FSMS Concrete2 Calculation

Simplified calculation methodsfollowing the EN 1992-1-2 standard

Patrick Bamonte

Department of Civil and Environmental Engineering - DICAPolitecnico di Milano

Introduction

many civil structures of great importance are at risk of firethe development of fire and the consequent rise oftemperature brings in two harmful effects:

I decay of stiffness and strength of the building materialsI thermal dilations of the structural members

these circumstances oblige the designer to take into dueaccount the possibility of a fire event when evaluating thebearing capacity

⇒ EN 1992-1-2 (EC2 Fire Design)

The EN 1992-1-2 standard Introduction 2 / 55

Some examples

New York (USA), WTC, September 2001:structural collapse due to a variety of factors


Some examples

Gretzenbach (Switzerland), underground parking,November 2004: collapse due to punching


The EN 1992-1-2 standard

1 General2 Basis of design3 Material properties4 Design procedures5 Tabulated data6 High strength concrete (HSC)7 Annexes


General requirements

Structural elements can be checked in terms of their

separating function: E criterion (integrity) or I criterion(insulation)bearing capacity: R criterionthe combination of the three criteria: REI

In the following reference will be made only to the control of thebearing capacity (R criterion).

The EN 1992-1-2 standard Basis of design 6 / 55

Design values of material properties

The design values of the material properties (fk or Ek ) can bedetermined as follows:

Xd ,fi = kθXk/γM,fi

where

Xk is the characteristic value of a strength or deformationpropertykθ is the reduction factor dependent on the materialtemperature θγM,fi is the partial safety factor for the relevant materialproperty (recommended value = 1)


Thermal and physical properties at elevatedtemperaturesThe design thermal properties (λ and c) can be determined asfollows:

Xd ,fi = Xk ,θ/γM,fi if an increase in Xk ,θ is favourable

Xd ,fi = γM,fiXk ,θ if an increase in Xk ,θ is unfavourable

where

Xk ,θ is the characteristic value of the propertyγM,fi is the same coefficient used for the mechanicalproperties


Safety verification

The verification of the safety in terms of bearing capacity, at anygiven fire duration t , can be written as follows:

Ed ,fi ≤ Rd ,t ,fi

where

Ed ,fi is the design effect of actions for the fire situation,determined in accordance with EN 1991-1-2, includingeffects of thermal expansions and deformationsRd ,t ,fi is the corresponding design resistance in the firesituation


Design actions

Actions acting on the structure in fire condition can bedetermined as follows:

summing the permanent loads Gk and the variable loadsQk ,i amplified with the combination coefficients ψ1,1 and ψ1,2

reducing the combination at ambient temperature usingcoefficient ηfi :

Ed ,fi = ηfiEd where ηfi =Gk + ψfiQk ,1

γGGk + γQ,1Qk ,1

where Ed are the actions at ambient temeprature and ψfi isthe coefficient for a frequent combination.


Design actionsrailroad

bridges*

0.2

0.5

0.7

ηfi

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Qk,1 / Gk

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ψ1,1 = 0.9

civil

buildings

storage

building*

reduction coefficient ηfi for γG = 1.35 and γQ = 1.5


Concrete

The behaviour of concrete in compression is represented bymeans of a temperature-dependent stress-strain diagram, whichdepends on

peak stress fc,θdeformation at the peak stress εc1,θ

deformation at ultimate εcu,θ

The variation of these characteristics depends only upon thetype of aggregate in the concrete (siliceous or calcareous); noprovisions are given for light-weight concrete.

The EN 1992-1-2 standard Mechanical properties 12 / 55

Concrete

linear softening branch

cubic softening branch

εc [‰]

σc / fc

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1.0

Sargin

P-R

Fire Design

comparison between different σ-ε diagrams for concrete


Concrete

0 200 400 600 800 1000 1200

temperature θ [°C]

0.0

0.2

0.4

0.6

0.8

1.0

1.2f c

,θ /

fck

calcareous aggregate

siliceous aggregate

compressive strength decay for concrete


Concrete

0 100 200 300 400 500 600 700 800

temperature [°C]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

f c /

fc2

0

EN 1992-1-2

experimental strength decay for different concretes


Concrete

0 200 400 600 800 1000 1200


0

10

20

30

40

50

εc1

,θ ,

εcu

,θ [

‰]

εc1,θ

εcu,θ

peak and ultimate deformation of concrete as a function oftemperature


Concrete

T = 600 °C

T = 800 °C

T = 1000 °C

T = 400 °C

T = 200 °C

εc [‰]

σc / fc

T = 20 °C

0 5 10 15 20 25 30 35 40 45

0

0.2

0.4

0.6

0.8

1.0

temperature-dependent stress-strain curves


ConcreteThe tensile strength, where applicable (e.g. shear strength ofUNREINFORCED members), can be reduced by means of alinear decay law:

fck ,t(θ) = kck ,t(θ) · fck ,t

where

kck ,t(θ) = 1.0 for 20°C ≤ θ ≤ 100°C

kck ,t(θ) = 1.0− θ − 100500

for 100°C ≤ θ ≤ 600°C

Beyond 600°C no tensile stresses are allowed.


