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purposes and is not to be referenced

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NATIONAL BUREAU OF STANDARDS REPORT

h-85hTHEORETICAL HEAT TRANSFER SCALING RELATIONSHIPS

FOR

FIRE ENDURANCE OF REINFORCED CONCRETE

by

A, F. Robertson and S* M, Genensky

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Aslin, Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

THEORETICAL HEAT TRANSFER SCALING RELATIONSHIPS

FOR

FIRE ENDURANCE OF REINFORCED CONCRETE

by

A, F* Robertson and S« M* Genensky

The work reported was initiated on National Bureau of Standards ' funds and further financial support was furnished by

NBS PROJECT NBS REPORT

1002-12-4872 September 21, 1956 4854

Office of Chief of Engineers

Department of Army

Directorate of Construction,

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on October 9, 201 5.

THEORETICAL HEAT TRANSFER SCALING RELATIONSHIPS

FOR

FIRE ENDURANCE OF REINFORCED CONCRETE

ABSTRACT

An attempt has been made to achieve a better appreciation

of the relative fire enduran.ce performance of various reinforced concrete structures. This has been^done by solution

in an approximate manner of the heat flow in a two dimensional

model* Thermal constants for concrete were selected on the

basis of providing a reasonable fit with limited experimental

data.

The results are expressed in terms of the time required

to reach temperatures of 750°F and 1000°F at various points

within the model. These times are presented in the form of

charts which are suggested for their possible use in

estimating relative fire endurance of two structures which

differ in one material or design parameter but are otherwise

similar*

1* INTRODUCTION

Experimental determination of the fire resistance (1)

performance of a structure is an expensive and time consuming

procedure* It requires the construction of a large, often

full size, specimen representative of a portion of a

building* This structure is mounted either within or as a

portion of the confining walls of a furnace. Determination

of the fire performance of the specimen is defined as the

period of time throughout which the specimen remains

structurally sound and restricts heat transfer to a degree

such that the temperatures of its unexposed portions remain

below certain limits. Such determinations continue to be

necessary because of difficulty in both expressing and.

analyzing the many factors which determine the ability of a

structure to resist attack by fire* Even if it were possible

to express all the complex structural and restraint details

in precise technical fashion, it would still be extremely

difficult to accurately solve the resulting heat flow and

mechanics problem. Even if mechanical problems were neglected

exact solution of the thermal problem would be difficult,

because of the many material components, uncertainty as to

the magnitude of the surface-to-gas-f ilm heat transfer

coefficients, and the complex manner of moisture migration

in the specimen during test*
-2 -

In spite of these difficulties it seemed desirable to

attempt to develop some theoretical basis for predicting the

thermal behavior of specimens during fire tests. Initial

studies of the unsteady heat flow equations with the associated

boundry conditions indicated that a simple scaling relationship should not be expected to exist. It appeared therefore

that a relationship would only be developed as a result of

systematic experimental study of a homologous series of specimens, or as a result of a number of numerical calculations

of heat transfer in simple specimens. Menzel’s data (2)

seems to comprise one of the most carefully planned and

executed experimental studies on the fire performance of

simple slabs but the data are quite limited. Because of

the time and costs involved in planning and performing a series of fire tests such as this, it appeared desirable to

initiate the work by the alternative procedure, mathematical

means

This report therefore presents the results of a series

of numerical computations on transient heat flow through

simple structures.

2. ASSUMPTIONS

In order to facilitate solution of the problem, it

appeared desirable to make the following assumptions:

1. Heat transfer within the solid occurred by

conduction alone.

2. The length of the specimens were great compared

with lateral dimensions and in the case of simple

slabs both length and breadth were considered

large compared with thickness.

3. The fire exposed surface of the specimen experienced a temperature-time relationship as 'defined

by the standard curve for fire resistance tests

described in ASTM method of test E-119« (1)

4-. The solids involved were homogeneous and inert.

5 « The mechanical behavior and thermal properties of.

the structure were independent of the temperature

to which it was, exposed until some limiting temperature had been reached.

6. Heat loss from the unexposed surface took place at

a rate proportional to the temperature difference

between the surface and the ambient at 68°F.
- 3

-

The first and second assumptions are considered justified

for the range of materials and specimen sizes considered.

