Stabilitate en 2014

1. STABILITY GENERAL ASPECTS

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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; = 0) (Fig. 1.1a); torsion buckling (v = 0; 0) (Fig. 1.1b); flexural-torsion buckling (v 0; 0) (Fig. 1.1c); where v means the lateral displacement in the plane of the cross-section and 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 in the 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


3

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

Lateral-torsional buckling may be prevented either by performing checks using suitable 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 acting in 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 buckling of the flange

Torsion (twisting of the beam)


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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 (), of tangential ones (), 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 () 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 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 (ov = 1,40 S235; ov = 1,30 S275; ov = 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.

1.3. CLASSES OF CROSS-SECTIONS

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


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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 local buckling when determining their bending moment resistance or compression resistance.

For practical reasons, the limits among 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.

real

Prandtl fy

0

Y F

y u


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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.

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

class 4 class 3 class 2 class 1

max < fy max = fy max = fy max = fy

y

y

z

z

( )

( + )

max = 0

max < y max = y max = 0 max > y max >> y

x

x

x

x

y

y

z

z

y

fy


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

Class Wall in bending Wall in

compression Wall in bending and compression

Stress distribution

1

72tc

33tc

when > 0,5: 113396

tc

when 0,5:

36tc

2

83tc

38tc

when > 0,5: 113456

tc

when 0,5:

5,41tc

Stress distribution

3

124tc

42tc

when > 1: 33,067,0

42tc

+

when 1: ( ) ( )162tc

yf235

= fy (N/mm2) 235 275 355 420 460

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.

c c c c

c c c

c

t t t t

t t t

t

Bending axis

Bending axis

c c c c

fy

fy

fy

fy

fy

fy

c c c c/2

fy

fy

fy fy

fy


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

Class Compressed flange Tension and compressed flange

Compressed edge Tension edge

Stress distribution

1 9

tc

9tc

9tc

2 10

tc

10tc

10tc

Stress distribution

3 14

tc

k21

tc

yf235

= fy (N/mm2) 235 275 355 420 460

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 502

2 d/t 702

3 d/t 902

yf235

=

fy (N/mm2) 235 275 355 420 460 1,00 0,92 0,81 0,75 0,71 2 1,00 0,85 0,66 0,56 0,51

t t t t c c c

c

c c c

c c c

c c

d t

2. INSTABILITY OF BARS

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Chapter 2

INSTABILITY OF BARS

2.1. TORSION

Generally, torsion is avoided in structural metal (steel or aluminium alloy) members. There are basically two types of torsion:

St. Venant torsion (torsiunea cu deplanare liber); warping torsion (torsiunea cu deplanare mpiedicat).

As a simplification, in the case of a member with a closed hollow cross-section, such as a structural hollow section, it may be assumed that the effects of torsional warping can be neglected; similarly, in the case of a member with open cross section, such as I or H, it may be assumed that the effects of St. Venant torsion can be neglected.

2.1.1. St. Venant torsion

It occurs when all the following assumptions are accomplished (Fig. 2.1): the torsion moment is constant along the bar; the area of the cross-section is constant along the bar; there are no connections at the ends or along the bar that could prevent

warping.

Fig. 2.1. St. Venant torsion

the flanges remain rectangles


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2.1.1.1. Stress and strain state

The following aspects can be noticed: there is no increase or reduction of the length of the fibres (as there is no

longitudinal force): x = 0 x = 0 (2.1)

warping (deplanarea) of the cross-section is a result of the assumption x = 0 (in order to keep the geometry);

=A

Ed dArT (2.2)

Fig. 2.2. St. Venant torsion stress state

each cross-section rotates like a rigid disk (it goes out of plane but the shape does not change);

the rotation between neighbour cross-section is the same along the bar.

.constdxd

=

= (2.3)

2.1.2. Warping torsion

It occurs anytime when at least one of the St. Venant assumptions is not fulfilled (Fig. 2.3).


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Fig. 2.3. Warping torsion

2.1.2.1. Stress and strain state

The following aspects can be noticed: there are longitudinal stresses and strains (Fig. 2.4):

x 0 x 0 w; w (2.4) the rotation between neighbour cross-section is variable along the bar.

.constdxd

= (2.5)

Fig. 2.4. Warping torsion stress state


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2.1.2.2. Equilibrium equations

The following aspects can be noticed: there is no axial force acting on the bar:

===A

wEdi,Ed 0dA0N0X (2.6)

there are no bending moments acting on the bar:

===A

wEd,yi,Ed,y 0zdA0M0M (2.7)

===A

wEd,zi,Ed,z 0ydA0M0M (2.8)

in each cross-section, the torsion moment is the sum of the St. Venant component and the warping component (Fig. 2.5):

0hVdArT ewA

Ed =+= (2.9)

Ed,wEd,tEd TTT += (2.10) where: Tt,Ed the internal St. Venant torsion; Tw,Ed the internal warping torsion.

Fig. 2.5. St. Venant torsion and warping torsion

2.1.3. Torsion and bending

2.1.3.1. Bi-symmetrical cross-section subject to bending moment and shear force


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The force F, acting in the plane xOz, generates only bending moment about the y y axis (and shear force) and no torsion moment, as the resultant forces Vw on the flanges are balanced (Fig. 2.6).

