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EN 1992-1-1:2004 (E)

⎡ ⎤

n

ε

⎛ ⎞

⎜ ⎟ ε ε

⎢ ⎥

= − − ≤ ≤

c

σ f (3.17)

1 1 for 0

⎜ ⎟

ε

c cd c c2

⎢ ⎥

⎝ ⎠

⎣ ⎦

c2

ε ε ε

= ≤ ≤

σ f for (3.18)

c cd c2 c cu2

where:

n is the exponent according to Table 3.1

ε is the strain at reaching the maximum strength according to Table 3.1

c2

ε is the ultimate strain according to Table 3.1

cu2

σ

c

f

ck

f cd ε ε ε

0 c2 c

cu2

Figure 3.3: Parabola-rectangle diagram for concrete under compression.

(2) Other simplified stress-strain relationships may be used if equivalent to or more

conservative than the one defined in (1), for instance bi-linear according to Figure 3.4

ε ε

(compressive stress and shortening strain shown as absolute values) with values of and

c3 cu3

according to Table 3.1.

σ c

f ck

f cd ε

ε

ε

0 c

c3 cu3

Figure 3.4: Bi-linear stress-strain relation. λ

(3) A rectangular stress distribution (as given in Figure 3.5) may be assumed. The factor , 35

EN 1992-1-1:2004 (E) η

defining the effective height of the compression zone and the factor , defining the effective

strength, follow from:

λ ≤

= 0,8 for f 50 MPa (3.19)

ck

λ ≤

= 0,8 - (f -50)/400 for 50 < f 90 MPa (3.20)

ck ck

and

η ≤

= 1,0 for f 50 MPa (3.21)

ck

η ≤

= 1,0 - (f -50)/200 for 50 < f 90 MPa (3.22)

ck ck

If the width of the compression zone decreases in the direction of the extreme compression fibre, the

Note: η

value f should be reduced by 10%.

cd ε η f

cu3 cd F c

A λ x

x

c d

A F

s s

ε s

Figure 3.5: Rectangular stress distribution

3.1.8 Flexural tensile strength

(1) The mean flexural tensile strength of reinforced concrete members depends on the mean

axial tensile strength and the depth of the cross-section. The following relationship may be

used:

f = max {(1,6 - h/1000)f ; f } (3.23)

ctm,fl ctm ctm

where:

h is the total member depth in mm

f is the mean axial tensile strength following from Table 3.1.

ctm

The relation given in Expression (3.23) also applies for the characteristic tensile strength

values.

3.1.9 Confined concrete

(1) Confinement of concrete results in a modification of the effective stress-strain relationship:

higher strength and higher critical strains are achieved. The other basic material characteristics

may be considered as unaffected for design.

(2) In the absence of more precise data, the stress-strain relation shown in Figure 3.6

(compressive strain shown positive) may be used, with increased characteristic strength and

strains according to: σ σ ≤

= f (1,000 + 5,0 /f ) for 0,05f (3.24)

f ck,c ck 2 ck 2 ck

36 EN 1992-1-1:2004 (E)

σ σ

f = f (1,125 + 2,50 /f ) for > 0,05f (3.25)

ck,c ck 2 ck 2 ck

2

ε ε

= (f /f ) (3.26)

c2,c c2 ck,c ck

ε ε σ

= + 0,2 /f (3.27)

cu2,c cu2 2 ck

σ σ

where (= ) is the effective lateral compressive stress at the ULS due to confinement

2 3

ε ε

and follow from Table 3.1. Confinement can be generated by adequately closed

and c2 cu2

links or cross-ties, which reach the plastic condition due to lateral extension of the concrete.

σ

σ f

= c

1 ck,c f

ck,c

f

ck f

cd,c A - unconfined

A

σ σ σ

( = )

2 3 2 ε ε ε

ε

cu cu2,c c

0 c2,c

Figure 3.6: Stress-strain relationship for confined concrete

3.2 Reinforcing steel

3.2.1 General

(1)P The following clauses give principles and rules for reinforcement which is in the form of

bars, de-coiled rods, welded fabric and lattice girders. They do not apply to specially coated

bars.

(2)P The requirements for the properties of the reinforcement are for the material as placed in

the hardened concrete If site operations can affect the properties of the reinforcement, then

.

those properties shall be verified after such operations.

(3)P Where other steels are used, which are not in accordance with EN10080, the properties

shall be verified to be in accordance with 3.2.2 to 3.2.6 and Annex C.

(4)P The required properties of reinforcing steels shall be verified using the testing procedures

in accordance with EN 10080.

EN 10080 refers to a yield strength R , which relates to the characteristic, minimum and maximum

Note: e

values based on the long-term quality level of production. In contrast f is the characteristic yield stress based

yk

on only that reinforcement used in a particular structure. There is no direct relationship between f and the

yk

characteristic R . However the methods of evaluation and verification of yield strength given in EN 10080

e

provide a sufficient check for obtaining f .

yk

(5) The application rules relating to lattice girders (see EN 10080 for definition) apply only to

those made with ribbed bars. Lattice girders made with other types of reinforcement may be

given in an appropriate European Technical Approval.

3.2.2 Properties

(1)P The behaviour of reinforcing steel is specified by the following properties: 37

EN 1992-1-1:2004 (E)

- yield strength (f or f )

yk 0,2k

- maximum actual yield strength (f )

y,max

- tensile strength (f )

t

- ductility (ε and f /f )

uk t yk

- bendability

- bond characteristics (f See Annex C)

R:

- section sizes and tolerances

- fatigue strength

- weldability

- shear and weld strength for welded fabric and lattice girders

(2)P This Eurocode applies to ribbed and weldable reinforcement, including fabric. The

permitted welding methods are given in Table 3.4.

The properties of reinforcement required for use with this Eurocode are given in Annex C.

Note 1: The properties and rules for the use of indented bars with precast concrete products may be found in

Note 2:

the relevant product standard.

(3)P The application rules for design and detailing in this Eurocode are valid for a specified

yield strength range, f = 400 to 600 MPa.

yk

The upper limit of f within this range for use within a Country may be found in its National Annex.

Note: yk

(4)P The surface characteristics of ribbed bars shall be such to ensure adequate bond with the

concrete.

(5) Adequate bond may be assumed by compliance with the specification of projected rib area,

f .

R Minimum values of the relative rib area, f , are given in the Annex C.

Note: R

(6)P The reinforcement shall have adequate bendability to allow the use of the minimum

mandrel diameters specified in Table 8.1 and to allow rebending to be carried out.

For bend and rebend requirements see Annex C.

Note:

3.2.3 Strength

(1)P The yield strength f (or the 0,2% proof stress, f ) and the tensile strength f are defined

yk 0,2k tk

respectively as the characteristic value of the yield load, and the characteristic maximum load in

direct axial tension, each divided by the nominal cross sectional area.

3.2.4 Ductility characteristics

(1)P The reinforcement shall have adequate ductility as defined by the ratio of tensile strength

ε

/f ) and the elongation at maximum force, .

to the yield stress, (f t y k uk

(2) Figure 3.7 shows stress-strain curves for typical hot rolled and cold worked steel.

ε

Values of (f /f ) and for Class A, B and C are given in Annex C.

Note: t y k uk

38 EN 1992-1-1:2004 (E)

σ σ

f = kf

f = kf t 0,2k

t ykt f 0,2k

f yk ε ε

ε 0,2% ε

uk uk

a) Hot rolled steel b) Cold worked steel

Figure 3.7: Stress-strain diagrams of typical reinforcing steel (absolute values are

shown for tensile stress and strain)

3.2.5 Welding

(1)P Welding processes for reinforcing bars shall be in accordance with Table 3.4 and the

weldability shall be in accordance with EN10080.

Table 3.4: Permitted welding processes and examples of application

1 1

Loading case Welding method Bars in tension Bars in compression

flash-welding butt joint

φ ≥

manual metal arc welding butt joint with 20 mm, splice, lap, cruciform

and 3

, joint with other steel members

joints

Predominantly metal arc welding with filling

electrode

static 2 3

metal arc active welding splice, lap, cruciform joints & joint with other

(see 6.8.1 (2)) steel members φ ≥

- butt joint with 20 mm

friction welding butt joint, joint with other steels

4

resistance spot welding lap joint 2, 4

cruciform joint

flash-welding butt joint

Not predominantly φ ≥

manual metal arc welding - butt joint with 14mm

static (see 6.8.1 (2)) 2 φ ≥

metal arc active welding - butt joint with 14mm

4

resistance spot welding lap joint 2, 4

cruciform joint

Notes:

1. Only bars with approximately the same nominal diameter may be welded together.

2. Permitted ratio of mixed diameter bars 0,57

φ ≤

3. For bearing joints 16 mm

φ ≤

4. For bearing joints 28 mm

(2)P All welding of reinforcing bars shall be carried out in accordance with EN ISO 17760. 39

EN 1992-1-1:2004 (E)

(3)P The strength of the welded joints along the anchorage length of welded fabric shall be

sufficient to resist the design forces.

(4) The strength of the welded joints of welded fabric may be assumed to be adequate if each

welded joint can withstand a shearing force not less than 25% of a force equivalent to the

specified characteristic yield stress times the nominal cross sectional area. This force should

be based on the area of the thicker wire if the two are different.

3.2.6 Fatigue

(1)P Where fatigue strength is required it shall be verified in accordance with EN 10080.

Information is given in Annex C.

Note :

3.2.7 Design assumptions

(1) Design should be based on the nominal cross-section area of the reinforcement and the

design values derived from the characteristic values given in 3.2.2.

(2) For normal design, either of the following assumptions may be made (see Figure 3.8):

ε γ ε

a) an inclined top branch with a strain limit of and a maximum stress of kf / at ,

ud yk s uk

where k = (f /f ) ,

t y k

b) a horizontal top branch without the need to check the strain limit.

ε

The value of for use in a Country may be found in its National Annex. The recommended value is

Note 1: ud

ε

0,9 uk (f /f )

The value of is given in Annex C.

Note 2: t y k A

σ kf

kf yk

yk γ

kf /

yk s

f

yk

γ

f f /

yd = yk s k = (f /f )

t y k

B A Idealised

B Design

ε

ε

ε

f / E uk

ud

yd s

Figure 3.8: Idealised and design stress-strain diagrams for reinforcing steel (for

tension and compression) 3

(3) The mean value of density may be assumed to be 7850 kg/m .

40 EN 1992-1-1:2004 (E)

(4) The design value of the modulus of elasticity, E may be assumed to be 200 GPa.

s

3.3 Prestressing steel

3.3.1 General

(1)P This clause applies to wires, bars and strands used as prestressing tendons in concrete

structures.

(2)P Prestressing tendons shall have an acceptably low level of susceptibility to stress

corrosion.

(3) The level of susceptibility to stress corrosion may be assumed to be acceptably low if the

prestressing tendons comply with the criteria specified in EN 10138 or given in an appropriate

European Technical Approval.

(4) The requirements for the properties of the prestressing tendons are for the materials as

placed in their final position in the structure. Where the methods of production, testing and

attestation of conformity for prestressing tendons are in accordance with EN 10138 or given in

an appropriate European Technical Approval it may be assumed that the requirements of this

Eurocode are met.

(5)P For steels complying with this Eurocode, tensile strength, 0,1% proof stress, and

elongation at maximum load are specified in terms of characteristic values; these values are

designated respectively f , f and ε .

pk p0,1k uk

EN 10138 refers to the characteristic, minimum and maximum values based on the long-term quality

Note:

level of production. In contrast f and f are the characteristic proof stress and tensile strength based on

p0,1k pk

only that prestressing steel required for the structure. There is no direct relationship between the two sets of

values. However the characteristic values for 0,1% proof force, F divided by the cross-section area, S

p0,1k n

given in EN 10138 together with the methods for evaluation and verification provide a sufficient check for

obtaining the value of f .

p0,1k

(6) Where other steels are used, which are not in accordance with EN 10138, the properties

may be given in an appropriate European Technical Approval.

(7)P Each product shall be clearly identifiable with respect to the classification system in 3.3.2

(2)P.

(8)P The prestressing tendons shall be classified for relaxation purposes according to

3.3.2 (4)P or given in an appropriate European Technical Approval.

(9)P Each consignment shall be accompanied by a certificate containing all the information

necessary for its identification with regard to (i) - (iv) in 3.3.2 (2)P and additional information

where necessary.

(10)P There shall be no welds in wires and bars. Individual wires of strands may contain

staggered welds made only before cold drawing.

(11)P For coiled prestressing tendons, after uncoiling a length of wire or strand the maximum

bow height shall comply with EN 10138 unless given in an appropriate European Technical

Approval. 41

EN 1992-1-1:2004 (E)

3.3.2 Properties

(1)P The properties of prestressing steel are given in EN 10138, Parts 2 to 4 or European

Technical Approval.

(2)P The prestressing tendons (wires, strands and bars) shall be classified according to:

(i) Strength, denoting the value of the 0,1% proof stress (f ) and the value of the ratio of

p0,1k ε

/f ) and elongation at maximum load ( )

tensile strength to proof strength (f pk p0,1k uk

(ii) Class, indicating the relaxation behaviour

(iii) Size

(iv) Surface characteristics.

(3)P The actual mass of the prestressing tendons shall not differ from the nominal mass by

more than the limits specified in EN 10138 or given in an appropriate European Technical

Approval.

(4)P In this Eurocode, three classes of relaxation are defined:

- Class 1: wire or strand - ordinary relaxation

- Class 2: wire or strand - low relaxation

Class 3: hot rolled and processed bars

- Class 1 is not covered by EN 10138.

Note:

(5) The design calculations for the losses due to relaxation of the prestressing steel should be

ρ , the relaxation loss (in %) at 1000 hours after tensioning and at a

based on the value of 1000

mean temperature of 20 °C (see EN 10138 for the definition of the isothermal relaxation test).

ρ

The value of is expressed as a percentage ratio of the initial stress and is obtained for an initial

Note: 1000

stress equal to 0,7f , where f is the actual tensile strength of the prestressing steel samples. For design

p p

calculations, the characteristic tensile strength (f is used and this has been taken into account in the following

)

pk

expressions. ρ

(6) The values for can be either assumed equal to 8% for Class 1, 2 5% for Class 2, and

,

1000

4% for Class 3, or taken from the certificate.

(7) The relaxation loss may be obtained from the manufacturers test certificates or defined as

the percentage ratio of the variation of the prestressing stress over the initial prestressing

stress, should be determined by applying one of the Expressions below. Expressions (3.28)

and (3.29) apply for wires or strands for ordinary prestressing and low relaxation tendons

respectively, whereas Expression (3.30) applies for hot rolled and processed bars.

− µ

0,75 ( 1 )

∆ σ ⎛ ⎞

t

pr −

= µ

6,7 5

5,39 e 10

Class 1 (3.28)

ρ ⎜ ⎟

1000 1000

σ ⎝ ⎠

pi − µ

0,75 ( 1 )

∆ σ ⎛ ⎞

t −

pr = µ

9,1 5

Class 2 (3.29)

0,66 ρ e 10

⎜ ⎟

1000

σ 1000

⎝ ⎠

pi − µ

0,75 ( 1 )

∆ σ ⎛ ⎞

t −

pr = µ

8 5

Class 3 (3.30)

1,98 ρ e 10

⎜ ⎟

1000

σ 1000

⎝ ⎠

pi

Where

σ

∆ is absolute value of the relaxation losses of the prestress

pr

σ σ σ σ

For post-tensioning is the absolute value of the initial prestress = (see

pi pi pi pm0

also 5.10.3 (2));

42 EN 1992-1-1:2004 (E)

σ

For pre-tensioning is the maximum tensile stress applied to the tendon minus

pi

the immediate losses occurred during the stressing process see 5.10.4 (1) (i)

t is the time after tensioning (in hours)

µ σ

= /f , where f is the characteristic value of the tensile strength of the

pi pk pk

prestressing steel

ρ is the value of relaxation loss (in %), at 1000 hours after tensioning and at a mean

1000 temperature of 20°C.

Where the relaxation losses are calculated for different time intervals (stages) and greater accuracy is

Note:

required, reference should be made to Annex D.

(8) The long term (final) values of the relaxation losses may be estimated for a time t equal to

500 000 hours (i.e. around 57 years).

(9) Relaxation losses are very sensitive to the temperature of the steel. Where heat treatment

is applied (e.g. by steam), 10.3.2.2 applies. Otherwise where this temperature is greater than

50°C the relaxation losses should be verified.

3.3.3 Strength

(1)P The 0,1% proof stress (f ) and the specified value of the tensile strength (f ) are

p0,1k pk

defined as the characteristic value of the 0,1% proof load and the characteristic maximum load

in axial tension respectively, divided by the nominal cross sectional area as shown in Figure

3.9. σ

f pk

f

p0,1k ε

0,1% ε uk

Figure 3.9: Stress-strain diagram for typical prestressing steel (absolute values

are shown for tensile stress and strain)

3.3.4 Ductility characteristics

(1)P The prestressing tendons shall have adequate ductility, as specified in EN 10138.

(2) Adequate ductility in elongation may be assumed if the prestressing tendons obtain the

specified value of the elongation at maximum load given in EN 10138.

(3) Adequate ductility in bending may be assumed if the prestressing tendons satisfy the 43

EN 1992-1-1:2004 (E)

requirements for bendability of EN ISO 15630.

(4) Stress-strain diagrams for the prestressing tendons, based on production data, shall be

prepared and made available by the producer as an annex to the certificate accompanying the

consignment (see 3.3.1 (9)P). ≥

(5) Adequate ductility in tension may be assumed for the prestressing tendons if f /f k.

pk p0,1k

The value of k for use in a Country may be found in its National Annex. The recommended value is 1,1.

Note:

3.3.5 Fatigue

(1)P Prestressing tendons shall have adequate fatigue strength.

(2)P The fatigue stress range for prestressing tendons shall be in accordance with EN 10138

or given in an appropriate European Technical Approval.

3.3.6 Design assumptions

(1)P Structural analysis is performed on the basis of the nominal cross-section area of the

prestressing steel and the characteristic values f , f and ε .

p0,1k pk uk

(2) The design value for the modulus of elasticity, E may be assumed equal to 205 GPa for

p

wires and bars. The actual value can range from 195 to 210 GPa, depending on the

manufacturing process. Certificates accompanying the consignment should give the

appropriate value.

(3) The design value for the modulus of elasticity, E may be assumed equal to 195 GPa for

p

strand. The actual value can range from 185 GPa to 205 GPa, depending on the manufacturing

process. Certificates accompanying the consignment should give the appropriate value.

(4) The mean density of prestressing tendons for the purposes of design may normally be

3

taken as 7850 kg/m

(5) The values given above may be assumed to be valid within a temperature range between

-40°C and +100°C for the prestressing steel in the finished structure.

γ

, is taken as f / (see Figure 3.10).

(6) The design value for the steel stress, f pd p0,1k S

(7) For cross-section design, either of the following assumptions may be made (see Figure

3.10): ε . The design may also be based on the actual

- an inclined branch, with a strain limit ud

stress/strain relationship, if this is known, with stress above the elastic limit reduced

analogously with Figure 3.10, or

- a horizontal top branch without strain limit.

ε

The value of for use in a Country may be found in its National Annex. The recommended value is

Note: ud

ε ε

0,9 . If more accurate values are not known the recommended values are = 0,02 and f /f = 0,9.

uk ud p0,1k pk

44 EN 1992-1-1:2004 (E)

A

σ

f pk γ

f /

pk s

f p 0,1k

γ

f f /

pd = p 0,1k s B A Idealised

B Design

ε

ε

ε

f / E uk

ud

pd p

Figure 3.10: Idealised and design stress-strain diagrams for prestressing steel

(absolute values are shown for tensile stress and strain)

3.3.7 Prestressing tendons in sheaths

(1)P Prestressing tendons in sheaths (e.g. bonded tendons in ducts, unbonded tendons etc.)

shall be adequately and permanently protected against corrosion (see 4.3).

(2)P Prestressing tendons in sheaths shall be adequately protected against the effects of fire

(see EN 1992-1-2).

3.4 Prestressing devices

3.4.1 Anchorages and couplers

3.4.1.1 General

(1)P 3.4.1 applies to anchoring devices (anchorages) and coupling devices (couplers) for

application in post-tensioned construction, where:

(i) anchorages are used to transmit the forces in tendons to the concrete in the anchorage

zone

(ii) couplers are used to connect individual lengths of tendon to make continuous tendons.

(2)P Anchorages and couplers for the prestressing system considered shall be in accordance

with the relevant European Technical Approval.

(3)P Detailing of anchorage zones shall be in accordance with 5.10, 8.10.3 and 8.10.4.

3.4.1.2 Mechanical properties

3.4.1.2.1 Anchored tendons

(1)P Prestressing tendon anchorage assemblies and prestressing tendon coupler assemblies

shall have strength, elongation and fatigue characteristics sufficient to meet the requirements of

the design. 45

EN 1992-1-1:2004 (E)

(2) This may be assumed provided that:

(i) The geometry and material characteristics of the anchorage and coupler components

are in accordance with the appropriate European Technical Approval and that their

premature failure is precluded.

(ii) Failure of the tendon is not induced by the connection to the anchorage or coupler.

(iii) The elongation at failure of the assemblies 2%.

(iv) Tendon-anchorage assemblies are not located in otherwise highly-stressed zones.

(v) Fatigue characteristics of the anchorage and coupler components are in accordance

with the appropriate European Technical Approval.

3.4.1.2.2 Anchorage devices and anchorage zones

(1)P The strength of the anchorage devices and zones shall be sufficient for the transfer of the

tendon force to the concrete and the formation of cracks in the anchorage zone shall not impair

the function of the anchorage.

3.4.2 External non-bonded tendons

3.4.2.1 General

(1)P An external non-bonded tendon is a tendon situated outside the original concrete section

and is connected to the structure by anchorages and deviators only.

(2)P The post-tensioning system for the use with external tendons shall be in accordance with

the appropriate European Technical Approval.

(3) Reinforcement detailing should follow the rules given in 8.10.

3.4.2.2 Anchorages

(1) The minimum radius of curvature of the tendon in the anchorage zone for non- bonded

tendons should be given in the appropriate European Technical Approval.

46 EN 1992-1-1:2004 (E)

SECTION 4 DURABILITY AND COVER TO REINFORCEMENT

4.1 General

(1)P A durable structure shall meet the requirements of serviceability, strength and stability

throughout its design working life, without significant loss of utility or excessive unforeseen

maintenance (for general requirements see also EN 1990).

(2)P The required protection of the structure shall be established by considering its intended

use, design working life (see EN 1990), maintenance programme and actions.

(3)P The possible significance of direct and indirect actions, environmental conditions (4.2) and

consequential effects shall be considered.

Note: Examples include deformations due to creep and shrinkage (see 2.3.2).

(4) Corrosion protection of steel reinforcement depends on density, quality and thickness of

concrete cover (see 4.4) and cracking (see 7.3). The cover density and quality is achieved by

controlling the maximum water/cement ratio and minimum cement content (see EN 206-1) and

may be related to a minimum strength class of concrete.

Note: Further information is given in Annex E.

(5) Where metal fastenings are inspectable and replaceable, they may be used with protective

coatings in exposed situations. Otherwise, they should be of corrosion resistant material.

(6) Further requirements to those given in this Section should be considered for special

situations (e.g. for structures of temporary or monumental nature, structures subjected to

extreme or unusual actions etc.).

4.2 Environmental conditions

(1)P Exposure conditions are chemical and physical conditions to which the structure is

exposed in addition to the mechanical actions.

(2) Environmental conditions are classified according to Table 4.1, based on EN 206-1.

(3) In addition to the conditions in Table 4.1, particular forms of aggressive or indirect action

should be considered including:

chemical attack, arising from e.g.

- the use of the building or the structure (storage of liquids, etc)

- solutions of acids or sulfate salts (EN 206-1, ISO 9690)

- chlorides contained in the concrete (EN 206-1)

- alkali-aggregate reactions (EN 206-1, National Standards)

physical attack, arising from e.g.

