EC3 & EC4 Worked Examples

Project Title: EC3 & EC4 Worked Examples

Project Number:

Sheet 1 of 6

Rev: 02a

Subject: Base plate without bending moment

Made by/date: GHC/December 2004

Client:

Checked/date: CR/December 2004

Base plate without bending moment The following Codes have been used for this worked example: BS EN 1990, Basis of Structural Design, July 2002, with UK National Annex, March 2004 BS EN 1991-1-1, Eurocode 1 – Actions on structures – Part 1.1: General actions – Densities, self-weight, imposed loads for buildings, July 2002 BS EN 1992-1-1, Eurocode 2 – Design of concrete structures – Part 1.1: General rules and rules for buildings, April 2003 prEN 1993-1-8, Eurocode 3 – Design of steel structures – Part 1.8: Design of joints, December 2003 prEN 10025-2, Hot rolled products of non-alloy structural steels – Part 2: Technical delivery conditions for flat products, March 1998 Notes on European Standards BSEN denotes a European Standard that has been published by BSI prEN denotes a draft European standard that is not publicly available The following design guidance documents have been used for this worked example: SCI and BCSA, Joints in Steel Construction – Simple Connections, P212, 2002, SCI

Note on values contained in this worked example The computer software used to calculate the expressions given in this worked example does not round the values at intermediate stages during the calculation. Therefore some values given on the following sheets may appear to be ‘incorrect’ when determined using ‘rounded’ input values.

BRE and Buro Happold have made every effort to ensure the accuracy and quality of all the information in this document when first published. However, they can take no responsibility for the subsequent use of this information, nor for any errors or omissions it may contain. © Queen's Printer and Controller of Her Majesty's Stationery Office 2005


Project Number:

Sheet 2 of 6

Rev: 02a



Client:


1. Introduction The method given in Eurocode 3 uses the same approach as BS5950-1: 2000, that is the ‘effective are method’. The ‘T-stub model’ is used to determine the resistance in compression of the base plate and underlying ground / concrete. Design the base plate for a 254 x 254 x 89 UC in grade S275 steel (shown in Figure 1). leff.2

tf

c c

leff.1 tw

c

c

beff.2

Minimum 70 mm to centreline of bolt hole beff.1

c

c

Figure 1. Base plate dimensions 1.1.

Column section dimensions

Section: UC 254 x 254 x 89

z

tf r y

y

d

h

tw z b Figure 2. Section dimensions h = 260.30 mm

b = 256.30 mm

d = 200.30 mm

tw = 10.30 mm

tf = 17.30 mm

r = 12.70 mm

2

A = 113.31 cm


Project Number:

Sheet 3 of 6

Rev: 02



Client:


2. Loading

Table & clause numbers given

The design axial column load, NEd = 1000 kN

relate to

Note: The above value has been determined using the partial loading factors, load

EN1993-1-8

combination and combination factors given in Annex A of EN 1990. For brevity this

unless stated

process has not be included in this worked example (see other worked examples in this

otherwise.

series for the methodology to use).

3. Material properties 3.1.

Steel

For steel grade S275 with a thickness of between 16 mm and 40 mm 2

Yield strength is fy = 265 N/mm

prEN 10025-2 7.3 & Table 4

Note: The guidance given in prEN 10025-2 has been used to determine the yield strength for steel instead of that given in Table 3.1 of EN1993-1-1, as it is assumed that the UK National Annex to EN 1993-1-1 will specify the use of that standard instead of the values given in Table 3.1.

3.2.

Concrete

Compressive strength equals the Characteristic cylinder strength (fck) For concrete grade C40/50

EN 1992-1-1

2

fck = 40 N/mm

Design compressive strength is determined from f cd = " cc x ( f ck / ! C )

3.1.6(1)

Where: γC is the partial safety factor for concrete

EN1992-1-1

γC = 1.5 (For the persistent and transient design situations)

2.4.2.4(1) &

Note: Recommended value used for γC. This value may be altered by the UK National

Table 2.1

Annex to EN1992-1-1.

αcc is the coefficient taking account of the long term effects on the compressive strength and unfavourable effects resulting from the way the load is applied.

EN1992-1-1

αcc = 1.0

3.1.6(1)

Note: Recommended value used for αcc. This value may be altered by the UK National Annex to EN1992-1-1.