Fig. 2.6. Shear stresses in a bisymmetrical cross-section in bending

2.1.3.2. Mono-symmetrical cross-section subject to bending moment and shear force

A force F, acting in the plane xOz in the centre of gravity of a mono-symmetrical cross-section, generates not only bending moment about the y y axis (and shear force) but torsion moment too (Fig. 2.7).

Fig. 2.7. Shear stresses for force acting in the centre of gravity

eFhFT wefEd += (2.11)

The shear centre (centrul de tiere, centrul de ncovoiere-rsucire) is the point through which the applied loads must pass to produce bending without twisting. A force F, acting in the plane xOz in the shear centre of a mono-symmetrical cross-section, generates only bending moment about the y y axis (and shear force) and no torsion moment (Fig. 2.8).


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Fig. 2.8. Shear stresses for force acting in the shear centre

cVT EdEd = (2.12) eFhFcV wefEd += (2.13)

Ed

wef

VeFhF

c+

= (2.14)

Notations: EdwEd

f VF;VF

== (2.15)

Ed

EdeEd

VeVhV

c+

= (2.16)

ehc e += (2.17)

F acting in the centre of gravity F acting in the shear centre Fig. 2.9. Effects of a force acting in or outside of the shear centre


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2.1.4. Torsion calculation

2.1.4.1. St. Venant torsion

The case of open cross-sections a) Rectangular cross-section

t

Edmax I

tT = t = minimum edge (2.18)

3t tb3

1I = (2.19)

.constIG

Tdxd

t

Ed=

=== (2.20)

= tEd IGT (2.21) b) Cross-section made of several rectangles

Rigid disk assumptions (simplifying assumptions): 1. each cross-section rotates one about the other; 2. the rotation varies from one cross-section to the other but it is constant

for all the points on the same cross-section; the cross-section does not change its shape in plane but it can go out of plane;

3. the rotation occurs around an axis parallel to the axis of the bar. As a result of assumption 2,

t

Edn

1i,t

n

ii,Ed

n,t

n,Ed

1,t

1,Ed

IGT

IG

T

IGT

...

IGT

=

=

==

=

(2.22)

=n

1

3iit tb3

1I (2.23)

Remark: For hot-rolled shapes,

=

n

1

3iit tb3

I = 1,1 1,3 (2.24)

t

maxEdmax I

tT = tmax = maximum thickness (2.25)

= tEd IGT (2.26)

t

b

1

i

n


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The case of hollow sections (Fig. 2.10) aVbVT baEd += (2.27)

Fig. 2.10. Torsion of hollow sections

It is accepted that: (Bredt relation)

2T

aVbV Edba == (2.28)

b2TV Eda

= ; a2

TV Edb

= (2.29)

a

Ed

a

aa tab2

Tta

V

=

= (2.30)

b

Ed

b

bb tba2

Ttb

V

=

= (2.31)

min

Edmax tA2

T

= (2.32)

2.1.4.2. Warping torsion

An exact calculation would consider the bar as a sum of shells (Fig. 2.11).

Fig. 2.11. Shell modelling of a bar in torsion


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In daily practice a simplified approach is used, based on the Vlasov theory. The simplifying assumptions are the following ones:

1. rigid disk behaviour: each cross-section rotates one about the other; the rotation varies from one cross-section to the other but it is constant

for all the points on the same cross-section; the rotation occurs around an axis parallel to the axis of the bar (Fig.

2.12);

Fig. 2.12. Axis of rotation of the bar

2. the shear deformations are zero in the mean axis of the cross-section (Fig. 2.13);

Fig. 2.13. Mean axis of the cross-section

3. w and w are constant on the thickness of the cross-section, because it is thin (the mean axis is representative for the cross-section);

4. when calculating w, it is assumed that w = 0.


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Based on these assumptions, the cross-section of the bar is reduced to its mean axis (Fig. 2.14) and the following relations can be written between in-plane strains and longitudinal ones (Fig. 2.15), considering rotation around point C: dv'nn = (2.33)

dxdv

dsdu

= (2.34)

= cosnn'nn (2.35) == cosnn'nndv (2.36)

Fig. 2.14. Mean surface of the member

= dCnnn (2.37) == cosdCn'nndv (2.38) = cosCnr (2.39) = drdv (2.40)

== ddsrdudxdr

dsdu

(2.41)

By definition (Fig. 2.15),

( )triangletheofarea22dsr2ddsr == (2.42)

Notation (Fig. 2.15):

[ ]2s0

s

0

Lddsr = sectorial area (coordonat sectorial) (2.43)

=== uddsrdu (2.44)

==dxdu

(2.45)


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Fig. 2.15. Geometric relations

Expressing w and w === EEwx (2.46) dAEdA 2w = (2.47) ==

A

2

Aw dAEdAB (bimoment) (2.48)

(bimoment de ncovoiere-rsucire) =

A

2w dAI (warping constant [L6]) (2.49)

(moment de inerie sectorial) Parallel between bending moment and warping torsion

zI

M

y

Ed,yx = =

w

w IB

(2.50)

y

yEd,zz It

SV

= w

wEd,ww It

SM

= (2.51)

=A

w dAS (first sectorial moment [L4]) (2.52)

