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# Eurocode 3 - General rules and ruled for buildings

Eurocode 3 applies to the design of buildings and civil engineering works in steel. It complies with the principles and requirements for the safety and serviceability of structures, the basis of their design and verification that are given in EN 1990 – Basis of structural design. Eurocode 3 is concerned only with requirements for resistance, serviceability, durability and fire... Vedi di più

Esame di Tecnica delle costruzioni docente Prof. F. D'assisi Ricciardelli

Anteprima

### ESTRATTO DOCUMENTO

EN 1993-1-1: 2005 (E)

h vertical depth of the un-haunched section

s

L length of haunch within the length L

h y

length between restraints

L y

1.7 Conventions for member axes

(1) The convention for member axes is:

x-x - along the member

y-y - axis of the cross-section

z-z - axis of the cross-section

(2) For steel members, the conventions used for cross-section axes are:

generally:

– y-y - cross-section axis parallel to the flanges

z-z - cross-section axis perpendicular to the flanges

for angle sections:

– y-y - axis parallel to the smaller leg

z-z - axis perpendicular to the smaller leg

where necessary:

– u-u - major principal axis (where this does not coincide with the yy axis)

v-v - minor principal axis (where this does not coincide with the zz axis)

(3) The symbols used for dimensions and axes of rolled steel sections are indicated in Figure 1.1.

(4) The convention used for subscripts that indicate axes for moments is: "Use the axis about which the

moment acts."

NOTE All rules in this Eurocode relate to principal axis properties, which are generally defined by

the axes y-y and z-z but for sections such as angles are defined by the axes u-u and v-v.

20 EN 1993-1-1: 2005 (E)

Figure 1.1: Dimensions and axes of sections 21

EN 1993-1-1: 2005 (E)

2 Basis of design

2.1 Requirements

2.1.1 Basic requirements

(1)P The design of steel structures shall be in accordance with the general rules given in EN 1990.

(2) The supplementary provisions for steel structures given in this section should also be applied.

(3) The basic requirements of EN 1990 section 2 should be deemed be satisfied where limit state design is

used in conjunction with the partial factor method and the load combinations given in EN 1990 together with

the actions given in EN 1991.

(4) The rules for resistances, serviceability and durability given in the various parts of EN 1993 should be

applied.

2.1.2 Reliability management

(1) Where different levels of reliability are required, these levels should preferably be achieved by an

appropriate choice of quality management in design and execution, according to EN 1990 Annex C and

EN 1090.

2.1.3 Design working life, durability and robustness

2.1.3.1 General

(1) Depending upon the type of action affecting durability and the design working life (see EN 1990) steel

structures should be

designed against corrosion by means of

– suitable surface protection (see EN ISO 12944)

– the use of weathering steel

– the use of stainless steel (see EN 1993-1-4)

detailed for sufficient fatigue life (see EN 1993-1-9)

– designed for wearing

– designed for accidental actions (see EN 1991-1-7)

– inspected and maintained.

2.1.3.2 Design working life for buildings

(1)B The design working life should be taken as the period for which a building structure is expected to be

used for its intended purpose.

(2)B For the specification of the intended design working life of a permanent building see Table 2.1 of

EN 1990.

(3)B For structural elements that cannot be designed for the total design life of the building, see

2.1.3.3(3)B.

2.1.3.3 Durability for buildings

(1)B To ensure durability, buildings and their components should either be designed for environmental

actions and fatigue if relevant or else protected from them.

22 EN 1993-1-1: 2005 (E)

(2)B The effects of deterioration of material, corrosion or fatigue where relevant should be taken into

account by appropriate choice of material, see EN 1993-1-4 and EN 1993-1-10, and details, see

EN 1993-1-9, or by structural redundancy and by the choice of an appropriate corrosion protection system.

(3)B If a building includes components that need to be replaceable (e.g. bearings in zones of soil

settlement), the possibility of their safe replacement should be verified as a transient design situation.

2.2 Principles of limit state design

(1) The resistance of cross-sections and members specified in this Eurocode 3 for the ultimate limit states

as defined in EN 1990, 3.3 are based on tests in which the material exhibited sufficient ductility to apply

simplified design models.

(2) The resistances specified in this Eurocode Part may therefore be used where the conditions for

materials in section 3 are met.

2.3 Basic variables

2.3.1 Actions and environmental influences

(1) Actions for the design of steel structures should be taken from EN 1991. For the combination of

actions and partial factors of actions see Annex A to EN 1990.

NOTE 1 The National Annex may define actions for particular regional or climatic or accidental

situations.

NOTE 3B For simplified load arrangement, see Annex AB.2.

(2) The actions to be considered in the erection stage should be obtained from EN 1991-1-6.

(3) Where the effects of predicted absolute and differential settlements need to be considered, best

estimates of imposed deformations should be used.

(4) The effects of uneven settlements or imposed deformations or other forms of prestressing imposed

during erection should be taken into account by their nominal value P as permanent actions and grouped

k

with other permanent actions G from a single action (G + P ).

k k k

(5) Fatigue actions not defined in EN 1991 should be determined according to Annex A of EN 1993-1-9.

2.3.2 Material and product properties

(1) Material properties for steels and other construction products and the geometrical data to be used for

design should be those specified in the relevant ENs, ETAGs or ETAs unless otherwise indicated in this

standard.

2.4 Verification by the partial factor method

2.4.1 Design values of material properties

(1) For the design of steel structures characteristic values X or nominal values X of material properties

K n

should be used as indicated in this Eurocode.

2.4.2 Design values of geometrical data

(1) Geometrical data for cross-sections and systems may be taken from product standards hEN or

drawings for the execution to EN 1090 and treated as nominal values. 23

EN 1993-1-1: 2005 (E)

(2) Design values of geometrical imperfections specified in this standard are equivalent geometric

imperfections that take into account the effects of:

geometrical imperfections of members as governed by geometrical tolerances in product standards or the

– execution standard;

structural imperfections due to fabrication and erection;

– residual stresses;

– variation of the yield strength.

2.4.3 Design resistances

(1) For steel structures equation (6.6c) or equation (6.6d) of EN 1990 applies:

R 1 ( )

= = η η

k

R R X ; X ; a (2.1)

d k 1 k 1 i ki d

γ γ

## M M

where R is the characteristic value of the particular resistance determined with characteristic or nominal

k values for the material properties and dimensions

γ is the global partial factor for the particular resistance

M η η

NOTE For the definitions of , , X , X and a see EN 1990.

1 i k1 ki d

2.4.4 Verification of static equilibrium (EQU)

(1) The reliability format for the verification of static equilibrium in Table 1.2 (A) in Annex A of

EN 1990 also applies to design situations equivalent to (EQU), e.g. for the design of holding down anchors

or the verification of uplift of bearings of continuous beams.

2.5 Design assisted by testing

(1) The resistances R in this standard have been determined using Annex D of EN 1990.

k

(2) In recommending classes of constant partial factors γ the characteristic values R were obtained from

Mi k

R = R γ (2.2)

k d Mi

are design values according to Annex D of EN 1990

where R

d

γ are recommended partial factors.

Mi

NOTE 1 The numerical values of the recommended partial factors γ have been determined such that

Mi

R represents approximately the 5 %-fractile for an infinite number of tests.

k

NOTE 2 For characteristic values of fatigue strength and partial factors γ for fatigue see

Mf

EN 1993-1-9.

NOTE 3 For characteristic values of toughness resistance and safety elements for the toughness

verification see EN 1993-1-10.

(3) Where resistances R for prefabricated products should be determined from tests, the procedure in (2)

k

should be followed.

24 EN 1993-1-1: 2005 (E)

3 Materials

3.1 General

(1) The nominal values of material properties given in this section should be adopted as characteristic

values in design calculations.

(2) This Part of EN 1993 covers the design of steel structures fabricated from steel material conforming to

the steel grades listed in Table 3.1.

NOTE For other steel material and products see National Annex.

3.2 Structural steel

3.2.1 Material properties

(1) The nominal values of the yield strength f and the ultimate strength f for structural steel should be

y u

obtained

a) either by adopting the values f = R and f = R direct from the product standard

y eh u m

b) or by using the simplification given in Table 3.1

NOTE The National Annex may give the choice.

3.2.2 Ductility requirements

(1) For steels a minimum ductility is required that should be expressed in terms of limits for:

the ratio f / f of the specified minimum ultimate tensile strength f to the specified minimum yield

– u y u

strength f ;

y A

the elongation at failure on a gauge length of 5,65 (where A is the original cross-sectional area);

– 0

o

ε ε

the ultimate strain , where corresponds to the ultimate strength f .

– u u u ε

NOTE The limiting values of the ratio f / f , the elongation at failure and the ultimate strain may

u y u

be defined in the National Annex. The following values are recommended:

f / f 1,10;

– u y

elongation at failure not less than 15%;

– ε ≥ ε

15ε , where is the yield strain (ε = f / E).

