United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory

Research

Paper

FPL 386

January 1981

Reliability Analysis

of Fire-Exposed

Light-Frame Wood

Floor Assemblies
Abstract

A reliability analysis using secondmoment approximations is conducted

on two types of fire-exposed, unprotected wood floors-conventional

wood joist and floor-truss assemblies.

A methodology is illustrated by which

the probability of structural failure of

a wood floor assembly can be

evaluated. This probability, with the

probabilities of failure of other

system components, can be used in a

systematic analysis that apparently is

a viable approach to realistic analysis

of building fire safety.

The use of reliability analysis to

compare relative fire safety of different floor components is

demonstrated. A procedure for introducing new components into the

market, based on a concept of an

equal safety index calculated for a

component with a proven inservice

record, is discussed.
United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory1

Research

Paper

FPL 386

Reliability Analysis

of Fire-Exposed

Light-Frame Wood

Floor Assemblies

By

F. E. Woeste,2Assistant Professor

and

E. L. Schaffer, Engineer

Introduction

The effectiveness of structural

assemblies-walls, floors, ceiling

roofs- to act as barriers to fire

growth is measured by using the

American Society for Testing and

Materials (ASTM) E-119 (3)3 fire endurance test. This method requires a

typical 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 fireendurance rating. The rating procedure is then used by codes to

regulate the assembly designs

satisfactory for buildings of various

occupancies. The Department of

Housing and Urban Development

(HUD), for example, specifies exterior

walls shall have a 20-minute endurance rating and floors a 10-minute

rating in one- and two-family

residences (41). This is believed to

protect occupants and 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 that of another by using a single-performance measure-a

single test of each assembly to

measure its fire endurance under a

standard fire condition. If safety is

discussed, soon the realization is of a

need for a measure of the risk

associated with an event occurring.

Hence, the present fire-endurance

rating system provides little information on the safety provided by structural assemblies. This lack is

somewhat unfortunate because of the

potential benefits to resident owners,

developers of new assemblies, and

the firefighting and code enforcement

community that could be realized by

having this information.

Approach

In this paper a risk-based

methodology is presented as well as

its application for assessing the fireendurance safety of two unprotected

light-frame assemblies-the conventional joist assembly and a floor-truss

assembly.

In 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

criteria of structural assembly failure

are evaluated (41).

One objective of this research was

to propose and illustrate a methodology by which the probability of

structural failure of wood floor

assemblies could be evaluated.

Another objective, perhaps more

important for the immediate future,

was a method by which new components could 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” if

exposed to the natural elements such

as live and dead load, loads due to

earthquakes, fire, and so on. The investigations reported here address

the fire-performance aspects of floor

assemblies.

1

 Maintained at Madison, Wis. 53705, in

cooperation with the University of Wisconsin.

2

 Agricultural Engineering Department, Virginia

Tech., Blacksburg, Va. 24061.

3

 Italicized numbers in parentheses refer to

literature cited at end of report.

1
(M 145 178) S

Figure 1.–Family of curves for fire endurance of wood floors (20)

a1/2 = 1/2(1 - TÖ

–

s

-

)[1 - (T/2Ö

–

s

-

)12 where

a =

applied load

breaking load (extreme fiber stress of 11,000 pounds per square inch at maximum load.)

= 

design allowable stress

ultimate stress

s = D/B (depth/breadth),

T = t/20Ö

– A

-

,

t = fire endurance (min), and

A = cross-section area (in.2).

Background

One rational approach to accomplishing both objectives is an

analysis using probabilistic engineering methods. Probabilistic engineering is by no means a new discipline.

Ang and others, as of 1972 in a review

of literature, reported on some 355

research papers on structural reliability (4). Textbooks have been published

on the subject, but research publications with application to steel, concrete, and wood engineering number

only about 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 reinforced concrete and steel

design areas, the concept of fire

design engineering as opposed to

strictly fire tests is gaining acceptance (1,2,12,34,40). Wood 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 (13).

Various researchers have been promoting a combination of design and

testing resembling the manner in

which the field of structural engineering emerged.

Sunley (38) reports the variability of

strength of timber assemblies, when

2
Figure 2.–An idealized exposed floor

joist subjected to fire on three sides.

Subfloor protects top side of joist.

Although it is known bottom corners

round, straight boundaries are used

as an approximation. After time,

t, and char rate, C, depth of char

equals C • t on a joist of height, d,

and thickness, h.

(M 148 531)

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 findings of Schaffer (33) and

Knudsen and Schniewind (19) on the

strength of small, clear specimens of

wood at elevated temperature. 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 (20,21). 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 predicting

strength at elevated temperatures

would be most desirable, and a considerable amount of basic input data

exists (34). However, this type of

modeling has not been accomplished;

thus, simplified strength models will

be used in the analysis that follows.

