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1. STABILITY – GENERAL ASPECTS

1

Chapter 1

STABILITY – GENERAL ASPECTS

1.1. EXAMPLES OF INSTABILITY

In many cases, instability is the most important limit state for structural members

made of steel or aluminium alloys. This phenomenon can affect a part of the

member, the entire member, a part of the structure or the entire structure.

1.1.1. Instability of bars

This is the type of loss of stability that affects bars in compression or compressed

parts of bars. At a certain value of the load, the bar in compression or the

compressed part of the bar (involving the un-compressed part too) finds equilibrium

in a deformed shape, in the neighbourhood of the straight one.

1.1.1.1. Straight bars in compression

This problem is probably the most studied one in the history of stability problems.

Beginning with Euler, 1744 [1], different researchers tried to express the equilibrium

and the failure mode of a perfectly straight member subject to axial compression [2].

When subject to an axial compression force, a straight member may lose its stability

in one of the following forms (Fig. 1.1):

• flexural buckling (v ≠ 0; ϕ = 0) (Fig. 1.1a);

• torsion buckling (v = 0; ϕ ≠ 0) (Fig. 1.1b);

• flexural-torsion buckling (v ≠ 0; ϕ ≠ 0) (Fig. 1.1c);

where v means the lateral displacement in the plane of the cross-section and ϕϕϕϕ is the

rotation of the cross-section in its plane.

Symbols – Widgit Symbols (c) Widgit Software 2002-2013 www.widgit.com

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( a ) ( b ) ( c )

Fig. 1.1. Forms of buckling of a straight bar in compression [2]

The buckling load is the critical force Fcr at which a perfectly straight member in

compression assumes a deflected position (Fig. 1.1). Buckling is a limit state, in the

meaning that once the force Fcr is reached, the deflection increases until the collapse

of the bar is reached. The member should be subjected only to loads inferior to the

critical force (F < Fcr) [2].

1.1.1.2. Straight bars in bending

In the same way as for any member in compression, the buckling problem appears

for the compressed flange of a beam. Generally, buckling (Fig. 1.2) may not occur inthe plane of the web, as the compressed flange is continuously connected through

the web material to the tensioned part of the cross-section, the tension flange. The

stabilizing effect of the tension zone transforms free transverse buckling into lateral-

torsional buckling, causing lateral bending and twisting of the beam [2].

Fcr Fcr Fcr

v v

ϕ ϕ

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Fig. 1.2. Lateral-torsional buckling of a beam [2]

Lateral-torsional buckling may be prevented either by performing checks usingsuitable relations, or by introducing lateral bracings whose purpose is to reduce the

distance on which this phenomenon may occur.

1.1.1.3. Curved bars in compression and bending

The arch is a very efficient structural member for resisting symmetrical loads actingin its plane, normally to the line of its supports. Its important strength comes from the

arm lever exiting between the compressed part (the arch) and the tensioned part (the

line of supports) (Fig. 1.3), which is much bigger than the distance between the

compressed flange and the tension one in the case of a beam.

Fig. 1.3. The working principle of an arch

span

Lateral bucklingof the flange

Torsion(twistingof thebeam)

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The horizontal stiffness of the supports (the abutment stiffness) is vital for the

performance of the arch; the smaller this one is, the closer the behaviour is to a

“curved beam” and strong values of the bending moment can be found along the

arch.

There are several forms of buckling of an arch (Fig. 1.4):

• in-plane symmetrical (Fig. 1.4a);

• in-plane asymmetrical (Fig. 1.4b);

• out-of-plane (Fig. 1.4c).

(a) (b) (c)

Fig. 1.4. Forms of buckling of an arch

1.1.2. Instability of plates and shells

1.1.2.1. Local buckling of plates

Local buckling of a plate may occur as a result of the action of in-plane normal

compression stresses (σ), of shear ones (ττττ), or of their combination (Fig. 1.5).

Fig. 1.5. Local buckling of plates [3]

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1.1.2.2. Local buckling of shells

Similarly to the case of plates, in-plane compression stresses (σ) can lead to local

buckling of shells (Fig. 1.6).

Fig. 1.6. Local buckling of shells [3]

1.1.3. Instability of structures

1.1.3.1. “Snap-through” buckling

Because of the big values of the angles among bars (Fig. 1.7), strong compression

forces result in the bars. Because of the strains in these bars, the geometry of the

structure changes and, in some circumstances, equilibrium can be found in tension.