- temperature change

- abrasion (see 4.4.1.2 (13))

- water penetration (EN 206-1). 47

EN 1992-1-1:2004 (E)

Table 4.1: Exposure classes related to environmental conditions in accordance

with EN 206-1

Class Description of the environment Informative examples where exposure classes

designation may occur

1 No risk of corrosion or attack

For concrete without reinforcement or

X0 embedded metal: all exposures except where

there is freeze/thaw, abrasion or chemical

attack

For concrete with reinforcement or embedded

metal: very dry Concrete inside buildings with very low air humidity

2 Corrosion induced by carbonation

XC1 Dry or permanently wet Concrete inside buildings with low air humidity

Concrete permanently submerged in water

XC2 Wet, rarely dry Concrete surfaces subject to long-term water

contact

Many foundations

XC3 Moderate humidity Concrete inside buildings with moderate or high air

humidity

External concrete sheltered from rain

XC4 Cyclic wet and dry Concrete surfaces subject to water contact, not

within exposure class XC2

3 Corrosion induced by chlorides

XD1 Moderate humidity Concrete surfaces exposed to airborne chlorides

XD2 Wet, rarely dry Swimming pools

Concrete components exposed to industrial waters

containing chlorides

XD3 Cyclic wet and dry Parts of bridges exposed to spray containing

chlorides

Pavements

Car park slabs

4 Corrosion induced by chlorides from sea water

XS1 Exposed to airborne salt but not in direct Structures near to or on the coast

contact with sea water

XS2 Permanently submerged Parts of marine structures

XS3 Tidal, splash and spray zones Parts of marine structures

5. Freeze/Thaw Attack

XF1 Moderate water saturation, without de-icing Vertical concrete surfaces exposed to rain and

agent freezing

XF2 Moderate water saturation, with de-icing agent Vertical concrete surfaces of road structures

exposed to freezing and airborne de-icing agents

XF3 High water saturation, without de-icing agents Horizontal concrete surfaces exposed to rain and

freezing

XF4 High water saturation with de-icing agents or Road and bridge decks exposed to de-icing agents

sea water Concrete surfaces exposed to direct spray

containing de-icing agents and freezing

Splash zone of marine structures exposed to

freezing

6. Chemical attack

XA1 Slightly aggressive chemical environment Natural soils and ground water

according to EN 206-1, Table 2

XA2 Moderately aggressive chemical environment Natural soils and ground water

according to EN 206-1, Table 2

XA3 Highly aggressive chemical environment Natural soils and ground water

according to EN 206-1, Table 2

48 EN 1992-1-1:2004 (E)

Note: The composition of the concrete affects both the protection of the reinforcement and the resistance of the

concrete to attack. Annex E gives indicative strength classes for the particular environmental exposure classes.

This may lead to the choice of higher strength classes than required for the structural design. In such cases

f

the value of should be associated with the higher strength in the calculation of minimum reinforcement and

ctm

crack width control (see 7.3.2 -7.3.4).

4.3 Requirements for durability

(1)P In order to achieve the required design working life of the structure, adequate measures

shall be taken to protect each structural element against the relevant environmental actions.

(2)P The requirements for durability shall be included when considering the following:

Structural conception,

Material selection,

Construction details,

Execution,

Quality Control,

Inspection,

Verifications,

Special measures (e.g. use of stainless steel, coatings, cathodic protection).

4.4 Methods of verification

4.4.1 Concrete cover

4.4.1.1 General

(1)P The concrete cover is the distance between the surface of the reinforcement closest to the

nearest concrete surface (including links and stirrups and surface reinforcement where

relevant) and the nearest concrete surface.

(2)P The nominal cover shall be specified on the drawings. It is defined as a minimum cover,

∆c

c (see 4.4.1.2), plus an allowance in design for deviation, (see 4.4.1.3):

min dev

∆c

c c

= + (4.1)

nom min dev

4.4.1.2 Minimum cover, c

min c

(1)P Minimum concrete cover, , shall be provided in order to ensure:

min

- the safe transmission of bond forces (see also Sections 7 and 8)

- the protection of the steel against corrosion (durability)

- an adequate fire resistance (see EN 1992-1-2)

c

(2)P The greater value for satisfying the requirements for both bond and environmental

min

conditions shall be used.

c c ∆c ∆c ∆c

= max {c ; + - - ; 10 mm} (4.2)

min min,b min,dur dur,γ dur,st dur,add

where:

c minimum cover due to bond requirement, see 4.4.1.2 (3)

min,b

c minimum cover due to environmental conditions, see 4.4.1.2 (5)

min,dur

∆c additive safety element, see 4.4.1.2 (6)

dur,γ

∆c reduction of minimum cover for use of stainless steel, see 4.4.1.2 (7)

dur,st

∆c reduction of minimum cover for use of additional protection, see 4.4.1.2 (8)

dur,add 49

EN 1992-1-1:2004 (E)

(3) In order to transmit bond forces safely and to ensure adequate compaction of the concrete,

c given in table 4.2.

the minimum cover should not be less than min,b

Table 4.2: Minimum cover, c , requirements with regard to bond

min,b

Bond Requirement c

Arrangement of bars Minimum cover *

min,b

Separated Diameter of bar φ

Bundled Equivalent diameter ( )(see 8.9.1)

n c

*: If the nominal maximum aggregate size is greater than 32 mm, should be increased by 5 mm.

min,b

c

Note: The values of for post-tensioned circular and rectangular ducts for bonded tendons, and pre-

min,b

tensioned tendons for use in a Country may be found in its National Annex. The recommended values for post-

tensioned ducts are:

circular ducts: diameter

rectangular ducts: greater of the smaller dimension or half the greater dimension

There is no requirement for more than 80 mm for either circular or rectangular ducts.

The recommended values for pre-tensioned tendon:

1,5 x diameter of strand or plain wire

2,5 x diameter of indented wire.

(4) For prestressing tendons, the minimum cover of the anchorage should be provided in

accordance with the appropriate European Technical Approval.

(5) The minimum cover values for reinforcement and prestressing tendons in normal weight

c

concrete taking account of the exposure classes and the structural classes is given by .

min,dur

Note: Structural classification and values of c for use in a Country may be found in its National Annex.

min,dur

The recommended Structural Class (design working life of 50 years) is S4 for the indicative concrete strengths

given in Annex E and the recommended modifications to the structural class is given in Table 4.3N. The

recommended minimum Structural Class is S1.

The recommended values of c are given in Table 4.4N (reinforcing steel) and Table 4.5N (prestressing

min,dur

steel).

Table 4.3N: Recommended structural classification

Structural Class Exposure Class according to Table 4.1

Criterion / XS2 / XS3

X0 XC1 XC2 / XC3 XC4 XD1 XD2 / XS1 XD3

Design Working Life of increase increase increase increase increase increase increase class

100 years class by 2 class by 2 class by 2 class by 2 class by 2 class by 2 by 2

≥ ≥ ≥ ≥ ≥ ≥ ≥

1) 2)

Strength Class C30/37 C30/37 C35/45 C40/50 C40/50 C40/50 C45/55

reduce reduce reduce reduce reduce reduce reduce class by

class by 1 class by 1 class by 1 class by 1 class by 1 class by 1 1

Member with slab reduce reduce reduce reduce reduce reduce reduce class by

geometry class by 1 class by 1 class by 1 class by 1 class by 1 class by 1 1

(position of reinforcement

not affected by construction

process)

Special Quality reduce reduce reduce reduce reduce reduce reduce class by

Control of the concrete class by 1 class by 1 class by 1 class by 1 class by 1 class by 1 1

production ensured

Notes to Table 4.3N

1. The strength class and w/c ratio are considered to be related values. A special composition (type of

cement, w/c value, fine fillers) with the intent to produce low permeability may be considered.

2. The limit may be reduced by one strength class if air entrainment of more than 4% is applied.

50 EN 1992-1-1:2004 (E)

Table 4.4N: Values of minimum cover, c , requirements with regard to durability for

min,dur

reinforcement steel in accordance with EN 10080.

Environmental Requirement for c (mm)

min,dur

Structural Exposure Class according to Table 4.1

Class X0 XC1 XC2 / XC3 XC4 XD1 / XS1 XD2 / XS2 XD3 / XS3

S1 10 10 10 15 20 25 30

S2 10 10 15 20 25 30 35

S3 10 10 20 25 30 35 40

S4 10 15 25 30 35 40 45

S5 15 20 30 35 40 45 50

S6 20 25 35 40 45 50 55

Table 4.5N: Values of minimum cover, c , requirements with regard to durability for

min,dur

prestressing steel

Environmental Requirement for c (mm)

min,dur

Structural Exposure Class according to Table 4.1

Class X0 XC1 XC2 / XC3 XC4 XD1 / XS1 XD2 / XS2 XD3 / XS3

S1 10 15 20 25 30 35 40

S2 10 15 25 30 35 40 45

S3 10 20 30 35 40 45 50

S4 10 25 35 40 45 50 55

S5 15 30 40 45 50 55 60

S6 20 35 45 50 55 60 65

∆c

(6) The concrete cover should be increased by the additive safety element .

γ

dur,

∆c

Note: The value of for use in a Country may be found in its National Annex. The recommended value is

γ

dur,

0 mm.

(7) Where stainless steel is used or where other special measures have been taken, the

∆c

minimum cover may be reduced by . For such situations the effects on all relevant

dur,st

material properties should be considered, including bond.

∆c

Note: The value of for use in a Country may be found in its National Annex. The recommended value,

dur,st

without further specification, is 0 mm.

(8) For concrete with additional protection (e.g. coating) the minimum cover may be reduced by

∆c .

dur,add ∆c

Note: The value of for use in a Country may be found in its National Annex. The recommended value,

dur,add

without further specification, is 0 mm.

(9) Where in-situ concrete is placed against other concrete elements (precast or in-situ) the

minimum concrete cover of the reinforcement to the interface may be reduced to a value

corresponding to the requirement for bond (see (3) above) provided that:

- the strength class of concrete is at least C25/30,

- the exposure time of the concrete surface to an outdoor environment is short (< 28 days),

- the interface has been roughened.

(10) For unbonded tendons the cover should be provided in accordance with the European

Technical Approval.

(11) For uneven surfaces (e.g. exposed aggregate) the minimum cover should be increased by

at least 5 mm. 51

EN 1992-1-1:2004 (E)

(12) Where freeze/thaw or chemical attack on concrete (Classes XF and XA) is expected

special attention should be given to the concrete composition (see EN 206-1 Section 6). Cover

in accordance with 4.4 will normally be sufficient for such situations.

(13) For concrete abrasion special attention should be given on the aggregate according to EN

206-1. Optionally concrete abrasion may be allowed for by increasing the concrete cover

c k

(sacrificial layer). In that case the minimum cover should be increased by for Abrasion

min 1

k k

Class XM1, by for XM2 and by for XM3.

2 3

Note: Abrasion Class XM1 means a moderate abrasion like for members of industrial sites frequented by

vehicles with air tyres. Abrasion Class XM2 means a heavy abrasion like for members of industrial sites

frequented by fork lifts with air or solid rubber tyres. Abrasion Class XM3 means an extreme abrasion like for

members industrial sites frequented by fork lifts with elastomer or steel tyres or track vehicles.

k k

k , and for use in a Country may be found in its National Annex. The recommended values

The values of 1 2 3

are 5 mm, 10 mm and 15 mm.

4.4.1.3 Allowance in design for deviation

c

(1)P To calculate the nominal cover, , an addition to the minimum cover shall be made in

nom

). The required minimum cover shall be increased by the

design to allow for the deviation (∆c dev

absolute value of the accepted negative deviation.

∆c

Note: The value of for use in a Country may be found in its National Annex. The recommended value is

dev

10 mm.

(2) For Buildings, ENV 13670-1 gives the acceptable deviation. This is normally also sufficient

for other types of structures. It should be considered when choosing the value of nominal cover

for design. The nominal value of cover for design should be used in the calculations and stated

on the drawings, unless a value other than the nominal cover is specified (e.g. minimum value).

∆c

(3) In certain situations, the accepted deviation and hence allowance, , may be reduced.

dev

∆c

Note: The reduction in in such circumstances for use in a Country may be found in its National Annex.

dev

The recommended values are:

- where fabrication is subjected to a quality assurance system, in which the monitoring includes

∆c

measurements of the concrete cover, the allowance in design for deviation may be reduced:

dev

∆c

≥ ≥

10 mm 5 mm (4.3N)

dev

- where it can be assured that a very accurate measurement device is used for monitoring and non

∆c

conforming members are rejected (e.g. precast elements), the allowance in design for deviation dev

may be reduced:

∆c ≥

≥ 0 mm (4.4N)

10 mm dev

(4) For concrete cast against uneven surfaces, the minimum cover should generally be

increased by allowing larger deviations in design. The increase should comply with the

k

difference caused by the unevenness, but the minimum cover should be at least mm for

1

k

concrete cast against prepared ground (including blinding) and mm for concrete cast directly

2

against soil. The cover to the reinforcement for any surface feature, such as ribbed finishes or

exposed aggregate, should also be increased to take account of the uneven surface (see

4.4.1.2 (11)). k k

Note: The values of and for use in a Country may be found in its National Annex. The recommended

1 2

values are 40 mm and 75 mm.

52 EN 1992-1-1:2004 (E)

SECTION 5 STRUCTURAL ANALYSIS

5.1 General

5.1.1 General requirements

(1)P The purpose of structural analysis is to establish the distribution of either internal forces

and moments, or stresses, strains and displacements, over the whole or part of a structure.

Additional local analysis shall be carried out where necessary.

In most normal cases analysis will be used to establish the distribution of internal forces and moments,

Note:

and the complete verification or demonstration of resistance of cross sections is based on these action effects;

however, for certain particular elements, the methods of analysis used (e.g. finite element analysis) give

stresses, strains and displacements rather than internal forces and moments. Special methods are required to

use these results to obtain appropriate verification.

(2) Local analyses may be necessary where the assumption of linear strain distribution is not

valid, e.g.:

- in the vicinity of supports

- local to concentrated loads

- in beam-column intersections

- in anchorage zones

- at changes in cross section.

(3) For in-plane stress fields a simplified method for determining reinforcement may be used.

A simplified method is given in Annex F.

Note:

(4)P Analyses shall be carried out using idealisations of both the geometry and the behaviour

of the structure. The idealisations selected shall be appropriate to the problem being

considered.

(5) The geometry and the properties of the structure and its behaviour at each stage of

construction shall be considered in the design.

(6)P The effect of the geometry and properties of the structure on its behaviour at each stage

of construction shall be considered in the design

(7) Common idealisations of the behaviour used for analysis are:

- linear elastic behaviour (see 5.4)

- linear elastic behaviour with limited redistribution (see 5.5)

- plastic behaviour (see 5.6), including strut and tie models (see 5.6.4)

- non-linear behaviour (see 5.7)

(8) In buildings, the effects of shear and axial forces on the deformations of linear elements

and slabs may be ignored where these are likely to be less than 10% of those due to bending.

5.1.2 Special requirements for foundations

(1)P Where ground-structure interaction has significant influence on the action effects in the

structure, the properties of the soil and the effects of the interaction shall be taken into

account in accordance with EN 1997-1.

For more information concerning the analysis of shallow foundations see Annex G.

Note: 53

EN 1992-1-1:2004 (E)

(2) For the design of spread foundations, appropriately simplified models for the description of

the soil-structure interaction may be used.

For simple pad footings and pile caps the effects of soil-structure interaction may usually be ignored.

Note:

(3) For the strength design of individual piles the actions should be determined taking into

account the interaction between the piles, the pile cap and the supporting soil.

(4) Where the piles are located in several rows, the action on each pile should be evaluated by

considering the interaction between the piles.

(5) This interaction may be ignored when the clear distance between the piles is greater than

two times the pile diameter.

5.1.3 Load cases and combinations

(1)P In considering the combinations of actions, see EN 1990 Section 6, the relevant cases

shall be considered to enable the critical design conditions to be established at all sections,

within the structure or part of the structure considered.

Where a simplification in the number of load arrangements for use in a Country is required, reference is

Note:

made to its National Annex. The following simplified load arrangements are recommended for buildings:

γ γ

(a) alternate spans carrying the design variable and permanent load ( Q + G + P ), other spans carrying

Q k G k m

γ

only the design permanent load, G + P and

G k m γ γ

(b) any two adjacent spans carrying the design variable and permanent loads ( Q + G + P ). All other

Q k G k m

γ

spans carrying only the design permanent load, G + P .

G k m

5.1.4 Second order effects

(1)P Second order effects (see EN 1990 Section 1) shall be taken into account where they are

likely to affect the overall stability of a structure significantly and for the attainment of the

ultimate limit state at critical sections.

(2) Second order effects should be taken into account according to 5.8.

(3) For buildings, second order effects below certain limits may be ignored (see 5.8.2 (6)).

5.2 Geometric imperfections

(1)P The unfavourable effects of possible deviations in the geometry of the structure and the

position of loads shall be taken into account in the analysis of members and structures.

Deviations in cross section dimensions are normally taken into account in the material safety factors.

Note:

These should not be included in structural analysis. A minimum eccentricity for cross section design is given

in 6.1 (4).

(2)P Imperfections shall be taken into account in ultimate limit states in persistent and

accidental design situations.

(3) Imperfections need not be considered for serviceability limit states.

(4) The following provisions apply for members with axial compression and structures with

vertical load, mainly in buildings. Numerical values are related to normal execution deviations

54 EN 1992-1-1:2004 (E)

(Class 1 in ENV 13670). With the use of other deviations (e.g. Class 2), values should be

adjusted accordingly. θ

(5) Imperfections may be represented by an inclination, given by:

I,

θ θ α α

⋅ ⋅

= (5.1)

i 0 h m

where

θ is the basic value:

0

α α α

≤ ≤

is the reduction factor for length or height: = 2/ ; 2/3 1

l

h h h

α α +

is the reduction factor for number of members: = 0

,

5

(

1 1 / m )

m m

l is the length or height [m], see (4)

m is the number of vertical members contributing to the total effect

θ

The value of for use in a Country may be found in its National Annex. The recommended value is

Note: 0

1/200 l m

(6) In Expression (5.1), the definition of and depends on the effect considered, for which

three main cases can be distinguished (see also Figure 5.1):

l m

- Effect on isolated member: = actual length of member, =1.

l m

- Effect on bracing system: = height of building, = number of vertical members

contributing to the horizontal force on the bracing system. l

- Effect on floor or roof diaphragms distributing the horizontal loads: = storey height,

m = number of vertical elements in the storey(s) contributing to the total horizontal force on

the floor. (see 5.8.1), the effect of imperfections may be taken into account in

(7) For isolated members

two alternative ways a) or b):

e

a) as an eccentricity, , given by

i

θ

e = l / 2 (5.2)

i i 0

l

where is the effective length, see 5.8.3.2

0 e l

For walls and isolated columns in braced systems, = /400 may always be used as a

i 0

α

simplification, corresponding to = 1.

h

H

b) as a transverse force, , in the position that gives maximum moment:

i

for unbraced members (see Figure 5.1 a1):

θ

H = N (5.3a)

i i

for braced members (see Figure 5.1 a2):

θ

H = 2 N (5.3b)

i i

N

where is the axial load

Eccentricity is suitable for statically determinate members, whereas transverse load can be used for

Note:

both determinate and indeterminate members. The force H may be substituted by some other equivalent

i

transverse action. 55

EN 1992-1-1:2004 (E) e

i

e

i N

N N

N H

i H l = l

i

l = l / 2 0

0

a1) Unbraced a2) Braced

a) Isolated members with eccentric axial force or lateral force

θ N

i a

H i N θ /2 θ

b l i i

N a

H i N b

θ /2

i

b) Bracing system c1) Floor diaphragm c2) Roof diaphragm

Figure 5.1: Examples of the effect of geometric imperfections

θ

(8) For structures, the effect of the inclination may be represented by transverse forces, to be

i

included in the analysis together with other actions.

Effect on bracing system, (see Figure 5.1 b):

θ

H = (N - N ) (5.4)

i i b a

Effect on floor diaphragm, (see Figure 5.1 c1):

θ

H = (N + N ) / 2 (5.5)

i i b a

Effect on roof diaphragm, (see Figure 5.1 c2):

θ ⋅

H = N (5.6)

i i a

N N H

where and are longitudinal forces contributing to .

a b i

(9) As a simplified alternative for walls and isolated columns in braced systems, an eccentricity

e l

= /400 may be used to cover imperfections related to normal execution deviations (see 5.2(4)).

i 0

56 EN 1992-1-1:2004 (E)

5.3 Idealisation of the structure

5.3.1 Structural models for overall analysis

(1)P The elements of a structure are classified, by consideration of their nature and function, as

beams, columns, slabs, walls, plates, arches, shells etc. Rules are provided for the analysis of

the commoner of these elements and of structures consisting of combinations of these

elements.

(2) For buildings the following provisions (3) to (7) are applicable:

(3) A beam is a member for which the span is not less than 3 times the overall section depth.

Otherwise it should be considered as a deep beam.

(4) A slab is a member for which the minimum panel dimension is not less than 5 times the

overall slab thickness.

(5) A slab subjected to dominantly uniformly distributed loads may be considered to be one-

way spanning if either:

- it possesses two free (unsupported) and sensibly parallel edges, or

- it is the central part of a sensibly rectangular slab supported on four edges with a ratio of

the longer to shorter span greater than 2.

(6) Ribbed or waffle slabs need not be treated as discrete elements for the purposes of

analysis, provided that the flange or structural topping and transverse ribs have sufficient

torsional stiffness. This may be assumed provided that:

- the rib spacing does not exceed 1500 mm

- the depth of the rib below the flange does not exceed 4 times its width.

- the depth of the flange is at least 1/10 of the clear distance between ribs or 50 mm,

whichever is the greater.

- transverse ribs are provided at a clear spacing not exceeding 10 times the overall depth of

the slab.

The minimum flange thickness of 50 mm may be reduced to 40 mm where permanent blocks

are incorporated between the ribs.

(7) A column is a member for which the section depth does not exceed 4 times its width and

the height is at least 3 times the section depth. Otherwise it should be considered as a wall.

5.3.2 Geometric data

5.3.2.1 Effective width of flanges (all limit states)

(1)P In T beams the effective flange width, over which uniform conditions of stress can be

depends on the web and flange dimensions, the type of loading, the span, the

assumed,

support conditions and the transverse reinforcement. l

(2) The effective width of flange should be based on the distance between points of zero

0

moment, which may be obtained from Figure 5.2. 57

EN 1992-1-1:2004 (E) l =

0 l = 0,15 l + l

l = 0,7 l

l = 0,85 l 0,15(l + l ) 0 2 3

0 2

0 1 1 2

l l l

1 2 3

Figure 5.2: Definition of l , for calculation of effective flange width

0

The length of the cantilever, l , should be less than half the adjacent span and the ratio of adjacent

Note: 3

spans should lie between 2/3 and 1,5.

b L

for a T beam or

(3) The effective flange width beam may be derived as:

eff

= + ≤

b b b (5.7)

b

eff eff,

i w

where = + ≤

b 0 ,

2 b 0 ,

1

l 0 ,

2 l (5.7a)

eff,

i i 0 0

and ≤

b b (5.7b)

eff,

i i

(for the notations see Figures 5.2 above and 5.3 below).

b

eff b

b eff,2

eff,1

b

w

b w b b

b

b 2 2

1 1 b

Figure 5.3: Effective flange width parameters

(4) For structural analysis, where a great accuracy is not required, a constant width may be

assumed over the whole span. The value applicable to the span section should be adopted.

5.3.2.2 Effective span of beams and slabs in buildings

The following provisions are provided mainly for member analysis. For frame analysis some of these

Note:

simplifications may be used where appropriate.

(1) The effective span, l , of a member should be calculated as follows:

eff

+ a

l = l + a (5.8)

eff n 1 2

where:

is the clear distance between the faces of the supports;

l

n

values for a and a , at each end of the span, may be determined from the appropriate a

1 2 i

values in Figure 5.4 where t is the width of the supporting element as shown.

58 EN 1992-1-1:2004 (E)

h

h a = h; t

{ }

min 1/2 1/2

a = h; t

{ }

min i

1/2 1/2 l

i n

l n l eff

l eff t

t

(a) Non-continuous members (b) Continuous members centreline

h l eff

a = min { h; t }

1/2 1/2

i l n l

a n

i

l eff

t

(c) Supports considered fully restrained (d) Bearing provided

h

a = min { h; t }

1/2 1/2

i l n

l eff

t

(e) Cantilever

Figure 5.4: Effective span (l ) for different support conditions

eff

(2) Continuous slabs and beams may generally be analysed on the assumption that the

supports provide no rotational restraint.

(3) Where a beam or slab is monolithic with its supports, the critical design moment at the

support should be taken as that at the face of the support. The design moment and reaction

transferred to the supporting element (e.g. column, wall, etc.) should be generally taken as the

greater of the elastic or redistributed values.

The moment at the face of the support should not be less than 0,65 that of the full fixed end moment.

Note: 59

EN 1992-1-1:2004 (E)

(4) Regardless of the method of analysis used, where a beam or slab is continuous over a

support which may be considered to provide no restraint to rotation (e.g. over walls), the design

support moment, calculated on the basis of a span equal to the centre-to-centre distance

∆M as follows:

between supports, may be reduced by an amount Ed

∆M = F t / 8 (5.9)

Ed,sup

Ed

where: is the design support reaction

F

Ed,sup

t is the breadth of the support (see Figure 5.4 b))

Where support bearings are used t should be taken as the bearing width.

Note:

5.4 Linear elastic analysis

(1) Linear analysis of elements based on the theory of elasticity may be used for both the

serviceability and ultimate limit states.

(2) For the determination of the action effects, linear analysis may be carried out assuming:

i) uncracked cross sections,

ii) linear stress-strain relationships and

iii) mean value of the modulus of elasticity.

(3) For thermal deformation, settlement and shrinkage effects at the ultimate limit state (ULS),

a reduced stiffness corresponding to the cracked sections, neglecting tension stiffening but

including the effects of creep, may be assumed. For the serviceability limit state (SLS) a

gradual evolution of cracking should be considered.