– u y y y y

(2) Steel conforming with one of the steel grades listed in Table 3.1 should be accepted as satisfying these

requirements.

3.2.3 Fracture toughness

(1) The material should have sufficient fracture toughness to avoid brittle fracture of tension elements at

the lowest service temperature expected to occur within the intended design life of the structure.

NOTE The lowest service temperature to be adopted in design may be given in the National Annex.

(2) No further check against brittle fracture need to be made if the conditions given in EN 1993-1-10 are

satisfied for the lowest temperature. 25

EN 1993-1-1: 2005 (E)

(3)B For building components under compression a minimum toughness property should be selected.

NOTE B The National Annex may give information on the selection of toughness properties for

σ

members in compression. The use of Table 2.1 of EN 1993-1-10 for = 0,25 f (t) is recommended.

Ed y

(4) For selecting steels for members with hot dip galvanized coatings see EN 1461.

Table 3.1: Nominal values of yield strength f and ultimate tensile strength f for

y u

hot rolled structural steel

Nominal thickness of the element t [mm]

Standard

and t ≤ 40 mm 40 mm < t ≤ 80 mm

steel grade 2 2 2 2

f [N/mm ] f [N/mm ] f [N/mm ] f [N/mm ]

y u y u

EN 10025-2

S 235 235 360 215 360

S 275 275 430 255 410

S 355 355 510 335 470

S 450 440 550 410 550

EN 10025-3

S 275 N/NL 275 390 255 370

S 355 N/NL 355 490 335 470

S 420 N/NL 420 520 390 520

S 460 N/NL 460 540 430 540

EN 10025-4

S 275 M/ML 275 370 255 360

S 355 M/ML 355 470 335 450

S 420 M/ML 420 520 390 500

S 460 M/ML 460 540 430 530

EN 10025-5

S 235 W 235 360 215 340

S 355 W 355 510 335 490

EN 10025-6

S 460 Q/QL/QL1 460 570 440 550

26 EN 1993-1-1: 2005 (E)

Table 3.1 (continued): Nominal values of yield strength f and ultimate tensile

y

strength f for structural hollow sections

u Nominal thickness of the element t [mm]

Standard

and t ≤ 40 mm 40 mm < t ≤ 80 mm

steel grade 2 2 2 2

[N/mm ] f [N/mm ] f [N/mm ] f [N/mm ]

f y u y u

EN 10210-1

S 235 H 235 360 215 340

S 275 H 275 430 255 410

S 355 H 355 510 335 490

S 275 NH/NLH 275 390 255 370

S 355 NH/NLH 355 490 335 470

S 420 NH/NHL 420 540 390 520

S 460 NH/NLH 460 560 430 550

EN 10219-1

S 235 H 235 360

S 275 H 275 430

S 355 H 355 510

S 275 NH/NLH 275 370

S 355 NH/NLH 355 470

S 460 NH/NLH 460 550

S 275 MH/MLH 275 360

S 355 MH/MLH 355 470

S 420 MH/MLH 420 500

S 460 MH/MLH 460 530

3.2.4 Through-thickness properties

(1) Where steel with improved through-thickness properties is necessary according to EN 1993-1-10, steel

according to the required quality class in EN 10164 should be used.

NOTE 1 Guidance on the choice of through-thickness properties is given in EN 1993-1-10.

NOTE 2B Particular care should be given to welded beam to column connections and welded end

plates with tension in the through-thickness direction.

NOTE 3B The National Annex may give the relevant allocation of target values Z according to

Ed

3.2(2) of EN 1993-1-10 to the quality class in EN 10164. The allocation in Table 3.2 is recommended

for buildings:

Table 3.2: Choice of quality class according to EN 10164

Required value of Z expressed

Target value of Rd

according to in terms of design Z-values

Z Ed

EN 1993-1-10 according to EN 10164

Z ≤ 10 —

Ed

10 < Z ≤ 20 Z 15

Ed

20 < Z ≤ 30 Z 25

Ed

Z > 30 Z 35

Ed 27

EN 1993-1-1: 2005 (E)

3.2.5 Tolerances

(1) The dimensional and mass tolerances of rolled steel sections, structural hollow sections and plates

should conform with the relevant product standard, ETAG or ETA unless more severe tolerances are

specified.

(2) For welded components the tolerances given in EN 1090 should be applied.

(3) For structural analysis and design the nominal values of dimensions should be used.

3.2.6 Design values of material coefficients

(1) The material coefficients to be adopted in calculations for the structural steels covered by this

Eurocode Part should be taken as follows: = 2

E 210 000 N / mm

modulus of elasticity

– E

= ≈

G 81 000 N / mm ²

shear modulus

– + ν

2 (

1 )

ν = 0

,

3

Poisson’s ratio in elastic stage

– α −

= × 6 ≤

12 10 perK

coefficient of linear thermal expansion (for T 100 °C)

– NOTE For calculating the structural effects of unequal temperatures in composite concrete-steel

α −

= × 6

10 10 per K

structures to EN 1994 the coefficient of linear thermal expansion is taken as .

3.3 Connecting devices

3.3.1 Fasteners

(1) Requirements for fasteners are given in EN 1993-1-8.

3.3.2 Welding consumables

(1) Requirements for welding consumables are given in EN 1993-1-8.

3.4 Other prefabricated products in buildings

(1)B Any semi-finished or finished structural product used in the structural design of buildings should

comply with the relevant EN Product Standard or ETAG or ETA.

4 Durability

(1) The basic requirements for durability are set out in EN 1990.

(2) The means of executing the protective treatment undertaken off-site and on-site should be in

accordance with EN 1090.

NOTE EN 1090 lists the factors affecting execution that need to be specified during design.

(3) Parts susceptible to corrosion, mechanical wear or fatigue should be designed such that inspection,

maintenance and reconstruction can be carried out satisfactorily and access is available for in-service

inspection and maintenance.

28 EN 1993-1-1: 2005 (E)

(4)B For building structures no fatigue assessment is normally required except as follows:

a) Members supporting lifting appliances or rolling loads

b) Members subject to repeated stress cycles from vibrating machinery

c) Members subject to wind-induced vibrations

d) Members subject to crowd-induced oscillations

(5) For elements that cannot be inspected an appropriate corrosion allowance should be included.

(6)B Corrosion protection does not need to be applied to internal building structures, if the internal relative

humidity does not exceed 80%.

5 Structural analysis

5.1 Structural modelling for analysis

5.1.1 Structural modelling and basic assumptions

(1) Analysis should be based upon calculation models of the structure that are appropriate for the limit

state under consideration.

(2) The calculation model and basic assumptions for the calculations should reflect the structural

behaviour at the relevant limit state with appropriate accuracy and reflect the anticipated type of behaviour of

the cross sections, members, joints and bearings.

(3) The method used for the analysis should be consistent with the design assumptions.

(4)B For the structural modelling and basic assumptions for components of buildings see also EN 1993-1-5

and EN 1993-1-11.

5.1.2 Joint modelling

(1) The effects of the behaviour of the joints on the distribution of internal forces and moments within a

structure, and on the overall deformations of the structure, may generally be neglected, but where such

effects are significant (such as in the case of semi-continuous joints) they should be taken into account, see

EN 1993-1-8.

(2) To identify whether the effects of joint behaviour on the analysis need be taken into account, a

distinction may be made between three joint models as follows, see EN 1993-1-8, 5.1.1:

simple, in which the joint may be assumed not to transmit bending moments;

– continuous, in which the behaviour of the joint may be assumed to have no effect on the analysis;

– semi-continuous, in which the behaviour of the joint needs to be taken into account in the analysis

(3) The requirements of the various types of joints are given in EN 1993-1-8.

5.1.3 Ground-structure interaction

(1) Account should be taken of the deformation characteristics of the supports where significant.

NOTE EN 1997 gives guidance for calculation of soil-structure interaction. 29

EN 1993-1-1: 2005 (E)

5.2 Global analysis

5.2.1 Effects of deformed geometry of the structure

(1) The internal forces and moments may generally be determined using either:

first-order analysis, using the initial geometry of the structure or

– second-order analysis, taking into account the influence of the deformation of the structure.

(2) The effects of the deformed geometry (second-order effects) should be considered if they increase the

action effects significantly or modify significantly the structural behaviour.

