Post on 04-Jun-2018
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Universitatea Politehnica TimioaraFacultatea de Construcii
Departamentul de Construcii Metalice i Mecanica Construciilor
PL CI CURBE SUBIRI
- CURS 5 -
. .
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INSTABILITY PHENOMENON: GENERAL
What is buckling?
Bucklin is a rocess b which a structure cannot withstand loads
with its original shape, so that it changes this shape in order to find a
new equil ibrium configuration. This is an undesired process (from ,
of the load.
The consequences of buckling are basically geometric:
There are large displacements in the structure
There may also be consequences for the material, in the sense that deflections in
the tank may induce plasticity in the walls of the structure
Local buckling Global buckling of a
of a tank wind turbine tower
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INSTABILITY PHENOMENA: GENERAL
What is buckling?
Stability and instabil ity: Behavior of a given structure (slender!) can
be control led by design if the three characteristic ranges of load-
deformation curve are correctl defined:
Pre-crit ical range
Critical point (or range)
P
P(0, Pcr] Structural stability
-
Pcr
Critical point
Post-criticalrange
P > Pcr Structural instability
Pre-critical rangeBuckling: elastic
plastic
d namic
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INSTABILITY PHENOMENON: MILESTONES
Probably first example of loss of stability is the collapse of
. . . ,
Heron of Alexandria (c. 100 B.C.) observed thast the strength
of a piece of wood reduces as its length increases
Leonardo da Vinci (1452-1519) and Gali leo Galilei (1564-
1642) provided empirical rules for the strength of column in
Marsenne M. (1588-1648) observed that iron, copper andother metal members, when subjected to a force or weight,
.
Musschebroek (1692-1791) confirmed by systematic tests
observations of Marsenne and proposed a qualitative low forfailure in compression of a wood parallelepiped.
Jacob Bernoull i (1654-1705) assumed the parabolic shape of
bending moment and curvature
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INSTABILITY PHENOMENON: MILESTONES
cr
2 =
cr
L
Pin ended column
eonar u er -
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INSTABILITY PHENOMENON: MILESTONES
Lagrange (1736-1813) applied variation principle to elastic
Tomas Young (1773 - 1829): lateral buckling of column of variable
cross-section and influence of imperfections
Hppl (1854-1924) and V. Karmann (1881-1963): equations for
large deflections of thin plates with in-plane stress
. . -
axial load of cylindrical shells
Donnel L.H: A new theory for the buckling of thin cylinders under
axial compression and bending (1934)
Koiter W. (1914-1997): post-buckling theory, in 1945
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INSTABILITY PHENOMENON: MILESTONES
Bifurcation instability
When the re-critical deformations do not corres ond with the
instabil ity deformations. For instance, the portal frame (a) has a
symmetrical pre-critical deformation, while the post-critical deformation is
When the structure is an ideal one, without geometrical imperfections,
as in case of compression bar (b), when the bar in the pre-critical state is
, -
When the structure is an actual one with geometrical imperfections, but
these are not similar with the post-critical deformations. As example, the
arch can have a symmetrical deviation from the designed form, but the
post-critical deformation is asymmetrical one (c).P
Pcr Bufurcation point
Post-critical path
u
Pre-critical path
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INSTABILITY PHENOMENON:Bifurcation Instability Of Cylinders
N
inf,x crN
,x cr
L
L
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INSTABILITY PHENOMENA:BASIC TYPES AND MODELS
Divergence of equilibrium
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INSTABILITY PHENOMENA:BASIC TYPES AND MODELS
Instability by limitation
The post critical path is an extension of the
pre-critical one and the load-deformation curve
presents a maximum value. It might be for: The shallow structures with transversal loads as shallow
arches (a)
The actual structure with geometrical imperfections with
the same form as the post-critical deformations. For
ns ance, e ac ua e as c-p as c compresse ar
The structure for which the pre-critical deformations
contain as well the instability deformations (c)
P
PlimPul
,
case (no imperfections) buckles by bifurcation, in
almost all real cases (with imperfections) may
Instability by limitation
uc e y m a on
Thick short cylinders may buckle by limitation
in post-elastic range
u
Elephant foot buckling of cylinders can beregarded as limitation buckling
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INSTABILITY PHENOMENA:BASIC TYPES AND MODELS
Jump of Equil ibrium or Snap Through Instabili ty
,
reticulated shells
EREN Exhibition hall,,
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INSTABILITY PHENOMENA:BASIC TYPES AND MODELS
Dynamic Instability
-
Static equivalent loads
Dynamic instability under dif ferent load types: step load,impulsive load, periodic loads