Concrete

0 100 200 300 400 500 600


0.0

0.2

0.4

0.6

0.8

1.0

1.2

f ctk

,θ /

fctk

kck,t(θ)

tensile strength decay for concrete


Reinforcing steel

The behaviour of reinforcing steel can be determined on thebasis of three parameters:

elastic modulus Es,θ

proportional limit fsp,θ

maximum stress level fsy ,θ

The decay of the parameters is different depending on

the ductility class (X or N)the production process (hot rolled or cold worked)


Reinforcing steel

class N

0 200 400 600 800 1000 1200


0.0

0.2

0.4

0.6

0.8

1.0

1.2f s

y,θ /

fyk

class X

hot rolled

cold worked

decay of the maximum stress for reinforcing steel


Prestressing steel

The behaviour of prestressing steel can be determined on thebasis of five parameters:

elastic modulus Ep,θ

proportional limit fpp,θ

peak stress fpy ,θ

deformation at peak stress εpt ,θ and at ultimate εpu,θ

The decay of the parameters is different depending on theproduction process.


Prestressing steel

0 200 400 600 800 1000 1200


0.0

0.2

0.4

0.6

0.8

1.0

1.2f p

y,θ /

(0

.9f p

k)

quenched and tempered

(bars)

cold worked

(wires and strands)

decay of the maximum stress for prestressing steel


Calculation methodsDifferent calculation methods are allowed by the standard. Theyare characterized by an increasing computational effort and thusby an increasing degree of accuracy:

tabulated datasimplified calculation methods for specific structuralelementsadvanced calculation methods, to simulate the behaviour of

I structural elementsI parts of the structureI the entire structure

Specific provisions are given for particular phenomena, such asexplosive spalling, the falling-off of concrete chips anddiscontinuity zones.

The EN 1992-1-2 standard Calculation methods 24 / 55

Tabulated data

The tabulated data have been developed on an empirical basisconfirmed by experience and theoretical evaluation of tests. Themain assumptions are:

standard fire of limited duration (t ≤ 240’)concrete density between 2000 and 2600 kg/m3

siliceous aggregate

The tables can be used also for other types of concrete(calcareous or light-weight); using tabulated data also allows aproper control of spalling.

The EN 1992-1-2 standard Tabulated data 25 / 55

First example: square columncivil builidingfire exposure on all sidesQk1/Gk = 0.3 ⇒ ηfi = 0.7axial force at ambient temperature: NSd = 560 kNfcd = fck = 30 MPa, fyd = fyk = 500 MPa

pinned

pinned

h = 3.00 m

30 x 30 cm

4 bars 16 mm

net cover 30 mm


Calculation of NRd at 20°C

Design concrete strength:

fcd = 0.85fck

γc= 19 MPa fsd =

fsk

γs= 435 MPa

Bearing capacity in pure compression:

NRd = fcdAc + fsdAs = 19 · 90000 + 435 · 804 = 1784 kN

Safety verification at 20°C:

NRd = 1784 kN ≥ NSd = 560 kN


Method A

the axial force in fire conditions is evaluated:

NSd ,fi = ηfi · NSd = 0.7 · 560 = 393 kN

the ratio µfi between NSd ,fi and NRd is evaluated:

µfi = NSd ,fi/NRd = 0.22

the minimum dimensions of the structural element are takenfrom the tables, or, in our case, the R-class:

µfi ≈ 0.20,bmin = 300 mm,a = 38 mm ⇒ R 90


Method A

determination of the R-class of the element using Table 5.2a(applicable for rectangular and circular columns)


Method B

the load level n in fire condition is evaluated:

n =NEd ,fi

0.7 (fcdAc + fsdAs)= 0.31

the mechanical reinforcement ratio ω is evaluated:

