Analytical Methods for Determining Fire Resistance of Steel Members

4–209

IntroductionTraditionally, fire resistance has been evaluated by

subjecting a structural member to a standard test for aspecified duration.1 All members performing acceptablyare rated and listed for the duration period of the test (e.g.,1 hr, 2 hr). Assemblies not listed are assumed to be unableto meet the test criteria and, thus, have no rating, unlessproved otherwise. Providing proof of acceptable perfor-mance can be accomplished in one of three manners:

1. Conduct the standard test.12. Conduct a special experiment.23. Apply an analytical technique.3

The standard test can involve an appreciable turn-around time in order to specify, schedule, and analyze theresults of the test. An experimental program can require asubstantial amount of effort in order to obtain accuratedata. The costs involved in sponsoring a standard test orexperimental program can be appreciable. In the case ofarchaic structural assemblies, materials may no longer beavailable to reconstruct the design for possible testing.

Because of these drawbacks, calculation methodshave been developed to analyze structural designs for fireconditions. The calculation methods have been formu-lated based on analyses of data from standard tests, exper-imental programs, and theoretically based investigations.

Analytical methods for fire resistance must considerthree basic aspects of the problem:

1. Fire exposure2. Heat transfer3. Structural response

The fire exposing the structure must be characterizedusing methods described in other chapters of this hand-

book for the case of a real fire, or by assuming the fire ex-posure specified in the standard test. The thermal re-sponse of the structural member can be addressed usingprinciples of heat transfer. Heating within the member istreated by conduction heat transfer analysis (radiation andconvection heat transfer may also need to be considered, ifvoids or porous insulation materials are present within theassembly). Typically, radiative and convective boundaryconditions are present. Finally, the structural response isexamined by comparing some or all of the following: de-flections, strains, and stress levels to established limits.

The following types of calculation methods are avail-able to assess the fire resistance of steel structural members:

1. Empirical correlations2. Heat transfer analyses3. Structural analyses

Empirical correlations are based on the analysis ofdata resulting from performing the standard test numer-ous times. A limitation of the empirical correlations is thatthey can only be applied when considering the fire expo-sure, loading, and span provided in the standard test. Ifother conditions apply, then another approach is needed.

The second group of calculation methods consists ofheat transfer analyses. The heat exposure conditions maybe those associated with the standard test or a specifiedfire. The purpose of the heat transfer analysis is to deter-mine the time required for the structural member to attaina predetermined critical temperature or to provide inputto a structural analysis. The temperature endpoint criteriacited by ASTM E1191 are often accepted as the criticaltemperatures. Typically, inaccuracies of this method arerelated to the temperature dependence of the materialproperties or the description of the heating conditions.

Many of the structural analysis–based calculationsare similar to those conducted for structural engineeringpurposes, except the material properties are evaluated atelevated temperatures and thermal expansion is consid-ered. In structural analyses, the loading and end condi-tions must be known or assumed. Limitations result from

SECTION FOUR

CHAPTER 9

Analytical Methods forDetermining Fire Resistance

of Steel Members

James A. Milke

James A. Milke is an associate professor in the Department of FireProtection Engineering at the University of Maryland. His recent re-search activities have included the impact of fires on the structuralresponse of steel and advanced composite members.

uncertainties in characterizing the end conditions and thematerial properties at elevated temperatures.

This chapter provides an overview of the availablecalculation methods for determining the fire resistance ofsteel structural members. The basis of each method willbe presented along with sample applications.

Standard Test for Fire Resistance of Structural Members

The standard test method in the United States for de-termining the fire resistance of columns, floor and roof as-semblies, and walls is ASTM E119.*1 Basically, the testinvolves subjecting the structural component to a heatedfurnace environment for the desired duration. If the end-point criteria are not reached prior to the end of the testperiod, the assembly passes the test and is rated.

Gas burners are used to heat the furnace in testinglaboratories throughout North America. The furnace isheated so that the temperature inside the furnace followsthe time-temperature curve illustrated in Figure 4-9.1. Inprinciple, the time-temperature curve is intended to relateto a severe exposure from a room fire. Thus, the applica-bility of the test method to examine the fire resistance ofexterior structural members exposed to fires outside ofthe building is questionable.4

Assemblies may be tested with or without load. Iftested under load, the assembly is subjected to maximum

design stress levels, based on common structural analysisprocedures for ambient temperature design. Floor androof assemblies and bearing walls are always tested un-der load. Columns are tested with or without a loading.Steel beams and girders may be tested without load if thedesign loading cannot be achieved in the laboratory.

Structural assemblies may be restrained or unre-strained against thermal expansion. The effect of restrainton the fire resistance of assemblies has been investigatedby Bletzacker.4 The degree of restraint in structural mem-bers varies with the geometry, connection method, andframing system, among other factors. The descriptionspresented in Table 4-9.1 relate actual construction condi-tions to the restrained and unrestrained designationnoted in the ASTM E119 test method.

The minimum dimensions of the structural compo-nents for testing are specified in ASTM E119. A maxi-mum set of dimensions is established by the size ofavailable test furnaces. While the test is large-scale, thetest cannot be considered full-scale, given the stipulationof the maximum permissible dimensions. The conse-quence of not testing full-scale members means that con-tinuous beams, actual floor/roof ASTM assemblies, andlong columns are not tested. Consequently, this test isonly comparative in nature and cannot be used to assessactual performance.

The ASTM E119 endpoint criteria for building assem-blies consider structural integrity, temperature, passage offlame, ignition of cotton waste, and in some cases, re-sponse to the hose stream. For the tests without loading,the structural integrity endpoint criterion is relaxed to re-quire that the component only remains in place. The struc-tural integrity criterion addresses the need for members toremain in place (supporting self-weight of member) and tocontinuously support any applied loads. The ignition-of-cotton-waste endpoint addresses the ability of the struc-tural assembly to prevent the transmission of flame andhot gases to the side not exposed to the furnace fire.

The temperature endpoint criteria are noted in Table4-9.2. In principle, the endpoint temperatures are basedon the maximum allowable reduction in load-bearingcapacity of the structural member, based on the reductionin strength experienced at elevated temperature and themaximum permissible loads stipulated by structural de-sign standards.

Fire Resistance of Steel Members

Several calculation techniques are available to deter-mine the fire resistance of steel members, including steelcolumns, beams in floor and roof assemblies, andtrusses.7–10 Three types of techniques are available: empir-ically derived correlations, heat transfer analyses, andstructural analyses.

The equations and models do not eliminate the needfor all future testing. Testing is still required, at least tovalidate the calculation techniques and assess the interac-tion and mechanical behavior of the constituents of theassembly, such as the steel structural member, insulatingmaterials, or other components. However, the calculationtechniques can be used to extend the application of testresults and reduce the number of required tests. In addi-

4–210 Design Calculations

*Versions of the test method are also published as NFPA 2515 and UL263.6

2400 1315.6

1093.3

1204.4

982.2

871.1

760.0

648.8

537.8

426.7

315.6

204.4

93.3

–17.8

1200

1400

1600

1800

2000

2200

1000

800

600

400

200

00 1 2

Time (hr)

83 4 5 6 7

Tem

pera

ture

(°F

)

Tem

pera

ture

(°C

)

Figure 4-9.1. ASTM E119 standard time-temperaturecurve.1

tion, experimental methods are essential in determiningthe material properties at elevated temperature of theprotection materials.

Steel Material Properties

The principal material properties of interest are yieldstrength, ultimate strength, modulus of elasticity, coeffi-cient of thermal expansion, density, specific heat, andthermal conductivity. The effect of temperature on steelproperties has been examined by many researchers.11 Forsteel, all of the properties, except for density, are stronglyinfluenced by temperature.

The thermal properties of ASTM A36 steel are pro-vided in the following correlations: 7,12,13,14

k C >0.022T = 48 for 0 D T D 900ÜCk C 28.2 for 900ÜC A Tcs C 0.51T = 420 for 0 D T D 650ÜCcs C 8.65T = 4870 for 650ÜC A T D 725ÜCcs C >10.9T = 9340 for 725ÜC A T D 800ÜCcs C 579 for 800ÜC A T

: C 7860 kg/m3

Analytical Methods for Determining Fire Resistance of Steel Members 4–211

Wall bearing:Single span and simply supported end spans of multiple bays:a

Open-web steel joists or steel beams, supporting concrete slab, precast units, or metal decking unrestrainedConcrete slabs, precast units, or metal decking unrestrained

Interior spans of multiple bays:Open-web steel joists, steel beams or metal decking, supporting continuous concrete slab restrainedOpen-web steel joists or steel beams, supporting precast units or metal decking unrestrainedCast-in-place concrete slab systems restrainedPrecast concrete where the potential thermal expansion is resisted by adjacent constructionb restrained

Steel framing:Steel beams welded, riveted or bolted to the framing members restrainedAll types of cast-in-place floor and roof systems (such as beam-and-slabs, flat slabs, pan joists, and waffle slabs) where the floor or roof system is secured to the framing members restrainedAll types of prefabricated floor or roof systems where the structural members are secured to the framing members and the potential thermal expansion of the floor or roof system is resisted by the framing system or the adjoining floor or roof constructionb restrained

Concrete framing:Beams securely fastened to the framing members restrainedAll types of cast-in-place floor or roof systems (such as beam-and-slabs, flat slabs, pan joists, and waffle slabs) where the floor system is cast with the framing members restrainedInterior and exterior spans of precast systems with cast-in-place joints resulting in restraint equivalent to that which would exist in condition III(1) restrainedAll types of prefabricated floor or roof systems where the structural members are secured to such systems and the potential thermal expansion of the floor or roof systems is resisted by the framing system or the adjoining floor or roof constructionb restrained

Wood construction:All types unrestrained

aFloor and roof systems can be considered restrained when they are tied into walls with or without tie beams, the walls being designed and detailed to resist thermalthrust from the floor or roof system.bFor example, resistance to potential thermal expansion is considered to be achieved when:1. Continuous structural concrete topping is used2. The space between the ends of precast units or between the ends of units and the vertical face of supports is filled with concrete or mortar, or3. The space between the ends of precast units, and the vertical faces of supports, or between the ends of solid or hollow core slab units, does not exceed 0.25%

of the length for normal-weight concrete members or 0.1% of the length for structural lightweight concrete members.