5.5 Linear elastic analysis with limited redistribution

(1)P The influence of any redistribution of the moments on all aspects of the design shall be

considered.

(2) Linear analysis with limited redistribution may be applied to the analysis of structural

members for the verification of ULS.

(3) The moments at ULS calculated using a linear elastic analysis may be redistributed,

provided that the resulting distribution of moments remains in equilibrium with the applied loads.

(4) In continuous beams or slabs which:

a) are predominantly subject to flexure and

b) have the ratio of the lengths of adjacent spans in the range of 0,5 to 2,

redistribution of bending moments may be carried out without explicit check on the rotation

capacity, provided that:

δ ≥ ≤

k + k x /d for f 50 MPa (5.10a)

1 2 u ck

δ ≥ k + k x /d for f > 50 MPa (5.10b)

3 4 u ck

≥ k where Class B and Class C reinforcement is used (see Annex C)

5

≥ k where Class A reinforcement is used (see Annex C)

6

Where:

δ is the ratio of the redistributed moment to the elastic bending moment

is the depth of the neutral axis at the ultimate limit state after redistribution

x u

60 EN 1992-1-1:2004 (E)

d is the effective depth of the section

The values of k k k k k and k for use in a Country may be found in its National Annex. The

Note: 1, 2, 3 , 4, 5 6 ε ε

is 0,44 for k is 1,25(0,6+0,0014/ ) for k = 0,54, for k = 1,25(0,6+0,0014/ ), for

recommended value for k 1 , 2 cu2 , 3 4 cu2

ε

k = 0,7 and k = 0,8. is the ultimate strain according to Table 3.1.

5 6 cu2

(5) Redistribution should not be carried out in circumstances where the rotation capacity cannot

be defined with confidence (e.g. in the corners of prestressed frames).

(6) For the design of columns the elastic moments from frame action should be used without

any redistribution.

5.6 Plastic analysis

5.6.1 General

(1)P Methods based on plastic analysis shall only be used for the check at ULS.

(2)P The ductility of the critical sections shall be sufficient for the envisaged mechanism to be

formed.

(3)P The plastic analysis should be based either on the lower bound (static) method or on the

upper bound (kinematic) method.

A Country’s National Annex Guidance may refer to non-contradictory complementary information.

Note:

(4) The effects of previous applications of loading may generally be ignored, and a monotonic

increase of the intensity of actions may be assumed.

5.6.2 Plastic analysis for beams, frames and slabs

(1)P Plastic analysis without any direct check of rotation capacity may be used for the ultimate

limit state if the conditions of 5.6.1 (2)P are met.

(2) The required ductility may be deemed to be satisfied without explicit verification if all the

following are fulfilled:

i) the area of tensile reinforcement is limited such that, at any section

≤ ≤

/d 0,25 for concrete strength classes C50/60

x u ≤ ≥

x /d 0,15 for concrete strength classes C55/67

u

ii) reinforcing steel is either Class B or C

iii) the ratio of the moments at intermediate supports to the moments in the span should be

between 0,5 and 2.

(3) Columns should be checked for the maximum plastic moments which can be transmitted by

connecting members. For connections to flat slabs this moment should be included in the

punching shear calculation.

(4) When plastic analysis of slabs is carried out account should be taken of any non-uniform

reinforcement, corner tie down forces, and torsion at free edges.

(5) Plastic methods may be extended to non-solid slabs (ribbed, hollow, waffle slabs) if their

response is similar to that of a solid slab, particularly with regard to the torsional effects. 61

EN 1992-1-1:2004 (E)

5.6.3 Rotation capacity

(1) The simplified procedure for continuous beams and continuous one way spanning slabs is

based on the rotation capacity of beam/slab zones over a length of approximately 1,2 times the

depth of section. It is assumed that these zones undergo a plastic deformation (formation of

yield hinges) under the relevant combination of actions. The verification of the plastic rotation in

the ultimate limit state is considered to be fulfilled, if it is shown that under the relevant

θ , is less than or equal to the allowable plastic

combination of actions the calculated rotation, s

rotation (see Figure 5.5).

0,6h 0,6h θ h

s

θ

Figure 5.5: Plastic rotation of reinforced concrete sections for continuous

s

beams and continuous one way spanning slabs.

(2) In regions of yield hinges, x /d shall not exceed the value 0,45 for concrete strength classes

u

less than or equal to C50/60, and 0,35 for concrete strength classes greater than or equal to

C55/67. θ

(3) The rotation should be determined on the basis of the design values for actions and

s

materials and on the basis of mean values for prestressing at the relevant time.

(4) In the simplified procedure, the allowable plastic rotation may be determined by multiplying

θ

the basic value of allowable rotation, , by a correction factor k that depends on the shear

λ

pl,d

slenderness. θ

Values of for use in a Country may be found in its National Annex. The recommended values for

Note: pl,d

steel Classes B and C (the use of Class A steel is not recommended for plastic analysis) and concrete

strength classes less than or equal to C50/60 and C90/105 are given in Figure 5.6N.

The values for concrete strength classes C 55/67 to C 90/105 may be interpolated accordingly. The values

λ θ

apply for a shear slenderness = 3,0. For different values of shear slenderness should be multiplied by

pl,d

k :

λ λ

=

k / 3 (5.11N)

λ λ

Where is the ratio of the distance between point of zero and maximum moment after redistribution and

effective depth, d. λ

As a simplification may be calculated for the concordant design values of the bending moment and shear :

λ ⋅

= M / (V d) (5.12N)

Sd Sd

62 EN 1992-1-1:2004 (E)

θ (mrad)

pl,d

35

30 ≤ C 50/60

25

20 Class C

C 90/105

15 Class B

10 ≤ C 50/60

5 C 90/105

0 0,35

0,15

0,10 0,40 0,45

0,30

0,25

0,20

0,05

0 (x /d)

u

θ

Figure 5.6N: Basic value of allowable rotation, , of reinforced concrete sections for Class B

pl,d λ = 3,0

and C reinforcement. The values apply for a shear slenderness

5.6.4 Analysis with strut and tie models

(1) Strut and tie models may be used for design in ULS of continuity regions (cracked state of

beams and slabs, see 6.1 - 6.4) and for the design in ULS and detailing of discontinuity regions

(see 6.5). In general these extend up to a distance h (section depth of member) from the

discontinuity. Strut and tie models may also be used for members where a linear distribution

within the cross section is assumed, e.g. plane strain.

(2) Verifications in SLS may also be carried out using strut-and-tie models, e.g. verification of

steel stresses and crack width control, if approximate compatibility for strut-and-tie models is

ensured (in particular the position and direction of important struts should be oriented according

to linear elasticity theory)

(3) Strut-and-tie models consist of struts representing compressive stress fields, of ties

representing the reinforcement, and of the connecting nodes. The forces in the elements of a

strut-and-tie model should be determined by maintaining the equilibrium with the applied loads

in the ultimate limit state. The elements of strut-and-tie models should be dimensioned

according to the rules given in 6.5.

(4) The ties of a strut-and-tie model should coincide in position and direction with the

corresponding reinforcement.

(5) Possible means for developing suitable strut-and-tie models include the adoption of stress

trajectories and distributions from linear-elastic theory or the load path method. All strut-and-tie

models may be optimised by energy criteria.

5.7 Non-linear analysis

(1) Non-linear methods of analysis may be used for both ULS and SLS, provided that

equilibrium and compatibility are satisfied and an adequate non-linear behaviour for materials is

assumed. The analysis may be first or second order. 63

EN 1992-1-1:2004 (E)

(2) At the ultimate limit state, the ability of local critical sections to withstand any inelastic

deformations implied by the analysis should be checked, taking appropriate account of

uncertainties.

(3) For structures predominantly subjected to static loads, the effects of previous applications

of loading may generally be ignored, and a monotonic increase of the intensity of the actions

may be assumed.

(4)P The use of material characteristics which represent the stiffness in a realistic way but take

account of the uncertainties of failure shall be used when using non-linear analysis. Only those

design formats which are valid within the relevant fields of application shall be used.

(5) For slender structures, in which second order effects cannot be ignored, the design method

given in 5.8.6 may be used.

5.8 Analysis of second order effects with axial load

5.8.1 Definitions

Biaxial bending: simultaneous bending about two principal axes

Braced members or systems: structural members or subsystems, which in analysis and

design are assumed not to contribute to the overall horizontal stability of a structure

Bracing members or systems: structural members or subsystems, which in analysis and

design are assumed to contribute to the overall horizontal stability of a structure

Buckling: failure due to instability of a member or structure under perfectly axial compression

and without transverse load

“Pure buckling” as defined above is not a relevant limit state in real structures, due to imperfections and

Note.

transverse loads, but a nominal buckling load can be used as a parameter in some methods for second order

analysis.

Buckling load: the load at which buckling occurs; for isolated elastic members it is

synonymous with the Euler load

Effective length: a length used to account for the shape of the deflection curve; it can also

be defined as buckling length, i.e. the length of a pin-ended column with constant normal

force, having the same cross section and buckling load as the actual member

First order effects: action effects calculated without consideration of the effect of structural

deformations, but including geometric imperfections

Isolated members: members that are isolated, or members in a structure that for design

purposes may be treated as being isolated; examples of isolated members with different

boundary conditions are shown in Figure 5.7.

Nominal second order moment: a second order moment used in certain design methods,

giving a total moment compatible with the ultimate cross section resistance (see 5.8.5 (2))

Second order effects: additional action effects caused by structural deformations

64 EN 1992-1-1:2004 (E)

5.8.2 General

(1)P This clause deals with members and structures in which the structural behaviour is

significantly influenced by second order effects (e.g. columns, walls, piles, arches and shells).

Global second order effects are likely to occur in structures with a flexible bracing system.

(2)P Where second order effects are taken into account, see (6), equilibrium and resistance

shall be verified in the deformed state. Deformations shall be calculated taking into account the

relevant effects of cracking, non-linear material properties and creep.

In an analysis assuming linear material properties, this can be taken into account by means of reduced

Note.

stiffness values, see 5.8.7.

(3)P Where relevant, analysis shall include the effect of flexibility of adjacent members and

foundations (soil-structure interaction).

(4)P The structural behaviour shall be considered in the direction in which deformations can

occur, and biaxial bending shall be taken into account when necessary.

(5)P Uncertainties in geometry and position of axial loads shall be taken into account as

additional first order effects based on geometric imperfections, see 5.2.

(6) Second order effects may be ignored if they are less than 10 % of the corresponding first

order effects. Simplified criteria are given for isolated members in 5.8.3.1 and for structures in

5.8.3.3.

5.8.3 Simplified criteria for second order effects

5.8.3.1 Slenderness criterion for isolated members λ

(1) As an alternative to 5.8.2 (6), second order effects may be ignored if the slenderness (as

λ

defined in 5.8.3.2) is below a certain value .

lim

λ

The value of for use in a Country may be found in its National Annex. The recommended value

Note: lim

follows from:

λ = 20⋅A⋅B⋅C/√n (5.13N)

lim

where: ϕ ϕ

) (if is not known, A = 0,7 may be used)

A = 1 / (1+0,2 ef ef

ω

+ ω

B = (if is not known, B = 1,1 may be used)

1 2

C = 1,7 - r (if r is not known, C = 0,7 may be used)

m m

ϕ effective creep ratio; see 5.8.4;

ef

ω = A f / (A f ); mechanical reinforcement ratio;

s yd c cd

A is the total area of longitudinal reinforcement

s

n = N / (A f ); relative normal force

Ed c cd

r = M /M ; moment ratio

m 01 02 ⏐M ⏐ ≥ ⏐M ⏐

M M are the first order end moments,

01, 02 02 01 ≤

If the end moments M and M give tension on the same side, r should be taken positive (i.e. C 1,7),

01 02 m

otherwise negative (i.e. C > 1,7).

In the following cases, r should be taken as 1,0 (i.e. C = 0,7):

m

for braced members in which the first order moments arise only from or predominantly due to imperfections

- or transverse loading

for unbraced members in general

- 65

EN 1992-1-1:2004 (E)

(2) In cases with biaxial bending, the slenderness criterion may be checked separately for each

direction. Depending on the outcome of this check, second order effects (a) may be ignored in

both directions, (b) should be taken into account in one direction, or (c) should be taken into

account in both directions.

5.8.3.2 Slenderness and effective length of isolated members

(1) The slenderness ratio is defined as follows:

λ = / i (5.14)

l 0

where: is the effective length, see 5.8.3.2 (2) to (7)

l 0

i is the radius of gyration of the uncracked concrete section

(2) For a general definition of the effective length, see 5.8.1. Examples of effective length for

isolated members with constant cross section are given in Figure 5.7. θ l

θ M

a) l = l b) l = 2l c) l = 0,7l d) l = l / 2 e) l = l f) l /2 <l < l g) l > 2l

0 0 0 0 0 0 0

Figure 5.7: Examples of different buckling modes and corresponding effective

lengths for isolated members

(3) For compression members in regular frames, the slenderness criterion (see 5.8.3.1) should

determined in the following way:

be checked with an effective length l

0

Braced members (see Figure 5.7 (f)):

⎛ ⎞ ⎛ ⎞

k k

⎜ ⎟ ⎜ ⎟

+ ⋅ +

⋅ 1 2

1 1 (5.15)

l = 0,5l ⎜ ⎟ ⎜ ⎟

0 + +

0

,

45 k 0

,

45 k

⎝ ⎠ ⎝ ⎠

1 2

Unbraced members (see Figure 5.7 (g)):

⎧ ⎫

⎛ ⎞ ⎛ ⎞

⎪ ⎪

k k k k

⎜ ⎟ ⎜ ⎟

+ ⋅ + ⋅ +

1 2 1 2

max 1 10 ; 1 1

l l⋅ (5.16)

⎨ ⎬

= ⎜ ⎟ ⎜ ⎟

0 + + +

⎪⎩ ⎪⎭

k k k k

1 1

⎝ ⎠ ⎝ ⎠

1 2 1 2

where:

k , k are the relative flexibilities of rotational restraints at ends 1 and 2 respectively:

1 2

66 EN 1992-1-1:2004 (E)

θ ⋅ Ι

k = ( / M) (E / l)

θ is the rotation of restraining members for bending moment M;

see also Figure 5.7 (f) and (g)

Ι is the bending stiffness of compression member, see also 5.8.3.2 (4) and (5)

E

l is the clear height of compression member between end restraints

k = 0 is the theoretical limit for rigid rotational restraint, and k = represents the limit for no restraint at

Note:

all. Since fully rigid restraint is rare in practise, a minimum value of 0,1 is recommended for k and k .

1 2

(4) If an adjacent compression member (column) in a node is likely to contribute to the rotation

Ι Ι Ι

at buckling, then (E /l) in the definition of k should be replaced by [(E / l) +(E / l) ], a and b

a b

representing the compression member (column) above and below the node.

(5) In the definition of effective lengths, the stiffness of restraining members should include the

effect of cracking, unless they can be shown to be uncracked in ULS.

(6) For other cases than those in (2) and (3), e.g. members with varying normal force and/or

cross section, the criterion in 5.8.3.1 should be checked with an effective length based on the

buckling load (calculated e.g. by a numerical method):

Ι

= π

l Ε / N (5.17)

0 B

where:

E is a representative bending stiffness

I

N is buckling load expressed in terms of this E I

B (in Expression (5.14), i should also correspond to this )

EI

(7) The restraining effect of transverse walls may be allowed for in the calculation of the

β

effective length of walls by the factor given in 12.6.5.1. In Expression (12.9) and Table 12.1, l

w

is then substituted by l determined according to 5.8.3.2.

0

5.8.3.3 Global second order effects in buildings

(1) As an alternative to 5.8.2 (6), global second order effects in buildings may be ignored if

∑ Ι

E

n cd c

≤ ⋅ ⋅

s

F k (5.18)

V,Ed 1 + 2

n L

1,6

s

where:

F is the total vertical load (on braced and bracing members)

V,Ed

n is the number of storeys

s

L is the total height of building above level of moment restraint

E is the design value of the modulus of elasticity of concrete, see 5.8.6 (3)

cd is the second moment of area (uncracked concrete section) of bracing member(s)

I c The value of k for use in a Country may be found in its National Annex. The recommended value is

Note: 1

0,31.

Expression (5.18) is valid only if all the following conditions are met:

- torsional instability is not governing, i.e. structure is reasonably symmetrical

global shear deformations are negligible (as in a bracing system mainly consisting of shear

- walls without large openings)

bracing members are rigidly fixed at the base, i.e. rotations are negligible

- 67

EN 1992-1-1:2004 (E)

- the stiffness of bracing members is reasonably constant along the height

- the total vertical load increases by approximately the same amount per storey

(2) k in Expression (5.18) may be replaced by k if it can be verified that bracing members

1 2

are uncracked in ultimate limit state.

The value of k for use in a Country may be found in its National Annex. The recommended value is

Note 1: 2

0,62. For cases where the bracing system has significant global shear deformations and/or end rotations,

Note 2:

see Annex H (which also gives the background to the above rules).

5.8.4 Creep

(1)P The effect of creep shall be taken into account in second order analysis, with due

consideration of both the general conditions for creep (see 3.1.4) and the duration of different

loads in the load combination considered.

(2) The duration of loads may be taken into account in a simplified way by means of an

ϕ

effective creep ratio, , which, used together with the design load, gives a creep deformation

ef

(curvature) corresponding to the quasi-permanent load:

ϕ ϕ ⋅M

= / M (5.19)

ef (∞,t0) 0Eqp 0Ed

where:

ϕ is the final creep coefficient according to 3.1.4

(∞,t0)

M is the first order bending moment in quasi-permanent load combination (SLS)

0Eqp

M is the first order bending moment in design load combination (ULS)

0Ed ϕ

It is also possible to base on total bending moments M and M , but this requires iteration and a

Note. ef Eqp Ed

ϕ ϕ

verification of stability under quasi-permanent load with = .

ef (∞,t0)

(3) If M / M varies in a member or structure, the ratio may be calculated for the section

0Eqp 0Ed

with maximum moment, or a representative mean value may be used.

ϕ

(4) The effect of creep may be ignored, i.e. = 0 may be assumed, if the following three

ef

conditions are met:

ϕ ≤ 2

- (∞,t0)

λ ≤ 75

- ≥

M /N h

- 0Ed Ed

Here M is the first order moment and h is the cross section depth in the corresponding

0Ed

direction.

If the conditions for neglecting second order effects according to 5.8.2 (6) or 5.8.3.3 are only just

Note.

achieved, it may be too unconservative to neglect both second order effects and creep, unless the mechanical

ω

reinforcement ratio ( , see 5.8.3.1 (1)) is at least 0,25.

5.8.5 Methods of analysis

(1) The methods of analysis include a general method, based on non-linear second order

analysis, see 5.8.6 and the following two simplified methods:

(a) Method based on nominal stiffness, see 5.8.7

(b) Method based on nominal curvature, see 5.8.8

68 EN 1992-1-1:2004 (E)

The selection of Simplified Method (a) and (b) to be used in a Country may be found in its National

Note 1:

Annex. Nominal second order moments provided by the simplified methods (a) and (b) are sometimes

Note 2:

greater than those corresponding to instability. This is to ensure that the total moment is compatible with the

cross section resistance.

(2) Method (a) may be used for both isolated members and whole structures, if nominal

stiffness values are estimated appropriately; see 5.8.7.

(3) Method (b) is mainly suitable for isolated members; see 5.8.8. However, with realistic

assumptions concerning the distribution of curvature, the method in 5.8.8 can also be used for

structures.

5.8.6 General method

(1)P The general method is based on non-linear analysis, including geometric non-linearity i.e.

second order effects. The general rules for non-linear analysis given in 5.7 apply.

(2)P Stress-strain curves for concrete and steel suitable for overall analysis shall be used. The

effect of creep shall be taken into account.

(3) Stress-strain relationships for concrete and steel given in 3.1.5, Expression (3.14) and 3.2.3

(Figure 3.8) may be used. With stress-strain diagrams based on design values, a design value

of the ultimate load is obtained directly from the analysis. In Expression (3.14), and in the k-

value, f is then substituted by the design compressive strength f and E is substituted by

cm cd cm

γ

E = E / (5.20)

cd cm cE γ

The value of for use in a Country may be found in its National Annex. The recommended value is

Note: cE

1,2.

(4) In the absence of more refined models, creep may be taken into account by multiplying all

ϕ

strain values in the concrete stress-strain diagram according to 5.8.6 (3) with a factor (1 + ),

ef

ϕ

where is the effective creep ratio according to 5.8.4.

ef

(5) The favourable effect of tension stiffening may be taken into account.

This effect is favourable, and may always be ignored, for simplicity.

Note:

(6) Normally, conditions of equilibrium and strain compatibility are satisfied in a number of

cross sections. A simplified alternative is to consider only the critical cross section(s), and to

assume a relevant variation of the curvature in between, e.g. similar to the first order moment or

simplified in another appropriate way.

5.8.7 Method based on nominal stiffness

5.8.7.1 General

(1) In a second order analysis based on stiffness, nominal values of the flexural stiffness

should be used, taking into account the effects of cracking, material non-linearity and creep on

the overall behaviour. This also applies to adjacent members involved in the analysis, e.g.

beams, slabs or foundations. Where relevant, soil-structure interaction should be taken into

account. 69

EN 1992-1-1:2004 (E)

(2) The resulting design moment is used for the design of cross sections with respect to

bending moment and axial force according to 6.1, as compared with 5.8.6 (2).

5.8.7.2 Nominal stiffness

(1) The following model may be used to estimate the nominal stiffness of slender compression

members with arbitrary cross section:

= K E + K E (5.21)

EI I I

c cd c s s s

where:

E is the design value of the modulus of elasticity of concrete, see 5.8.6 (3)

cd is the moment of inertia of concrete cross section

I c

E is the design value of the modulus of elasticity of reinforcement, 5.8.6 (3)

s is the second moment of area of reinforcement, about the centre of area of the

I s concrete

K is a factor for effects of cracking, creep etc, see 5.8.7.2 (2) or (3)

c

K is a factor for contribution of reinforcement, see 5.8.7.2 (2) or (3)

s ρ ≥

(2) The following factors may be used in Expression (5.21), provided 0,002:

K = 1

s (5.22)

ϕ

K = k k / (1 + )

c 1 2 ef

where:

ρ is the geometric reinforcement ratio, A /A

s c

A is the total area of reinforcement

s

A is the area of concrete section

c

ϕ is the effective creep ratio, see 5.8.4

ef

k is a factor which depends on concrete strength class, Expression (5.23)

1

k is a factor which depends on axial force and slenderness, Expression (5.24)

2

k = f / 20 (MPa) (5.23)

ck

1 λ

⋅ ≤

n

k = 0,20 (5.24)

2 170

where:

n is the relative axial force, N / (A f )

Ed c cd

λ is the slenderness ratio, see 5.8.3

λ

If the slenderness ratio is not defined, k may be taken as

2

k = n⋅0,30 0,20 (5.25)

2 ρ ≥

(3) As a simplified alternative, provided 0,01, the following factors may be used in

Expression (5.21):

K = 0

s (5.26)

ϕ

K = 0,3 / (1 + 0,5 )

c ef

. The simplified alternative may be suitable as a preliminary step, followed by a more accurate calculation

Note

according to (2).

70 EN 1992-1-1:2004 (E)

(4) In statically indeterminate structures, unfavourable effects of cracking in adjacent members

should be taken into account. Expressions (5.21-5.26) are not generally applicable to such

members. Partial cracking and tension stiffening may be taken into account e.g. according to

7.4.3. However, as a simplification, fully cracked sections may be assumed. The stiffness

should be based on an effective concrete modulus:

ϕ

E = E /(1+ ) (5.27)

cd,eff cd ef

where:

E is the design value of the modulus of elasticity according to 5.8.6 (3)

cd

ϕ is the effective creep ratio; same value as for columns may be used

ef

5.8.7.3 Moment magnification factor

(1) The total design moment, including second order moment, may be expressed as a

magnification of the bending moments resulting from a linear analysis, namely:

⎡ ⎤

β

= + (5.28)

M M 1

⎢ ⎥

( )

Ed 0Ed −

N N

/ 1

⎣ ⎦

B Ed

where:

M is the first order moment; see also 5.8.8.2 (2)

0Ed st nd

β is a factor which depends on distribution of 1 and 2 order moments, see 5.8.7.3

(2)-(3)

N is the design value of axial load

Ed

N is the buckling load based on nominal stiffness

B

(2) For isolated members with constant cross section and axial load, the second order moment

may normally be assumed to have a sine-shaped distribution. Then

2

β π

= / c (5.29)

0

where:

c is a coefficient which depends on the distribution of first order moment (for instance,

0 = 8 for a constant first order moment, c = 9,6 for a parabolic and 12 for a

c 0 0

symmetric triangular distribution etc.).

(3) For members without transverse load, differing first order end moments M and M may

01 02

according to 5.8.8.2 (2).

be replaced by an equivalent constant first order moment M 0e

Consistent with the assumption of a constant first order moment, c = 8 should be used.