Fig. 1.7. “Snap-through” buckling [4]

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1.1.3.2. Instability of structures containing members in compression

A part of a structure or the entire structure can go unstable, as the end supports or

the joints of members in compression or in compression and bending have limited

stiffness. Figure 1.8 shows the case of a plane frame and of the reticulated structure

of a roof.

Fig. 1.8. Instability of frames or of roof structures [4]

1.1.3.3. Instability of tension structures

Tension members do not generate static instability of structures. However, they can

be subject of dynamic instability of structures like vibrations, resonance, “flutter” etc.

A very well-known example is the failure of Tacoma Narrows Bridge on November 7,

1940 (Fig. 1.9).

Fig. 1.9. Tacoma Narrows Bridge

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1.2. COMMENTS

Important

Instability is a more severe problem than overcoming strength:

1. Generally speaking, in the case of most strength checks, the

capable loads are expressed based on the yielding limit of the

material (except for checks for phenomena that involve brittle

fracture). Compared to the behaviour of the actual member, at

least two conservative aspects can be noticed:

• The nominal yielding limit, used in calculation, is smaller than

the average value (γov = 1,40 – S235; γov = 1,30 – S275; γov =

1,25 – S355 according to P100-2013 [5]);

• There is a certain reserve from the yielding limit till the

ultimate strength (about 1,53 – S235; 1,56 – S275; 1,44 –

S355 according to EN 1993-1-1 [6]).

2. In the case of stability checks, the capable loads are expressed

based on the yielding limit and, depending on the values of the

slenderness, failure can occur even before reaching the yielding

limit.

3. Using a material with higher strength increases the strength

capacity but, in some cases, it has no influence on the stability

capacity.

Example

There are well known situations when the recorded value of the

snow load, for instance, were much higher than the design ones and

no failure was noticed, as no instability problems were involved.

Such an example occurred on February 1st 2014 in Austria, when a

snow load of 4,35kN/m2 was measured.

1.3. CLASSES OF CROSS-SECTIONS

1.3.1. Definition of the classes of cross-sections

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Generally, given the strength of steel and aluminium alloys, failure of a metal

member subjected to loads other than tension occurs by buckling or by local

buckling. Depending on the slenderness of the element, this can happen either in the

elastic range (0 – Y in figure 1.10) or in the plastic range (Y – F in figure 1.10). To

manage this, EN 1993-1-1 [6] defines four classes of cross-sections of structural

members. They are best expressed for members in bending. In these definitions, the

behaviour of the material is presumed perfectly elastic up to the yielding limit and

perfectly plastic for elongations superior to the strain corresponding to the yielding

limit (Fig. 1.10). This model is known as the Prandtl model.

Fig. 1.10. The Prandtl model for steel behaviour

Depending on the stress state that causes local buckling, cross-sections of structural

members are classified as [6] (Fig. 1.11):

Class 1 – cross-sections that can form a plastic hinge with sufficient rotation

capacity to allow redistribution of bending moments. Only class 1 cross-

sections may be used for plastic design.

Class 2 – cross-sections that can reach their plastic moment resistance but local

buckling may prevent development of a plastic hinge with sufficient

rotation capacity to permit plastic design (redistribution of bending

moments).

Class 3 – cross-sections in which the calculated stress in the extreme

compression fibre can reach the yield strength but local buckling may

prevent development of the full plastic bending moment.

Class 4 – cross-sections in which it is necessary to take into account the effects of

ε

real

Prandtlfy

0

Y F

σ

εy εu

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local buckling when determining their bending moment resistance or

compression resistance.

For practical reasons, the limits between these classes are expressed in terms of

slenderness. Tables 1.1, 1.2, 1.3 show the requirements for different cross-sectional

classes. The class of a cross-section is the maximum among the classes of its

components.