Analysis of Time-toFailure

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, if fire endurance is

associated with load S and time-tofailure of the component is

associated with resistance R, the

stated requirements are met. 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

safety of assemblies.

Fire Duration

The prediction of fire duration, and

more generally fire severity, is itself a

research topic (6,11,23,29,30,39). For

this analysis, it will suffice to use the

approach reviewed by Lie (21). For

ventilation-controlled fires, in which

the duration would be the longest, the

equation that relates fire duration to

available ventilation (e.g., window

area and window height) is given by

t d =

WAF

5.5 AWH1⁄2

(min)

(1)

where W, AF, AW, and H will be treated

as random and defined as

W = fuel load density (kg/m2)

AF = floor area (m2)

AW = window area (m2

)

H = window height (m)

The constant 5.5 has units

kg min -1 m -5/2

Model Building for an

Exposed Floor Joist

The failure during fire exposure is

assumed to be caused by charring of

the three exposed sides of a joist;

this loss of section, coupled with the

strength-reducing influence of

elevated temperature, causes rupture

of the joist. Although burn-through

and elevated temperatures of the

unexposed surface can be additional

failure criteria, they are not considered in this analysis. (These failure

criteria relate directly to the floorsubfloor design that can be analyzed

separately.) Load sharing and composite action are not accounted for

directly in the analysis; however, they

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 (34) reports bottom corners

round when charring occurs; furthermore, the radius of the corners can

be approximated by the depth of char.

To account for this rounding by the

moment of inertia would 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 flexure formula, an

equation can be written to quantify

failure in a fire situation as

M Y (tf

,C)

 = aB

l(tf,C) (2)

where

M

t

 f

Y(tf

,C)

l(tf,C)

a

B

= applied moment caused by

both dead and live loads

(in.-lb)

= time-to-failure (min)

= distance to extreme fiber

being a function of time-tofailure and char rate (in.)

= moment of inertia about

an axis midheight the remaining uncharred section (in.4

)

= an exposed joist performance factor that relates

normal-temperature strength

to high-temperature strength

= joist modulus of rupture at

room temperature (Ib/in.2)

3
This model is similar to that used

by others for large beams under fire

exposure (20,21); it also neglects any

contribution to strength by the flooring itself.

The selection of the model needs

some justification. At room

temperature a = 1, and the model is

exact with the thought that modulus

of rupture is an idealized linear state.

The actual state of stress is

nonlinear, but the model is adequate

for design especially if “depth effect”

is considered. In a fire situation, the

nonlinearities are expected to worsen

because the joist cross section will

not maintain a uniform temperature.

The movement of the neutral axis

caused by a nonuniform modulus of

elasticity (MOE) resulting from a nonuniform temperature profile can be

expected. At the same time, the compressive and the tensile strengths of

the wood fibers are reduced because

of elevated temperatures [Schaffer

(33)]. It may be possible to model the

net effect of the mentioned behavior

if coupled with the normaltemperature nonlinearities of bending,

but this 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

flooring-joist assembly.

By referring to figure 2, it can be

seen that equation (2) can be rewritten as

M (d - C tf) /2

= aB

(b - 2C tf) (d -C tf)

3

 /12

where

(3)

b = initial joist width (in.)

d = initial joist depth (in.)

The remaining variables have been

defined with equation (2). All of the factors except b and d are treated as random variables. By rearranging equation (3), a cubic equation in time, t

f

results

6M = bd2

 - 2Cd (d + b) t

f

aB + C2

 (b + 4d) tf

2

 - 2C3 tf

3 (4)

Whereas cubic equations can be

readily solved by hand calculation or

computer, the derivation of the

statistics of the variable tf would be

cumbersome. This led the authors to

investigate the error introduced in tf

by dropping the cubic term and simply solving the quadratic equation for

tf. Fortunately, the errors introduced

are negligible, ranging from 1.58 percent for a nominal 2 by 6 (38 by 140

mm) to 0.35 percent for a nominal 2

by 12 (38 by 286 mm). It must be emphasized that this approximation was

shown to be adequate for only

nominal 2 by 6 (38 by 140 mm),

nominal 2 by 8 (38 by 184 mm),

nominal 2 by 10 (38 by 235 mm), and

nominal 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 (20) were used to study the

applicability of the model. A question

immediately arises regarding the appropriate value of the modulus of rupture, B. Because the test for the

measurement of B and the fire test

are destructive, it is impossible to

have knowledge of both properties for

a single piece. At this impasse the

alternative was to investigate the

mean and the variability of the product of the two variables a and B.