ω =Asfsd

Acfcd= 0.24

the minimum dimensions of the structural element (or theR-class) are taken from the tables:

n = 0.31, ω = 0.24 ⇒ R 90


Method B

Standard fi re

Mec hanic al reinforce ment

Minimum dimensions (mm). Colu mn width bmin/axis distanc e a

resistance ratio n = 0,15 n = 0 ,3 n = 0,5 n = 0,7

1 2 3 4 5 6

R 30

R 60

R 9 0

R 1 20

R 1 80

R 2 40

0,100 0,500 1,000

0,100 0,500 1,000

0,100 0,500 1,000

0,100 0,500 1,000

0,100 0,500 1,000

0,100 0,500 1,000

150/25* 150/25* 150/25*

150/30:2 00/25*

150/25* 150/25*

200/40:2 50/25* 150/35:2 00/25*

200/25*

250/50:3 50/25* 200/45:3 00/25* 200/40:2 50/25*

400/50:5 00/25* 300/45:4 50/25* 300/35:4 00/25*

500/60:5 50/25* 450/45:5 00/25* 400/45:5 00/25*

150/25* 150/25* 150/25*

200/40:3 00/25* 150/35:2 00/25* 150/30:2 00/25*

300/40:4 00/25* 200/45:3 00/25* 200/40:3 00/25*

400/50:5 50/25* 300/45:5 50/25* 250/50:4 00/25*

500/60:5 50/25* 450/50:6 00/25* 450/50:5 50/25*

550/40:6 00/25* 550/55:6 00/25* 500/40:6 00/30

200/30:2 50/25*

150/25* 150/25*

300/40:5 00/25* 250/35:3 50/25* 200/40:4 00/25*

500/50:5 50/25* 300/45:5 50/25* 250/40:5 50/25*

550/25*

450/50:6 00/25* 450/45:6 00/30

550/60:6 00/30 500/60:6 00/50 500/60:6 00/45

600/75 600/70 600/60

300/30:3 50/25* 200/30:2 50/25* 200/30:3 00/25*

500/25*

350/40:5 50/25* 300/50:6 00/30

550/40:6 00/25* 500/50:6 00/40 500/50:6 00/45

550/60:6 00/45 500/60:6 00/50

600/60

(1) 600/75

(1)

(1) (1) (1)

* Normall y the co ver re quired by EN 1992-1-1 will control.

determination of the R-class of the element using Table 5.2b(applicable for rectangular and circular columns)


Other elements

Tabulated data are available for other elements:

load-bearing wallsflat and ribbed slabs (continuous or simply-supported)beams (continuous or simply-supported)non load-bearing walls

For the first two elements, which are bidimensional, the tablesspecify the REI-class; for beams only the parameter R isspecified, whereas for non load-bearing walls the EI-class isgiven.


The 500°C isotherm method

This is a very popular and simple method (also called “swedish”method), based on the following assumptions:

standard fire curve (or equivalent)dimensions greater than a minimumreduction of the effective section:

I concrete with θ ≥ 500°C has fc = 0I the decay of the steel strength is explicitly evaluated, on the

basis of the temperature θ

The EN 1992-1-2 standard Simplified calculation methods 33 / 55

The 500°C isotherm methody

x

T = T(x=0, y, t)b

d

fc = fc (20°C)

T fc

y y

effective section

500°C

representation of the 500°C isotherm method


The 500°C isotherm method

The calculation procedure can be summarized as follows:

the position of the 500°C isotherm on the section is workedoutthe width bfi and the effective depth dfi of the reduced (or“effective”) section (enveloped by the 500°C isotherm) areworked outthe temperature θsi of each rebar is worked outthe decay ks(θsi) of the strength of each rebar is worked outthe bearing capacity is evaluated, by means of the usualcalculation procedures

It is worth noting that all bars can be taken into account, alsothose falling outside the effective section.


Decay of the reinforcing steel

0 200 400 600 800 1000 1200


0.0

0.2

0.4

0.6

0.8

1.0

1.2f s

y,θ /

fyk

εs,fi < 2%

εs,fi > 2%

decay of the reinforcing steel in simplified methods


Worked-out examplebeam in pure bending (no axial force and shear)three sides exposed to the fire

fcd = fck = 25 MPa, fyd = fyk = 500 MPa

240 3030

450

300

400

16 mm bars


Evaluation of MRd at 0’ (20°C)The calculation is performed under the assumption that

concrete has reached its ultimate strain εcu = 3.5‰all the rebars are yielded (σsi = fyk )

As2

As3

d2 = d3

T2

T1

C

b

d1

MRd

As1

εcu

εs1

εs2

x


Evaluation of MRd at 0’ (20°C)total compression and tension:

C = 0.8fckbx e T =3∑1

fykiAsi

equilibrium in the horizontal direction:

T = C ⇒ x =

∑31 fykiAsi

0.8fckb= 50 mm

rotation equilibrium with respect to the centroid of thecompression zone:

MRd =3∑1

fykiAsi(di − 0.4x) = 206 kNm


Reduced section at 60’ fire durationReference is made to the thermal profiles of the standard fort = 60’ and rectangular section 300 × 600. . .