Sw = [L4] (moment static sectorial) The coordinates of the shear centre about the centre of gravity are:

y

AC I

dAzy

= (2.53)

z

AC I

dAyz

= (2.54)

du


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2.2. BUCKLING LENGTH

The first known theoretical approach for solving a bar in compression belongs to Euler (1744) [1]. He started by writing the following equilibrium equation (Fig. 2.16) for a pin connected bar axially loaded in compression:

==

+

1EIM

dxdv1

dxvd

232

2

2

(2.55)

where: vFM = (2.56)

Fig. 2.16. The equilibrium of a pin connected bar in compression

The solution he obtained is the very well known:

2

2

cr LEIF pi= (2.57)

for the critical force that generates buckling of the bar and:

Lx

sinez 0pi

= (2.58)

for the deformed shape of the bar.

This relation was then extended to other types of restraints at the ends, by inscribing the bar on an equivalent pin-connected bar (Fig. 2.17). To allow this, the buckling length was defined as a concept. All these theoretical approaches are based on the theory of bifurcation of equilibrium.

Definition

The system length (EN 1993-1-1 [6] def. 1.5.5) is the distance in a given plane between two adjacent points at which a member is braced against lateral displacement in this plane, or between one such point and the end of the member.

L

e0

F x


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Definition

The buckling length (Lcr) (EN 1993-1-1 [6] def. 1.5.6) is the system length of an otherwise similar member with pinned ends, which has the same buckling resistance as a given member or segment of member. It is also defined as the distance between two consecutive inflection points along the deformed shape of a bar. Sometimes, in practice, it is replaced by the system length.

Eulers relation is then expressed as:

2cr

2

cr LEIF pi= (2.59)

where Lcr = kL is the buckling length (Fig. 2.17). k end fixity condition.

k = 1,0 k = 0,7 k = 2,0 k = 0,5 k = 1,0 Fig. 2.17. Different values of the buckling length factor

2.2.1. Buckling length of columns

In everyday situations, bars are part of a structure, they are connected to other bars and so the joints are not purely fixed or purely pinned. As a result, the buckling length of an element depends on its loading state and on the stiffness of the neighbour bars. Relations for calculating it are given in different books and were given in Annexe E (informative) of the previous version of Eurocode 3 ENV 1993-


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1-1 [7]. For defining the buckling length of a column, (parts of) structures are separated in sway and non-sway, depending whether the (lateral) displacements of the joints at the end of the bar are permitted or not. This separation is done by means of stiffness criteria that will be presented later. Usually, the non-sway behaviour is guaranteed by means of bracings. The distribution factors used in figure 2.3 2.6 are calculated using the following relations:

1211C

C1 KKK

K =

++ (ENV 1993-1-1 [3], rel. (E.1)) (2.60)

2221C

C2 KKK

K =

++ (ENV 1993-1-1 [3], rel. (E.2)) (2.61)

where: KC stiffness of the column (I/L); Kij stiffness of the beam ij. Remark: A more precise formulation for Kij would be stiffness of the connection between beam ij and column, as semi-rigid connections could be used. In this case a more careful analysis should be carried out.

The buckling length for non-sway buckling mode is presented in figure 2.18 [7].

Fig. 2.18. Non-sway buckling mode (ENV 1993-1-1 [7] Fig. E.2.3)


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The buckling length for sway buckling mode is presented in figure 2.19 [7].

Fig. 2.19. Sway buckling mode (ENV 1993-1-1 [7] Fig. E.2.3)

Fig. 2.20. End fixity condition, k, for non-sway buckling (ENV 1993-1-1 [7] Fig. E.2.1)


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Fig. 2.21. End fixity condition, k, for sway buckling (ENV 1993-1-1 [7] Fig. E.2.2)

This model can be expanded to continuous columns, presuming the loading factor N/Ncr is constant on their entire length. If this does not happen (which is the actual case) the procedure is conservative for the most critical part of the column [7]. In this case, the distribution factors are calculated using the following relations:

12111C

1C1 KKKK

KK =

+++

+ (ENV 1993-1-1 [3], rel. (E.3)) (2.62)

22212C

2C2 KKKK

KK =

+++

+ (ENV 1993-1-1 [3], rel. (E.4)) (2.63)

where K1 and K2 are the values of the stiffness of the neighbour columns (Fig. 2.22).


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Fig. 2.22. Distribution factors for continuous columns (ENV 1993-1-1 [7] Fig. E.2.4)

2.2.2. Buckling length of beams

Presuming the beams are not subject to axial forces, their stiffness can be taken from table 2.1, as long as they remain in the elastic range [7].

Table 2.1. Stiffness of a beam in the elastic range (ENV 1993-1-1 [7] Tab. E.1) Connection at the other end of the beam Stiffness K of the beam

Fixed 1,0 I/L Pinned 0,75 I/L

Rotation equal to the adjacent one (double curvature) 1,5 I/L Rotation equal and opposite to the adjacent one

(simple curvature) 0,5 I/L

General case: a rotation at the adjacent end and b rotation at the opposite end

(1,0 + 0,5 a/b) I/L


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For regular buildings with rectangular frames and reinforced concrete floors, subject to uniform loads, it is accepted to consider the stiffness of the beams given in table 2.2.

