AlllDS fiSlfiie ,, PUeUWiONS

NBSIR 80-2188

Joint US-USSR Seminar on

Mathematical Methods for

Estimating the Fire Endurance of

Structural Assemblies

R. S. Levine, Editor

I

Center for Fire Research

National Engineering Laboratory

U S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

December 1980

Final Report

]^QQ U S. DEPARTMENT OF COMMERCE

U56

80-2188

1980

c. 2

NATIONAL BUREAU OF STANDARDS

NBSIR 80-2188

JOINT US-USSR SEMINAR ON

MATHEMATICAL METHODS FOR

ESTIMATING THE FIRE ENDURANCE OF

STRUCTURAL ASSEMBLIES

NATIONAL BURBAO

or 8TANOABDS

UBRART

MAY i 1981

noi

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no, 7 >

C. 7^

R. S. Levine, Editor

Center for Fire Research

National Engineering Laboratory

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

December 1980

Final Report

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Jordan J. Baruch, Assistant Secratary for Productivity. Technoiogy. and innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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Joint US-USSR Seminar on Mathematical Methods for

Estimating the Fire Endurance of Structural Assemblies

R. S, Levine, Editor

FOREWORD

This publication is a compilation of papers presented May 14, 1980,

at a Joint US-USSR Seminar on "Mathematical Methods for Estimating the

Fire Resistance of Structural Assemblies". The seminar was arranged by

the US-USSR Panel on Fire Resistance of Buildings and Structures as a part of their continuing protocol (agreement) to cooperate in this

field. This panel is organized within the framework of the US-USSR

agreement on cooperation in the field of housing and other construction,

for which the Department of Housing and Urban Development serves as the

U.S. Executive Agency. Dr. Robert S. Levine of the National Bureau of

Standards, Center for Fire Research and Dr. Igor G. Romanenkov, Laboratory

Chief, Central Research Institute of Building Construction, GOSSTROY,

are the Co-Chairmen of the Fire Panel.

This is the first of a series of planned yearly joi it seminars on specific applied fire safety research topics. The Soviec papers were

translated into English by the Soviets, but editorial changes have been

made by U.S. personnel who are expert in the particular subject. It is

intended that the original authors’ meanings have not been changed. In some cases, the U.S. presentations are not yet available as published

papers, or were summaries of work that has already been published elsewhere. Hence, one of the U.S. presentations is in the form of slid.es

and commentary. These presentations, in the form given, are intended to

represent up-to-date U.S. technology.

The agenda of the Joint Seminar is presented on pages v and vi.

The papers are published here in the order they appear in the agenda.

The positions and affiliations of the Soviet speakers are given on page

vii.

ill
TABLE OF CONTENTS

Page

FOREWORD iii

AGENDA V

POSITIONS AND AFFILIATIONS OF THE SOVIET SPEAKERS vii

REPORTS

:

"Basic Principles of Calculation of Building Structures* Fire

Resistance Limits", A. I. Yakovlev 1

"Behavior of Inorganic Materials In Fire", M. S. Abrams,

Portland Cement Association 13

"Calculation of the Fire Resis'iance for Flexural Reinforced

Concrete Structures", V. V. Zhukov and V. N. Samoylenko . . 44

"A Mathematical Model of Heat Flow in Concrete Beams Based on the Results of Thirty-two Full Scale Fire Tests,

T. D. Lin and M. S. Abrams, Portland Cement Association . . 50

"Calculation Method for Determining the Possibility of

Concrete Spalling Due to Fires", V, V. Zhukov 75

"Calculating Fire Endurance for Steel Structural Members

Located on the Outer Walls of Buildings", R. G. Gewain,

American Iron and Steel Institute 82

"Fire Resistance Calculation Methods for Steel Structures",

A. I. Yakovlev 110

"Reliability Analysis of Fire-Exposed Light-Frame Wood

Floor Assemblies", E. Schaffer, Forest Products

Laboratory. Presented by F. Woeste, Virginia Institute

of Technology 128

"Basic Principles of Calculating the Fire Resistance of

Timber Structures", G. M. Klrpichenkov and I. G.

RomanenkoV 181

"Approach to the Fire Resistance Design for Light Wall

Panels Using Effective Materials", V. N. Zlgern-Korn

and I. G. Romanenkov 190

"Sprinkler Systems: An Approach to Active Fire Resistance",

J. Hankins, Factory Mutual Research Corporation 197

IV
u.s.-u.s.s

May 14, 1980

9:00 a.m.

9:10

9:15-9:45

9:45-10:15

10:15-10:45

10:45-11:00

11:00-11:30

11:30-12:00

12 : 00- 1:00

/ V \

•*<«$ o»

UNITED STATES DEPARTMENT DF COMMERCE

National Bureau of Standards

Washington, D.C. 20234

AGENDA

R. PANEL ON FIRE RESISTANCE OF BUILDINGS AND STRUCTURES

Joint Seminar

9:00 a.m. B119 Polymers (Bldg. 224)

"MATHEMATICAL METHODS FOR ESTIMATING

THE FIRE RESISTANCE OF STRUCTURAL ASSEMBLIES"

Announcements and Plans

Dr. Robert S. Levine, Center for Fire Research,

National Bureau of Standards

Welcome

Ms. Nina Shafran, Office of International Affairs,

Department of Housing and Urban Development

Basic Principles of Calculation of Building Structures*

Fire Resistance Limits , A. I. Yakovlev

Behavior of Inorganic Materials in Fire , M. S. Abrams,

Portland Cement Association

Calculation of the Fire Resistance for Flexural Reinforced

Concrete Structures , V. V. Zhukov and V. N. Samoylenko

Coffee Break

A Mathematical Model of Heat Flow in Concrete Beams Based

on the Results of Thirty-two Full Scale Fire Tests , T. D.

Lin and M. S. Abrams, Portland Cement Association

Calculation Method for Determining the Possibility of

Concrete Spalling Due to Fires , V. V. Zhukov

Lunch

V
1:G0--1:30 Calculating Fire Endurance for Steel Structural Members

Located on the Outer Walls of Buildings , R. G. Gewain,

American Iron and Steel Institute

1:30-2:00 Fire Resistance Calculation Methods for Steel Structures

A. I. Yakovlev

2:00-2:30 Reliability Analysis of Fire-Exposed Light-Frame Wood

Floor Assemblies, E. Schaffer, Forest Products Laboratory.

Presented by F. Woeste, Virginia Institute of Technology.

2:30-3:00 Basic Principles of Calculating the Fire Resistance of

Timber Structures , G. M. Klrpichenkov and I. G. Romanenkov

3:00-3:15 Coffee Break

3:15-3:45 Approach to the Fire Resistance Design for Light Wall

Panels Using Effective Materials , V, N. Zigern-Korn

and I. G. Romanenkov

3:45-4:15 Sprinkler Systems: An Approach to Active Fire Resistance , J. Hankins, Factory Mutual Research Corporation

Adj ourn
POSITIONS AND AFFILIATIONS OF THE SOVIET SPEAKERS

I . G . Romanenkov

Laboratory Chief at Central Research Institute of Building

Constructions, USSR State Construction Committee

A. I. Yakovlev

Department Chief at All-Union Research Institute of Antifire

Defense Institute, USSR Ministry of Internal Affairs

V . V . Zhukov

Laboratory Chief of Research Institute of Concrete and

Ferro-Concrete, USSR Construction Committee

vii
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BASIC PRINCIPLES OF CALCULATION OF BUILDING

STRUCTURES' FIRE RESISTANCE LIMITS

Dr. A. I. Yakovlev

VNIIPO, Ministry of Interior of the USSR

Fire resistance limits of building structures are calculated according to

two indexes: (a) load bearing capacity failure due to strength reduction

temperature creep development, thermal expansion and pyrolysis of structural materials in the process of heating; (b) heating of the unexposed

surface to 140°C due to heat transfer process.

Generally, the calculation process consists of two parts: heat transfer

analysis and stress analysis.

The heat transfer calculations aim at temperature field determining the

different sections of the structure and is performed by solving the

problem as a case of unsteady state conductive heat transfer. It goes

without saying that the heating index of the fire resistance limit of

structures is determined solely by the heat transfer calculation.

The stress analysis determines the load bearing capacity of a structure

under heating on the basis of ultimate equilibrium equations and deformations considering the variation of physical-mechanical properties with

temperature and the pyrolysis rate of building materials under high

temperatures

.

In general outline, the concept of calculating the fire resistance limit

with regard to load bearing capacity failure is shown in figure 1. In

figure 1 the abscissa indicates heating time, t. The ordinate indicates

the durability of the heated structure (for bending elements, M

bending moment; or for compression and tension numbers, Np longitudinal force). On the right vertical axis is shown the nominal load

value (Mjj and Ny, respectively) required for the structure.

The temperatures in the sections of the structure are determined for

certain periods by heat transfer calculations. Then the stress analysis

calculation is made, the results of which form the load bearing capacity

reduction curve of the structure exposed to heating. The calculated

time during which the load bearing capacity of structure is reduced to

the working load value is the calculated fire resistance of the structure

This is the general principle of fire resistance calculation with regard

to load bearing capacity failure. It is applicable to every structure.

However, there are particular cases when there is no necessity of determining the load bearing capacity reduction curve, but it is possible to

determine the fire endurance by merely calculating a section temperature.

The temperature so determined is termed "critical" and it causes the

load bearing capacity failure, i.e. fire resistance limit. Then the

time necessary for heating the structure to this critical temperature Is

1
calculated by solving the inverse heat transfer problem. The time

mentioned is considered to be the fire resistance limit. This is the

way to calculate the bending capacity of reinforced concrete elements

and of steel structures. Their failure takes place due to heating of

the reinforcement and flange to the critical temperature that is regarded

as uniformly distributed over the sections of these elements due to high

heat conductivity of metal. As the fire resistance estimation generally

starts with a heat transfer calculation, we shall first examine the

calculation methods.

I. Heat Transfer Calculation

The initial and boundary conditions of the calculation are as follows:

Initial conditions:

the temperature over the structure's section is uniform and

identical with ambient temperature - t„ = 20°C. n Boundary conditions:

a) compartment fire temperature - t^ varies according to the

"standard curve":

tg = 345 log (8 T + 1), “C,

where t - in min.

b) the coefficient of heat transfer from the fire to the exposed

surface of the structure - a - is estimated according to the

formula:

^B-^273y ft^ + 273

a = y 25 + 4.96 S \ 100 /

"

\ 100 / np 7 -V *

kcal, m ,

where S

”p + i_ . 1

S_ s B o

- the given emissivity of the system

between the furnace and the surface of the structure;

Sg = 0.85 - the emissivity of the furnace fire box ambient;

- the emissivity of the structure's exposed surface;

t - the temperature of the structure's exposed surface;

c) the coefficient of heat transfer from the unexposed surface of

the structure to the ambient - a' - with t^ = 20®C is estimated

according to the formula:

2
4

t„ + 273

4

= 1.3^v/t» - t„ + 4.96 S’

H o

t' + 273

o

V 100 100

t' - t o H

1 1 “2 +.“1 Op-1

kcal, m , t , C

where t' - the exposed surface temperature;

o - the emissivity of the unexposed surface.

The heat transfer calculation also takes into account the nonlinearity of

the internal heat transmission and heat absorption for evaporation of free

water in the pores of material (negative heat source)

The nonlinear heat transmission is taken into account by the straight line

dependence of the thermal conductivity - and heat capacity - - on temperature - t

’

X^ = A + B. t, = C + D.t

where A and C are the initial values of thermal conductivity and

heat capacity;

B and D are the coefficients of thermal conductivity and

heat capacity change with regard to temperature.

The influence of material moisture content on the change of the temperature

field in the exposed structure section takes place due to moisture transmission and water evaporation in the pores of the material.

As it is necessary to provide the coincidence of experimental and calculated

curves of structure's heating at critical temperatures 100°C and higher,

to simplify the process, the calculation is made without taking moisture

transmission into consideration. It is assumed that the initial moisture

content of the layers does not change and water evaporation takes place in

one layer after another. The evaporation starts at 100°C and requires

539 kcal/kg. As a result, the process of heating the structure is retarded.

The phenomenon mentioned above is taken into account in the heat transfer

equations used in the calculation.

The majority of building structures have one dimension that is significantly

more or less than the other two. If the dimension in question is less

than two others, for example, the width of flat enclosing structures, the

temperature variations are considered to occur solely in the direction of

the dimension in question, i.e. a one-dimension analysis is solved. In case one dimension is more than two others, for example, the length of

beams, columns and cross-bars, the temperature calculation is made in the

cross-section in the other two directions, i.e. a two-dimensional analysis

is solved.

The above mentioned heat transfer evaluation conditions indicate the

necessity of taking into account the internal and external nonlinearity

plus the internal (negative) heat sources. Therefore, it is practically

3
impossible to obtain analytical solutions of the equations even for a single-dimension heat flow. The similar solutions available for

engineering design are likely to be obtained only with certain simplifying assumptions, for example, introduction of constant (taken at some average temperature) heat transfer coefficients and application of first

order boundary condition. With these assumptions the analytical solution for heating time - t - of the exposed surface Is possible for one- dimensional heat flow. To calculate the heating time - t - of the

reinforcement of concrete solid slabs to a critical temperature for

one-dimensional heat flow the following analytical solution is used:

where k is a temperature variation coefficient of the exposed

surface dependent on the density - ” of dry concrete (table 1).

Table 1

kg*m ^

100

and less

1000 1500 2000 2300 2450

1, V, 1/2 k,hr 0.46 0.55 0.58 0.60 0.62 0.65

y

-

d

- ^1

•

is the thickness of

is the reinforcement

- is a coefficient of

heating, depending

(table 2)

the concrete cover bar’s diameter

the rod's metal mass influence on on the dry concrete density - y

c

its

Table 2

1 -3

kg*m

500

and less

800 1100 1400 1700 2000

and more

^1 1.0 0.9 0.8 0.7 0.6 0.5

A

np

- is the given thermal diffusivlty coefficient of concrete.

m hr

-1

A = np (C + 0.012 p) Y t,cp 'c

Y and C are the average values of concrete thermal conductivity

t , cp t , cp

and heat capacity, evaluated at 450°C;

4
p - concrete moisture content, weight %;

X - Gauss' error function evaluated according to the formula:

1250 - T

H

Ta - is the reinforcement critical temperature, °C.

The analytical time determination - x - for the unexposed surface of the

structure to reach 140°C (fire resistance limit on the basis of heating)

,

applying the same simplification, may be introduced as

T = 2.3

(6 + k/a' ) H2_ log

140

hrs.

np 1 + B.

X

1250 - tH

where 6 is the structure depth, m;

B. - Biot number

^ a' n - cp

i A

t,cp

a’ =4+7. 4S' is the average heat transfer coefficient of

the unexpSsed surface, to the ambient air t^ = 20°C;

including radiation

(6 + k /U')

np

t,cp

is the average heat thermal conductivity, evaluated

at t = 250°C

A^p is the given thermal diffusivity coefficient at t = 250“C

y and A are the first characteristic equations’ roots

^ ^ (table 3).

5
Table 3

B. y

A.

X X X

0 1.5708 1.2735

0.1 1.632 1.1865

0.2 1.6887 1.1037

0.3 1.7414 1.0329

0.4 1.7906 0.9758

0.5 1.8366 0.9246

0.6 1.8798 0.8812

0.7 1.9203 0.8406

0.8 1.9586 0.8038

0.9 1.9947 0.7710

1.0 2.0288 0.7415

1.5 2.1746 0.6253

2.0 2.2889 0.5435

2.5 2.3723 0.4889

3.0 1.4557 0.4342

4.0 2.5704 0.3587

5.0 2.6537 0.3065

6.0 2.7165 0.2692

7.0 2.6558 0.238

8.0 2.8044 0.2133

9.0 2.8363 0.1934

10.0 2.8628 0.1763

The differential numerical methods utilized by means of a computer are the universal ones for solving the unsteady state heat transfer tasks

without simplifying preconditions, and are applicable for practically

any structure. The method of elementary heat balance is one of them.

According to this method, the heat balance equations are derived for a series of ultimate elements that are chosen arbitrarily in the structure

by means of a grid attached to its section. Square-, rectangular- or triangular-celled grids are used subject to the section configuration.

The heat balance equation of the ultimate element 1 m long and AX. AY

square with a node XY (fig. 2) for the rectangular grid is given below:

6
t 1 + t \ ft T - t 'jAyAfl

. R x-l,y x,y

) V x-l,y

2 / Ax

^x+l,y^~'' ^x,yj

(

^x+l^y + B

,^x,y

^ A + B

ft T~t '\Ax*At* 1 / t ,1+t \/ft - t Ax-At*1\

V x,y-i x,y; + ( A + B

x? y+y

Ay \ 2 / \ Ay /

Y Ax*Ayl(C + D*t ) (t . - t ) +

c

^ x,y^ x,y,Ax x,y

Y^* Ax* Ay p*r*l

100

where r is the latent heat, kcal/kg

The temperature - t certain time interval.At - evaluation formula at the node point

At, is derived by the present equation.

in a

The formulas for all the nodes as well as edge conditions are the computer

algorithms for structural fire resistance evaluation on the basis of the

heat transfer analysis. In case of a one-dimensional heat flow, e.g.

along the y-axis, the nodal temperatures along the x-axis become equal to

each other

(t = t

x,y x-l,y ^x+l,y^ »

and the evaluation algorithm is simplified.

Of concern is that the nodal locations are properly specified and the

heat balance equations properly derived for reinforced concrete and

protected steel structures with distinctly variable material heat transfer

analysis characteristics. The section temperature of bent reinforcement

bars and of straight reinforcement profiles walls thickness is considered

constant in reinforced concrete structures. Therefore, the bent reinforcement

bars' nodes are located in their section's center.

The nodes of the straight reinforcement profiles are located along the

length of the walls and flanges in their thickness center. The application

of the grid to coated steel structure profiles is carried out analogously.

The methods of locating grid nodes in reinforced concrete, steel and

laminar nonbearing structure sections as well as heat balance equation

types for the nodes are more precisely described in the reports in this

seminar dedicated to the fire resistance calculation of the structures

mentioned above.

7
II. Statilc Stress Analysis Calculation

As !ias been already mentioned, this type of calculation is carried out in case the fire resistance of the structure is determined by bearing capacity

loss (collapse)

As follows from the basic equilibrium equations, the load-bearing capacity

(durability) is affected by the fire destruction according to loss of

strength of the structural materials under high temperature conditions.

The section diminution due to pyrolysis is taken into consideration in

structures made of combustible materials.

a) Reinforced Concrete Structures

The heating of statically determinate building structures at the

bottom results in their failure; as a rule the failure takes place at the

cross-section of the maximum bending moment. Sheer failure by diagonal

tension is a rare phenomenon and can be prevented by stirrup reinforcement

near the bearing points.

The destruction in the cross-section of almost all the slabs and beams

takes place due to the tensile reinforcement's heating to its critical

temperature. At this temperature a rapid increase in creep occurs due

to the working stresses. Thus the reinforcement capacity at the critical

temperature is reduced to the value of the working stresses. Therefore,

the critical temperature - T ^

- for the fire endurance estimate of a standard bending structure ac&o?ding to the heat transfer calculation is

derived from the strength vs. temperature diagram of the reinforcement

steel. The working stress values are calculated from the equilibrium of

the cross-section.

In the case of concrete flexural members, such as beams, where the

tension reinforcement is by means of bars manufactured from various

classes of steel, and the concrete in the compression zone may be heated

to high temperatures, the fire endurance time is estimated according to

figure 1 for both the tension and compression zones. For all these

cases the load bearing capacity of the structure is estimated for definite time periods by substituting the concrete strength values in the

ultimate equilibrium equations.

Statically Indeterminate flexural structures are subjected to stress

redistribution under heating. In this regard the beam members with

support restraint undergo a reduction of the bending moments at midspan and an increase of bending moments at the supports. In the ultimate

equilibrium conditions the bearing capacity is based on the residual

strength of the supporting and mid-span sections.

8
When slabs supported along four sides with approximately equal sides sag

significantly, their compressed area moves to the sides and the compressive

area becomes extended over the width of the slab. Owing to considerable

stress reduction the tensile reinforcement's critical temperature is

assumed to be 800°C. If the ratio of the sides increases, the reinforcement critical temperature falls to the value noted for simple beam

members

The brittle failure of columns with axial compression is accompanied by

concrete crushing over the whole of the cross-section and also by

bulging of the longitudinal reinforcement.

The columns load bearing capacity in the ultimate equilibrium conditions

is found by summing up the load bearing capacity of the concrete section

and of the reinforcement at high temperatures.

Due to considerable temperature variation through the solid column

cross-section (more than 200 x 200 mm) the deformation of the central

slightly heated layers is significantly less than the intensely heated

periphery layers. Therefore, the columns maximum deformation in the

ultimate equilibrium condition takes place solely in the cooler concrete

at the center of the section. Because of this load transfer, the strength

of the outer concrete layers is only partially available.

In this connection the concrete section bearing capacity of solid

columns should be estimated considering the temperature creep deformations of loaded concrete. The data may be obtained by testing concrete

samples in an apparatus capable of imposing constant load while a specimen

is heated according to the concrete heating curve for the fire resistance

test on structures.