(3) First order analysis may be used for the structure, if the increase of the relevant internal forces or

moments or any other change of structural behaviour caused by deformations can be neglected. This

condition may be assumed to be fulfilled, if the following criterion is satisfied:

## F

α = ≥

cr 10 for elastic analysis

cr F

Ed (5.1)

## F

α = ≥

cr 15 for plastic analysis

cr F

Ed

α is the factor by which the design loading would have to be increased to cause elastic instability

where cr in a global mode

Ed is the elastic critical buckling load for global instability mode based on initial elastic

F cr stiffnesses α

NOTE A greater limit for for plastic analysis is given in equation (5.1) because structural

cr

behaviour may be significantly influenced by non linear material properties in the ultimate limit state

(e.g. where a frame forms plastic hinges with moment redistributions or where significant non linear

deformations from semi-rigid joints occur). Where substantiated by more accurate approaches the

α

National Annex may give a lower limit for for certain types of frames.

cr

(4)B Portal frames with shallow roof slopes and beam-and-column type plane frames in buildings may be

checked for sway mode failure with first order analysis if the criterion (5.1) is satisfied for each storey. In

α

these structures may be calculated using the following approximative formula, provided that the axial

cr

compression in the beams or rafters is not significant:

 

 H h

 

α = Ed (5.2)

  

cr δ

V 

  

Ed H , Ed

where H is the design value of the horizontal reaction at the bottom of the storey to the horizontal loads

Ed and fictitious horizontal loads, see 5.3.2(7)

V is the total design vertical load on the structure on the bottom of the storey

Ed

δ is the horizontal displacement at the top of the storey, relative to the bottom of the storey,

which are applied at each floor level

h is the storey height

30 EN 1993-1-1: 2005 (E)

Figure 5.1: Notations for 5.2.1(2)

NOTE 1B For the application of (4)B in the absence of more detailed information a roof slope may

be taken to be shallow if it is not steeper that 1:2 (26°).

NOTE 2B For the application of (4)B in the absence of more detailed information the axial

compression in the beams or rafters may be assumed to be significant if

A f y

λ ≥ 0

,

3 (5.3)

N Ed

where N is the design value of the compression force,

Ed

λ is the inplane non dimensional slenderness calculated for the beam or rafters considered

as hinged at its ends of the system length measured along the beams of rafters.

(5) The effects of shear lag and of local buckling on the stiffness should be taken into account if this

significantly influences the global analysis, see EN 1993-1-5.

NOTE For rolled sections and welded sections with similar dimensions shear lag effects may be

neglected.

(6) The effects on the global analysis of the slip in bolt holes and similar deformations of connection

devices like studs and anchor bolts on action effects should be taken into account, where relevant and

significant.

5.2.2 Structural stability of frames

(1) If according to 5.2.1 the influence of the deformation of the structure has to be taken into account (2)

to (6) should be applied to consider these effects and to verify the structural stability.

(2) The verification of the stability of frames or their parts should be carried out considering imperfections

and second order effects.

(3) According to the type of frame and the global analysis, second order effects and imperfections may be

accounted for by one of the following methods:

a) both totally by the global analysis,

b) partially by the global analysis and partially through individual stability checks of members according to

6.3,

c) for basic cases by individual stability checks of equivalent members according to 6.3 using appropriate

buckling lengths according to the global buckling mode of the structure. 31

EN 1993-1-1: 2005 (E)

(4) Second order effects may be calculated by using an analysis appropriate to the structure (including

step-by-step or other iterative procedures). For frames where the first sway buckling mode is predominant

first order elastic analysis should be carried out with subsequent amplification of relevant action effects (e.g.

bending moments) by appropriate factors.

(5)B For single storey frames designed on the basis of elastic global analysis second order sway effects due

to vertical loads may be calculated by increasing the horizontal loads H (e.g. wind) and equivalent loads

Ed

φ

V due to imperfections (see 5.3.2(7)) and other possible sway effects according to first order theory by

Ed

the factor:

1 (5.4)

1

1 α cr α ≥

provided that 3,0,

cr

α may be calculated according to (5.2) in 5.2.1(4)B, provided that the roof slope is shallow and

where cr that the axial compression in the beams or rafters is not significant as defined in 5.2.1(4)B.

α

NOTE B For < 3,0 a more accurate second order analysis applies.

cr

(6)B For multi-storey frames second order sway effects may be calculated by means of the method given in

(5)B provided that all storeys have a similar

– distribution of horizontal loads and

– distribution of frame stiffness with respect to the applied storey shear forces.

– NOTE B For the limitation of the method see also 5.2.1(4)B.

(7) In accordance with (3) the stability of individual members should be checked according to the

following:

a) If second order effects in individual members and relevant member imperfections (see 5.3.4) are totally

accounted for in the global analysis of the structure, no individual stability check for the members

according to 6.3 is necessary.

b) If second order effects in individual members or certain individual member imperfections (e.g. member

imperfections for flexural and/or lateral torsional buckling, see 5.3.4) are not totally accounted for in the

global analysis, the individual stability of members should be checked according to the relevant criteria in

6.3 for the effects not included in the global analysis. This verification should take account of end

moments and forces from the global analysis of the structure, including global second order effects and

global imperfections (see 5.3.2) when relevant and may be based on a buckling length equal to the system

length

(8) Where the stability of a frame is assessed by a check with the equivalent column method according to

6.3 the buckling length values should be based on a global buckling mode of the frame accounting for the

stiffness behaviour of members and joints, the presence of plastic hinges and the distribution of compressive

forces under the design loads. In this case internal forces to be used in resistance checks are calculated

according to first order theory without considering imperfections.

NOTE The National Annex may give information on the scope of application.

5.3 Imperfections

5.3.1 Basis

(1) Appropriate allowances should be incorporated in the structural analysis to cover the effects of

imperfections, including residual stresses and geometrical imperfections such as lack of verticality, lack of

32 EN 1993-1-1: 2005 (E)

straightness, lack of flatness, lack of fit and any minor eccentricities present in joints of the unloaded

structure.

(2) Equivalent geometric imperfections, see 5.3.2 and 5.3.3, should be used, with values which reflect the

possible effects of all type of imperfections unless these effects are included in the resistance formulae for

member design, see section 5.3.4.

(3) The following imperfections should be taken into account:

a) global imperfections for frames and bracing systems

b) local imperfections for individual members

5.3.2 Imperfections for global analysis of frames

(1) The assumed shape of global imperfections and local imperfections may be derived from the elastic

buckling mode of a structure in the plane of buckling considered.

(2) Both in and out of plane buckling including torsional buckling with symmetric and asymmetric

buckling shapes should be taken into account in the most unfavourable direction and form.

(3) For frames sensitive to buckling in a sway mode the effect of imperfections should be allowed for in

frame analysis by means of an equivalent imperfection in the form of an initial sway imperfection and

individual bow imperfections of members. The imperfections may be determined from:

a) global initial sway imperfections, see Figure 5.2:

φ φ α α

= (5.5)

0 h m

φ φ

where is the basic value: = 1/200

0 0

α is the reduction factor for height h applicable to columns:

h 2

2 ≤ α ≤

α = 1

,

0

but h

h 3

h

h is the height of the structure in meters  

1

α = +

α  

0

,

5 1

is the reduction factor for the number of columns in a row:

m m m

 

m is the number of columns in a row including only those columns which carry a vertical load

N not less than 50% of the average value of the column in the vertical plane considered

Ed Figure 5.2: Equivalent sway imperfections

b) relative initial local bow imperfections of members for flexural buckling

e / L (5.6)

0

where L is the member length

NOTE The values e / L may be chosen in the National Annex. Recommended values are given in

0

Table 5.1. 33

EN 1993-1-1: 2005 (E)

Table 5.1: Design values of initial local bow imperfection e / L

0

Buckling curve elastic analysis plastic analysis

acc. to Table 6.1 e / L e / L

0 0

a 1 / 350 1 / 300

0

a 1 / 300 1 / 250

b 1 / 250 1 / 200

c 1 / 200 1 / 150

d 1 / 150 1 / 100

(4)B For building frames sway imperfections may be disregarded where

H 0,15 V (5.7)

Ed Ed

(5)B For the determination of horizontal forces to floor diaphragms the configuration of imperfections as

φ

given in Figure 5.3 should be applied, where is a sway imperfection obtained from (5.5) assuming a single

storey with height h, see (3) a). φ

Figure 5.3: Configuration of sway imperfections for horizontal forces on floor

diaphragms

(6) When performing the global analysis for determining end forces and end moments to be used in

member checks according to 6.3 local bow imperfections may be neglected. However for frames sensitive to

second order effects local bow imperfections of members additionally to global sway imperfections (see

5.2.1(3)) 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

– A f y

λ > 0

,

5 (5.8)

– N Ed

where N is the design value of the compression force

Ed

λ

and is the in-plane non-dimensional slenderness calculated for the member considered as hinged at

its ends

NOTE Local bow imperfections are taken into account in member checks, see 5.2.2 (3) and 5.3.4.

34 EN 1993-1-1: 2005 (E)

(7) The effects of initial sway imperfection and local bow imperfections may be replaced by systems of

equivalent horizontal forces, introduced for each column, see Figure 5.3 and Figure 5.4.

initial sway imperfections initial bow imperfections

Figure 5.4: Replacement of initial imperfections by equivalent horizontal forces

(8) These initial sway imperfections should apply in all relevant horizontal directions, but need only be

considered in one direction at a time.

(9)B Where, in multi-storey beam-and-column building frames, equivalent forces are used they should be

applied at each floor and roof level.

(10) The possible torsional effects on a structure caused by anti-symmetric sways at the two opposite faces,

should also be considered, see Figure 5.5.