Table 4-9.1 Restrained and Unrestrained Construction Systems (from ASTM E119 Table X3.1)1

StructuralMember

Walls/partitions(bearing and nonbearing)

Steel columns

Floor/Roofassemblies andloaded beams

Steel beams/girders (notloaded)

Location

1. Unexposed sideAverageSingle point

1. Average Single point


1. Unexposed side Average Single point

2. Steel beamAverageSingle point

3. Pre-stressing steel4. Reinforcing steel5. Open-web steel joists


MaximumTemperature ÜC (ÜF)*

139 (250)a

181 (325)a

538 (1000)649 (1200)

538 (1000)649 (1200)

139 (250)a

181 (325)a

593 (1100)704 (1300)426 (800)593 (1100)593 (1100)

538 (1000)649 (1200)

Table 4-9.2 ASTM E119 Temperature Endpoint Criteria1

*Maximum temperature cited refers to the maximum temperature rise aboveinitial conditions

The influence of temperature on the mechanicalproperties of A36 steel is presented in Figure 4-9.2. At538ÜC (1000ÜF), the yield strength is approximately 60 per-cent of the value at normal room temperature. The Amer-ican Institute for Steel Construction’s Specification for theDesign, Fabrication, and Erection of Structural Steel for Build-ings16 limits the maximum permissible design stress toapproximately 60 percent of the yield strength. Thus, forstructural members at 538ÜC (1000ÜF) designed to carrythe maximum permissible stress, the applied stress is ap-proximately the same as the strength of the member. Itshould also be noted that at 538ÜC (1000ÜF) the modulus ofelasticity has decreased appreciably from the value atnormal room temperature.

Mathematical expressions describing the relationshipof the yield strength, modulus of elasticity, and coefficientof thermal expansion on temperature are7,17,18

For 0 A T D 600ÜC,

;yT C “ —1 = T

900 ln (T/1750) ;y0

ET C “ —1 = T

2000 ln (T/1100) E0

For T B 600ÜC,

;yT C 340 > 0.34TT > 240 ;y0

ET C 690 > 0.69TT > 53.5 E0

*T C (0.004T = 12) ? 10>6

where;yT C yield strength at temperature T (MPa) (psi);y0 C yield strength at 20ÜC (68ÜF) (MPa) (psi)ET C modulus of elasticity at temperature T (MPa) (psi)E0 C modulus of elasticity at 20ÜC (68ÜF) (MPa) (psi)

*T C coefficient of thermal expansion at temperature T(m/mÜC)

T C steel temperature (ÜC)

1 CT′> 68

1800 T′ in ÜFT′> 20

1000 T′ in ÜCT′C steel temperature

In addition to the changes in material properties thatoccur at elevated temperatures, the crystalline structure ofsteel also changes, as noted in Figure 4-9.3.19 However, forthe low-carbon steels typically used in building construc-tion, significant changes in crystalline structure only beginto occur at temperatures in excess of 650ÜC (1200ÜF),20

above the temperature typically associated with failure.Creep, the time-dependent deformation of a material,

may be significant in structural steel at temperatures in ex-cess of 460ÜC (860ÜF).21 The rate of creep increases approxi-mately 300 times for ASTM A36 structural steel, when thesteel temperature is increased from 460 to 520ÜC (860 to968ÜF). Since creep is a complex phenomenon dependingon the stress level, rate of heating, and other factors, often itis included implicitly in the mechanical properties to sim-plify the fire resistance calculations.15,20 In-depth discus-sions of creep have been prepared by Harmathy.22,23

Methods of ProtectionThe basic intent of the various methods of protection

is to reduce the rate of heat transfer to the structural steel.This is accomplished by using insulation, membranes,flame shielding, and heat sinks.

Insulation

Insulation of the steel is achieved by surrounding thesteel with materials that preferably have the followingcharacteristics:24

1. Noncombustibility and the added attribute of not pro-ducing smoke or toxic gases when subjected to ele-vated temperatures

2. Thermal protective capability when tested in accor-dance with the standard fire test ASTM E119

3. Product reliability giving positive assurance of consis-tent uniform protection characteristics

4. Availability in a form that permits efficient and uni-form application

5. Sufficient bond strength and durability to prevent ei-ther dislodgement or surface damage during normalconstruction operations

6. Resistance to weathering or erosion resulting from at-mospheric conditions

In addition to the insulating qualities of the protec-tion materials, chemical reactions may occur in the insu-lation, further reducing the rate of heat transfer. Thechemical reactions include calcination, ablation, intumes-cence, thermal hydrogeneration, and sublimation.


0.4

0.2

0

0.8

0.6

1.0

Pro

port

ion

of p

rope

rty

at a

mbi

ent t

empe

ratu

re

Temperature (°C)

100 200 300 400 500 600 7000

Modulus of elasticity

Yield strength

Figure 4-9.2. Temperature effects on properties ofASTM A36 steel.12,15

Insulating methods include the use of board prod-ucts, spray-applied materials, and concrete encasement.A brief review of each method is presented below.

Board products: Four types of board products are com-monly used to protect structural steel: gypsum board, fiber-

reinforced calcium silicate board, vermiculite-sodium sili-cate board, and mineral fiber board. In each case, the meansof attachment of the boards surrounding the steel is a criti-cal parameter affecting the performance of the assembly.Two commonly used methods of attachment of gypsumwallboard with and without steel covers are illustrated in


00 0.5 0.8 1 1.5 2

0 5 10 15

Carbon content in weight percent

Cementite content in weight percent

20 25 30

100

200

300

400

500

600

700

800

900911

1000

1100

1200

1300

13921400

15001536

1600

C

Steel part

0 2.3 3.6 4.5

Carbon content in atomic percent

6.6

F

2912

2732

2552

2372 Yellowish white

2192 Light yellow

1146 C

2012 Yellow

1832 Light yellowish red

Reddish yellow

Very reddish yellow

Light red

Light cherry redCherry red

Dark cherry red

Dark red

Brown red

Dark brown

GrayBlack grayLight blueDark blueVioletRedYellowish brownStrong yellowVery pale yellow

1652

1472

(K)1292

1112

932

752

572

392

212

32

A

1493CH

Figure 4-9.3. Influence of elevated temperatures versus carbon content in steel.19

Figure 4-9.4. Detailed descriptions of the attachment mech-anisms for the other board products are provided else-where.25–27 Also, board products can be used in wallassemblies to provide an envelope around steel trusses.

Spray-applied materials: Several types of spray-appliedmaterials are commonly used. These include cementitiousplasters, mineral fibers, magnesium oxychloride cements,and intumescents. Sufficient data has been obtained tocharacterize spray-applied cementitious and mineral fibermaterials for the purpose of estimating the fire enduranceof structural steel protected with these materials. An illus-

tration of a steel column protected by a spray-applied ma-terial is presented in Figure 4-9.5.

Concrete encasement: Concrete encasement of steelmembers to surround and insulate the steel is illustratedin Figure 4-9.6. As indicated in Figure 4-9.6, the concrete iscast to fill in all re-entrant spaces. Alternatively, concretecolumn covers may be used, as illustrated in Figure 4-9.7.The concrete is assumed to act only to thermally protectthe steel. Some empirical correlations implicitly accountfor the load-bearing capacity of the concrete and possiblesteel-concrete composite action.


A

A

123

A

A

123

5/8 in.(16 mm)

3/8 in.(9 mm)

3/8 in.(9 mm)

Snap-lock

5/16 in. min.(8 mm)

Pittsburg seam

Corner Joint Details (A)

3/4 in.(19 mm)

No. 8 × 12 in. sheetsteel screws.Spaced 12 in. (0.3 m) O.C.

Lap

42 3

7 A

1 Layer = 5/8 in. (16 mm) or 1/2 in. (13 mm)

1

4

2

7 B

2 Layers = 1-1/4 in. (32 mm) or 1 in. (25 mm)

1

3

4

7 C

3 Layers = 1-7/8 in. (48 mm) or 1-1/2 in. (38 mm)

52

1

3

4

7 C

4 Layers = 2-1/2 in. (64 mm) or 2 in. (50 mm)

5

62 3

1

Figure 4-9.4. Attachment mechanisms of gypsum wallboard to steel columns.25

Membrane: Suspended ceiling assemblies are used asmembranes to protect structural steel in floor and roof as-semblies. The ceiling panels and tiles comprising the ceil-ing assembly may consist of gypsum, perlite, vermiculite,or mineral fibers.

The membrane method of protection is illustrated inFigure 4-9.8. Heat transfer to the structural steel is re-duced due to the air space above the membrane and theinsulating characteristics of the membrane. Also, mem-branes help prevent the direct impingement of flame onthe structural steel.

Flame shield: Flame shields are intended to reduce theincident radiant heat flux on the steel by preventing directflame impingement. The effectiveness of flame shields toprotect exposed spandrel beams was first examined bySeigel.2,28 In this instance, 14-gage sheet steel was used asthe flame shield.

Heat sinks: The heat sink approach delays the heating ofsteel by absorbing heat transferred through the steel. Theheat sink approach usually involves liquid- or concrete-

filling of the interior of hollow steel members (tubular andpipe sections). Liquid-filling can be used to provide a suf-ficient level of protection for the columns, without any ex-ternally applied coating. The liquid used for protection isan aqueous solution. Additives are provided primarily forantifreeze, corrosion protection, and biological reasons.

A diagram of a typical design for a liquid-filled col-umn fire protection system is presented in Figure 4-9.9.The components of this system include the hollow struc-tural steel columns, piping to connect the columns, a wa-ter storage tank, and associated valves.

The system operates on the principle that heat inci-dent on the column is removed by circulation of the liq-uid. If sufficient heat is delivered to the liquid,boiling canbe expected, which enhances the efficiency of the heat-removal process. In many tests with liquid-filling, steeltemperatures have been observed to be well below thoserequired for failure, as long as the column remains full ofthe liquid.

Another heat-sink approach consists of filling the inte-rior of hollow steel columns with concrete. If the concrete isreinforced, load transfer from the steel to the concrete canbe expected as the steel weakens with increasing tempera-ture. Calculation methods to determine the fire resistanceof concrete-filled steel columns are available.10,12


Figure 4-9.6. Steel column with concrete encasement.24

(a) (b)

Figure 4-9.5. (a) Sprayed insulation; (b) Metal lath andplaster encasement.24

(a)

h

h

L

L

(b)

h1

h2

L

L2

(c)

h1

h2

bf

d

Figure 4-9.7. Concrete-protected structural steel columns. (a) Squareshape protection with a uniform thickness of concrete cover on allsides; (b) Rectangular shape with varying thickness of concrete cover;and (c) Encasement having all re-entrant spaces filled with concrete.