The temperature variations through thin-walled columns sections having

dimensions 200 x 200 mm or less are insignificant. The concrete deformation of the section of the thin-walled columns at ultimate loading may

be considered to be the same, and concrete ultimate strength for all the

layers is reached. That results in nearly equal fire endurance limits

(approximately 1 hr.) for columns with section dimensions 200 x 200 ram and, for example, double-T columns with walls thickness of 60 mm. Columns

with section dimensions 300 x 300 mm and double-T columns with walls

thickness of 120 mm have an endurance of approximately 2 hrs.

b) Steel Structures

To calculate the fire resistance limit of steel structures it is

necessary to determine their heating time to critical temperature. The

temperature value is evaluated according to point of plasticity or to

the elastic modulus curve of steel structures when heated depending on their mode of failure. Bearing capacity loss of compressed elements may

take place due to loss of strength and also loss of integrity. The

failure mode depends on the type of structure, steel characteristics,

element elasticity, type of support and working load value, /vccordingly

,

the plastic limit and the modulus of elasticity of the steel are used in

the ultimate equilibrium equations. The critical temper tures are determined according to the curves with regard to these characteristics.

9
c) Timber Structures

Timber structure failure at fire exposure takes place due to

functional section reduction resulting from timber charring and also due

to reduction of mechanical properties of the heated timber cross-section.

The time of complete failure is determined according to general case scheme (figure 1). Of concern is that in the calculated heating time

periods the structure's bearing capacity depends on the cross-section's

size and its durability under heated condition. The reduction in the

modulus of elasticity of the timber in the compression zone of the heated

cross-section is also taken into consideration with the view of determining

the critical force affecting the structure's stability and failure.

10
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43
CALCULATION OF FIRE RESISTANCE LIMITS FOR

FLEXURAL REINFORCED CONCRETE STRUCTURES

Dr. V. V. Zhukov, PhD. and Dr. V. N. Samoylenko, PhD.

NIIZhB, Gosstroy USSR

At present the fire resistance of reinforced concrete structures,

especially of those newly being designed is mainly determined experimentally. The Experimental method for determining the fire resistance

limits of structures is rather labor-consuming and expensive. Therefore

both in our country and abroad one should search for ways of determining

the fire resistance limits of structures without performing fire tests.

On the basis of numerous test analyses on fire resistance of reinforced

concrete members tables and diagrams were made to give the fire resistance

limits of structures by their sizes, thickness of concrete cover; reinforcing

steel, types of concrete, etc. These methods of determining the fire

resistance limits are given in SNIP 11-A. 5-70(1) and in Recommendations

(2) . It should be noted that according to this method heating of elements

will be regulated to follow the curve given in ISO DIS 834.

For structures of new materials, different cross-sections as well as other fire exposures, this method of determining the fire resistance

limits for reinforced concrete structures is unusable. For this reason

a calculation method for the fire resistance limits of reinforced concrete

structures has been developed.

The calculation approach consists of two stages; the first is the

determination of transient heat flow in cross-sections of structures,

and the second one is an analysis of structure under static load,

considering the change of concrete and steel properties with respect to

temperature during their heating period.

The transient heat flow computation method in our country is based on A.

V. Lykov's theory of total heat mass exchange (3). In general it is

necessary to solve a system of partial differential equations in order

to find the interrelationship between temperature, moisture content and

steam pressure in concrete.

For determining the fire resistance limits of reinforced concrete

structures, this system is considerably simplified to solving two (4, 5) or one (6, 7) differential equation.

Calculation of temperatures in reinforced concrete structures is a nonlinear differential equation, which is written as

c . p

. = div (A . grad T) + l (1) da

where T - temperature;

A - thermal conductivity;

L - heat of liberation and absorption due to phase transition

of water in concrete;

C - specific heat;

p - density of concrete.

44
This is the Fourier's equation in its general form. A digital computer

was used to solve this equation considering the transformation of mechanically

combined water in concrete structure into steam only through an adjustment

in tna value of "C" (6). Another way for thermal calculations (4, 7) is

based on the method of smoothed heat transfer coefficients that takes

into account different phase transitions of cement paste in concrete.

This was done by introducing the latent heat, H(T) as a temperature

function to justify the effect of phase transitions. In reference (5),

equation (1) is solved directly with data measured during a test of a column that was exposed to a non-isothermic fire.

It is natural that all heat-flow calculations are performed on the basis

of correct initial and boundary conditions. However, one may encounter

difficulties of physical sense and mathematical order in specific solution

steps.

The above basic calculation methods of transient heat flow for fire

resistance limits give a reasonable agreement between calculated and

measured temperatures in concrete members exposed to fire. This agreement

is attributed to the right formulation of the problem, and accurate

experimental data on the thermodynamic parameters of material. Recently

there has emerged a better method that accounts for phase transition of

water in the concrete capillary system at any calculated temperature

level.

The calculation of flexural reinforced concrete structures subjected to

static load and high temperature mainly involves the computation of

moment capacity and the associated deflections.

The moment capacity of the structure is computed at the stage under • which the strength of the most stressed cross-section decreases so much

that it is unable to bear the load and dead weight any longer.

Calculation methods for the strength of heated flexural members have

been elaborated rather well in our country and abroad (6, 8). For

calculations one may use formulas under ambient temperature. Nevertheless the coefficients that account for a decrease of yield stress of

reinforcement due to heating must be introduced.

The influence of temperature on reinforcement yield stress has been

investigated by many experimenters. The published data provide accurate

values of the above-mentioned coefficients.

Decrease of concrete strength in the compressed zone is not usually

taken into account due to its low temperature. It should be noted that

Investigations, carried out by A. I. Yakovlev (6), have shown that

consideration must be given to the decrease of cross-sectional area in

the compression zone caused by stress relaxation of reinforcement due to

the high thermal strains of reinforcing steel.

45
The time shortly after a considerable increase of deflection in the test

members is often assumed to be the end point of flexural reinforced

concrete structures exposed to fire. Therefore, for the sake of accuracy, one should calculate the fire resistance of flexural reinforced concrete

structures throughout the fire test. Calculations for thermal strains

in a reinforced concrete member are considerably more complicated than

the determination of bearing capacities of that member and have not been

elaborated in depth. For example, in Recommendations (2) there are no proposals for strains calculation for flexural members.

Experimental and theoretical investigations on the development of calculational methods for strains in reinforced concrete flexural members,

subjected to fire have been conducted for some years in NIIZhB, Gosstroy

USSR. These calculation methods given in Manual (9) , are assumed as a basis for computing thermal strains according to the following formula:

1/p = 1/p^ + 1/p^ ( 2 )

where l/p„ and 1/p. - axis curvatures of member due to load and temperature

n L respectively.

The curvature of heated reinforced concrete flexural members due to

external load and dead weight is determined from the formula

1/Ph (3)

where M - bending moment due to dead weight and external load;

b - width of the flexural member

h - distance from extreme compression fiber to centroid of

tensile reinforcement;

- distance from centre of gravity of reinforcing steel to

the reference axis;

E^, Eg - elastic moduli of reinforcement and concrete respectively;

3 “ coefficients, accounting for decrease in elastic moduli

of reinforcement and concrete under heating, respectively;

V ,v - coefficients, accounting for development of plastic deformations in reinforcement and in compressed concrete of

member

;

*^a’'^6*^ - coefficients, determined in accordance with Manual (10);

F - area of tensile reinforcement,

a

46
The Axis of curvature of a reinforced concrete member due to heating,

is determined from:

a ^

• t at a 6t 6 h

o

(4)

where

'^at’°^6t ~ coefficients of linear thermal expansion of

reinforcement and concrete, correspondingly;

t ,t^ - temperatures of tensile reinforcement and extreme

compression fiber of concrete, correspondingly.

In order to solve equations (1) through (3) for the deflections of flexural

reinforced concrete members under fire according to proposed methods, one should know the temperature distribution on the member cross-section and

the temperature related coefficients effect of

temperature on physical and mechanical properties of reinforcement and

concrete has been determined in numerous experiments (6,8,10). On the

basis of test results analyzed for the coefficients 3^,3g,v various

types of steel and concrete, it is possible to form mithemafical^relatlonships between the temperature and the above coefficients.

Formulas (3) and (4) were used in the computer program for calculating

the strains in flexural reinforced concrete members under temperature

and load action. The program is also used to determine the strength of

members due to heating.

Figure 1 shows deflections obtained by the methods developed by A. F.

Milovanov and experimental results, obtained by Kh. U. Kambarov on ceramsite concrete slabs with steel percentage of y = 0.44% (10).

Comparing calculated and experimental values one find that calculated

values are in a good agreement with the measured ones throughout the

fire test. As the member reaches its fire resistance limit (x = 95

min), the calculated deflection sharply increases.

The above-mentioned methods may also be used for determining the

rigidity and temperature effect on statically indeterminate reinforced

concrete structures.

References

1. SNIP II A. 5-70 "Fire-prevention design codes for buildings".

2. FIP/CEB Recommendations for the Design of Reinforced and Prestressed

Concrete Structural Members for Fire Resistance", FIP/CEB 1975.

3. Lykov, A. V., "Heat Exchange", Publishing House "Energiya" ("Energy"),

M., 1972.

47
4. Robsman, V. A. , Investigation of heat and mass transfer in refractory

capillary porous bodies. Candidate thesis, M. , 1975.

5. Roltman, V. M. , Lyrina, G. N., Solution of heat and technical

problem of structure fire resistance considering the processes of

moisture transfer on digital computers according to implicit finitseparatlon diagram. Collection N 2, "Fire resistance of building

structures", VNIIPO MVD USSR, M., 1974.

6. Byshev, V. P., Pchelintsev, V. A., Fedorenko, V. S., Yakovlev,

A. I., "Fire resistance of buildings", Stroyizdat, M. , 1970.

7. Zhukov, V. V., Robsman, V. A., Shevchenko, V. I., Tsygankova, L. R.,

Heat and physical calculations of structures from refractory

concrete on digital computers, Volgograd, 1971.

8. Report on methods of assessment of the fire resistance of concrete

structural members, FIP/CEB, 1978.

9. Design Manual for Concrete and Reinforced Concrete Structures for

Fires, Stroyizdat, 1977.

10. Milovanov, A. F., Kambarov, Kh. V., Malkina, T. N., Fire resistance

of ceramsite concrete structures, FIP Notes 76, 1978.

48
6S 7^vut^7y.^^ tj^9eu/A:e2o

i£x> du^UZ^hA kA

/

C.TtxAT^T'

O

49
A MATHEMATICAL MODEL OF HEAT FLOW IN CONCRETE BEAMS

BASED ON THE RESULTS OF THIRTY-TWO FULL SCALE TESTS

T. D. Lin and M. S. Abrams

Portland Cement Association

Editors * Note : This presentation was given by

Dr. T. D. Lin from slides and notes. A final

text has not yet been released by the Portland

Cement Association. The text has been prepared

from several of Dr. Lin's slides and his notes,

and constitutes an "extended abstract" of the

presentation. A full publication will be

released subsequently by the Portland Cement

Association.

In the 1960's, the Fire Research Laboratory of the Portland Cement

Association carried out an extensive program to determine isotherms in

concrete beams in a furnace 13 m long, 3.5 m deep, and 2 m wide.

Figure 1 shows six of the 32 specimens used in the program. A beam

being placed in the furnace is shown on figure 2.

The temperature information from the test program was used to

develop and validate a mathematical model of heat flow in concrete beams.

The major steps in the computer program, outlined in figure 3, include

input, nodal points, the time-temperature curve, boundary conditions,

assignment of volume specific heat and thermal conductivity, and then the

iterative heat transfer calculation.

Input data include the dimensions of the beam. The time-temperature

curve used was the standard ASTM E 119, which was presented by parabolic

equations as indicated in figures 4 and 5, where y is the temperature and

X is the test time. There are (n-1) parabolic equations for n control

points

.

Figure 1 = slide 4

Figure 2 = slide 36

Figure 3 = slide 6

Figure 4 = slide 10

Figure 5 = slide 11

The general solution for the unknown constants of the nth parabolic

equation can be expressed as shown in figure 6, and the temperature com- puted as shown in figure 7.

The temperatures on the exposed surfaces were prescribed according

to the measured data. They varied with position on the beam surface

because some of the heat transfer from the furnace to the beam was by

50
radiation, and the radiative view factor varied with position as shown

in figure 8.

Figure 9 indicates the increases in surface temperature, AUi, AU 2 , etc., at a point on the beam surface for time intervals, AT^, AT2 , etc.

Combining these measurements, one obtains temperature boundary conditions

shown in figure 10.

The initial volume specific heat data (specific heat x density)

used in the program were taken from a paper by Harmathy, and are shown in

figure 11. The high values at temperatures near 500°C are associated

with heat absorption due to chemical decomposition. The symbol e represents the extent of the chemical reaction.

Figure 6 = slide 13 Figure 9 = slide 18

Figure 7 = slide 14 Figure 10 = slide 19

Figure 8 = slide 17 Figure 11 = slide 21

The thermal conductivity as a function of temperature, shown in figure 12, was also taken from Harmathy 's data.

The equation used for temperature calculation in the computer program

is shown in figure 13. Using At steps of 1/2 minute, convergence was obtained without difficulty. The value of temperature at a central nodal

point is taken as the arithmetic mean of the values at four neighboring

points

.

The specimen used in the computation to be shown is a T beam. Only

half of the cross section is shown in figure 14. For simplicity, only

temperatures in the circle are shown at t = 1/2 hour in figure 15. The

numbers in parentheses are measured data. Those not in parentheses are

computed using a program developed at the University of California; the

computed results under-estimated the true temperature as much as 220°C.

Similar errors were calculated at other times.

Figure 16 shows a measured temperature-position curve along the

centerline of a rectangular beam after 1 hr exposure. The dashed line

would be expected if the heat flow reached a steady state. Because of

the steep slopes of both lines, the temperature difference between the

two can be substantial; on the order of 100°C.

Figure 12 = slide 23 Figure 15 = slide 34

Figure 13 = slide 24 Figure 16 = slide 37

Figure 14 = slide 33

To obtain better agreement between calculated temperatures and

the data, the literature values of the volume specific heat were adjusted,

as a function of temperature, by the empirical correction factors shown

in figure 17

.

51
I

Typical results at 1/2 hr, 1 hr, 2 hr, and 4 hrs of test are shovm in figures 18, 19, 20, and 21, showing reasonable agreement.

Calculated temperatures at 1 hr in a T beam (that was not tested) are

shown in figure 22.

Figure 17 == slide 42

Figure 18 = slide 44

Figure 19 = slide 45

Figure 20 = slide 46

Figure 21 = slide 47

Figure 22 = slide 50

52
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53

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57

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General

Solution

I

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58

,

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SLIDE

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60

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i

61

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Measured

Calculated

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Measured

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4

SLIDE-50
CALCULATION METHOD FOR DETERMINING THE

POSSIBILITY OF CONCRETE SPALLING DUE TO FIRES

Dr. V. V. Zhukov, PhD

NIIZhB, Gosstroy USSR

Spalling may take place in reinforced concrete structures exposed to

fire. It occurs suddenly with a great speed and therefore is a most

dangerous phenomenon. As a rule, spalling begins 5-20 min. after the

start of fire and manifests itself as splitting of concrete pieces in

the form of slab rocks about 1 cm^ to 0.5 -1 m^ in area and from 1 mm to

5 cm in thickness from the heated surface. Spalling may continue during

the entire period of fire test until complete destruction of the structure, accompanied by a sound like a light "slap”, cracks of different

intensities or "explosions". Scattering of pieces up to some kilograms

as far as 10-20 m may take place.

Concrete spalling due to fire may quickly lead to failure of reinforced

concrete structures. In this case the limit of structural fire resistance may be considerably less than the required one due to decrease in

size of the concrete cross-section of the structure; decrease in thickness or complete elimination of concrete cover as well as the formation of a penetration.

Decrease in cross-section and increase of bending moment due to unsymmetric

heating of columns or panels subjected to vertical loads, results in

stress concentration in the remaining part of cross-section (figure 1)

.

Thickness decrease or splitting of concrete cover in reinforced concrete

beams results in a quick rise of steel temperature up to a criticallevel (500-700°C) that causes structural failure (figure 2)

.

Decrease in slab thickness leads to a sharp temperature rise on its

unheated surface to a critical temperature (180-220°C) . As a result the

structural fire resistance limit is met.

Spalling may cause a through-hole (penetration) in a slab system at once or after a certain period of fire exposure. Therefore, the slab will no longer be a barrier preventing fire passing from one room to another

(figure 3). The fire resistance limit in any structure is reached as soon as a penetration forms.

Under fire conditions, the onset of spalling often does not take place

simultaneously with collapse of the entire structure, which occurs considerably later. Sometimes, despite the occurrence of spalling, the

structure can resist fire for a long time. In this case its fire resistance

may be quite sufficient to meet the standard requirements.

75
When evaluating the effect of spalling on the f ire-reiiutance limits of

concrete and reinforced concrete structures one should (consider every

possibility that may cause a structural failure.

Spalling on load-bearing structures under fire conditions is most

dangerous, especially for those with a small cross-section, carrying

heavy loads. Their unpredictable failure may result in a collapse of

other parts of the structure, or the entire building. For example

concrete spalling in columns and panels on ground floors and basements

of multi-story buildings is impermissible.

The cause of concrete spalling under fire conditions is the transformation

of cracks formed before heating or during heating, from equilibrium

state into a non-equilibrium one, and the spontaneous development

(figure 4) of high stresses in a local area.

When working out new mix designs or those already available, one should

evaluate the possibility of concrete spalling due to fire by a special

computational method. This computational method determines the value of

a spalling criterion, F, as shown in the following equation:

F = a

g •

. p K • A . n

* o where a - proportionality factor;

( 1 )

g - coefficient of linear thermal expansion of concrete;

E - elastic modulus of heated concrete;

p - density of dry concrete;

K - tensile strength of concrete;

A - thermal conductivity of concrete;

n - porosity of concrete;

£ W - moisture content of concrete by volume before fire,

o

’

Table 1

Value Comments on spalling and necessary measures

1 ^

> 4; < 6

> 6

Concrete will not spall. No additional check or structural

protection are required.

Concrete may spall in the compression block due to long-term

standard load or in members which have thickness less than

4 cm. Here one needs to further control concrete from

spalling.

Concrete will fail. Special measures should be applied

for its protection.

76
The possibility of concrete spalling due to fire in a non-load bearing

structural element may also be estimated on the basis of mean critical

moisture of concretes by weight ; 3% - for heavy concrete with

coarse aggregate of granite; 4% - for heavy concrete with coarse aggregate

of limestone rocks; 5-10% - for lightweight structural concrete with

coarse porous aggregate; 2-3% - for heavy silicate concretes.

If the moisture content of concrete by volume (W ) is less than the mean

° -3

critical moisture by weight, multiplied by the concrete density and 10

i.e. • p 10 then the concrete will not spall.

Concrete, having ”F" values more than 4 or less than 6, should be checked on the possibility of spalling due to fire in the compression block near

the edge fibre on the exposed surface of the concrete member that carries

the long term design load. In this case the compressive stresses (a) in an independent member due to fire are determined by the long-term design

load. The critical volumetric moisture content of concrete is

calculated fron the formula

c • • n

E

A (1+0, 1 + 0.15 a ^/R ) sh p

( 2 )

where C - proportionality factor;

JJ Rp - allowable tensile strength of concrete.

For steam cured concretes, the value of derived from formula (2) * o

• should be reduced by a factor of 1.4.

Value w^P is compared with the moisture content W^.

o o If

: ^ then the concrete will be subject to spalling due to fire.

One should take special measures to protect the concrete against spalling

or should decrease the design compressive stresses in the concrete.

To assess concrete spalling in load bearing members under fire conditions one may use the mean critical concrete moisture in connection with a reduced factor "n" that depends on relative compressive stresses in the

compression block of the load bearing members. Fire resistant limits of

reinforced concrete structures based on spalling may be also determined

by a conventional method that gives due regard to the rate of decrease

in cross section with a maximum speed of 2.6 mm/min.

77
Application of these design methods to assess the possibility of concrete

spalling due to fire can provide the following solutions: first - to

eliminate spalling completely; second - to decrease the probability of

manifesting spalling; or third, which serve as a preventative indicator

of the spalling.

Among the first measures are the following: use of a water screen,

sprinkler and other devices to protect the concrete and reinforced

concrete structures from fire; by lowering the room humidity to dry the

concrete to a moisture value which will decrease the possibility of

spalling; the use of wire mesh with openings measuring from 3 to 15 mm

and with the wire diameter from 0.7 to 1 mm at a distance of 0.5 cm from

the heated surface to control the spalling; applying insulating plaster

1-2 cm thick on the concrete to retard spalling on the structure surface;

application of refractory concretes made with chamotte aggregate to

minimize the possibility of spalling; spreading reinforcing fibers such

as admixtures (5% of the mass for binding) of materials like asbestos,

glass fibre or metal fibre in refractory concrete layers about 1-2 cm thick on the concrete to prevent the concrete from spalling.

These measures may be used for concretes with F ^ A.