However, due to confinement of the concrete it is permissible to consider an enhanced characteristic strength. This is a function of the confinement stress σ2,

EN1992-1-1

however EN1992-1-1 gives no guidance on calculating the latter. Future guidance

3.1.9(2)

may be given in the National Annex to EN1992-1-1. For this example it is assumed that the enhanced compressive strength (fck.c) is: 2

fck.c = 1.2 × fck = 48 N/mm


Project Number:

Sheet 4 of 6

Rev: 02



Client:


Therefore the enhanced design compressive strength is: fcd.c = αcc × (fck.c / γC) = 32 N/mm

2

4. Design 4.1.

Size of effective area

Determine the required dimension by considering the axial load and the strength of the grout / concrete. The design bearing strength of the ‘support’, considering a flange is determined from:

f jd = (! j x FRdu ) /(b eff .1 x l eff .1 )

6.2.5(7) Eq. 6.6

Where: beff.1 and leff.1 are shown in Figure 1. βj is the foundation joint material coefficient βj = 2/3 Assuming that the characteristic strength of the grout will not be less than 0.2 times that of the concrete and the thickness of the grout will not be greater than 0.2 times the smallest width of the base plate. FRdu is the concentrated design resistance force given in EN1992,

FRdu = A c 0 x f cd x ( A c1 / A c 0 ) ! 3 x f cd x A c 0

6.7(2) Eq. 6.63

Where: Ac0 is the loaded area (taken as beff x leff for base plate) Ac1 is the maximum design distribution area with a similar shape to Ac0 (defined in Figure 6.29 of EN1992-1-1). However, for the case of a base plate A c1 = A c 0 in order to satisfy the criteria: •

The centre of Ac1 should be in the line of action passing through the centre of Ac0

•

Areas should not overlap

As A c1 = A c 0 and A c 0 = b eff .1 x l eff .1

FRdu = b eff .1 x l eff .1 x f cd Substituting this into equation 6.6 of EN1993-1-8 gives the design bearing strength of the ‘support’ as 2

fjd = 2/3 × fcd.c = 21.33 N/mm

6.2.5(7) Eq. 6.6

Therefore the bearing area required is: 2

NEd / fjd = 46875 mm Where:

NEd is the design axial column load, NEd = 1000 kN

Sheet 3


Project Number:

Rev: 02



Client:


The bearing area provided is:

(2 x (l eff .1 x b eff .1 ) + (l eff .2 x b eff .2 ) Where: beff.1, beff.2, leff.1 and leff.2 are defined in Figure 1.

l eff .1 x b eff .1 = (256.3 + (2 x c )) x (17.3 + (2 x c )) = ( 4 x c 2 ) + (547.2 x c ) + 4433.99 l eff .2 x b eff .2 = (260.3 ! (2 x 17.3) ! (2 x c )) x (10.3 + (2 x c )) = ( !4 x c 2 ) + ( 430.8 x c ) + 2327.71 Substituting the expressions for (leff.1 x beff.1) and (leff.2 x beff.2) into the above equation gives the bearing area of (978 x c ) + 6761.7 2

For the area required (46875 mm ) the dimension c equals: c = (46875 – 6761.7) / 978 = 41.02 mm This will provide a bearing area that is sufficiently large to avoid crushing of the concrete under the applied axial load. 4.2.

Sheet 5 of 6

Plan dimensions of the base plate

c = 41.02 mm

≥ 49.5 mm

≥ 70 mm ≥ 49.5 mm

≥ 70 mm c = 41.02 mm ≥ 50 mm

≥ 50 mm Figure 2. Corner dimensions of base plate Therefore the size of the base plate is not governed by dimension c, rather by the detailing requirements shown in Figure 2. Choose plate Width 260.3 + (49.5 x 2) + (50 x 2) = 459.3 mm say 460 mm Length 256.3 + ( 49.5 x 2) + (50 x 2) = 455.3 mm say 460 mm


Project Number:

Subject: Simply supported beam with full lateral restraint – Fire Limit State Client:

Sheet 1 of 13

Rev: 02

Made by/date: TL / August 2004 Checked/date: YW/ October 2004

Simply supported beam with full lateral restraint – Fire Limit State The following Eurocodes and pre-Eurocodes have been used for this worked example: BS EN 1990, Basis of Structural Design, July 2002, with UK National Annex, March 2004 BS EN 1991-1-1, Eurocode 1 – Actions on structures – Part 1.1: General actions – Densities, self-weight, imposed loads for buildings, July 2002 BS EN 1991-1-2, Eurocode 1 – Actions on structures – Part 1.2: General actions – Actions on structures exposed to fire, November 2002. prEN 1993-1-1, Eurocode 3 – Design of steel structures – Part 1.1: General rules and rules for buildings, December 2003 prEN 1993-1-2, Eurocode 3 – Design of steel structures – Part 1.2: General rules structural fire design, June 2004 Notes on European Standards BSEN denotes a European Standard that has been published by BSI prEN denotes a draft European standard that is not publicly available

Note on values contained in this worked example The computer software used to calculate the expressions given in this worked example does not round the values at intermediate stages during the calculation. Therefore some values given on the following sheets may appear to be ‘incorrect’ when determined using ‘rounded’ input values.