Table 2.2. Stiffness K of beams structures with reinforced concrete floors ([7] Tab. E.2)

Loading condition of the beam Non-sway buckling mode

Sway buckling mode

Beams supporting directly the reinforced concrete slabs

1,0 I/L 1,0 I/L

Other beams under direct loads 0,75 I/L 1,0 I/L Beams subjected only to

bending moments at the ends 0,50 I/L 1,5 I/L

When the beams are subject to axial forces, stability functions must be used for expressing their stiffness. A simplified conservative approach is proposed in ENV 1993-1-1 [7], neglecting the increase of stiffness generated by tension and considering only compression in the beams. Based on these assumptions, the values in table 2.3 can be considered.

Table 2.3. Stiffness of beams in compression (ENV 1993-1-1 [7] Tab. E.3) Connection at the other end of the beam Stiffness K of the beam

Fixed 1,0 I/L (1,0 0,4 N/NE) Pinned 0,75 I/L (1,0 1,0 N/NE)

Rotation equal to the adjacent one (double curvature)

1,5 I/L (1,0 0,2 N/NE)

Rotation equal and opposite to the adjacent one (simple curvature)

0,5 I/L (1,0 1,0 N/NE)

where:

2

2

E LEIN pi= (2.64)

2.2.3. Empirical relations for the buckling length of columns


30

ENV 1993-1-1 [7] provides empirical expressions as safe approximations that can be used as an alternative to the values from figures 2.20 and 2.21. The k coefficient for the buckling length can be calculated by the following relations: a. for non-sway buckling mode (Fig. 2.20) ( ) ( )22121 055,014,05,0 =k ++++ ([7], rel. (E.5)) (2.65)

or, alternatively,

( )( ) 2121

2121

247,0364,00,2265,0145,00,1

=k +++

([7], rel. (E.6)) (2.66)

b. for sway buckling mode (Fig. 2.21)

( )( )

5,0

2121

2121

60,08,00,112,02,00,1

=k

+++

([7], rel. (E.7)) (2.67)

2.2.4. Comments on the buckling length of beams

If the buckling length is generally easy to identify for members subject to axial compression forces, the effective lateral buckling length is a more delicate subject, given the complexity of the deformed shape (at the same time buckling and torsion). This leads to a temptation to simplified approaches, like considering the effective lateral buckling length as equal to the distance between points of zero (Fig. 2.23) in the bending moment diagram, or between inflection points of the strong axis deformed shape [8].

Important

In order to prevent this, the American code [9] states in the 6.3 commentary: In members subjected to double curvature bending, the inflection point shall not be considered a brace point.


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Fig. 2.23. Zero bending moment points along a beam [8]

2.2.5. Present day practical buckling design

The previous approaches are valid only in the elastic range, using the theory of bifurcation of equilibrium. Several researchers tried to express buckling when at least one of Eulers requirements for the bifurcation of equilibrium: the axis of the member is rigorously straight; the compression load acts strictly in the centre of gravity of the cross-section; the cross-section is bi-symmetrical; the moment of inertia of the cross-section is constant all along the bar; the deflected shape is a sinusoid; the material is homogenous and has a perfectly elastic behaviour (E = constant). is not fulfilled.

Several researchers focused on buckling in the elasto-plastic range; the following ones can be mentioned [2]: Engesser and Considre (1889), Tetmayer (1890), von Krmn and Iassinski at the same time with Engesser (1910), Shanley (1946).

A different approach, based on the theory of divergence of equilibrium, was considered by ECCS (European Convention for Steel Structures) which conducted an experimental analysis in seven countries (Belgium, France, Germany, Great


32

Britain, Italy, Netherlands and Yugoslavia) during the decade 1960 1970. Following this, at the beginning one buckling curve and then several ones were drawn, depending on: the shape of the cross-section; the axis of the cross-section (the plane of buckling); the yield limit of the steel grade.

Present day practical buckling design codes approach is a compromise of two distinct concepts:

buckling length, usually based on the equilibrium bifurcation model, built around the perfect member idea;

buckling (reduction) factors, based on the equilibrium divergence model, taking into account the actual imperfect member.

In fact, the old classical stability analysis consists of two main steps: a stress and deflections analysis (often using a computer program) and a conventional code analysis on isolated members [8].

2.3. STABILITY BRACING

Bracings are essential components for the structural stability control. They can provide the lateral supports that are needed for preventing buckling.

Various criteria are used for classifying bracings, as: nodal or relative; punctual or continuous; against translation, against rotation or against both, specific for buckling or for lateral buckling etc. Usually, the main supports have also stability functions. A standard case is that of the classical fork support (Fig. 2.24). Some usual bracing cases are presented in figure 2.24.

Two main requirements apply on bracing systems: strength the bracing system must be able to resist the forces (or moments)

generated by the buckling trend;


33

stiffness the displacements of the supports provided by the bracing system must be low, otherwise buckling occurs.

In many cases, the stiffness requirement is more severe than the strength one.

Fig. 2.24. Usual bracing systems [8]

Complex bracings systems could be investigated using the spring composition procedure. Two strategies are available in dealing with stability bracings: requiring minimum values for the strength and the stiffness of the bracings to

allow the consideration of the analysis unbraced length as the distance between two consecutive braces ([9], appendix 6);

establishing the load capacity of the structure taking into account the bracing system as it is [6].