Fig. 1.11. Possible stress distribution, depending on the cross-section class

The plastic hinge is a concept. It is a model of a cross-section where all the fibres

reached the yielding limit in tension or compression (Fig. 1.11) generated by a

bending moment, presuming a Prandtl behaviour diagram for the material, while in

the neighbour cross-sections the stress state is elastic. In reality, the stress and

strain state is more complex (Fig. 1.12): the material behaviour is not ideally elasto-

plastic and the plastic deformations extend on a certain length.

class 4 class 3 class 2 class 1

σmax < fy σmax = fy σmax = fy σmax = fy

y y

z

z

( – )

( + )

σmax = 0

εmax < εy εmax = εyεmax = 0 εmax > εy εmax >> εy

x x

x x y y

z

z

εy

fy

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Fig. 1.12. The stresses in the region of a plastic hinge

Table 1.1. Limitations for the slenderness of internal walls [6]

ClassWall in

bendingWall in

compressionWall in bending and compression

Stress distribution

1ε72

t

c≤ ε33

t

c≤

when α > 0,5:1α13

ε396

t

c

−≤

when α ≤ 0,5:α

ε36

t

c≤

2ε83

t

c≤ ε38

t

c≤

when α > 0,5:1α13

ε456

t

c

−≤

when α ≤ 0,5: α

ε5,41

t

c

≤

c c c c

c c c c

t t t t

t tt

t

Bending axis

Bending axis

cc cαc

fy

fy

fy

fy

fy

fy

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Stress distribution

3ε124

t

c≤ ε42

t

c≤

when ψ > –1:ψ33,067,0

ε42

t

c

+≤

when ψ ≤ –1: ( ) ( )ψψ1ε62t

c−−≤

yf

235ε =

fy (N/mm2) 235 275 355 420 460

ε 1,00 0,92 0,81 0,75 0,71

Note: (+) means compression

In many cases instability is the most important limit state for structural members.

Table 1.2. Limitations for the slenderness of flanges [6]

ClassCompressed

flangeTension and compressed flange

Compressed edge Tension edge

Stress distribution

1ε9

t

c≤

α

ε9

t

c≤

αα

ε9

t

c≤

2ε10

t

c≤

α

ε10

t

c≤

αα

ε10

t

c≤

Stress distribution

c c c

c/2

fy

fy

fyfy

ψfy

t t t t

c c cc

c c c

c c c

αc αc

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3ε14

t

c≤ σkε21

t

c≤

y

f

235ε =

fy (N/mm2) 235 275 355 420 460

ε 1,00 0,92 0,81 0,75 0,71

Note: (+) means compression

Table 1.3. Limitations for the slenderness of the walls of round tubes [6]

Class Cross-section in bending and/or compression

1 d/t ≤ 50ε2

2 d/t ≤ 70ε2

3 d/t ≤ 90ε2

yf

235ε =

fy (N/mm2) 235 275 355 420 460

ε 1,00 0,92 0,81 0,75 0,71

ε2 1,00 0,85 0,66 0,56 0,51

In a similar manner, the American code ANSI/AISC 360-10 [7] distinguishes between

compact, noncompact and slender cross-sections. The Japanese code JSCE [8]

makes reference to EN 1993-1-1 [6] and associates compact to class 1 and class 2

cross-sections, noncompact to class 3 and slender to class 4 ones, using its own

limits.

1.3.2. Practical procedure for determining the class of the cross-section

Establishing the class of a given cross-section might generate discussions: except

for the case of round tubes (Tab. 1.3), where it depends only on the slenderness of

the walls, it involves the loading state too. The stress state that defines the limits

between the classes of cross-sections is associated to the resistance ultimate limit

state. In practical situations, the stress state on the cross-section is inferior to the

limit one. The question that arises is how to reach the limit, as several options are

possible (Fig. 1.13).

dt

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Fig. 1.13. Establishing the class of the cross-section

Given a loading state on the cross-section, the limit can be reached by:

• increasing the axial force (Fig. 1.13(a));

• increasing the axial force and the bending moment according to a variation

law (Fig. 1.13(b));

• increasing the bending moment (Fig. 1.13(c));

The interaction curve in figure 1.13 is obtained using the following relations:

• for class 1 and class 2 cross-sections:

Rd,NEd MM ≤ (EN 1993-1-1 [6] rel. (6.31)) (1.1)

where MN,Rd is the design plastic moment resistance reduced due to the axial

force NEd;

For doubly symmetrical I- and H- sections or other flanges sections, no

reduction of the plastic resistance moment MN,y,Rd about the y-y axis needs to

be done when both the following criteria are satisfied:

Rd,plEd N0,25N ⋅≤ (EN 1993-1-1 [6] rel. (6.33)) and (1.2)