Project Number:


Sheet 2 of 13

Rev: 02


1. Introduction The principles of the design of steel structures for the fire limit state are set out in EN1993-1-2. Many of the concepts will be familiar to UK designers as the equivalent National standard, BS5950 Part 8, is, like EN1993-1-2, a limit state code which takes into account uncertainties in material strength and load distribution. It recognises the important influence of applied load on the performance in fire of structural steel elements. A number of routes of various degrees of complexity are available to the designer in order to provide the required performance. These range from a simple reliance on the results from standard fire tests on isolated members to a consideration of the physical parameters influencing fire development coupled with an analysis of the entire building. The design procedure is summarised in Figure 4. Any fire design must take into account the following three inter-dependent relationships: •

The fire model used to assess the structural performance

•

The thermal response of the structure

•

The response of the structure

1.1.

Choice of fire model

The fire model (thermal actions) adopted for design may be either nominal or physically based. Examples of nominal fire models are the standard (ISO 834, BS476 Part 20) fire curve used for furnace tests on structural elements, the external fire curve used for members subject to external flaming from openings and the hydrocarbon curve used for offshore, petrochemical or other extreme exposure conditions. Examples of physically based thermal actions include empirically based parametric fire curves, localised fires or mathematically based simulations of the anticipated thermal exposure. For the purpose of this document the thermal exposure will be restricted to the familiar standard time-temperature response. 1.2.

Thermal response of the structure

Once the fire model has been chosen consideration should be given to the temperature development within the structural member(s). Heat transfer to structural elements is a complex process which requires a solution of the equations for energy conservation and mass balance. However, for structural steel the situation can be simplified to an assumption of uniform temperature through the cross-section. Tabulated values are available in National standards based on specific fire resistance periods. Alternatively iterative simple calculation models (suitable for use in a spreadsheet) are presented for unprotected and protected steel to enable the designer to calculate the temperature of the member at a specific time period dependent on the fire model used. Advanced calculation models are available to determine more accurately the heat transfer to structural members. However, the use of such advanced methods is beyond the scope of this simplified guidance. 1.3.

Structural response

The basic concept of the simplified structural fire engineering design procedure in the Eurocode is to provide data on the reduction in strength and stiffness at elevated temperature to enable the designer to utilise familiar methods to assess the resistance at specified time or temperature steps. Reduced partial


Project Number:


Sheet 3 of 13

Rev: 02


factors for loading at the fire limit state are used to assess performance against the reduced resistance to determine whether additional protection or an alternative design (such as the use of a larger section than required for ambient conditions) should be used. Again advanced non-linear methods are available to determine more precisely the response of a frame or an entire building to the effects of fire but the use of such techniques is outside the scope of this guidance. For the purpose of this document the worked examples considered will follow the simplified design procedure summarised in Figure 5. For step 1 the fire resistance requirements will be as specified in Approved Document B and related to building occupancy and height above ground. The fire resistance requirements will be provided as a specified time to failure under standard test conditions. It is important to note that there are alternative (physically based) methods for determining required performance not considered here. The calculation of the load effects at the fire limit state is similar to the procedure adopted in the latest version of BS5950 Part 8. The designer must be familiar with both EN1990 (Basis of Structural Design) which provides the required load combinations (as for ambient temperature design) and with EN1991-1-2 (the fire part of the Actions code) which in addition to specifying the available options for thermal actions for temperature analysis (see above) also specifies the mechanical actions for structural analysis. In particular EN 1991-1-2 specifies the partial factor for imposed (assuming leading variable action) loading for the fire limit state. The value chosen for use in the UK is ψ1 as detailed in Table 1 below. Table 1. ψfi values for the UK Action Imposed loads in buildings, Category A: Domestic, residential Category B: Office areas Category C: Congregation areas Category D: Shopping areas Category E: Storage areas Category F: Traffic areas, ≤ 30 kN Category G: Traffic areas, 30 - 160 kN Category H: Roofs Snow load: H 1000m a.m.s.l Wind loads on buildings

ψ1

ψ1

0.5 0.5 0.7 0.7 0.9 0.7 0.5 0

0.3 0.3 0.6 0.6 0.8 0.6 0.3 0

0.2 0.2

0 0

The partial factor for imposed loading at the fire limit state is a recognition of the small probability that the full design load will be in place at the time of a fire. The relatively large influence of imposed loads on steel construction provides benefits in terms of the reduction factor for the design load level for the fire limit state. This relationship is illustrated in Figure 6. The procedure is illustrated with reference to a simple worked example illustrating the use of unprotected structural steel in fire.


Project Number:


Sheet 4 of 13

Rev: 02


Laterally restrained simply supported secondary beams are located at 3m centres.