Assumption three is considered desirable for the sake of

simplicity and lack of detailed information on the magnitude

of the heat transfer coefficients involved. It seems

justified to a certain extent by the fact that furnace

temperature is monitored by means of thermocouples enclosed

within iron pipes.

The fourth assumption, that of a homogeneous and inert

solid, presents a considerable simplification of the true

condition. It seemed necessary however to permit practical

solution of the problem with available computer facilities

and time. This assumption requires neglecting the effects

of absorbed moisture and the thermal discontinuities produced

by reinforcing steel necessary in concrete construction. The

first of these, moisture, is an important factor in determining the fire resistance behavior of many materials. The

need to neglect the inclusion of its behavior in the calculations limits the usefulness of the results and requires

that they serve only as indications of relative behavior

of similar structures. The problem of the effects of reinforcing steel on heat flow within the structure was considered as a desirable but not essential refinement of the

proposed method of analysis.

Assumption five was a necessary one. There is at

present no known way of predicting the effects of high

temperatures on the mechanical integrity of materials such

as concrete and plasters. The effects of temperature on

thermal properties of materials are probably better known.

However neglect of the latter effect seems justified for the

sake of simplicity in view of the other more restrictive

assumptions which were found necessary.

The assumption that heat losses from the unexposed

surface could be considered as Newtonian in character was

considered necessary because of limitations in the storage

system of the computer. It does not appear to limit the

usefulness of the data developed for temperatures near the

exposed surface of the specimens or the bottom of deep beams.

For the case of very shallow beams and simple slabs the

assumption is not well justified and does limit usefulness

of the data on temperatures within the slab when unexposed

surface temperatures become elevated.

These assumptions must therefore be considered as

restricting the applicability of the data which have been

developed. They should be considered in any use to which the

data are put.
J+.

3, STATEMENT OF THE PROBLEM

A "T" beam formed the prototype of the construction

considered for analysis* Figure 1 presents the geometry

of the cross section of this beam* The upper plane surface

represents the unexposed floor* Heat loss was assumed to

take place from this surface according to Newton's law of

cooling** The lower surface of the slab and beam together

with the two sides of the beam were assumed to be exposed

to the furnace fires* These surfaces were made to follow

the time temperature relationship as defined for fire

resistance tests (1)* The two extreme ends of the slab

were assumed to be perfectly insulated*

These conditions may be expressed in mathematical

symbolic form as follows:

9© = oi(

at

V 3X2

32 © +

3

2 ©

) dyc

-&o<y<£c»o, 0<fx<a, t > 0

-b<Ty<fb, a < x + d, t>0

with the following boundry equations and conditions:

9 = f (t)

9©

-ct<y<t-b, x - a

b^ty<.+
o, x = a

-b ^ y^ b, x = d + a

y

= -b, a^.x^a + d)

y

= b, a<; x ^ a + d r t£0

k|^ = -h© -°°<y<oo, x = 0, t^O

0 as y —

3© __ v 9y

9=0

2^ = ° dy

+oo)

fO ^x ial a , t ^ 0

as y —

- oo< y <y oo, 0 ^x <fa , t = 0

-b^y<b, a^x^a + d, t = 0

0 x^ a + dj y

= 0, t^O

( 1 )

( 2 )

(3)

J

( 5

)

7

( 6 )

*The heat transfer coefficient entering this equation was

altered to account for heat loss by radiation*
-5

where

x and y are rectangular coordinates in the plane of

heat flow,

t is the time measured from some initial time t = 0.

© = Q(x,y,t) is the temperature a function of two

rectangular coordinates x and y and also the time t.

k is the thermal conductivity of the material forming

the beam,

p is the density of the material forming the beam,

c is the specific heat of the material forming the beam,

°v. = k/pc is the diffusivity of the material forming the beam,

h is the heat transfer coefficient between the slab

material and air.

f(t) is the temperature of the exposed face of the T-beam

and depends upon the time t.

Although common engineering units were used- for the present

study it is obvious that any consistent set of units may be used.