0

c

The value of = 8 also applies to members bent in double curvature. It should be noted that in some

Note: 0

cases, depending on slenderness and axial force, the end moments(s) can be greater than the magnified

equivalent moment β

(4) Where 5.8.7.3 (2) or (3) is not applicable, = 1 is normally a reasonable simplification.

Expression (5.28) can then be reduced to:

M

= 0Ed

M (5.30)

( )

Ed −

1 N / N

Ed B 71

EN 1992-1-1:2004 (E)

5.8.7.3 (4) is also applicable to the global analysis of certain types of structures, e.g. structures braced

Note:

by shear walls and similar, where the principal action effect is bending moment in bracing units. For other

types of structures, a more general approach is given in Annex H, Clause H.2.

5.8.8 Method based on nominal curvature

5.8.8.1 General

(1) This method is primarily suitable for isolated members with constant normal force and a

defined effective length (see 5.8.3.2). The method gives a nominal second order moment

l 0

based on a deflection, which in turn is based on the effective length and an estimated maximum

curvature (see also 5.8.5(4)).

(2) The resulting design moment is used for the design of cross sections with respect to

bending moment and axial force according to 6.1.

5.8.8.2 Bending moments

(1) The design moment is:

M = M + M (5.31)

Ed 0Ed 2

where: st

M is the 1 order moment, including the effect of imperfections, see also 5.8.8.2 (2)

0Ed nd

M is the nominal 2 order moment, see 5.8.8.2 (3)

2

The maximum value of M is given by the distributions of M and M ; the latter may be taken

Ed 0Ed 2

as parabolic or sinusoidal over the effective length.

M

For statically indeterminate members, is determined for the actual boundary conditions, whereas

Note: 0Ed

M will depend on boundary conditions via the effective length, cf. 5.8.8.1 (1).

2

(2) Differing first order end moments M and M may be replaced by an equivalent first order

01 02

:

end moment M 0e ≥

M = 0,6 M + 0,4 M 0,4 M (5.32)

0e 02 01 02

M and M should have the same sign if they give tension on the same side, otherwise

01 02 ⏐M ⏐≥⏐M ⏐.

opposite signs. Furthermore, 02 01

(3) The nominal second order moment M in Expression (5.31) is

2

M = N e (5.33)

2 Ed 2

where:

N is the design value of axial force

Ed o2

e is the deflection = (1/r) l / c

2

1/r is the curvature, see 5.8.8.3

is the effective length, see 5.8.3.2

l

o

c is a factor depending on the curvature distribution, see 5.8.8.2 (4)

2

π

(4) For constant cross section, c = 10 (≈ ) is normally used. If the first order moment is

constant, a lower value should be considered (8 is a lower limit, corresponding to constant total

moment).

72 EN 1992-1-1:2004 (E)

2

π

The value corresponds to a sinusoidal curvature distribution. The value for constant curvature is 8.

Note. c total c

Note that depends on the distribution of the curvature, whereas in 5.8.7.3 (2) depends on the

0

curvature corresponding to the first order moment only.

5.8.8.3 Curvature

(1) For members with constant symmetrical cross sections (incl. reinforcement), the following

may be used:

⋅K ⋅1/r

1/r = K (5.34)

ϕ

r 0

where:

K is a correction factor depending on axial load, see 5.8.8.3 (3)

r

K is a factor for taking account of creep, see 5.8.8.3 (4)

ϕ ε

1/r = / (0,45 d)

0 yd

ε = f / E

yd yd s

d is the effective depth; see also 5.8.8.3 (2)

(2) If all reinforcement is not concentrated on opposite sides, but part of it is distributed parallel

to the plane of bending, d is defined as

d = (h/2) + i (5.35)

s

where i is the radius of gyration of the total reinforcement area

s

(3) K in Expression (5.34) should be taken as:

r ≤

K = (n - n) / (n - n ) 1 (5.36)

r u u bal

where:

n = N / (A f ), relative axial force

Ed c cd

N is the design value of axial force

Ed ω

n = 1 +

u

n is the value of n at maximum moment resistance; the value 0,4 may be used

bal

ω = A f / (A f )

s yd c cd

A is the total area of reinforcement

s is the area of concrete cross section

A

c

(4) The effect of creep should be taken into account by the following factor:

βϕ ≥

K = 1 + 1 (5.37)

ϕ ef

where:

ϕ is the effective creep ratio, see 5.8.4

ef

β λ

= 0,35 + f /200 - /150

ck

λ is the slenderness ratio, see 5.8.3.1

5.8.9 Biaxial bending

(1) The general method described in 5.8.6 may also be used for biaxial bending. The following

provisions apply when simplified methods are used. Special care should be taken to identify

the section along the member with the critical combination of moments. 73

EN 1992-1-1:2004 (E)

(2) Separate design in each principal direction, disregarding biaxial bending, may be made as a

first step. Imperfections need to be taken into account only in the direction where they will have

the most unfavourable effect.

(3) No further check is necessary if the slenderness ratios satisfy the following two conditions

λ λ λ λ

≤ ≤

/ 2 and / 2 (5.38a)

y z z y

and if the relative eccentricities e /h and e /b (see Figure 5.7) satisfy one the following

y z

conditions:

e / h e / b

y eq z eq

≤ ≤

0,2 or 0,2 (5.38b)

e h

e / b /

z eq y eq

where:

b, h are the width and depth of the section

⋅ i 12

b = i 12 and h = for an equivalent rectangular section

eq eq

y z

λ λ

, are the slenderness ratios l /i with respect to y- and z-axis respectively

y z 0

i , i are the radii of gyration with respect to y- and z-axis respectively

y z

e = M / N ; eccentricity along z-axis

z Edy Ed

e = M / N ; eccentricity along y-axis

y Edz Ed

M is the design moment about y-axis, including second order moment

Edy

M is the design moment about z-axis, including second order moment

Edz

N is the design value of axial load in the respective load combination

Ed z e

y

N Ed

i e

y z y

b i

y i

i z

z h

Figure 5.8. Definition of eccentricities e and e .

y z

(4) If the condition of Expression (5.38) is not fulfilled, biaxial bending should be taken into

nd

account including the 2 order effects in each direction (unless they may be ignored according

to 5.8.2 (6) or 5.8.3). In the absence of an accurate cross section design for biaxial bending, the

following simplified criterion may be used:

a

a ⎛ ⎞

⎛ ⎞ M

M Edy

+ ≤

Edz ⎜ ⎟ 1,0 (5.39)

⎜ ⎟ ⎜ ⎟

M M

⎝ ⎠ ⎝ ⎠

Rdz Rdy

74 EN 1992-1-1:2004 (E)

where: nd

is the design moment around the respective axis, including a 2 order moment.

M Edz/y is the moment resistance in the respective direction

M Rdz/y

a is the exponent;

for circular and elliptical cross sections: a = 2

for rectangular cross sections: N /N 0,1 0,7 1,0

Ed Rd

a = 1,0 1,5 2,0

with linear interpolation for intermediate values

is the design value of axial force

N

Ed = A f + A f , design axial resistance of section.

N

Rd c cd s yd

where: is the gross area of the concrete section

A

c is the area of longitudinal reinforcement

A

s

5.9 Lateral instability of slender beams

(1)P Lateral instability of slender beams shall be taken into account where necessary, e.g. for

precast beams during transport and erection, for beams without sufficient lateral bracing in the

finished structure etc. Geometric imperfections shall be taken into account.

(2) A lateral deflection of / 300 should be assumed as a geometric imperfection in the

l = total length of beam. In finished

verification of beams in unbraced conditions, with l

structures, bracing from connected members may be taken into account

(3) Second order effects in connection with lateral instability may be ignored if the following

conditions are fulfilled: l 50

≤ ≤

0t and h/b 2,5 (5.40a)

- persistent situations: ( )

13

b h b

l 70

≤ ≤

0t and h/b 3,5 (5.40b)

- transient situations: ( )

13

b h b

where: is the distance between torsional restraints

l 0t

h is the total depth of beam in central part of l 0t

b is the width of compression flange

(4) Torsion associated with lateral instability should be taken into account in the design of

supporting structures.

5.10 Prestressed members and structures

5.10.1 General

(1)P The prestress considered in this Standard is that applied to the concrete by stressed

tendons.

(2) The effects of prestressing may be considered as an action or a resistance caused by

prestrain and precurvature. The bearing capacity should be calculated accordingly. 75

EN 1992-1-1:2004 (E)

(3) In general prestress is introduced in the action combinations defined in EN 1990 as part of

the loading cases and its effects should be included in the applied internal moment and axial

force.

(4) Following the assumptions of (3) above, the contribution of the prestressing tendons to the

resistance of the section should be limited to their additional strength beyond prestressing. This

may be calculated assuming that the origin of the stress/strain relationship of the tendons is

displaced by the effects of prestressing.

(5)P Brittle failure of the member caused by failure of prestressing tendons shall be avoided.

(6) Brittle failure should be avoided by one or more of the following methods:

Method A: Provide minimum reinforcement in accordance with 9.2.1.

Method B: Provide pretensioned bonded tendons.

Method C: Provide easy access to prestressed concrete members in order to check and

control the condition of tendons by non-destructive methods or by monitoring.

Method D: Provide satisfactory evidence concerning the reliability of the tendons.

Method E: Ensure that if failure were to occur due to either an increase of load or a

reduction of prestress under the frequent combination of actions, cracking would

occur before the ultimate capacity would be exceeded, taking account of moment

redistribution due to cracking effects.

The selection of Methods to be used in a Country may be found in its National Annex.

Note:

5.10.2 Prestressing force during tensioning

5.10.2.1 Maximum stressing force

(1)P The force applied to a tendon, P (i.e. the force at the active end during tensioning) shall

max

not exceed the following value

:

σ

P = A (5.41)

max p p,max

where: is the cross-sectional area of the tendon

A

p

σ is the maximum stress applied to the tendon

p,max · f ; k · f }

= min { k 1 pk 2 p0,1k

k k

The values of and for use in a Country may be found in its National Annex. The recommended

Note: 1 2

k k

values are = 0,8 and = 0,9

1 2 ±

(2) Overstressing is permitted if the force in the jack can be measured to an accuracy of 5 %

of the final value of the prestressing force. In such cases the maximum prestressing force P

max

· f (e.g. for the occurrence of an unexpected high friction in long-line

may be increased to k 3 p0,1k

pretensioning). k

The values of for use in a Country may be found in its National Annex. The recommended value is

Note: 3

0,95.

76 EN 1992-1-1:2004 (E)

5.10.2.2 Limitation of concrete stress

(1)P Local concrete crushing or splitting at the end of pre- and post-tensioned members shall

be avoided.

(2) Local concrete crushing or splitting behind post-tensioning anchors should be avoided in

accordance with the relevant European Technical Approval.

(3) The strength of concrete at application of or transfer of prestress should not be less than

the minimum value defined in the relevant European Technical Approval.

(4) If prestress in an individual tendon is applied in steps, the required concrete strength may

(t) at the time t should be k [%] of the required concrete

be reduced. The minimum strength f cm 4

strength for full prestressing given in the European Technical Approval. Between the minimum

strength and the required concrete strength for full prestressing, the prestress may be

[%] and 100% of the full prestressing.

interpolated between k 5

k k

The values of and for use in a Country may be found in its National Annex. The recommended

Note: 4 5

k k

value for is 50 and for is 30.

4 5

(5) The concrete compressive stress in the structure resulting from the prestressing force and

other loads acting at the time of tensioning or release of prestress, should be limited to:

σ ≤ 0,6 f (t) (5.42)

c ck

where f (t) is the characteristic compressive strength of the concrete at time t when it is

ck

subjected to the prestressing force.

For pretensioned elements the stress at the time of transfer of prestress may be increased to

· f (t), if it can be justified by tests or experience that longitudinal cracking is prevented.

k 6 ck k

The value of for use in a Country may be found in its National Annex. The recommended value is

Note: 6

0,7.

If the compressive stress permanently exceeds 0,45 f (t) the non-linearity of creep should be

ck

taken into account.

5.10.2.3 Measurements

(1)P In post-tensioning the prestressing force and the related elongation of the tendon shall be

checked by measurements and the actual losses due to friction shall be controlled.

5.10.3 Prestress force

(1)P At a given time t and distance x (or arc length) from the active end of the tendon the mean

(x) is equal to the maximum force P imposed at the active end, minus the

prestress force P

m,t max

immediate losses and the time dependent losses (see below). Absolute values are considered

for all the losses. (x) (at time t = t0) applied to the concrete

(2) The value of the initial prestress force P

m0

immediately after tensioning and anchoring (post-tensioning) or after transfer of prestressing

the immediate

(pre-tensioning) is obtained by subtracting from the force at tensioning P

max

∆P

losses (x) and should not exceed the following value:

i 77

EN 1992-1-1:2004 (E)

σ

P (x) = A (x) (5.43)

m0 p pm0

where:

σ (x) is the stress in the tendon immediately after tensioning or transfer

pm0 · f ; k f }

= min { k 7 pk 8 p0,1k

k k

The values of and for use in a Country may be found in its National Annex. The recommended

Note: 7 8

k k

value for is 0,75 and for is 0,85

7 8 ∆P

(3) When determining the immediate losses (x) the following immediate influences should

i

be considered for pre-tensioning and post-tensioning where relevant (see 5.10.4 and 5.10.5):

∆P

losses due to elastic deformation of concrete

- el

∆P

losses due to short term relaxation

- r

∆P

losses due to friction (x)

- µ ∆P

losses due to anchorage slip

- sl >

(4) The mean value of the prestress force P (x) at the time t t0 should be determined with

m,t

respect to the prestressing method. In addition to the immediate losses given in (3) the time-

∆P

dependent losses of prestress (x) (see 5.10.6) as a result of creep and shrinkage of the

c+s+r (x)

concrete and the long term relaxation of the prestressing steel should be considered and P

m,t

∆P

= P (x) - (x).

m0 c+s+r

5.10.4 Immediate losses of prestress for pre-tensioning

(1) The following losses occurring during pre-tensioning should be considered:

(i) during the stressing process: loss due to friction at the bends (in the case of curved wires

or strands) and losses due to wedge draw-in of the anchorage devices.

(ii) before the transfer of prestress to concrete: loss due to relaxation of the pretensioning

tendons during the period which elapses between the tensioning of the tendons and

prestressing of the concrete.

In case of heat curing, losses due to shrinkage and relaxation are modified and should be assessed

Note:

accordingly; direct thermal effect should also be considered (see Annex D)

(iii) at the transfer of prestress to concrete: loss due to elastic deformation of concrete as

the result of the action of pre-tensioned tendons when they are released from the

anchorages.

5.10.5 Immediate losses of prestress for post-tensioning

5.10.5.1 Losses due to the instantaneous deformation of concrete

(1) Account should be taken of the loss in tendon force corresponding to the deformation of

concrete, taking account the order in which the tendons are stressed.

∆P

(2) This loss, , may be assumed as a mean loss in each tendon as follows:

el ( )

⎡ ⎤

σ

j t

∆ = ⋅ ⋅ c (5.44)

P A E ⎢ ⎥

( )

el p p E t

⎣ ⎦

cm

where:

σ

∆ (t) is the variation of stress at the centre of gravity of the tendons applied at time t

c

78 EN 1992-1-1:2004 (E)

j is a coefficient equal to

(n -1)/2n where n is the number of identical tendons successively

prestressed. As an approximation j may be taken as 1/2

1 for the variations due to permanent actions applied after prestressing.

5.10.5.2 Losses due to friction

∆P

(1) The losses due to friction (x) in post-tensioned tendons may be estimated from:

µ (5.45)

µ θ

− +

∆ = − ( k x )

P ( x ) P ( 1 e )

µ max

where:

θ is the sum of the angular displacements over a distance x (irrespective of direction or

sign)

µ is the coefficient of friction between the tendon and its duct

k is an unintentional angular displacement for internal tendons (per unit length)

x is the distance along the tendon from the point where the prestressing force is equal

(the force at the active end during tensioning)

to P

max

µ µ

The values and k are given in the relevant European Technical Approval. The value

depends on the surface characteristics of the tendons and the duct, on the presence of rust, on

the elongation of the tendon and on the tendon profile.

The value k for unintentional angular displacement depends on the quality of workmanship, on

the distance between tendon supports, on the type of duct or sheath employed, and on the

degree of vibration used in placing the concrete. µ

(2) In the absence of data given in a European Technical Approval the values for given in

Table 5.1 may be assumed, when using Expression (5.45).

(3) In the absence of data in a European Technical Approval, values for unintended regular

displacements for internal tendons will generally be in the range 0,005 < k < 0,01 per metre.

(4) For external tendons, the losses of prestress due to unintentional angles may be ignored.

µ

Table 5.1: Coefficients of friction of post-tensioned internal tendons and external

unbonded tendons External unbonded tendons

1)

Internal tendons Steel duct/ non HDPE duct/ non Steel duct/ HDPE duct/

lubricated lubricated lubricated lubricated

Cold drawn wire 0,17 0,25 0,14 0,18 0,12

Strand 0,19 0,24 0,12 0,16 0,10

Deformed bar 0,65 - - - -

Smooth round bar 0,33 - - - -

1) for tendons which fill about half of the duct

HPDE - High density polyethylene

Note:

5.10.5.3 Losses at anchorage

(1) Account should be taken of the losses due to wedge draw-in of the anchorage devices,

during the operation of anchoring after tensioning, and due to the deformation of the anchorage

itself. 79

EN 1992-1-1:2004 (E)

(2) Values of the wedge draw-in are given in the European Technical Approval.

5.10.6 Time dependent losses of prestress for pre- and post-tensioning

(1) The time dependent losses may be calculated by considering the following two reductions

of stress:

(a) due to the reduction of strain, caused by the deformation of concrete due to creep and

shrinkage, under the permanent loads:

(b) the reduction of stress in the steel due to the relaxation under tension.

The relaxation of steel depends on the concrete deformation due to creep and shrinkage. This

Note:

interaction can generally and approximately be taken into account by a reduction factor 0,8.

(2) A simplified method to evaluate time dependent losses at location x under the permanent

loads is given by Expression (5.46). E ϕ σ

+ ∆ + p

ε E σ

0

,

8 ( , ). ,

t t 0 c QP

cs p pr E

∆ = ∆ = cm (5.46)

P A σ A

+ + + +

c s r p p ,

c s r p E A A ϕ

+ + +

p p 2

1 (

1 z ) [

1 0

,

8 ( t , t )]

c cp 0

E A Ι

cm c c

where:

∆σ is the absolute value of the variation of stress in the tendons due to creep,

p,c+s+r shrinkage and relaxation at location x, at time t

ε is the estimated shrinkage strain according to 3.1.4(6) in absolute value

cs is the modulus of elasticity for the prestressing steel, see 3.3.3 (9)

E

p is the modulus of elasticity for the concrete (Table 3.1)

E

cm

σ

∆ is the absolute value of the variation of stress in the tendons at location x, at

pr time t, due to the relaxation of the prestressing steel. It is determined for a

σ σ ψ

= (G+P + Q)

stress of p p m0 2

σ σ ψ

= (G+P + Q) is the initial stress in the tendons due to initial

where p p m0 2

prestress and quasi-permanent actions.

ϕ (t,t ) is the creep coefficient at a time t and load application at time t

0 0

σ is the stress in the concrete adjacent to the tendons, due to self-weight and

c,QP initial prestress and other quasi-permanent actions where relevant. The value

σ may be the effect of part of self-weight and initial prestress or the effect

of c,QP σ ψ

(G+P + Q)), depending on

of a full quasi-permanent combination of action ( c m0 2

the stage of construction considered.

is the area of all the prestressing tendons at the location x

A

p is the area of the concrete section.

A

c

Ι is the second moment of area of the concrete section.

c is the distance between the centre of gravity of the concrete section and the

z cp tendons

Compressive stresses and the corresponding strains given in Expression (5.46) should be used

with a positive sign.

(3) Expression (5.46) applies for bonded tendons when local values of stresses are used and

for unbonded tendons when mean values of stresses are used. The mean values should be

calculated between straight sections limited by the idealised deviation points for external

tendons or along the entire length in case of internal tendons.

80 EN 1992-1-1:2004 (E)

5.10.7 Consideration of prestress in analysis

(1) Second order moments can arise from prestressing with external tendons.

(2) Moments from secondary effects of prestressing arise only in statically indeterminate

structures.

(3) For linear analysis both the primary and secondary effects of prestressing should be applied

before any redistribution of forces and moments is considered (see 5.5).

(4) In plastic and non-linear analysis the secondary effect of prestress may be treated as

additional plastic rotations which should then be included in the check of rotation capacity.

(5) Rigid bond between steel and concrete may be assumed after grouting of post-tensioned

tendons. However before grouting the tendons should be considered as unbonded.

(6) External tendons may be assumed to be straight between deviators.

5.10.8 Effects of prestressing at ultimate limit state

(1) In general the design value of the prestressing force may be determined by P (x) =

d,t

γ γ

P (x) (see 5.10.3 (4) for the definition of P (x)) and 2.4.2.2 for .

P, m,t m,t p

(2) For prestressed members with permanently unbonded tendons, it is generally necessary to

take the deformation of the whole member into account when calculating the increase of the

stress in the prestressing steel. If no detailed calculation is made, it may be assumed that the

increase of the stress from the effective prestress to the stress in the ultimate limit state is

σ

∆ .

p,ULS σ

The value of for use in a Country may be found in its National Annex. The recommended value

Note: p,ULS

is 100 MPa.

(3) If the stress increase is calculated using the deformation state of the whole member the

mean values of the material properties should be used. The design value of the stress increase

σ σ γ γ γ

∆ ⋅

∆ = should be determined by applying partial safety factors and

∆P ∆P,sup ∆P,inf

pd p

respectively. γ γ

The values of and for use in a Country may be found in its National Annex. The

Note: ∆P,sup ∆P,inf

γ γ

recommended values for and are 1,2 and 0,8 respectively. If linear analysis with uncracked

∆P,sup ∆P,inf γ

sections is applied, a lower limit of deformations may be assumed and the recommended value for both ∆P,sup

γ

and is 1,0.

∆P,inf

5.10.9 Effects of prestressing at serviceability limit state and limit state of fatigue

(1)P For serviceability and fatigue calculations allowance shall be made for possible variations

in prestress. Two characteristic values of the prestressing force at the serviceability limit state

are estimated from:

P = r P (x) (5.47)

k,sup sup m,t

P = r P (x) (5.48)

k,inf inf m,t

where: is the upper characteristic value

P

k,sup is the lower characteristic value

P

k,inf 81

EN 1992-1-1:2004 (E)

r r

The values of and for use in a Country may be found in its National Annex. The recommended

Note: sup inf

values are: r r

- for pre-tensioning or unbonded tendons: = 1,05 and = 0,95

sup inf

r r

- for post-tensioning with bonded tendons: = 1,10 and = 0,90

sup inf r r

- when appropriate measures (e.g. direct measurements of pretensioning) are taken: = = 1,0.

sup inf

5.11 Analysis for some particular structural members

(1)P Slabs supported on columns are defined as flat slabs.

(2)P Shear walls are plain or reinforced concrete walls that contribute to lateral stability of the

structure.

For information concerning the analysis of flat slabs and shear walls see Annex I.

Note:

82 EN 1992-1-1:2004 (E)

SECTION 6 ULTIMATE LIMIT STATES (ULS)

6.1 Bending with or without axial force

(1)P This section applies to undisturbed regions of beams, slabs and similar types of members

for which sections remain approximately plane before and after loading. The discontinuity

regions of beams and other members in which plane sections do not remain plane may be

designed and detailed according to 6.5.

(2)P When determining the ultimate moment resistance of reinforced or prestressed concrete

cross-sections, the following assumptions are made:

- plane sections remain plane.

- the strain in bonded reinforcement or bonded prestressing tendons, whether in tension or

in compression, is the same as that in the surrounding concrete.

- the tensile strength of the concrete is ignored.

- the stresses in the concrete in compression are derived from the design stress/strain

relationship given in 3.1.7.

- the stresses in the reinforcing or prestressing steel are derived from the design curves in

3.2 (Figure 3.8) and 3.3 (Figure 3.10).

- the initial strain in prestressing tendons is taken into account when assessing the

stresses in the tendons. ε ε

(3)P The compressive strain in the concrete shall be limited to , or , depending on the

cu2 cu3

stress-strain diagram used, see 3.1.7 and Table 3.1. The strains in the reinforcing steel and the

ε

prestressing steel shall be limited to (where applicable); see 3.2.7 (2) and 3.3.6 (7)

ud

respectively.

(4) For cross-sections with symmetrical reinforcement loaded by the compression force it is

necessary to assume the minimum eccentricity, e = h/30 but not less than 20 mm where h is

0

the depth of the section.

(5) In parts of cross-sections which are subjected to approximately concentric loading (e/h <

0,1), such as compression flanges of box girders, the mean compressive strain in that part of

ε ε

(or if the bilinear relation of Figure 3.4 is used).

the section should be limited to c2 c3

(6) The possible range of strain distributions is shown in Figure 6.1.

(7) For prestressed members with permanently unbonded tendons see 5.10.8.