This was done in the following manner:

The 42 Douglas-fir floor assemblies

tested by Lawson were divided into

Table 1 .–Apparent high-temperature modulus of rupture, aB, for each grade group

Grade

Sample

size

N

Apparent

rupture

strength

a B

1

Coefficient

of variation

(COV)

Wa B

800

1,200-1,600

1,600

Clear

Combined

20

15

2

5

42

Lb/in.2

kPa

8,715

8,729

8,150

1,126 7,764

1.126 8,384

1,264

1,266

1,182

0.238

.320

.442

.199

.271

four grade groups–800, 1,200-1,600,

1,600, and clear-as shown in table 1.

The data shown in the table indicate

little difference in the apparent hightemperature modulus of rupture for

the four different grades. More surprising are the calculated coefficients

of variation (COV). From data collected by Hoyle (15) in a world search

of lumber data, a value of 0.45 for the

COV of B is typical for Construction

grade joist. With all grades combined,

the Ω was only 0.271, which shows

E-119 exposure (3) apparently does

have a variance-reducing influence on

B just as Sunley (38) suggested.

To scrutinize the model further, an

attempt was made to predict the fire

endurance of two Douglas-fir

assemblies tested by Son (35). 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 of similar quality as those of

the Lawson report, i.e., aB = 1,165

pounds per square inch (Ib/in.2) (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

the live load levels used by Lawson

(20) and Son. In Son’s tests, 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

ranged from only 200 to 917 lb/in.2

(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 minutes,

respectively, for a nominal 2 by 8 joist

floor as Son tested. Hence, as expected, the lower the load level, the

greater is the time-to-failure; thus,

more thermal degradation is allowed

to occur. Beams more heavily loaded

will have shorter times-to-failure and

less thermal degradation. The right

side of equation (3), representing the

strength change as influenced by

temperature rise, cannot account for

heat accumulation degrading the

cross-sectional strength unless some

measure of time dependence is introduced. By making several data

plots, the following model was

developed to include this kind of time

dependence in which the variables

have been previously defined.

1

 Constant char rate, C, of 0.025 in./min (0.0635 mm/min) was used in calculation; theretore, Ω values

reported are an upper bound because they include variability of char rate.

4
M(d - Ctf

) /2

=

(b - 2Ctf) (d - Ctf

)

3

 /12

B

(5)

The [(b + 2d) γt

f

]/(bd) term may be

viewed as a time-dependent geometric factor to account for heat flowing into the cross section, bd, through

the perimeter, b + 2d. Although

visual inspection of the data plots

suggests using this model, some

variation of the model may be more

suitable. Again, as before, the cubic

term tf 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 (20).

The second model, a single

parameter of table 2, is seen as the

preferred model because the residual

standard deviation is only slightly

greater than that of the twoparameter 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 the

Table 2.– Least-squares nonlinear regression analysis on five different variations of a

time-to-structural-failure model1

Residual

Model2 Parameter standard estimates deviation

Min

1

 In each variation, 42 data points were used to estimate either one or two parameters, depending

on form of the model. Parameters must carry necessary units to make equations dimensionally

homogeneous.

2

 K = (b + 2 * d)/(b * d).

selected model is a reliable predictor

if it is recognized that the residual

standard deviation of 2.57 minutes includes the variability of aB as shown

in table 1.

Solving equation (5) for t

f by omitting

the cubic, results in

tf =

2Cd(d + b) + 6MKγ /B (6)

2C2 (b + 4d)

where K = (b + 2d)/bd, which is an

explicit expression for the time-tofailure. Time-to-failure, tf

, is compatible to the previously defied fire

duration td, which is the load

variable.

Results predicted by equation (5)

were compared with results of four

floor fire-endurance tests obtained by

the NFPA (National Forest Products

Association) (24,25,26,27). The actual

times-to-failure to carry load versus

those predicted are given in table 3;

the predicted times are consistently

and significantly less than the timesto-failure of the whole floor assembly.

This deviation can be explained by

the model parameter derivation based

on the results of Lawson (20). Lawson

conducted fire-endurance tests of

paired Douglas-fir joists with essentially noncontinuous floor sheathing.

The NFPA tests are of assemblies of

many joists and a more or less continuous floor sheathing. These

assemblies result in load-sharing between joists and increased loadcarrying capacity of the sheathing

that is not similarly reflected in the

Table 3.–Predicted and actual times-to-failure for NFPA unprotected floor fire-endurance tests (24,25,26,27) 1

Sample Nominal

size

Applied

joist

moment

B

2

- / Predicted Assembly– Joist3

C

tf

observed observed

tf tf

In. In.-lb Lb/in.2

In./min Min Min

No. 2 Douglas-fir “S-dry” w/vinyl tile,

19/32 in. plywood 2 x 8 19,054 4,308 0.0245 5.42 10.2

No. 2 Douglas-fir “S-dry” w/nylon

carpet, 19/32 in. plywood 2 x 8 19,054 4,308 0.0245 5.42 12.86

No. 2 MG southern pine “S-dry”

w/vinyl tile, 23/32 in. plywood 2 x 10 31,017 5,730 0.03 7.01 13.34

No. 2 MG southern pine “S-dry”

w/nylon carpet, 23/32 in. plywood 2 x 10 31,017 5,730 0.03 7.01 12.06

5.0

Min

11.5

9.0

12.06

1

 NFPA, National Forest Products Association; italicized numbers in parentheses refer to Literature Cited at end of paper.