Since in practice condition (4) is satisfied if the slab

extends laterally about six times its thickness to the right and

left of vertical faces of the beam, that condition may be

relaxed to read:

2© =o y

= + (J?+b), 0 < x<^ a, t^O (

1+')-/

^

Jb6a

and conditions (1), (2), (3) and (5) may be modified by

replacing the symbol "©o" by "A+b" . Let these modified

equations be known as equations (l 1 ), (2'), (3') and (5')

respectively.

Consideration of figure 1 and equations (l')-(5') and (6)

indicates that the problem is symmetric about the ordinate

y

- 0 and thus only the inverted L section to the right of

this line need be considered.

Since a high speed digital machine such as SEAC is most

useful for the elementary operations of addition, subtraction,
-bmulti plication and division the equations (

1 ) - ( 5

' ) and (6)

must be so approximated that only these arithmetic operations

are involved. The methods of finite differences afford

sufficient approximating relationships.

Let

negative

T?

j = 9(iAx, jAy, pAt) where i,

integers

j and p may be nonThen 3© T

P+1 -

1 i,J TP . lv,l (7)

at At

3© oT i+ljJ_ - pP . i.l (8)

9x Ax

3© TP •

. n O' x i. 3+1 - TP . 21.1 (9)

ay

' V/ Ay

mP

tv i+i,.i + tL, . - 2T? , i l ^ i i* (10)

"<)x

2 (4x) 2

3

2 q

rpP

<v/ 1 i , j +1 + T h.i-1 - 2T?,i (11)

9y2 (Ax) 2

Assuming Ay = Ax,

* = 31113 1

approximated by;

T

P+1 = *ZT? 1>J i +l, J

using (6), (7)? (10), and (11), letting

- b A and rearranging terms, (!') may be

+ Tf_n , + TP , + TP 1-L^J 1 } J l ,5-i-7+ ^

T?

1 j J

i l,2,6oo, mp j

_L

, 2 ,6oo, Up 1 p 0,l,2,ooe

where mp Ax = a and np Ay = JL + b

and

l mp ? mp l,oo©, nip 1 j l,2,ooo, Up —1 p 0,1,2, o where mpAx = a + d n

2 Ay - b (12)

It is thus seen that the inverted L-section has been replaced

by a rectangular array of points (see figure 2) and equation
-7

-

(12) may be used to compute

interior points at any time

temperature at all the grid

In difference notation

Tp

? j = X(pAt) i = mp

i = m2

i = mp,

The values of f(t) for t = pAt were tabulated before the

problem was run and were stored on an auxiliary tape unit*

In the experimental setup, the furnace temperature is caused

to follow a curve adopted by the ASTM for use in fire tests (1).

Using equation (8), boundry condition (3 ! ) becomes after

rearranging terms:

thp temperature at any of the

(p + 1) At provided that the

points is known at time p At.

boundry condition (2') becomes:

j = n2 , n2 + 1 , 0 *., n

j — 0,1,2,»»«, n2

mp + 1 , . • •

,

2 j = n2

P~0,l,2

( 13 )

pP 1

(1 + £ 4x)

k

kT

laJ.

k~hAx

j 0,1,2,#®*, np p 1 2 j © © ©

(i^

)

At the extremities of the slab (y = b +JL) heat flow is

assumed normal to the y coordinate as required by boundry

condition (

1+')« Equation 12 then simplifies to that for one

dimensional heat flow:

T Cj

= l(Ti+l,d +

where yt/p = 1-2 X and

i = 1 ,2 , . .

. ,mp-l j = np p

= 0,1,2,

( 15 )

Boundry condition (6) is accounted for by a simple device.

An extra line of points is placed below the abscissa

y

- 0 (i.e. along the line y

= - Ay) and the points in

this column are assigned the values, assumed or computed, for the

points in the column y = Ay. Thus:
-8

T i,l = T i,-1 1 = l,2,...,m2 -l j = 1 P = 1,2,... (17)

and temperatures along the centerline of the beam (j = 0) may

be expressed as follows:

T P+1 =

i +l, J TP . i-l, J + tP^J+l + T

(18)

where i = 1 ,2 , , „

. ,m2 -1 j - 0 p - 1,2,...