(8) For external prestressing tendons the strain in the prestressing steel between two

subsequent contact points (anchors or deviation saddles) is assumed to be constant. The strain

in the prestressing steel is then equal to the initial strain, realised just after completion of the

prestressing operation, increased by the strain resulting from the structural deformation

between the contact areas considered. See also 5.10. 83

EN 1992-1-1:2004 (E) ε ε

(1- / )h

c2 cu2

or

ε ε

(1- / )h

c3 cu3 B

A s2

d C

h ∆ε ε

p p(0)

A A

p

A

s1

ε ε ε

s , p c

ε

ε ε ε

0

y

ud c2 cu2

ε

ε ( )

( ) cu3

c3

A - reinforcing steel tension strain limit

B - concrete compression strain limit

C - concrete pure compression strain limit

Figure 6.1: Possible strain distributions in the ultimate limit state

6.2 Shear

6.2.1 General verification procedure

(1)P For the verification of the shear resistance the following symbols are defined:

is the design shear resistance of the member without shear reinforcement.

V

Rd,c

V is the design value of the shear force which can be sustained by the yielding shear

Rd,s reinforcement.

is the design value of the maximum shear force which can be sustained by the

V

Rd,max member, limited by crushing of the compression struts.

In members with inclined chords the following additional values are defined (see Figure 6.2):

V is the design value of the shear component of the force in the compression area, in

ccd the case of an inclined compression chord.

V is the design value of the shear component of the force in the tensile reinforcement,

td in the case of an inclined tensile chord.

V

ccd

V

td

Figure 6.2: Shear component for members with inclined chords

84 EN 1992-1-1:2004 (E)

(2) The shear resistance of a member with shear reinforcement is equal to:

V = V + V + V (6.1)

Rd Rd,s ccd td ≤V no calculated shear reinforcement is necessary.

(3) In regions of the member where V

Ed Rd,c

V is the design shear force in the section considered resulting from external loading and

Ed

prestressing (bonded or unbonded).

(4) When, on the basis of the design shear calculation, no shear reinforcement is required,

minimum shear reinforcement should nevertheless be provided according to 9.2.2. The

minimum shear reinforcement may be omitted in members such as slabs (solid, ribbed or

hollow core slabs) where transverse redistribution of loads is possible. Minimum reinforcement

may also be omitted in members of minor importance (e.g. lintels with span 2 m) which do not

contribute significantly to the overall resistance and stability of the structure.

(5) In regions where V > V according to Expression (6.2), sufficient shear reinforcement

Ed Rd,c ≤

should be provided in order that V V (see Expression (6.8)).

Ed Rd - V - V ,

(6) The sum of the design shear force and the contributions of the flanges, V

Ed ccd td

should not exceed the permitted maximum value V (see 6.2.3), anywhere in the member.

Rd,max

(7) The longitudinal tension reinforcement should be able to resist the additional tensile force

caused by shear (see 6.2.3 (7)).

(8) For members subject to predominantly uniformly distributed loading the design shear force

need not to be checked at a distance less than d from the face of the support. Any shear

reinforcement required should continue to the support. In addition it should be verified that the

shear at the support does not exceed V (see also 6.2.2 (6) and 6.2.3 (8).

Rd,max

(9) Where a load is applied near the bottom of a section, sufficient vertical reinforcement to

carry the load to the top of the section should be provided in addition to any reinforcement

required to resist shear.

6.2.2 Members not requiring design shear reinforcement

is given by:

(1) The design value for the shear resistance V

Rd,c

ρ σ

1/3

V = [C k(100 f ) + k ] b d (6.2.a)

Rd,c Rd,c l ck 1 cp w

with a minimum of

σ

V = (v + k ) b d (6.2.b)

Rd,c min 1 cp w

where:

f is in MPa

ck 200

+ ≤

k = 1 2

,

0 with d in mm

d

A sl

ρ ≤ 0

,

02

=

l b d

w ≥

A is the area of the tensile reinforcement, which extends (l + d) beyond the

sl bd

section considered (see Figure 6.3). 85

EN 1992-1-1:2004 (E)

b is the smallest width of the cross-section in the tensile area [mm]

w

σ = N /A < 0,2 f [MPa]

cp Ed c cd

is the axial force in the cross-section due to loading or prestressing [in N] (N >0

N

Ed Ed

may be ignored.

for compression). The influence of imposed deformations on N

E

2

is the area of concrete cross section [mm ]

A

C is [N]

V

Rd,c

The values of C , v and k for use in a Country may be found in its National Annex. The

Note: Rd,c min 1

γ

is 0,18/ , that for v is given by Expression (6.3N) and that for k is 0,15.

recommended value for C Rd,c c min 1

3/2 ck1/2

v =0,035 k f (6.3N)

min l

l A

A

V bd

bd V sl

Ed Ed

o

o 45

45

d d

o

45 V

A

A A l

A Ed

sl

sl bd

A - section considered

Figure 6.3: Definition of A in Expression (6.2)

sl

(2) In prestressed single span members without shear reinforcement, the shear resistance of

the regions cracked in bending may be calculated using Expression (6.2a). In regions

γ

/ ) the shear

uncracked in bending (where the flexural tensile stress is smaller than f ctk,0,05 c

resistance should be limited by the tensile strength of the concrete. In these regions the shear

resistance is given by:

Ι ⋅ b ( )

2 α σ

= +

w

V f f (6.4)

Rd,c ctd l cp ctd

S

where

Ι is the second moment of area

b is the width of the cross-section at the centroidal axis, allowing for the presence of

w ducts in accordance with Expressions (6.16) and (6.17)

S is the first moment of area above and about the centroidal axis

α ≤

l l

= / 1,0 for pretensioned tendons

I x pt2

= 1,0 for other types of prestressing

l is the distance of section considered from the starting point of the transmission

x length

l is the upper bound value of the transmission length of the prestressing element

pt2 according to Expression (8.18).

σ is the concrete compressive stress at the centroidal axis due to axial loading

cp σ N A N

and/or prestressing ( = / in MPa, > 0 in compression)

cp Ed c Ed

For cross-sections where the width varies over the height, the maximum principal stress may

occur on an axis other than the centroidal axis. In such a case the minimum value of the shear

at various axes in the cross-section.

resistance should be found by calculating V Rd,c

86 EN 1992-1-1:2004 (E)

(3) The calculation of the shear resistance according to Expression (6.4) is not required for

cross-sections that are nearer to the support than the point which is the intersection of the o .

elastic centroidal axis and a line inclined from the inner edge of the support at an angle of 45

(4) For the general case of members subjected to a bending moment and an axial force, which

can be shown to be uncracked in flexure at the ULS, reference is made to 12.6.3. M

(5) For the design of the longitudinal reinforcement, in the region cracked in flexure, the -

Ed

a d

= in the unfavourable direction (see 9.2.1.3 (2)).

line should be shifted over a distance l ≤ ≤

d a d

(6) For members with loads applied on the upper side within a distance 0,5 2 from

v

the edge of a support (or centre of bearing where flexible bearings are used), the contribution

β

V a / d.

may be multiplied by = 2 This reduction may be

of this load to the shear force Ed v

V in Expression (6.2.a). This is only valid provided that the

applied for checking Rd,c ≤

a d a d

longitudinal reinforcement is fully anchored at the support. For 0,5 the value = 0,5

v v

should be used. β

V , calculated without reduction by , should however always satisfy the

The shear force Ed

condition ν

V b d f

0,5 (6.5)

Ed w cd

ν

where is a strength reduction factor for concrete cracked in shear

ν

The value for use in a Country may be found in its National Annex. The recommended value follows

Note:

from: ⎡ ⎤

f

ν = − ck

0

,

6 1 (f in MPa) (6.6N)

⎢ ⎥⎦ ck

250

⎣ a

v

d d

a

v

(a) Beam with direct support (b) Corbel

Figure 6.4: Loads near supports

(7) Beams with loads near to supports and corbels may alternatively be designed with strut and

tie models. For this alternative, reference is made to 6.5. 87

EN 1992-1-1:2004 (E)

6.2.3 Members requiring design shear reinforcement

(1) The design of members with shear reinforcement is based on a truss model (Figure 6.5).

θ

Limiting values for the angle of the inclined struts in the web are given in 6.2.3 (2).

In Figure 6.5 the following notations are shown:

α is the angle between shear reinforcement and the beam axis perpendicular to the

shear force (measured positive as shown in Figure 6.5)

θ is the angle between the concrete compression strut and the beam axis

perpendicular to the shear force

F is the design value of the tensile force in the longitudinal reinforcement

td

F is the design value of the concrete compression force in the direction of the

cd longitudinal member axis.

b is the minimum width between tension and compression chords

w

z is the inner lever arm, for a member with constant depth, corresponding to the

bending moment in the element under consideration. In the shear analysis of

z d

= 0,9 may normally

reinforced concrete without axial force, the approximate value

be used.

In elements with inclined prestressing tendons, longitudinal reinforcement at the tensile chord

should be provided to carry the longitudinal tensile force due to shear defined in (3).

A B θ α )

V(cot - cot

F

cd M

α z

½ N

θ

d 0.9d

z =

V z

½ V

F

td

C

s

D

A - compression chord, B - struts, C - tensile chord, D - shear reinforcement

b b

w w

Figure 6.5: Truss model and notation for shear reinforced members

θ

(2) The angle should be limited.

θ

The limiting values of cot for use in a Country may be found in its National Annex. The recommended

Note:

limits are given in Expression (6.7N).

θ

≤ ≤

1 cot 2,5 (6.7N)

88 EN 1992-1-1:2004 (E)

V

(3) For members with vertical shear reinforcement, the shear resistance, is the smaller

Rd

value of: A θ

= sw

V z f cot (6.8)

Rd,s ywd

s

If Expression (6.10) is used the value of f should be reduced to 0,8 f in Expression (6.8)

Note: ywd ywk

and α ν θ θ

V b z f

= /(cot + tan ) (6.9)

Rd,max cw w 1 cd

where:

A is the cross-sectional area of the shear reinforcement

sw

s is the spacing of the stirrups

f is the design yield strength of the shear reinforcement

ywd

ν is a strength reduction factor for concrete cracked in shear

1

α is a coefficient taking account of the state of the stress in the compression chord

cw ν α

The value of and for use in a Country may be found in its National Annex. The recommended

Note 1: 1 cw

ν ν

is (see Expression (6.6N)).

value of 1 ν

If the design stress of the shear reinforcement is below 80% of the characteristic yield stress f ,

Note 2: yk 1

may be taken as:

ν = 0,6 for f ≤ 60 MPa (6.10.aN)

1 ck

ν = 0,9 – f /200 > 0,5 for f ≥ 60 MPa (6.10.bN)

1 ck ck α

The recommended value of is as follows:

Note 3: cw

1 for non-prestressed structures

σ σ ≤

(1 + /f ) for 0 < 0,25 f (6.11.aN)

cp cd cp cd

σ ≤

< 0,5 f (6.11.bN)

1,25 for 0,25 f

cd cp cd

σ σ

/f ) for 0,5 f < < 1,0 f (6.11.cN)

2,5 (1 - cp cd cd cp cd

where:

σ is the mean compressive stress, measured positive, in the concrete due to the design axial force.

cp

This should be obtained by averaging it over the concrete section taking account of the reinforcement.

σ θ

The value of need not be calculated at a distance less than 0.5d cot from the edge of the support.

cp θ

The maximum effective cross-sectional area of the shear reinforcement, A , for cot =1 is given

Note 4: sw,max

by:

A f α ν

sw,max ywd ≤ f

1 (6.12)

1

cw cd

2

b s

w

(4) For members with inclined shear reinforcement, the shear resistance is the smaller value of

A θ α α

= +

sw

V z f (cot cot ) sin (6.13)

Rd,s ywd

s

and α ν θ α θ

= + + 2

V b z f (cot cot )/(1 cot ) (6.14)

Rd,max cw w 1 cd θ

The maximum effective shear reinforcement, A for cot =1 follows from:

Note: sw,max

α ν

A f f

1

sw,max ywd ≤ cw 1 cd

2 (6.15)

α

b s sin

w 89

EN 1992-1-1:2004 (E) V

(5) In regions where there is no discontinuity of (e.g. for uniformly distributed loading) the

Ed θ α

l z +

shear reinforcement in any length increment = (cot cot ) may be calculated using the

V in the increment.

smallest value of Ed φ b V

(6) Where the web contains grouted ducts with a diameter > /8 the shear resistance

w Rd,max

should be calculated on the basis of a nominal web thickness given by:

φ

Σ

b b

= - 0,5 (6.16)

w,nom w

φ φ

Σ

where is the outer diameter of the duct and is determined for the most unfavourable

level. φ ≤ b b b

For grouted metal ducts with /8, =

w w,nom w

For non-grouted ducts, grouted plastic ducts and unbonded tendons the nominal web thickness

is: φ

Σ

b b

= - 1,2 (6.17)

w,nom w

The value 1,2 in Expression (6.17) is introduced to take account of splitting of the concrete

struts due to transverse tension. If adequate transverse reinforcement is provided this value

may be reduced to 1,0. ∆F V

(7) The additional tensile force, , in the longitudinal reinforcement due to shear may be

td Ed

calculated from: θ α

∆F V

= 0,5 (cot - cot ) (6.18)

td Ed

∆F

M z M z M

( / ) + should be taken not greater than / , where is the maximum

Ed td Ed,max Ed,max

moment along the beam. ≤ ≤

d a d

(8) For members with loads applied on the upper side within a distance 0,5 2,0 the

v

β

V a d

contribution of this load to the shear force may be reduced by = /2 .

Ed v

V , calculated in this way, should satisfy the condition

The shear force Ed

α

≤ ⋅f

V A sin (6.19)

Ed sw ywd

⋅f

A

where is the resistance of the shear reinforcement crossing the inclined shear crack

sw ywd

between the loaded areas (see Figure 6.6). Only the shear reinforcement within the central

β

a

0,75 should be taken into account. The reduction by should only be applied for

v

calculating the shear reinforcement. It is only valid provided that the longitudinal

reinforcement is fully anchored at the support. 0,75a

0,75a v

v α α

a v a v

Figure 6.6: Shear reinforcement in short shear spans with direct strut action

90 EN 1992-1-1:2004 (E)

a d a d

For < 0,5 the value = 0,5 should be used.

v v β

V

The value calculated without reduction by , should however always satisfy Expression

Ed

(6.5).

6.2.4 Shear between web and flanges of T-sections

(1) The shear strength of the flange may be calculated by considering the flange as a system

of compressive struts combined with ties in the form of tensile reinforcement.

(2) A minimum amount of longitudinal reinforcement should be provided, as specified in 9.3.1.

v

(3) The longitudinal shear stress, at the junction between one side of a flange and the web

Ed,

is determined by the change of the normal (longitudinal) force in the part of the flange

considered, according to:

∆F ⋅ ∆x

v h

= /( ) (6.20)

Ed d f

where:

h is the thickness of flange at the junctions

f

∆x is the length under consideration, see Figure 6.7

∆F ∆x

is the change of the normal force in the flange over the length .

d A

F d

b

F eff

d ∆x

s

θ f

f

A A h f

B ∆F

F +

d d

A sf ∆F

F +

d d b w

A - compressive struts B - longitudinal bar anchored beyond this projected point

(see 6.2.4 (7))

Figure 6.7: Notations for the connection between flange and web

∆x

The maximum value that may be assumed for is half the distance between the section

where the moment is 0 and the section where the moment is maximum. Where point loads are

∆x should not exceed the distance between point loads.

applied the length A s

(4) The transverse reinforcement per unit length / may be determined as follows:

sf f

θ

≥ ⋅

A f s v h

( / ) / cot (6.21)

sf yd f Ed f f 91

EN 1992-1-1:2004 (E)

To prevent crushing of the compression struts in the flange, the following condition should be

satisfied:

ν θ θ

v f sin cos (6.22)

Ed cd f f θ

The permitted range of the values for cot for use in a country may be found in its National Annex.

Note: f

The recommended values in the absence of more rigorous calculation are:

θ θ

≤ ≥ ≥

≤ 2,0 for compression flanges (45° 26,5°)

1,0 cot f f

θ θ

≤ ≤ ≥ ≥

1,0 cot 1,25 for tension flanges (45° 38,6°)

f f

(5) In the case of combined shear between the flange and the web, and transverse bending,

the area of steel should be the greater than that given by Expression (6.21) or half that given by

Expression (6.21) plus that required for transverse bending.

v kf

(6) If is less than or equal to no extra reinforcement above that for flexure is required.

Ed ctd

The value of k for use in a Country may be found in its National Annex. The recommended value is 0,4.

Note:

(7) Longitudinal tension reinforcement in the flange should be anchored beyond the strut

required to transmit the force back to the web at the section where this reinforcement is

required (See Section (A - A) of Figure 6.7).

6.2.5 Shear at the interface between concrete cast at different times

(1) In addition to the requirements of 6.2.1- 6.2.4 the shear stress at the interface between

concrete cast at different times should also satisfy the following:

v v

≤ (6.23)

Edi Rdi

v is the design value of the shear stress in the interface and is given by:

Edi β

v V z b

= / ( ) (6.24)

Edi Ed i

where:

β is the ratio of the longitudinal force in the new concrete area and the total

longitudinal force either in the compression or tension zone, both calculated for the

section considered

V is the transverse shear force

Ed

z is the lever arm of composite section

b is the width of the interface (see Figure 6.8)

i

v is the design shear resistance at the interface and is given by:

Rdi µ σ ρ µ α α ν

v c f f f

= + + ( sin + cos ) ≤ 0,5 (6.25)

Rdi ctd n yd cd

where: µ

c and are factors which depend on the roughness of the interface (see (2))

f is as defined in 3.1.6 (2)P

ctd

σ stress per unit area caused by the minimum external normal force across the

n interface that can act simultaneously with the shear force, positive for

σ σ

f c

< 0,6 , and negative for tension. When is tensile

compression, such that n cd n

f should be taken as 0.

ctd

ρ A A

= /

s i

92 EN 1992-1-1:2004 (E)

b i

b i

b i

Figure 6.8: Examples of interfaces

A is the area of reinforcement crossing the interface, including ordinary shear

s reinforcement (if any), with adequate anchorage at both sides of the interface.

A is the area of the joint

i

α α

° ≤ ≤ °

is defined in Figure 6.9, and should be limited by 45 90

ν is a strength reduction factor (see 6.2.2 (6))

α

≤ ≤

45 90 N

h 10 d Ed

2 C

A α V Ed d 5 mm

h 10 d

1 V

B C Ed

≤ 30

A - new concrete, B - old concrete, C - anchorage

Figure 6.9: Indented construction joint

(2) In the absence of more detailed information surfaces may be classified as very smooth,

smooth, rough or indented, with the following examples:

Very smooth: a surface cast against steel, plastic or specially prepared wooden moulds:

- µ

c = 0,25 and = 0,5

Smooth: a slipformed or extruded surface, or a free surface left without further treatment

- µ

c

after vibration: = 0,35 and = 0,6

Rough: a surface with at least 3 mm roughness at about 40 mm spacing, achieved by

- c

raking, exposing of aggregate or other methods giving an equivalent behaviour: = 0,45

µ

and = 0,7 µ

c

- Indented: a surface with indentations complying with Figure 6.9: = 0,50 and = 0,9

(3) A stepped distribution of the transverse reinforcement may be used, as indicated in Figure

6.10. Where the connection between the two different concretes is ensured by reinforcement 93

EN 1992-1-1:2004 (E) v

(beams with lattice girders), the steel contribution to may be taken as the resultant of the

Rdi α

° ≤ ≤ °

135 .

forces taken from each of the diagonals provided that 45

(4) The longitudinal shear resistance of grouted joints between slab or wall elements may be

calculated according to 6.2.5 (1). However in cases where the joint can be significantly cracked,

c should be taken as 0 for smooth and rough joints and 0,5 for indented joints (see also 10.9.3

(12)). c

(5) Under fatigue or dynamic loads, the values for in 6.2.5 (1) should be halved.

ρ µ α α

f ( )

sin + cos

v yd

Edi µ σ

c f +

ctd n

Figure 6.10: Shear diagram representing the required interface reinforcement

6.3 Torsion

6.3.1 General

(1)P Where the static equilibrium of a structure depends on the torsional resistance of

elements of the structure, a full torsional design covering both ultimate and serviceability limit

states shall be made.

(2) Where, in statically indeterminate structures, torsion arises from consideration of

compatibility only, and the structure is not dependent on the torsional resistance for its stability,

then it will normally be unnecessary to consider torsion at the ultimate limit state. In such cases

a minimum reinforcement, given in Sections 7.3 and 9.2, in the form of stirrups and longitudinal

bars should be provided in order to prevent excessive cracking.

(3) The torsional resistance of a section may be calculated on the basis of a thin-walled closed

section, in which equilibrium is satisfied by a closed shear flow. Solid sections may be modelled

by equivalent thin-walled sections. Complex shapes, such as T-sections, may be divided into a

series of sub-sections, each of which is modelled as an equivalent thin-walled section, and the

total torsional resistance taken as the sum of the capacities of the individual elements.

(4) The distribution of the acting torsional moments over the sub-sections should be in

proportion to their uncracked torsional stiffnesses. For non-solid sections the equivalent wall

thickness should not exceed the actual wall thickness.

(5) Each sub-section may be designed separately.

94 EN 1992-1-1:2004 (E)

6.3.2 Design procedure

(1) The shear stress in a wall of a section subject to a pure torsional moment may be

calculated from:

T

τ = Ed

t (6.26)

t,i ef,i 2A

k V i

The shear force in a wall due to torsion is given by:

Ed,i

τ

=

V t z (6.27)

Ed,i t,i ef,i i

where

T is the applied design torsion (see Figure 6.11)

Ed A z

i A - centre-line

C B

T B - outer edge of effective cross-

Ed u

section, circumference ,

t /2

ef C - cover

t

ef

Figure 6.11: Notations and definitions used in Section 6.3

A is the area enclosed by the centre-lines of the connecting walls, including inner

k hollow areas.

τ i

is the torsional shear stress in wall

t,i

t A u,

is the effective wall thickness. It may be taken as / but should not be taken as

ef,i less than twice the distance between edge and centre of the longitudinal

reinforcement. For hollow sections the real thickness is an upper limit

A is the total area of the cross-section within the outer circumference, including inner

hollow areas

u is the outer circumference of the cross-section

z i

is the side length of wall defined by the distance between the

i intersection points with the adjacent walls

(2) The effects of torsion and shear for both hollow and solid members may be superimposed,

θ θ

. The limits for given in 6.2.3 (2) are also

assuming the same value for the strut inclination

fully applicable for the case of combined shear and torsion.

The maximum bearing capacity of a member loaded in shear and torsion follows from 6.3.2 (4).

ΣA

(3) The required cross-sectional area of the longitudinal reinforcement for torsion may be

sl

calculated from Expression (6.28): 95

EN 1992-1-1:2004 (E)

∑ A f T

sl yd θ

= Ed cot (6.28)

u A

2

k k

where

u A

is the perimeter of the area

k k

f A

is the design yield stress of the longitudinal reinforcement

yd sl

θ is the angle of compression struts (see Figure 6.5).

In compressive chords, the longitudinal reinforcement may be reduced in proportion to the

available compressive force. In tensile chords the longitudinal reinforcement for torsion should

be added to the other reinforcement. The longitudinal reinforcement should generally be

z

distributed over the length of side, , but for smaller sections it may be concentrated at the ends

i

of this length.

(4) The maximum resistance of a member subjected to torsion and shear is limited by the

capacity of the concrete struts. In order not to exceed this resistance the following condition

should be satisfied: ≤

T T V V

/ + / 1,0 (6.29)

Ed Rd,max Ed Rd,max

where:

T is the design torsional moment

Ed

V is the design transverse force

Ed

T is the design torsional resistance moment according to

Rd,max ν α θ θ

=

T f A t

2 sin cos (6.30)

Rd,max cw cd k ef,i

ν α

where follows from 6.2.2 (6) and from Expression (6.9)

c

V is the maximum design shear resistance according to Expressions (6.9) or

Rd,max (6.14). In solid cross sections the full width of the web may be used to determine

V

Rd,max

(5) For approximately rectangular solid sections only minimum reinforcement is required (see

9.2.1.1) provided that the following condition is satisfied:

T T V V

/ + / 1,0 (6.31)

Ed Rd,c Ed Rd,c

where τ

T f

is the torsional cracking moment, which may be determined by setting =

Rd,c t,i ctd

V follows from Expression (6.2)

Rd, c

6.3.3 Warping torsion

(1) For closed thin-walled sections and solid sections, warping torsion may normally be

ignored.

(2) In open thin walled members it may be necessary to consider warping torsion. For very

slender cross-sections the calculation should be carried out on the basis of a beam-grid model

and for other cases on the basis of a truss model. In all cases the design should be carried out

according to the design rules for bending and longitudinal normal force, and for shear.

96 EN 1992-1-1:2004 (E)

6.4 Punching

6.4.1 General

(1)P The rules in this Section complement those given in 6.2 and cover punching shear in solid

slabs, waffle slabs with solid areas over columns, and foundations.