Table & clause

Carry out the design checks for a 406 x 178 x 54 UB in grade S355 steel for a fire

numbers given

resistance period of 30 minutes under a uniform temperature distribution.

relate to EN1993-1-2

Uniformly distributed load

unless stated otherwise.

6m Figure 1. Overview of simply supported beam Span of beam L = 6.00m Secondary beams at centres l = 3.00 m 2. Loading 2.1.

Permanent actions (G)

Uniformly Distributed Load over whole floor area

2

Gk.area = 3.00 kN/m

Uniformly Distributed Load along beam (UDL) Gk = Gk.area × l = 9.00 kN/m 2.2.

Variable actions (Q)


2

Qk.area = 3.50 kN/m

Uniformly Distributed Load along beam (UDL) Qk = Qk.area × l = 10.50 kN/m 2.3.

Loading factors

For the fire limit state partial loading factors (γi) are not applied to either permanent

EN 1990 Table

actions or variables actions.

A1.3

Combination coefficient for variable action

ψ1 = 0.50

Table A1.3 & A1.1

Note: EN 1990 allows use of either ψ1 or ψ2 with the main variable action. The National

& UK National

Annex will specify which coefficient to use. EN 1991-1-2 ‘Actions on structures exposed

Annex

to fire’ recommends the use of ψ2, however it is expected that the UK National Annex will specify the use of ψ1.

3. Design values of actions – Ultimate Limit State Accidental design situation UDL

FEd.fi = Gk + (ψ1 × Qk) = 14.25 kN/m

Note: EN 1990 includes Ad (design value of an accidental action)in Eq. 6.11b. In this example Ad is the effect of the fire itself on the structure i.e. the effects of the restrained thermal expansion, thermal gradients etc. However, EN1991-1-2, 4.1(4) states that ‘Indirect actions from adjacent members need not be considered when fire safety requirements refer to members under standard fire conditions’. Furthermore 4.1(1) states ‘Imposed and constrained expansions and deformations caused by temperature changes due to fire exposure results in effects of actions, e.g. forces and moments which shall be considered with the exception of those where they: – May be recognised a priori to be negligible or favourable – Are accounted for by conservatively chosen support models and boundary conditions and/or implicitly considered by conservatively specified fire safety requirements.’

EN 1990 Table A1.3 & Eq. 6.11b

Mfi.θ.Rd = 59 kNm < Mfi.d (64.13 kNm), therefore beam needs fire protection or designer could try a larger beam section. In this case a 457 x 152 x 82 UB is used.


Project Number:


Sheet 7 of 13

Rev: 02


Therefore beam may remain unprotected for 30 minutes fire exposure As an alternative the check may be carried out in the temperature domain. 6.2.

Repeat check in the temperature domain for the 457 x 178 x 54 UB

For the original member selection (406 x 178 x 82 UB). The degree of utilisation (µ0) is determined from: Efi.d / Rfi.d.0 For this check the effect of actions (Efi.d) is the bending moment at the fire limit state: Mfi.d = 64.13 kNm

Sheet 5

and Rfi.d.0 is the design moment resistance at time t = 0 is equal to the plastic moment capacity: MRd = Mpl.Rd = 374.40 kNm

µ0 = Mfi.d / MRd =0.171

4.2.4(4)

Note: This value is lower than the lowest tabulated value in Table 4.1 of EN1993-1-2 therefore it is necessary to calculate θa.cr explicitly. 3.833

θa.cr = 39.19 × ln((1 / (0.9674 × µ0

)) – 1) + 482 = 748 °C

4.2.4(2)

Note: This temperature is slightly less than the design temperature previously adopted (760°C) and therefore some minor fire protection would be required. The time-temperature relationship for the unprotected section is illustrated in Figure 7.

7. Temperature-time response It is possible to calculate the temperature-time response of the bare steel member using the formula given in EN 1993-1-2. The steel temperature difference (Δθa.t) for the specific time step is determined from: ksh × ((Am / V) / (ca × pa)) × hnet.d × Δt

4.2.5.1(1)

Where: ksh is the shadow factor (for I sections under nominal fire actions, determined from: 0.9 × [Am / V]b / [Am / V]

4.2.5.1(2)

Am / V is the profiled section factor for unprotected members [Am / V]b is the boxed value of the section factor Am is the surface area of the member per unit length (m²) ρa is the unit mass of steel (7850 kg/m³) ca is the specific heat of steel (600 J/kgK) hnet,d is the design value of the net heat flux per unit area (W/m²) – from EN1991-1-2 Δt is the time interval (seconds) The net heat flux is composed of radiative and convective components of which the rediative (hnet.r) is determined from: -8