2

## A B A B

1

(a) Faces A-A and B-B sway (b) Faces A-A and B-B sway

in same direction in opposite direction

1 translational sway

2 rotational sway

Figure 5.5: Translational and torsional effects (plan view) 35

EN 1993-1-1: 2005 (E) η

(11) As an alternative to (3) and (6) the shape of the elastic critical buckling mode of the structure may

cr

be applied as a unique global and local imperfection. The amplitude of this imperfection may be determined

from: N e N

η η η

= = (5.9)

0

cr Rk

e η η

0

init cr cr

" 2 "

## EI EI

λ

,max ,max

cr cr

where: 2

χλ

1 γ

( ) M

α λ λ

= − > (5.10)

1

Rk M

0,2 0,2

e for

0 2

N χλ

1

Rk

α ult , k

λ = is the relative slenderness of the structure (5.11)

and α cr

α is the imperfection factor for the relevant buckling curve, see Table 6.1 and Table 6.2;

χ is the reduction factor for the relevant buckling curve depending on the relevant cross-section, see

6.3.1;

α is the minimum force amplifier for the axial force configuration N in members to reach the

ult,k Ed

characteristic resistance N of the most axially stressed cross section without taking buckling

Rk

into account

α is the minimum force amplifier for the axial force configuration N in members to reach the

cr Ed

elastic critical buckling

M is the characteristic moments resistance of the critical cross section, e.g. M or M as

Rk el,Rk pl,Rk

relevant

N is the characteristic resistance to normal force of the critical cross section, i.e. N

Rk pl,Rk

η

η is the bending moment due to at the critical cross section

"

EI cr

,max

cr

η is the shape of elastic critical buckling mode

cr α α

NOTE 1 For calculating the amplifiers and the members of the structure may be considered

ult,k cr

to be loaded by axial forces N only that result from the first order elastic analysis of the structure for

Ed

NOTE 2 The National Annex may give information for the scope of application of (11).

5.3.3 Imperfection for analysis of bracing systems

(1) In the analysis of bracing systems which are required to provide lateral stability within the length of

beams or compression members the effects of imperfections should be included by means of an equivalent

geometric imperfection of the members to be restrained, in the form of an initial bow imperfection:

α

e = L / 500 (5.12)

0 m

where L is the span of the bracing system

 

1

α = +

 

0

,

5 1

and m m

 

in which m is the number of members to be restrained.

(2) For convenience, the effects of the initial bow imperfections of the members to be restrained by a

bracing system, may be replaced by the equivalent stabilizing force as shown in Figure 5.6:

δ

+

e

∑ 0 q

= (5.13)

q 8

d Ed 2

## L

36 EN 1993-1-1: 2005 (E)

δ

where is the inplane deflection of the bracing system due to q plus any external loads calculated from

q first order analysis

δ

NOTE may be taken as 0 if second order theory is used.

q

(3) Where the bracing system is required to stabilize the compression flange of a beam of constant height,

the force N in Figure 5.6 may be obtained from:

Ed

N = M / h (5.14)

Ed Ed

is the maximum moment in the beam

where M Ed

and h is the overall depth of the beam.

NOTE Where a beam is subjected to external compression N should include a part of the

Ed

compression force.

(4) At points where beams or compression members are spliced, it should also be verified that the bracing

α

system is able to resist a local force equal to N / 100 applied to it by each beam or compression member

m Ed

which is spliced at that point, and to transmit this force to the adjacent points at which that beam or

compression member is restrained, see Figure 5.7.

(5) For checking for the local force according to clause (4), any external loads acting on bracing systems

should also be included, but the forces arising from the imperfection given in (1) may be omitted.

e imperfection

0

q equivalent force per unit length

d

1 bracing system

The force N is assumed uniform within the span L of the bracing system.

Ed

For non-uniform forces this is slightly conservative.

Figure 5.6: Equivalent stabilizing force 37

EN 1993-1-1: 2005 (E) N

Ed Φ N Ed

Φ 1 Φ

2 N Ed 2

Φ Φ N Ed

N Ed Φ α Φ Φ

= : = 1 / 200

m 0 0

α

2ΦN = N / 100

Ed m Ed

1 splice

2 bracing system

Figure 5.7: Bracing forces at splices in compression elements

5.3.4 Member imperfections

(1) The effects of local bow imperfections of members are incorporated within the formulas given for

buckling resistance for members, see section 6.3.

(2) Where the stability of members is accounted for by second order analysis according to 5.2.2(7)a) for

compression members imperfections e according to 5.3.2(3)b), 5.3.2(5) or 5.3.2(6) should be considered.

0

(3) For a second order analysis taking account of lateral torsional buckling of a member in bending the

imperfections may be adopted as ke , where e is the equivalent initial bow imperfection of the weak axis

0,d 0,d

of the profile considered. In general an additional torsional imperfection need not to be allowed for.

NOTE The National Annex may choose the value of k. The value k = 0,5 is recommended.

5.4 Methods of analysis considering material non-linearities

5.4.1 General

(1) The internal forces and moments may be determined using either

a) elastic global analysis

b) plastic global analysis.

NOTE For finite element model (FEM) analysis see EN 1993-1-5.

(2) Elastic global analysis may be used in all cases.

38 EN 1993-1-1: 2005 (E)

(3) Plastic global analysis may be used only where the structure has sufficient rotation capacity at the

actual locations of the plastic hinges, whether this is in the members or in the joints. Where a plastic hinge

occurs in a member, the member cross sections should be double symmetric or single symmetric with a plane

of symmetry in the same plane as the rotation of the plastic hinge and it should satisfy the requirements

specified in 5.6. Where a plastic hinge occurs in a joint the joint should either have sufficient strength to

ensure the hinge remains in the member or should be able to sustain the plastic resistance for a sufficient

rotation, see EN 1993-1-8.

(4)B As a simplified method for a limited plastic redistribution of moments in continuous beams where

following an elastic analysis some peak moments exceed the plastic bending resistance of 15 % maximum,

the parts in excess of these peak moments may be redistributed in any member, provided, that:

a) the internal forces and moments in the frame remain in equilibrium with the applied loads, and

b) all the members in which the moments are reduced have Class 1 or Class 2 cross-sections (see 5.5), and

c) lateral torsional buckling of the members is prevented.

5.4.2 Elastic global analysis

(1) Elastic global analysis should be based on the assumption that the stress-strain behaviour of the

material is linear, whatever the stress level is.

NOTE For the choice of a semi-continuous joint model see 5.1.2(2) to (4).

(2) Internal forces and moments may be calculated according to elastic global analysis even if the

resistance of a cross section is based on its plastic resistance, see 6.2.

(3) Elastic global analysis may also be used for cross sections the resistances of which are limited by local

buckling, see 6.2.

5.4.3 Plastic global analysis

(1) Plastic global analysis allows for the effects of material non-linearity in calculating the action effects

of a structural system. The behaviour should be modelled by one of the following methods:

by elastic-plastic analysis with plastified sections and/or joints as plastic hinges,

– by non-linear plastic analysis considering the partial plastification of members in plastic zones,

– by rigid plastic analysis neglecting the elastic behaviour between hinges.

(2) Plastic global analysis may be used where the members are capable of sufficient rotation capacity to

enable the required redistributions of bending moments to develop, see 5.5 and 5.6.

(3) Plastic global analysis should only be used where the stability of members at plastic hinges can be

assured, see 6.3.5.

(4) The bi-linear stress-strain relationship indicated in Figure 5.8 may be used for the grades of structural

steel specified in section 3. Alternatively, a more precise relationship may be adopted, see EN 1993-1-5.

Figure 5.8: Bi-linear stress-strain relationship 39

EN 1993-1-1: 2005 (E)

(5) Rigid plastic analysis may be applied if no effects of the deformed geometry (e.g. second-order

effects) have to be considered. In this case joints are classified only by strength, see EN 1993-1-8.

(6) The effects of deformed geometry of the structure and the structural stability of the frame should be

verified according to the principles in 5.2.

NOTE The maximum resistance of a frame with significantly deformed geometry may occur before

all hinges of the first order collapse mechanism have formed.

5.5 Classification of cross sections

5.5.1 Basis

(1) The role of cross section classification is to identify the extent to which the resistance and rotation

capacity of cross sections is limited by its local buckling resistance.

5.5.2 Classification

(1) Four classes of cross-sections are defined, as follows:

Class 1 cross-sections are those which can form a plastic hinge with the rotation capacity required from

– plastic analysis without reduction of the resistance.

Class 2 cross-sections are those which can develop their plastic moment resistance, but have limited

– rotation capacity because of local buckling.

Class 3 cross-sections are those in which the stress in the extreme compression fibre of the steel member

– assuming an elastic distribution of stresses can reach the yield strength, but local buckling is liable to

prevent development of the plastic moment resistance.

Class 4 cross-sections are those in which local buckling will occur before the attainment of yield stress in

– one or more parts of the cross-section.