(2)P Punching shear can result from a concentrated load or reaction acting on a relatively small

A

area, called the loaded area of a slab or a foundation.

load

(3) An appropriate verification model for checking punching failure at the ultimate limit state is

shown in Figure 6.12. h

d

θ

θ A

2d

θ = arctan (1/2) A - basic control

= 26,6° section

c

a) Section

B D A

B - basic control area cont u

C - basic control perimeter, 1

2d A

D - loaded area load

r further control perimeter

cont

r

cont C

b) Plan

Figure 6.12: Verification model for punching shear at the ultimate limit state

(4) The shear resistance should be checked at the face of the column and at the basic control

perimeter u . If shear reinforcement is required a further perimeter u should be found where

1 out,ef

shear reinforcement is no longer required. 97

EN 1992-1-1:2004 (E)

(5) The rules given in 6.4 are principally formulated for the case of uniformly distributed

loading. In special cases, such as footings, the load within the control perimeter adds to the

resistance of the structural system, and may be subtracted when determining the design

punching shear stress.

6.4.2 Load distribution and basic control perimeter

u d

(1) The basic control perimeter may normally be taken to be at a distance 2,0 from the

1

loaded area and should be constructed so as to minimise its length (see Figure 6.13).

The effective depth of the slab is assumed constant and may normally be taken as:

( )

+

d d

y z

=

d (6.32)

eff 2

where d and d are the effective depths of the reinforcement in two orthogonal directions.

y z 2d 2d

2d u u

1 1

u 1 2d

b

z b

y

Figure 6.13: Typical basic control perimeters around loaded areas

d

(2) Control perimeters at a distance less than 2 should be considered where the concentrated

force is opposed by a high pressure (e.g. soil pressure on a base), or by the effects of a load or

d

reaction within a distance 2 of the periphery of area of application of the force.

(3) For loaded areas situated near openings, if the shortest distance between the perimeter of

d

the loaded area and the edge of the opening does not exceed 6 , that part of the control

perimeter contained between two tangents drawn to the outline of the opening from the centre

of the loaded area is considered to be ineffective (see Figure 6.14).

2d l l

6 d l > l

1 2 1 2

l √ (l .l )

2 1 2 A - opening

A

Figure 6.14: Control perimeter near an opening

(4) For a loaded area situated near an edge or a corner, the control perimeter should be taken

as shown in Figure 6.15, if this gives a perimeter (excluding the unsupported edges) smaller

98 EN 1992-1-1:2004 (E)

than that obtained from (1) and (2) above. u

1

2d

2d 2d

u

u 1

1 2d

2d

2d

Figure 6.15: Basic control perimeters for loaded areas close to or at edge or corner

d

(5) For loaded areas situated near an edge or corner, i.e. at a distance smaller than , special

edge reinforcement should always be provided, see 9.3.1.4.

(6) The control section is that which follows the control perimeter and extends over the

d

effective depth . For slabs of constant depth, the control section is perpendicular to the middle

plane of the slab. For slabs or footings of variable depth other than step footings, the effective

depth may be assumed to be the depth at the perimeter of the loaded area as shown in Figure

6.16. A A - loaded area

d

θ θ ≥ arctan (1/2)

Figure 6.16: Depth of control section in a footing with variable depth

u

(7) Further perimeters, , inside and outside the basic control area should have the same

i

shape as the basic control perimeter. l h

(8) For slabs with circular column heads for which < 2 (see Figure 6.17) a check of the

H H

punching shear stresses according to 6.4.3 is only required on the control section outside the

r

column head. The distance of this section from the centroid of the column may be taken as:

cont

r d l c

= 2 + + 0,5 (6.33)

cont H

where:

l is the distance from the column face to the edge of the column head

H

c is the diameter of a circular column 99

EN 1992-1-1:2004 (E) r

r cont

cont A

θ θ d

h h

θ θ

H H A - basic control section

θ = arctan (1/2) B - loaded area A

B

= 26,6° load

c l < 2,0 h

l < 2,0 h H H

H H

Figure 6.17: Slab with enlarged column head where l < 2,0 h

H H

l h

< 2,0 (see Figure 6.17) and overall

For a rectangular column with a rectangular head with H H

l l l c l l c l l l r

dimensions and ( = + 2 , = + 2 , ≤ ), the value may be taken as the lesser

1 2 1 1 H1 2 2 H2 1 2 cont

of: r d l l

= 2 + 0,56 (6.34)

cont 1 2

and

r d I

= 2 + 0,69 (6.35)

cont 1 l h

(9) For slabs with enlarged column heads where > 2 (see Figure 6.18) control sections

H H

both within the head and in the slab should be checked. d

(10) The provisions of 6.4.2 and 6.4.3 also apply for checks within the column head with

d

taken as according to Figure 6.18.

H

(11) For circular columns the distances from the centroid of the column to the control sections

in Figure 6.18 may be taken as:

r l d c

= + 2 + 0,5 (6.36)

cont,ext H

r d h c

= 2( + ) +0,5 (6.37)

cont,int H r r

cont,ext cont,ext

r r

cont,int cont,int θ

θ d

d d

d H

H

h h

θ

θ

H H A - basic control

A

B sections for

circular columns

θ = 26,6° c A

B - loaded area

l > 2(d + h ) l > 2(d + h ) load

H H

H H

Figure 6.18: Slab with enlarged column head where l > 2(d + h )

H H

100 EN 1992-1-1:2004 (E)

6.4.3 Punching shear calculation

(1)P The design procedure for punching shear is based on checks at the face of the column

u u

and at the basic control perimeter . If shear reinforcement is required a further perimeter

1 out,ef

(see figure 6.22) should be found where shear reinforcement is no longer required. The

following design shear stresses (MPa) along the control sections, are defined:

v is the design value of the punching shear resistance of a slab without punching shear

Rd,c reinforcement along the control section considered.

v is the design value of the punching shear resistance of a slab with punching shear

Rd,cs reinforcement along the control section considered.

v is the design value of the maximum punching shear resistance along the control

Rd,max section considered.

(2) The following checks should be carried out:

(a) At the column perimeter, or the perimeter of the loaded area, the maximum punching

shear stress should not be exceeded:

v v

<

Ed Rd,max

(b) Punching shear reinforcement is not necessary if:

v v

<

Ed Rd,c v v

(c) Where exceeds the value for the control section considered, punching shear

Ed Rd,c

reinforcement should be provided according to 6.4.5.

(3) Where the support reaction is eccentric with regard to the control perimeter, the maximum

shear stress should be taken as:

V

β

= Ed

v (6.38)

Ed u d

i

where

d d d

is the mean effective depth of the slab, which may be taken as ( + )/2 where:

y z

d d

, is the effective depths in the y- and z- directions of the control section

y z

u is the length of the control perimeter being considered

i

β is given by:

M u

β = + ⋅

Ed 1

1 k (6.39)

V W

Ed 1

where

u is the length of the basic control perimeter

1

k c c

is a coefficient dependent on the ratio between the column dimensions and : its

1 2

value is a function of the proportions of the unbalanced moment transmitted by

uneven shear and by bending and torsion (see Table 6.1).

W corresponds to a distribution of shear as illustrated in Figure 6.19 and is a function of

1 u

the basic control perimeter :

1

u i

= (6.40)

W e dl

1 0 101

EN 1992-1-1:2004 (E)

l

d is a length increment of the perimeter

e l M

is the distance of d from the axis about which the moment acts

Ed

Table 6.1: Values of k for rectangular loaded areas

≤ ≥

c c 1,0 2,0

/ 0,5 3,0

1 2 0,45 0,60 0,70 0,80

k 2d

c

1 c 2d

2

Figure 6.19: Shear distribution due to an unbalanced moment at a slab-internal

column connection

For a rectangular column:

2

c π

= + + + +

2

1

W c c c d d dc

4 16 2 (6.41)

1 1 2 2 1

2

where:

c is the column dimension parallel to the eccentricity of the load

1

c is the column dimension perpendicular to the eccentricity of the load

2 β

For internal circular columns follows from:

e

β π

= + (6.42)

1 0

,

6 +

D 4

d

D

where is the diameter of the circular column

For an internal rectangular column where the loading is eccentric to both axes, the following

β

approximate expression for may be used:

2

2 ⎛ ⎞

⎛ ⎞

e e

⎜ ⎟

⎜ ⎟

β y

= + + (6.43)

z

,

1 1 8 ⎜ ⎟ ⎜ ⎟

b b

⎝ ⎠ ⎝ ⎠

z y

where:

e e M V

and are the eccentricities / along y and z axes respectively

y z Ed Ed

b b

and is the dimensions of the control perimeter (see Figure 6.13)

y z

e results from a moment about the z axis and e from a moment about the y axis.

Note: y z

(4) For edge column connections, where the eccentricity perpendicular to the slab edge

(resulting from a moment about an axis parallel to the slab edge) is toward the interior and there

is no eccentricity parallel to the edge, the punching force may be considered to be uniformly

u

distributed along the control perimeter as shown in Figure 6.20(a).

1*

102 EN 1992-1-1:2004 (E)

≤ 1,5d

≤ 0,5c

1 ≤ 1,5d

c

2d 2 ≤ 0,5c

2

c 1

c 2d

u

2 *

1

u *

1 2d

≤ 1,5d

c 2d

1 ≤ 0,5c

1

a) edge column b) corner column

Figure 6.20: Reduced basic control perimeter u

1* β

Where there are eccentricities in both orthogonal directions, may be determined using the

following expression:

u u

β = + (6.44)

1 1

k e

par

u W

1

* 1

where:

u is the basic control perimeter (see Figure 6.15)

1

u is the reduced basic control perimeter (see Figure 6.20(a))

1*

e is the eccentricity parallel to the slab edge resulting from a moment about an axis

par perpendicular to the slab edge.

k c c c c

may be determined from Table 6.1 with the ratio / replaced by /2

1 2 1 2

W u

is calculated for the basic control perimeter (see Figure 6.13).

1 1

For a rectangular column as shown in Figure 6.20(a):

2

c π

= + + + +

2

2

W c c c d d dc

4 8 (6.45)

1 1 2 1 2

4

If the eccentricity perpendicular to the slab edge is not toward the interior, Expression (6.39)

W e

applies. When calculating the eccentricity should be measured from the centroid of the

1

control perimeter.

(5) For corner column connections, where the eccentricity is toward the interior of the slab, it is

assumed that the punching force is uniformly distributed along the reduced control perimeter

u β

, as defined in Figure 6.20(b). The -value may then be considered as:

1* u

β = (6.46)

1

u

1

*

If the eccentricity is toward the exterior, Expression (6.39) applies. 103

EN 1992-1-1:2004 (E)

(6) For structures where the lateral stability does not depend on frame action between the

slabs and the columns, and where the adjacent spans do not differ in length by more than 25%,

β may be used.

approximate values for

β

Values of for use in a Country may be found in its National Annex. Recommended values are given in

Note:

Figure 6.21N.

C

β = 1,5 A - internal column

B - edge column

B A

β = 1,4 β = 1,15 r column

β

Figure 6.21N: Recommended values for

(7) Where a concentrated load is applied close to a flat slab column support the shear force

reduction according to 6.2.2 (6) and 6.2.3 (8) respectively is not valid and should not be

included. V

(8) The punching shear force in a foundation slab may be reduced due to the favourable

Ed

action of the soil pressure. V

(9) The vertical component resulting from inclined prestressing tendons crossing the

pd

control section may be taken into account as a favourable action where relevant.

6.4.4 Punching shear resistance of slabs and column bases without shear reinforcement

slab

(1) The punching shear resistance of a should be assessed for the basic control section

according to 6.4.2. The design punching shear resistance [MPa] may be calculated as follows:

( )

ρ

= + ≥ +

1/ 3

v C k (100 f ) k σ v k σ (6.47)

Rd,c Rd,c l ck 1 cp min 1 cp

where:

f is in MPa

ck 200

= + ≤

k d

1 2

,

0 in mm

d

ρ ρ

ρ = ⋅ ≤ 0 ,

02

l ly lz

ρ ρ

, relate to the bonded tension steel in y- and z- directions respectively. The values

ly lz ρ ρ

and should be calculated as mean values taking into account a slab width

ly lz

104 EN 1992-1-1:2004 (E)

d

equal to the column width plus 3 each side.

σ σ σ

= ( + )/2

cp cy cz

where

σ σ

, are the normal concrete stresses in the critical section in y- and z-

cy cz directions (MPa, positive if compression):

N N

σ σ

Ed,y

= = Ed,z

and

c,y c,z A

A cz

cy

N N

, are the longitudinal forces across the full bay for internal columns and the

Edy Edz longitudinal force across the control section for edge columns. The force

may be from a load or prestressing action.

A N

is the area of concrete according to the definition of

c Ed

The values of C , v and k for use in a Country may be found in its National Annex. The

Note: Rd,c min 1

γ

is 0,18/ , for v is given by Expression (6.3N) and that for k is 0,1.

recommended value for C Rd,c c min 1 d

(2) The punching resistance of column bases should be verified at control perimeters within 2

from the periphery of the column.

For concentric loading the net applied force is

∆V

V V

= - (6.48)

Ed,red Ed Ed

where:

V is the applied shear force

Ed

∆V is the net upward force within the control perimeter considered i.e. upward

Ed pressure from soil minus self weight of base.

v V ud

= / (6.49)

Ed Ed,red d

2

ρ

= ≥

1 / 3

v C k f d a v (6.50)

(

100 ) x 2 / x a

Rd Rd

,

c ck min

where

a is the distance from the periphery of the column to the control perimeter considered

C is defined in 6.4.4(1)

Rd,c

v is defined in 6.4.4(1)

min

k is defined in 6.4.4(1)

For eccentric loading

⎡ ⎤

V M u

= +

Ed,red Ed

v k

1 (6.51)

⎢ ⎥

Ed ud V W

⎣ ⎦

Ed

,

red

k W W

Where is defined in 6.4.3 (3) or 6.4.3 (4) as appropriate and is similar to but for

1

u

perimeter .

6.4.5 Punching shear resistance of slabs and column bases with shear reinforcement

(1) Where shear reinforcement is required it should be calculated in accordance with

Expression (6.52): α

v v d s A f u d

= 0,75 + 1,5 ( / ) (1/( )) sin (6.52)

Rd,cs Rd,c r sw ywd,ef 1 105

EN 1992-1-1:2004 (E)

where 2

A is the area of one perimeter of shear reinforcement around the column [mm ]

sw

s is the radial spacing of perimeters of shear reinforcement [mm]

r

f is the effective design strength of the punching shear reinforcement, according to

ywd,ef ≤

f d f

= 250 + 0,25 [MPa]

ywd,ef ywd

d is the mean of the effective depths in the orthogonal directions [mm]

α is the angle between the shear reinforcement and the plane of the slab

d s

If a single line of bent-down bars is provided, then the ratio / in Expression (6.52) may be

r

given the value 0,67.

(2) Detailing requirements for punching shear reinforcement are given in 9.4.3.

(3) Adjacent to the column the punching shear resistance is limited to a maximum of:

β V

= ≤

Ed

v v (6.53)

Ed Rd,max

u d

0

where

u u

for an interior column = length of column periphery [mm]

0 0 ≤

u c d c c

= + 3 + 2 [mm]

for an edge column 0 2 2 1

u d c c

= 3 + [mm]

for a corner column 0 1 2

c c

, are the column dimensions as shown in Figure 6.20

1 2

ν see Expression (6.6)

β see 6.4.3 (3), (4) and (5)

The value of v for us in a Country may be found in its National Annex. The recommended value is

Note: Rd,max

.

ν

0,5 f

cd u u

(4) The control perimeter at which shear reinforcement is not required, (or see Figure

out out,ef

6.22) should be calculated from Expression (6.54):

β

u V v d

= / ( ) (6.54)

out,ef Ed Rd,c

The outermost perimeter of shear reinforcement should be placed at a distance not greater than

kd u u

within (or see Figure 6.22).

out out,ef B

A > 2d

2d kd d

kd d

u u

A Perimeter B Perimeter

out out,ef

Figure 6.22: Control perimeters at internal columns

The value of k for use in a Country may be found in its National Annex. The recommended value is 1,5.

Note:

106 EN 1992-1-1:2004 (E)

V

(5) Where proprietary products are used as shear reinforcement, should be determined

Rd,cs

by testing in accordance with the relevant European Technical Approval. See also 9.4.3.

6.5 Design with strut and tie models

6.5.1 General

(1)P Where a non-linear strain distribution exists (e.g. supports, near concentrated loads or

plain stress) strut-and-tie models may be used (see also 5.6.4).

6.5.2 Struts

(1) The design strength for a concrete strut in a region with transverse compressive stress or

no transverse stress may be calculated from Expression (6.55) (see Figure 6.23).

σ Rd,max A transverse compressive stress or

no transverse stress

A

Figure 6.23: Design strength of concrete struts without transverse tension

σ f

= (6.55)

cd

Rd,max

It may be appropriate to assume a higher design strength in regions where multi-axial

compression exists.

(2) The design strength for concrete struts should be reduced in cracked compression zones

and, unless a more rigorous approach is used, may be calculated from Expression (6.56) (see

Figure 6.24). σ Rd,max

Figure 6.24: Design strength of concrete struts with transverse tension

σ ν f

= 0,6 ’ (6.56)

Rd,max cd ν

The value of ’ for use in a Country may be found in its National Annex. The recommended value is

Note:

given by equation (6.57N).

ν ’ = 1 - f /250 (6.57N)

ck

(3) For struts between directly loaded areas, such as corbels or short deep beams, alternative

calculation methods are given in 6.2.2 and 6.2.3.

6.5.3 Ties

(1) The design strength of transverse ties and reinforcement should be limited in accordance

with 3.2 and 3.3.

(2) Reinforcement should be adequately anchored in the nodes. 107

EN 1992-1-1:2004 (E)

(3) Reinforcement required to resist the forces at the concentrated nodes may be smeared

over a length (see Figure 6.25 a) and b)). When the reinforcement in the node area extends

over a considerable length of an element, the reinforcement should be distributed over the

T

length where the compression trajectories are curved (ties and struts). The tensile force may

be obtained by: ⎛ ⎞

H

b

a) for partial discontinuity regions , see Figure 6.25 a:

⎜ ⎟

2

⎝ ⎠

1 b a

=

T F (6.58)

4 b ⎛ ⎞

H

>

b

b) for full discontinuity regions , see Figure 6.25 b:

⎜ ⎟

2

⎝ ⎠

⎛ ⎞

1 a

= −

⎜ ⎟

T ,

1 0 7 F (6.59)

4 h

⎝ ⎠

b b

ef ef

a a

F F

D h = b z = h/2 h = H/2

B H B Continuity region

D D Discontinuity region

F

F

b b ≤

b = 0,5H + 0,65a; a h

b = b ef

ef

a) Partial discontinuity b) Full discontinuity

Figure 6.25: Parameters for the determination of transverse tensile forces in a

compression field with smeared reinforcement

6.5.4 Nodes

(1)P The rules for nodes also apply to regions where concentrated forces are transferred in a

member and which are not designed by the strut-and-tie method.

(2)P The forces acting at nodes shall be in equilibrium. Transverse tensile forces

perpendicular to an in-plane node shall be considered.

(3) The dimensioning and detailing of concentrated nodes are critical in determining their load-

bearing resistance. Concentrated nodes may develop, e.g. where point loads are applied, at

supports, in anchorage zones with concentration of reinforcement or prestressing tendons, at

bends in reinforcing bars, and at connections and corners of members.

(4) The design values for the compressive stresses within nodes may be determined by:

a) in compression nodes where no ties are anchored at the node (see Figure 6.26)

108 EN 1992-1-1:2004 (E)

σ ν

k f

= ’ (6.60)

Rd,max 1 cd

The value of k for use in a Country may be found in its National Annex. The recommended value is 1,0.

Note: 1

σ

where is the maximum stress which can be applied at the edges of the node. See

Rd,max ν ’.

6.5.2 (2) for definition of F

F cd,3

σ a

cd,2 3

c0

a

2 σ

σ Rd,3

F

Rd,2 cd,0 σ

Rd,1

F F

cd,1l cd,1r

F = F + F

cd,1 cd,1r cd,1l

a

1

Figure 6.26: Compression node without ties

b) in compression - tension nodes with anchored ties provided in one direction (see Figure

6.27),

σ ν

k f

= ’ (6.61)

2 cd

Rd,max

σ σ σ ν

where is the maximum of and , See 6.5.2 (2) for definition of ’.

Rd,max Rd,1 Rd,2 a 2

F σ

cd2 Rd,2

s

0 F

u s td

s

0 σ

Rd,1

F

cd1

a

2s 1

0 l

bd

Figure 6.27: Compression tension node with reinforcement provided in one

direction 109

EN 1992-1-1:2004 (E)

The value of k for use in a Country may be found in its National Annex. The recommended value is

Note: 2

0,85.

c) in compression - tension nodes with anchored ties provided in more than one direction

(see Figure 6.28), F

td,1

σ

Rd,max

F

cd

F

td,2

Figure 6.28: Compression tension node with reinforcement provided in two

directions

σ ν

k f

= ’ (6.62)

3 cd

Rd,max

The value of k for use in a Country may be found in its National Annex. The recommended value is

Note: 3

0,75.

(5) Under the conditions listed below, the design compressive stress values given in 6.5.4 (4)

may be increased by up to10% where at least one of the following applies:

- triaxial compression is assured, ≥

- all angles between struts and ties are 55°,

- the stresses applied at supports or at point loads are uniform, and the node is confined

by stirrups,

- the reinforcement is arranged in multiple layers,

the node is reliably confined by means of bearing arrangement or friction.

-

(6) Triaxially compressed nodes may be checked according to Expression (3.24) and (3.25)

σ ν

≤ k f

‘ if for all three directions of the struts the distribution of load is known.

with Rd,max 4 cd

The value of k for use in a Country may be found in its National Annex. The recommended value is 3,0.

Note: 4

(7) The anchorage of the reinforcement in compression-tension nodes starts at the beginning

of the node, e.g. in case of a support anchorage starting at its inner face (see Figure 6.27). The

anchorage length should extend over the entire node length. In certain cases, the reinforcement

may also be anchored behind the node. For anchorage and bending of reinforcement, see 8.4

to 8.6.

(8) In-plane compression nodes at the junction of three struts may be verified in accordance

σ σ σ σ

, , , ) should be

with Figure 6.26. The maximum average principal node stresses ( c0 c1 c2 c3

checked in accordance with 6.5.4 (4) a). Normally the following may be assumed:

σ σ σ σ

F /a = F /a = F /a = = =

resulting in

cd,1 1 cd,2 2 cd,3 3 cd,1 cd,2 cd,3 cd,0.

(9) Nodes at reinforcement bends may be analysed in accordance with Figure 6.28. The

average stresses in the struts should be checked in accordance with 6.5.4 (5). The diameter of

the mandrel should be checked in accordance with 8.4.

110 EN 1992-1-1:2004 (E)

6.6 Anchorages and laps

(1)P The design bond stress is limited to a value depending on the surface characteristics of

the reinforcement, the tensile strength of the concrete and confinement of surrounding

concrete. This depends on cover, transverse reinforcement and transverse pressure.

(2) The length necessary for developing the required tensile force in an anchorage or lap is

calculated on the basis of a constant bond stress.

(3) Application rules for the design and detailing of anchorages and laps are given in 8.4 to 8.8.

6.7 Partially loaded areas

(1)P For partially loaded areas, local crushing (see below) and transverse tension forces (see

6.5) shall be considered. A

(2) For a uniform distribution of load on an area (see Figure 6.29) the concentrated

c0

resistance force may be determined as follows:

= ⋅ ⋅ ≤ ⋅ ⋅

F A f A A f A

/ 3

,

0 (6.63)

Rdu c 0 cd c 1 c 0 cd c 0

where:

A is the loaded area,

c0

A A

is the maximum design distribution area with a similar shape to

c1 c0

A F

(3) The design distribution area required for the resistance force should correspond to

c1 Rdu

the following conditions:

- The height for the load distribution in the load direction should correspond to the

conditions given in Figure 6.29 A

- the centre of the design distribution area should be on the line of action passing

c1

A

through the centre of the load area .

c0

- If there is more than one compression force acting on the concrete cross section, the

designed distribution areas should not overlap.

F A

The value of should be reduced if the load is not uniformly distributed on the area or

Rdu c0

if high shear forces exist.

A c0 b

1

d

1 A A - line of action

h d 3d

2 1 ≥

h b - b

( ) and

2 1

≥ d - d

( )

2 1

b 3b

2 1 A c1

Figure 6.29: Design distribution for partially loaded areas 111

EN 1992-1-1:2004 (E)

(4) Reinforcement should be provided for the tensile force due to the effect of the action.

6.8 Fatigue

6.8.1 Verification conditions

(1)P The resistance of structures to fatigue shall be verified in special cases. This verification

shall be performed separately for concrete and steel.

(2) A fatigue verification should be carried out for structures and structural components which

are subjected to regular load cycles (e.g. crane-rails, bridges exposed to high traffic loads).

6.8.2 Internal forces and stresses for fatigue verification

(1)P The stress calculation shall be based on the assumption of cracked cross sections

neglecting the tensile strength of concrete but satisfying compatibility of strains.