4

4

5.67 × 10 × Φεres × ((θr + 273) - (θm + 273) ) Where: -8

5.67 × 10 is the Stefan-Boltzmann constant

EN1991-1-2, 3.1(6)


Project Number:


Sheet 8 of 13

Rev: 02


Φ is the configuration facture (1.0) εres is determined from εm - εf εm is the emissivity of the material (0.8) εf is the emissivity of the fire (1.0) θr is the radiation temperature of the fire environment θm is the member surface temperature The convective heat flux (hnet.c) is determined from: αc × (θg - θm)

EN1991-1-2, 3.1(3)

Where: 2

αc is the convective heat transfer coefficient (25 W/m K for the standard time-temperature curve) θg is the gas temperature θm is the member temperature


Project Number:


Sheet 9 of 13

Rev: 02


The solution is solved iteratively using a spreadsheet and a time step equal in this case to 5 seconds. The steel temperature using the above equations is calculated as 833°C at 30 minutes and the time/temperature relationship for the steel and atmosphere temperature is illustrated in Figure 7. Although the method is illustrated here with reference to the standard time-temperature curve one of the main advantages is that the procedure can be adapted to cover any known time-temperature relationship. Clearly the examples above has been chosen to illustrate certain concepts such as: •

The potential use of unprotected steel

•

The use of two simple verification methods (based on resistance and temperature respectively)

•

The importance of load ratio (degree of utilisation) on the performance in fire of structural steel members

The example chosen is not particularly practical as the very low level of utilisation means that it is very inefficient at the ultimate limit state. For steel construction a more realistic case would be to consider the same section (406 x 178 x 54 UB) for an increased fire resistance period. The same condition will be assessed for a required period of fire resistance of 60 minutes. It is clear from the above that the section will require fire protection. In this case a number of alternatives are available to the designer, they may: •

Determine the section factor according to the Eurocode classification and utilise tabulated values such as those in the “Yellow Book”

•

Calculate the critical temperature for the appropriate fire resistance period and specify this as a target value for fire protection

•

Calculate the thickness of fire protection required using the formula in the Eurocode.

Section Factor The section factor (Am / V) is the ratio between the exposed surface area and the volume of steel. This is synonymous with the Hp/A value familiar to UK designers. For UK sections the section factor is tabulated in -1

the “Yellow Book” and, for the example here the relevant figure for four sided exposure is 215m . The value can be easily calculated using the tables in the Eurocode. As an example for the fire resistance period required a thickness of 1.15mm of a particular water based thin film intumescent coating would provide the required level of fire resistance. The steel temperature at 60 minutes is approximately 937°C for an unprotected section. Clearly this is way above the critical temperature of 748°C. The critical temperature can then be used as the target value for fire protection manufacturers to demonstrate that the steel temperature can be kept below it for the specified period.


Project Number:


Sheet 10 of 13

Rev: 02


For members with passive fire protection the method of calculating the heat transfer is similar to that above for unprotected steel. The use of a highly insulating layer considerably reduces the heating rate of the member. The appropriate formula is:

!% a.t = ((& p x A p /( V x (% g.t " % a.t ))) / (dp x c a x $ a x (1 + (# / 3)))) x ( ! t " ((e ( # / 10 ) " 1) x ! g.t )) With:

! = ((c p x p p ) / (c a x p a )) x d p x ( A p / V ) Where Ap / V is the section factor for protected steel member ca is the specific heat of the steel cp is the specific heat of the protective material dp is the thickness of fire protection θa,t is the temperature of the steel at time t θg,t is the temperature of the gas at time t Δg,t is the increase in gas temperature over the time step t λp is the thermal conductivity of the fire protection material ρa is the density of the steel ρp is the density of the protection material As an example the use of 20mm Gypsum boarding to the section: Thickness dp = 20 mm Density ρp = 800 kg/m³ Specific heat cp = 1700 J/kg°K Thermal conductivity λp = 0.2 W/m°K Section factor for boxed protection 4 sided A p / V = 70 m -1 Therefore φ = 0.9817 Using an iterative spreadsheet calculation the temperature of the insulated steelwork is calculated as 496°C after 60 minutes of the standard fire. This is within the critical temperature for the member and therefore the protection is adequate. (Note: there is still a requirement to demonstrate the “stickability” of the fire protection material). The results are illustrated in Figure 8. The example above has been used to illustrate the potential use of unprotected steel and to illustrate verification in the terms of resistance and temperature. The alternative calculation procedures are related as the former requires a knowledge of the temperature of the member at the fire resistance period (used to derive the reduction factor ky,θ) while the latter requires a knowledge of the degree of utilisation (μfi).