Among the second group of measures are the following: placing wire mesh

made with 3 mm diameter wires having 10 x 10 cm openings around the main

reinforcement at a distance of not more than 0.5-1 cm from the heated

surface in the concrete cover of beams and slabs; application of polymer

admixture in amounts of 5-10% by weight in the heated surface layer of

concrete; application of coarse aggregates that have low coefficients of

linear thermal expansion, for example, limestone, basalt, diabases,

blast furnace slag instead of granite or natural quartz sand; substitution

of a part (not less than 1/3) or the whole natural sand by the sand from

limestone, basalt, diabases, slennit or diorite; use of concrete coroposition

with a limited amount of cement paste (portland cement of not more than

400 kg per 1 m^ of concrete mix) and with high values of W/C ratio (not

less than 0,5); use of concretes with lightweight aggregates; application

of cement concretes with lightweight aggregates; application of cement

with air entraining agent; application of cements with lower specific

surface and slag portland cement; use of fine-ground admixtures from

blast-furnace and cast slags.

All these measures are applied to the case when "F" is less than 6 or more than 4, but their full efficiency still requires spot checks by

fire tests.

Among the measures of the third group are the following: application

(in reinforced concrete structures) of reinforcement of the same area

but of smaller diameter; use of structures with cross-sections without

irregular angles, for instance, round columns without cut-off angles

instead of columns of rectangular or square cross section.

78
Conclusion

This calculation method allows us to evaluate the possibility of concrete

spalling due to fires and to plan measures to prevent structural failures

without conducting fire tests on reinforced concrete structures.

Captions

For Recommendations on the Protection of Concrete and Reinforced Concrete

Structures Against Spalling Due to Fires

Figure 1. Effect of spalling on the fire resistance limit of supporting

reinforced concrete panels exposed to one-sided fire

a - panel scheme with load (P)

;

b - time (t) - temperature (T ) curve under fire conditions;

1 - panel; 2 - spalled piece of concrete;

T- - fire-resistance limit of panel under spalling; - fire-resistance limit of panel without spalling; “t

fire temperature at time x, ; T

time X 2

« 1’ n,2

1 -

- fire temperature at n

Figure 2. Effect of spalling on fire resistance limit of reinforced

concrete beam

a - scheme of beam; b - time-temperature curves for room

atmosphere and reinforcement; 1 - beam; 2 - reinforcement;

3 - spalled piece of concrete; 4 - room time-temperature curve;

5 - time-temperature curve for reinforcement under spalling;

6 - time-temperature curve for reinforcement without spalling

Figure 3. Effect of concrete spalling on fire resistance limit of

reinforced concrete enclosure structure

a - scheme of enclosure structure; b - time-temperature curve,

5 - fire exposure; 1 - ceiling (floor); 2 - floor; 3 - enclosure

structure; 4 - spalled piece of concrete; 5 and 6 - rooms which

are divided by the structure; 7 - time-temperature curve in

room 5; 8 - the same on surface of enclosure structure from

the side of room 6, where concrete spalling does not occur;

9 - the same under concrete spalling.

Figure 4. Trajectory of crack, resulting in concrete splitting under

spalling

1. Cross-section of concrete member;

2. crack;

3. trajectory;

4. temperature diagram.

79


CALCULATING FIRE RESISTANCE FOR STEEL STRUCTURAL MEMBERS

LOCATED ON THE OUTER WALLS OF BUILDINGS

Richard G. Gewaln

American Iron and Steel Institute

Abstract

In 1964, American Iron and Steel Institute (AISI) fire

research programs were initiated to develop data on factors

influencing fire intensity, flame projection and other

aspects of fire behavior relevant to a fire-safe design

method for structural members located outside the exterior

walls of buildings. The methods also were developed in

such a manner so as to be appropriate, with modifications,

for structural members inside a building.

Margaret Law of Ove Arup and Partners, under

joint contract with AISI and the Constructional Steel

Research and Development Organization (Constrado) of

England, prepared a detailed technical report setting

out the most plausible models of flame projection from

openings in building walls. The University of Maryland

Department of Fire Protection Engineering used the

technical report to prepare the AISI Design Guide

arranged for ready reference by fire protection officials

and designers. It presents a step by step analysis and

procedures for designing fire safe exterior members.

This paper deals with the establishment of the

conditions of fire exposure (fire temperature in the

fire compartment, flame geometry and temperature). This

paper is an explanation of Miss Law’s work, for which

she was nominated for Engineering News-Record's Engineer

of the Year Award in 1979. The conditions of heat

transfer from the fire exposure to the structural

members are not covered in this paper but are contained

in the Design Guide available from AISI.

Introduction

"Fire exposure arises from interior and exterior origins. The

evaluation of the exterior exposure can be done only with difficulty in

quantitative terms, and the gradual accumulation of data from actual

fires will probably continue as the main guidance in providing the proper

protection.

82
This quote by Mr. S. H, Ingberg, then Senior Engineer, U.S. Bureau

of Standards, introduced his paper, "The Severity of Fire in Buildings,"

published in the NFPA Quarterly, July 1928 (Vol. 22, No. 1, pp. 43-61).

This indicated that the development of the ASTM E 119 fire test and fire

protection requirements in building codes focused attention on structural

elements located inside a building exposed to a fire inside a building.

This is still true today; however, engineering methods continue to be

developed to place structural fire resistance on the same basis as seismic

design, and the design for structural performance when exposed to external

loads such as wind and snow.

Building codes require fire resistance ratings for specific portions

of a building. These ratings are based on a specific exposure period

based on the Standard Fire Test, ASTM E 119. As a result of studies in

the United States and abroad, it has been shown that the fire exposure

in building fires (fig. 1) may not be the same as the standard fire

exposure; little has been done except to modify the code-required fire

endurance time for structural members.

Building members located dn the interior of a building are surrounded

by flames and by heated surfaces (walls, ceiling, and floor) of the fire

compartment. These heating conditions are similar to those in the Standard

Fire Test where the member is located within the test furnace.

Exterior structural members are exposed to two conditions; radiation

through a window from the fire in the compartment, and radiation and con- vection from flames and hot gases issuing from windows. The value of the

intensity of the radiation received by the exterior member varies with

the relative position of the window and flame (radiator) and the surface

of the structural member (receiver) . Also the structural member loses

heat to the air surrounding it at normal ambient temperature. Depending

on the size and location of the structural member and the behavior of the

fully developed fire, structural members may not need any fire protection.

Early attempts to assess the external fire exposure involved experiments with ASTM E 119 furnace tests which had a window in the furnace

wall [1]1. This effort showed that no matter how long the E 119 fire test is

conducted, the flame exposure is generally less severe than from a more realistic building fire. As a result, there have been full-scale experimental fire tests with exterior structural elements exposed to flames and

radiation from "real" fires. As a result of these tests, approvals have

been obtained for specific buildings [2] (fig. 2).

A design approach has been developed to assess the fire exposure

from a building fire including fires of a size beyond the size limits of

practical fire tests. This approach will permit analysis of the external

heat transfer to structural members and enables engineers to calculate

the amount of fire protection if any is needed.

Figures in brackets refer to references at the end of this paper.

83
For structural steel elements, critical conditions can be defined

in terms of a critical steel temperature and given the heat transfer

conditions, calculation of the steel temperatures is relatively simple.

The main problem is to define the external heat transfer. A large

body of data on building fire and flame behavior exists. It has been

collected and analyzed by Margaret Law at Ove Arup and Partners, London,

England, to estimate external heat transfer for practical designs in

buildings [3]. Correlations have been derived, not only from model-scale

experiments but also from measurements for a wide range of experimental

fires in large-scale building compartments.

The Designers Manual prepared by Ove Arup and Partners was used by

the University of Maryland, Fire Protection Engineering Department to

develop a Design Guide [4] for Exposed Exterior Steel,

The principal elements needed to make this design approach useful

are:

(1) Flame shapes must be based on parameters which can be

readily identified by the designer,

(2) Temperatures of the room fire and the external flame

plume must be determined by relatively simple equations,

(3) Heat transfer must be simplified to facilitate

calculation.

Research

Most flame and structural damage occurs in the post flashcver period.

This phase of the fire has been studied by many researchers and Dr. Philip

Thomas [5] has summarized this work. Analysis of a fully developed fire is

complex and the model developed used a number of simplifying conclusions

principally, a uniform temperature throughout the compartment and that

fuel burns in a uniform way. Important parameters involving the way a fully developed fire burns have been defined by Margaret Law and the

large amount of test data has been used to show how these parameters

interact

.

Ingberg was the first to study the relationships between fire load

density and fire severity and these studies have formed the basis for

building code requirements for the fire resistance of structures. Later,

Fujita [6] quantified the effect of ventilation in terms of the area and

height of ventilation openings, usually the window. Later work on models

was carried out cooperatively by the Counseil International de Batiment

(GIB) [7] which modified the Fujita relationship on ventilation by introducing the size and shape of the fire compartment (fig. 3). In addition

to using scale models in the GIB program, the various factors were studied

in full scale studies at Borehamwood [8], Maisieres-les-Metz [9] and Carteret,

New Jersey [10].

84
INTERIOR FIRE BEHAVIOR

A number of full scale experiments have been carried out using wood

cribs as fuel; in still or lightly moving air, although some tests were conducted with additional air supply.

Two of the most important factors affecting exterior fire exposure

from a fully developed interior fire are rate of burning , which affects

flame size and fire duration; and fire temperatures , which affects the

radiation from the window.

Rate of Burning

In the case of natural draft conditions where the fire behavior and

compartment dimensions control the air flow , continuous weighing of the

fire load during tests have shown that the rate of weight loss is fairly

steady over the fully developed fire period when the weight of the fire

load falls from 80 percent to 30 percent of its initial value. This rate

of burning R is defined as;

R = I

(1)

Where T is the effective fire duration and L is the total fire load in

the fire compartment.

For a free burning condition the value of T' is and is determined

by the characteristics of the fire load—thin fuels with large surface

areas give a smaller value for T . r Where ventilation is restricted, as is generally the case, there is an upper limit for R regardless of how large the fire load. It has been

shown that the important parameters are the window area A , and its

height h, and the area of the heat absorbing surfaces of ?he fire com- partment A , and the ratio of the depth D to width W of the fire com- partment (fig. 4).

Thomas has drawn a line through the points in figure 3 which include

measurements for R which have been reported for large-scale, ventilation

restricted fires. The values for R are in reasonable agreement with the

CIB data.

The following equation for R results from the Thomas Line:

= 1»22

E

, -065n 1-e ( 2 )

where the ventilation factor is:

n att: (3)

85
EXTERIOR FLAME BEHAVIOR

Yokol [11] made the first comprehensive study of flame projection from

windows to evaluate the risk of vertical fire spread. He demonstrated

several influences on the trajectory of the flame plume and temperature

of the flame plume. The wall above the window absorbs heat but restricts

air from entering at the flame edges which will force the flame plume

away from the wall; the wider the window, the closer the flame plume

remains to the wall. Yokoi defined the shape of the window as the ratio

of the width to the height of the upper half of the window and derived

a series of plume shapes for different window shapes. In checking the

results of his experiments against large scale tests using wood cribs

he obtained good agreement. He also observed that where ventilation is

restricted the emerging gases will continue to burn after leaving the

window and will affect correlation.

Thomas [12] correlated results of model tests by Webster [13-15]

using dimensional analysis similar to that used by Yokoi, By assigning

a flame tip temperature of 1000°F or 1460°R reasonable agreement is

obtained between these data and those of Yokoi.

Additional work by Seigel [16] on tests conducted at Underwriters

Laboratories in which he treated the flames as forced horizontal jets,

defined their projection by a temperature of 1000°F at the flame tip

and recorded flame temperature in the exterior flame plume.

Other work was conducted at Borehamwood and full scale tests in

the United States, All of this work was used to establish the relationships defining the flame plume dimensions outside of windows.

Flame Plume Dimensions

No Draft Condition—The flame dimensions are an important factor in

calculating radiation from the flame to the receiver or exposed structural

member. Thomas and Law [17] analyzing the data of Yokoi, Webster et al.

showed a correlation that takes into account the dominant role of buoyancy

and the turbulent mixing of hot gases in a flame emerging from a window.

The recommended correlation shown in figure 8 is expressed as follows:

Z+h

h

16

R

A/z(gh)

2/3

1/2 ( 11 )

This may be written for the flame height (Z) as follows;

2/3

Z = 3.55 -h [i] ( 12 )

This distance of the center of the flame tip away from the exterior face

of the building (x) depends on the shape of the window and if there is a wall above the window. A wall is defined as a vertical surface which

86
the fire compartment geometry is:

E = (A )"^

w ^ hW (4)

For a particular fire load and compartment, R should

for both equations (1) and (2) and the lower value used.

be calculated

Fire Temperature

There is a maximum temperature developed within the fire compartment

depending on the fire load and compartment dimensions. Thomas gives the

correlation of the GIB measurements of average fire temperature rise 0^

over the fully developed fire period as a function of n as shown in

figure 5. The significance of this data is that the fire temperature

rises to a maximum for p = 5 to 10 then declines. The average fire

temperature also depends on the fire load as shown in figure 6 for plots

of large scale tests with low fire load densities which fall well below

the Thomas curve. There is justification to assume that the following

equation for a no draft or natural draft condition gives a maximum or upper limit fire temperature (6^ or T^) for a given value of pt

T^ = 8025 (Function of (Function of p) + 520 (5)

where the fire load and fire compartment is

^ w t

(6)

the function of \p and p:

_ , . -0.25ii; , f \b = 1 - e and

X

(7)

- -0.18p

r l“e

(8)

There is limited information in the rate of burning under forced

draft conditions. The results of fire tests using excess air at

Underwriters Laboratories show no significant variation of temperature

with n or air supply but fire temperature (6 or T^) can be related to

^ as shown in figure 7 with the curve shown having the following equation

T^ = 2160 (Function of i|^) + 520 (9)

where the function of is:

f ip = 1-e

X

-O.ZOip

( 10 )

87
retains its integrity exceeding two-thirds the flame height. A flame

will be projected away from the wall surface if air can get behind it.

In situations where there is either a narrow window or a wide window

with no wall above, air can move more easily between the flame and the

wall thereby deflecting the flame or flames outward. The work of Yokoi was studied and a regression analysis for flame tip projection gives:

X ^ 0.454

h 0.53

n

(13)

where n is the ratio of width to height of the upper half of the windows.

Data plotted in figure 9 indicates the flame tip projection decreases

with n and is less than half the window height for values of n exceeding

unity, which includes most situations. Therefore the value of x is

independent of n and may be represented by the following equation if

there is no wall above:

fz “1 X = 0. 6h I — J (14)

If there is a wall above the window and where h (window height) is

less than 1.25 times the window width

h

X

2

(15)

Where there is a wall above the window and where h is greater than

1.25 times the window width and the distance to any other window on the same floor exceeds four widths of an individual window, use the following

equation:

0.54

X = 0.3hf — ^ (16) (k)

Forced Draft

The effect of a forced draft on a fire is to increase the rate of

burning of a ventilation controlled fire. Also for a given rate of burning, wind or a draft blowing flames out a window may also affect the

flame size and direction. Experimental data from tests at Underwriters

Laboratories is plotted in figure 10 and indicates that the correlation

proposed by Seigel with R/A 1/2 raised to the power of unity, is reasonable provided the wind effect (u) is included. The following solves

for the height of the flame above the bottom of the window:

0.43

Z-f-h = 17.7 I - I Q (17)

where

0 = R

(18) w

88
Analysis of experimental data shown in figure 11 shows a correlation

between horizontal projection, flame height, window height and wind

speed. Comparison with this correlation shows that as wind speed

increases, flame height decreases, but that the horizontal projection

increases. The horizontal projection (x) of the flame tip away from the

building is given thus:

X = 0.077 J®*^^(Z+h) (19)

where

2

J = -^ (20)

The maximum width w of a flame emerging from a window usually exceeds

the window width. uTie angle made by the emerging flame as shown in

figure 12 averages 11 degrees giving:

W —

w

-|^ = 0.194 (21)

therefore

w — w+0.4x (22)

EFFECTIVE FLAME BOUNDARY

In order to estimate heat transfer from a flame plume, boundaries

must be defined. The temperature distribution across a flame section

as shown in figure 13 shows one approach is to define the flame boundary

by the 1000°F contour. Margaret Law proposed that since radiative transfer will be an integrated effect, which can be made equivalent to a. uniform effective temperature, an equivalent step function distribution

be used. Since radiant transfer is so sensitive to the value of temperature, she suggested adoption of the axial temperature (maximum) for the

step function with a defined effective thickness. The problem is to

define this effective thickness bearing in mind an assumption of maximum

temperature across the flame thickness.

Natural draft : In figure 14 the flame emerges above the neutral

plane from the upper two-thirds of the window. The flame width varies

little with the distance from the window plane. Therefore it is reason- able to assume the step function remains the same size throughout the

trajectory (wx2h/^) . This assumption is consistent with estimated values

of emissivity. Wxnd may deflect the flame sideways and it is assumed

that the deflection will not exceed 45 degrees.

Forced draft : The flame shown in figure 14 can emerge from the

entire window opening. The width of the flame does increase with distance out from the window. It is also reasonable to assume the vertical

dimension increases, however, the upper vertical increase is already

contained in the value for Z, it is proposed that the size of the flame

89
at the window opening be h x w, increasing to h x (w + 0.4x) at the flame

tip as shown in figure 14.

TEMPERATURE AT FLAME AXIS

Correlation of test data to analyze the temperature distribution

in flames for natural draft or no draft condition is illustrated in

figure 15. The line in figure 15 has the equation:

e o 1-0. 33

£w

R

(23)

A similar correlation for the forced draft data is shown in figure 16

and the line has the following equation: ~ = 1-0.33

O

JIA 1/2

w

R

(24)

Note in equations (23) and (24) that the decrease in flame temperature is directly proportional to the distance along the center line of

the flame. By substituting 0^ or T = 940°F (1000°F-60°F) and 5, = X in

equations (23) and (24) the value or 0 or T may be derived. For fires

with natural draft , this may give values of 0 or T (at the window)

greater than the fire temperature 9^ or T^ in the fire compartment.

This results from ignition of unbumed gases outside the fire compartment.

Where forced draft exists the opposite can be expected.

Conclusion

In order to be able to calculate heat transfer to a structural

member from flames, it is necessary to establish the shape and size of

flame emerging from the window, and the temperature distribution within

the flame. The flame height and its horizontal projection from the

building can be calculated, as can the temperature at any point on the

flame axis. From this information an "idealized" flame is deterained

so that heat transfer calculations can be made.

The temperature of the fire within the building is determined.

Although the fire temperature in a building fire will vary with time,

the maximum temperature reached by the fire is used since this will

result in the highest temperature in the material exposed.

The assessment of fire exposure using the procedures given in this

paper will provide a conservative basis for calculating temperatures of

exterior structural members. In some building fires the duration of

the fire may be sufficiently long that bare or unprotected structural

members will be heated to "steady-state-conditions," In other words

the material may reach an equilibrium temperature such that the heat

falling on the material will be balanced by the heat losses to the sur- roundings. The steady state condition assumes the most conservative

approach to heat transfer.

90
REFERENCES

[1] Pryor, A.J. Fire Exposure of Exterior Structural Members

Final Report, July 1965, Southwest Research Institute, San

Antonio, Texas, 1965.

[2] Seigel, L.G. Fire Test of an Exposed Steel Spandrel Girder

Materials Research and Standards, MTRSA 10(2), 1970, pp. 10-13.

[3] Ove Arup and Partners. Design Guide for Fire Safety of

Bare Exterior. Structural Steel --Technical Reports and Designers

Manual^ Report for American Iron and Steel Institute, Ove Arup

and Partners, London, England. January 1977.

ij] American Iron and Steel Institute. Design Guide for Fire

are Structural Steel. March 1979

[5] Thomas, P.H. Fires in Model Rooms: CIB Research Programmes

Building Research Establishment Current Paper CP 32/74

BREy Borehamwood, 1974.

[6] Fujita, K. Characteristics of Fire Inside a Non-combustible

Room and Prevention of Fire Damage Report2 (2 ) . Japanese Ministry

of Construction, Building Research Institute, Tokyo.

[7] Heselden, A.J.M. and P.J. Thomas. Fully Developed Fires in

Single Compartments, CIB Report No. 20 Fire Research Note 923/1972,

Joinst Fire Research Organization, Borehamwood, 1972.

[8] Heselden, A. J.M. , P.G. Smith, and C.R. Theobald. Fires in a Large Compartment Containing Structural Steelv7ork -- Detailed

Measurements of Fire Behavior. Fire Research Note 646/1966, Joint

Fire Research Organization, Borehamv/ood , 1966.

[9] Arnault, P. , H. Ehm and J. Kruppa. Rapport Experimental sur les Essais avec de Feux Naturels Executes dans la Petite Installation,

CECH 3/73-11-FCTICM , Puteaux, France, June 1973 ' ' *

[10] Gross, D. Field Burnout Tests of Apartment Dwelling Units

Building Science Series 10, National Bureau of Standards,

September 29, 1967.

[11] Yqkoi, S. Study on the Prevention of Fire-spread Caused by Hot

Upward Current. Report No. 34, Japanese Building Research Institute

Tokyo, 1960.

[12] Thomas, P.H. On the Heights of Buoyant Flames Fire Research

Note 489/1961, Joint Fire Research Orangization, Borehamwood, 1961.

91
[13] Webster, C.T. , Monica M. Raftery, and P.G. Smith

The Burning of Fires in Rooms Part II . Fire Research Note 401/1959

Joint Fire Research Organization. Borehamwood, 1959.

[14] Webster, C.T. , Monica M. Raftery, and P.G. Smith

The Burning of Fires in Rooms, Part III . Fire Research Note

474/1961, Joint Fire Research Organization, Borehamwood, 1961.