Equations (12) -(18) were thus used in the preparation of a

code for the numerical calculations instead of equations

(l)-(6) or (1» )-(5' ) and (6).

In using the digital computer for solution of this type

of problem complete temperature data for all grid points were

recorded at periodic intervals during the computation process.

The computation process was terminated when the temperature at

grid point y

= 0 and x = (mp-^) Ax reached 1000°F. This

point was selected because it was located on a plane through

the vertical centerline of the beam and was equally remote

from both sides and bottom surface of the beam and therefore

represented the position within the lower portion of the beam

at which reinforcement could be placed in such a manner to

secure the maximum amount of "cover" or protection from fire.

The temperature of 1000°F corresponds to that at which the

elastic limit of the steel reinforcement becomes approximately

equal to the working stresses customarily used in such

construction.

4-. PARAMETERS INVESTIGATED

Original plans for this study included the solution of

the heat transfer problem for all combinations of three sets

of thermal and physical properties representative of a wide

range of concretes, five different beam designs as represented

by the ratio of beam depth to slab thickness, (B/S ratio) and eight

different beam sizes as determined by linear scale ratio

of model to prototype. This program would require the solution

and analysis of 120 problems. For purposes of conserving time

only thirty-three of these problems have been solved, see

Table I. These were selected to provide as comprehensive an

investigation as practicable of the full range of variables ' involved
-9-

5. THERMAL PROPERTIES

As mentioned earlier there was considerable uncertainty

as to the values of the thermal properties of materials

during performance of a fire resistance test. The problem

was chiefly one of physical and chemical phase changes together with the migration behavior of gases and vapors.

Water absorbed within the specimen was one important factor

influencing uncertainties of these properties. Because

of this, rather arbitrary methods were necessary in selecting

the thermal properties to be used in this study. Data on dry

concretes presented in reference 4 served as a guide in this

selection. However, conductivity values were increased by

about 25 percent to be more representative of conditions

existing at elevated temperatures. The values selected

were those for densities of about l40, 120, and 3*+ lb/ft3

concrete as shown below for concrete A, B and D.

Assumed Thermal Properties of Concrete

Concrete P k dp

lb/ft3 BTU/hr ft°F f

2 /hr ft2/hr

A 140 0*0 >905 .0323 .0112

B 120 Of*) .580 . 024-2 .0084

C 80 0*^ . 101 * .014-9 .0056

D 34 Sf

r

—

*> .0149 .0056

*As will be explained later this value of k though

not exact may be used without serious error. It

is not however consistent with the value of c*L

tabulated.

In this table oL‘ is the diffusivity based on a specific

heat of 0.2 x BTU/lb°F. The values shown were used in preliminary trial solutions of the problem for simple slabs

similar to those for which a few experimental results were • available. It was found that the calculated fire endurance

periods as indicated by temperature rise within the slab

were considerably shorter than indicated by experiment.

Because of this the values of c/J were reduced by a factor
-10

approximating 3 resulting in the values of cL tabulated above.

These values of oL and k were then used in performing the

calculations. These values are shown plotted in figure 3.

As indicated by equations (1) and (3) of the previous

section, oL is important in controlling the rate with which

heat is distributed within the solid and k functions to

control the rate of heat loss and thus the temperature of

the unexposed surface of the specimen. For the case of

relatively deep beams the temperature of reinforcement is

almost independent of the conditions at the unexposed surface.

The fact that the curve of thermal diffusivity as a function

of density of concrete seems to have a minimum value at a

density of about 60 lb/ft3, permits the use of calculations

for 3k lb fto concrete as indicative of performance of

concrete of 80 lb/ft 3f Thus under these limitations it is

possible to produce data for four concrete densities by the

use of only three sets of calculations.

6. CONVECTION AND RADIATION LOSSES

At the time this problem was prepared for solution by

means of SEAC, limitations in storage capacity prevented

the use of a temperature dependent heat transfer coefficient

(h) at the unheated surface. It was necessary therefore to

consider these losses as Newtonian for any particular problem.