2

 Douglas-fir rupture strength, from (76); southern pine rupture strength, from (7).

3

 Failure time for first joists to fail; not assembly failure time.
paired joist tests by Lawson. If the

time is noted when the first joist ruptures in the NFPA tests, the difference between the model-predicted

results and those actually observed

are closer in two of the tests in which

such failure was observed. This,

again, illustrates the need for a

degrade parameter, g, or other

parameters that include both the effect of load-sharing and that of floor

sheathing. As a result, these types of

replicate experiments are planned.

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.

(7)

Here fb and f

t denote applied

stresses; Fb and Ft , allowable design

stresses in bending and tension,

respectively.

As in previous reliability work at

the Forest Products Laboratory (37),

this interaction equation can be

modified to indicate failure (with

some reservations discussed in the

report). However, in a fire-exposure

case, one parameter needs to be

estimated; thus some slight inaccuracy in the neighborhood of the

combined stresses associated with a

floor truss will be corrected. The

failure equation for fire exposure

would read

(8)

where the right side of the equation

has a form similar to that for the exposed floor joist. Function g accounts

for the thermal degrade of the section, and its arguments are later

defined. B is the modulus of rupture

from which Fb was derived; and T, the

ultimate tensile strength property

from which the design value Ft was

derived.

Because four-sided fire exposure of

the lower chord in a floor truss is

critical, expansion of the interaction

formula for this case is given in the

following equation. (It is assumed

that the mode of failure is rupture of

the lower chord. The upper chord has

Table 4.—Fire endurance of 2 by 4’s under constant tensile load1

Douglas-fir, coast Southern pine

Test Failure

time Test

Min

P = 6,100 lb, 90 pct

1

2

Ft

11.20

7.67

3

11.25

9.35

4

5 8.74

Mean

Standard deviation

P = 4,960 lb, 73 pct Ft

6

7

8

9

Mean

Standard deviation

9.64

1.57

9.24

9.96

13.36

13.92

11.62

2.36

Failure

time

Min

P = 6,100 lb, 83 pct Ft

1

2

3

4

5

Mean

Standard deviation

10.00

11.61

12.85

12.34

11.80

11.72

1.08

1

 Select Structural allowable stresses for Douglas-fir and southern pine from 1977 National Design

Specifications.

only three-side exposure, and the

webs are only stressed to approximately one-half the level of the lower

chord.)

P

(b - 2C tf)(d - 2C tf)

T

M(d - 2C tf) /2

(b - 2C tf)(d - 2C tf)

3

+ /12

B

1

=

1 + γK tf

(9)

where

K = 2(b + d)/(bd)

γ = thermal degrade factor

P = axial tensile force due to dead

plus live load

b = width

d = depth

C = char rate

t

f = time-to-failure

M = maximum bending moment

caused by dead plus live load

B = modulus of rupture

T = 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, a cubic equation

results:

(10)

This equation will later be solved for

tf 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, no data are available for

this purpose. Schaffer (31), however,

has conducted fire-exposure tests of

constantly tension-loaded Select

Structural coast Douglas-fir and

southern pine 2 by 4 members. The

time-to-failure was recorded (table 4).

6
For pure tension, the failure model

reduces to

P(1 + γ K t

f

)

= T(b - 2C t

f

)(d - 2C t

f

) (11)

This expression is easy to deal with,

because it is only a quadratic in tf

.

After solving it for tf, the results are

t =

2T C(b + d) + PγK

(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 γ for the available tension fire test data,

An estimate of T is available for inland Douglas-fir Select Structural

[Hoyle (14)]. Based on a sample size

of 30, the average tensile strength

value was 5,020 lb/in.2 (34,600 kPa)

with a COV of 0.388. This value can

be compared to what might be

calculated using normal distribution

theory and the allowable tensile

stress. From the 1977 National

Design Specification (28) the

allowable tension value for Select

Structural Douglas-fir/larch is

1,200 lb/in2 Using the calculated COV

of 0.388, the mean value is determined to be 6,811 lb/in.2

 by the following formula:

(13)

This value is significantly larger than

the actual mean of 5,020 that illustrates the non-normal nature of tension data. This shows, therefore, the

mean value obtained from lumber

tests should be used whenever possible for a grade and a species in question.

The nine Douglas-fir fire and tension test results can then be used toestimate γ for Douglas-fir. By using T

= 5,020 and C = 0.0245 inch per

minute, γ was estimated to be 0.113

with a residual standard deviation of

the time-to-failure of 1.829 minutes.