[15] VJebster, C.T.'. and P.G. Smith. The Burning of Fires in Rooms , Part IV. Fire Research Note 574/1964, Joint Fire Research Organization,

Borehamwood, 1964.

[X 5 ] Seigel, L.G. The Projection of Flames from Burning Buildings

Fire Technology, 5(1), pp. 43-51, 1969.

[17] Thomas, P.H. and Margaret Law. The Projection of Flames

from Biiildings on Fire Fire Prevention Science and Technology,

No. 10, pp. 19-26, December 1974.

92
Explanation of Symbols

SYMBOL MEANING

UNITS

AMERICAN METRIC

Total room surface area

= 2Aj. + 2H(DxW)

ft2 m^

.Aw Window area m^

D Depth of compartment ft m

E 6«) Ib/min ft-^^^ kg/sec m'

• e Base of Natural logs

H Height of compartment ft m

h Height of winddw ft m

L Fire load lb kg

i Distance along flame centerline

from window ft m

Q

R

Ib/min/ft kg/sec/m

R Rate of burning Ib/min kg/sec

T Absolute temperature R K

u Velocity f t/min m/sec.

W VJidth of compartment ft m

w Width of window ft m

X Centerline distance of flame tip

from window ft m

X Horizontal distance of center of

flame tip from window ft m

z Vertical distance of flame above

top of window ft m

p Density Ib/ft^ kg/m^

T Fire duration min sec Free burning fire duration min sec f,0,2 These subscripts denote fire, window plane, and flame respectively.

-

93
Summarized

Relationships

94

0.077
Flame

w_

w

+

0.4x

X

• o

+

CN

rHs<

OJo• o

I rH

Pi

•&- O

St

^1^

0

1 o

o

CM

CMrH4

Pi

COCO

•9- o

CMo

I (U

o

I

I—

I

o

CD

CM

N I O

o I a> Pi 4h o

0)P

•H

fu

95
2400

2000

10

Test Number

Fire Load

(PSFt

20

. Time,

Forced Air (CFM)

30

dinutes Windows

40

. Number Size. Ft X Ft Areo.Sg Ft

1 10 0 1 2X6 12 2 10 O 1 2X6 16 3 10 0 2 2X6 24

4 lO 0 2 2X8 32

£0

Figure 1 -- Effect of window area on fire

temperatures during burnout tests (natural air flow).

96
Figure 2 — One Liberty Plaza, N.Y.C.,

exterior surfaces of steel spandrel beams

have the web bare and the flanges flameshielded .

97

i
Large-scale data

Borehamwood

Meti

G Carteret

Fig. 3 . Variation of R/A^h with compartment size and

ventilation, as given by Thomasfor CIB data

i

i

98
“ w.h At « 2At+2H(D + W )-Aw

Fig. 4 . Simple fire compartment

99
2000 -

1500

e,“F 1000 -

500

T,- ’/i

Fig. 5 . Variation of average fire temperature rise with

compartment size and window area, natural draft

100
9i Max.ei

10

0-8

0-6

0-4

0-2

8 10 12 14

-iy(AwAT)V».|b/lt*

Fig. 6 . Variation ofaverage fire temperature rise with fire

load, compartment size, and window area, natural draft

101
Fig. 1 . Variation of average fire temperature rise with fire

load, compartment size, and window area, forced draft.

Underwriters* Laboratories data
10

(2 h)/h

5

V VotUM A TnHtton OOisnay \Mixtd

BomKamxivood

O Underwriter*

X Trartas C Carteret K Kordina

Fig. 8 . Flame heights for large-scale tests with natural draft

(Note: Range of n: 0.5-18.7)

Eq.(8) Thomas and Law U«W ~~\

Eq.(n) y >

001 002 005

lVAwp,(gh)’^

9

103
X

Yokoi

Tr«nton OOisfwy VYorid O Bofhtwwood

OLktd*rwnt*rs X Tnnas

C Carteret K Kordina

n Fig. 9 . Horizontal projection offlame tip for large-scale tests

with natural draft, wall above
R/A^’* Ib/ftmin

Fig.io. Flame heights for large-scale tests with forced drafts

Underwriters* Laboratories data

(Note: Range ofu: 700-367ft/min)

105
(u»/h)»“ ii*h) fl’»Vmin®”

Fig. 11 . Horizontal projection offlame tip for large-scale tests

with forced draft. Underwriters* Laboratories data

(Note: Range of vP-/h: 1000-22,500ft/min^)

\44

106
-«I

« j

Fig. 12 . Plan view ofemergingflames with forced draft

. lkmp*r«tur«

Fig. 13 . Temperature distribution across flame section

107
X - * + I,

Natural draught w > 0-8h wall abowa

X - {i>-^ I,

Natural draught w < O Bh wall abova

or no wall abova

X- «*

Forcad draught

wall or no wall aboua

I

Natural draugnt

Plan

Natural draught

Plan

w, - w + 0‘4*

Forcad draught

Ptan Fig. 14 Assumed trajectories of emergingflames

»

V,

108
Fig. 15. Flame temperature distribution for large-scale tests

with natural draft

Fig. 16 . Flame temperature distribution for large-scale tests

with forced draft, Underwriters' Laboratories data

109
FIRE RESISTANCE CALCULATION METHODS

FOR STEEL STRUCTURES

Dr. A. I. Yakovlev

(VNIIPO, Ministry of Interior of the USSR)

The fire resistance limit of steel structures is defined as the time for the

bearing member or its separate parts to be heated to a critical temperature.

I. Critical Temperatures* Determination

The critical temperature for structure members subjected to a standard

load is:

500° C - for steel members,

250° C - for aluminum alloy members.

In case the structure is subjected to considerably less load the

critical temperature calculations take into consideration the type of structure, the support conditions, load level, and material characteristics in

order to more precisely estimate the fire resistance limit.

The critical temperature of the bearing member is determined according

to the table (for steel structures) with regard to coefficients Ym and y„.

1 £i The coefficients characterize the reduction in bearing capacity for the

heated element: y„ - the stability failure of the compression member under

ill elastic deformation; Yrp ~ the durability failure under plastic deformation.

The critical temperature of the compression element is defined as the

least of the two values taken from the table with regard to Yt7 and Ym

£j 1 coefficients.

Y^ and Ym coefficients are derived from the formulas (l)-(4).

ht i

For compressed elements:

2

^E

H o Si -E^-T . H min

( 1 )

110
where N - is the working load, kg; " -2

E - is the metal elasticity initial modulus, kg •cm ; “ 4

T , - is the least moment of inertia, cm ; min ’ - is the design bar length, cm.

Table

Yt, and Y_ Coefficients Relative to Nominal

i H^ Strength (R ) and Elastic Modulus (E ) for Hot-Rolled

n Steel Grades C^.3 and C^.5 Depending on Temperature

Temperature - °C Yt ^E

20 1.0 1.0

100 0.99 0.96

150 0.93 0.95

200 0.85 0.94

250 0.81 0.92

300 0.77 0.90

350 0.74 0.88

400 0.70 0.86

450 0.65 0.84

500 0.58 0.80

550 1.45 0,77

600 0.34 0.72

650 0.22 0.68

700 0,11 0.59

value is equal to:

- Z (£ - is the bar's length) for hinged end conditions;

- 0.5)1 for fixed end conditions;

- 2Z for one end fixed and another end loose (cantilevered);

- 0.7)1 for one end fixed and the other end hinged.

Ill
For members with axial compression and tensile loads

where F -

Yt

d) • F • R

^t

H

( 2 )

is the cross-section, cm ; -2

is the standard initial metal strain, kg* cm ; is the longitudinal bending ratio in heating conditions;

for compressed members with slenderness ratio A < 40 and for

tensile bars, c))^ = 1.0;

for compressed members with slenderness ratio A ^ 40,

= 0,95.

For members with eccentric compression or tensile loads

YT V T ^ F/ (3)

where e - is the eccentricity of the applied load, cm;

4

T - is the moment of inertia for bending in one plane, cm ; y

- is the distance between the central axis and the section

extreme fiber, cm.

For bending elements

Yt

W -R

ttJI H

(4)

where

w

ttZ

is the maximum bending moment under working load, kg* cm;

3

is the section modulus, cm .

The given static calculation formulas are simple enough to use.

Having determined the critical temperature the thermotechnical

calculation is carried out aimed at estimating the heating time until

critical temperatures are reached, e.g. structure's fire resistance limit.

112
2. Thermotechnical Calculation

The boundary conditions for the thermotechnical calculation are given

in the report "The General Fire Resistance Calculation Principles of Building

Structures" and are not referred to in this paper.

The thermotechnical calculation for steel structures is carried out

by the elementary heat balance method using computers.

a) Exposed Steel Structures

Steel is known to have great thermal conductivity resulting in a rather

quick temperature distribution over the structural section. That makes it

possible to assume a uniform temperature distribution over the sections of

steel structures exposed to heating.

Thus the heat balance equation for the unit length of the infinitely

long member exposed to heating over the whole of its surface is as follows;

a(t„ - t-) .tt.1.At=y~*F* 1(C - + D - . t -) . B cm cm cm cm cm

(t - . - t -)

cm,

A

t cm

(5)

Making the necessary transformations we obtain a computer formula for

the exposed member temperature determination:

At . a . (t - t -) _

B cm

^cm,AT ^'cm 60 . ^ cm

. (C - + D - . t -)

TTp cm cm cm

( 6 )

where t - - is the temperature of the member at a given moment, °C; cm

^cm,AT - the temperature of member in a definite interval - At, °C;

t„ - the "standard fire" temperature at a given moment, °C;

D

a - thermal conductivity ratio from the "standard fire" region

-2 -1 -1

to the surface of the member, kcal*m *hr • °C ; -3

Y

- - the specific weight of metal, kg*m ; -1 o “1

C - - initial heat capacity of the metal, kcal*kg • °C ; cm

113
D -cm

Trp 100* TT

change in heat capacity of steel with regard to temperature

“1 o “2

increase, kcal'kg *°C ; the given steel thickness, m;

2

is the member cross-sectional area, cm ; is the heated perimeter of the section, cm.

It follows from (6) that in order to determine the temperature of

the unprotected member its geometry is characterized by the parameter

6 - the equivalent metal thickness equal to the ratio of the cross- sectional area to its perimeter.

The stability of the algorithm for a calculation according to (6) is

dependent on the choice of a At value that must not exceed At derived ^ max

from the following formula:

60. Y

- .(C-+D-.t- ) At cm TTp cm max

cm cm, max * minutes (7)

max

where a and t - - are the maximum allowable values in the calculation

max cm, max

process.

The heating curves for steel members with various 6 values are given

in Fig. 1. The curves are drawn with regard to temperatures calculated

according to (6) . Computer program "Nairy-2" is used for the calculation

process making use of the following data:

At = 0.1 min; Y “ = 7800 kg*m C.- = 0,105 kcal*kg ^*°C ^ cm & » tm * ^

D - = 0.000114 kcal*kg"^.°C"^; cm ’

= 0.85 - is the emissivity of the surrounding

fire box;

S = 0.69 - is the emissivity of the surface of

o the steel member

114
b) Structures with a Fire Protection Coating

Coated structures exposed to fire are subjected to considerable

temperature variation over the thickness of the protection. Besides this, some fire protected elements (e.g. of I-shaped section) also suffer temperature variation over the depth of the web and width of the flanges. In

this case the thermotechnical calculation is carried out with regard to

two-dimensional heat flow by the elementary heat balance method.

For this purpose a net (mesh) is attached to the section. The

temperatures in the net (nodes) are determined in successive time

intervals - At with the help of a computer.

An example of the computer mesh for a fire protected concrete column

or a wide-flanged double-T beam is given in FLg. 2. Due to symmetry of the

section about the "y" axis the temperatures are only estimated in one half

of the section.

The temperatures at the nodes are estimated according to formulas

derived from the heat balance equations of finite element analysis. The

latter are formed by dotted lines drawn through the centers of the net cells.

To compose the heat balance equations it is necessary to consider temperature variation over the flange width and the web depth only; temperature

variation depending on the flange and web thicknesses is assumed to be zero. In addition, the heat balance equations of elements include the heat of

free water evaporation in pores of the coating.

Considering these peculiarities, the design formulas for temperature

estimation in the most typical net nodes take the form (see Fig. 2)

:

Node 9, Fig. 2

^9, At ^9 30(AX^ + AX2)*AY^.y(C + Dtg)

+

ay

AX,

1

2

1

115

+
+

00 8 p

X

<3X

<1

CM as uI

CM

PQ

LT| o+

Os a o 8 o^ p

<uM0)

CM X< + Xo

LO

CM iH P M

•Hm

tn

CM

CM

tH

<3

CM +m

I—( p

II

H

<1m

iH P +

CM X

<3 CM X

<3

CM m

CM

pI

CM CM P

PQLOO+m

CM p

cnp

CO p

CO (P

?*-CM T3I

CM

<3

X +<o pI so

CM iH P

pqlOO+

inI—! P

vO(—I Pc

CM

>-< <

iH

><

<\

LO

CM iH P

CM tH p

PQLOO+

lO I

-CP rH P

.<! .

Os

SO

CM iH P

lO

rH PI vOP

CO

116 +

0.5B
o

(U

(U

ol X

<1

QOO CM + <

CO

u

ISo

CM X< <

CM

O.

X<+X<

LOoQoo+uoo

t—I m 8 iH u

<N X

<3

00

CM

H4J

CM

CM

CM

tJO •HPH

00

00

pH

4J O+u

J>- 1“^o

T3

X)I CM X<

t= t3CM X<I

CM X<+<

CM X<

00H<

001—I +

CM X<X<m

CM

00

CM pH u

CM pH

+J

PPmo+

00

pH

•M VwP

<ti

00

pH

4-1

4J

CO P+

4-> CO XI CM X

<]I—I

t=

CM X

<3

O

'O T3m

CM X

<3LO CM X<

00

CM pH 4J

<r

CM CM

4J«

LO0+

00

pH

4J1

'd' CM 4J

CM X

<3

00

CM pH

CM pH M

UO• O+^—s

00

pH 4JI

r>- pH U

4-1 CO

00

CM CM

4-1

4-1 CD

mm

CD+

00

pH

4-1I MT

CM

4-1

4J

CO

(U

$-1 (U

poo XoT3

•n

u

4J

CO

Xo

’TO

'OI

CM X

<3

CM X

<1

X<

t= TJ

CM X

<3

X<3X<m+Poo+uoo

00

117
where: t - is the latent heat of water, kcal*kg ;

A, B, Y» C, D - are the thermophysical characteristics of the

coating;

A , B , Y , C , D - are the thermophysical characteristics of the

St st* 'st’ St’ St

steel;

d - is the flange thickness of profile, m;

IT

2dc - is the web thickness of profile, m;

p

- is the moisture content of coating, wt. %.

The maximum design° time interval - Ax - is calculated according to max

the following formula:

At

max

30 . Y .. AX . C ^ St st

St

min utes ( 11 )

where AX - is the net cells’ minimum dimension within the steel profile

limits, m.

It is possible to simplify the thermotechnical calculation of some coated

steel elements with rectangular or round sections and also double-T sections

with contour protection by reducing these sections to a non-dimensional plate

coated on one side and having perfect heat isolation on the other side

(Fig. 3).

The temperature nomogram for the steel plate with various thickness -

6^- and coated with cement-sand plaster 30 mm in thickness is given in Fig. 4.

The nomogram was prepared based on computations using the following baseline

data:

118
At

A

Z

0.1 min,

0,72

539,

Y = 7800 kg*m

-3

St

AX

B

P

=st

0.005 m,

-0.00038,

2 %,

0.105 D

Y =

C =

S =

o

>st =

1930 kg’m

0.184,

0.625,

0.000114

-3

D = 0.00015

Sg = 0.85

The computation was carried out with an algorithm developed according

to Fig. 3 and including the following formulas:

- the temperature - t^ of the exposed coating surface:

t A = t +

o,At o

I

Ax a( {(tg “ + — [A(tj^ - t^) + 0.5B (t

30 . Y • AX(C + Dt^)

; - ') t

( 12 )

- the temperatures - t^ of "n"-coating middle layer:

= t +

At

[A(t , + t - 2 ) +

n,Ax n Av-2/r. L ^ tn * 60 . Y • (C + D . t )

2 2 2 00 0.5B (t^ . + t^.T - 2tn - t n-1 n+i n n

(13)

where

^o(n) - P-T100(C + lOOD)

- the temperature t^^ of steel plate;

^st,Ax ^st 60 . AX[0.5 . y . AX(C + D . t ^) + 6 ^

. Y . (C , + Dt ^) ] * L , X lgj_ g^. g^. - t St

(14)

where

i.T St

[

2 . 6 100 (C + lOOD) +

St Y _(C ^ + lOOD ) St St s*- AX . Y

119
The data formulas were derived from the heat balance equations for the

elementary layers shown in Fig. 3 by dotted lines.

The reduction of the actual cross-section of the member to a non-dimensional plate is carried out according to formulas based on the

following conditions.

The box-shaped section of a coated member is presented in Fig. 5.

Let us compose the heat balance equation for mnm^n^-element restricted

by the axes of x and y and mm^-line, crossing the middle of coating thickness,

11 11

Heat transfer is not observed through the lines mn , nn and m n as: mn lies on the axis of section's symmetry, nn^ adjoins the closed air

space with the symmetry axis - "y" and m^n^ - is the symmetry axis for

isotherms in the section corners of a structure.

Consequently, only heat passing through the mm -line to the

restricted element results in the element's heating.

Therefore, the heat balance equation for mnm^n^-element takes the form

A + B

t + t^\ (t - t ^)(0.5B + 0.56 )At . 1 o St \ o St o

(C + D . t . Y . ^ (0.5B + 0.256 ) +

St z o (15)

(C + D

St St St 6^ (0.5B - 0.56y) (t , - t ) St,

A

t St

Having derived the left and the right parts of the equation by 0,5

(b + 6^) and setting the derived expression in square brackets equal to

the heat capacity value of an element of non-dimensional coated plane with

the given thickness of 6 we may derive the following design formula:

120
6np,x

b - 6 b + 6^ - 0.25

(C + D .

(C ^ + D

St St ^st> St

b + 6 (16)

In this expression the heat capacity of the coating and of the steel at

high temperature may be substituted for the initial heat capacities of these

materials

.

Therefore:

b - 6 = 6 y - 0.25

npx X b + 6 C ^ Y ^ b + 6 St St o

(17)

Analogously for 6 we have:

np,y

a - 6 = 6 . np,y y a + 6 0.25

c y ^

' a + 6 St St o (18)

In the given formulas the second member of the right part having the

coating thickness of 6^ _< 20 mm has insignificant value and may be neglected

in the process of practical calculations. Thus, the design formulas for

the given thickness will take the forms:

6 np,x

6

X

b - 6

b + 6 (19)

a - 6

6 = 6 . -

np,y y ^ + 6

The general plate’s thickness -

6 . b + 6 .a

5 =

npj.x gp.ii

np a + b

Of concern is that the value of

equal to:

( 20 )

6 is

:

np

( 21 )

6 (at 6 =0 and lack of coating) np o is

121
1

b - 5 a - 6 6 . . b + « . —-Ji

anp

=

a + b ^ s—. a

2b6 + 2a . 6 - 46 . 6 „ X X y _ £

2(a + b) n

i.e. equal to the given thickness value for the exposed bars.

According to the curves given in Fig. 4 the heating time of the plate

to the critical temperature is determined for 6 = 6 . The time

np St

determined will be the fire resistance limit of the member with a boxshaped section.

For a circular section;

- 6 np St d - 6 H

gt

- 0.25 „ ° • X. St St Si

( 22 )

where 6 ^ - is the thickness of section wall;

St

- is the external diameter of the section,

In case of a solid circular section:

0.25

np + 6^ 4 C

c -.X

St St

. 6

‘ T, ^ o (23)

122
123

Figure
Ox^ 2~5 ^

I 2lXf

^ AXz

^ A A Yt

^

4- -i- — —|-

8 I g I no. _4 1-

,/^-i- I --I—I- J3_

-H 1-

zs--f - -f

-+ — +-

is

Mil

JL

Z6

_j— -I ,2?

y Y

'

//

+

£_

i-H

I tz

+ 4-§

4-

—}- 1 1- — 1— •—

S0\ \H\ \5Z,

4

ZZ> Zd

Tf

^^

1 „ ^| M\ ^.. f

Z,

J6, .Jn Ji

h

'A.

a

*4

N

3>

<\ Ln^Z~\5\

it

. \

Figure 2

124

1
125

\

f

127
RELIABILITY ANALYSIS OF FIRE-EXPOSED LIGHT-FRAME

WOOD FLOOR ASSEMBLIES

by

E. L. Schaffer

Forest Products Laboratory

presented by

F. E. Woeste

Virginia Institute of Technology

Abstract

A reliability analysis using second-moment approximations

is conducted on two types of fire-exposed, unprotected wood

floors—conventional wood joist and floor truss assemblies.

A methodology is illustrated by whi( h the probability of

structural failure of a wooden floor assembly can be evaluated. This probability, along with the probabilities of

failure of other system components, can be used in a systematic analysis which appears to be a viable approach to

realistic analysis of building fire safety.

The use of reliability analysis to compare the relative

fire safety of different floor components is demonstrated.

A procedure for introducing new components into the market,

based on the concept of an equal safety index as calculated

for a component with a proven inservice record, is discussed.