Dynamic loads
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INSTABILITY PHENOMENA:INFLUENCE OF INPERFECTIONS
Dynamic propagation of instability or progressive instability
om no e ec ou e ayer
grids)
Instabil ity propagation (single
layer reticulated shells)
Domino effect for double-layer grids
Instability propagation for single-layer
reticulated shells
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INSTABILITY PHENOMENA:INFLUENCE OF INPERFECTIONS
Agreement of theoretical and
experimental values: a) bars; b) shells
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INSTABILITY PHENOMENA:INFLUENCE OF INPERFECTIONS
Types of bifurcation and influence of imperfections: a) Eulers type; b)
uns mmetrical c stable-s mmetrical d unstable-s mmetrical
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INSTABILITY PHENOMENA: Structure
Classification Accordin to the Instabilit T e
Symmetrical stable bifurcation
Double - symmetrical cross-section
compressed bar with fixed ends Lateral buckling of bending members
Symmetrical frames
,
constant direction pressure Rectangular and circular compressed
Sphere with concentrated loads
Stable post-critical behavior
Low sensitivity to imperfections
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INSTABILITY PHENOMENA: Structure
Classification Accordin to the Instabilit T e
Symmetrical unstable bifurcation
Bar with elastically fixed ends
Bar and rin on elastic bed
Ring under variable pressure
Double-hinged and double-restrained
Axially symmetrical buckling of the
axially loaded cylinder
Axially symmetrical buckling of the
sphere under uniform pressure
Unstable post-critical behavior
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INSTABILITY PHENOMENA: Structure
Classification Accordin to the Instabilit T e
Unsymmetrical bifurcation
Unsymmetrical cross-section
compressed bar with fixed ends
Three-hinged arch
Unsymmetrical frame
Latticed planar space structure
ery uns a e pos -cr ca e av or
Very high sensitivity to imperfections and
their sign
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POST-CRITICAL BEHAVIOUR OF ELASTIC
P
u Plength
unloaded
u
Perfect bar
length
(unloaded)
P
w
w
erfect
cylinder
length
(unloaded)
Pu
Perfectplate
w
P
Pcr
P
A
Perfect
bar
Imperfect
bar
Pcr
PPerfect cylindrical
shell
Im erfec t la te
Perfect plateP
Pcr
ww0
w0w
Imperfect cylindrical
shell
w0 w
Post-critical behaviour of elastic structures: a) columns: indifferent post-critical
path; b) cylinders: unstable post-critical path; c) plates: stable post-critical path
a)
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FAVORABLE AND UNFAVORABLE
( ) 222
2
2
2
,13
=b
tEpcr
( ) 22, 413
+
=rb
ccr
component
component
Curvature effect in axial compression: a) increase in critical load; b) increase in
sensitivity to geometrical imperfections
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FAVORABLE AND UNFAVORABLE EFFECTS
Curvature effect in axis-symmetrical compression: a) increase in critical load; b)
increase in sensitivit to eometrical im erfections
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FACTORS INTRODUCING UNSTABLE
Factors which introduce unstable components: a) extensional deformations;
b) lateral movable supports; c) coupled instabilities; d) plastic deformations
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SINGLE AND COUPLED BUCKLING MODES.IMPERFECTION SENSITIVITY.
Erosion of theoretical buckling strength
Single and multi-modal buckling modes. Interactive buckling
m=1
xt
S.S
S.S
S.S a
y
12
8
m=2
m=3
m=4k
x
b
a
xt
S.S
bx
4
01 2 3 4
k
a/b
x
Rectangular simply supported thin plate
subjected to compression stress
Multimodal buckling of rectangular
plate (Garland curve)
Pattern change for plate
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SINGLE AND COUPLED BUCKLING MODES.IMPERFECTION SENSITIVITY.
Erosion of theoretical buckling strength
Sin le and multi-modal bucklin modes. Interactive bucklinBuckling modes for a lipped channel in
compression:
Sin le modes:
(a) local (L);
(b) distortional (D);
(c) flexural (F);
(d) torsional (T);
a) b) c) d) e)
(e) flexural-torsional (FT).
Coupled (interactive) modes:
(f) L + D;
(h) F + D;
(i) FT + L;
(j) FT + D;
+
f) g) h) i) j) k)
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COUPLED INSTABILITIES FOR PLATE ANDSHELL ELEMENTS
W weak interaction
M moderate interaction
S strong interaction
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OVERALL AND LOCAL BUCKLING FOR
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EROSION CLASSES AND IMPERFECTIONS
Class 1 stron interaction hi h erosion 0.5 Class 2 moderate interaction moderate erosion 0.3
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MODELS AND METHODS
Generic classification of structures in terms of characteristic
nsta ty types an sens t v ty to mper ect ons
Linear, nonlinear, elastic, plastic models
Linear bucklin anal sis ei en-bucklin LBA
Geometrical nonlinear imperfection analysis GNIA
Geometrical material nonlinear imperfection analysis GMNIA
- -
critical solver methods (Arc-length); Designed load checkingor load-deformation curve