34

2.3.1. The bow imperfection (imperfeciunea iniial n arc)

To take into account the fact that the bar is not perfectly straight, EN 1993-1-1 [6] uses an equivalent bow imperfection (Fig. 2.25).

Fig. 2.25. Bow imperfection for a pinned member [8]

Considering a sine-function for the buckling shape (Fig. 2.25):

Lx

sinez 0pi

= (2.68)

The associated shear force on the supports is:

Lx

cosL

eNdxdzNH 0

pi

pi== (2.69)

LeNH 0max pi= (2.70)

If a parabola is considered as initial deformed shape, the value of Hmax increases a little and the relation in figure 5.4 from EN 1993-1-1 [6] can be obtained:

LeN4H 0max = (2.71)

To manage this bow imperfection, EN 1993-1-1 [6] uses an equivalent lateral load (Fig. 2.25) that would produce a bending moment equal to N e0.

q

P

e0 L

H

H N P

H

P

P

L

q

e0


35

8Lq

eNM2

0max

== (2.72)

The values recommended for e0 are given in table 2.4.

Table 2.4. Recommended values for bow imperfection (EN 1993-1-1 [6] Tab. 5.1) Buckling curve Elastic analysis Plastic analysis

e0 / L e0 / L a0 1 / 350 1 / 300 a 1 / 300 1 / 250 b 1 / 250 1 / 200 c 1 / 200 1 / 150 d 1 / 150 1 / 100

For the buckling curves given in EN 1993-1-1 [6], Hmax results as (0.9...2.7)% of the compression force.

2.3.2. The sway imperfection (imperfeciunea iniial datorat abaterii de la axa vertical)

To take into account the fact that the force does not act exactly on the centroid line of the bar, EN 1993-1-1 [6] uses an equivalent sway imperfection (Fig. 2.26).

P P

P

H

H

L


36

Fig. 2.26. Sway imperfection for a pinned member [8]

PH = (2.73) where: mh0 = (EN 1993-1-1 [6], rel. (5.5)) (2.74) 0 basic value, 0 = 1/200; h the reduction factor for height h (h = L in Fig. 2.26) applicable to columns:

h2

= h but 1 32

h (2.75)

h the height of the structure in meters; in this case, h = L; m the reduction factor for the number of columns in a row;

+=

m

115,0m (2.76)

m the number of columns in a row including only those columns which carry a vertical load NEd not less than 50% of the average value of the column in the vertical plane considered (Fig. 2.27).

Fig. 2.27. Equivalent sway imperfections (EN 1993-1-1 [6] Fig. 5.2)

2.4. STRUCTURAL STABILITY ANALYSES

The present day computer programs for structural analysis and computing devices used to run these applications allow analysing a structure by writing the equilibrium equations on its deformed shape. However, this type of analysis is not always necessary. In general, two types of analyses can be carried out on a structure:


37

first order analysis equilibrium is expressed on the initial shape of the structure (efforts increase because of the displacements);

second order analysis equilibrium is expressed on the deformed shape of the structure;

According to EN 1993-1-1 [6], it is not necessary to perform a second order analysis in situations when:

analysisplasticfor15FF

analysiselasticfor10FF

Ed

crcr

Ed

crcr

=

= (EN 1993-1-1 [6], rel. (5.1)) (2.77)

where: cr the factor by which the design loading would have to be increased to cause

elastic instability in a global mode; FEd the design loading on the structure; Fcr the elastic critical buckling load for global instability mode based on initial

values of the elastic stiffness. These requirements must be fulfilled on each floor of a building.

Provided that the compression forces in the beams are not important, cr for portal frames with shallow roof slopes can be calculated with the following approximative formula (Fig. 2.28):

=

Ed,HEd

Edcr

hVH

(EN 1993-1-1 [6], rel. (5.2)) (2.78)

Fig. 2.28. Notations for determining cr (EN 1993-1-1 [6] Fig. 5.1)


38

where: HEd the design value of the horizontal reaction at the bottom of the storey to the

horizontal loads and fictitious horizontal loads; VEd the total design vertical load on the structure on the bottom of the storey; h the storey height; H,Ed the horizontal displacement at the top of the storey, relative to the bottom of

the storey, when the frame is loaded with horizontal loads and fictitious horizontal loads which are applied at each floor level.

The previous limitation for the compression force in the beams is considered to be satisfied if:

Ed

y

NfA

3,0

(EN 1993-1-1 [6], rel. (5.3)) (2.79)

where:

the in-plane non dimensional slenderness calculated for the beam considered as hinged at its ends of the system length measured along the beam;

NEd the design value of the compression force.

For single storey frames, in the elastic range, if cr 3,0, second order sway effects due to vertical loads may be calculated by increasing the horizontal loads HEd and equivalent loads VEd due to imperfections by the factor:

cr

11

1

(EN 1993-1-1 [6], rel. (5.4)) (2.80)

When performing the global analysis for determining end forces and end moments to be used in member checks, local bow imperfections may be neglected. However for frames sensitive to second order effects local bow imperfections of members additionally to global sway imperfections should be introduced in the structural analysis of the frame for each compressed member where the following conditions are met:

at least one moment resistant joint at one member end;


39

Ed

y

NfA

5,0> (EN 1993-1-1 [6], rel. (5.8)) (2.81)

where:

the in-plane non-dimensional slenderness calculated for the member considered as hinged at its ends;

NEd the design value of the compression force.