128
Reliability Analysis of Fire-Exposed

Light-Frame Wood Floor Assemblies

By

F. E. Woeste—

and

E. L. Schaffer, Engineer

2 / Forest Products Laboratory,— Forest Service

U.S. Department of Agriculture

Introduct ion

The effectiveness of structural assembJ ies--walls , floors, ceiling roofs--

to act as barriers to fire growth is currently measured by using the American

3/ Society for Testing and Materials (ASTM) E-119 (3)— fire endurance test. This

method requires that a t3rpical loaded assembly be exposed to fire and the time

to failure recorded. This time is then employed to "rate” the assembly. An

assembly that maintains its integrity for 30 minutes (or more) is said to have

a 30-minute fire endurance rating. The rating procedure is then used by codes

to regulate which assembly designs are satisfactory for buildings of various

occupancies. The Department of Housing and Urban Development (HUD), for

example, specifies that exterior walls shall have a 20-minute and floors a

10-minute endurance rating in one- and two-family residences (^) . This is

Agricultural Engineering Department, Virginia Polytechnic Institute

and State University, Blacksburg, Va. 24061.

2/ Maintained at Madison, Wis., in cooperation with the University of

Wisconsin.

3/ Underlined numbers in parentheses refer to literature cited at end of

report.

129
believed to protect occupants cind enable firefighters to safely combat dwelling

fires

One of several shortcomings of the present fire endurance evaluation

and rating system is that it does not realistically measure the fire safety

afforded by assemblies. It simply compares the fire performance of one assembly with another through the use of a single performance measure--that of a

single test of each to measure an assembly's fire endurance under a standard

fire condition. When one discusses safety, one is soon led to the need for a

measure of risk associated with an event occurring. Hence, the present fire

endurance rating system provides little information on the safety provided by

structural assemblies. This is somewhat unfortunate becauLe of the potential

benefits accruing to resident owners, developers of new assemblies, and the

firefighting and code enforcement community.

Approach

This paper illustrates a risk-based methodology and its application to

assessing the fire endurance safety of two unprotected light-frame assemblies--

the conventional joist assembly and a floor-truss assembly.

In the case of a floor, there are three primary modes of failure: structural collapse, flame penetration, and excessive temperature rise on the unexposed surface (3). In a system analysis the probability of occurrence of each

of these events would be required. In one- and two-family dwellings, only

structural assembly failure criteria are evaluated (42)

.

One objective of this research was to propose and illustrate a methodology by which the probability of structural failure of wooden floor assemblies can be evaluated.

130
A second objective, perhaps more important for the immediate future, is a

method by which new components can be introduced into the market. The underlying premise is that a new component can be substituted for a conventional

and code acceptable component if it provides the same "degree of safety" when

exposed to the natural elements such as live and dead load, loads due to earthquakes, fire, and so on. The investigations of this research will address the

fire performance aspects of floor assemblies only.

Background

One rational approach to the accomplishment of both objectives is an

analysis using probabilistic engineering methods. Probabilistic engineering

is by no means a new disciplxne. Ang, as of 1972 in a review of literature,

reports on some 355 research papers on structural reliability (4). Textbooks

have been published on the subject, and the number of research publications

dealing with steel, concrete, applications in wood engineering number as few

as 20. The use of this same theory in fire situations has been suggested and

illustrated by researchers (5,9,18,22)

.

Fire Endurance of Floor Assemblies

(Deterministic)

In the areas of reinforced concrete and steel design, the concept of fire

design engineering as opposed to strictly fire tests is gaining acceptance (1^,

2,13,^,^). Wooden floor assemblies are qualified or fire rated based on test

only, and these data have been published for assemblies with at least a 1-hour

fire rating (1^). Various researchers have been promoting a combination of

design and testing much in the way the field of struc ural engineering emerged

throughout the years. Sunley reports that the variability of ;trength of

131
timber assemblies, when exposed to ASTM E-119 conditions, is less than the

variability of assembly strength at normal temperature (^) . Additional support for this is evident in the results of work of Schaffer, and Knudsen and

Schniewind on the strength of small, clear specimens of wood at elevated temperature (]^,34) . Sunley argues further that fire design engineering should

be no more difficult than designing for other load types.

Equations have been developed to predict the fire endurance of fireexposed joist floor (^,^). The method of analysis was empirical, having as

a starting point the flexure formula for pure bending. In each case, the solutions to these equations are obtained by graphical methods (fig. 1). These

equations cannot be used conveniently in a probabilistic analysis, thus additional modeling is needed. A purely analytical approach to the prediction of

strength at elevated temperatures would be most desirable, and a considerable

amount of basic input data exists (^) . However, such modeling has not been

accomplished, thus simplified strength models will be used in the analysis to

follow.

Analysis

The first step of a probabilistic solution to an engineering problem is

to identify two (preferably independent) random variables, one of which represents a load effect and the other a resistance effect. These variables are

normally denoted by load S and resistance R. When R and S have the same units

and the probability of R being less than S can be interpreted physically as

failure, the stage is set for a meaningful solution. In a fire situation,

when fire endurance is associated with load S and time to failure of the component is associated with resistance R, the above requirements are met.

132
Modeling the time to failure of various floor assemblies and the fire severity

to which they may be exposed is the first required step to further estimate

' dfety of the assemblies.

The prediction of fire duration, and more generally fire severity, is a

research topic in itself (6,]^,^,^,^,^). For this analysis, it will suffice here to use the approach reviewed by Lie (^) . For ventilation controlled

fires, in which case the duration would be the longest, the relation which

relates fire duration to available ventilation (e.g., window area and window

height) is given by

WA,

'd 5.5 _ A

. XU (min) ( 1 )

where W, A^, and H will be treated as random and defined as

2

W = fuel load density (kg/ra ) 2

Aj. = floor area (m ) 2 A^ = window are (m )

H = window height (m)

-1 -5/2 The constant 5.5 has units kg min m

Model Building for an Exposed

Floor Joist

It is assumed that the failure during fire exposure is due to charring of

the three exposed sides of the joist, and thif loss of section coupled with

the strength reducing influence of elevated temperature causes rupture of the

joist. While burn- through and elevated temperatures of the unexposed surface

133
can be additional failure criteria, they are not considered in this analysis.

(These failure criteria relate directly to the floor-subfloor design which can

be analyzed separately.) Load sharing and composite action is not accounted

for directly in the analysis; however, it should eventually be included in an

experimental verification of the model.

A typical floor joist section is shown in figure 2. The shaded region

shows an idealized charred area. Schaffer (^) reports that the bottom corners

round when charring occurs, and furthermore, the radius of the corners can be

approximated by the depth of char. To account for this rounding by means of

the moment of inertia, would extremely complicate the computations in the

analysis, and it is clear that the error involved by assuming straight boundaries is of minor concern.

By use of the flexur^ formula, an equation can be written to quantify

failure in a fire situation as

M Y (t^,C)

( 2 )

where

M the applied moment due to both dead and live loads (in. -lb)

t^ = time to failure (min)

Y(t^,C) = the distance to the extreme fiber being a function of the time

to failure and char rate (in.)

I(t^,C) = the moment of inertia about an axis located midheight the

4 remaining unci arrtd section (in. )

a an exposed joist performance factor which relates normal

temperature strength to high temperature strength

134
2

B = the joist modulus of rupture at room temperature (Ib/in. ),

This model is similar to that used by others for large beams under fire exposure It also neglects any contribution to strength by the flooring

itself

.

Ihe selection of the above model needs some justification. At room

temperature a = 1 and the model is exact with the thought in mind that the

modulus of rupture is an idealized linear state. The actual state of stress

is nonlinear, but the model is adequate for design especially when the "depth

effect" is taken into account. In a fire situation, the nonlinearities are

expected to worsen since the joist cross section will not maintain a uniform

temperature. The movement of the neutral axis, due to a nonuniform modulus of

elasticity (MOE) resulting from a nonuniform temperature profile, can be

expected. At the same time, the compressive and tensile strengths of the wood

fibers are reduced due to elevated temperatures (Schaffer (^)). It may be

possible to model the net effect of the above-mentioned behavior when coupled

with the normal temperature nonlinearities of bending, but it has not been

reported in the literature. By introducing the exposed joist performance

factor, a, these unknowns will be accounted for in an empirical sense. In

addition, the factor a will account for the rounding of the corners and to

some degree, load sharing and composite action of the f looring-joist assembly.

Referring to figure 2, it can be seen that equation (2) can be rewritten

as

M (d - C t )/2

r = «B (3)

(b - 2C t ) (d - C t ) /12

135
where

b = the initial joist width (in.)

d = the initial joist depth (in.)

The remaining variables have been defined with equation (2) . All of the above

except b and d are treated as random variables. It is assumed that the failure

is due to charring of the three exposed sides of the joist and this loss of

section coupled with the influence of elevated temperature on strength properties causes rupture of the joist. By rearranging (3), there results a cubic

equation in time, t^

II

= bd^ - 2Cd(d + b) t^ + C^(b + 4d) t^ - 2C^ t^ (4)

While cubic equations can be readily solved by hand calculation and computer,

the derivation of the statistics of the variable t^ would be cumbersome. This

fact led the authors to investigate the error introduced in t^ by dropping the

cubic term and simply solving the quadratic equation for t^. Fortunately, the

errors introduced are negligible, ranging from 1.58 percent for a 2 by 6 (38 by

140 mm) to 0.35 percent for a 2 by 12 (38 by 286 mm). It must be emphasized

that this approximation was shown to be adequate for only 2 by 6 (38 by 140 mm),

2 by 8 (38 by 184 mm), 2 by 10 (38 by 235 mm), and 2 by 12 's (38 by 286 mm).

The data and results of full-scale floor section (two Douglas-fir joists

per assembly) tests reported by Lawson were used to study the applicability of

the model (^) . A question immediately arises with regard to the appropriate

value of the modulus of rupture, B. Since the test for the measurement of B

and the fire test are both destructive tests, it is impossible to have knowledge of both properties for a single piece. At this impasse the alternative

136
was to investigate the mean and variability of the product of the two variables

a and B. This was done in the following way:

The 42 Douglas-fir floor assemblies tested by Lawson were divided into

four grade groups as shown in table 1. These data indicate that there is

little difference in the apparent high temperature modulus of rupture for the

four different grades. More surprising are the calculated coefficients of

variation (COV) . From data collected by Hoyle in a world search of lumber

data, a value of 0.45 for the COV of B is typical for Construction grade

joist (15). With all grades combined, the Q was only 0.271 which shows that

E-119 exposure appears to have a variance-reducing influence on B just as

suggested by Sunley (39)

.

To scrutinize the model further, an attempt was made to predict the fire

endurance of two Douglas-fir assemblies tested by Son (36). Structural failures occurred at 11.63 and 13.00 minutes for nominal 2 by 10 and 2 by 8 joist

floors respectively. By assuming the 2 by 10 and 2 by 8 joist floors to be

of similar quality as of the Lawson report, i.e., ^ =, 1,165 pounds per square

2 inch (Ib/in. ) (8,032 kPa), the predicted time to failure was 2.58 minutes.

The large discrepancy between actual and predicted time is attributable to the

difference between live load levels used by Lawson and Son. In the tests of

Son, the joists were stressed to 100 percent of the allowable design stress,

as is specified by ASTM E-119, whereas the load levels used in the Lawson tests

2

ranged from only 200 to 917 Ib/in. (1,378 to 6,322 kPa), which is approximately 16 to 75 percent of the allowable design stress. This results in failure times calculated using figure 1 of 23 and 11.4 miautes respectively for a

nominal 2 by 8 joist floor as Son (^) tested. Henct , as one expects, the

lower the load level, the greater is the time to failure and thus more thermal

137
degradation is allowed to occur. Beams more heavily loaded will have shorter

times to failure and less thermal degradation. The right-hand side of

equation (3), representing the retained strength due to temperature rise, cannot account for the influence of heat accumulation degrading the crosssectional strength unless some measure of time dependence is introduced. By

making several data plots, the following model was developed to include such

time dependence where the variables have been previously defined.

M(d - Ct )/2 B

± = ( 5 ) (b - 2Ct^) (d - Ct^)^/12 1 + Ytj

The [ (b + 2'0 yt^]/(bd) term may be viewed at. a time-de jendent geomet ic f ictor

to account for h-at flov ing into the cross section, bd, through the perimeter,

b + 2d

.

While visual inspection of the data plots suggest the use of the above

model, some variation of the model may be more suitable. Again, as before,

the cubic term t^ of equation (4) is negligible. A least-squares nonlinear

regression analysis was conducted on five variations of equation (4) as shown

in table 2. The models were fitted to the 42 full-scale floor section tests

of Lawson.

It is seen that the second single-parameter model of the table is the

preferred model, since the residual standard deviation is only slightly more

than the residual standard deviation of the two-parameter model listed first

in the table. The range of predicted times-to-failure for the 42 floor assemblies was 7 to 29 minutes. This shows that the selected model is a reliable

predictor when recognition is made of the fact that the residual standard

deviation of 2.57 minutes includes the variability of aB as shown in table 1.

138
Solving equation (5) for by omitting the cubic, results in

_ 2Cd(d + b) + 6MKV/B

f

- V {2Cd(d + b) + 6MKy/B}^ - 4C^(b + 4d)(bd^ - 6n/^) (6)

2C^(b + 4d)

where K = (b + 2d)/bd, which is an explicit expression for the time to failure.

Time to failure, t^, is compatible to the previously defined fire duration t^,

which is the load variable.

Results predicted by equation (5) were compared to four floor fire endurance test results obtained by National Forest Products Association (44,45,46,

47) . The actual times to failure to carry load versus those predicted are

given in table 3. It is observed that the predicted times are consistently

and significantly less than the time;- to failure of !,he whole floor assembly.

This deviation can be explained by the model parameter derivation based upon

Lawson's (^) results. Lawson conducted fire endurance test of paired

Douglas-fir joists with essentially noncontinuous floor sheathing. The NFPA

tests are of assemblies consisting of many joists and a more or less continuous

floor sheathing. Such assemblies result in load-sharing between joists and

increased load-carrying capacity of the sheathing which is not similarly

reflected in the paired joist tests by Lawson. If one takes note of the time

when the first joist ruptures in the NFPA tests, the difference between the

model predicted results and actually observed are closer in two of the tests

where it was observed. This, again, illustrates the need to have a degrade

parameter, y, or other parameters which include both the effect of loadsharing and floor sheathing. As a result, such replicate experiments are

planned.

139
Model for Exposed Floor Truss

The lower chord of a floor truss is subjected to both bending and tension

and the well-known interaction equation is used for design purposes.

I < 1 (7)

Here f^ and f^ denote applied stresses, and F^ denote allowable design

stresses in bending and tension, respectively.

As in previous reliability work (^) , this interaction equation can be

modified to indicate failure (with some reservations as discussed in the

Forest Products Laboratory report No. 302). However, in a fire exposure case,

we need to estimate one parameter, thus some slight inaccuracy in the neighbor

hood of the combined stresses associated with a floor truss will be corrected.

The failure equation for fire exposure would read

Of =

1

1 + g(b,d,t^,Y) ( 8 )

where the right-hand side of the equation has a form similar to that for the

exposed floor joist. Function g

accounts for the therrmal degrade of the

section, and its arguments are later defined. B is the modulus of rupture

from which F^ was derived and T is the ultimate tensile strength property from

which the design value F^ was derived.

140
Because four-sided fire exposure of the lower chord in a floor truss is

4/ critical,— expansion of the interaction formula for this case is as follows:

P

M(d - 2C t^)/2

(b - 2C t^)(d - 2C t^)

+

(b - 2C t^)(d - 2C t^)^/12

1 (9) T B 1 + YK t^

where

K = 2(b + d)/(bd)

Y = the thermal degrade factor

P = the axial tensile force due to dead plus live load

b = the width

d = the depth

C = the char rate

t^ = the time to failure

M = the maximum bending moment due to dead plus live load

B = the modulus of rupture

T = the ultimate tensile stress.

Analogous to the floor joist case, K is the ratio of the lower chord perimeter

(or surface area for heat transfer) to the cross-sectional area (or volume for

heat storage). After some manipulation, there results the cubic equation

4/ It is assumed that the mode of failure is rupture of the lower chord.

The upper chord has only three-side exposure, and the webs are only stressed

to approximately one-half the level of the lower chord.

141
(10)

{-8C^} + {AC^(b + 2d) + + {-2dC(d + 2b)

Pd „ . 2PC 6M yK, . . Pd 6M, _ ^ ^ yk + - —^) + {bd = 0

This equation will later be solved for t^ and used to estimate assembly

reliability.

Model Parameters for an Exposed Floor Truss

To properly estimate model parameters, test data on 2 by 4 assembly

members under combined tension and bending are required. Unfortunately, none

are available for this purpose. Schaffer (^) , however, has conducted fire

exposure tests of constantly tensile loaded Select Structural coast Douglas-fir

and southern pine 2 by 4 members. The time to failure was recorded (table 4).

For pure tension, the failure model reduces to

P(1 + Y K t^) = T(b - 2C t^)(d - 2C t^) (11)

This expression is easy to deal with, since it is only a quadratic in t^.

Solving it for t^, there results

t = 2T C(b + d) + PyK -sf {2T C(b + d) + PyK}^ - 16T C^(Tbd - P)

2

8T C

( 12 )

where the minus root is the meaningful root.

An estimate is needed for the mean tensile strength, T, and char rate, C,

to determine y for fhe available tensile fire test data.

142
An estimate of T is available for inland Douglas-fir Select Structural

[Hoyle (26)]. Based on a sample size of 30, the average tensile strength value

was 5,020 Ib/in.^ (34,600 kPa) with a COV of 0.388.

This average tensile strength value can be compared to what might be

calculated using normal distribution theory and the allowable tensile stress.

From NDS 77 (26) the allowable tensile value for Select Structural Douglas-fir/

2 larch is 1,200 Ib/in. . Using the calculated COV of 0.388, the mean value is

2 determined to be 6,811 Ib/in. by the following formulae:

1,200 = (T - 1.645 * COV * T)/2.1

This value is significantly larger than the actual mean of 5,020 which illustrates the non-normal nature of tensile data This shows, therefore, that one

should use the mean value obtained from lumber tests whenever possible for the

grade and species in question.

The nine Douglas-fir fire and tension test results can then be used to

estimate ^ for Douglas-fir. By using T = 5,020 and C = 0.0245 inch per minute,

was estimated to be 0.113 with a residual standard deviation of the time to

failure of 1.829 minutes.

A similar estimate of the fire exposure reduction factor ^ can be done for

the southern pine test results of table 4. The southern pine lumber tested was

nearly clear of defects, and it was imgraded. The lumber appeared to have a

quality at least as good as Select Structural. There are data [Hoyle (17)

]

that show that high quality southern pine (2400f machine stress rated (MSR))

is stronger in tension than high quality Douglas-fir (2400f MSR) --the ratio

in strengths being 1.24. To arrive at a mean value for the tensile strength

2

T of Select Structural southern pine, the 5,020 Ib/in. value for Douglas-fir

143
was multiplied by 1.24 to yield 6,233 Ib/in. for southern pine. Using a mean

char rate for southern pine of 0.03 inch per minute (^) in a nonlinear regression analysis of the fire test results, the yield is an ^

of 0.0839 inch per

minute with a residual standard deviation of the time to failure of

1.077 minutes.

Both species value of the reduction factor, (0.113 for Douglas-fir and

0.0839 for southern pine) are substantially lower than that of 0.17 obtained

for the floor joist assembly in the preceding section. The reasons for this

are not clear at present. It is known, however, that smaller sections (2 by 4,

as compared to 2 by 8, or 2 by 10' s) are likely to suffer a more rapid reduction in cross section than larger ones. The difference for 2 by 4 sections ol

southern pine and Douglas-fir are shown in ^^igures 3 a id 4 as a f motion of

fire exposure time.

The developed model and parameters may be used to estimate the structural

failure of a given floor truss assembly. This was done for the truss shown in

figure 5. The lumber of the floor truss is No. 1 Dense KD southern pine.

B for No. 1 Dense southern pine was obtained from table 2 of Doyle and

Markwardt (7). As tensile strength data for Dense No. 1 was not available,

data for No. 1 KD southern pine was taken from Doyle and Markwardt (8). The

published value of 5,706 was adjusted to reflect current standards by Forest

2 Products Laboratory personnel to 5,646 Ib/in. .

The truss was then analyzed with Purdue Plane Structures Analyzer (37),

and as normally is the case, the center pan^l of the lower chord was most

highly stressed with an axial force of 4,209 pounds and a bending moment of

140 inch-pounds. A summary of the input parameters can now be given for the

fire tested floor truss.

144
y = 0.0839 inch per minute

P = 4,209 pounds

b = 3.5 inches

d = 1.5 inches

C = 0.03 inch per minute

M = 140 inches-pounds

B = 9,410 Ib/in.^

T = 5,646 Ib/in.^

Substitution into and solution of the failure model equation (1^) results in a

time to failure estimate of 11.2 minutes. The actual failure time was estimated

to be at 10.2 minutes (1^). This test continued to be conducted under reduced

load until 14.6 minutes, where fire exposure was terminated without occurrence

of collapse. The predicted time to failure falls within this 10- to 15-minute

lange. This result is most promising for future use of the model.