Project Number:

Sheet 11 of 13

Rev: 02

Subject: Simply supported beam with full lateral restraint – Fire Limit State

Made by/date: TL / August 2004

Client:

Checked/date: YW/ October 2004

Project Design

Prescriptive Rules (Thermal Actions given by Nominal Fire

Tabulated Data

Performance-Based Code (Physically based Thermal Actions)

Member Analysis

Analysis of Part of the Structure

Analysis of Entire Structure

Calculation of Mechanical Actions at Boundaries


Selection of Mechanical Actions

Member Analysis

Analysis of Part of the Structure

Analysis of Entire Structure

Advanced Calculation Models



Selection of Mechanical Actions



Simple Calculation Models


Simple Calculation Models (if available)


Selection of Simple or Advanced Fire Development Models

SimpleCalculation Models (if available)

Figure 4. Fire Design routes available



Project Number:


Step 1: Determine fire resistance requirements (National regulations (ADB), Fire engineering design)

Step 2: Calculate load effects at the fire limit state (EN1990 / EN1991-1/ EN1991-1-2 / EN1992-1)

Step 3: Choose the relevant section / protection to meet the requirements obtained in 1 (EN1993-1-2)

Figure 5. Simplified design procedure

Figure 6. Relationship between reduction factor and ratio of dead and imposed load

Sheet 12 of 13

Rev: 02



Project Number:


Sheet 13 of 13


Unprotected steel temperature for 30 minute fire resistance period 406x178UB54 900

800

700

temperature (deg C)

600

500

400

300

200

100

0 0

5

10

15

20

25

30

35

time (mins) furnace temperature

steel temperature

Figure 7. Unprotected steel temperature for 30 minute fire resistance period 406x178UB54

Protected steel temperature for 60 minute exposure 1000 900 800

temperature (deg C)

700 600 500 400 300 200 100 0 0

10

20

30

40

50

60

time (mins) furnace temperature

Rev: 02

steel temperature

Figure 8. Protected steel temperature for 60 minute fire resistance period 406x178UB54

70


Project Number:

Sheet 1 of 9

Rev: 02

Subject: Simply supported composite beam – Fire Limit State

Made by/date: TL / October 2004

Client:

Checked/date: YW / November 2004

Simply Supported Composite beam – Fire Limit State The following Codes have been used for this worked example: BS EN 1990, Basis of Structural Design, July 2002, with UK National Annex, March 2004 BS EN 1991-1-1, Eurocode 1 – Actions on structures – Part 1.1: General actions – Densities, self-weight, imposed loads for buildings, July 2002 BS EN 1991-1-2, Eurocode 1 – Actions on structures – Part 1.2: General actions – Actions on structures exposed to fire, November 2002. prEN1992-1-1, Eurocode 2 – Design of concrete structures – Part 1.1: General rules and rules for buildings, April 2003 prEN 1993-1-1, Eurocode 3 – Design of steel structures – Part 1.1: General rules and rules for buildings, December 2003 prEN 1994-1-2, Eurocode 4 – Design of composite steel and concrete structures – Part 1.2: General rules structural fire design, October 2003 Notes on European Standards BSEN denotes a European Standard that has been published by BSI prEN denotes a draft European standard that is not publicly available DDENV denotes a European Prestandard that was made available for provisional application, but does not have the status of a European Standard Note on values contained in this worked example The computer software used to calculate the expressions given in this worked example does not round the values at intermediate stages during the calculation. Therefore some values given on the following sheets may appear to be ‘incorrect’ when determined using ‘rounded’ input values.



Project Number:

Sheet 3 of 9

Rev: 02



Client:


1. Loading 1.1.

Permanent actions (G)


2

Gk.area = 3.92 kN/m

Uniformly Distributed Load along beam (UDL) Gk = Gk.area × l = 11.76 kN/m 1.2.

Variable actions (Q)


2

Qk.area = 5.00 kN/m

Uniformly Distributed Load along beam (UDL) Qk = Qk.area × l = 15.00 kN/m 1.3.

Loading factors – Ambient temperature

Partial loading factor for permanent actions

γG = 1.35 γQ = 1.50

Partial loading factor for variable actions

EN 1990 Table A1.2(B) & N.A

Note: For strength / capacity check on a structural member EN 1990 recommends the use of STR checks given in Table A1.2(B).

1.4.

Loading factors – Fire limit state

For the fire limit state partial loading factors (γi) are not applied to either permanent

EN 1990 Table

actions or variables actions.

A1.3

Combination coefficient for variable action

ψ1 = 0.50

Table A1.3 & A1.1

Note: EN 1990 allows use of either ψ1 or ψ2 with the main variable action. The National

& UK National

Annex will specify which coefficient to use. EN 1991-1-2 ‘Actions on structures exposed

Annex

to fire’ recommends the use of ψ2, however it is expected that the UK National Annex will specify the use of ψ1.For a more detailed explanation of the choice of partial load factors see simply supported beam example for fire design.