Figure 3.—Residual cross-sectional area divided by initial area (pct) of coast

Douglas-fir (top) and southern pine (bottom) nominal 2 by 4 (1- 5/8 by 3- 5/8)

members with duration of exposure to ASTM E- 119 fire conditions. Top

curves result from using the mean species charring rate, CM, of 0.025 and

of 0.030 in./min for large sections of coast Douglas-fir and southern pine,

respectively. Bottom curves from using an effective charring rate, Cf of 0.038

and of 0.45 in./min for coast Douglas-fir and southern pine, respectively.

Straight lines are linear regression fits to actual data in which lines were

forced through 100 percent at time equal to zero. (Model Y = bX was fitted

to one residual area/initial area.) Equation of the curves is given by

100(b -2Ct)(d -2Ct)/(bd).

(M 148 528)

(M 148 529)
A similar estimate of the fireexposure reduction factor γ can be

made for the southern pine test

results of table 4. The southern pine

lumber tested was almost clear of

defects and ungraded. The lumber appeared to be of a quality at least as

good as Select Structural. There are

data [Hoyle and Maloney (17)] that

show high-quality southern pine

(2400f machine stress rated (MSR)) is

stronger in tension than is highquality Douglas-fir (2400f MSR)—the

ratio in strengths being 1.24. To arrive

at a mean value for the tensile

strength T of Select Structural

southern pine, the 5,020 Ib/in.2 value

for Douglas-fir was multiplied by 1.24

to yield 6,233 Iblin.2 for southern pine.

Using a mean char rate for southern

pine of 0.03 inch per minute (32) in a

nonlinear regression analysis of the

fire test results, the yield is a γ of

0.0839 inch per minute with a residual

standard deviation of the time-tofailure of 1.077 minutes.

Both species values of the reduction factor, γ, (0.113 for Douglas-fir,

0.0839 for southern pine) are substantially lower than that of 0.17 for the

floor-joist assembly described in the

preceding section. The reasons for

this are not clear. It is known,

however, that smaller sections (2 by

4, compared to 2 by 8 or 2 by 10’s) are

likely to undergo more rapid reduction

in cross section than are larger. The

difference for 2 by 4 sections of

Douglas-fir and southern pine are

shown in figure 3 (top and bottom,

respectively) as a function of fireexposure time.

The developed model and the parameters may be used to estimate the

structural failure of a given floor-truss

assembly. This was done for the truss

shown in figure 4; 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).

Because tensile strength data for

Dense No. 1 were not available, data

for No. 1 KD southern pine were

taken from Doyle and Markwardt (8).

The published value of 5,706 Ib/in.2

was adjusted to 5,646 Ib/in.2 to reflect

current standards by Forest Products

Laboratory personnel.

The truss was-then analyzed with a

Purdue Plane Structures Analyzer (36),

and as normally done, the center

panel of the lower chord was most

highly stressed with an axial force of

4,209 pounds and a bending moment

8

Figure 4.—Floor-truss design subjected to test conditions of ASTM E- 119. Upper chord was loaded with tanks simulating a uniform load of 55.1 Ib/ft2

that resulted in a combined live and dead load of 60 Ib/ft2.

(M 148 527)

Figure 5.—Relationship of applied load to time-to-failure for floor-truss

assembly of figure 4.

(M 148 530)

of 140 inch-pounds. A summary of the

input parameters can now be given

for the fire-tested floor truss.

γ = 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 pounds per square inch

T = 5,646 pounds per square inch

Substitution into and solution of the

failure-model equation (10) results in

a time-to-failure estimate of

11.2 minutes; the actual failure time

resulting in a fire endurance test was

estimated at 10.2 minutes (10). This

test continued to be conducted under

reduced load until 14.6 minutes, when

fire exposure was terminated without

collapse occurring. The predicted

time-to-failure falls within this 10- to
15- minute range. This result is most

promising for future use of the model.

An interesting examination is how

time-to-failure is altered for the same

truss by reducing the applied load.

The failure-model equation predicts

times-to-failure as a function of applied load as shown in figure 5. For

reduction in load to 50 percent of full

design, it is seen 5 minutes is added

to the predicted time under full

design load. If there were no load on

the floor assembly except the dead

weight (4.9 Ib/ft2) of the assembly

itself, a failure time of 21.1 minutes

could result. Hence, failure times

greater than this are theoretically impossible for this truss design.

Estimating FloorAssembly Safety

In the preceding paragraphs the

authors developed models for predicting the time-to-failure of two floorassembly types, and have given an

accepted model to predict severity of

fire exposure. They are given in units

of time.