(2) In Class 4 cross sections effective widths may be used to make the necessary allowances for

reductions in resistance due to the effects of local buckling, see EN 1993-1-5, 5.2.2.

(3) The classification of a cross-section depends on the width to thickness ratio of the parts subject to

compression.

(4) Compression parts include every part of a cross-section which is either totally or partially in

compression under the load combination considered.

(5) The various compression parts in a cross-section (such as a web or flange) can, in general, be in

different classes.

(6) A cross-section is classified according to the highest (least favourable) class of its compression parts.

Exceptions are specified in 6.2.1(10) and 6.2.2.4(1).

(7) Alternatively the classification of a cross-section may be defined by quoting both the flange

classification and the web classification.

(8) The limiting proportions for Class 1, 2, and 3 compression parts should be obtained from Table 5.2. A

part which fails to satisfy the limits for Class 3 should be taken as Class 4.

(9) Except as given in (10) Class 4 sections may be treated as Class 3 sections if the width to thickness

ε

ratios are less than the limiting proportions for Class 3 obtained from Table 5.2 when is increased by

γ

f /

y M 0 σ

, where is the maximum design compressive stress in the part taken from first order or

com , Ed

σ com , Ed

where necessary second order analysis.

40 EN 1993-1-1: 2005 (E)

(10) However, when verifying the design buckling resistance of a member using section 6.3, the limiting

proportions for Class 3 should always be obtained from Table 5.2.

(11) Cross-sections with a Class 3 web and Class 1 or 2 flanges may be classified as class 2 cross sections

with an effective web in accordance with 6.2.2.4.

(12) Where the web is considered to resist shear forces only and is assumed not to contribute to the bending

and normal force resistance of the cross section, the cross section may be designed as Class 2, 3 or 4

sections, depending only on the flange class.

NOTE For flange induced web buckling see EN 1993-1-5.

5.6 Cross-section requirements for plastic global analysis

(1) At plastic hinge locations, the cross-section of the member which contains the plastic hinge should

have a rotation capacity of not less than the required at the plastic hinge location.

(2) In a uniform member sufficient rotation capacity may be assumed at a plastic hinge if both the

following requirements are satisfied:

a) the member has Class 1 cross-sections at the plastic hinge location;

b) where a transverse force that exceeds 10 % of the shear resistance of the cross section, see 6.2.6, is

applied to the web at the plastic hinge location, web stiffeners should be provided within a distance along

the member of h/2 from the plastic hinge location, where h is the height of the cross section at this

location.

(3) Where the cross-section of the member vary along their length, the following additional criteria should

be satisfied:

a) Adjacent to plastic hinge locations, the thickness of the web should not be reduced for a distance each

way along the member from the plastic hinge location of at least 2d, where d is the clear depth of the web

at the plastic hinge location.

b) Adjacent to plastic hinge locations, the compression flange should be Class 1 for a distance each way

along the member from the plastic hinge location of not less than the greater of:

2d, where d is as defined in (3)a)

– the distance to the adjacent point at which the moment in the member has fallen to 0,8 times the

– plastic moment resistance at the point concerned.

c) Elsewhere in the member the compression flange should be class 1 or class 2 and the web should be class

1, class 2 or class 3.

(4) Adjacent to plastic hinge locations, any fastener holes in tension should satisfy 6.2.5(4) for a distance

such as defined in (3)b) each way along the member from the plastic hinge location.

(5) For plastic design of a frame, regarding cross section requirements, the capacity of plastic

redistribution of moments may be assumed sufficient if the requirements in (2) to (4) are satisfied for all

members where plastic hinges exist, may occur or have occurred under design loads.

(6) In cases where methods of plastic global analysis are used which consider the real stress and strain

behaviour along the member including the combined effect of local, member and global buckling the

requirements (2) to (5) need not be applied. 41

EN 1993-1-1: 2005 (E)

Table 5.2 (sheet 1 of 3): Maximum width-to-thickness ratios for compression

parts

Internal compression parts

c

c c c Axis of

bending

t

t

t t t

t

t

t Axis of

c

c c bending

c

Part subject to Part subject to

Class Part subject to bending and compression

bending compression

f f f

y y y

Stress

distribution + + + αc

in parts c c c

(compression - -

-

positive) f

f f

y y y ε

396

α > ≤

when 0

,

5 : c / t α −

13 1

≤ ε ≤ ε

c / t 72 c / t 33

1 ε

36

α ≤ ≤

when 0

,

5 : c / t α ε

456

α > ≤

when 0

,

5 : c / t α −

13 1

≤ ε ≤ ε

c / t 83 c / t 38

2 ε

41

,

5

α ≤ ≤

when 0

,

5 : c / t α

f f

f

y y

Stress y

+ +

distribution

in parts c c

c

+

c/

(compression 2

- -

positive) ψ f

f y

y ε

42

ψ > − ≤

when 1 : c / t + ψ

0

,

67 0

,

33

≤ ε ≤ ε

c / t 124 c / t 42

3 ψ ≤ − ≤ ε − ψ − ψ

*)

when 1 : c / t 62 (

1 ) ( )

f 235 275 355 420 460

ε = 235 / f y

y ε 1,00 0,92 0,81 0,75 0,71

ψ ≤ σ ≤ ε

*) -1 applies where either the compression stress f or the tensile strain > f /E

y y y

42 EN 1993-1-1: 2005 (E)

Table 5.2 (sheet 2 of 3): Maximum width-to-thickness ratios for compression

parts

Outstand flanges

c c

c t

t

t t c

Rolled sections Welded sections

Part subject to bending and compression

Class Part subject to compression Tip in compression Tip in tension

αc

αc

Stress +

distribution +

+

in parts c

(compression - -

c c

positive) ε

ε 9

9 ≤

≤ c / t

≤ ε c / t

c / t 9

1 α α α

ε

ε 10

10 ≤

≤ c / t

≤ ε c / t

c / t 10

2 α α α

Stress + +

distribution -

in parts c c c

(compression

positive) ≤ ε

c / t 21 k

≤ ε σ

c / t 14

3 For k see EN 1993-1-5

σ

f 235 275 355 420 460

ε = 235 / f y

y ε 1,00 0,92 0,81 0,75 0,71 43

EN 1993-1-1: 2005 (E)

Table 5.2 (sheet 3 of 3): Maximum width-to-thickness ratios for compression

parts

Angles

h Does not apply to angles in

t b

Refer also to “Outstand flanges” continuous contact with other

(see sheet 2 of 3) components

Class Section in compression

Stress

distribution f

+ y

across +

section

(compression

positive) +

b h

≤ ε ≤ ε

h / t 15 : 11

,

5

3 2 t

Tubular sections

t d

Class Section in bending and/or compression

≤ ε 2

1 d / t 50

≤ ε 2

2 d / t 70

≤ ε 2

d / t 90

3 > ε 2

d / t 90

NOTE For see EN 1993-1-6.

f 235 275 355 420 460

y

ε = 235 / f ε 1,00 0,92 0,81 0,75 0,71

y 2

ε 1,00 0,85 0,66 0,56 0,51

44 EN 1993-1-1: 2005 (E)

6 Ultimate limit states

6.1 General γ

(1) The partial factors as defined in 2.4.3 should be applied to the various characteristic values of

## M

resistance in this section as follows: γ

resistance of cross-sections whatever the class is:

– M0

γ

resistance of members to instability assessed by member checks:

– M1

γ

resistance of cross-sections in tension to fracture:

– M2

see EN 1993-1-8

resistance of joints:

– NOTE 1 For other recommended numerical values see EN 1993 Part 2 to Part 6. For structures not

γ

covered by EN 1993 Part 2 to Part 6 the National Annex may define the partial factors ; it is

Mi

γ

recommended to take the partial factors from EN 1993-2.

Mi

γ

NOTE 2B Partial factors for buildings may be defined in the National Annex. The following

Mi

numerical values are recommended for buildings:

γ = 1,00

M0

γ = 1,00

M1

γ = 1,25

M2

6.2 Resistance of cross-sections

6.2.1 General

(1) The design value of an action effect in each cross section should not exceed the corresponding design

resistance and if several action effects act simultaneously the combined effect should not exceed the

resistance for that combination.

(2) Shear lag effects and local buckling effects should be included by an effective width according to

EN 1993-1-5. Shear buckling effects should also be considered according to EN 1993-1-5.

(3) The design values of resistance should depend on the classification of the cross-section.

(4) Elastic verification according to the elastic resistance may be carried out for all cross sectional classes

provided the effective cross sectional properties are used for the verification of class 4 cross sections.

(5) For the elastic verification the following yield criterion for a critical point of the cross section may be

used unless other interaction formulae apply, see 6.2.8 to 6.2.10.