2. Ambient temperature design values of actions Ultimate limit state

EN1990 Table

Design UDL FEd = (γG × Gk) + (γQ × Qk) = 38.38 kN/m

A1.2(B) & Eq. 6.10

2.1.

Design Moment

Maximum moment occurs at mid-span of beam 2

MEd = (FEd × L ) / 8 = 479.70 kNm 3. Fire limit state design values of actions Ultimate Limit State Accidental design situation

EN1990 Table

Design UDL

A1.3 & Eq. 6.11b

FEd.fi = Gk + (ψ1 × Qk) = 19.26 kN/m

Note: EN 1990 includes Ad (design value of an accidental action)in Eq. 6.11b. In this example Ad is the effect of the fire itself on the structure i.e. the effects of the restrained thermal expansion, thermal gradients etc. However, EN1991-1-2, 4.1(4) states that ‘Indirect actions from adjacent members need not be considered when fire safety requirements refer to members under standard fire conditions’. Furthermore 4.1(1) states ‘Imposed and constrained expansions and deformations caused by temperature changes due to fire exposure results in effects of actions, e.g. forces and moments which shall be considered with the exception of those where they: – May be recognised a priori to be negligible or favourable


Project Number:

Sheet 4 of 9

Rev: 02



Client:


– Are accounted for by conservatively chosen support models and boundary conditions and/or implicitly considered by conservatively specified fire safety requirements.’

3.1.

Design Moment – Fire limit state 2

Mfi.d = (FEd.fi × L ) / 8 = 240.75 kNm 4. Section Classification Section: UB 406 x 178 x 60

tf

z

1000 130

y

y

d

tw

r

7.9

h

12.8 406

z b

178

(all dimensions in mm)

Figure 3. Section dimensions

Figure 4. Beam cross-section

h = 406.40 mm

b = 177.90 mm

d = 360.40 mm

tw = 7.90 mm

tf = 12.80 mm

r = 10.20 mm

From the ambient temperature design worked example the cross-section is Class 1. prEN 10025-2 2

For tf = 12.80 mm Yield strength isfy = 355 N/mm

7.3 & Table 4 2

Normal weight concrete strength class C25/30, cylinder strength fck = 25 N/mm

prEN1992-1-1 Table 3.1

5. Ambient temperature moment resistance From the ambient temperature design worked example the moment resistance is: MRd = Mpl.Rd = 800.94 kNm 6. Fire limit state – Critical temperature model Note: When using the critical temperature model the temperature of the steel section is considered to be uniform.

Check model limits:

4.3.4.2.3(2)

Depth of steel cross-section h = 406.40 mm < 500 mm Depth of concrete hc = 130 mm > 120 mm Beam is simply supported and subject to only sagging bending moments Therefore OK to use critical temperature model The critical temperature is related to the load level and the strength of the steel at elevated temperature by the relationship:

4.3.4.2.3(3)

1.0" fi.t = f ay.!cr / f ay (for fire resistance periods other than 30 minutes)

Eq. 4.10b

Where: fay. θcr is the strength of the steel section at the critical temperature


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fay is the strength of the steel section at ambient temperature (fay = fy)

! fi.t = E fi.d.t / R d (as defined in clause 4.1(7)P) Efi.d.t is the design effect of actions in the fire situation at time t (E fi.d.t = ! fi x E d ) Ed is the design effect of actions at ambient temperature Therefore: ηfi = FEd.fi / ((γG × Gk) + (γQ × Qk)) = 0.502 Efi.d.t = ηfi × MEd = 240.75 kNm ηfi.t = Efi.d.t / MRd = 0.301 Note: The use of very similar symbols for ηfi and ηfi,t is confusing. The former is the relationship between the load (or actions) under fire conditions and the corresponding load under normal conditions while the latter is the relationship between the effects of actions (in this case bending moment) under fire conditions and the resistance at ambient temperature. This is a similar concept to the load ratio as defined in BS5950: Part 8

Therefore the strength of the steel section at the critical temperature is: 2

fay. θcr = ηfi.t × fay = 106.71 N/mm

4.3.4.2.3(3)

The strength reduction coefficient at time t is: ky. θ.max = fay. θcr / fay = 0.301

Table 3.2

Note: For this example the strength reduction coefficient is equal to the load level for fire design (ηfi.t), however, it should be noted that this is dependant on the fire resistance period.