Empirically Derived CorrelationsNumerous, easy-to-use, empirically derived correla-

tions are available to calculate the fire resistance of steelcolumns, beams, and trusses. The correlations are basedon data from performing the standard test numeroustimes on variations of a particular assembly. Curve-fittingtechniques are used to establish the various correlations.In some cases, a best-fit line has been drawn for the datapoints, whereas in other cases, lines were placed to pro-vide conservative estimates of the fire endurance by con-necting the two lowest points.29

Steel Columns

The correlations to estimate the fire endurance of un-protected and protected steel columns are given in Table4-9.3. Present in each of the equations is W/D for wide-flange sections and A/P for hollow sections. The W/D

and A/P ratios are comparable. The W/D ratio is theweight per lineal foot to the heated perimeter of the steelat the protection interface (or the perimeter of the steel ifunprotected). The A/P ratio is the cross-sectional area di-vided by the heated perimeter. Essentially, the W/D ratiorelates to the product of the density of the steel and theA/P ratio.

The relevance of the W/D and A/P ratios was firstnoted by Lie and Stanzak.30 W/D ratios for commonlyused wide-flange and tubular shapes for columns andbeams are available elsewhere.25,31,32 The two factors inthe W/D ratio that affect the rate of heat transfer to thesteel (and consequently the rise in temperature of thesteel) are (1) shape of the fire protection system, D, and(2) steel mass per unit of length, W.

The parameter that characterizes the shape of the fireprotection system is D, the heated perimeter expressed ininches, which is defined as the inside perimeter of thesteel at the fire protection material interface. Figure 4-9.10illustrates the method for determining D in four typical


8 J2 joists24″ (610 mm)O.C.

12 SWGhanger wire

Duct support1-1/2″ (38 mm)C.R. channel

3/4″ × 12″ × 12″ K & RFire-rated mineral tile

24-gage galvanized steelAir duct 18″ w. × 6″ dp. × 5′–0″ lg.

Damper protected with 1/16″ (1.6 mm) asbestos paperboth sides & held open with160F fusible link

1-1/2″ (38 mm) C.R. channel4′ (1.2 m) O.C.

12″(13 mm)

12″ × 24″ air diffuser Concealed Z runner12″ (0.3 m) O.C.

Border channel 1-3/8″ (35 mm) high with 3/4″ (19 mm) flanges

A

A

2-1/2″ (63.5 mm)

1-1/2″(38 mm)

12″(0.3 m)

7/8″(22 mm)

7/8″ (22 mm)

7/8″ (22 mm)

1/2″(12.7 mm)

1/2″(12 mm)

1/2″(13 mm)

Section A–A

4-1/2″(114 mm)

(a)

(b)

Figure 4-9.8. Membrane method of protection,24 (a) Cross-section of a floor-ceiling system with conventionalsheet steel fusible-link damper for protecting typical ceiling outlets in galvanized sheet ducts; (b) Sprayed con-tact fireproofing applied directly to the underside of formed-steel decking and to a supporting steel beam.

cases. As can be seen from the figure, the heated perime-ter depends on the size of the column and also on the pro-file of the protection system. Two different commonlyused profiles are: (1) contour profile, where all surfaces ofthe steel column are in contact with the protection mater-ial and (2) box profile, where a rectangular box of protec-tion material is built around the column.

A large value of W refers to a column with a largeweight per lineal foot. A given amount of energy willraise the temperature of the massive column to a lesserdegree than that of a light column. Less surface area isavailable for heat transfer if the heated perimeter, D, issmall, thereby inhibiting the temperature rise in the steel.The greater the W/D ratio, the greater the inherent fire re-sistance of the assembly is.

Because steel elements with larger W/D ratios are in-herently more fire resistant, substituting shapes withgreater W/D ratios for shapes identified in the listed de-signs in the UL Fire Resistance Directory3 is permittedwhile maintaining the same thickness of protection. How-ever, such substitution yields inefficient designs, becauseshapes with large W/D ratios actually require less fireprotection material than shapes with small W/D ratiosfor the same level of fire resistance.

The equation for gypsum wallboard protection is non-linear. The weight of the gypsum wallboard is includedbecause the heat capacity of gypsum has a considerable


Open vent

Zone waterstorage tank

Pipe loop attop of zone

Solid diaphragmbetween zones

May be interioror exterior

Pipe loop at bottom of zone

Figure 4-9.9. Schematic layout of a typical piping arrangementused in a liquid-filled column fire-protection system.4

a

D = 2(a + b)

b

a

D = 4a + 2b – 2c

b

a

D = 2(a + b)

b

D = 4b

b

c

Figure 4-9.10. Heated perimeter for steel columns.25

impact on the fire resistance of the assembly. The thicknessof wallboard required to achieve a particular level of fire re-sistance as a function of the W/D ratio of the column is pre-sented in Figure 4-9.10.

Based on an elementary heat transfer analysis, Stan-zak and Lie conducted a parametric analysis that resultedin correlations of the following form to estimate the thick-ness of material required to achieve a particular level offire resistance:25,27

R C (C1W/D= C2)h

whereR C fire endurance (hr)WC steel weight per lineal foot (lb/ft)D C heated perimeter of the steel at the insulation inter-

face (in.)h C thickness of insulation (in.)


Member/Protection

Column/Unprotected

Column/Gypsum Wallboard

Column/Spray-applied materialsand board products—wideflange shapes

Column/Spray-applied materialsand board products—hollowsections

Column/Concrete cover

Column/Concrete encased

Solution

R C 10.3 (W/D)0.7, for W/D A 10R C 8.3 (W/D)0.8, for W/D E 10(for critical temperature of 1000ÜF)

R C 130‹ �

0.75

where

W ′ C W =‹ �

R C [C1(W/D) = C2]h

R C C1

‹ �h = C2

R C R0(1 = 0.03m)

where

R0 C 10(W/D)0.7 = 17‹ �

Ý 41 + 26

“ —0.8

8

D C 2(bf = d)

for concrete-encased columns use

H C 0.11W = (bfd > As)

D C 2(bf = d )L C (bf = d )/2

:ccc

144

H:ccch(L = h)

h1.6

kc0.2

AP

50hD144

hW ′/D2

Symbols

R C fire endurance time (min)W C weight of steel section per linear foot (lb/ft)D C heated perimeter (in.)

h C thickness of protection (in.)W ′C weight of steel section and gypsum wallboard

(lb/ft)

C1 & C2 C constants for specific protection material

C1 & C2 C constants for specific protection material

The A/P ratio of a circular pipe is determined by

A/P pipe Cwhered C outer diameter of the pipe (in.)t C wall thickness of the pipe (in.)

The A/P ratio of a rectangular or square tube isdetermined by

A/P tube Cwherea C outer width of the tube (in.)b C outer length of the tube (in.)t C wall thickness of the tube (in.)

R0 C fire endurance at zero moisture content ofconcrete (min.)

m C equilibrium moisture content of concrete (% byvolume)

bf C width of flange (in.)d C depth of section (in.)kc C thermal conductivity of concrete at ambient

temperature (Btu/hrÝftÝÜF)h C thickness of concrete cover (in.)

H C thermal capacity of steel section at ambienttemperature (C 0.11W Btu/ftÝÜF)

cc C specific heat of concrete at ambient temperature (Btu/lbÝÜF)

L C inside dimension of one side of square concretebox protection (in.)

As C cross-sectional area of steel column (in.2)

t(a = b > 2t)a = b

t(d – t )d

Table 4-9.3 Empirical Equations for Steel Columns20,26,27

The constants C1 and C2 need to be determined foreach protection material. The constants take into accountthe thermal conductivity and heat capacity of the insula-tion material. Constants for some materials are includedin listings in the UL Fire Resistance Directory.3

Considering the equation for the concrete cover col-umn protection method (see Table 4-9.3), R0 is the fire en-durance of the assembly if the concrete has no moisturecontent. However, because the fire resistance of concrete-cover over steel columns is known to increase by approx-imately 3 percent for each 1 percent of moisture, R0 ismultiplied by the (1 = 0.03m) factor where m is the equi-librium moisture content of concrete. The parameters hand L noted in the equation are shown in Figure 4-9.7. Ifthe protection thickness or column dimensions are not thesame in the vertical and horizontal directions, averagevalues are used for h and L.

The heat capacity of the concrete must be accountedfor in the determination of H if all re-entrant spaces arefilled. (See Figure 4-9.7.) If specific data on the concrete’sthermal properties are not available, values given in Table4-9.4 may be used. Typical densities for normal-weightand lightweight concrete are 145 and 110 lb/ft3 (2320 and1760 kg/m3). Also, the typical equilibrium moisture con-tent (by volume) for normal-weight concrete is 4 percentand lightweight concrete is 5 percent.

Many of the equations cited in Table 4-9.3 are limitedto a range of shapes or protection thickness. Before apply-ing any equation from this table, users should consult theoriginal reference and confirm that the equation is beingapplied properly.

EXAMPLE 1:Determine the thickness of spray-applied cementi-

tious material to obtain a 2-hr fire endurance when ap-plied to a W 12 ? 106 column.

SOLUTION:From UL X772, the applicable equation is

R C (63W/D = 36)h

Solving for h,

h C R63W/D = 36

whereR C 2 hrs C 120 min

W/D C 1.44 lb/ftÝin. for a W 12 ? 106 with contour pro-file protection

Substituting,

h C 12063 ? 1.44 = 36 C 0.95 in.

EXAMPLE 2:Determine the fire endurance of a W 8 ? 28 column

encased in lightweight concrete (density of 110 lb/ft3

[176.2 kg/m3]) with all re-entrant spaces filled. The con-crete cover thickness is 1.25 in. (31.8 mm).

SOLUTION:From Table 4-9.3, the appropriate equation is

R C R0(1 = 0.03m)

where

R0 C 10(W/D)0.7 = 17‰h1.6/k0.2

c

�21 = 26[H/:ccch(L = h)]0.8

6Referring to Figure 4-9.7,

h2 C h1 C h C 1.25 in. (31.8 mm)

bf C 6.535 in. (166 mm)

d C 8.060 in. (204.7 mm)

W/DC 0.67 lb/ftÝin. (39.3 kg/m2)(contour profile)

A C 8.25 in.2 (0.0053 m2)

From Table 4-9.4,

kc C 0.35 Btu/hrÝftÝÜFcc C 0.20 Btu/lbÝÜF:c C 110 lb/ft3

L C 12 (bf = d) C 7.30 in.