It is interesting to examine how time to failure is altered for the same

truss with a reduction in applied load. The failure model equation predicts

times-to-failure as a function of applied load as shown in figure 6. For

reduction in load to 50 percent of full design, it is seen that 5 minutes is

added to the predicted time under full design load. If there were no load on

2

the floor assembly except the dead weight (4.9 Ib/ft ) of the assembly itself,

a failure time of 21.1 minutes results. Hence, failure times greater than

this are theoretically impossible for this truss design.

145
Estimating the Safety

In the preceding paragraphs we have developed models for predicting the

time-to-failure of two floor assembly types and have given an accepted model

to predict severity of fire exposure. Both are given in units of time.

Using the standard reliability notation, the assembly time to failure and

the time duration of fire exposure, t^ and t^ are denoted as follows:

R = (14)

and

S = t^ (15)

With R and S defined, the fire reliability (Rel) of an exposed joist or floor

truss (as a part of a floor), g, ven the occurrence of a fully developed fire,

is given by

Rel = Pr(R > S) (16)

which reads the probability (Pr) that the fire resistance is greater than the

fire load. It is convenient, from a calculations standpoint, that R and S be

independent. In other words, the amount of fire load and associated parameters cannot be correlated to the members resistance, char rate, and so on.

Conversely, the structural load, member resistance, and char rate cannot be

influenced by the fire load. Unfortunately, it is known that the char rate

is correlated with the variables of equation (1) which is discussed by

Schaffer (^) . However, in practice, the joist or component will be exposed

to a standard fire condition such as ASTM E-119 where the char rate will not

be influenced by the variables W, F, A, and H. The real problem lies in the

146
fact that the fire severity of E-H9 may not be representative of the fir--

severities associated with actual fire situations. It is thus appropriate in

this analysis to treat R and S as independent random variables.

Using the approach of Zahn (^) ,

Rel = Pr(R/S > 1) (17)

and taking the logarithm of the arguments

Rel = Pr(ln(R/S) > 0). (18)

By making the following definition

J = ln(R/Sj (19)

there results

Rel = Pr(J > 0) = 1 - Fj (0) ( 20 )

where F is the cumulative density function of the variable J.

Using first order, second moment approximations, the mean and variance

are given by

Mt = In

J MR

( 21 )

(22

147
Standardizing J,

\ E (23)

and

Rel = Pr \ > (24)

If the distribution of A is similar in all applications, then the variable

In

R

2 2

(25)

is a consistent measure of fire safety, p is normally called the safety index.

The next step is to estimate the means and variances of the resistance and

2 2 load, |jj^ and [Jg, and Og . A first order approximation of the mean, E, of

a function, Z, where

Z = h(X^, X^, ..., X^j (26)

is given by

E(Z) = h[E(X^),E(X)2),... ,E(Z^)] (27)

Performing this operation on equation (6) for the conventional joist floor

assembly and replacing the expected values, E, of the compone it variables by

their statistical estimates denoted by a superscript bar, the result is

148
2Cd(d b) + 6MKY/B

where Y is treated as a random variable and B will be treated as a constant

equal to the average strength of the joist grade. In otherwords, ^

will be

used to account for the total variability associated with ^ ^nd B.

For a floor truss assembly, the result for the mean time to failure is

similarly obtained by dropping the cubic term in equation (10), solving the

quadratic, and by placing the expected values of the component variables by

their statistical estimates

(29)

2a

1

where

a = 4C^(b + 2d) + 2PC y K/T (29a)

b^ = 2PC/T - 2dC(d + 2b) - Pd y S/T - 6M y K/B (29b)

c^ = bd2 Pd/T - 6M/B

(29c)

149
Again, ^ is treated as a random variable, and B and T are treated as constants

equal to the average bending and tensile strength of the lumber grade.

The same operation is performed on the fire severity equation, which

results in

W Aj.

5.5 A^^/2

(30)

The first order, second moment approximation of the variance of Z defined

by equation (26) is given by

(31)

provided the X^'s are uncorrelated. The partial are evaluated at their

respective mean values.

In the expression for the load variable S, the floor area ^ and window

area ^ could

to the perimeter which involves the same variables as the floor area. However,

since data are not available to substantiate this correlation, a zero correlation will be assumed. In the case of light-frame construction, the other pairs

of variables lack an obvious cause for correlation. Without showing the

computations

be correlated since one would expect ^ to be some way related

2 -2 2 2 2 2

(32)

150
(33)

where the s are the respective COV.

In the equation for resistance, or time to failure of the floor joist,

there does not appear to be any obvious confounding correlations. Using

equation (31) to estimate the variance of equation (6), three partial derivatives must be calculated. These derivatives are lengthy so they will only be

substituted into equation (31) symbolically, which results in

Again, the partial derivatives are evaluated at the mean values of the component variables.

For the case of the floor truss, component variables M and P are perfectly

correlated with a correlation of +1. Therefore, equation (31) must be altered

to include correlations as

where p.. is the correlation coefficient between variables X. and X.. By

ij 1 J assuming zero correlation for the other variables, as for the floor joist case,

application of equation (35) to equation (29) yields

(34)

X. X.

1 J

(35)

151
ac

2

9Y

2

(36)

2 \ 2 ^ /aR\^ 2 ^ /8R\^ : j (aMj (^apj

^ ^'i ik <^p

The variance of the char rate, , has been estimated by reanalyzing

data previously developed (^) . The mean charring rate, C, and estimated

2 variance, o under ASTM E-119 fire exposure for coast Douglas-fir and

^ > southern pine are

coast Douglas-fir C = 0.0245 in. /min, cr^ = 5.56 x 10

— 'y • southern pine C = 0.0299 in. /min, a = 3.80 x 10

The variance in the strength reduction factor, is unavailable.

Substantial information is available on tlie variation of the applied load

which defines the variation of M and P. Eventually, all the variances denoted

2 by a will be replaced by statistical estimates from the data available or

being researched now. Finally, the COV of R can be obtained and is given by

a

% MR (37)

As statistical data become available for the various parameters, the components

of the safety index equation (25) may be defined and comparisons of assembly

safety accomplished.

Discussion

Probability of Failure

Knowing the distribution of X of equation (23), the probability of structural failure of an exposed floor assembly, given the occurrence of a fully

152
developed fire, can be calculated. In general, the distribution of A is not

known, and some assumption about the distribution of A must be made. It is

common practice to assume that A follows a normal distribution. In this

analysis, the assumption of normality will be used realizing that the probability estimated will be in the neighborhood of 0.1; thus deviations from

normality from one application to the next will not be amplified. Under these

assumptions, the probability of failure, P^, is given by

= (j)(-p) = 1 - (j)0)

where (|) is the cumulative area under the standard normal curve.

Code Calibration

In recent years the use of engineering components has increased dramatitally. Often these components are fabricated wi.,h manmade materials, and the

variability of the mechanical properties of these materials is substantially

less than the variability of a natural material such as wood. The shortcoming

of the present '*fire rating" system is that while it may account for the average response of a component to fire, it does not account for the variability

of the component response. Quite simply, the present system of fire rating

allows for the use of two different components with an equal "fire rating" of,

say, 1 hour, but at the same time have unequal safety levels. This situation

can best be illustrated by a hypothetical example using equation (25). The

example involves the calculation of the safety index for two different components, the only difference being the variability of the time to failure or

resistance of the component.

153
For the example, assume that the following data in table 4 applies to two

component types A and B.

Each column is identical except for the column indicating the COV of fire

resistance. Upon applying equation (25) to each case, the result is = 0.98

and = 1.24, which indicates an unequal level of safety. Converting these

^'s by equation (38) to probabilities, the results are 0.163 for component A

and 0.107 for component B. It may be hastily argued that there is no difference between a 16.3 percent and 10.7 percent failure rate, but upon closer

examination, it can be calculated that component B could have a fire endurance

time of 51.9 minutes and have the same relative safety as component A at

60 minutes.

The above hypothetical situation illustrates just one possibility of how

the safety index could be used to obtain equal safety and give a "fair shake"

to new materials and new components

.

In the future it may be possible to identify where the fire load, and

hence the fire duration, have different expected values and different levels

of variability. In such a case, a lower or higher fire-rated component may

be needed.

Summary and Conclusions

1. Time to structural failure prediction models, based upon the residual

load-carrying capacity of fire-exposed floor elements, are given for two

unprotected light-frame floor assemblies. A reduction in strength factor, a.

was calculated from the limited fire exposure test data according to two

equations

:

154
Joist floor

Floor truss

MY(t^,C)

a

The thermal reduction factor is further a function of fire exposure time

and the geometry of small cross sections:

Joist floor

a = 1/(1 + yKt^)

where

Y

= 0.170 (in. /min)

Floor truss

a = 1/(1 + YKt^)

where for

Douglas-fir y = 0.113 in. /min

Southern pine y = 0.0839 in. /min

2. A comparison of predicted times to failure versus those actually

observed for four unprotected joist floor assemblies results in predicted

times consistently less than that observed. The difference is attributed to

three factors:

--The model has parameters quantified on the basis of fire endurance

tests of paired joists with negligible floor sheathing.

--The actual floors consist of many joists with load-sharing likely.

155
--The actual floors have floor sheathing that contributes to increased

load-carrying capacity.

Predicted times to failure for two of the floor assemblies were similar to

that of observed times of failure of the first joist (not total floor assembly

failure)

.

3. The time-to-failure model with input parameters for a southern pine

floor truss assembly that had been fire endurance tested, resulted in a predicted time to failure of 11.2 minutes. The fire test had been concluded at

10.2 minutes due to excessive deflection of the floor assembly, without collapse evidenced.

4. The time-to-failure model for a loaded and fire-exposed floor truss

assembly was employed to examine the influence of various floor loads on the

time to failure. At full design load, failure was predicted to be 11.2 minutes,

but with dead load alone, the period was extended to 21.1 minutes. This indicates the sensitivity of such a fire-exposed assembly to the load applied

during test. This analysis did critically assume that failure of the lower

chord, rather than connectors, would result in collapse of the assembly.

5. The procedure to be used to calculate the safety of unprotected

light-frame floor assemblies is given. An example is provided to show how

variability in assemblies can affect the comparative safety of assemblies.

6. The predictive capabilities for both proposed assembly models requires

further fire exposure experiments for independent validation and parameter

refinement.

156
Nomenclature

Member or structural resistance

Applied "load"

Fire duration

Fuel load density

Floor area

Window or opening area

Window or opening height

Applied moment

Distance from beam centroid to outer fiber

Moment of inertia

Ratio of high temperature to normal temperature strength

Modulus of rupture at room temperature

Char rate

Modulus of elasticity

Beam breadth

Beam depth

Standard deviation

Coefficient of variation

Thermal degrade factor

Allowable stress

Axial load

Ultimate tensile stress

Statistical mean

Safety index
p Correlation coefficient

4> Accumulative density function for standard normal distribution

K Ratio of perimeter to area of cross section.

Subscripts

b Bending

t Tension

f Failure

d Duration

R Resistance

S Load

158
Literature Cited

1. Abrams, M. S.

1978. Behavior of inorganic materials in fire. ASTM E-5 Symp. on design of buildings for fire safety. Boston Park Plaza, Boston, Mass.

2. American Iron and Steel Institute.

1978. Designing fire protection for steel columns. American Iron and

Steel Inst., Washington, D.C.

3. American Society for Testing and Materials.

1977. Standard methods of fire tests of building construction and

materials (E-119). Annu. Book of ASTM Standards, Part 18, p. 739-757.

4. Ang, A. , et al

.

1972. Structural safety—A literature review. J. Struc. Div.

ASCE 98 (ST4), p. 845-884.

5. Burros, R. H.

1975. Probability of failure of building from fire. J. Struc. Div.

Proc. ASCE 101 (ST9), p. 1947-1960.

6. Coward, S.K.D.

1975. A simulation method for estimating the distribution of fire

severities in office rooms. Build. Res. Estab. Current Pap.

No. 31/75. Fire Res. Stn. , Borehamwood Hertfordshire. WD6 2BL.

7. Doyle, D. V., and L. J. Markwardt.

1966. Properties of southern pine in relation to strength grading of

dimension lumber. U.S. Dep. Agric. , For. Serv. Res. Pap. FPL 64.

For. Prod. Lab., Madison, Wis.

8. Doyle, D. V., and L. J. Markwardt.

1967. Tension parallel-to-grain properties of southern pine dimension

lumber. U.S. Dep. Agric., For. Serv. Res. Pap. FPL 84. For. Prod.

Lab., Madison, Wis.

9. Ellingwood, B. , and J. R. Shaver.

1977. Reliability of RC beams subjected to fire. J. Struc. Div. Proc.

ASCE 103 (ST5), p. 1047-1059.

10. Factory Mutual Research.

1977. Fire endurance test of floor-ceiling assembly. Wood trusses with

plywood floor design. ASTM E-119-76, FC-250. Factory Mutual Res.

11. General Services Administration.

1972. Building fire safety criteria. PBS P 5920.9 CHGE 2. Gen. Serv.

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159
12.

13.

14.

15.

16.

17.

18.

19.

20 .

21 .

22 .

23.

Gross, D.

1977. Measurements of fire loads and calculations of fire severity.

Wood and Fiber 9(1): 72-85. Wood Sci. and Tech.

Gustaferro, A. H. , and D. P. Jenny.

1978. Design concrete structures for fire endurance. South. Build.,

p. 24-28. June-July.

Gypsum Association.

1978. Fire resistance design manual. Gypsum Assoc., Evanston, 111.

Hoyle, R. J.

1977. Review of world literature on characteristic distribution of

properties of 2-inch softwood dimension lumber. Res. Rep.

No. 77/57-11. Washington State Univ. , Col. of Eng. Res. Div.

,

Pullman, Wash.

Hoyle, R. J.

1976. Digest of tension parallel-to-grain strength of Inland North

Douglas-fir. Res. Rep. No. 76/57-22. Washington State Univ.,

Pullman, Wash.

Hoyle, R. J. , and T. M. Maloney.

1976. Tension strength and bending elastic modulus of truss joist

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Kameda, H. , and T. Koike.

1975. Reliability theory of deteriorating structures. J. Struc. Div.,

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Knuds en, R. M. , and A. P. Schniewind.

1975. Performance of structural wood members exposed to fire. For.

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Lawson, D. I., C. T. Webster, and L. A. Ashton.

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1972. Fire and buildings. Appl. Sci. Pub., Ltd. Ripple Road, Barking,

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Lie, T. T.

Optimum fire resistance of structures. J. Struc. Div. ASCE 98(ST1),

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1974. Characteristic temperature curves for various fire severities.

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24.

25.

26.

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31.

32.

33.

34.

Ligtenberg, F. K.

1971. Structural safety and fire (1.3). Build. Int., p. 271-272.

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.

Malhotra, H. L.

1977. The philosophy and principles of fire tests. Fire prevention

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1977. Design values for wood construction--a supplement to the 1977

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1977. Directions to improve application of systems approach to fire

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1977. The influence of a building's construction and fire load on the

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1977. The definition and evaluation of fire safety. Fire Prevention

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1970. Fire load, fire severity, and fire endurance. Special Tech.

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1978. Toward a knowledge-based fire safety system. Amer. Soc. for

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Schaffer, E. L.

1973. Effect of pyrolytic temperatures on the longitudinal strength of

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.

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35.

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38.

39.

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41.

42.

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45.

46.

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1977. State of structural timber fire endurance. Wood and

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1973. Fire endurance tests on unprotected wood-floor constructions for

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Suddarth, S. K.

1972. A computerized wood engineering system: Purdue plane structures

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Sunley, J. G.

Design concepts for fire-resisting constructions. Behavior of wood

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1977. Reliability-based design procedures for wood structures.

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National Forest Products

1974. ASTM E-119 fire

assembly. Design FC

National Forest Products

1974. ASTM E-119 fire

assembly. Design FC

National Forest Products

1974. ASTM E-119 fire

assembly. Design FC

Association,

endurance tesL--2- by

212. Nat. For. Prod.

Association,

endurance test--2- by

209. Nat. For. Prod.

Association

.

endurance test--2- by

216. Nat. For. Prod.

10-inch wood joi t floor

Assoc., Washington, D.C.

10-inch wood joist floo»

Assoc., Washington, D.C.

8-inch wood joist floor

Assoc., Washington, D.C.

162
47. National Forest Products Association.

1974. ASTM E-119 fire endurance test--2- by 8-inch wood joist floor

assembly. Design FC 213. Nat. For. Prod. Assoc., Washington, D.C.

48. Hoyle, R. J. , and T. M. Maloney.

1976. Bending strength tests of visually graded 2-inch dimension lumber

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,

Pullman, Wash.

163
Table 1.- -The apparent high temperature modulus of

' rupture , aB, for each grade group

Grade

Sample

size

Apparent rupture

strength

Coefficient of

variation (COV)

N aB1/ aB 1

Lb/ in.kPa

800 20 1,126 7,764 0.238 1

1,200-1,600 15 1,126 8,384 .320 1

1,600 2 1,264 8,715 .442 i Clear 5 1,266 8,729 .199

1 Combined 42 1,182 8,150 .271 1

1 V A constant char rate. C, of 0. 025 in. /min

l (0.0635 mm/min)

was used in the calculation; therefore, the Q values reported

are an upper bound since they include the variability of the

char rate.

164
Table 2. A least-squares non-linear regression analysis was conducted on five different variations of a time

to-structural-failure model. In each case,

42 data points were used to estimate either one or two parameters, depending upon the form of

the model. The parameters must carry the

necessary units to make the equations dimensionally homogeneous

Model Parameter

estimates

Residual

standard

deviation

Min

M(d - Ct^)/2

B

(b - 2Ct^)(d - Ct^) /12

= 0.296

'^o'^ Y, = 0.206

2.54

1 + K Y t,

Y

= 0.170 2.57

^o = 2.60

1 + K "Y^t^ Yj = 0.160

1 + Y t,

Y

= 0.216 2.74

B Y -0.324

1 + K(y^ + Y2 t/) Y2 = 0.0328

= (b + 2 * 1/ K d)/(b * d).
T3OJ UU

OJ 4JO

iHa-d3<

OiS <r\

iH vD| o<4H ml 0) '^1

iH3 -^1 <t 1 •H ' > 3

CO 4-> o CO +j 0) 4J

CO

01 01 E U

'H

d

+J cO

iHd

CO ’O 3 d

+J o; u

CO OJ

iH T3 *H d m

<0

iH T3 o

o; o 4J rH U 4H •HT3

(U

iH Oh1

CO

01 hQ

CO H

(U>

01 ,nO

4J iH TJ ^ U E >

o; >-i CO OJ

CO CO < O cs o

i/^

m o(3^

VOO

<N

CvJ o

VO

00

CM

'i- COcn

VOo

CN

OJ 1 CM CM •iH 4-> 1 d Hf O O CM U 1 •H • • • • 'rl 1 s m m rn. C'' • d

TJ 1 0) iH U 1 E OJ iH 1 -H X5 CM 1 H-t 4-) 1 3 1 o; o 1 iH Vi 1 d m m 3 1 •H hT rH 1 E CM CM CO CO •iH • CJ 1 o o O o CO 00 1 • • • • • •4H HT 1 d o o o o 1 M >v 01 1 iH U 1 XI d 1 E 01 1 CM 01 iH 1 • 00 00 o o CO 01 1 d o o CO CO CO Mh

<N| 1 *rH CO CO r-r fH. CO 01 ICQ 1 r. iH 1 HQ •«d- MT m m 4J 1 O E 1 d 0 1 IH 1 •r MH 1 Ho- Hi- |H. rH

OJ 4J 4J 1 hQ rn

m rH rH •H -O

•H to

d 1 rH o o o o CO 01 iH -tH 0) 1 •V r MH >

d,

O S 1 • OV as rH •iH CM •'O O 1 d rH rH CO CO O Ml < E 1 l-H 4-> 01 1 T3 1 to 1 4J X3 r*H 1 o O CO 4-) cd 1 00 00 rH rH •H 00 d 0) 1 o d

•H

N 1 X

XXX •I-} D E ‘H 1 Ml O CO 1 CM CM CM CM U 4J a 1 CO to

>» -oI CO

>-l TOI OT

(U (U C3 rH 4J

(U

OJ (Oh d m 1 Zi T) r -o CM 4-» -3 CM O -3 E iH 1 O o o O •H •H 1 u o Ml r O d rH o d d o MH 1 •rH -rl 4-) a iH a Ml o a 1 OJ 1 MH MH 0) ^ 01 d 3 rH ^ d) CO rH 1 1 Oj rH 1 CM rH xi • H rH Si ^ rH U CO 1 to iH CM to Ml CM 4-> > CM 4J d CM 3 rH a 1 CO »H CO CO 3 3 rH 00 • CQ 1 rH • rH O • O a • o a • •H 3

C/^ 1 00

d 00d (0d tod

O 1 3 iH •rH 3 d -H z •H r -rH Ph Q 3 1 o o o O >s (J3 >» U 1 CD d CVJ CD iH CM Z Ml CM z Ml CM d 1 CO >> CO -o CO -3 CO rH CMl 3 1 CM > CM d \ CM 1 (N 1 \ Ml 1 as as Vi CO Vi CO 3 1 • rH • a c-c • z CM • : CM MH 1 O O O O 3 1 Iz; s; iz. Ml

TD

01>

•Hu

<u T3xiu

00d

OJ

Ul u

CO

01

S-l d

4-> o<3ud

r^

VMo

OJud

4-) CX3

iH

166
Table 4.

Fire endurance of 2 by 4*s under constant tensile

load. Select Structural allowable stresses for

Douglas-fir and southern pine are taken from

1977 National Design Specifications (NDS)

Douglas-fir, coast Southern pine

Test

Failure

time Test

Failure

time

Min Min

P = 6,100 lb, 90 pet F

1 11.20

P = 6,100 lb, 83 pet F

1 ^ 10.00

2 7.67 2 11.61

3 9.35 3 12.85

4 11.25 4 12.34

5 8.74 5 11.80

Mean 9.64 Mean 11.72-

Standard deviation 1.57 Standard deviation 1.08

P = 4,960 lb, 73 pet F

6 ^ 9.24

7 9.96

8 13.36

9 13.92

Mean 11.62

Standard deviation 2.36

167
Table 5.

Two components, A and B, have the same average

fire endurance time (60 min) but different

variabilities (COV of 0.25 versus 0.5).