2.5. THE UNIQUE GLOBAL AND LOCAL IMPERFECTION

EN 1993-1-1 [6] accepts a unique imperfection approach, as an alternative to the two types of imperfections bow and sway ones, using the following relation:

cr

max,cr

Rk20

cr

max,cr

cr0init IE

NeIEN

e

=

= (EN 1993-1-1 [6], rel. (5.9)) (2.82)

where:

( ) 2,0for1

1

NM2,0e 2

1M

2

Rk

Rk0 >

= (EN 1993-1-1 [6], rel. (5.10)) (2.83)

cr

k,ult

= is the relative slenderness (EN 1993-1-1 [6], rel. (5.11)) (2.84)

the imperfection factor for the relevant buckling curve; the reduction factor for the relevant buckling curve for that cross-section; ult,k the minimum force amplifier for the axial force configuration NEd in members

to reach the characteristic resistance NRk of the most axially stressed cross section without taking buckling into account;

cr the minimum force amplifier for the axial force configuration NEd in members to reach the elastic critical buckling;

MRk the characteristic moment resistance of the critical cross section (Mel,Rd or Mpl,Rd);

NRk the characteristic resistance to axial force;

max,crIE the bending moment due to cr at the critical cross section;

cr the shape of elastic critical buckling mode.


40

2.6. LATERAL-TORSIONAL BUCKLING OF BEAMS

Lateral-torsional buckling may occur in the case of beams or lattice girders. It can be prevented either by performing checks using suitable relations, or by introducing lateral bracings whose purpose is to reduce the distance on which this phenomenon may occur.

The checking relation given in EN 1993-1-1 [6] for lateral buckling check is:

1M

yyLTRd,b

fWM

= (EN 1993-1-1 [6], rel. (6.55)) (2.85)

where:

LT the reduction factor for lateral-torsional buckling; Wy the appropriate section modulus:

Wy = Wpl,y for Class 1 or 2 cross-sections Wy = Wel,y for Class 3 cross-sections Wy = Weff,y for Class 4 cross-sections

fy the yielding limit; M1 the partial safety factor for resistance of members to instability assessed by

member checks (M1 = 1,0 in the National Annex of EN 1993-1-1 [6]); LT is taken form the appropriate buckling curve, based on the non-dimensional slenderness:

cr

yyLT

MfW

= (2.86)

where Mcr is the elastic critical moment for lateral-torsional buckling, whose expression is not given in EN 1993-1-1 [6]. Under these circumstances, it can be taken from recognised sources like publications, or computer programs. Some examples of such sources are the following ones:

the previous version of Eurocode 3 - ENV 1993-1-1 [7]; published books (ex. Timoshenko, Gere [10] etc.); Non-Contradictory Complementary Contribution documents (NCCI), ex.

SN003b-EN-EU [11];


41

computer programs like LTBEAM [12], developed at CTICM (Centre Technique Industriel de la Construction Mtallique).

The relation recommended by SN003b-EN-EU [11] is:

( )( ) ( )

+

pi

+

pi= g2

2g2

z

2t

2z

z

w

2

w

z

2z

z

2

1cr zCzCIEIGLk

II

kk

LkIECM (2.87)

where: E modulus of elasticity (Youngs modulus E =210000N/mm2); G shear modulus (G =81000N/mm2); Iz the second moment of the area about the weak axis (z z); It the torsion constant; Iw the warping constant; L the beam length between points which have lateral restraints; kz the effective length factor that refers to the end rotation about the z z axis; kw the effective length factor that refers to the end warping; zg the distance between the point of load application and the shear centre; C1, C2 coefficients depending on the loading and end restraint conditions. It is to note that the value of the critical moment is influenced by the position of the loading point. The load can have a stabilizing or a destabilizing effect (Fig. 2.29).

Fig. 2.29. Influence of the position of the loading point

compression flange

tension flange

destabilizing effect

stabilizing effect


42

The checking relation (2.85), given in EN 1993-1-1 [6], can be used by means of three methods given in the code.

2.6.1. The general method

For all cross-sections, unless otherwise specified, the following relation can be used:

0,1but1 LT2LT

2LTLT

LT +

= (EN 1993-1-1 [6], rel. (6.56)) (2.88)

where:

( )[ ]2LTLTLTLT 2,015,0 ++= (2.89) LT imperfection factor, given in the (Romanian) National Annex of EN 1993-1-1 [6]; the recommended values can be taken from table 2.5.

Table 2.5. Values for lateral torsional buckling (EN 1993-1-1 [6] Tab. 6.3) Buckling curve

a b c d Imperfection factor LT 0,21 0,34 0,49 0,76

The proper buckling curve, depending on the type of cross-section, is chosen based on the recommendations given in table 2.6.