2 2 2

     

   

σ σ σ σ τ

     

   

x , Ed z , Ed x , Ed z , Ed

+ − + ≤

Ed

3 1 (6.1)

     

   

γ γ γ γ γ

f f f f f

     

   

y M 0 y M 0 y M 0 y M 0 y M 0

σ

where is the design value of the local longitudinal stress at the point of consideration

x , Ed

σ is the design value of the local transverse stress at the point of consideration

z , Ed

τ is the design value of the local shear stress at the point of consideration

Ed

NOTE The verification according to (5) can be conservative as it excludes partial plastic stress

distribution, which is permitted in elastic design. Therefore it should only be performed where the

interaction of on the basis of resistances N , M , V cannot be performed.

Rd Rd Rd 45

EN 1993-1-1: 2005 (E)

(6) The plastic resistance of cross sections should be verified by finding a stress distribution which is in

equilibrium with the internal forces and moments without exceeding the yield strength. This stress

distribution should be compatible with the associated plastic deformations.

(7) As a conservative approximation for all cross section classes a linear summation of the utilization

ratios for each stress resultant may be used. For class 1, class 2 or class 3 cross sections subjected to the

combination of N , M and M this method may be applied by using the following criteria:

Ed y,Ed z,Ed

N y , Ed z , Ed

+ + ≤

Ed 1 (6.2)

## N M M

Rd y , Rd z , Rd

, M and M are the design values of the resistance depending on the cross sectional

where N Rd y,Rd z,Rd

classification and including any reduction that may be caused by shear effects, see 6.2.8.

NOTE For class 4 cross sections see 6.2.9.3(2).

(8) Where all the compression parts of a cross-section are at least Class 2, the cross-section may be taken

as capable of developing its full plastic resistance in bending.

(9) Where all the compression parts of a cross-section are Class 3, its resistance should be based on an

elastic distribution of strains across the cross-section. Compressive stresses should be limited to the yield

strength at the extreme fibres.

NOTE The extreme fibres may be assumed at the midplane of the flanges for ULS checks. For

fatigue see EN 1993-1-9.

(10) Where yielding first occurs on the tension side of the cross section, the plastic reserves of the tension

zone may be utilized by accounting for partial plastification when determining the resistance of a Class 3

cross-section.

6.2.2 Section properties

6.2.2.1 Gross cross-section

(1) The properties of the gross cross-section should be determined using the nominal dimensions. Holes

for fasteners need not be deducted, but allowance should be made for larger openings. Splice materials

should not be included.

6.2.2.2 Net area

(1) The net area of a cross-section should be taken as its gross area less appropriate deductions for all

holes and other openings.

(2) For calculating net section properties, the deduction for a single fastener hole should be the gross

cross-sectional area of the hole in the plane of its axis. For countersunk holes, appropriate allowance should

be made for the countersunk portion.

(3) Provided that the fastener holes are not staggered, the total area to be deducted for fastener holes

should be the maximum sum of the sectional areas of the holes in any cross-section perpendicular to the

member axis (see failure plane in Figure 6.1).

NOTE The maximum sum denotes the position of the critical fracture line.

46 EN 1993-1-1: 2005 (E)

(4) Where the fastener holes are staggered, the total area to be deducted for fasteners should be the greater

of:

a) the deduction for non-staggered holes given in (3)

 

2

s

 

t nd (6.3)

b)  

0 4 p

 

where s is the staggered pitch, the spacing of the centres of two consecutive holes in the chain measured

parallel to the member axis;

p is the spacing of the centres of the same two holes measured perpendicular to the member axis;

t is the thickness;

n is the number of holes extending in any diagonal or zig-zag line progressively across the member

or part of the member, see Figure 6.1.

d is the diameter of hole

0

(5) In an angle or other member with holes in more then one plane, the spacing p should be measured

along the centre of thickness of the material (see Figure 6.2).

Figure 6.1: Staggered holes and critical fracture lines 1 and 2

Figure 6.2: Angles with holes in both legs

6.2.2.3 Shear lag effects

(1) The calculation of the effective widths is covered in EN 1993-1-5.

(2) In class 4 sections the interaction between shear lag and local buckling should be considered according

to EN 1993-1-5.

NOTE For cold formed thin gauge members see EN 1993-1-3. 47

EN 1993-1-1: 2005 (E)

6.2.2.4 Effective properties of cross sections with class 3 webs and class 1 or 2 flanges

(1) Where cross-sections with a class 3 web and class 1 or 2 flanges are classified as effective Class 2

cross-sections, see 5.5.2(11), the proportion of the web in compression should be replaced by a part of 20εt w

adjacent to the compression flange, with another part of 20εt adjacent to the plastic neutral axis of the

w

effective cross-section in accordance with Figure 6.3. f

1 y

-

ε

20 t w

4 -

ε

20 t

3 w +

2 f y 2 1

1 compression

2 tension

3 plastic neutral axis

4 neglect

Figure 6.3: Effective class 2 web

6.2.2.5 Effective cross-section properties of Class 4 cross-sections

(1) The effective cross-section properties of Class 4 cross-sections should be based on the effective widths

of the compression parts.

(2) For cold formed thin walled sections see 1.1.2(1) and EN 1993-1-3.

(3) The effective widths of planar compression parts should be obtained from EN 1993-1-5.

(4) Where a class 4 cross section is subjected to an axial compression force, the method given in

EN 1993-1-5 should be used to determine the possible shift e of the centroid of the effective area A

N eff

relative to the centre of gravity of the gross cross section and the resulting additional moment:

∆ =

M N e (6.4)

Ed Ed N

NOTE The sign of the additional moment depends on the effect in the combination of internal forces

and moments, see 6.2.9.3(2).

(5) For circular hollow sections with class 4 cross sections see EN 1993-1-6.

48 EN 1993-1-1: 2005 (E)

6.2.3 Tension

(1) The design value of the tension force N at each cross section should satisfy:

Ed

N ≤

Ed 1

,

0 (6.5)

N t , Rd should be taken as the smaller of:

(2) For sections with holes the design tension resistance N

t,Rd

a) the design plastic resistance of the gross cross-section

A f y

=

N (6.6)

pl , Rd γ M 0

b) the design ultimate resistance of the net cross-section at holes for fasteners

0,9 A f

= net u

N (6.7)

u , Rd γ M 2 (as given in

(3) Where capacity design is requested, see EN 1998, the design plastic resistance N

pl,Rd

6.2.3(2) a)) should be less than the design ultimate resistance of the net section at fasteners holes N (as

u,Rd

given in 6.2.3(2) b)).

(4) In category C connections (see EN 1993-1-8, 3.4.2(1), the design tension resistance N in 6.2.3(1) of

t,Rd

the net section at holes for fasteners should be taken as N , where:

net,Rd

A f

net y

=

N (6.8)

net , Rd γ M 0

(5) For angles connected through one leg, see also EN 1993-1-8, 3.6.3. Similar consideration should also

be given to other type of sections connected through outstands.

6.2.4 Compression

(1) The design value of the compression force N at each cross-section should satisfy:

Ed

N ≤

Ed 1

,

0 (6.9)

N c , Rd

(2) The design resistance of the cross-section for uniform compression N should be determined as

c,Rd

follows: A f y

=

N for class 1, 2 or 3 cross-sections (6.10)

c , Rd γ M 0

A f

eff y

=

N for class 4 cross-sections (6.11)

c , Rd γ M 0

(3) Fastener holes except for oversize and slotted holes as defined in EN 1090 need not be allowed for in

compression members, provided that they are filled by fasteners.

(4) In the case of unsymmetrical Class 4 sections, the method given in 6.2.9.3 should be used to allow for

## ∆M

the additional moment due to the eccentricity of the centroidal axis of the effective section, see

Ed

6.2.2.5(4). 49

EN 1993-1-1: 2005 (E)

6.2.5 Bending moment

(1) The design value of the bending moment M at each cross-section should satisfy:

Ed

M ≤

Ed 1

,

0 (6.12)

M c , Rd

is determined considering fastener holes, see (4) to (6).

where M c,Rd

(2) The design resistance for bending about one principal axis of a cross-section is determined as follows:

W f

pl y

= =

M M for class 1 or 2 cross sections (6.13)

c , Rd pl , Rd γ M 0

W f

el , min y

= =

M M for class 3 cross sections (6.14)

c , Rd el , Rd γ M 0

W f

eff , min y

=

M for class 4 cross sections (6.15)

c , Rd γ M 0

where W and W corresponds to the fibre with the maximum elastic stress.

el,min eff,min

(3) For bending about both axes, the methods given in 6.2.9 should be used.

(4) Fastener holes in the tension flange may be ignored provided that for the tension flange:

A f

A 0

,

9 f f y

f , net u ≥ (6.16)

γ γ

M 2 M 0

is the area of the tension flange.

where A f

NOTE The criterion in (4) provides capacity design (see 1.5.8) in the region of plastic hinges.

(5) Fastener holes in tension zone of the web need not be allowed for, provided that the limit given in (4)

is satisfied for the complete tension zone comprising the tension flange plus the tension zone of the web.