H C 0.11W = :ccc144 (bf D> As)

H C 0.11 ? 28 = 110 ? 0.20144 (6.535 ? 8.060 > 8.25)

C 9.87

R0 C 10(0.67)0.7 = 17‹ �

1.251.6

0.350.2

?™§›

š̈œ1 = 26

“ —9.87

110 ? 0.2 ? 1.25(7.30 = 1.25)

0.8

R0 C 99 min

Assuming a moisture content of 5 percent for light-weight concrete,

R C 99(1 = 0.03 ? 5) C 114 min

Steel Beams

As in the case of columns, the W/D ratio is an impor-tant parameter affecting the fire resistance of a beam.Beams with larger W/D ratios may be substituted for


Thermal conductivity (k)a

Specific heat (c)b

Normal-WeightConcrete

0.950.20

Structural Lightweight Concrete

0.350.20

aExpressed as Btu/hrÝftÝÜFbExpressed as Btu/lbÝÜF

Table 4-9.4 Thermal Properties of Concrete at 70êF

beams with lesser W/D ratios for an equivalent ratingwith no change in the protection thickness. However, aswith columns, designs resulting from the direct substitu-tion of larger beams without reducing the protectionthickness may be inefficient.

In 1984, an empirically derived correlation was de-veloped to calculate the required thickness of spray-applied material protection.31 Correlations of the formfor steel columns are not possible, given the deck’s roleas a heat sink. Thus, the thickness of protection forsteel beams is determined based on the following scalingrelationship:

h1 C Œ �W2/D2 = 0.6W1/D1 = 0.6 h2 (1)

whereh C thickness of spray-applied fire protection (in.)WC weight of steel beam (lb/ft)D C heated perimeter of the steel beam (in.) (See Figure

4-9.11)

and where the subscripts1 C substitute beam and required protection thickness2 C the beam and protection thickness specified in the ref-

erenced tested design or tested assembly

Limitations of this equation are noted as follows:

1. W/D E 0.372. h E 3/8 in. (9.5 mm)3. The unrestrained beam rating in the referenced tested

design or tested assembly is at least 1 hr

It should be noted that the above equation only per-tains to the determination of the protection thickness for abeam in a floor or roof assembly. All other features of theassembly, including the protection thickness for the deck,must remain unaltered.

EXAMPLE 3:Calculate the thickness of spray-applied fire pro-

tection required to provide a 2-hr fire endurance for a W 12

? 16 beam to be substituted for a W 8 ? 17 beam requiring1.44 in. (36.6 mm) of protection for the same rating.

SOLUTION:The beam substitution correlation, presented as

Equation 1, is used.

h1 C Œ �W2/D2 = 0.6W1/D1 = 0.6 h2

where

W2/D2 C 0.54 for W 8 ? 17

W1/D1 C 0.45 for W 12 ? 16

h2 C 1.44

h1 C ‹ �0.54 = 0.60.45 = 0.6 ? 1.44 C 1.6 in.

Steel Trusses

There are three types of trusses used in buildings:transfer, staggered, and interstitial trusses. Because of theinherent features of each type of truss, some fire protec-tion systems are more appropriate than others.33

A load-transfer truss (see Figure 4-9.12) supports loadsfrom more than one floor. The loads may be suspended


16

17

15

14

13

12

11

10

9

8

7

6

5

4

3

2

Grade

Existing

Suspended

Interior columnsomitted

Roof

Newcaisson

Newcaisson

Existingsubway structure

Truss

Supported

Figure 4-9.12. Vierendell truss providing support fromabove and below.33

bf

D = 2d + bf

d

bf

D = 3bf + 2d – 2tw

(b) Box protection(a) Contour protection

dtw

Figure 4-9.11. Heated perimeter for steel beams.31

from a transfer truss or the transfer truss can be used toeliminate columns on lower floors.

A staggered truss is illustrated in Figure 4-9.13. Gen-erally, staggered trusses are used in residential occupancybuildings. Staggered trusses carry loads from two floors.

Interstitial trusses are used to create deep floor/ceil-ing concealed spaces containing mechanical and electricalequipment, as shown in Figure 4-9.14. Interstitial trussessupport only those loads from the equipment enclosureand the floor above. Interstitial trusses are typically usedin health-care facilities with heavy mechanical equipmentneeds.

Three methods of fire protection are often used fortrusses: membrane, envelope, and individual elementprotection. Some fire protection methods are more appro-priate than others for the specific truss types. The fire pro-tection methods typically used for each truss type areindicated in Table 4-9.5. Membrane protection is accom-plished through the use of a fire-resistant ceiling assem-bly. Design parameters for such an assembly can bedetermined from listings of fire-rated designs.3,34 No em-pirical correlations are available to assess the design ofmembrane protection systems.

The envelope means of protection is illustrated inFigure 4-9.15. The truss is enclosed in layers of a boardproduct, with the number of layers determined by the re-quired fire endurance. Some practical rules of thumbbased on test results are noted in Table 4-9.6.

Individual element protection is generally accom-plished using a spray-applied material. Since critical trusselements perform structurally as columns, that is, intension or compression (as opposed to bending), theapplicable equations for determining the thickness ofspray-applied material for columns is used. In order touse these equations, the W/D ratio must be calculatedfor each element. Unlike columns and beams, the ratiomay not be readily available. The diagrams in Figure4-9.16 are provided for assistance in calculating theheated perimeter.


Section A–A

Section B–B

Truss

Truss

Truss

Truss

Truss framing plan

A

A

B B

Figure 4-9.13. A typical truss and positionings in astaggered truss system.33

Figure 4-9.14. Hospital interstitial truss system.33

Fire Protection Method

Truss Type

TransferStaggeredInterstitial

Membrane

X

Envelope

XXX

IndividualElement

XXX

Table 4-9.5 Typical Fire Protection Methods for SteelTrusses

FireEndurance

60120180

GypsumX

5/8" (16 mm)11/4" (32 mm)

—

WallboardType

5/8" (16 mm)—

11/2" (38 mm)

Table 4-9.6 Practical Guidelines for Thickness ofGypsum Wallboard for Steel TrussEnvelope Protection33

Heat Transfer AnalysesHeat transfer analyses are applied to determine the

time period required to heat structural members to aspecified critical temperature or to provide temperaturedata as input to the structural analysis of the heated mem-ber. The time required to heat the member to a specifiedcritical temperature is often defined as the fire endurancetime of the member.

The critical temperature of a structural member canbe determined by referring to the temperature endpointcriteria cited in ASTM E1191 or by a structural assessment,as is discussed later in this chapter.

The available types of heat transfer analyses can begrouped into the following categories:

1. Numerical methods2. Graphical solutions3. Computer programs

Numerical Methods

Many numerical methods are available to estimate thetemperature rise in steel structural elements. The equa-tions are derived from simplified heat transfer approaches.

Unprotected steel members: The temperature in an un-protected steel member can be calculated using a quasi-steady-state, lumped heat capacity analysis. This methodassumes that the steel member is at a uniform tempera-ture. The equation for temperature rise during a shorttime period, !t, is21

!Ts C *cs(W/D) (Tf > Ts)!t (2)

where!Ts C temperature rise in steel (ÜF)(ÜC)

* C heat transfer coefficient from exposure to steelmember (Btu/ft2ÝsÝR)(W/m2ÝK),

D C heat perimeter (ft)(m) (see Figure 4-9.16)cs C steel specific heat (Btu/lbÝÜF)/(J/kgÝÜC)W C steel weight per lineal foot (lb/ft)/(kg/m)Tf C fire temperature (R)(K)Ts C steel temperature (R)(K)!t C time step (s)

where

* C *r = *c

*r C radiative portion of heat transfer

*r C C1.f

Tf > Fs

‰T4

f > T4s

�where C1 C 4.76 ? 10>13 Btu/sÝft2ÝR4 (5.77 ? 10>8 W/m3ÝK4)and .f , the effective emissivity, can be evaluated fromTable 4.97.

*c C convective portion of heat transfer

9.8 ? 10>4 to 1.2 ? 10>3 Btu/ft2Ýs (20 to 25 W/m2ÝK)


Top chordof trust

Cont-horizontalsteel studat mid-height

Gussetplate

Secondarytrussmembers

Gussetplate

Tapejoints

Bottom chordof truss

Required number of layers of fire-resistant gypsum wallboard

Steel studs

Third layer may beplaced horizontally

Figure 4-9.15. Staggered truss protection with envelope protection.33

The quasi-steady assumption dictates that the timestep should be small, that is, on the order of 10 s.59

Equation 2 is successively applied up to the time du-ration of interest. Correlations for the time-temperaturecurve associated with standard fire resistance tests are in-cluded by Lie, in another chapter of this handbook. Forthe ISO 834 test, Tf at any time, t, can be estimated by thefollowing expression:21

Tf C CT log10 (0.133t = 1) = T0 (3)

where

CT C 620 with Tf , T0 in ÜF345 with Tf , T0 in ÜC

Protected steel members: For protected members, thethermal resistance provided by the insulating materialmust be considered. If the thermal capacity of the insula-tion layer is neglected,21

!Ts C kicshW/D (Tf > Ts)!t (4)

where all parameters are as defined in Equation 2, and

ki C thermal conductivity of insulation material (Btu/ftÝsÝÜF) (W/mÝÜC)

h C protection thickness (ft) (m)

Malhotra suggests that the thermal capacity of the in-sulation material may be neglected if the following in-equality is true (see parameter definitions for Equation 2):21

csW/DB 2ci:ih

If the thermal capacity must be accounted for, as in thecase of gypsum and concrete insulating materials, then

!Ts C kih

” ˜Tf > Ts

cs(W/D) = 1/2ci:ih!t (5)

where all parameters are as defined for Equation 2, andci C specific heat of insulating material (Btu/lbÝÜF) (J/kgÝÜC):iC density of insulating material (lb/ft3) (kg/m3)

An evaluation of the predictive capability of thelumped heat capacity approach using Equation 5 for pro-tected steel sections was conducted by Berger for steelcolumns protected with a spray-applied cementitious ma-terial.36 The analysis consisted of comparing predictedversus measured temperatures for steel columns exposedto the standard fire exposure. A comparison of the pre-dicted versus measured times for the steel column toreach 538ÜC is provided in Table 4-9.8. A comparison ofthe predicted temperature with that measured for oneprotected steel column assembly is provided in Figure4-9.17.