T = 420°C, fyk1 = 480 MPa

T = 520°C, fyk2 = 365 MPa

T < 400°C, fyk3 = 500 MPa

bfi = 260 mm


Evaluation of MRd at 60’ fire durationtotal compression and tension:

C = 0.8fckbfix e T =3∑1

fykiAsi


T = C ⇒ x =

∑31 fykiAsi

0.8fckbfi= 52 mm


MRd =3∑1



Reduced section at 120’ fire durationThe thermal profiles t = 120’ lead to hgiher temperatures in therebars, and thus to lower values of the yield strength.

T = 630°C, fyk1 = 200 MPa

T = 740°C, fyk2 = 90 MPa

T = 450°C, fyk3 = 440 MPa

bfi = 220 mm


Evaluation of MRd at 120’ fire durationtotal compression and tension:

C = 0.8fckbfix e T =3∑1

fykiAsi


T = C ⇒ x =

∑31 fykiAsi

0.8fckbfi= 33 mm


MRd =3∑1



Worked-out example

The geometry is modified, by reducing the concrete cover:

100

150 7575

450

300

400

16 mm bars


Reduced section at 60’ fire duration

T < 400°C, fyk1 = 500 MPa

T < 400°C, fyk2 = 500 MPa

T < 400°C, fyk3 = 500 MPa

bfi = 260 mm

The calculation of the resisting moment yields MRd = 206 kNm(= MRd at 0’).


Comparisons

100%

0 30 60 90 120 150 180

time [min]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

MR

d,t /

MR

d,0 80%87%

42%

The increase of the concrete cover has beneficial effects,because of the enhanced protection for the rebars.


Role of the steel

fym,t / fym,0

MRd,t / MRd,0

0 30 60 90 120 150 180

time [min]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

MR

d,t /

MR

d,0 ,

fym

,t /

fym

,0

The reduction of MRd is due solely to the steel, when thebehaviour is governed by steel yielding.


The zone method

This method, also called “Hertz” method, is based on thefollowing assumptions:

the section is divided in several zones, parallel to theisothermal linesthe damage of the superficial zone is evaluated, on thebasis of the average temperatures of the different zonesthe reduced section is identifiedthe usual procedures are applied to the reduced section


The zone method

T-beamrectangular beam

wall

slab

typical zones for common elements


Temperatures at 60’ fire durationThe average temperature is evaluated in the different zones(here there are 15), that are taken parallel to the isothermal linesin the compressed part of the section:

4-5 cm: Tm = 78°C

9-10 cm: Tm = 220°C

14-15 cm: Tm = 760°C


Damage of the concreteThe damage of the superficial layers of concrete is evaluated,on the basis of the temperatures.

kc(60)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

cm

0

200

400

600

800

1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

kc(120)

T(120)

T(60)

tem

pe

ratu

re [

°C]


Evaluation of the reduced section

The average damage is evaluated:

kcm =(1− 0.2/n)

n

n∑i=1

kc (θi)

The reduced width is calculated:

az = w[1− kcm

kc(θM)

]where M is usually on the axis of simmetry.


Reduced sectionThe reduced width, as a function of time, can be plotted:

0 30 60 90 120 150 180

time [min]

15

20

25

30

35

500°C isotherm method

zone method

eff

ective

wid

th [

cm

]


Another simplified method

In Appendix E a third method is proposed:

it applies to elements with distrinuted loads, designed bymeans of linear elastic calculation without or with limitedredistributioit can be applied whenever the tabulated data are notapplicablethe steel decay is evaluated as given before


Simply-supported beams and slabsThe verification requires that:

MEd ,fi ≤ MRd ,fi

The acting moment MEd ,fi is calculated as

MEd ,fi = wEd ,fi l2eff/8

The internal moment MRd ,fi is evaluated as

MRd ,fi = (γs/γs,fi)× ks(θ)×MEd(As,prov/As,req)

MEd is the moment acting on the section at 20°CAs,prov is the total area of steelAs,req is the minimum area required at room temperature


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FSMS Concrete2 Calculation