(6) Fastener holes except for oversize and slotted holes in compression zone of the cross-section need not

be allowed for, provided that they are filled by fasteners.

6.2.6 Shear

(1) The design value of the shear force V at each cross section should satisfy:

Ed

V ≤

Ed 1

,

0 (6.17)

## V

c , Rd is the design shear resistance. For plastic design V is the design plastic shear resistance V

where V c,Rd c,Rd pl,Rd

as given in (2). For elastic design V is the design elastic shear resistance calculated using (4) and (5).

c,Rd

(2) In the absence of torsion the design plastic shear resistance is given by:

( )

A f / 3

v y

=

V (6.18)

pl , Rd γ M 0

where A is the shear area.

v

50 EN 1993-1-1: 2005 (E)

(3) The shear area A may be taken as follows:

v ( )

− + + η

A 2 bt t 2 r t h t

but not less than

a) rolled I and H sections, load parallel to web f w f w w

( )

− + +

A 2 bt t r t

b) rolled channel sections, load parallel to web f w f

( )

0

,

9 A bt

c) rolled T-section, load parallel to web f

( )

η h t

d) welded I, H and box sections, load parallel to web w w ( )

∑ h t

e) welded I, H, channel and box sections, load parallel to flanges A- w w

f) rolled rectangular hollow sections of uniform thickness:

g) circular hollow sections and tubes of uniform thickness 2A/π

where A is the crosssectional area;

h is the overall depth;

is the depth of the web;

h w

is the flange thickness;

t f

t is the web thickness (If the web thickness in not constant, t should be taken as the minimum

w w

thickness.).

η see EN 1993-1-5.

η

NOTE may be conservatively taken equal 1,0.

(4) For verifying the design elastic shear resistance V the following criterion for a critical point of the

c,Rd

cross section may be used unless the buckling verification in section 5 of EN 1993-1-5 applies:

τ ≤

Ed

( ) 1

,

0 (6.19)

γ

f 3

y M 0 V S

τ = Ed

τ may be obtained from: (6.20)

where Ed Ed I t

is the design value of the shear force

where V Ed

S is the first moment of area about the centroidal axis of that portion of the cross-section between

the point at which the shear is required and the boundary of the cross-section

I is second moment of area of the whole cross section

t is the thickness at the examined point

NOTE The verification according to (4) is conservative as it excludes partial plastic shear

distribution, which is permitted in elastic design, see (5). Therefore it should only be carried out where

the verification on the basis of V according to equation (6.17) cannot be performed.

c,Rd 51

EN 1993-1-1: 2005 (E)

(5) For I- or H-sections the shear stress in the web may be taken as:

V ≥

τ = Ed A / A 0

,

6

if (6.21)

f w

Ed A w

is the area of one flange;

where A f is the area of the web: A = h t .

## A

w w w w

(6) In addition the shear buckling resistance for webs without intermediate stiffeners should be according

to section 5 of EN 1993-1-5, if

ε

h >

w 72 (6.22)

η

t w

η

For see section 5 of EN 1993-1-5.

η

NOTE may be conservatively taken equal to 1,0.

(7) Fastener holes need not be allowed for in the shear verification except in verifying the design shear

resistance at connection zones as given in EN 1993-1-8.

(8) Where the shear force is combined with a torsional moment, the plastic shear resistance V should

pl,Rd

be reduced as specified in 6.2.7(9).

6.2.7 Torsion

(1) For members subject to torsion for which distortional deformations may be disregarded the design

value of the torsional moment T at each cross-section should satisfy:

Ed

T ≤

Ed 1

, 0 (6.23)

## T

Rd is the design torsional resistance of the cross section.

where T Rd

(2) The total torsional moment T at any cross- section should be considered as the sum of two internal

Ed

effects:

T = T + T (6.24)

Ed t,Ed w,Ed

is the internal St. Venant torsion;

where T t,Ed

T is the internal warping torsion.

w, Ed

(3) The values of T and T at any cross-section may be determined from T by elastic analysis,

t,Ed w,Ed Ed

taking account of the section properties of the member, the conditions of restraint at the supports and the

distribution of the actions along the member.

(4) The following stresses due to torsion should be taken into account:

τ

the shear stresses due to St. Venant torsion T

– t,Ed t,Ed

σ τ

the direct stresses due to the bimoment B and shear stresses due to warping torsion T

– w,Ed Ed w,Ed w,Ed

(5) For the elastic verification the yield criterion in 6.2.1(5) may be applied.

(6) For determining the plastic moment resistance of a cross section due to bending and torsion only

torsion effects B should be derived from elastic analysis, see (3).

Ed

(7) 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. Also as a

simplification, 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.

52 EN 1993-1-1: 2005 (E)

(8) For the calculation of the resistance T of closed hollow sections the design shear strength of the

Rd

individual parts of the cross section according to EN 1993-1-5 should be taken into account.

(9) For combined shear force and torsional moment the plastic shear resistance accounting for torsional

effects should be reduced from V to V and the design shear force should satisfy:

pl,Rd pl,T,Rd

V ≤

Ed 1

,

0 (6.25)

## V

pl , T , Rd may be derived as follows:

in which V

pl,T,Rd

for an I or H section:

– τ t,Ed

= − ( )

V 1 V (6.26)

pl , T , Rd pl , Rd

1

, 25 f / 3 /γ

y M 0

for a channel section:

–  

τ

τ

 

t,Ed w , Ed

= − −

( ) ( )

V 1 V (6.27)

pl , T , Rd pl , Rd

 

1

, 25 f / 3 /γ f / 3 /γ

 

y M 0 y M 0

for a structural hollow section:

– 

 τ t,Ed

= − 

 ( )

V 1 V (6.28)

pl , T , Rd pl , Rd

f / 3 /γ 

 

 y M 0

is given in 6.2.6.

where V pl,Rd

6.2.8 Bending and shear

(1) Where the shear force is present allowance should be made for its effect on the moment resistance.

(2) Where the shear force is less than half the plastic shear resistance its effect on the moment resistance

may be neglected except where shear buckling reduces the section resistance, see EN 1993-1-5.

(3) Otherwise the reduced moment resistance should be taken as the design resistance of the cross-section,

calculated using a reduced yield strength

ρ)

(1 – f (6.29)

y

for the shear area, 2

 

2 V

 

ρ = −

Ed 1

where and V is obtained from 6.2.6(2).

  pl,Rd

## V

 

pl , Rd

 

2 V

 

ρ = −

Ed

ρ 1

(4) When torsion is present should be obtained from , see 6.2.7, but should be taken

 

## V

 

pl , T , Rd

as 0 for V 0,5V .

Ed pl,T,Rd 53

EN 1993-1-1: 2005 (E)

(5) The reduced design plastic resistance moment allowing for the shear force may alternatively be

obtained for I-cross-sections with equal flanges and bending about the major axis as follows:

 

2

ρ A

− w

W f

 

pl , y y

4 t

 

 

w

= ≤

## M M M

but (6.30)

y , V , Rd y , V , Rd y , c , Rd

γ M 0

where M is obtained from 6.2.5(2)

y,c,Rd

= h t

and A

w w w

(6) For the interaction of bending, shear and transverse loads see section 7 of EN 1993-1-5.

6.2.9 Bending and axial force

6.2.9.1 Class 1 and 2 cross-sections

(1) Where an axial force is present, allowance should be made for its effect on the plastic moment

resistance.

(2) For class 1 and 2 cross sections, the following criterion should be satisfied:

M M (6.31)

Ed N,Rd

is the design plastic moment resistance reduced due to the axial force N .

where M N,Rd Ed

(3) For a rectangular solid section without fastener holes M should be taken as:

N,Rd

[ ]

( )

2

= −

M M 1 N / N (6.32)

N , Rd pl , Rd Ed pl , Rd

(4) For doubly symmetrical I- and H-sections or other flanges sections, allowance need not be made for

the effect of the axial force on the plastic resistance moment about the y-y axis when both the following

criteria are satisfied:

N 0,25 N and (6.33)

Ed pl , Rd

0,5 h t f

w w y

N (6.34)

Ed γ M 0

For doubly symmetrical I- and H-sections, allowance need not be made for the effect of the axial force on the

plastic resistance moment about the z-z axis when:

h t f

w w y

N (6.35)

Ed γ M 0

(5) For cross-sections where fastener holes are not to be accounted for, the following approximations may

be used for standard rolled I or H sections and for welded I or H sections with equal flanges:

M = M (1-n)/(1-0,5a) but M M (6.36)

N,y,Rd pl,y,Rd N,y,Rd pl,y,Rd

≤ = M (6.37)

for n a: M

N,z,Rd pl,z,Rd 

 2

 

n a

−  

1 

for n > a: M = M (6.38)

N,z,Rd pl,z,Rd −

1 a

  

 

/ N

where n = N

Ed pl.Rd ≤

)/A but a 0,5

a = (A-2bt f

54 EN 1993-1-1: 2005 (E)

For cross-sections where fastener holes are not to be accounted for, the following approximations may be

used for rectangular structural hollow sections of uniform thickness and for welded box sections with equal

flanges and equal webs: ≤

M = M (1 - n)/(1 - 0,5a ) but M M (6.39)

N,y,Rd pl,y,Rd w N,y,Rd pl,y.Rd

= M (1 - n)/(1 - 0,5a ) but M M (6.40)

M N,z,Rd pl,z,Rd f N,z,Rd pl,z,Rd

where a = (A - 2bt)/A but a 0,5 for hollow sections

w w ≤

= (A-2bt )/A but a 0,5 for welded box sections

a

w f w ≤

a = (A - 2ht)/A but a 0,5 for hollow sections

f f ≤

= (A-2ht )/A but a 0,5 for welded box sections

a

f w f

(6) For bi-axial bending the following criterion may be used:

α β

  

y , Ed z , Ed ≤

+ 1

   (6.41)

## M M

 

  

 N , y , Rd N , z , Rd

α β

in which and are constants, which may conservatively be taken as unity, otherwise as follows:

I and H sections:

– α = β = β ≥

2 ; 5 n but 1

circular hollow sections:

– α = β =

2 ; 2

rectangular hollow sections:

– 1

,

66

α = β = α = β ≤

but 6

− 2

1 1

,

13 n

where n = N / N .