Table 2.6. Recommended values for lateral torsional buckling curves for cross-sections using relation (2.88) (EN 1993-1-1 [6] Tab. 6.4)

Cross-section Limits Buckling curve Rolled I-sections

h/b 2 h/b > 2

a b

Welded I-sections h/b 2 h/b > 2

c d

Other cross-sections - d

For values of the slenderness 0,LTLT or for 2

0,LTcr

Ed

MM , lateral torsional buckling

effects may be ignored and only cross sectional checks apply. The maximum


43

recommended value for 0,LT , which was adopted in the Romanian National Annex of EN 1993-1-1 [6] is 0,4.

2.6.2. The specific method for rolled sections or equivalent welded sections

This method is a particular case for rolled sections or equivalent welded section. In this case,

+=

2LT

LT

LT

2LT

2LTLT

LT 10,1

but1 (EN 1993-1-1 [6], rel. (6.57)) (2.90)

where:

( )[ ]2LT0LTLTLTLT 15,0 ++= (2.91) LT imperfection factor.

4,00LT the value in the Romanian National Annex of EN 1993-1-1 [6] 4,00LT = 75,0 the value in the Romanian National Annex of EN 1993-1-1 [6] 75,0=

The values of LT are taken from table 2.5, depending on the proper buckling curve, chosen based on the recommendations given in table 2.7.

Table 2.7. Recommended values for lateral torsional buckling curves for cross-sections using relation (2.90) (EN 1993-1-1 [6] Tab. 6.5)

Cross-section Limits Buckling curve Rolled I-sections

h/b 2 h/b > 2

b c

Welded I-sections h/b 2 h/b > 2

c d

2.6.3. The modified specific method

This method is also specific for rolled sections or equivalent welded section. A correction is introduced, to take into account the bending moment diagram along the bar. In this idea, a modified reduction factor is calculated, as follows:


44

0,1butf mod,LTLT

mod,LT

= (EN 1993-1-1 [6], rel. (6.57)) (2.92)

where:

LT the reduction factor obtained at 2.6.2, using relation 2.90;

( ) ( )[ ] 0,1fbut8,00,21k15,01f 2LTc = (2.93) kc is given in table 2.8, depending on the bending moment diagram.

Table 2.8. Correction factor kc (EN 1993-1-1 [6] Tab. 6.6) Moment distribution kc

= 1 1,0

-1 1 33,033,1

1

0,94

0,90

0,91

0,86

0,77

0,82

2.6.4. Simplified assessment methods for beams with restraints in buildings

In the case where the compressed flange is provided with discrete lateral restraints (the recommended situation) at a distance Lc between two consecutive ones, the beam is not susceptible to lateral-torsional buckling if:

Ed,y

Rd,c0c

1z,f

ccf

MM

iLk

= (EN 1993-1-1 [6], rel. (6.59)) (2.94)

where: My,Ed the maximum design value of the bending moment within the restraint spacing;


45

1M

yyRd,c

fWM

= ;

kc a slenderness correction factor for moment distribution between restraints, given in table 2.8; if,z the radius of gyration of the equivalent compression flange composed of the compression flange plus 1/3 of the compressed part of the web area, about the minor axis of the section;

0c a slenderness limit of the equivalent compression flange defined as follows; it is given in the National Annex of EN 1993-1-1 [6]; the recommended value is

1,00,LT0c += ;

=pi= 9,93fE

y1 ;

yf235

= (fy in N/mm2).

If the requirements from relation (2.94) are not fulfilled, the design buckling resistance moment is expressed as: Rd,cfRd,b MkM = l but Rd.cRd.b MM (EN 1993-1-1 [6], rel. (6.60)) (2.95) where:

the reduction factor of the equivalent compression flange determined with f ; kf the modification factor accounting for the conservatism of the equivalent

compression flange method. The recommended value is kf = 1,10 and it was adopted in the National Annex of

EN 1993-1-1 [6]. is determined based on f , using curve d for welded sections

having 44thf

and curve c for all other situations, where:

h the overall depth of the cross-section; tf the thickness of the compression flange.

2.7. STABILITY REQUIREMENTS FOR BRACING SYSTEMS


46

Buckling and lateral-torsional buckling checks strongly depend on the position of the bracing points along the structural member in discussion. The requirements for a bracing system are not only in terms of strength but stiffness too. In fact, if a system needs important displacements to resist against forces generated by the buckling trend of an element, buckling of the braced element could occur.

Following the strength calculation according to relations given in EN 1993-1-1 [6], it could be considered that, globally, a mean force of about 2% of the axial load (1.6% corresponding to the bow imperfection and 0.5% for the sway imperfection) is going to the support of a pin-connected member. On this basis, it could be appreciated that some required strengths for bracings in ANSI/AISC 360-10 [9] are under-evaluated by EN 1993-1-1 [6], as in the American code requirements are expressed not only as strength but as stiffness too. Winter, Yura and others [9] showed that the minimum theoretical stiffness of bracings, evaluated for the ideal member, is not enough for the actual imperfect member. Figure 2.30 and table 2.9 show examples [8] of strength and stiffness requirements for bracing systems in ANSI/AISC 360-10 [9].