Predictions of temperature rise in steel beams by thelumped heat capacity approach are prone to be inherentlyless accurate than those for steel columns.37 As noted pre-viously, a steel beam in contact with a slab only has threesides exposed to a fire and also will lose heat to the slab.38

Consequently, the temperature of a steel beam exposed tofire is likely to vary appreciably from the bottom flange tothe top flange, stretching the validity of the uniform tem-perature assumption. Nonetheless, for many engineeringapplications, the lumped heat capacity approach can pro-vide a conservative estimate of the average temperature


D = 3bf + 2d – 2tw

tw

bf

d

D = bf + 2d

bf

d

D = 4bf + 2d – 2tw

tw

bf

d

D = 8bf + 2d + 2a – 4tw

tw d

abf

D = 4a + 2b + 2c

ca

tb

D = 4bf + 2d + 2a

d

abf

Figure 4-9.16. Heated perimeter for steel truss shapes.33

Type of Construction

1. Column exposed to fire on all sides2. Column outside facade3. Floor girder with floor slab of concrete, only

the underside of the bottom flange beingdirectly exposed to fire

4. Floor girder with floor slab on the top flangeGirder of 1 section for which the width-depth

ratio is not less than 0.5Girder of 1 section for which the width-depth

ratio is less than 0.5Box girder and lattice girder

ResultantEmissivity

0.70.3

0.5

0.5

0.70.7

Table 4-9.7 Effective Emissivity35

rise of a steel beam.39 Heat losses to the slab may be com-pensated for by reducing the effective flame emissivity to0.5.35 However, if the temperature gradient across thebeam is important, another analytical approach will needto be applied.37

Exterior steel columns and steel spandrel beams: Adesign guide is available for calculating the exposure ofexterior steel columns and steel spandrel beams.40 Theguide is based on research by Law and basic radiationheat transfer principles.41 A similar calculation procedureis available in the Eurocodes.9

The temperature of the steel member is calculatedfrom a steady-state conduction analysis. The exposureboundary conditions consist of radiant heating from afully developed room fire and flames emitting from win-dows near the steel member. For this method, a specificdesign is considered unacceptable if the steel temperatureexceeds 1000ÜF.

Liquid-filled columns: The design calculations for liq-uid-filled columns are based on the thermal capacity ofthe liquid. The design of a liquid-filled column fire pro-tection system consists of three major steps:

1. Heat transfer analysis2. Determination of volume of liquid required3. Pipe network design

The heat transfer analysis is used to assess the impactof fire exposure on the liquid-filled column. The heat trans-fer analysis considers radiation and convection heat trans-fer from the fire to the column surface, conduction throughthe column wall, and convection with localized boiling intothe liquid. Both temperature of the steel column and totalamount of heat transferred to the liquid causing evapora-tion are determined as a result of this analysis.

The liquid volume calculation is important to ensurethe column remains full of liquid for the entire fire expo-sure period. Since heat transferred to the liquid will causesome evaporation, a supplemental amount of liquid mustbe provided in a storage tank.

The final step in the design method is a hydraulicanalysis of the tubular column and pipe network. Thisanalysis assesses the ability of the liquid to circulate basedon friction losses, elevation changes, and buoyancy of theheated liquid.

A comprehensive design aid for liquid-filled columnsis available.42 Since the procedure is rather lengthy, it willnot be reviewed here.

Graphical Solutions

Because heat transfer analyses can be very tediousand may involve the use of complex computer programs,graphic solutions have been formulated to simplify theestimation of steel temperature. Graphs of the tempera-ture of protected steel members have been developed byMalhotra,21 Jeanes,12 Lie,15 and others.

The series of graphs developed by Malhotra,21 pre-sented in Figure 4-9.18, for estimating the temperature ofsteel members exposed to the standard exposure arebased on the lumped heat capacity approach described inthe previous section. Steel temperatures are plotted ver-sus the D/A ratio (analogous to the inverse of W/D) forselected time periods of exposure and thermal resistancesof the insulating material. Time periods of 30 to 120 minare noted in the graphs. The range of thermal resistancesof the insulating material covered by these graphs is 0.01to 0.30 (W/m2ÝÜC)–1 (0.003 to 0.10) (Btu/ft2ÝhrÝÜF)–1.


Shape

W 6 ? 16

W 8 ? 28

W 10 ? 49

W 12 ? 106

W 14 ? 228

W 14 ? 233

h(cm)

1.93.87.6

3.58.39.5

1.95.6

3.8

1.4

2.9

Test(min)

58112210

122291355

70217

200

123

225

Calc.(min)

56119251

121298352

62220

203

140

251

Table 4-9.8 Comparison of Predicted Time fromLumped Heat Capacity Analysis andMeasurements for Protected Steel Columnto Reach 538êC

200

100

0

400

500

600

300

700

Tem

pera

ture

(°C

)

Time (min)

10 20 30 40 50 60 700

Test

Calc

Figure 4-9.17. Predicted steel column temperature.36

Based on the application of FIRES-T3, a heat transfercomputer program which will be described in the next sec-tion, Jeanes formulated a series of time-temperature graphsof protected steel beams.12 The steel beams are protected bya proprietary specific spray-applied cementitious materialwith a range of thicknesses of 0.5 to 1.5 in. (12.7 to 38.1mm). Graphs are available for a variety of common wide-flange beam shapes.12 Examples of these graphs are pre-sented in Figure 4-9.19 with graphs addressing the averageand single-point steel temperatures relating to the maxi-mum endpoint criteria from ASTM E119.1 Average and sin-gle-point steel temperatures are represented by the dashedlines. These graphs can be used to determine the thicknessof protection material required to provide a desired level offire resistance. Alternatively, the fire endurance can be esti-mated for a particular steel beam and insulation thicknessdesign which has not been tested.12

Information from numerous applications of FIRES-T3 examining the time-temperature response of steelbeams protected with a spray-applied cementitious mate-rial exposed to the standard fire exposure is summarizedin Figure 4-9.20. Using this graph, the fire endurance ofprotected steel beams with a W/D ratio of 0.4 to 2.5 lb/ft-in. can be determined for thicknesses of the spray-appliedprotection between 1.3 to 3.8 cm (0.5 to 1.5 in.).

Lie provides graphical representations of the exactsolutions of the governing differential equations for thetemperature of protected steel members exposed to the

standard fire.15 The heat transfer is assumed to be one-dimensional through the insulation layer. A uniformtemperature throughout the steel cross section is assumed.The two graphs presented in Figure 4-9.21 are applicableto a wide range in the Fourier number, Fo, for the insula-tion layer. In order to use the graphs, the following dimen-sionless parameters must be defined:

Fo C *th2

N C :icihcs(W/D)

1 C T > T0Tm > T0

where* C thermal diffusivity of insulation (ft2/hr) (m2/hr)t C heating time (hr)h C thickness of insulation (ft) (m):i C density of insulation (lb/ft3) (kg/m3)ci C specific heat of insulation (Btu/lbÝÜF)cs C specific heat of steel (Btu/lbÝÜF)T C temperature of steel at time t (ÜF) (ÜC)T0 C initial temperature of steel (ÜF) (ÜC)Tm C mean fire temperature (ÜF) (ÜC)


t = 30 min

t = 120 min

t = 90 min

t = 60 min

1100

900

1000

800

700

600

500

400

300

200

100

050 100 150 200 250 300

D/A

Tem

pera

ture

(°C

)

1000

900

800

700

600

500

400

300

200

100

050 100 150 200 250 300

D/A

Tem

pera

ture

(°C

)

t = 120 mint = 90 min

t = 60 min

t = 30 min

t = 30 min

hki⎯ = 0.2

hki⎯ = .05

hki⎯ = 0.1

hki⎯ = 0.3

t = 120 min

t = 90 min

t = 60 min

D/A

1000

900

800

700

600

500

400

300

200

100

050 100 150 200 250 300

D/A

Tem

pera

ture

(°C

)

t = 30 min

t = 120 min

t = 90 min

t = 60 min

1000

900

800

700

600

500

400

300

200

100

050 100 150 200 250 300

Tem

pera

ture

(°C

)

Figure 4-9.18. Relationship of heated area to steel weight with temperature.21


0

W/D of beam

Fire

end

uran

ce (

hr)

2.52.01.51.00.50.0

4

3

2

1

1/2″

1″

1–1/2″

Figure 4-9.20. Fire endurance of steel beams versusfire protection thickness for average section tempera-ture of 1000êF (538êC). (Based on FIRES-T3 analysis ofASTM E119 fire exposure.)12

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.4 0.8 1.4 1.8

Fo

2.2 2.6 3.0

1.0

θ

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Fo

1.0

θ

N = 10

52

1

0.6

0.4

0.3

0.2

0.15

0.10.08

0.060.04

0.02

0.03

0.040.05

0.06

0.08

0.10

0.15

0.2

0.3

0.61N = 5

0.02

Figure 4-9.21. Dimensionless steel temperature versusFourier numbers.15

01 2

Time (hr)

Tem

pera

ture

(°F

)

3 4

500

1000

1500

Average Section Temperature of Steel Beam,W12 × 14 (W/D = 0.40), for Various Thicknesses of Direct-Applied Fire Protection

01 2

Time (hr)

Tem

pera

ture

(°F

)

3 4

500

1000

1500

Maximum Steel Beam Temperature,W12 × 14 (W/D = 0.40), for Various Thicknessesof Direct-Applied Fire Protection

1/2″ 1″ 1–1/2″

1/2″ 1″ 1–1/2″

Maximum

Average

FIRES-T3 analysisbased on theASTM E-119 exposure

Maximum

Average

FIRES-T3 analysisbased on theASTM E-119 exposure

Figure 4-9.19. Predicted steel beam temperature byFIRES-T3.12

The mean fire temperature associated with a heatingtime, t, for these graphs is calculated from the standardtime-temperature curve, where

Tm C 5150(ln 480t > 1) > 30/t, TÜC270(ln 480t) > 238 > 54/t, TÜF

EXAMPLE 4:Determine the fire resistance of a W 24 ? 76 steel

beam based on the temperature endpoint criteria notedin ASTM E119. The beam is protected with 0.50 in. (12.7mm) of spray-applied cementitious material, by threemethods:

1. Graphical approach from Jeanes12

2. Graphical approach by Lie15

3. Quasi-steady-state approach by Malhotra21

SOLUTION:A W 24 ? 76 steel beam has a W/D ratio of 1.03 lb/ftÝ

in. or 12.36 lb/ft2. The material properties are evaluated atmean temperatures expected during the exposure. Thefire resistance can be assessed using the temperature end-point criteria in ASTM E119. Mean temperatures of 500ÜF(260ÜC) and 750ÜF (400ÜC) are selected (arbitrarily) for thesteel and insulation, respectively, to determine the ther-mal properties. The following material property valuesare assumed:12

Steel Insulation

Thermal conductivity (Btu/ftÝhrÝÜF) 25.6 0.067Specific heat (Btu/lbÝÜF) 0.132 0.304Density (lb/ft3) 490 15

Jeanes’s graph: Using Figure 4-9.21 with a W/D of1.03 lb/ftÝin. and an insulation thickness of 0.50 in. (12.7mm), the fire endurance is estimated to be 1.33 hr or 80 min.