Using the standard reliability notation, the assembly time-to-failure and

the time duration of fire exposure, t

f

and td, are denoted as follows:

R = t

f (14)

and

S = td (15)

With R and S defined, the fire

reliability (Rel) of an exposed joist or

floor truss (as a part of a floor), given

a fully developed fire, is given by

Rel = Pr(R > S) (16)

which reads the probability (Pr) that

the fire resistance is greater than that

of the fire load. It is convenient for

calculations R and S to be independent. In other words, the fire load and

associated parameters cannot be correlated to the members’ resistance,

char rate, and other factors. Conversely, structural load, member

resistance, and char rate cannot be

influenced by fire load. Unfortunately,

it is known that char rate is correlated with the variables of equation

(1) that have been discussed by

Schaffer (34). However, in practice,

the joist or component will be exposed to a standard fire condition

such as ASTM E-119 in which char

rate will not be influenced by

variables W, F, A, and H. The real

problem lies in the fire severity of

E-119 not being representative of the

fire 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 (42),

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 = In(R/S) (19)

there results

Rel = Pr(J > 0) = 1 - FJ (0) (20)

where FJ the cumulative density

function of the variable J.

Using first-order, second-moment

approximations, the mean and the

variance are given by

Standardizing J,

(22)

(23)

and

(24)

(30)

If the distribution of A is similar in all

applications, then the variable

(25)

is a consistent measure of fire safety.

b is normally called the safety index.

The next step is to estimate the

means and the variances of the

resistance and the load, μR and μS, σ

2

R

and σ2S A first-order approximation of

the mean, E, of a function, Z, where

Z = h(X1, X2, . . . Xn)

is given by

(26)

E(Z) = h[E(X1),E(X)2), . . . , E(Zn)] (27)

Performing this operation on equation

(6) for the conventional joist-floor

assembly and replacing the expected

values, E, of the component variables

by their statistical estimates denoted

by a superscript bar, the result is

(28)

(29)

where

(29a)

(29b)

(29c)

Again, γ 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

The first-order, second-moment approximation of the variance of Z

defined by equation (26) is given by

provided the Xi’s are uncorrelated.

The partials are evaluated at their

respective mean values.

(31)

9
In the expression for the load

variable S, the floor area AF and window area AW could be correlated

because it would be expected AW be

some way related to the perimeter

that involves the same variables as

the floor area. However, because data

are not available to substantiate this

correlation, a zero correlation will be

assumed. In light-frame construction,

the other pairs of variables lack an

obvious cause for correlation.

Without showing the computations

(32)

(33)

where the Ω’s are the respective COV.

In the equation for resistance, or

time-to-failure, of the floor joist, no

obvious confounding correlations appear. 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 resultsin

(37)

(34)

Again, the partial derivatives are

evaluated at the mean values of the

component variables.

For 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

(35)

where ρ

ij is the correlation coefficient

between variables Xi and Xj. By

assuming zero correlation for the

other variables, as for the floor joist,

application of equation (35) to equation (29) yields

(36)

The variance of the char rate, σc

2

,

has been estimated by reanalyzing

data previously developed (32). The

mean charring rate, C, and estimated

variance, σc

2

, under ASTM E-119 fire

exposure for coast Douglas-fir and

southern pine are

The variance in the strengthreduction factor, γ, is unavailable.

Substantial information is available

on the variation of the applied load

that defines the variation of M and P.

Eventually, all the variances denoted

by σ

2 will be replaced by statistical

estimates from the data available or

from the results of present research.

Finally, the COV of R can be obtained, and is given by

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 γ of

equation (23), the probability of structural failure of an exposed floor

assembly, given the occurrence of a

fully developed fire, can be

calculated. In general, the distribution

of γ is not known, and some assumption about it must be made. Common

practice is to assume γ follows a normal distribution. In this analysis, the

assumption of normality will be used

realizing 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, Pf, is given

by

(38)

where φ is the cumulative area under

the standard normal curve.

Code Calibration

In recent years the use of engineering components has increased

dramatically. Often these components

are fabricated with 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 it

does not account for the variability of

the component response, although it

may account for the average

response of a component to fire.

Quite simply, the present system of

fire rating allows using two different

components with an equal “fire

rating” of, say, 1 hour, but at the

same time the components have unequal safety levels.

The situation of two components

with unequal safety levels can best

be illustrated by a hypothetical example using equation (25). The example

involves calculating the safety index

for two different components, the only difference being the variability of

the time-to-failure or resistance of the

component.

For the example, assume that the

following data in table 5 apply to two

component types A and B. Each column is identical except for the column indicating the COV of fire

resistance. On applying equation (25)

to each type, the resulting safety indices are bA = 0.98 and bB = 1.24,

which indicates an unequal level of

safety. Converting these b’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 no difference exists between a 16.3 percent and a 10.7 percent failure rate, but on closer examination, calculations show 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 hypothetical situation illustrates just one possibility of how

the safety index could be used to obtain equal safety and give a “fair

10
Table 5.—Level of safety of components, A and B, with same average fire-endurance time

(60 min) but different variabilities (COV of 0.25 versus 0.5)

Component

A

B

μR

WR

μ

S

WS

Min Min

60

60

0.5 30 0.5

.25 30 .5

1

 Each component is exposed to an identically distributed fire load, as illustrated by last two columns; result is an unequal level of safety, as calculated by the safety index equation (25).

shake” to new materials and new

components.