Each component is exposed to an identically

distributed fire load, as illustrated by

the last two columns of the table. The re- sult is an unequal level of safety, as calculated by the safety index equatiog (25)

Component

Min Min

A 60 0.5 30 0.5

B 60 .25 30 .5

168
Figure 1. —The family of curves for fire endurance of wood floors

- 1/2(1 - TVs)[1 -(T/2Vi)]^ ((20)) where

applied load , breaking load—

design allowable stress

ultimate stress

s = D/B (depth/breadth)

,

T = t/20VA,

t = fire endurance (min) , and

2

A = cross-section area (in. ).

\/ Extreme fiber stress of 11,000 pounds per square inch at

maximum load.

(M 145 178)

169
0.6

170

i45

178
Figure 2. —An idealized exposed floor joist when subjected to fire chars on

three sides. The subfloor protects the top side of the joist. While it

is known that the bottom corners round, straight boundaries are used as

an approximation. After time, t, and char rate, C, the depth of char

equals C*t.

(M 148 531)
M 148 531

172
Figure 3

. --Residual cross-sectional area divided by the initial area in percent

of coast Douglas-fir nominal 2 by 4 (1-5/8 by 3-5/8) members with duration of

exposure to ASTM E-119 fire conditions. The top curve is the result of using

the mean species charring rate, of 0.025 in. /min for large sections. The

bottom curve is the result of using an effective charring rate, C^, of

0.038 in. /min. The straight line is the linear regression fit to the actual

data where the line was forced through 100 percent at time equal to zero.

(The model Y = pX was fitted to one residual area/initial area.) The

equation of the curves is given by 100(b - 2Ct)(d - 2Ct)/(bd).

(M 148 528)

I

173

Figure A. --Residual cross-sectional area divided by the initial area in percent

of southern pine nominal 2 by 4 (1-5/8 by 3-5/8) members with duration of

exposure to ASTM E-119 fire conditions. The top curve is the result of using

the mean species charring rate, of 0.030 in. /min. The bottom curve is

the result of using an effective charring rate, C^, of 0.45 in. /min. The

straight line is the linear regression fit to the actual data where the line

was forced through 100 percent at time equal to zero. (The model Y pX was

fitted to the residual area/initial area). The equation of the curves is

given by 100(b - 2Ct)(d - 2Ct)/(bd).

(M 148 529)

175
SOUTHERN

PINE

2x4

176

TIME,

t

(MINS)
Figure 5. --Floor truss design subjected to test conditions of ASTM E-119. The

upper chord was loaded with tanks simulating a uniform load of 55.1 Ib/ft^

which resulted in a combined live and dead load of 60 Ib/ft^.

(M 148 527)

177
N.

CM

00 I

178
Figure 6. —Time to failure as a function of applied load for the floor truss

assembly of figure 5.

(M 148 530)

179
001

180

TIME

TO

FAILURE

(MINS)
BASIC PRINCIPLES OF CALCULATING THE

FIRE RESISTANCE OF TIMBER STRUCTURES

Dr. G. M. Kirpichenkov

and

Dr . I . G . Romanenkov

(CNIISK, GOSSTROY USSR)

Timber structures are a tradition in the Soviet Union. The

nomenclature of timber structures includes beams of constant depth,

lean-to, gable, triangular and segmental trusses, triangular arches,

curved glue-laminated beams, three-hinged arches, etc. When exposed to

fire, timber structures burn and contribute to fire spread over the

surface of the structure. The latest fire tests show that timber

structures of massive cross-sections are highly resistant to fire

exposure. In 1976, building regulations were supplemented with an additional requirement concerning the values of fire resistance limits

for timber structures. Depending on the structure type the required

fire resistance is 0. 5-2.0 hours. The introduction of fire protection

requirements has stimulated the development of a fire resistance calculation method for timber structures.

The pecularity of timber structure performance in fires consists

in the loss by timber surface layers of their initial physical properties

and alterations in the mechanical properties of the unexposed timber

cross-sections. The stresses of the imposed load reach their extreme

values and there is a loss of strength of the structure due to the

reduction of timber cross-section and the temperature increases.

The main factors that influence the fire resistance limit are as follows: timber carbonization rate and depth, changes in the timber

structures cross-sections, stress values in the structural members before

fire exposure and stress increase in the working sections at fire exposure,

changes in the elastic properties of timber in fire conditions, and

changes in load. Other factors are also known that are capable of

exerting influence on the fire resistance value, dependent on definite

types of stresses: change in elasticity and lateral stability; the

correlation of tangential and normal stresses in bending members, etc.

The principal aspects of timber structure fire resistance design based

on the solution of the equations for heat transfer and statics are con- sidered in this report; the results of fire tests are also given. The

structures are subjected to "standard" fire exposure. The structure's

fire resistance is determined by the structural member with the least

resistance to fire exposure. The failure of a structural member results

in a functional loss of the structure.

181
. The heat transfer solution consists of establishing the temperature

distribution in the cross-section of a structural member and determining

the change in the geometrical characteristics of a member's cross-section

due to fire exposure.

Timber is considered to be a combustible material; it ignites and

burns on exposure to high temperatures. The timber ignition temperature

may be assumed to be 280®C. According to the standard fire, the time

from the fire exposure until timber ignition (tj^) takes 3 min. In case of a fire retardant coating the time to timber ignition is determined by

the coating's effectiveness and its thickness. The fire exposure results

in the partial carbonization (combustion) of a member's cross-section

which, however, remains partially undamaged. The carbonized area of a cross-section loses the physical properties of timber. The thickness of

the carbonized layer depends on the fire exposure duration. The temperature of the carbonized region is assxamed to correspond to the fire

exposure temperature.

The temperature of internal (undamaged) timber layers changes with

the fire exposure duration, so for this reason the cross section dimensions are of great importance. The isotherms are distributed over the

cross-section in such a way that the timber temperature does not exceed

100°C after an hour-long fire or high temperature exposure.

Timber carbonization is one of the main factors to be considered

in the process of stress and deformation analysis of timber structures

with respect to time. Timber carbonization takes place successively

from the surface layer at an equal rate for similar timber species of

the same moisture content. The timber carbonization rate (Vj^) is influenced by moisture content, timber density and air supply for combustion.

Carbonization rate is determined experimentally by timber carbonization

depth for rather large and straight members. The carbonization rate at

90 min. fire exposure is assumed to be constant. The carbonization rate

for members with a minimiam cross-section dimension of 120 mm made from

air-dried solid timber is assvimed to be 0.8 mm/min; for members with the

cross-section dimension less than 120 mm-1.0 mm/min. Fire-retardant

treatment is assumed not to have an influence on timber carbonization

rate. Timber carbonization depth (Z) is a linear function of a fire

exposure duration ( Tn) , i.e. Z - Vn(Xn-T]3 ).

The change in the geometry of a cross-section takes place due to

timber carbonization. The change in the geometric properties included

in the design formulas is stated according to "time-carbonization depth"

dependence by reducing the cross-section dimensions from each fire exposed

side. Thus the fire resistance value of bending structures depends on

the cross-section area, the resisting moment and the profile factor (the

relation of the beams' side surfaces to the cross-section area).

182
More intensive timber carbonization at the corners of the

section results in changes in the geometry of a cross-section and causes an additional alteration in the geometric parameters. A marked

difference between the actual and an assumed rectangular cross-section

after 10-15 min. of carbonization is noted. After a long-term fire

exposure period the uncharred portion of a rectangular cross-section

tends to acquire a rovind shape. Timber beams of rectangular shape with

the initial cross-section dimensions 12 x 40 cm subjected to a 60 min.

fire exposure on the lower exposed surface took an approximately round

shape with the diameter of a circle equal to the width of the carbonized

member

.

Depending on the cross-section height to width ratio and on the

fire exposure duration, its area may be reduced by 22% and the resisting

moment of the cross section by 32% compared with the residual cross- section area estimated from the carbonization depth on each side. The

additional reduction in the resisting moment and the cross-section area

is taken into account by multiplying the coefficient r\ which depends on

the relationship between the cross-section dimensions (height and width)

and carbonization time.

The fire resistance limit determination for timber structures is

calculated by one of three approximate methods: "time cross section

area", "time resisting moment", "time profile factor". Each of these

methods estimates the unknown value differently, and their correlation

has no causal relation. To determine the fire resistance limit it is

necessary to take the alterations in geometric properties into account

.as well as variation of stresses in a fire exposed structure.

The static problem consists of the calculation of the structures'

bearing capacity under the conditions of fire exposure and working load.

The load-bearing capacity of a structure is insured by the existing

safety factor. The problem solution is based on calculation methods

introduced in the building regulations for timber structure design.

Variations in mechanical and strength characteristics of timber members

at fire and load exposure are analyzed. Reduction of timber strength

properties occurs upon heating. Timber strength reduction under high

temperature conditions is considered to occur due to the changes in

moisture and change in size due to shrinkage. The fire exposure is

characterized by partial moisture transfer from the region of carbonization and an increase of moisture content in the residual cross-section.

The phenomenon of increase in temperature and moistufe content causes a simultaneous timber strength reduction. It is difficult to estimate each

of these factors separately from the methodical point of view in the

process of fire tests.

Temperature and strength are considered, as a rule, to be parameters

to be estimated. The influence of heating on timber and glue-line

strength under different moisture conditions is examined. In the

structural design process (for normal conditions of load) the ultimate

183
strength estimation at the datum temperature is substituted for the timber

strength at the standard temperature of 20°C. In case of a temperature

increase from 20°C to 100°C (at an initial moisture content of 12%) the

magnitude of the timber strength reduction depends on the type of stress.

In compression conditions the strength is reduced by 20%; in conditions

of tension, by 15%; in bending, by 20%; at fracture along the glue line,

by 22%. The stated reduction of timber mechanical properties when exposed

to fire is for short-term load exposure and the possibility of a safety

factor reduction of the material taken into consideration. The design

resistance value is stated for each stress type based on the nature of

destruction due to both fire and load exposure

.

The changes of stress in timber members at constant carbonization

rate and external load are examined with regard to fire exposure

conditions

.

The given fire and static tests allowed us to specify the modes of

failure and transfer regularities of glued timber structures to an ultimate state in fire and load exposure conditions. The experimental

stress values were determined from the residual cross section and the

actual load at the time of failure.

Two failure schemes for bending members were described. They

depend on the relationship between the bending moment value and axial

force and various combinations of beam length and depth. The beams of

rectangular section with dimensions 12 x 40 cm including the control

ones, were tested without fire exposure up to failure. The beams were

subjected to shear failure in the maximum tangential stress area. The

average value of tangential stress was 11 kg/cm^. Normal stresses did

not reach ultimate values at the failure point since the beams' failure

took place at the shear plane. The tests on a special series of beams

with non-standard dimensions provided the researchers with useful information concerning the failure resulting from rupture on the tension side

of the beam. The average value of normal stresses at failure was

245 kgF/cm^. In case the load is distributed simultaneously, and assuming

the given experimental values of normal and tangential stresses for the

corresponding failure stages, their correlation makes 22.2. Therefore,

the destruction of beams with dimensions £ ^ 22.2h at fire and load

exposure should occur due to shear at bending. The given relation (or

the relation of design resistance) may be used as a boundary condition

for the design of a bending point at fire exposure.

The analysis of the experiments carried on for the compression

members allows one to consider their load-bearing capacity and stress

increase with regard to reduction of a cross-section, flexibility alterations, and lateral bending coefficient. The average strength value at

compression before failure was 250 kgF/cm^, taking into account lateral

stability and more intensive carbonization of members at their corners.

184
The fire resistance value (T) of timber structure is determined

by the time from the fire exposure until timber ignition (tj^) and by

the time from timber ignition until the bearing capacity loss by the

design member (Xj^) , i.e. T = x^ + Xj^. The value of x^^ is determined by

the ultimate admissible reduction in the dimensions of the member ' s cross section (Ai) :Xn = Ai/Vn- The timber carbonization depth should not

exceed the admissible reduction in the dimensions of the cross section,

i.e. Z ^ Aj^. The admissible Aj^ value is determined by the condition

that the stress value (a^) in the non-carbonized cross-section part of

the structural member at fire exposure does not exceed the design timber

resistance (R^) for fire resistance design, i.e. Rq.

The determination of time until failure of the design member is

carried on for the various types of stress conditions. In case of tension

members the determination of value in question takes place with regard to

the reduction of the cross-section area. The determination of time to

failure of bending members in which the stability failure of a flat shape

is not provided, is performed taking the reduction of resisting moment

and the cross-sectional area into consideration. The x^ estimation for

the members designed for combined compression and bending stress is

performed taking the area reduction of the cross-section, the resisting

moment and g ratio into account. The n ratio accounts for the secondary

moment's influence of the axial force due to member deformation. To

determine the compressed members' strength, time to failure estimation

is carried out by taking the cross-sectional area reduction into account.

To calculate the members' stability, the cross-sectional area reduction

is determined as well as a longitudinal bending ratio \p .

In the process of the structural fire resistance determination the

service load value is estimated on the basis of real load data. If this

is not possible, the calculation is performed using standard, dead and

live loads. The transient loads excluded are; people's weight, assem- bling loads, the loads transported by means of a hoisting crane. The snow load may be reduced in single-floor industrial, store, sport and

other buildings without suspended ceilings and garrets. The snow load

reduction is estimated depending on the quantity of snow melted in the

process of fire exposure. The real stresses in timber members before

fire exposure are estimated with regard to load according to the building

regulations

.

Taking the given preconditions into consideration, the nomograms

and design formulas are worked out for the engineering fire resistance

determination of timber structures (or their members) for the principle

types of stress conditions: compression (central or eccentric) , tension,

bending, and compression with bending.

185
CONTROL CALCULATIONS

COMPRESSION

N

nF

.F'o

< R° — c N

Hr-rF^

R°

c where

:

N

r'

iP’n

- longitudinal force at standard load;

- design resistance at compression in fire conditions;

- cross-section reduction ratio due to carbonization of angles;

- cross-sectional area exposed to carbonization;

- lateral bending ratio of a carbonized member.

; at X <75 ip' = ; at X > 75 o — n z o

X o TENSION

N

n^.F'

F o

< R°

P

central tension; ^

•W'.EC w n H

eccentric

tension;

where

:

R° - design resistance at tension in fire conditions;

P

- resisting moment of a cross-section exposed to carbonization;

n - resisting moment reduction ratio due to carbonization of

angles;

e' - eccentricity of normal force determined by alteration in

gravity center of a cross-section at 3-sides carbonization.

AT 3-SIDES FIRE EXPOSURE;

F^ = (b - 2A) (h - A) Tip

(b - 2A) (h - A)^ „ w = . n b w

AT 4-SIDES FIRE EXPOSURE;

F' = (b - 2A) (h - 2A) n„

° 2 ^

186
CHANGES IN THE GEOMETRICAL CHARACTERISTICS OF

RECTANGULAR CROSS-SECTIONS IN FIRE CONDITIONS

F = b-h - rectangular cross-sectional area before fire exposure;

“ carbonized area of rectangular cross-section,

where A - ultimately allowable reduction value of cross-sectional

F' = (b - 2A) (h - 2A)n„

dimensions (b,h)

,

ratio of area reduction resulted from rounding of a rectangular section.

at X > 60 min.

F^ = (h - b) (b - 2A) - n/4 (b - 2A)

CHANGES IN THE GEOMETRICAL CHARACTERISTICS OF

::ec'l;ional to itc

width

187
BENDING

i2_<

2Fn — shear at bending; ——r < R - bending;

n W — u w n

where; M - lateral force at standard load;

Q

- design resistance to shear in fire conditions;

R° - design resistance to bending in fire conditions;

OiC n - coefficient of changes in geometrical characteristics

of sections due to more intensive carbonization of

angles.

COMPRESSION WITH BENDING.

N

r, .p'

F n

where

:

N4 R'

?T| ‘W

w n

< R

R — c u

C = 1 - o

N

3100R”F'

c n

\ o

ao

no

- member's design length;

- gyration radius of a carbonized cross-section.

no V

n *T

_y n

n„-F'

F n

where: T - gyration moment of a cross-section subjected to

carbonization (angles' carbonization is not taken into

account)

;

n - gyration moment reduction ratio of a cross-section due

^ to angles' carbonization.

FOR MEMBERS OF RECTANGULAR CROSS-SECTIONS.

T = 0.29b

o

n° = T . + T . oi ni

R° - timber resistance for fire resistance calculation.

Z . ni

ni V

o

' m

< R

Tn —

188
DURABILITY OF STRETCHED, COMPRESSED AND COMPRESSED-CURVED

ELEMENTS

square section

rectangular section

4-SIDES FIRE EXPOSURE

= I

(1 - v/M)

square section

rectangular section

Ai

c^(p)

“3

c, (p)

R

o

3-SIDES FIRE EXPOSURE

^ b+2h ^//b+2h\ bh

z = —-yl—) - ^ (1 -

“ stress in the element due to standard load

before carbonization

Ai

'1^013

ij; R° n c

COMPRESSED ELEMENTS' STABILITY CALCULATION

=

1

b

In

- is the shortest side of a section

- is the longitudinal bending ratio

before carbonization and destruction due to fire and load.

COMPRESSED-CURVED ELEMENTS' CALCULATION

Ai = e “3 10 =

+ n ^n

P R

C

1 e

e

P-C

eccentricity of normal force — S

N cores radii of carbonized and

and non-carbonized sections

the influence ratio of a longitudinal force's secondary moment.

189
APPROACH TO THE FIRE RESISTANCE DESIGN FOR LIGHT WALL PANELS

USING EFFECTIVE MATERIALS

V. N. Zigern-Korn

and

I . G . Romanenkov

Light enclosing prefabricated wall panels called "light structures"

have been successfully used in the Soviet Union for the last ten years.

The structures are characterized by the high degree of technological and

installation effectiveness, transportability and good utilization qualities. The weight of these structures does not exceed 75 kg/m^. Shaped

and sheet, flat and corrugated steel, aluminum, asbestos-cement, plastics,

timber, and other materials, as well as effective heat insulating materials

of 200 kg/m^ density, are used in the process of manufacturing. The words

"light structures" include complex steel panels without framework intended

for industrial buildings, asbestos-cement panels with or without framework

for agricultural buildings, and partitioning panels using asbestos-cement,

timber, plastics, etc.

In terms of fire security, the structures should meet the fire

resistance and ignition requirements. Depending on the building's class

of fire resistance the required fire resistance of wall panels, cap plates,

and partitions is 0.25 hr to 0.5 hr (1)^. According to the combination

of materials used, the structures are divided into non-combustible,

combustible with difficulty, and combustible.

The actual fire resistances of light enclosing structures are estimated by means of fire tests carried on according to standard methods.

The results of the tests cover a wide range of values.

According to their ignitability , the light enclosing structures are considered to be combustible or combustible with difficulty. Light enclosing structures possess specific properties that determine their behavior

in fire conditions. But insufficient information concerning these properties together with the imperfection of fire protection requirements and

estimation methods for the fire security level of buildings are likely to

result in irrational usage of these structures. The structures' behavior

in fire includes: 1) the resistance to fire exposure, expressed as the

time that bearing and enclosing functions are maintained in a structure

fire; 2) the degree of the structure's contribution to fire propagation

and the accumulation of dangerous fire factors (heat, smoke, toxic products

of combustion, etc., or fire hazard). Each of the two properties mentioned

above is detemined by a group of parameters. In the process of designing

buildings for fire security, each group of parameters should be considered

separately.

^Figures in parentheses refer to references at the end of this paper.

190
In terms of the terminology of 2) above, the structure reaches its

limit with regard to fire resistance when the predicted failures take

place, and its fire risk—if unanticipated events occur. Really, the

load-bearing capacity loss of the structure or failure as an enclosing

structure is seen as spasmodic fire development or by sudden accumulation

of material losses due to fire propagation.

The increase of fire risk factors, i.e., fire propagation, results

in the gradual accumulation of material loss. Besides this, losses of

non-economic character, i.e., injuries and deaths of people, particularly

during initial fire stages results from temperature increases or from the

high concentration of toxic combustion products

.

Of concern is that the methods of determining the fire resistance

criteria for light enclosing structures are unsettled. The fire risk

determination of these structures is limited, as a rule, to a fire propagation test.

The regulations (3) state three criteria of the structure’s limiting

condition regarding fire resistance: 1) the load-bearing capacity loss,

failure or deflection; 2) heating to a definite temperature on the

unexposed side; 3) the appearance of fire through cracks or openings.