The critical temperature at which the yield strength will reduce to a value of 2

106.5 N/mm must be determined and compared with the required fire resistance period (60 minutes). Steel temperature θa = 600°C ky. θ = 0.47

Table 3.2

Steel temperature θa = 700°C ky. θ = 0.23 From interpolation when ky. θ = ky. θ.max = 0.301 θa.max = 600 + ((100 × (0.47 – 0.301)) / (0.47 – 0.23)) = 670 °C The increase in temperature of the various parts of an unprotected steel beam during the time interval Δt is given by: •

!# a.t = k shadow x (1 / (c a x " a ) x ( A / Vi ) x h net x !t Where: kshadow is the correction factor for the shadow effect ca is the specific heat of steel (600 J/kgK) 3

ρa is the density of steel (700 kg/m )

4.3.4.2.2(3) Eq 4.6


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Ai is the exposed surface area of the part i of the steel cross-section per unit 2

length (m /m) -1

Ai / Vi is the section factor of the part I of the steel cross section (m ) 3

Vi is the volume of the part I of the steel cross-section per unit length (m /m) Δt is the time interval (seconds) •

2

h net is the design value of the net heat flux per unit area (W/m ) (obtained from EN1991-1-2) •

•

Rev: 02

•

h net = h net.c + h net.r

EN1991-1-2 3.1 (2)

•

h net.r = # m x # f x 5.67 x 10 "8 x ((! t + 273 ) 4 " (! a.t + 273 ) 4 ) -8

Where 5.67 x 10 is the Stefan-Boltzmann constant θt is the ambient gas temperature at time t(°C) θa.t is the steel temperature at time t (assumed uniform in each part of the cross-section) (°C) εm is the emissivity of the material (0.7)

2.2 (2) EN1991-1-2

εf is the emissivity of the fire (1.0) •

h net.c = # c x (! g " ! m )

k shadow = 0.9 x ((e 1 + e 2 + (b 1 / 2) + (h 2w + (b 1 ! b 2 ) 2 / 4))) / (h w + b 1 + (b 2 / 2) + e 1 + e 2 ! e w ) Where the dimensions are given in figure 4.3 of EN1994-1-2 (see Figure 5) Alternatively the configuration factor approach from EN1991-1-2 can be used – In this case the section factor used is that for the lower flange of the steel section.

3.1 (6)


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e1

hw

y

h

ew r

z b1 Figure 5. Steel beam cross-section dimensions for calculation of correction factor for the shadow effect Here the correction factor for the shadow effect is 0.736 And the section factor assuming 4 sided exposure is 167.5m-1 An iterative method using an excel spread sheet is used to calculate the increase in temperature of the uninsulated steel section. The time-temperature response is illustrated in Figure 6.

1000

900

800

700

600

500

400

300

200

100

0 0

10

20

Rev: 02


z

y

Sheet 7 of 9

30

40 FURNACE

50

60

70

steel

Figure 6. Time-temperature response for unprotected steel beam From Figure 6 it can be seen that the critical temperature of 670 °C corresponding to a reduction in the effective yield stress to a value of 106.5 N/mm² occurs after


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approximately 16 minutes. Therefore the steel member will require protection to achieve the 60 minute fire resistance period required. This can be achieved either by applying a sprayed or boarded fire passive fire protection system, an intumescent paint or by providing partial protection by filling between the flanges with reinforced concrete. In this instance a sprayed passive fire protection system is used. As with the EN1993 example the iterative calculation procedure for determining the rise in the steel temperature needs to be carried out taking into account the properties of the fire protection system. For protected members the relevant formula is:

"! a.t = (((% p / d p ) /(c a x $ a )) x ( A p.i / Vi ) x (1 / (1 + ( w / 3))) x (! g.t # ! a.t ) x "t ) # ((e w / 10 # 1) x "! t ) Where

w = 0.419 = (c p / ! p / (c a x ! a )) x d p x (A p.i / Vi ) where: λp is the thermal conductivity of the fire protection material (0.174 W/mK) dp is the thickness of the fire protection material (0.025m) Api is the area of the inner surface of the fire protection material per unit length of the relevant part of the steel member cp is the specific heat of the fire protection material (1200 J/kgK) Δθa.t is the increase in the ambient gas temperature during time interval t (°C) ρp is the density of the fire protection material ( 430 kg/m³) Therefore: w = 0.419 As sprayed protection is applied directly to the surface of the steel member the -1

section factor remains unchanged at 167.5m . For a similar time step the temperature rise is similarly calculated for the protected section using a spreadsheet. The results are illustrated in Figure 7 (on sheet 9). In this instance the temperature at 60 minutes is just over 450°C and the critical temperature is not exceeded even for the 90 minute period. Consequently the design is acceptable for the fire resistance period. However, the design is not particularly efficient and the designer may wish to complete the calculation using a smaller section size.

4.3.4.2.2(6) Eq4.8


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1200

1000

800

600

400

200

0 0

10

20

Rev: 02

30

40 FURNACE

50 steel temperature

Figure 7. Time-temperature response for insulated beam

60

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80

90

EC3 & EC4 Worked Examples

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