipation
Rolling deformation and flow over the toroidal surface (inextensible in rolling/flow direction)
SFE – Elementary Solutions
Right angle (F = 90 deg) Elements
Symmetric (extensional) Asymmetric (quasi-inextensional)
SFE – Effective crushing distance (square column case)
b
a
C
5 ...
2 , 1 ),
, (
} 2 ) ( )
(
) ( )
) ( { (
4
*
5 5 4 4
3 3 2 2
1 1
2
F
i A
A where
A H t
A H
r A H H
A C t
A r P t
i i
eff N
o M
o
M o M
o N
m o
Standing alone SFEuniform thickness
eff N
o M
o M
o
M b o M a
o N
m o
A H t t
A H r t
A H t
H t b H
t A a
t A r P t
) 2 ( )
( )
(
) ( )
) ( 4 (
1
} ) (
{
5 5 2
5 4 4 4
2 3 4
3 2
3
2 2 2
2 2 1 1 2
1 1
Standing alone SFE
different thicknesses of flanges
m i
J
i N i
i o
M i i
o
M i
o
M i
i o
N i
eff
P t
A r
t A C
H A H
r A H
t A H
12
0 1 1 2 2 3 3
4 4 5 5
4
2
{ ( ) ( ) ( )
( ) ( ) }
Assembly of SFE’s
P H
P r
P
m m m
0 ;
0 ;
0
*
SFE – Governing Functional for Arbitrary Cross Section
Prediction of leading folding mode based on the level of mean crushing force
The “energy barrier” concept accounts for initial imperfection and spontaneous change of folding
mode
a m
a m s
m
P P q P
q q cr
Crushing response of an isolated SFE
) 120 90
(
0
0
F F ( 90
0 120
0)
Natural folding modes of a cross-section and Leading Corner concept
Leading Corner Concept – deformation transfer and generation of a natural folding mode
Axial Crushing Response of Cross Section
(synergy of analytical, semi-analytical solutions and experimental data)
Super Folding Element
•Folding modes
•Mean crushing force
•Deep collapse response
Amplitude of progressive buckling – experimental formula
Von Karman/Stowell - Iliushyn effective width formula
(unified approach)
Effective Width Formula – Unified Approach
ELASTICITY (Euler, Timoshenko etc)
PLASTICITY (bilinear material representation ES, ET) (Stowell - Iliushyn) General equilibrium equation
D w N w N w N w p
w w w w
x xx xy xy y yy
xxxx xxyy yyyy
4
4
2 2
( , , , )
, , ,
Example – in-plane loading of the plate Governing equilibrium equations
D4wN wx ,xx0 (1 )
4 3
4 2 0
4 4
4
2 2
4 4
2
E 2
E w x
w x y
w y
N D
w x
T S
x
Critical stress for simply supported loaded ends
2 2 2
) 1 (
412
b t E
cr
S T p S
cr E
E b
t
E 2 1 3
9
2 2
Abramowicz’s critical strain approach
2 2 2
) 1 (
3
b
t E
cr
cr
S T s
p cr p
cr E
E b
t
E 2 1 3
9
2 2
Consider power type material representation:
n
y
y
( )
then:
1
1
n E
n n E
y y y S y
y y
T
2 2 2
) 1 (
3
b
t E
cr
cr
does not depend on ‘n’
n
b t
p
cr 2 1 3
9
2 2
The graph on the left hand side shows that the critical elastic and plastic strains differ slightly in a wide range of hardening exponent ‘n’. It follows that the simple expression for elastic strain can be used to estimate critical strains in both elastic as well as plastic plates. This approach is used throughout all the calculation modules of VCS. E.q. in torsion
2 2 2
) 1 (
12
b ks t
cr
for elastic material. This expression is also used to estimate crpl. It is very useful procedure since closed-form solution does not exist for plastic torsion buckling.
Leading Corner Concept – diversity of folding modes
Different folding modes initialize in
leading SFE
Deformation Transfer - Connection Concept
Unconnected SFE fold freely and do not create compatible folding
Connections force connected element to fold in
compatible manner regardless of the type of physical connection