In the future it may be possible to

identify at what point fire load, hence

fire duration, have different expected

values and different levels of variability. In this type of situation, a lower or

a higher fire-rated component may be

needed.

Summary and

Conclusions

1. Time-to-structural-failure prediction models, based on the residual

load-carrying capacity of fire-exposed

floor elements, are given for two unprotected light-frame wood floor

assemblies. A reduction in strength

factor, α, was calculated from the

limited fire-exposure test data according to two equations:

Joist floor

Floor truss

The thermal reduction factor is further a function of fireexposure time

and the geometry of small cross sections:

Joist floor

where

Floor truss

where for

Douglas-fir

γ = 0.113 in./min

Southern pine

γ = 0.0839 in./min

2. A comparison of predicted

times-to-failure versus those actually

observed for four unprotected joistfloor assemblies results in predicted

times consistently less than those

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.

—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

those of observed times-to-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 because of excessive deflection of the floor

assembly, without evidence of collapse.

4. The time-to-failure model for a

loaded and a fire-exposed floor-truss

assembly was employed to examine

the influence of various floor loads on

time-to-failure. At full-design load,

failure was predicted to be 11.2

minutes; but with only dead load, the

period was extended to 21.1 minutes.

This indicates the sensitivity of this

type of fire-exposed assembly to the

load applied during test. In this

analysis it was critically assumed

that failure of the lower chord, rather

than connectors, would result in collapse of the assembly.

5. The procedure to use for

calculating safety of unprotected

light-frame floor assemblies is given.

An example is provided to

demonstrate how variability in

assemblies can affect the comparative safety of assemblies.

6. The predictive capabilities for

both of the proposed assembly

models require further fire-exposure

experiments for independent validation and parameter refinement.

11
Nomenclature

Floor area

Window or opening area

Beam breadth

Modulus of rupture at room

temperature

Char rate

Beam depth

Allowable stress

Window or opening height

Moment of inertia

Ratio of perimeter to area of

cross section.

Applied moment

MOE Modulus of elasticity

Axial load

Member or structural resistance

Applied “load”

Ultimate tensile stress

Fire duration

Fuel load density

Distance from beam centroid to

outer fiber

Ratio of high-temperature to

normal-temperature strength

Safety index

Thermal degrade factor

Statistical mean

Correlation coefficient

Standard deviation

Accumulative density function

for standard normal distribution

Coefficient of variation

Subscripts

Bending

Duration

Failure

Resistance

Load

Tension

12
Literature Cited

1. Abrams, M. S.

1978. Behavior of inorganic materials in fire. ASTM E-5 Symposium 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). Annual Book of ASTM Standards, Part 18. ASTM,

Philadelphia, Pa. p. 739-757.

4. Ang, A. H.-S., and others.

1972. Structural safety-A literature review. J. Structural Division, Proc.

ASCE 98 (ST4):845-884.

5. Burros, R. H.

1975. Probability of failure of building from fire. J. Structural Division,

Proc. ASCE 101 (ST9):1947-1960.

6. Coward, S.K.D.

1975. A simulation method for estimating the distribution of fire severities

in office rooms. Build. Res. Establ. Current Pap. 31/75. Fire Res. Stn.,

Borehamwood Hertfordshire, England, 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. Forest Products

I

Laboratory, 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. Forest Products

Laboratory, Madison, Wis.

9. Ellingwood, B., and J. R. Shaver.

1977. Reliability of RC beams subjected to fire. J. Structural Division,

Proc. ASCE 103 (ST5):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. Gross, D.

1977. Measurements of fire loads and calculations of fire severity. Wood

and Fiber 9(1):72-85.

12. Gustaferro, A. H., and D. P. Jenny.

1978. Alternative to fire testing-Design of concrete structures for fire

endurance. Southern Building. June-July:24-28.

13. Gypsum Association.

1978. Fire resistance design manual. Gypsum Assoc., Evanston, III.

14. Hoyle, R. J.

1976. Digest of tension parallel-to-grain strength of Inland North Douglasfir. Res. Rep. 76/57-22. Coll. of Eng. Res. Div., Washington State Univ.,

Pullman, Wash.

15. Hoyle, R. J.

1977. Review of world literature on characteristic distribution of properties of 2-inch softwood dimension lumber. Res. Rep. 77157-11. Coll. of Eng.

Res. Div., Washington State Univ., Pullman, Wash.