(2)P The effect of different bond behaviour of prestressing and reinforcing steel shall be taken

into account by increasing the stress range in the reinforcing steel calculated under the

η

, given by

assumption of perfect bond by the factor,

+

A A

η = S P (6.64)

( )

ξ φ φ

+

A A /

S P S P

where:

A is the area of reinforcing steel

s

A is the area of prestressing tendon or tendons

P

φ is the largest diameter of reinforcement

S

φ is the diameter or equivalent diameter of prestressing steel

P φ √A

=1,6 for bundles

P P

φ φ φ

=1,75 for single 7 wire strands where is the wire diameter

P wire wire

φ φ φ

=1,20 for single 3 wire strands where is the wire diameter

P wire wire

ξ is the ratio of bond strength between bonded tendons and ribbed steel in

concrete. The value is subject to the relevant European Technical Approval. In

the absence of this the values given in Table 6.2 may be used.

ξ

,

Table 6.2: Ratio of bond strength, between tendons and reinforcing steel

ξ

prestressing steel bonded, post-tensioned

pre-

tensioned ≤ ≥

C50/60 C70/85

smooth bars and wires Not 0,3 0,15

applicable

strands 0,6 0,5 0,25

indented wires 0,7 0,6 0,3

ribbed bars 0,8 0,7 0,35 .

For intermediate values between C50/60 and C70/85 interpolation may be used

Note:

112 EN 1992-1-1:2004 (E)

θ

(3) In the design of the shear reinforcement the inclination of the compressive struts may be

fat

calculated using a strut and tie model or in accordance with Expression (6.65).

θ θ

= ≤

tan tan 1,0 (6.65)

fat

where:

θ is the angle of concrete compression struts to the beam axis assumed in ULS design

(see 6.2.3)

6.8.3 Combination of actions

(1)P For the calculation of the stress ranges the action shall be divided into non-cycling and

fatigue-inducing cyclic actions (a number of repeated actions of load).

(2)P The basic combination of the non-cyclic load is similar to the definition of the frequent

combination for SLS:

ψ ψ

= ≥ >

E E {

G ; P ; Q ; Q } j 1

; i 1 (6.66)

d k , j 1

,

1 k ,

1 2

,

i k ,

i

The combination of actions in bracket { }, (called the basic combination), may be expressed as:

ψ ψ

+ + +

∑ ∑ (6.67)

G " " P " " Q " " Q

k , j 1

,

1 k ,

1 2 ,

i k ,

i

≥ >

j 1 i 1

Q and Q are non-cyclic, non-permanent actions

Note: k,1 k,I

(3)P The cyclic action shall be combined with the unfavourable basic combination:

ψ ψ

= ≥ >

{{ ; ; ; } ; } 1; 1

E E G P Q Q Q j i (6.68)

d k,j 1,1 k,1 2,i k,i fat

The combination of actions in bracket { }, (called the basic combination plus the cyclic action),

can be expressed as:

⎛ ⎞

∑ ∑

⎜ ⎟

ψ ψ

+ + + + (6.69)

" " " " " " " "

G P Q Q Q

⎜ ⎟

k , j 1

,

1 k ,

1 2

,

i k ,

i fat

⎝ ⎠

≥ >

1 i 1

j

where: is the relevant fatigue load (e.g. traffic load as defined in EN 1991 or other cyclic

Q fat load)

6.8.4 Verification procedure for reinforcing and prestressing steel

∆σ

(1) The damage of a single stress amplitude may be determined by using the

corresponding S-N curves (Figure 6.30) for reinforcing and prestressing steel. The applied load

γ ∆σ

should be multiplied by . The resisting stress range at cycles obtained should be

N*

F,fat Rsk

γ

divided by the safety factor .

S,fat

γ

values of for use in a Country may be found in its National Annex. The recommended value

Note 1: The F,fat

is 1,0. 113

EN 1992-1-1:2004 (E)

∆σ

log A b = k A reinforcement at yield

1

Rsk 1 b = k

2 1

N* N

log

Figure 6.30: Shape of the characteristic fatigue strength curve (S-N-curves for

reinforcing and prestressing steel)

The values of parameters for reinforcing steels and prestressing steels S-N curves for use in a Country

Note 2:

may be found in its National Annex. The recommended values are given in Table 6.3N and 6.4N which apply

for reinforcing and prestressing steel respectively.

Table 6.3N: Parameters for S-N curves for reinforcing steel σ

Type of reinforcement stress exponent (MPa)

Rsk

N* k k at N* cycles

1 2

6

1 10 5 9 162,5

Straight and bent bars 7

10 3 5 58,5

Welded bars and wire fabrics 7

10 3 5 35

Splicing devices σ

Values for are those for straight bars. Values for bent bars should be obtained using a

Note 1: Rsk ζ φ

= 0,35 + 0,026 D / .

reduction factor

where:

D diameter of the mandrel

φ bar diameter

Table 6.4N: Parameters for S-N curves of prestressing steel σ

S-N curve of prestressing steel stress exponent (MPa)

Rsk

used for k at N* cycles

N* k 1 2

6

10 5 9 185

pre-tensioning

post-tensioning

− single strands in plastic ducts 6

10 9 185

5

− straight tendons or curved 6

10 10 150

5

tendons in plastic ducts 6

− 10 5 7 120

curved tendons in steel ducts

− splicing devices 6

10 5 5 80

(2) For multiple cycles with variable amplitudes the damage may be added by using the

Palmgren-Miner Rule. Hence, the fatigue damage factor of steel caused by the relevant

D

Ed

fatigue loads should satisfy the condition:

114 EN 1992-1-1:2004 (E)

σ

( )

n

∑ <

= i (6.70)

1

D

Ed σ

( )

N

i i

where:

∆σ ∆σ

( ) is the applied number of cycles for a stress range

n i i

∆σ ∆σ

( ) is the resisting number of cycles for a stress range

N i i

(3)P If prestressing or reinforcing steel is exposed to fatigue loads, the calculated stresses

shall not exceed the design yield strength of the steel.

(4) The yield strength should be verified by tensile tests for the steel used.

(5) When the rules of 6.8 are used to evaluate the remaining life of existing structures, or to

assess the need for strengthening, once corrosion has started the stress range may be

determined by reducing the stress exponent for straight and bent bars.

k 2

The value of k for use in a Country may be found in its National Annex. The recommended values is 5.

Note: 2

(6)P The stress range of welded bars shall never exceed the stress range of straight and bent

bars.

6.8.5 Verification using damage equivalent stress range

(1) Instead of an explicit verification of the damage strength according to 6.8.4 the fatigue

verification of standard cases with known loads (railway and road bridges) may also be

performed as follows:

− by damage equivalent stress ranges for steel according to 6.8.5 (3)

− damage equivalent compression stresses for concrete according to 6.8.7

(2) The method of damage equivalent stress range consists of representing the actual

operational loading by cycles of a single stress range. EN 1992-2 gives relevant fatigue

N* σ

loading models and procedures for the calculation of the equivalent stress range for

S,equ

superstructures of road and railway bridges.

(3) For reinforcing or prestressing steel and splicing devices adequate fatigue resistance

should be assumed if the Expression (6.71) is satisfied:

( )

∆ σ N *

( )

γ Rsk

⋅ ≤ (6.71)

∆ σ N* γ

F,fat S,equ s,fat

where:

σ

∆ (N*) is the stress range at N* cycles from the appropriate S-N curves given in

Rsk Figure 6.30.

See also Tables 6.3N and 6.4N.

Note:

σ

∆ (N*) is the damage equivalent stress range for different types of reinforcement

S,equ and considering the number of loading cycles N*. For building construction

σ

∆ ∆σ

(N*) may be approximated by

S,equ S,max .

σ

∆ is the maximum steel stress range under the relevant load combinations

S,max 115

EN 1992-1-1:2004 (E)

6.8.6 Other verifications

(1) Adequate fatigue resistance may be assumed for unwelded reinforcing bars under tension,

σ

∆ ≤

if the stress range under frequent cyclic load combined with the basic combination is k .

S 1

The value of k for use in a Country may be found in its National Annex. The recommended value is

Note: 1

70MPa.

For welded reinforcing bars under tension adequate fatigue resistance may be assumed if the

σ ≤

∆ k .

stress range under frequent load combined with the basic combination is S 2

The value of k for use in a Country may be found in its National Annex. The recommended value is

Note: 2

35MPa.

(2) As a simplification to (1) above verification may be carried out using the Frequent load

combination. If this is satisfied then no further checks are necessary.

(3) Where welded joints or splicing devices are used in prestressed concrete, no tension

should exist in the concrete section within 200 mm of the prestressing tendons or reinforcing

steel under the frequent load combination together with a reduction factor of k for the mean

3

value of prestressing force, P

m,

The value of k for use in a Country may be found in its National Annex. The recommended value is 0,9.

Note: 3

6.8.7 Verification of concrete under compression or shear

(1) A satisfactory fatigue resistance may be assumed for concrete under compression, if the

following condition is fulfilled:

+ − ≤ (6.72)

E 0

, 43 1 R 1

cd , max, equ equ

where: E cd,min,equ

=

R (6.73)

equ E cd,max,equ

σ cd,min,equ

= (6.74)

E cd,min,equ f

cd,fat

σ cd,max,equ

=

E (6.75)

cd,max,equ f

cd,fat

where :

R is the stress ratio

equ

E is the minimum compressive stress level

cd,min,equ

E is the maximum compressive stress level

cd,max,equ

f is the design fatigue strength of concrete according to (6.76)

cd,fat

σ is the upper stress of the ultimate amplitude for N cycles

cd,max,equ

σ is the lower stress of the ultimate amplitude for N cycles

cd,min,equ 6

The value of N (≤ 10 cycles) for use in a Country may be found in its National Annex. The

Note: 6

recommended value is N = 10 cycles.

116 EN 1992-1-1:2004 (E)

⎛ ⎞

f

( )

= − ck

1

f k β t f (6.76)

⎜ ⎟

cd,fat 1 cc 0 cd 250

⎝ ⎠

where:

β (t ) is a coefficient for concrete strength at first load application (see 3.1.2 (6))

cc 0

t is the time of the start of the cyclic loading on concrete in days

0

The value of k for use in a Country may be found in its National Annex. The recommended value for N

Note: 1

6

= 10 cycles is 0,85.

(2) The fatigue verification for concrete under compression may be assumed, if the following

condition is satisfied:

σ σ

c ,

max c ,

min

≤ +

0

,

5 0

,

45 (6.77)

f f

cd

,

fat cd

,

fat

≤ ≤

0,9 for 50 MPa

f ck

≤ 0,8 for > 50 MPa

f ck

where:

σ is the maximum compressive stress at a fibre under the frequent load combination

c,max (compression measured positive)

σ σ σ

is the minimum compressive stress at the same fibre where occurs. If

c,min c,max c,min

σ

is a tensile stress, then should be taken as 0.

c,min

(3) Expression (6.77) also applies to the compression struts of members subjected to shear. In

should be reduced by the strength reduction factor (see

this case the concrete strength f

cd,fat

6.2.2 (6)).

(4) For members not requiring design shear reinforcement for the ultimate limit state it may be

assumed that the concrete resists fatigue due to shear effects where the following apply:

V ≥

Ed,min

- for 0 :

V

Ed,max ≤

⎧ 0,9 up to C50 / 60

| | | |

V V

≤ +

Ed,max Ed,min

0,5 0,45 (6.78)

⎨ ≤

| | | | 0,8 greater than C55 / 67

V V ⎩

Rd,c Rd,c

V <

Ed,min

- for 0 :

V

Ed,max

| | | |

V V

≤ −

Ed,max Ed,min

0,5 (6.79)

| | | |

V V

Rd,c Rd,c

where: is the design value of the maximum applied shear force under frequent load

V

Ed,max combination

is the design value of the minimum applied shear force under frequent load

V

Ed,min occurs

combination in the cross-section where V

Ed,max

is the design value for shear-resistance according to Expression (6.2.a).

V

Rd,c 117

EN 1992-1-1:2004 (E)

SECTION 7 SERVICEABILITY LIMIT STATES (SLS)

7.1 General

(1)P This section covers the common serviceability limit states. These are:

- stress limitation (see 7.2)

- crack control (see 7.3)

- deflection control (see 7.4)

Other limit states (such as vibration) may be of importance in particular structures but are not

covered in this Standard.

(2) In the calculation of stresses and deflections, cross-sections should be assumed to be

f f

uncracked provided that the flexural tensile stress does not exceed . The value of may

ct,eff ct,eff

f f

be taken as or provided that the calculation for minimum tension reinforcement is also

ctm ctm,fl

based on the same value. For the purposes of calculating crack widths and tension stiffening

f should be used.

ctm

7.2 Stress limitation

(1)P The compressive stress in the concrete shall be limited in order to avoid longitudinal

cracks, micro-cracks or high levels of creep, where they could result in unacceptable effects on

the function of the structure.

(2) Longitudinal cracks may occur if the stress level under the characteristic combination of

loads exceeds a critical value. Such cracking may lead to a reduction of durability. In the

absence of other measures, such as an increase in the cover to reinforcement in the

compressive zone or confinement by transverse reinforcement, it may be appropriate to limit

f

k in areas exposed to environments of exposure classes

the compressive stress to a value 1 ck

XD, XF and XS (see Table 4.1).

k

Note: The value of for use in a Country may be found in its National Annex. The recommended value is 0,6.

1 k f

(3) If the stress in the concrete under the quasi-permanent loads is less than , linear creep

2 ck

f

k , non-linear creep should be

may be assumed. If the stress in concrete exceeds 2 ck

considered (see 3.1.4)

k

Note: The value of for use in a Country may be found in its National Annex. The recommended value is

2

0,45.

(4)P Tensile stresses in the reinforcement shall be limited in order to avoid inelastic strain,

unacceptable cracking or deformation.

(5) Unacceptable cracking or deformation may be assumed to be avoided if, under the

characteristic combination of loads, the tensile stress in the reinforcement does not exceed

f

k . Where the stress is caused by an imposed deformation, the tensile stress should not

3 yk k f k f

exceed . The mean value of the stress in prestressing tendons should not exceed

4 yk 5 pk

k k k

,

Note: The values of and for use in a Country may be found in its National Annex. The

3 4 5

recommended values are 0,8, 1 and 0,75 respectively.

118 EN 1992-1-1:2004 (E)

7.3 Crack control

7.3.1 General considerations

(1)P Cracking shall be limited to an extent that will not impair the proper functioning or

durability of the structure or cause its appearance to be unacceptable.

(2) Cracking is normal in reinforced concrete structures subject to bending, shear, torsion or

tension resulting from either direct loading or restraint or imposed deformations.

(3) Cracks may also arise from other causes such as plastic shrinkage or expansive chemical

reactions within the hardened concrete. Such cracks may be unacceptably large but their

avoidance and control lie outside the scope of this Section.

(4) Cracks may be permitted to form without any attempt to control their width, provided they

do not impair the functioning of the structure.

w

(5) A limiting calculated crack width, , taking into account the proposed function and nature

max

of the structure and the costs of limiting cracking, should be established.

w

Note: The value of for use in a Country may be found in its National Annex. The recommended values

max

for relevant exposure classes are given in Table 7.1N.

Table 7.1N Recommended values of w (mm)

max

Exposure Reinforced members and prestressed Prestressed members with

Class members with unbonded tendons bonded tendons

Quasi-permanent load combination Frequent load combination

1

X0, XC1 0,4 0,2

2

XC2, XC3, XC4 0,2

0,3

XD1, XD2, XS1, Decompression

XS2, XS3

Note 1: For X0, XC1 exposure classes, crack width has no influence on durability and this limit

is set to guarantee acceptable appearance. In the absence of appearance conditions

this limit may be relaxed.

Note 2: For these exposure classes, in addition, decompression should be checked under the

quasi-permanent combination of loads.

In the absence of specific requirements (e.g. water-tightness), it may be assumed that limiting the calculated

w

crack widths to the values of given in Table 7.1N, under the quasi-permanent combination of loads, will

max

generally be satisfactory for reinforced concrete members in buildings with respect to appearance and

durability.

The durability of prestressed members may be more critically affected by cracking. In the absence of more

w

detailed requirements, it may be assumed that limiting the calculated crack widths to the values of given

max

in Table 7.1N, under the frequent combination of loads, will generally be satisfactory for prestressed concrete

members. The decompression limit requires that all parts of the bonded tendons or duct lie at least 25 mm

within concrete in compression.

(6) For members with only unbonded tendons, the requirements for reinforced concrete

elements apply. For members with a combination of bonded and unbonded tendons

requirements for prestressed concrete members with bonded tendons apply. 119

EN 1992-1-1:2004 (E)

(7) Special measures may be necessary for members subjected to exposure class XD3. The

choice of appropriate measures will depend upon the nature of the aggressive agent involved.

(8) When using strut-and-tie models with the struts oriented according to the compressive

stress trajectories in the uncracked state, it is possible to use the forces in the ties to obtain the

corresponding steel stresses to estimate the crack width (see 5.6.4 (2).

(9) Crack widths may be calculated according to 7.3.4. A simplified alternative is to limit the bar

size or spacing according to 7.3.3.

7.3.2 Minimum reinforcement areas

(1)P If crack control is required, a minimum amount of bonded reinforcement is required to

control cracking in areas where tension is expected. The amount may be estimated from

equilibrium between the tensile force in concrete just before cracking and the tensile force in

reinforcement at yielding or at a lower stress if necessary to limit the crack width.

(2) Unless a more rigorous calculation shows lesser areas to be adequate, the required

minimum areas of reinforcement may be calculated as follows. In profiled cross sections like T-

beams and box girders, minimum reinforcement should be determined for the individual parts of

the section (webs, flanges).

σ

A k k f A

= (7.1)

s,min s c ct,eff ct

where:

A is the minimum area of reinforcing steel within the tensile zone

s,min

A is the area of concrete within tensile zone. The tensile zone is that part of the

ct section which is calculated to be in tension just before formation of the first crack

σ is the absolute value of the maximum stress permitted in the reinforcement

s immediately after formation of the crack. This may be taken as the yield strength

f

of the reinforcement, . A lower value may, however, be needed to satisfy the

yk

crack width limits according to the maximum bar size or spacing (see 7.3.3 (2))

f is the mean value of the tensile strength of the concrete effective at the time

ct,eff when the cracks may first be expected to occur:

f

f = or lower, (f (t)), if cracking is expected earlier than 28 days

ct,eff ctm ctm

k is the coefficient which allows for the effect of non-uniform self-equilibrating

stresses, which lead to a reduction of restraint forces

h

= 1,0 for webs with 300 mm or flanges with widths less than 300 mm

h

= 0,65 for webs with 800 mm or flanges with widths greater than 800 mm

intermediate values may be interpolated

k is a coefficient which takes account of the stress distribution within the section

c immediately prior to cracking and of the change of the lever arm:

k

For pure tension = 1,0

c

For bending or bending combined with axial forces:

- For rectangular sections and webs of box sections and T-sections:

⎡ ⎤

σ

= ⋅ − ≤

c

k ⎢ ⎥

0

,

4 1 1 (7.2)

c ∗

k h h f

( / )

⎣ ⎦

1 ct,eff

- For flanges of box sections and T-sections:

120 EN 1992-1-1:2004 (E)

F

= ≥

k cr (7.3)

0

,

9 0

,

5

c A f

ct ct, eff

where

σ is the mean stress of the concrete acting on the part of the section under

c consideration:

N

σ = Ed (7.4)

c bh

N is the axial force at the serviceability limit state acting on the part of the

Ed N should

cross-section under consideration (compressive force positive). Ed

be determined considering the characteristic values of prestress and axial

forces under the relevant combination of actions

h* h* h h

= for < 1,0 m

h* h

= 1,0 m for 1,0 m

k is a coefficient considering the effects of axial forces on the stress

1 distribution: N

k = 1,5 if is a compressive force

1 Ed

h

2

= N

k if is a tensile force

Ed

1 h

3

F is the absolute value of the tensile force within the flange immediately prior

cr f

to cracking due to the cracking moment calculated with ct,eff

(3) Bonded tendons in the tension zone may be assumed to contribute to crack control within a

distance 150 mm from the centre of the tendon. This may be taken into account by adding the

ξ σ

A

term ‘∆ to the left hand side of Expression (7.1),

1 p p

where A

A ‘ is the area of pre or post-tensioned tendons within .

p c,eff

A is the effective area of concrete in tension surrounding the reinforcement or

c,eff h

h , where is the lesser of 2,5(h-d), (h-x)/3 or

prestressing tendons of depth, c,ef c,ef

h/2 (see Figure 7.1).

ξ is the adjusted ratio of bond strength taking into account the different diameters of

1 prestressing and reinforcing steel:

φ

ξ

= ⋅ s (7.5)

φ p

ξ ratio of bond strength of prestressing and reinforcing steel, according to Table 6.2

in 6.8.2.

φ largest bar diameter of reinforcing steel

s

φ equivalent diameter of tendon according to 6.8.2

p ξ

ξ = ⋅

If only prestressing steel is used to control cracking, .

1

σ

∆ Stress variation in prestressing tendons from the state of zero strain of the concrete

p at the same level

(4) In prestressed members no minimum reinforcement is required in sections where, under

the characteristic combination of loads and the characteristic value of prestress, the concrete is

σ .

compressed or the absolute value of the tensile stress in the concrete is below ct,p

σ

Note: The value of for use in a Country may be found in its National Annex. The recommended value is

ct,p

f in accordance with 7.3.2 (2).

ct,eff 121

EN 1992-1-1:2004 (E)

x ε = 0

2

d

h A A - level of steel centroid

h c,ef A

B - effective tension area, c,eff

ε 1

B

a) Beam

x ε = 0

2

d

h ε

h 1

c,ef B A

B - effective tension area, c,eff

b) Slab B h B - effective tension area for upper

ε

c,ef A

2 surface, ct,eff

d d

h C - effective tension area for lower

A

surface, cb,eff

ε

h 1

c,ef C

c) Member in tension

Figure 7.1: Effective tension area (typical cases)

7.3.3 Control of cracking without direct calculation

(1) For reinforced or prestressed slabs in buildings subjected to bending without significant

axial tension, specific measures to control cracking are not necessary where the overall depth

does not exceed 200 mm and the provisions of 9.3 have been applied.

(2) The rules given in 7.3.4 may be presented in a tabular form by restricting the bar diameter

or spacing as a simplification.

Note: Where the minimum reinforcement given by 7.3.2 is provided, crack widths are unlikely to be excessive

if:

- for cracking caused dominantly by restraint, the bar sizes given in Table 7.2N are not exceeded where the

σ in Expression (7.1)).

steel stress is the value obtained immediately after cracking (i.e. s

- for cracks caused mainly by loading, either the provisions of Table 7.2N or the provisions of Table 7.3N are

complied with. The steel stress should be calculated on the basis of a cracked section under the relevant

combination of actions.

122 EN 1992-1-1:2004 (E)

For pre-tensioned concrete, where crack control is mainly provided by tendons with direct bond, Tables 7.2N

and 7.3N may be used with a stress equal to the total stress minus prestress. For post-tensioned concrete,

where crack control is provided mainly by ordinary reinforcement, the tables may be used with the stress in this

reinforcement calculated with the effect of prestressing forces included.

φ * 1

Table 7.2N Maximum bar diameters for crack control

s

2

Steel stress Maximum bar size [mm]

[MPa] w = 0,4 mm w = 0,3 mm w = 0,2 mm

k k k

160 40 32 25

200 32 25 16

240 20 16 12

280 16 12 8

320 12 10 6

360 10 8 5

400 8 6 4

450 6 5 -

Notes: 1. The values in the table are based on the following assumptions:

c f h k k k k

= 25mm; = 2,9MPa; = 0,5; (h-d) = 0,1h; = 0,8; = 0,5; = 0,4; = 1,0;

ct,eff cr 1 2 c

k = 0,4 and k’ = 1,0

t

2. Under the relevant combinations of actions

1

Table 7.3N Maximum bar spacing for crack control

2

Steel stress Maximum bar spacing [mm]

[MPa] w =0,4 mm w =0,3 mm w =0,2 mm

k k k

160 300 300 200

200 300 250 150

240 250 200 100

280 200 150 50

320 150 100 -

360 100 50 -

For Notes see Table 7.2N

The maximum bar diameter should be modified as follows:

Bending (at least part of section in compression):

k h

φ φ

= c cr

(f /2,9) (7.6N)

s

s ct,eff h d

2 ( - )

Tension (uniform axial tension)

φ φ

= (f /2,9)h /(8(h-d)) (7.7N)

s

s ct,eff cr

where:

φ is the adjusted maximum bar diameter

s

φ is the maximum bar size given in the Table 7.2N

s

h is the overall depth of the section

h is the depth of the tensile zone immediately prior to cracking, considering the characteristic values

cr of prestress and axial forces under the quasi-permanent combination of actions

d is the effective depth to the centroid of the outer layer of reinforcement

h d

Where all the section is under tension - is the minimum distance from the centroid of the layer of

reinforcement to the face of the concrete (consider each face where the bar is not placed symmetrically). 123

EN 1992-1-1:2004 (E)

(3) Beams with a total depth of 1000 mm or more, where the main reinforcement is

concentrated in only a small proportion of the depth, should be provided with additional skin

reinforcement to control cracking on the side faces of the beam. This reinforcement should be

evenly distributed between the level of the tension steel and the neutral axis and should be

located within the links. The area of the skin reinforcement should not be less than the amount

σ f

k as . The spacing and size of suitable bars may

obtained from 7.3.2 (2) taking as 0,5 and s yk

be obtained from 7.3.4 or a suitable simplification (see 7.3.3 (2)) assuming pure tension and a

steel stress of half the value assessed for the main tension reinforcement.