Each of the criteria mentioned above 1) , 2) , and 3) , has been of

major importance for all types of buildings. Until 1978, the fire resistance of a structure was determined with regard to the minimum time

necessary for the structure to reach one of the criteria.

Now that new types of structures with the use of effective materials

have appeared, the determination of limiting conditions with regard to

fire resistance have required a more concrete approach with regard to

function. The structural design is likely to influence the method of

determining the time until a limiting condition is reached. A more con- crete approach to the limiting condition determination has already been

partially given in the Standard (3) . The limiting condition for external

walls, beams, frames, columns, poles, and coatings is solely the loadbearing capacity loss by the structures and joints. It is suggested that

the limiting condition of unloaded structures protected with fire resisting

coverings be the critical temperature of the structural material. The

criteria, 1), 2), and 3), are the limiting conditions for other types of

structures

.

While comparing the nomenclature of light structures used in national

building practice and the types of structures with their function (bearing

and non-bearing, external and internal, horizontal and vertical, etc.), it

is easy to note the existence of fire resistance design method development

for enclosing panels, cap plates, and external non-bearing walls.

As was stated, the fire resistance of light enclosing structures

(non-bearing walls and cap plates) is specified according to factor 1)

.

191
The failure of panels used as separating walls occurs at fire tests

after the insulation has burned out. However, it is impossible to determine the panel failure time analytically as the forces applied to it are close to zero.

The adopted list of possible structure limiting conditions with

regard to fire resistance (1,2,3) does not correspond to the variety of

the enclosing structure failures observed in practice. This circumstance

makes difficult the development of fire resistance design methods for the

enclosing structures useful for building fire security analysis. For

example, during fire tests on the external walls (fragments) manufactured

from light metal panels with the use of foam plastic insulation the steel

covered panels’ failure is not observed even after the insulation has

burned out. But holes occur in aluminum covered panels. The hole dimensions increase with time—the panel failure occurs when it burns out along

the whole of its width. The fire resistance remains undetermined in the

first case and depends on the fragment’s structural design in the second

case. In both cases, the fire resistance determination is of some difficulty. The information on the fire resistance in the second case is of no use for the analysis of fire process development.

The fire resistance design consists of the following: determination

of the unsteady state temperature field in the structure at a given temperature regime of fire exposure; bearing capacity estimation taking into

account the structure’s capacities during the fire and force exposure

process; determination of the time from the fire exposure beginning until

a limiting condition occurs (1,2,3). The structural load-bearing capacity

is not controlled if the fire resistance is stated according to factor 2)

.

The change in the thermal properties and physical states of materials

with temperature, the change in the geometrical dimensions of elements due

to the material burning process, and the change in the heat transfer con- ditions on the surface of a structure due to temperature and properties

alterations are taken into account in the unsteady state temperature field

determination

.

When determining the load-bearing capacity of a structure exposed to

a fire it is necessary to take into consideration the change in the mechanical properties of materials with temperature, the extra stresses and the

changes in the structures ' design scheme that occur due to the thermal

strains and the change in the structural members' properties.

There are no reasons for calculating the fire resistance of enclosing

structures according to factor 3) . It is possible to assume that concrete

rupture research will develop ways for determining the possible failure

of sheet material on the basis of mineral bending (4)

.

192
Also of importance is work on the burning out intensity of foam

plastic insulation in three-layer panels with steel coverings. The

results will help to estimate the possibility of open hole formation in

panels with aluminum cover plates.

In terms of an impartial fire resistance evaluation of buildings it

is of major importance to obtain information on the capacity of light

wall enclosures to resist vertical and horizontal fire spread to the

adjacent compartment, i.e. to high temperatures and fire spread along

the internal surface through partitions and ceilings, adjacent to these

walls. The formation of open orifices in the suspended external walls

may influence the ventilation of the fire and, consequently, the fire

temperature regime. This problem is likely to be of concern as well.

One-dimensional temperature fields for laminated enclosing structures using non-combustible materials are developed with regard to a non- stationary fire exposure regime (5) . The structural fire resistance

design criterion is the unexposed surface heating to 160° C (3) . The

calculation consists of solving the heat engineering problem of the

heating time of the unexposed surface to this limiting temperature. The

calculation is based on the elementary heat balance method permitting

one to achieve calculation results without requiring the solution of differential equations (6) . The non-stat ionary temperature field calculation

of a multilayer enclosing structure of non-combustible materials is

carried out using a numerical method. The structure cross-section is

divided into a number of elementary layers . Their width remains constant

within the limits of fixed material. The elementary layers and various

materials limits as well as the points situated on the structure's surface

and the air-exposed limits are considered to be points of calculation.

The problem is reduced to the determination of temperature values at the

structure's points of calculation at various time intervals until the

limiting temperature is reached on the unexposed surface. The calculation

is carried out by means of a step method. The heat balance equation is

set up for each elementary layer. The sectional and surface temperatures

of single-layer and multilayer structures are stated according to these

equations. The calculation equations are algorithms for computers. The

calculation permits determining the fire resistance of non-combustible

structures according to factor 2) . Of concern in this calculation process is the heat required to vaporize materials. The effect on the

heating of air circulation in the hollow structure's layers after the

structures have lost their tightness is also of concern. The problem

grows more complex if combustible materials are used in a structure.

There is the necessity of solving the static problem for external

horizontal bearing enclosures, e.g. membrane-type coverings, coverings

with steel corrugated plates, coverings from frame-type panels with

asbestos-cement, veneer, glass-fiber-reinforced plastics, and other types

of sheetings. The load-bearing capacity of the first two coverings is

determined solely by the bearing capacity of metal elements-membranes and

193
corrugated sheet. In this case, the method of calculating the enclosing

structures' fire resistance is not difficult, as it does not differ from

the calculation method for steel structures. The calculation of coverings

on the basis of frame-type panels is more difficult. As the load imposed

on these coverings during fire is significantly lower than the calculated

one they may retain their shape even after openings occur in the coverings.

Another phenomenon is observed in the process of calculating the fire

resistance of any multilayer compressed or deflected structure if the

layers and members are connected with each other mechanically and if these

connections provide their mutual resistance to the imposed load. In these

cases one, two, or three intermediate states with various sudden changes

of calculation scheme will take place before the limiting condition with

regard to structural bearing capacity is reached. The variety of the sudden

changes in the calculation scheme is explained, for example, by rupture or melting of one of the facing layers , the beginning or completion of the

burning process or the insulation melting.

The fire resistance of the structures mentioned is defined by the sum of time intervals corresponding to the definite states of the structure. The development of these calculation methods requires the accumu- lation and systematization of initial data both on the structure and

material behavior in fire conditions.

The problem of the load level imposed on the structures in fire is one of the basic problems in the process of structural fire resistance

evaluation. There is no unambiguous and well grounded opinion concerning

this problem. For example, the existing Standard (3) suggests that a load should be determined on the basis of detailed analysis of conditions

that may occur at fire. In case they cannot be defined, the load is

considered to be the standard. The possibility of a load level reduction

in fire in comparison with the calculated one is explained by the fact

that a fire is a rare phenomenon. The possibility of its coincidence in

time with the calculated load level is insignificant. The standard load

level means the major technological loads (equipment, installation,

materials, furniture, and people) and also on average annual unfavorable

atmospheric loads. Therefore, the standard load level is also overestimated for the fire exposure conditions.

At present, the requirements for the choice of loads imposed on a structure during fire are unlikely to be formulated. It is necessary to

study the statistical data concerning the fires that occur, comparing the

fire beginning time and the stage of the daily or annual functional cycle

of various types of buildings. The analysis of the data obtained will

promote the evaluation of possible various technological and atmosphere

load levels coincident with the time of the fire. In the process of

developing recommendations, it is suggested that the existing design and

calculation standards should be used for various types of structures

(steel, timber, reinforced concrete). Of concern are the methods of

investigation of possible various coincident loads (7)

.

194
The probabilistic-economic calculation methods of building

structures and the concept of "failure" are widely used at present and

aim at the optimization of failure possibility on the basis of the

minimization of expenditures directed to engineering and possible

losses resulted from failure.

To use the evaluation method of building structure fire resistance

successfully, it is necessary to be aware of the possible moment of

failure, which depends on the imposed load level, and take the physicomechanical properties of materials and their variation into account. In

this case only the development of analytical methods for the description

of structure behavior during fire together with the statistical modeling

methods will allow us to calculate the detailed characteristics of

structural fire resistance, while a large scale test will be aimed at

verifying the calculation methods.

Of major importance will be the accumulation of data concerning the

physical and mechanical properties of materials under fire conditions when

exposed to high temperatures and injury.

The fire security calculations of buildings will contain not a "standard" fire temperature regime but a real one. Therefore, it is

desirable to prefer fire resistance calculation methods that will yield

the working capacity at any temperature and fire exposure regime.

REFERENCES

(1) Fire safety norms of design of buildings and structures, SNiP

II-A.5-70.

(2) A. R. Rzhanitsyn, B. I. Snarskis, Yu. D. Sukhov, The main premises

of the probability-economic technique for calculation of construction structures. Construction mechanics and calculation of

structures, 1979, No. 3, pp. 67-71.

(3) Fire safety norms of construction design. A method of testing of

construction structures for fire resistance. ST COMECON 1000-78

standard

.

(4) Recommendations for protection of concrete and reinforced concrete

structures from brittle failure during fires. NIIZHB of the State

Construction Administration of the USSR, Moscow, 1979.

(5) N. F. Gavrikov: Calculation of the fire resistance of incombustible

protective structures. In the collection "Fire resistance of con- struction structures". Issue 3, 1975, pp. 100-108.

(6) N. A. Strel’chuk et al. Engineering-Physical Journal, Vol. VII,

No. 4, 1964, pp. 105-110.

195
(7) A. R. Rzhanitsyn, The theory of calculation of construction

structures for reliability. Construction Publishing House,

Moscow, 1978.

196
SPRINKLER SYSTEMS: AN APPROACH TO ACTIVE FIRE RESISTANCE

Joseph Hankins

Factory Mutual Research Corporation

In discussions of fire resistance, what is normally considered is the resistance

of structural components to the effects of heat produced by an assumed uncontrolled fire. A standard time-temperature curve was developed quite a number

of years ago, and fire resistance is typically specified in terms of the

period of time the component in question can withstand this standard fire exposure before exceeding some failure criteria. In more recent times, methods

have been developed to approximate the exposure presented by various types of

"real-world" fires, and relate those to the standard fire resistance measuring

techniques to allow more realistic evaluations of fire resistance. A key point

to note here is that the subject being dealt with is, in both cases, passive fire

resistance.

I would like to suggest an additional step. That is to consider the active

fire resistance offered by automatic sprinkler protection. In addition to the

well known extinguishing effects of sprinklers, they also serve to reduce the

temperature over a fire through the cooling effects of water. The combined

effects of cooling and the reduction in the rate of burning can result in a

marked reduction in temperatures above a fire. It is desirable to consider

this factor when designing fire resistance into a structure.

Given the fact that automatic sprinkler protection is provided in many structures for the purpose of controlling and extinguishing fires, it is reasonable

197
to include the effects of sprinklers in an evaluation of a building's fire

resistance. Also, trade-offs might be possible so that the fire extinguishing

benefits of sprinklers can be obtained at a lower cost as a result of the

reduced level of structural fire resistance needed.

It must be made clear at the outset that I am not proposing that automatic

sprinkler protection is a replacement for fire resistance. It is equally

important to note that fire resistance is not a final answer either. As in

any design process, goals must be set before choices are made.

For example, in a hotel or apartment building, provision of reliably fire

resistant walls, floors, and ceiling between each unit can assure that fire

will be contained in the area that it originates, thereby limiting damage and

risk to life. By the same token, provisions of a fire resistant structure

used as a large, open manufacturing area will simply assure that there is a

shell remaining after all contents have been consumed by fire.

Consideration must be given to reliability as well. Sprinkler systems are

subject to mechanical failure; any fire resistance scheme that depends on

doors, dampers, etc. is also subject to mechanical failure. There is no such

thing as "close enough" in the design of sprinklers or structural fire resistance.

If a sprinkler system is not designed to produce the discharge required to

control a fire in the area it protects, it will be no more effective than rain

I

I

I 198 3

i

}

1
on a volcano. By the same token, structural fire protection must be complete

in every detail. A fire resistant partition can be rendered useless by the

collapse of a pipe passing through it.

I would like to begin by showing you an example of the effect sprinklers have

on temperatures at the ceiling over a severe fire. This figure shows the

results of one of a series of tests where pairs of tests, one sprinklered and

one unsprinklered, were run for various plastic commodities. Also included in

the figure is the standard time-temperature curve. It is interesting to note,

by the way, that the unsprinklered test follows the standard time-temperature

curve very closely. On the other hand, in the sprinklered test, the most

important effect to note is the considerable reduction in maximum ceiling

temperature resulting from the operation of sprinklers. Results of the other

tests were similar: unsprinklered tests generally resulted in maximum temperatures well in excess of 1000°F for a considerable length of time, while

temperatures stayed below 1000“F for most of the sprinklered tests. The

1000®F mark is widely used as the rule-of-thumb critical temperature for

steel.

199
Temperature

Sprinklered Vs. Non Sprinklered Test

of 2 Pallet X 2 Pallet x lift. Array of

Polyethylene Bags in Cardboard Cartons

200
What I will attempt to do here today is present a simple method, developed by

our Basic Research Group, for estimating the area of sprinkler operation and

maximum anticipated ceiling temperature, given the type and configuration of

combustibles present and the design parameters of the sprinkler system.

Equation 1 gives us the radius of sprinkler operation in terms of the location

and freeburn heat release rate of the combustibles present; the distance

between the top of the combustibles and the ceiling; and the sprinkler density

and operating pressure. These relationships have proven to be accurate,

within practical limits, when compared to results of actual tests.

Equations 2 and 3 allow the temperature near the ceiling, within and outside

of the fire plume to be predicted, based on the above data. Again, these

relationships compare well with temperatures measured in actual tests.

These relationships apply only to 1/2 in. orifice sprinklers. A relationship

can also be developed for 17/32 in. orifice sprinklers.

The greatest problem in using these relationships comes in determining a

realistic heat release rate of the combustible in question. As there is more

easily accessible data on freeburn heat release rates, these are used in our

calculations. Use of the freeburn rate also introduces a safety factor into the

2

calculation. Freeburn heat release rates can range between 1,500 Btu/min/ft

2

and 96,000 Btu/min/ft . This chart shows some representative values.

201
(I)

(CB^ + l93CpDQl'2 -

* 96pD

Where

:

®

4.74

C Dimtntionlett location foctor

C I far from wolit

C« 2 directly adjacent to walls

€4 directly in corner 0 Sprinkler density (GPM/ft^)

H Ceiling height above top of combustibles (ft)

P • Pressure at operating sprinklers (psig)

Q * Free- burn heat release rate of combustibles

r « Radial distance from center of fire (ft.)

ro Radius of arc of sprinkler operation (ft.)

T Maximum gas temperature at ceiling at r (*F)

Temperature rating of sprinklers

Trm'Room temperature (*F)

Outside of Fire Plume (r 2 .I8H )

:

T-T„

Inside of Fire Plume (r < .I8H) =

T-T„ [CQ- 48pDro*]^

[4.74'

L H J

.I8H) :

ri4.87l

J

s 1

( Btu/min.)

( 2 )

(3)

202
TABLE

Heat-release rate per unit floor area of fully involved combustibles, bracketed between

discrete values. (PE = polyethylene; PS = polystyrene; PVC = polyvinyl chloride;

PP = pol)T)ropylene; PU = polyurethane)

2 Btu/min per ft of Floor Area

1.5K 3K 6K 12K 24K 48K 96K

1. Wood pallets, stack 1 1/2 ft high X

2. Wood pallets, stack 5 ft high X

3. Wood pallets, stack 10 ft high X

4. Wood pallets, stack 16 ft high X

5. Mail bags, filled, stored 5 ft high X

6. Cartons, compartmented, stacked 15 ft high X

7. PE letter trays, filled, stacked 5 ft high

on cart X

8. PE trash barrels in cartons, stacked

15 ft high X

9. PE-f iberglass shower stalls in cartons,

stacked 15 ft high X

10. PE bottles packed in item 6 X

11. PE bottles in cartons, stacked 15 ft high X

12. PU insulation board, rigid foam, stacked

15 ft high X

13. PS jars packed in item 6 X

14. PS tubs nested in cartons, stacked

14 ft high X

15. PS toy parts in cartons, stacked 15 ft high X

16. PS insulation board, rigid foam,

stacked 14 ft high X

17. PVC bottles packed in item 6 X

• 1 00

—

1

PP tubs packed in item 6 X

19. PP and PE film in rolls, stacked 14 ft high X

20. Methyl alcohol X

21. Gasoline X

22. Kerosine X

23. Diesel oil X

203
As an example, we will use the tests from which were derived the previously

shown time- temperature curve.

For this test, the height of the test array was 12 ft and the ceiling height

2 was 30 ft. The test array occupied 80 ft of floor area. Sprinklers were of

2

the 165°F rating and the sprinkler system maintained a density of .3 gpm/ft ,

requiring a pressure at the operating sprinklers of 28 psig. The ambient temperature at the test site was 55®F, and the test array was located well away

from walls.

As the commodity tested was essentially a solid block of polyethylene in a

cardboard box, it is reasonable to bracket the heat release rate somewhere

between polyethylene bottles and 10 ft stacks of pallets. We chose

18 K Btu/min/ft^.

Therefore:

H = 18 ft

P = 28 psig

Q

= 18 K Btu/min/ft^

C = 1

B = (. 165 - 55. ) (18) , ^^3 4.74

D = 0.3 gpm/ft^

T_ = 165“F

Li (80 ft^) = 1.44 X 10^ Btu/min

T = 55“F

rm

204
The radius of sprinkler operation is:

^ [(418)^ + 193 (1) (28) (.3) (1.44 x 10^)]^^^ - (418)^^^

o 96 (28) (.3)

= 50 ft

In the actual test, the radius of sprinkler operation was 40 ft.

If we assume no sprinklers, the temperature directly of the fire would be

predicted to be:

T - T = [1.33 X

10^]^'^^ rm

14.87

(18)

5/3

= 1425®F

T = 1480°F

The actual measured value was 1305°F.

For the case where sprinklers were provided, the predicted temperature would

be:

T - T = [1.44 X 10^ - 48 (28) (.3) (50)^]^^^ rm

14.87

(18)

5/3

= 643°F

T = 698°F

The actual measured value was 700“F.

205
Obviously, this is not a procedure that is in a final, useable form. Work

continues at Factory Mutual to develop a sophisticated and very accurate

computer model of sprinklered fires. In the meantime, anyone doing research

in numerical determinations of fire resistance should at least be aware that

methods do exist to allow the inclusion of the effects of sprinklers in their

calculations. Ultimately, it is hoped that a designer will have more flexibility in making choices among building materials or assemblies. One will be

able to select a material alone with a higher fire resistance rating, or

another material with a lower rating but with sprinkler protection provided to

bring the overall fire resistance up to the desired level.

206
NBS*114A (REV. S-7BI

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 80-2188

i.Gov*t Accmsion No.

-

Z, No,

, '

4. TITLE AND SUBTITLE

Joint US-USSR Seminar on Mathematical Methods for Estimating

the Fire Endurance of Structural Assemblies

5. Publication Date January 1981

i. Perfortning Organization Code

7. AUTHOR(S)

R. S. Levine, Editor

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS fSfreer, City, state, ZIP) 13. Type of Report 8* Period Covered

14. ^nsorlng Ag^cy Code

15. SUPPLEMENTARY NOTES

1

I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-worc/ or less factual summary of most significant information. If document includes a significant bibliography or I literature survey, mention it here.) This publication is a compilation of papers presented May 14, 1980, at a Joint US-USSR

Seminar on "Mathematical Methods for Estimating the Fire Resistance of Structural

Assemblies". The seminar was arranged by the US-USSR Panel on Fire Resistance of

Buildings and Structures as a part of their continuing protocol (agreement) to cooperate

in this field. In turn, this agreement is part of a wider US-USSR agreement to cooperate in the field of Housing and Other Construction. The responsible U.S. government

agency for the parent agreement is the Department of Housing and Urban Development

(HUD). Mr. Lawrence P. Simons, Assistant Secretary for Housing and Federal Housing

Commissioner is U.S. Deputy Co-Chairman for the Parent Agreement. Dr. Robert S. Levine

of the National Bureau of Standards, Center for Fire Research and Dr. Igor G. Romanenkov,

Laboratory Chief, Central Research Institute of Building Construction, GOSSTROY, are

the Co-Chairmen of the Fire Panel. This is the first of a series of planned yearly

joint seminars on specific applied fire safety research topics. The Soviet papers were

translated into English by the Soviets, but editorial changes have been made by U.S.

personnel who are expert in the particular subject. It is intended that the original

authors’ meanings have not been changed. In some cases, the U.S. presentations are not

yet available as published papers, or were summaries of work that has already been

published elsewhere. Hence, one of the U.S. presentations is in the form of slides and

commentary. These presentations, in the form given, are intended to represent up-toHarp n.c;. Tprhnningy.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Fires; fire endurance; mathematical models; reinforced concrete; steel structures; wood

structures

.

18. AVAILABILITY Unlimited

1 I For Official Distribution. Do Not Release to NTIS

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; Order From Sup. of Doc., U.S. Government Printing Office, Washington, DC

‘ 20402, SD Stock No. SN003-003-

1^ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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