(a) (b) (c) (d) (e) Fig. 2.30. Strength and stiffness requirements in ANSI/AISC 360-10 [9]

In table 2.9, the notations have the following meanings: Pr,b required brace strength (N); Pr required strength in axial compression (N); Lb unbraced length (mm); br required brace stiffness (N/mm);

Pr,b br

P

P

P

P

P

Pr,b br

Pr,b br

Pr,b br

Pr,b br

Pr,b br

Pr,b br

Lb

Lb

Lb

Lb

Lb Lb Lb Lb

Lb


47

resistance factor ( = 0,75); Mr required flexural strength (Nmm); Cd = 1,0, except in the following case; = 2,0, for the brace closest to the inflection point in a beam subject to double

curvature bending; h0 distance between flange (chord) centroids (mm).

Table 2.9. Strength and stiffness requirements in ANSI/AISC 360-10 [9] Case Required strength Required stiffness (a)

rb,r P004,0P = ([9], rel. (A-6-1))

= b

rbr L

P21 ([9], rel. (A-6-2))

(b) rb,r P004,0P = ([9], rel. (A-6-1))

= b

rbr L

P21 ([9], rel. (A-6-2))

c) rb,r P01,0P = ([9], rel. (A-6-3))

= b

rbr L

P81 ([9], rel. (A-6-4))

(d) 0

drb,r h

CM008,0P = ([9], rel. (A-6-5))

= 0bdr

br hLCM41

([9], rel. (A-6-6))

(e) 0

drb,r h

CM02,0P = ([9], rel. (A-6-7))

= 0bdr

br hLCM101

([9], rel. (A-6-8))

In figure 2.30 and table 2.9, case (d) refers to relative brace while case (e) is for nodal brace. Relative brace brace that controls the relative movement of two adjacent brace points along the length of a beam or column or the relative lateral displacement of two stories in a frame [9]. Nodal brace brace that prevents lateral movement or twist independently of other braces at adjacent brace points [9]. Bracing members or system that provides stiffness and strength to limit the out-of-plane movement of another member at a brace point [9].

2.8. THE USE OF BUCKLING LENGTH FOR ARCHES AND FRAMES

There are situations when simplified relations are useful for the design of arches or single storey frames. The following relations can be used [13].


48

Arches for two- and three-hinged arches (Fig. 2.31) with the ratio h/L between 0,15 and 0,5 and essentially uniform cross-section, the in-plane buckling length may be taken as:

s25,1Lcr = (2.96) where s is half of the arch length;

Fig. 2.31. Buckling length for a two-hinged arch [13]

Two- and three-hinged frames if the inclination of the columns is less than 15, the following relation can be used for the column buckling length (Fig. 2.32):

r0cr Kh

IE10hIsI2,34hL

+

+= (2.97)

where Kr is the stiffness of the semi-rigid connection in the joint;

Fig. 2.32. Buckling length for a three-hinged frame [13]

For the buckling length of the rafter, the following relation can be used:


49

0

0

r0cr NI

NIKhIE10

hIsI2,34hL

+

+= (2.98)

where N and N0 are the axial forces in the column and in the rafter; in the case of tapered cross-sections, the cross-sections at 0,65h or 0,65s (Fig. 2.32) respectively should be used;

Columns or rafters with knee bracing the in-plane buckling length for columns (Fig. 2.33(a)) and for rafters (Fig. 2.33(b)) can be estimated as:

0lcr s7,0s2L += (2.99)

( a ) ( b ) Fig. 2.33. Portal frame (a) and frame with V-shaped columns (b) [13]

Torsional buckling of spatial frames for the rotational buckling of axi-symmetrical spatial structures (Fig. 2.34), the following approximate relation can be used for determining the buckling length factor k, provided that 1 < k < 2 and a/s < 0,2.

Fig. 2.34. Rotationally symmetric spatial frame [13]


50

( ) r22

Ksa1s4IEa310

s

a21k+

pi++= (2.100)

where EI is the bending stiffness of the rafter for bending about the vertical axis and Kr is the rotational stiffness of the connection between the rafter and the compression ring for bending about the vertical axis.

Bibliography

1. Dalban C., Dima S., Chesaru E., erbescu C. Construcii cu structura metalic, Ed. Didactic i Pedagogic, 1997

2. Dima, ., tefnescu B. Steel Structures basic elements, Conspress Bucureti, 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 Partea 1 Prevederi de proiectare pentru cldiri

6. EN 1993-1-1 Eurocode 3: Design of Steel Structures Part 1-1: General rules and rules for buildings

7. ENV 1993-1-1 Eurocode 3: Design of Steel Structures and National Application Document Part 1-1: General rules and rules for buildings

8. Diacu I., tefnescu B. Stability bracing in practice, Steel A New and Traditional Material for Building, Proceedings of the International Conference in Metal Structures Poiana Braov, Romnia, September 20-22, 2006, D. Dubin, V. Ungureanu Editors, Taylor & Francis Group, 2006, pag. 109-117

9. ANSI/AISC 360-10. Specification for Structural Steel Buildings. AISC, USA, 2010

10. Timoshenko, S.P., Gere, J.M. Theory of elastic stability, 2nd Edition. McGraw-Hill, 1961

11. SN003b-EN-EU Elastic critical moment for lateral torsional buckling

12. LTBEAM http://www.cticm.com/content/ltbeam-version-1011


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13. Blass H.J. Timber Engineering STEP 1: Basis of design, material properties, structural components and joints, Almere: Centrum Hout, 1995

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