Lie’s graph: Evaluating the dimensionless parameters,

Fo C *th2

where

* C ki:ici

C 0.06715 ? 0.304 C 0.0147 ft2/hr

Fo C 0.0147t(0.5/12)2 C 8.47t (t in hr)

Referring to Figure 4-9.22 and using a trial and errorapproach with a critical temperature selected as 1000ÜF(538ÜC), the fire endurance time is estimated as approxi-mately 75 min.

Quasi-steady-state approach: First, a check is per-formed to determine if the thermal capacity of the insula-tion material must be considered.

cs W/DB 2ci:ih

0.132 ? 12.36 B 2 ? 0.304 ? 15 ? 0.50/121.63 B 0.38


600

500

400

300

200

0

100

Tem

pera

ture

(°C

)

0

20 × 20 cm steel core, insulated by heavy clay brick

20 40 60 80

Time (min)

100 120 140 160

ExperimentalCalculated

600

500

400

300

200

0

100

Tem

pera

ture

(°C

)

0

15 × 15 cm steel core, insulated by fire brick

20 40 60 80

Time (min)

100 120 140 160


1000

500

600

700

800

900

400

300

200

0

100

Tem

pera

ture

(°C

)

0

20 × 20 cm steel core, insulated by vermiculite boards

20 40 60 80

Time (min)

100 120 140 160


Steeltemperature

Furnacetemperature

Figure 4-9.22. Comparison of calculated and measuredsteel temperatures.43

Disregarding the thermal capacity of the insulation,Equation 4 is used to predict the steel temperature rise foreach time step.

!Ts C 0.067/36000.0132 ? 0.50/12 ? 12.36 (Tf > Ts)!t

C 2.74 ? 10>4(Tf > Ts)!t

Steel Fire Fire-SteelTemperature Temperature Temperature W/m2ÝK !Ts

Time (ÜC) (ÜC) (ÜC) k/h (ÜC)

10 20.0 46 26 9.13 0.120 20.1 72 51 9.13 0.230 20.3 96 76 9.13 0.340 20.5 120 99 9.13 0.350 20.8 143 122 9.13 0.4

3220 534.2 888 353 9.13 1.23230 535.3 888 353 9.13 1.23240 536.5 888 352 9.13 1.23250 537.7 889 351 9.13 1.23260 538.9 889 350 9.13 1.23270 540.1 890 350 9.13 1.23280 541.2 890 349 9.13 1.2

Thus, the fire endurance is 54 min.The fire endurances calculated by the three methods

can be compared as follows:

Jeanes (FIRES-T3) 80 minLie 75 minQuasi-steady-state 54 min

The agreement between the fire endurance times de-termined by Jeanes’s and Lie’s graphs is very good. Thesignificantly reduced fire endurance calculated using thequasi-steady-state approach is attributable to the approx-imate nature of the lumped heat capacity method assum-ing an adiabatic surface at the beam–slab interface.

Computer-Based Analyses

Several computer-based analyses are available to es-timate the temperature rise of steel members. The analy-ses range from a spreadsheet procedure to perform theiterative calculations for the quasi-steady-state approachto finite element models.

Spreadsheets are one example of providing a frame-work to perform the iterative, quasi-steady calcula-tions.36,37,44 Typically, the spreadsheet procedures mimicthe quasi-steady analysis procedure described previously,including the evaluation of material properties at a mid-range temperature for the exposure of interest. Althoughtemperature-dependent material properties can be in-cluded within the spreadsheet framework, the accuracyimplied by considering temperature-dependent proper-ties is not consistent with the first-order nature of thequasi-steady approach.

Another framework for conducting computer-basedanalyses includes the numerous mathematical-equation-solver software packages. This software can be used to

conduct the iterations associated with the quasi-steady ap-proach or to solve the partial differential equations exactly.

Harmathy and Lie developed a two-dimensional fi-nite difference model to predict the temperature rise inprotected steel columns.43 The two-dimensional networkis formulated over the cross-section of the insulationlayer, assuming the temperature to be independent oflength. The steel is assumed to be a perfect conductor (i.e.,the temperature is uniform throughout the steel). Heattransfer via radiation is considered across any air spacesenclosed by the insulation and steel.

The boundary conditions included by Harmathy arethose associated with the ASTM E119 test.1 To simplify themodel, convection is disregarded, since convection com-prises a minor portion of the heat transfer process in thefurnace test. A flame emissivity of 0.9 is selected. A com-parison between the calculated and experimental steeltemperatures is presented in Figure 4-9.22. As is evident,the agreement is very good for three insulating materials.

Pettersson et al.35 include a finite difference formula-tion to predict the temperature rise of steel beams pro-tected with a suspended ceiling exposed to a specifiedfire. The formulation uses a one-dimensional approxima-tion accounting for conduction through the suspendedceiling and floor slab (above the beam), and radiation andconvection in the air space between the slab and beam.The temperature of the steel is assumed to be uniform.The assembly is divided into several elements as depictedin Figure 4-9.23.

A system of simultaneous equations is derived for thetemperature rise in each of the assembly elements. Anumerical integration technique such as Runge-Kutta isused to obtain the solution. A comparison of the calcu-lated versus experimentally observed temperatures for asteel beam is presented in Figure 4-9.24.

General heat-transfer finite-element programs havebeen available for many years.45 FIRES-T3, TASEF-2,SAFIR, and SUPER-TEMPCALC, among others, have been


ϑn

ϑ1

ϑt

ϑo

ϑY1

ϑY2

ϑY4

ϑY3

α1

α2

αu

Figure 4-9.23. Division of the floor slab into elements.35

developed specifically to address the heating of assem-blies with steel structural members exposed to fire condi-tions.46–48

TASEF-2 examines the conduction heat transferthrough assemblies.46 Assemblies may include internalvoids, in which convection and radiation heat transfermodes are considered. Two time-temperature curves areavailable: (1) the ISO 834 standard time-temperaturecurve and (2) a time-temperature curve from a ventila-tion-controlled fire.

SUPER-TEMPCALC can also be used to analyze theconduction heat transfer through assemblies with air gaps.Numerous fire curves are included within the software.

FIRES-T3 was specifically developed to examine theheating of structural members exposed to fire conditions.47

FIRES-T3 has been applied successfully to predict thetemperature rise in protected steel beams and columns.12,49

Almand used a finite-difference heat-transfer model to es-timate the protection thickness of spray-applied cementi-tious material required for tubular steel columns.50

The input data requirements for the heat transfercomputer models can be grouped into two categories:

1. A description of the assembly2. A description of the fire exposure

The information necessary to describe the assemblyincludes geometric factors (dimensions, shape of mem-ber) and material property values (thermal conductivity,specific heat, and density). The fire exposure is character-ized in terms of the temperature of the surrounding envi-ronment and appropriate heat transfer coefficients. Thegeometry of the assembly is established by formulatingan element mesh for the assembly of interest. Requiredmaterial property data consists of the density, specificheat, and thermal conductivity of the steel and insulation.Material property data is available for a limited numberof insulation materials.12,51 The exposure associated withthe ASTM E119 test1 may be selected as the fire exposureto be simulated by FIRES-T3.12,49 A pre-processor routinefor FIRES-T3 was recently developed by Stubblefield andEdwards.52 TASEF and SUPER-TEMPCALC also includesa post-processor to generate graphs of the output.

For models using an explicit transient solution tech-nique, such as FIRES-T3, caution must be exercised in se-lecting the time step and mesh size to obtain correct resultsthat are numerically stable. TASEF-2 internally determinesa numerically stable time step. Most heat transfer modelsdo not address the effects of phase changes or chemical re-actions that may influence the heating process. Phasechanges and chemical reactions have been accounted forby altering the value of the material properties. Milke ad-dressed the evaporization of free water in a spray-appliedcementitious material by increasing the specific heat in anarrow temperature region around 100ÜC (212ÜF).49

Agreement between the predicted and experimentalaverage steel temperatures is quite good in both applica-tions of FIRES-T3 by Jeanes and Milke. A comparison ofthe temperature history for a steel column protected witha spray-applied cementitious material subjected to theASTM E119 test is presented in Figure 4-9.25. A similarcomparison is presented in Figure 4-9.26 for steel beamsprotected with the same material.12

FIRES-T3 has also been used to conduct a prelimi-nary analysis of the heating of partially protected steelcolumns (i.e., where a portion of the spray-applied pro-tection is missing).53 The analysis indicated that even asmall portion of missing protection significantly de-creased the fire resistance of the column, especially forcases involving small columns. Results of the analysis areindicated in Figure 4-9.27.

Structural AnalysesThe structural analysis methods calculate one of

three parameters: deflection, critical temperature, or criti-cal load. In several of the methods, all three of the para-meters may be considered, since they are interrelated.Algebraic equations, graphs, and computer programs areavailable to perform a structural analysis for the purposeof addressing fire resistance.

General Discussion of Three Parameters Addressedin Structural Analysis

Deflection: The total deflection and rate of deflectioncan be calculated for loaded and heated steel beams by


400

200

0.5

Time (hr)

1 1.5

Tem

pera

ture

(°C

)

Figure 4-9.24. Calculated (--) and measured (—) steeltemperature-time (šs – t) curve for a floor girder IPE 140with insulation in the form of a suspended ceiling of 40-mm-thick mineral wool slabs of density • C 150 kg/m3.The figure also gives the calculated (–ê–) and measured(–x–) temperature-time curve for the top of the 50-mm-thick concrete floor slab.35

considering all sources of strain. The total strain com-prises components of the elastic and plastic strains due tothe applied loads, thermal strain (due to thermal expan-sion), and creep stain.

The calculated deflection and rate of deflection canbe compared with established maximum limits of each.The Robertson-Ryan criteria have been widely acceptedfor this purpose.20,54,55 However, calculation of the de-flection of unheated beams is difficult except for simpleloadings, geometries, and end conditions. Adding thethermal expansion and creep components further com-plicates the calculation, virtually requiring computersolution.