(4) It should be noted that there are particular risks of large cracks occurring in sections where

there are sudden changes of stress, e.g.

- at changes of section

- near concentrated loads

- positions where bars are curtailed

- areas of high bond stress, particularly at the ends of laps

Care should be taken at such areas to minimise the stress changes wherever possible.

However, the rules for crack control given above will normally ensure adequate control at these

points provided that the rules for detailing reinforcement given in Sections 8 and 9 are applied.

(5) Cracking due to tangential action effects may be assumed to be adequately controlled if the

detailing rules given in 9.2.2, 9.2.3, 9.3.2 and 9.4.4.3 are observed.

7.3.4 Calculation of crack widths

w

(1) The crack width, may be calculated from Expression (7.8):

k,

ε ε

w s

= ( - ) (7.8)

k r,max sm cm

where

s is the maximum crack spacing

r,max

ε is the mean strain in the reinforcement under the relevant combination of loads,

sm including the effect of imposed deformations and taking into account the effects of

tension stiffening. Only the additional tensile strain beyond the state of zero strain

of the concrete at the same level is considered

ε is the mean strain in the concrete between cracks

cm

ε ε

(2) - may be calculated from the expression:

sm cm f ( )

− +

k α ρ

ct,eff

σ 1

s t e p,eff

ρ σ

− ≥

p,eff ,

ε = s

ε 0 6 (7.9)

sm cm E

E s s

where:

σ is the stress in the tension reinforcement assuming a cracked section. For

s σ σ

pretensioned members, may be replaced by the stress variation in

s p

prestressing tendons from the state of zero strain of the concrete at the same

level.

α E

is the ratio /E

e s cm

ρ ξ 12 A

(A + ’)/A (7.10)

p,eff s p c,eff

A

A ’ and are as defined in 7.3.2 (3)

p c,eff

ξ according to Expression (7.5)

1

k is a factor dependent on the duration of the load

t

124 EN 1992-1-1:2004 (E)

k = 0,6 for short term loading

t

k = 0,4 for long term loading

t

(3) In situations where bonded reinforcement is fixed at reasonably close centres within the

φ

tension zone (spacing 5(c+ /2), the maximum final crack spacing may be calculated from

Expression (7.11) (see Figure 7.2): A - Neutral axis

B - Concrete tension surface

C - Crack spacing predicted by

Expression (7.14)

D - Crack spacing predicted by

Expression (7.11)

E - Actual crack width

Figure 7.2: Crack width, w, at concrete surface relative to distance from bar

φ ρ

s k c + k k k

= / (7.11)

r,max 3 1 2 4 p,eff

where:

φ is the bar diameter. Where a mixture of bar diameters is used in a section, an

φ φ

n

equivalent diameter, , should be used. For a section with bars of diameter

eq 1 1

φ

n

and bars of diameter , the following expression should be used

2 2

φ φ

+

n n

2 2

φ = 1 1 2 2 (7.12)

φ φ

+

eq n n

1 1 2 2

c is the cover to the longitudinal reinforcement

k is a coefficient which takes account of the bond properties of the bonded

1 reinforcement:

= 0,8 for high bond bars

= 1,6 for bars with an effectively plain surface (e.g. prestressing tendons)

k is a coefficient which takes account of the distribution of strain:

2 = 0,5 for bending

= 1,0 for pure tension k should be

For cases of eccentric tension or for local areas, intermediate values of 2

used which may be calculated from the relation:

ε ε ε

k = ( + )/2 (7.13)

2 1 2 1

ε ε

Where is the greater and is the lesser tensile strain at the boundaries of the

1 2

section considered, assessed on the basis of a cracked section 125

EN 1992-1-1:2004 (E)

k k

Note: The values of and for use in a Country may be found in its National Annex. The recommended

3 4

values are 3,4 and 0,425 respectively. φ

Where the spacing of the bonded reinforcement exceeds 5(c+ /2) (see Figure 7.2) or where

there is no bonded reinforcement within the tension zone, an upper bound to the crack width

may be found by assuming a maximum crack spacing:

s x)

= 1,3 (h - (7.14)

r,max

(4) Where the angle between the axes of principal stress and the direction of the °

reinforcement, for members reinforced in two orthogonal directions, is significant (>15 ), then

s may be calculated from the following expression:

the crack spacing r,max

1 (7.15)

s = θ θ

r ,

max cos sin

+

s s

r ,

max,y r ,

max,z

where:

θ is the angle between the reinforcement in the y direction and the direction of the

principal tensile stress

s

s are the crack spacings calculated in the y and z directions respectively,

r,max,y r,max,z

according to 7.3.4 (3) A does not

(5) For walls subjected to early thermal contraction where the horizontal steel area, s

fulfil the requirements of 7.3.2 and where the bottom of the wall is restrained by a previously

s may be assumed to be equal to 1,3 times the height of the wall.

cast base, r,max

Note: Where simplified methods of calculating crack width are used they should be based on the properties

given in this Standard or substantiated by tests.

7.4 Deflection control

7.4.1 General considerations

(1)P The deformation of a member or structure shall not be such that it adversely affects its

proper functioning or appearance.

(2) Appropriate limiting values of deflection taking into account the nature of the structure, of

the finishes, partitions and fixings and upon the function of the structure should be established.

(3) Deformations should not exceed those that can be accommodated by other connected

elements such as partitions, glazing, cladding, services or finishes. In some cases limitation

may be required to ensure the proper functioning of machinery or apparatus supported by the

structure, or to avoid ponding on flat roofs.

Note: The limiting deflections given in (4) and (5) below are derived from ISO 4356 and should generally

result in satisfactory performance of buildings such as dwellings, offices, public buildings or factories. Care

should be taken to ensure that the limits are appropriate for the particular structure considered and that that

there are no special requirements. Further information on deflections and limiting values may be obtained from

ISO 4356.

(4) The appearance and general utility of the structure could be impaired when the calculated

sag of a beam, slab or cantilever subjected to quasi-permanent loads exceeds span/250. The

sag is assessed relative to the supports. Pre-camber may be used to compensate for some or

126 EN 1992-1-1:2004 (E)

all of the deflection but any upward deflection incorporated in the formwork should not generally

exceed span/250.

(5) Deflections that could damage adjacent parts of the structure should be limited. For the

deflection after construction, span/500 is normally an appropriate limit for quasi-permanent

loads. Other limits may be considered, depending on the sensitivity of adjacent parts.

(6) The limit state of deformation may be checked by either:

- by limiting the span/depth ratio, according to 7.4.2 or

- by comparing a calculated deflection, according to 7.4.3, with a limit value

Note: The actual deformations may differ from the estimated values, particularly if the values of applied

moments are close to the cracking moment. The differences will depend on the dispersion of the material

properties, on the environmental conditions, on the load history, on the restraints at the supports, ground

conditions, etc.

7.4.2 Cases where calculations may be omitted

(1)P Generally, it is not necessary to calculate the deflections explicitly as simple rules, for

example limits to span/depth ratio may be formulated, which will be adequate for avoiding

deflection problems in normal circumstances. More rigorous checks are necessary for members

which lie outside such limits, or where deflection limits other than those implicit in simplified

methods are appropriate.

(2) Provided that reinforced concrete beams or slabs in buildings are dimensioned so that they

comply with the limits of span to depth ratio given in this clause, their deflections may be

considered as not exceeding the limits set out in 7.4.1 (4) and (5). The limiting span/depth ratio

may be estimated using Expressions (7.16.a) and (7.16.b) and multiplying this by correction

factors to allow for the type of reinforcement used and other variables. No allowance has been

made for any pre-camber in the derivation of these Expressions.

⎡ ⎤

3

ρ ρ

⎛ ⎞

l 2

⎢ ⎥ ρ ρ

⎜ ⎟

= + + − ≤

K f f

0 0

11 1

,

5 3

,

2 1 (7.16.a)

if

⎜ ⎟ 0

ρ ρ

⎢ ⎥

ck ck

d ⎝ ⎠

⎣ ⎦

⎡ ⎤

ρ ρ

l 1 ' ρ ρ

= + +

K f f

⎢ ⎥

0

11 1

,

5 if > (7.16.b)

0

ρ ρ ρ

ck ck

d ' 12

⎣ ⎦

0

where:

l/d is the limit span/depth

K is the factor to take into account the different structural systems

ρ √ -3

f

is the reference reinforcement ratio = 10

0 ck

ρ is the required tension reinforcement ratio at mid-span to resist the moment due to

the design loads (at support for cantilevers)

ρ ´ is the required compression reinforcement ratio at mid-span to resist the moment due

to design loads (at support for cantilevers)

f is in MPa units

ck

Expressions (7.16.a) and (7.16.b) have been derived on the assumption that the steel stress,

under the appropriate design load at SLS at a cracked section at the mid-span of a beam or

f = 500 MPa).

slab or at the support of a cantilever, is 310 MPa, (corresponding roughly to yk 127

EN 1992-1-1:2004 (E)

Where other stress levels are used, the values obtained using Expression (7.16) should be

σ

multiplied by 310/ . It will normally be conservative to assume that:

s

σ A A

310 / = 500 /(f / ) (7.17)

s yk s,req s,prov

where:

σ is the tensile steel stress at mid-span (at support for cantilevers) under the design

s load at SLS

A is the area of steel provided at this section

s,prov

A is the area of steel required at this section for ultimate limit state

s,req

For flanged sections where the ratio of the flange breadth to the rib breadth exceeds 3, the

l/d

values of given by Expression (7.16) should be multiplied by 0,8.

For beams and slabs, other than flat slabs, with spans exceeding 7 m, which support partitions

l/d

liable to be damaged by excessive deflections, the values of given by Expression (7.16)

l (l in metres, see 5.3.2.2 (1)).

should be multiplied by 7 / eff eff

For flat slabs where the greater span exceeds 8,5 m, and which support partitions liable to be

l/d

damaged by excessive deflections, the values of given by Expression (7.16) should be

l (l in metres).

multiplied by 8,5 / eff eff

K K

Note: Values of for use in a Country may be found in its National Annex. Recommended values of are

σ

given in Table 7.4N. Values obtained using Expression (7.16) for common cases (C30, = 310 MPa, different

s

ρ ρ

= 0,5 % and = 1,5 %) are also given.

structural systems and reinforcement ratios

Table 7.4N: Basic ratios of span/effective depth for reinforced concrete members without axial

compression Concrete highly stressed Concrete lightly stressed

K ρ ρ

Structural System = 1,5% = 0,5%

Simply supported beam, one- or 20

14

two-way spanning simply 1,0

supported slab

End span of continuous beam or

one-way continuous slab or two- 26

18

1,3

way spanning slab continuous over

one long side

Interior span of beam or one-way 30

20

1,5

or two-way spanning slab

Slab supported on columns without 24

17

1,2

beams (flat slab) (based on longer

span) 8

6

0,4

Cantilever

Note 1: The values given have been chosen to be generally conservative and calculation may frequently

show that thinner members are possible.

Note 2: For 2-way spanning slabs, the check should be carried out on the basis of the shorter span. For

flat slabs the longer span should be taken.

Note 3: The limits given for flat slabs correspond to a less severe limitation than a mid-span deflection of

span/250 relative to the columns. Experience has shown this to be satisfactory.

The values given by Expression (7.16) and Table 7.4N have been derived from results of a parametric study

made for a series of beams or slabs simply supported with rectangular cross section, using the general

128 EN 1992-1-1:2004 (E)

approach given in 7.4.3. Different values of concrete strength class and a 500 MPa characteristic yield strength

were considered. For a given area of tension reinforcement the ultimate moment was calculated and the quasi-

permanent load was assumed as 50% of the corresponding total design load. The span/depth limits obtained

satisfy the limiting deflection given in 7.4.1(5).

7.4.3 Checking deflections by calculation

(1)P Where a calculation is deemed necessary, the deformations shall be calculated under

load conditions which are appropriate to the purpose of the check.

(2)P The calculation method adopted shall represent the true behaviour of the structure under

relevant actions to an accuracy appropriate to the objectives of the calculation.

(3) Members which are not expected to be loaded above the level which would cause the

tensile strength of the concrete to be exceeded anywhere within the member should be

considered to be uncracked. Members which are expected to crack, but may not be fully

cracked, will behave in a manner intermediate between the uncracked and fully cracked

conditions and, for members subjected mainly to flexure, an adequate prediction of behaviour is

given by Expression (7.18):

α ζα ζ α

= + (1 - ) (7.18)

II I

where α is the deformation parameter considered which may be, for example, a

α

strain, a curvature, or a rotation. (As a simplification, may also be taken as a

deflection - see (6) below)

α α

, are the values of the parameter calculated for the uncracked and fully

I II

cracked conditions respectively

ζ is a distribution coefficient (allowing for tensioning stiffening at a section) given by

Expression (7.19):

2

⎛ ⎞

σ

ζ β ⎜ ⎟

sr

= 1 - (7.19)

⎜ ⎟

σ

⎝ ⎠

s

ζ = 0 for uncracked sections

β is a coefficient taking account of the influence of the duration of the loading

or of repeated loading on the average strain

= 1,0 for a single short-term loading

= 0,5 for sustained loads or many cycles of repeated loading

σ is the stress in the tension reinforcement calculated on the basis of a

s

cracked section

σ is the stress in the tension reinforcement calculated on the basis of a

sr cracked section under the loading conditions causing first cracking

σ σ M N M

Note: / may be replaced by /M for flexure or /N for pure tension, where is the cracking moment

sr s cr cr cr

N

and is the cracking force.

cr

(4) Deformations due to loading may be assessed using the tensile strength and the effective

modulus of elasticity of the concrete (see (5)). 129

EN 1992-1-1:2004 (E)

Table 3.1 indicates the range of likely values for tensile strength. In general, the best estimate

f is used. Where it can be shown that there are no axial

of the behaviour will be obtained if ctm

tensile stresses (e.g. those caused by shrinkage or thermal effects) the flexural tensile strength,

f , (see 3.1.8) may be used.

ctm,fl

(5) For loads with a duration causing creep, the total deformation including creep may be

calculated by using an effective modulus of elasticity for concrete according to Expression

(7.20): E

=

E cm (7.20)

( )

ϕ

+ ∞

c ,

eff t

1 , 0

where:

ϕ ∞

( ,t ) is the creep coefficient relevant for the load and time interval (see 3.1.3)

0

(6) Shrinkage curvatures may be assessed using Expression (7.21):

S

1 ε α

= (7.21)

Ι

cs e

r cs

where: is the curvature due to shrinkage

1/r cs

ε is the free shrinkage strain (see 3.1.4)

cs

S is the first moment of area of the reinforcement about the centroid of the section

Ι is the second moment of area of the section

α is the effective modular ratio

e α E E

= /

e s c,eff

Ι

S and should be calculated for the uncracked condition and the fully cracked condition, the

final curvature being assessed by use of Expression (7.18).

(7) The most rigorous method of assessing deflections using the method given in (3) above is

to compute the curvatures at frequent sections along the member and then calculate the

deflection by numerical integration. In most cases it will be acceptable to compute the deflection

twice, assuming the whole member to be in the uncracked and fully cracked condition in turn,

and then interpolate using Expression (7.18).

Note: Where simplified methods of calculating deflections are used they should be based on the properties

given in this Standard and substantiated by tests.

130 EN 1992-1-1:2004 (E)

SECTION 8 DETAILING OF REINFORCEMENT AND PRESTRESSING TENDONS -

GENERAL

8.1 General

(1)P The rules given in this Section apply to ribbed reinforcement, mesh and prestressing

tendons subjected predominantly to static loading. They are applicable for normal buildings and

bridges. They may not be sufficient for:

- elements subjected to dynamic loading caused by seismic effects or machine vibration,

impact loading and

- to elements incorporating specially painted, epoxy or zinc coated bars.

Additional rules are provided for large diameter bars.

(2)P The requirements concerning minimum concrete cover shall be satisfied (see 4.4.1.2).

(3) For lightweight aggregate concrete, supplementary rules are given in Section 11.

(4) Rules for structures subjected to fatigue loading are given in 6.8.

8.2 Spacing of bars

(1)P The spacing of bars shall be such that the concrete can be placed and compacted

satisfactorily for the development of adequate bond.

(2) The clear distance (horizontal and vertical) between individual parallel bars or horizontal

⋅bar

k k

layers of parallel bars should be not less than the maximum of diameter, (d + mm) or

g

1 2

d

20 mm where is the maximum size of aggregate.

g k k

Note: The value of and for use in a Country may be found in its National Annex. The recommended

1 2

values are 1 and 5 mm respectively.

(3) Where bars are positioned in separate horizontal layers, the bars in each layer should be

located vertically above each other. There should be sufficient space between the resulting

columns of bars to allow access for vibrators and good compaction of the concrete.

(4) Lapped bars may be allowed to touch one another within the lap length. See 8.7 for more

details.

8.3 Permissible mandrel diameters for bent bars

(1)P The minimum diameter to which a bar is bent shall be such as to avoid bending cracks in

the bar, and to avoid failure of the concrete inside the bend of the bar.

(2) In order to avoid damage to the reinforcement the diameter to which the bar is bent

φ .

(Mandrel diameter) should not be less than m,min

φ

Note: The values of for use in a Country may be found in its National Annex. The recommended values

m,min

are given in Table 8.1N. 131

EN 1992-1-1:2004 (E)

Table 8.1N: Minimum mandrel diameter to avoid damage to reinforcement

a) for bars and wire Minimum mandrel diameter for

Bar diameter bends, hooks and loops (see Figure 8.1)

φ φ

≤ 16 mm 4

φ φ

> 16 mm 7

b) for welded bent reinforcement and mesh bent after welding

Minimum mandrel diameter

d or

or φ φ

d : 5

3

φ φ

d

5 < 3 or welding within the curved zone:

φ

20 φ

Note: The mandrel size for welding within the curved zone may be reduced to 5

where the welding is carried out in accordance with prEN ISO 17660 Annex B

(3) The mandrel diameter need not be checked to avoid concrete failure if the following

conditions exist: φ

- the anchorage of the bar does not require a length more than 5 past the end of the bend;

- the bar is not positioned at the edge (plane of bend close to concrete face) and there is a

φ

cross bar with a diameter inside the bend.

- the mandrel diameter is at least equal to the recommended values given in Table 8.1N.

φ

Otherwise the mandrel diameter, , should be increased in accordance with Expression

m,min

(8.1)

φ φ

≥ ))

F f

((1/a ) +1/(2 / (8.1)

m,min bt b cd

where:

F is the tensile force from ultimate loads in a bar or group of bars in contact at the

bt start of a bend

a for a given bar (or group of bars in contact) is half of the centre-to-centre distance

b between bars (or groups of bars) perpendicular to the plane of the bend. For a

a

bar or group of bars adjacent to the face of the member, should be taken as the

b

φ /2

cover plus

f

The value of should not be taken greater than that for concrete class C55/67.

cd

8.4 Anchorage of longitudinal reinforcement

8.4.1 General

(1)P Reinforcing bars, wires or welded mesh fabrics shall be so anchored that the bond forces

are safely transmitted to the concrete avoiding longitudinal cracking or spalling. Transverse

reinforcement shall be provided if necessary.

(2) Methods of anchorage are shown in Figure 8.1 (see also 8.8 (3)).

132 EN 1992-1-1:2004 (E)

φ

≥5 α l b,eq

o o

≤ α

90 < 150

l

a) Basic tension anchorage length, , b) Equivalent anchorage length for

b

for any shape measured along the standard bend

centreline

φ

≥ 5 φ φ

≥ φ

0.6 5

≥150 t

l l

l b,eq b,eq

b,eq

c) Equivalent anchorage d) Equivalent anchorage e) Equivalent anchorage

length for standard hook length for standard loop length for welded

transverse bar

Figure 8.1: Methods of anchorage other than by a straight bar

(3) Bends and hooks do not contribute to compression anchorages.

(4) Concrete failure inside bends should be prevented by complying with 8.3 (3).

(5) Where mechanical devices are used the test requirements should be in accordance with the

relevant product standard or a European Technical Approval.

(6) For the transmission of prestressing forces to the concrete, see 8.10.

8.4.2 Ultimate bond stress

(1)P The ultimate bond strength shall be sufficient to prevent bond failure.

f

(2) The design value of the ultimate bond stress, , for ribbed bars may be taken as:

bd

f η η f

= 2,25 (8.2)

bd 1 2 ctd

where:

f is the design value of concrete tensile strength according to 3.1.6 (2)P. Due to the

ctd f

increasing brittleness of higher strength concrete, should be limited here to the

ctk,0,05

value for C60/75, unless it can be verified that the average bond strength increases

above this limit

η is a coefficient related to the quality of the bond condition and the position of the bar

1 during concreting (see Figure 8.2):

η = 1,0 when ‘good’ conditions are obtained and

1 133

EN 1992-1-1:2004 (E)

η = 0,7 for all other cases and for bars in structural elements built with slip-forms,

1

unless it can be shown that ‘good’ bond conditions exist

η is related to the bar diameter:

2 η φ ≤

= 1,0 for 32 mm

2

η φ φ

= (132 - )/100 for > 32 mm

2 A A

α 250

α ≤

≤ h

90º c) > 250 mm A Direction of concreting

a) 45º A

A 300 h

h ≤

h h

b) 250 mm d) > 600 mm

a) & b) ‘good’ bond conditions c) & d) unhatched zone – ‘good’ bond conditions

for all bars hatched zone – ‘poor’ bond conditions

Figure 8.2: Description of bond conditions

8.4.3 Basic anchorage length

(1)P The calculation of the required anchorage length shall take into consideration the type of

steel and bond properties of the bars. σ

l A

(2) The basic required anchorage length, , for anchoring the force in a straight bar

b,rqd s. sd

f follows from:

assuming constant bond stress equal to bd

φ σ

l = f

( / 4) ( / ) (8.3)

b,rqd sd bd

σ

Where is the design stress of the bar at the position from where the anchorage is

sd

measured from.

f

Values for are given in 8.4.2.

bd l l

(3) For bent bars the basic anchorage length, , and the design length, , should be

b bd

measured along the centre-line of the bar (see Figure 8.1a). φ

(4) Where pairs of wires/bars form welded fabrics the diameter, , in Expression (8.3) should

φ φ √2.

=

be replaced by the equivalent diameter n

134 EN 1992-1-1:2004 (E)

8.4.4 Design anchorage length l

(1) The design anchorage length, , is:

bd

l = α α α α α l l (8.4)

bd 1 2 3 4 5 b,rqd b,min

α , α , α α α

where , and are coefficients given in Table 8.2:

1 2 3 4 5

α is for the effect of the form of the bars assuming adequate cover (see Figure 8.1).

1

α is for the effect of concrete minimum cover (see Figure 8.3)

2

c a a

1 c c

1

c

a) Straight bars b) Bent or hooked bars c) Looped bars

c c c) c c c c

= min (a/2, , = min (a/2, ) =

d 1 d 1 d

c

Figure 8.3: Values of for beams and slabs

d

α is for the effect of confinement by transverse reinforcement

3 φ φ

α )

is for the influence of one or more welded transverse bars ( > 0,6 along the design

4 t

l

anchorage length (see also 8.6)

bd

α is for the effect of the pressure transverse to the plane of splitting along the design

5 anchorage length ≥

α α

The product (α ) 0,7 (8.5)

2 3 5

l is taken from Expression (8.3)

b,rqd

l is the minimum anchorage length if no other limitation is applied:

b,min φ

l

- for anchorages in tension: > max{0,3l ; 10 ; 100 mm} (8.6)

b,min b,rqd φ

l > max{0,6l ; 10 ; 100 mm} (8.7)

- for anchorages in compression: b,min b,rqd

(2) As a simplified alternative to 8.4.4 (1) the tension anchorage of certain shapes shown in

l l

Figure 8.1 may be provided as an equivalent anchorage length, . is defined in this figure

b,eq b,eq

and may be taken as:

α α

l

- for shapes shown in Figure 8.1b to 8.1d (see Table 8.2 for values of )

1 b,rqd 1

α α

l

- for shapes shown in Figure 8.1e (see Table 8.2 for values of ).

4 b,rqd 4

where

α α

and are defined in (1)

1 4

l is calculated from Expression (8.3)

b,rqd 135


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DESCRIZIONE APPUNTO

P Eurocode 2 applies to the design of buildings and civil engineering works in plain, reinforced and prestressed concrete. 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.P Eurocode 2 is only concerned with the requirements for resistance, serviceability, durability and fire resistance of concrete structures. Other requirements, e.g. concerning thermal or sound insulation, are not considered.


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 vipviper 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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