0M

yww

Ed

f th0,5N

γ

⋅⋅⋅≤ (EN 1993-1-1 [6] rel. (6.34)) (1.3)

Otherwise, the following reduction is done:

a5,01

n1

MM Rd,y,plRd,y,N⋅−

−

⋅= but Rd,y,plRd,y,N MM ≤ (EN 1993-1-1 [6] rel. (6.36)) (1.4)

N

M

(a)

(b)

(c)

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where:

Rd,pl

Ed

N

Nn =

Atb2Aa f ⋅⋅−= but a ≤ 0,5

• for class 3 cross-sections:

0M

y

Ed,x

f

γ ≤σ (EN 1993-1-1 [6] rel. (6.42), (6.43)) (1.5)

The approach in figure 1.13(c), by increasing the bending moment only, is consistent

to the present day version of EN 1993-1-1 [6]. In this idea, using the notations and

the stress distributions given in tables 1.1 and 1.2, the values of α, for the distinction

between class 1 and class 2, and ψ, for the distinction between class 3 and class 4,

should be determined. In these tables, (+) means compression.

1.3.2.1. Determine α – the fraction of the web in compression – for classes 1 and 2

1. Determine the height of the web used by the compression force (considering

a plastic stress distribution)

yw

EdN

f t

Fh

⋅= (1.6)

2. If hN > c (for the web), the entire web is in compression and the values for

compression must be used (Tab. 1.1)

3. The height of the tensioned part (for plastic stress distribution) of the web is

2

hch NT

−= (1.7)

4. α – the fraction of the web that is in compression (Tab. 1.1)

c

hh NT +=α (1.8)

5. The obtained value for α is used for determining the limits (for class 1 and 2)

in table 1.1.

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6. If the slenderness of the web is bigger than the limit for class 2, the limit

between class 3 and class 4 must be checked.

1.3.2.1. Determine ψ – the stress in the extreme fibre opposite to the one (possibly)

yielding in compression – for class 3

7. Determine the stress generated by the compression force on the area of the

cross-section (A) (Tab. 1.1)

A

FEdN =σ (1.9)

8. Determine the stress range “available” for bending (Tab. 1.1)

NyM f σ−=σ (1.10)

9. ψ – the stress in the extreme fibre opposite to the one (possibly) yielding in

compression (Tab. 1.1)

y

NM

f

σ+σ−=ψ (1.11)

For each component of the cross-section, the check is done successively for class 1,

2 and 3, till the class is established.

Bibliography

1. Dalban C., Dima S., Chesaru E., Şerbescu C. – Construc ţ ii cu

structura metalic ă, Ed. Didactică si Pedagogică, 1997

2. Dima, Ş., Ştefănescu B. – Steel Structures – basic elements ,

Conspress Bucureşti, 2005

3. ESDEP – The European Steel Design Education Programme ,

http://www.haiyangshiyou.com/esdep/master/toc.htm

4. Tien T. Lan – Space Frame Structures, Structural Engineering

Handbook , Ed. Chen-Wai Fah, 1999, pag. 24-32

5. P100-1/2013 – Cod de proiectare seismic ă – Partea 1 –

Prevederi de proiectare pentru cl ădiri

6. EN 1993-1-1 – Eurocode 3: Design of Steel Structures – Part 1-1:

General rules and rules for buildings

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7. ANSI/AISC 360-10 – Specification for Structural Steel Buildings

8. JSCE – Standard Specifications for Steel and Composite

Structures

Summary

In many cases, instability is the most important limit state for

structural members made of steel or aluminium alloys. This

phenomenon can affect a part of the member, the entire member, a

part of the structure or the entire structure.

Instability is generally associated to compression but shear stresses

(associated or not with compression stresses) can lead to local

buckling too. Even members in tension can be subject of dynamic

instability.

Instability is a more severe problem than overcoming strength.

EN 1993-1-1 [6] defines four classes of cross-sections of structural

members to manage local buckling. Similar approaches can be

found in the American code ANSI/AISC 360-10 [7] and in the

Japanese code JSCE [8].

Keywords

Instability, buckling, flexural buckling, torsional buckling, lateral

torsional buckling, in-plane buckling, out-of-plane buckling, snap-

through buckling, local buckling, classes of cross-sections, beam,

arch, plate, shell, structure, frame

stab 1 e 2015

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