Critical temperature: As mentioned earlier in the chap-ter, the material properties of steel change with increasingtemperature. The most important material properties forcritical-temperature calculations are yield strength, ulti-mate strength, and modulus of elasticity. The criticaltemperature is defined as the temperature at which thematerial properties have decreased to the extent that thesteel structural member is no longer capable of carrying aspecified load or stress level. In this context, the factor ofsafety of the member is considered to be reduced if the

member reaches unacceptable stress levels, buckling be-comes imminent, or deflections exceed maximum limits.The critical temperature can be calculated as long as thedependence of the material properties with temperatureis known. There are numerous algebraic equations to cal-culate the critical temperature of steel structural mem-bers.56 Often, the critical temperature is defined based ontemperature limits stated in the standard test. However,in recent tests steel members experienced temperatures inexcess of 800ÜC (1470ÜF) without collapse.57


1500

1000

500

00.0 1.0

Time (hr)

Beam: W 8 × 28 (W/D = 0.80)Fireproofing: 1″ thick (monokote)

2.0 3.0

FIRES-T3 prediction

Test data

Ave

rage

ste

el te

mpe

ratu

re (

°F)

1500

1000

500

00.0 1.0

Time (hr)

Beam: W 12 × 27 (W/D = 0.63)Fireproofing: 7/8″ thick (monokote)

2.0 3.0

FIRES-T3 prediction

Test data

Ave

rage

ste

el te

mpe

ratu

re (

°F)

Figure 4-9.26. Comparison of experimental data andFIRES-T3 analysis.12

1000

900

750

600

450

300

150

25 50

Time (min)

Col

umn

tem

pera

ture

(°F

)

75 100

UL data

W 10 × 493/4 in. of protection

Fires-T3

Figure 4-9.25. Comparison of predicted and measuredaverage steel column temperature.47

Critical load: The critical load is defined as the mini-mum applied load that will result in failure if the struc-tural member is heated to a temperature, T. The criticalload can be expressed as a point load or distributed load.As with critical temperature, the critical load calculationrequires the material properties at elevated temperatures.Critical load calculations can be conducted with algebraicequations or with a computer program.

Algebraic Equations: Critical Temperature

Beams

The critical temperature of Grade 250 steel beams with anallowable stress of 20,000 psi (138 Mpa) can be determinedusing equations by Lie and Stanzak.30 The Lie and Stanzakequations account for creep strain and assume the beam issimply supported and thermally unrestrained.

Similar approaches have been developed by Malho-tra,21 Vinnakota,55 and Kruppa.58 Differences in the per-cent reduction in yield stress or modulus of elasticity arerelated to design method (elastic or plastic), factor ofsafety, and end conditions. Equations for the ratio of yieldstress at elevated temperature with yield stress at ordi-nary room temperature are presented in Table 4-9.9. Typi-cal values of Z/S are between 1.13 and 1.15 for I sections,21

and 1.5 for rectangular sections.Another example of the second approach is the

analysis of the critical temperature of beams by EuropeanConvention for Constructional Steelwork (ECCS).56,59 TheECCS guide addresses the maximum allowable reductionin yield strength by considering the applied loading,beam geometry, structural end conditions, and whetherthe applied loading results in stresses in the elastic orplastic range. Critical temperature calculations based onthe ECCS analysis are presented in Table 4-9.10.

EXAMPLE 5:Determine the critical temperature of a simply sup-

ported W 12 ? 26 steel beam supporting a 53-in.- (1.35-m-)

thick rectangular slab. The applied moment is 41,750 ftÝlb(15,480 NÝm). The rectangular slab is 8 ft (2.4 m) wide. Thesection properties of the beam are

Ze C 33.4 in.3 (547 ? 103 mm3)

I C 204 in.4 (84.9 ? 106 mm4)

Assume ;y C 36,000 psi (248 MPa).

SOLUTION:Using Lie and Stanzak’s equation for a beam,

Tcr C 70,00045.62 > 4.23(Id/I) > 460

Id C 33 ? 9612 C 216 in.4

T C 70,00045.62 > 4.23(216/204) > 460 C 1,240ÜF

Columns. Lie and Stanzak calculated a critical tempera-ture of 941ÜF (505ÜC) for slender, axially loaded col-umns.30 The calculation was based on the temperature forthe onset of elastic buckling for columns under maximumpermissible applied stress conditions.

The Euler buckling stress at which elastic buckling isimminent is given by

;cr C 92ET42 (6)

where

;cr C Euler buckling stress (MPa) (psi)

ET C modulus of elasticity at temperature T (MPa) (psi)

4 C slenderness ratio C Kl/r

r C radius of gyration (ft)(m)

Kl C effective length of column (ft)(m)

Included in the ECCS guide59 are dimensionlessbuckling curves for steel columns at elevated tempera-tures. These curves are presented in Figure 4-9.28.


Design Basics Critical Yield Stress

Elastic design CPlastic design Cwhere;YT C critical yield stress at elevated temperature, T;Y C yield stress at ordinary room temperatureFe C factor of safety, elastic designFp C factor of safety, plastic designZe C elastic section modulusZp C plastic section modulus

1Fp

;YT;Y

ZeZp

1Fe

;YT;Y

Table 4-9.9 Critical Stress Equations21

20

00 2 4 6 8 10 12 14 16 18

40

60

80

100

120

140

160

180

One hour

Two hour

Three hour

Fire

res

ista

nce

(min

) (A

ST

M E

-119

)

Figure 4-9.27. Fire resistance versus percent protec-tion loss for W 10 ? 49 column, flange exposure.

Equation 6 is only valid for columns that buckle inthe elastic range. Generally, slender columns having aslenderness ratio in excess of approximately 90 can be ex-pected to buckle elastically. Buckling stresses for stoutcolumns (slenderness ratio less than 90) are in the plasticrange, requiring a more complex analysis. The failuremode for columns with a slenderness ratio between 80and 100 cannot be reliably predicted.60 The tangent mod-

ulus can be used instead of the modulus of elasticity inEquation 6 for stout columns. However, predictions of thecritical temperature using Equation 6 may not be accu-rate, due to residual stresses from the steel fabricationprocess.60 Thus, for stout columns, a conservative esti-mate for the critical temperature of steel columns may beobtained by determining the temperature at which theyield stress is equal to the applied stress.

General

Malhotra has observed that critical temperatures de-termined from the structural analysis algebraic equationswill be somewhat low when compared to experimentaldata.21 Thus, the following correction factors, V, are sug-gested by Malhotra to improve the prediction capabilitiesof the approach:

1. Columns: V C 0.85

2. Statically determinate beams: V C 0.77 = 0.15PsPu

3. Statically indeterminate beams: V C 0.25 = 0.77PsPu

where

Ps C service (applied) load (N or N/m) (lb or lb/ft)

Pu C load to induce ultimate stress at midspan (N orN/m) (lb or lb/ft)


Table 4-9.10 Critical Temperature of Steel Beams37

0.3

585

605

640

605

650

615

650

0.4

540

565

605

565

615

580

615

0.5

490

525

575

525

590

545

590

0.6

430

475

545

475

560

505

560

0.7

360

425

510

425

535

465

535

Base of structural designat room temperature

qp = Ultimate plastic load

q* = Applied load

k = Load multiplier

Θ = Factor addressing plastic reserve of beam from redistribution of moments.

Theory of plasticity

Theory of elasticity

Statically determinateStatically indeterminate

Statically determinate

Statically indeterminate

Θ = 1.33

Θ = 1.0

Θ = 1.47

Θ = 1.12

Θ = 1.47

Static System

factor ⎯⎯ resp. ⎯⎯kq*qe

kq*qp

0

λ

Ncr θ

1.41.21.00.80.60.40.20

1.0

0.8

0.6

0.4

0.2

20 °C200 °C300 °C400 °C450 °C500 °C550 °C600 °C

Figure 4-9.28. Dimensionless buckling curves for steelcolumns.59

EXAMPLE 6:Determine if the following steel column is expected

to buckle if it achieves an average temperature of 1100ÜF(593ÜC). The column is simply supported, 15 ft (4.6 m)long and has an applied load of 12,000 psi (82.8 MPa). As-sume the yield stress is 36,000 psi (248.4 MPa) and themodulus of elasticity is 30,000,000 psi. The characteristicsof the column are

A C 8.23 in.2 (5310 mm2)I C 21.6 in.4 (8.99 ? 106 mm2)

Kl C 180 in. (4572 mm)

At 1100ÜF (593ÜC):

ET C 1 = T2000 ln (T/1100)

Eo C 15.6 ? 106

SOLUTION:Calculate the slenderness ratio to determine the fail-

ure mode.

4 C Klr C 110

Since the slenderness ratio exceeds 90, the column is sus-ceptible to buckling. The buckling stress at 1100ÜF (595ÜC)is 12,700 psi (87.6 MPa). Thus, the column does not buckledue to the applied load and elevated temperature.

Critical Stress

Columns: Sample expressions for determining the criti-cal stress for steel columns30 are noted below.

P2cr > Pcr

” ˜;yT = 92ET

‹ �4.8 ? 10>5 = 1

42 = ;yTA92Et42 C 0

wherePcr C critical point load (N) (lb);yT C yield stress at temperature T (Pa) (psi)ET C modulus of elasticity at temperature T (Pa) (psi)4 C Kl/r

In order to improve the prediction capabilities of thecritical stress approach for slender columns, the modulusof elasticity should be replaced by the reduced modulusof elasticity.15 The reduced modulus is defined as

Er C 4EET‰‚E = ƒ

Et

�2

whereEt C tangent modulus

In addition, the 0.2 percent proof stress may be re-placed by the 0.5 percent proof stress in the yield stressparameter.61

Results of a buckling analysis on concrete-filledsquare hollow sections are provided in Figure 4-9.29.

Beams: The expressions for the critical loads for beamsassume at failure that the beam is in a state of full plas-ticity at the location of the maximum moment.61 Obvi-ously, in order to calculate the critical stress, the materialproperty–temperature relationships must be known.

The critical distributed load for a simply supportedbeam is55

qcr C 8;yTZP

L2

whereqcr C critical distributed load (N/m) (lb/ft)ZP C plastic section modulus (m3) (in.3)

L C span of beam (m) (ft);yT C yield stress at elevated temperature (MPa) (psi)

Considering a cantilever beam with a point load ap-plied one-third of the span from the fixed end, plastichinges can be expected at the point of load application andat the fixed end. The critical load can be determined by

pcr

7.5;yTZL

The above equations in this section do not account forcreep strain. Based on an analysis of the deflection historyof heated, loaded beams, Pettersson et al. include a loadratio, +, to determine the critical distributed stress.35

qcr C +8;yTZ

L2

where the yield stress is evaluated at ordinary room tem-perature, relaxing the need to know the yield stress–temperature relations. + is defined as the ratio of the loadcausing a maximum allowable deflection under fire condi-tions to the load inducing stresses equal to the yield stressat ordinary room temperature. Thus, the parameter +takes into account the dependence of both the yield stressand creep on temperature. Graphs of + are available for avariety of thermal restraint and structural end conditions.