Ed pl,Rd

6.2.9.2 Class 3 cross-sections

(1) In the absence of shear force, for Class 3 cross-sections the maximum longitudinal stress should

satisfy the criterion:

f y

σ ≤ (6.42)

x , Ed γ M 0

σ is the design value of the local longitudinal stress due to moment and axial force taking account

where x , Ed

of fastener holes where relevant, see 6.2.3, 6.2.4 and 6.2.5

6.2.9.3 Class 4 cross-sections σ

(1) In the absence of shear force, for Class 4 cross-sections the maximum longitudinal stress x,Ed

calculated using the effective cross sections (see 5.5.2(2)) should satisfy the criterion:

f y

σ ≤ (6.43)

x , Ed γ M 0

σ

where is the design value of the local longitudinal stress due to moment and axial force taking account

x , Ed

of fastener holes where relevant, see 6.2.3, 6.2.4 and 6.2.5 55

EN 1993-1-1: 2005 (E)

(2) The following criterion should be met:

+ +

M N e M N e

N y , Ed Ed Ny z , Ed Ed Nz

+ ≤

+

Ed 1 (6.44)

γ γ

γ

A f / W f / W f /

eff y M 0 eff , y , min y M 0 eff , z , min y M 0

is the effective area of the cross-section when subjected to uniform compression

where A

eff

W is the effective section modulus (corresponding to the fibre with the maximum elastic stress)

eff,min of the cross-section when subjected only to moment about the relevant axis

e is the shift of the relevant centroidal axis when the cross-section is subjected to compression

N only, see 6.2.2.5(4) ∆M

NOTE The signs of N , M , M and = N e depend on the combination of the respective

Ed y,Ed z,Ed i Ed Ni

direct stresses.

6.2.10 Bending, shear and axial force

(1) Where shear and axial force are present, allowance should be made for the effect of both shear force

and axial force on the resistance moment.

(2) Provided that the design value of the shear force V does not exceed 50% of the design plastic shear

Ed

resistance V no reduction of the resistances defined for bending and axial force in 6.2.9 need be made,

pl.Rd

except where shear buckling reduces the section resistance, see EN 1993-1-5.

(3) Where V exceeds 50% of V the design resistance of the cross-section to combinations of moment

Ed pl.Rd

and axial force should be calculated using a reduced yield strength

(1-ρ)f (6.45)

y

for the shear area 2

ρ= / V -1) and V is obtained from 6.2.6(2).

where (2V Ed pl.Rd pl,Rd

NOTE Instead of reducing the yield strength also the plate thickness of the relevant part of the cross

section may be reduced.

6.3 Buckling resistance of members

6.3.1 Uniform members in compression

6.3.1.1 Buckling resistance

(1) A compression member should be verified against buckling as follows:

N ≤

Ed 1

,

0 (6.46)

N b , Rd

where N is the design value of the compression force;

Ed

N is the design buckling resistance of the compression member.

b,Rd

(2) For members with non-symmetric Class 4 sections allowance should be made for the additional

## ∆M

moment due to the eccentricity of the centroidal axis of the effective section, see also 6.2.2.5(4), and

Ed

the interaction should be carried out to 6.3.4 or 6.3.3.

56 EN 1993-1-1: 2005 (E)

(3) The design buckling resistance of a compression member should be taken as:

χ A f y

=

N for Class 1, 2 and 3 cross-sections (6.47)

b , Rd γ M 1

χ A f

eff y

=

N for Class 4 cross-sections (6.48)

b , Rd γ M

1

χ

where is the reduction factor for the relevant buckling mode.

NOTE For determining the buckling resistance of members with tapered sections along the member

or for non-uniform distribution of the compression force second order analysis according to 5.3.4(2)

(4) In determining A and A holes for fasteners at the column ends need not to be taken into account.

eff

6.3.1.2 Buckling curves χ

(1) For axial compression in members the value of for the appropriate non-dimensional slendernessλ

should be determined from the relevant buckling curve according to:

1 χ ≤

χ = 1

,

0

but (6.49)

2

Φ + Φ − λ

2

[ ]

( ) 2

Φ = + α λ − + λ

0

,

5 1 0

, 2

where Af y

λ = for Class 1, 2 and 3 cross-sections

N cr

A f

eff y

λ = for Class 4 cross-sections

N cr

α is an imperfection factor

is the elastic critical force for the relevant buckling mode based on the gross cross sectional

## N

cr properties. α

(2) The imperfection factor corresponding to the appropriate buckling curve should be obtained from

Table 6.1 and Table 6.2.

Table 6.1: Imperfection factors for buckling curves

a b c d

Buckling curve a

0

α 0,13 0,21 0,34 0,49 0,76

Imperfection factor

χ

(3) Values of the reduction factor for the appropriate non-dimensional slendernessλ may be obtained

from Figure 6.4. N

λ ≤ ≤

Ed

0

, 2 0

,

04

(4) For slenderness or for the buckling effects may be ignored and only cross

N cr

sectional checks apply. 57

EN 1993-1-1: 2005 (E)

Table 6.2: Selection of buckling curve for a cross-section Buckling curve

Buckling S 235

Cross section Limits about S 275 S 460

axis S 355

S 420

z

t f y – y a a

0

t 40 mm

1,2 f z – z b a

0

>

h/b y – y b a

sections ≤

40 mm < t 100

f z – z c a

y

h y y – y b a

Rolled ≤

t 100 mm

f z – z c a

1,2

h/b y – y d c

z t > 100 mm

f z – z d c

b y – y b b

t

t ≤

t 40 mm

f

f f z – z c c

I-sections

Welded y y

y y y – y c c

t > 40 mm

f z – z d d

z z hot finished any a a

0

sections

Hollow cold formed any c c

t

z f generally (except as any b b

box below)

sections y y

Welded h thick welds: a > 0,5t

t f

w any c c

b/t < 30

f

z h/t <30

b w

sections

and any c c

T- solid

## U-,

L-sections any b b

58 EN 1993-1-1: 2005 (E)

1,1

1,0 a 0

0,9 a

b

0,8 c

0,7

χ d

factor 0,6

Reduction 0,5

0,4

0,3

0,2

0,1

0,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0

λ

Non-dimensional slenderness

Figure 6.4: Buckling curves

6.3.1.3 Slenderness for flexural buckling

λ

(1) The non-dimensional slenderness is given by:

Af L 1

y

λ = = cr for Class 1, 2 and 3 cross-sections (6.50)

λ

N i

cr 1 A eff

A f L A

eff y

λ = = cr for Class 4 cross-sections (6.51)

λ

N i

cr 1

is the buckling length in the buckling plane considered

where L cr

i is the radius of gyration about the relevant axis, determined using the properties of the gross

cross-section

## E

λ = π = ε

93

,

9

1 f y

235

ε = 2

(f in N/mm )

y

f y

NOTE B For elastic buckling of components of building structures see Annex BB.

(2) For flexural buckling the appropriate buckling curve should be determined from Table 6.2. 59

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### DESCRIZIONE APPUNTO

Eurocode 3 applies to the design of buildings and civil engineering works in steel. It complies with the principles and requirements for the safety and serviceability of structures, the basis of their design and verification that are given in EN 1990 – Basis of structural design. Eurocode 3 is concerned only with requirements for resistance, serviceability, durability and fire resistance of steel structures. Other requirements, e.g. concerning thermal or sound insulation, are not covered.

DETTAGLI
Corso di laurea: Corso di laurea in ingegneria civile
SSD:
A.A.: 2011-2012

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Atreyu di informazioni apprese con la frequenza delle lezioni di Tecnica delle costruzioni e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università Mediterranea - Unirc o del prof D'assisi Ricciardelli Francesco.

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