16. Hoyle, R. J., and T. M. Maloney.

1976. Bending strength tests of visually graded 2-inch dimension lumber

from western Canada. Res. Rep. No. 76157-36. Coll. of Eng., Res. Div.,

Washington State Univ., Pullman, Wash.
17. Hoyle, R. J., and T. M. Maloney.

1976. Tension strength and bending elastic modulus of truss joist

machine stress rated 2 by 4 lumber. Res. Rep. 76/57-44. Coil. of Eng. Res.

Div., Washington State Univ., Pullman, Wash.

18. Kameda, H., and T. Koike.

1975. Reliability theory of deteriorating structures. J. Structural Division,

Proc. ASCE 101 (ST1):295-310.

19. Knudsen, R. M., and A. P. Schniewind.

1975. Performance of structural wood members exposed to fire. Forest

Products J. 25(2):23-32.

20. Lawson, D. I.

1952. The fire endurance of timber beams and floors. Structural Engineer,

30(3):27-33.

21. Lie, T. T.

1972. Fire and buildings. Applied Science Publishers, Ltd. Ripple Road,

Barking, Essex, England. 276 p.

22. Lie, T. T.

1972. Optimum fire resistance of structures. J. Structural Division, Proc.

ASCE 98(ST1):215-232.

23. Lie, T. T.

1974. Characteristic temperature curves for various fire severities. Fire

Technology 10(4):315-326.

24. National Forest Products Association.

1974. ASTM E-119 fire endurance test—2- by 10-inch wood joist floor

assembly. Design FC 209. Nat. For. Prod. Assoc., Washington, D.C.

25. National Forest Products Association.

1974. ASTM E-119 fire endurance test—2- by 10-inch wood joist floor

assembly. Design FC 212. Nat. For. Prod. Assoc., Washington, D.C.

26. 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.

27. National Forest Products Association.

1974. ASTM E-119 fire endurance test—2- by 8-inch wood joist floor

assembly. Design FC 216. Nat. For. Prod. Assoc., Washington, D.C.

28. National Forest Products Association.

1977. Design values for wood construction-a supplement to the 1977

edition of national design specification for wood construction.

Washington, D.C. 20036. 20 p.

29. Raes, H.

1977. The influence of a building’s construction and fire load on the intensity and duration of a fire. Fire Prevention Sci. and Technol. 16:4-16.

30. Robertson, A. F., and D. Gross.

1970. Fire load, fire severity, and fire endurance. Special Technical

Publication 464. American Society for Testing and Materials, Philadelphia,

Pa. p. 329.

31. Schaffer, E. L.

1961. The effects of fire on selected structural timber joints. M.S. thesis,

Univ. of Wis., Madison, Wis.

32. Schaffer, E. L.

1966. Charring rate of selected woods-transverse to grain. U.S. Dep.

Agric., For. Serv. Res. Pap. FPL 69. Forest Products Laboratory, Madison,

Wis.

33. Schaffer, E. L.

1973. Effect of pyrolytic temperatures on the longitudinal strength of dry

Douglas-fir. ASTM J. Test Evaluation 1(4):319-329.

14
34. Schaffer, E. L.

1977. State of structural timber fire endurance. Wood and Fiber

9(2):145-170.

35. Son, B. C.

1973. Fire endurance tests on unprotected wood-floor constructions for

single-family residences. Center for Build. Tech. Inst. for Applied Technol.,

National Bureau of Standards, NBSIR 734263. Washington, D.C.

36. Suddarth, S. K.

1972. A computerized wood engineering system: Purdue plane structures

analyzer. U.S. Dep. Agric., For. Serv. Res. Pap. FPL 168. Forest Products

Laboratory, Madison, Wis.

37. Suddarth, S. K., F. Woeste, and W. Galligan.

1978. Differential reliability: Probabilistic engineering applied to wood

members in bending/tension. U.S. Dep. Agric., For. Serv. Res. Pap. FPL

302. Forest Products Laboratory, Madison, Wis.

38. Sunley, J. G.

Design concepts for fire-resisting constructions. Behavior of wood products in fire. Pergamon Press, Oxford. p. 95-101.

39. Thomas, P. H., and C. R. Theobald.

1977. Part 2: The burning rates and durations of fires. Fire Prevention Sci.

and Technol. 17:15-16.

40. U.S. Department of Commerce/National Bureau of Standards.

1978. Building research translation-the behavior of concrete structures

in fire-a method for prediction by calculation. NBS Tech. Note

710-7110. U.S. Government Printing Office, Washington, D.C.

41. U.S. Department of Housing and Urban Development.

1973. HUD minimum property standards-one- and two-family dwellings.

U.S. HUD, Washington, D.C.

42. Zahn, J. J.

1977. Reliability-based design procedures for wood structures. Forest Products J. 27(3):21-28.
2.0-17-1/80

U.S. GOVERNMENT PRINTING OFFICE: 1980-750-027/31