The Eurocodes include a method of analysis usingalgebraic equations to consider the moment capacity ofsteel beams which have a temperature gradient through thedepth of the beam.9 The method involves dividing thebeam into small isothermal sections and treating theseisothermal sections as a composite beam (see Figure 4-9.30).In this case, the moment capacity of the beam is given as

Mcap C }n

iC1

;iAizi

whereMcap C moment capacity (NÝm) (lbÝft)

;i C applied stress in isothermal element (Pa) (psi)Ai C area of isothermal element (m2) (ft2)zi C distance from neutral axis to centroid of isother-

mal element (m) (ft)


Computer Programs

Several finite element computer models are availableto assess the structural response of fire-exposed structuralmembers or frames. Sullivan et al. indicate that most ofthe existing finite element models used for structural fireprotection analyses were developed originally for re-search applications.62

FASBUS-II is an example of a finite element modeldeveloped in the United States to evaluate the structuralresponse of complex building assemblies such as floor as-semblies consisting of a two-way concrete slab, steel deck,and steel beam.63 Input for FASBUS-II includes the tem-perature distribution, temperature-dependent mechani-cal properties, geometry, end conditions, and loading.The output of FASBUS-II includes deflections, rotations,


1000°F (538°C)

1025°F (552°C)

1060°F (571°C)

1110°F (600°C)

1150°F (620°C)

1200°F (650°C)

Figure 4-9.30. Isothermal sections of beam.

Hollowstructuralsection 400 × 10

Steelgrade Fe 360

Reinforcingbars S 400Fire class R60

7000

6000

5000

4000

3000

2000

1000

01 2

Buckling length Lcr, θ (m)3 4

(9)

(8)(6)

(5)

(3)

(2)

Buc

klin

g lo

ad N

cr, θ

(kN

)

(7)

(4)

(1)

Hollowstructuralsection 400 × 10

Steelgrade Fe 360

Reinforcingbars S 400Fire class R90

6000

5000

4000

3000

2000

1000

01 2

Buckling length Lcr, θ (m)3 4

(9)

(8)

(6)

(5)

(3)

(2)

Buc

klin

g lo

ad N

cr, θ

(kN

)

(4)

(1)

(7)

(1) C20 1.0

(2) C20 2.5

(3) C20 4.0

(4) C30 1.0

(5) C30 2.5

(6) C30 4.0

(7) C40 1.0

(8) C40 2.5

(9) C40 4.0

μ%

Concrete(C20, C30, C40)

Reinforcingbars S 400

Hollow structuralsection 400 × 10

Figure 4-9.29. Design graphs for ISO fire resistance requirements R60 and R90. For the concrete-filled square hollowstructural section 400 ? 10, the axial buckling load is a function of the buckling length, of the concrete quality, and ofthe percentage � of reinforcement; this design diagram is based on a simple calculation model.8

and stresses in the components of the assembly, whichthen need to be compared to performance limits.

Sullivan et al. and Franssen et al. provide extensivereviews and comparisons of existing finite element mod-els for structural fire protection applications.62,64 Accord-ing to Sullivan et al. all of the models make the followingassumptions:

• Planesectionsremainplane(Navier-Bernoullihypothesis).• Perfect composite action is assumed for steel-concrete

assemblies, disregarding any slippage between thesteel and concrete.

• Torsion is disregarded.• Moisture effects are disregarded.• Large displacements are not accurately modeled.

Traditionally, analysis of the response of the structureexposed to fire has been limited to an analysis of the re-sponse of single members. However, in structural framescomprising many members, load transfer or membraneaction may occur to permit the steel member to maintainits integrity, despite achieving a temperature in excess ofthat typically associated with failure.

Load transfer allows stronger members to support ad-ditional loads not capable of being carried by heated,weak members. In order to capture this phenomenon, aframe analysis is required.44 Numerous software packagesare available to conduct the frame analysis. Results of aframe analysis are presented in Figures 4-9.31 and 4-9.32.

The frame analyses range from algebraic-equation-based methods to finite element analyses. Pettersson et al.


6000 mm

4480 mm

2400 mm

P = 972 kN

q = 94.2 kN/m

HE 400 AA

HE 400 AA

1/2 HE 180M

ΔH

ΔH (mm)

Full scale compositeframe highly loaded/fire class F 150

150

100

50

0 30 60 90

ISO fire exposure time t (min)

120 150

Numericalsimulation

Fire test 3.10March 15, 1985

Testultimate

150 min

Horizontal displacementof frame column duringISO fire test

Simulationultimate

149 min

t

t

Figure 4-9.31. Deformations measured and calculated by a numerical model for a composite frame.8

include a frame analysis via algebraic equations used todetermine displacement.35 The frames consist of beamssupported by one or two columns at mid-span. The analy-sis assumes that each beam or column has a uniform tem-perature (though the temperature of the beam is notrequired to be that of a column). A pinned connection be-tween the structural members is assumed. The analysisconsiders the compatibility of the deformation of eachmember by requiring that the change in length of the col-umn is equal to the beam deflection at the point of contact.

Schleich et al. describe the application of CEFICOSSfor a frame analysis.65,66 The frame consists of a singlebeam and column, where one end of the column is con-nected to an end of the beam. Reasonable agreement is in-dicated between predicted and measured results.

El-Rimavi et al. describe the application of anotherfinite element model, NARR2, for the evaluation of alarge building frame involving numerous beams andcolumns.67 The large frame is divided into several sub-frames for computational ease. Good agreement is noted

between predictions of deflections and force resultantsobtained involving simulations of the full building frameand subframes. Slightly greater failure temperatures weredetermined for semi-rigid connections as compared torigid connections.

Nomenclaturea characteristic dimensionA cross-section area of steel tube, steel columnAs cross-section area of steel columnb characteristic dimensionbf width of flangec characteristic dimensioncc specific heat of concreteci specific heat of protection materialcs specific heat of steelC1 constant


1A2II

32

4

T3

2.14

3.10

1.7

1.3

1.6 AC

1.51.2

1.8

56

1.1

2.111.4

2.13

2.12

1.15

B

t

Simulationultimate

Testultimate

= 0.9

Simulation ultimate, t (min)

Test

ulti

mat

e, t

(min

)

t

Simulationultimate

Testultimate

= 1.1

240

210

180

150

120

90

60

30

0 30 60 90 120 150 180 210 240

Structureelement

Beam

GentB

Braun-schweig

D

Maiziereslès–Metz

F

Bore-hamwood

UK

Failurecriterion

Column

Frame

f ≤ L/30

Buckling

Buckling

Test station

Figure 4-9.32. Fire resistance times measured and calculated by a numerical model for columns, beams, or frames ofany cross section types (bare steel, protected steel, composite).6

C2 constantd outer diameter of steel piped depth of sectionD heated perimeter of steel sectionE0 modulus of elasticity at ambient temperatureEr reduced modulusEt tangent modulusET modulus of elasticity at temperature TF factor of safetyFe factor of safety, elastic designFp factor of safety, plastic designFo Fourier numberh thickness of protection materialH thermal capacity of steel section at ambient tempera-

turek thermal conductivity of steelkc thermal conductivity of concreteki thermal conductivity of protection materialK end condition factorl unsupported length of columnL inside dimension of one side of square concrete box

protectionL span of beamm moisture concrete of concreteN ratio of thermal capacity of protection material to that

of steelP perimeter of steel tubePcr critical point loadPs service (applied) loadPu ultimate loadqcr critical distributed loadr radius of gyrationR fire resistanceR0 fire resistance with zero moisture content of concretet wall thickness of steel pipet timetw width of web!t time stepT steel temperatureTf fire temperatureTm mean fire temperatureT0 ambient temperatureTs steel temperature!Ts change in steel temperatureV correction factorW weight of steel section per unit lengthZe elastic section modulusZP plastic section modulus

Greek

* thermal diffusivity (when used with Fourier number)* heat transfer coefficient

*c convective heat transfer coefficient*r radiative heat transfer coefficient*T coefficient of thermal expansion at temperature T+ ratio of distributed load causing maximum allowable

deflection to distributed load inducing yielding.f fire emissivity4 slenderness ratio1 dimensionless temperature: density:i density of insulation material;cr critical stress for buckling;y0 yield strength at ambient temperature;yT yield strength at temperature T

References Cited1. ASTM-119-98, Standard Test Methods for Fire Tests of Building

Construction and Materials, American Society for Testing andMaterials, Philadelphia (1998).

2. L.G. Seigel, “Fire Test of an Exterior Exposed Steel Span-drel,” Mtls. Res. and Standards, 10, 2, pp. 10–13 (1970).

3. Fire Resistance Directory, Underwriters Laboratories, North-brook, IL (2000).

4. R.W. Bletzacker, Effect of Structural Restraint on the Fire Resis-tance of Protected Steel Beam Floor and Roof Assemblies, OhioState University, Columbus, OH (1966).

5. NFPA 251, Standard Methods of Fire Tests of Building Construc-tion and Materials, National Fire Protection Association,Quincy, MA (1999).

6. UL 263, Fire Tests of Building Construction and Materials, Un-derwriters Laboratories, Northbrook, IL (1997).

7. T.T. Lie (ed.), Structural Fire Protection, American Society ofCivil Engineers, New York (1992).

8. International Fire Engineering Design for Steel Structures: Stateof the Art, International Iron and Steel Institute, Brussels, Bel-gium (1993).

9. Design of Steel Structures—Part 1–2, General Rules—StructuralFire Design, CEN, Eurocode 3, Brussels (1995).

10. ASCE/SFPE 29, Standard Calculation Methods for StructuralFire Protection, American Society of Civil Engineers (1999).

11. D. Boring, J. Spence, and W. Wells, Fire Protection throughModern Building Codes, American Iron and Steel Institute,Washington, DC (1981).

12. D.C. Jeanes, Technical Report 84-1, Society of Fire ProtectionEngineers, Boston (1984).

13. T.Z. Harmathy, NRCC 20956 (DBR Paper No. 1080), NationalResearch Council of Canada, Ottawa (1983).

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Analytical Methods for Determining Fire Resistance of Steel Members

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