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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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cr

2 =

cr

L

Pin ended column

eonar u er -


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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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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 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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Jump of Equil ibrium or Snap Through Instabili ty

,

reticulated shells

EREN Exhibition hall,,


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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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Agreement of theoretical and

experimental values: a) bars; b) shells


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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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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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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

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