However a method was devised for estimating maximum temperature

rise at the unexposed surface, a value for h was then selected

as representative of the average effective value over that

temperature range. Figure k presents these average values of

surface heat transfer coefficients as a function of unexposed

surface temperature rise at the end of the test. These

coefficients were based on radiation and convection losses

from the surface to the ambient. This ambient was assumed

constant at 68°F and the emissivity of the specimen surface

was assumed as unity. The convection losses were assumed to

follow Jakob's relationship reference k:

1/3

h c = 0.275 ©

7. RESULTS AND DISCUSSION

A typical solution of the T beam problem is presented

in figure 5» This shows temperatures within the specimen

representative of concrete A and having a beam depth to slab

thickness ratio of k.2 . This chart presents the data which

were periodically "read out" of the digital computer. - In
general such data were recorded from 6 to 15 times during the

course of solution of each problem® It obviously was necessary

to select another means of presentation of the data® The

method used consisted in plotting temperature-time data

for each of the encircled points near the lower corner of the

beam. These particular points were considered of primary

importance because they occur in positions frequently occupied

by the tensile reinforcing steel® Fire endurance times were

then read off these charts corresponding to the attainment of

temperatures of 1000°F and 750°F for each of the points

involved. These data together with similar data for beams

of other scale sizes were replotted to form the family of

curves shown in figures 6 and 7° Figures 8 and 9 present data

on the variations of endurance caused by changing the beam

to slab ratio. The data presented here are for concrete A,

scale ratio of one, and critical temperatures 0C of 1000° F and

750°F respectively® Figures 10 and 11 present similar data

for the case of simple slabs as exemplified by the extremities

of the top slab of the beam® Here heat flow is one dimensional

and follows the relationship shown in equation (15) together

with suitable boundry conditions® In deriving these charts a

series of computations were made for each specimen for a

range of unexposed surface temperature rise conditions® In

this manner the limitations involved by the assumption of

Newtonian heat flow at the unexposed surface were largely

avoided®

Original plans called for preparation and presentation

of similar charts for concrete of other thermal properties.

However analysis of the computations indicated that a simpler

method might be used® Figures 12, 13, and Ik together with

figures 6, 7, 10 and 11 present a means for estimating fire

endurance performance of specimens formed from concretes B,

D, and less accurately, those of concrete C on the basis of

results obtained with specimens fabricated with concrete A.

Figure 12 was developed from a plot of calculated results for

beams having scale ratios varying from 1/4- to 2 and beam to

slab ratios varying from 1.8 to 7®4-® From it the relative

performance can be estimated for beams which differ only in

thermal properties® Figures 13 and 14- were developed from

figures 8 and 9 for the scale ratio of 1® Because of the

generalization shown in figure 12, figures 13 and 14- may be

applicable to cone etes of a wide range of thermal properties.
-12 -

8. DISCUSSION

Little use would be served by lengthy speculation on the

effects of the various assumptions which were considered

necessary for solution of the problem* The present discussion

will be confined to a review of the data which have been

developed and brief comments on the manner in which theymight

be used.

The plot of isotherms within the beam shown in figure 5

is useful as an illustration of the temperature distribution

within a given beam at a particular time. The vulnerability

of exposed corners to thermal attack is evident. The selection of such corners for analysis of relative performance

based on heat transfer seems well justified. Consideration

of the triangular area A 0 B for the case of this and other

beams of large beam to slab depth ratio indicates that the

results obtained could be considered applicable to columns

too. Likewise the portions of the floor slab remote from

the beam are representative of conditions in a single slab.

As mentioned previously the data tabulated here are those

for points along the diagonal 0 A of the beam. Figures 6 and

7 present such data in the form of fire endurance as a function

of beam size or scale ratio. The latter has been selected

as a parameter for presentation of the data in preference to

use of an actual linear dimension in an effort to emphasize

the relative rather than absolute character of the fire endurance data plotted. The two curves differ only in the temperature which has been assumed as the criterion of failure at

the point in question* These correspond to 1000°F and

750°F respectively. The former is considered applicable to

steel as used in conventionally reinforced concrete construction.

The latter is presented as possibly of interest in connection

with studies of behavior of prestressed concrete structures.

Both curves are the result of calculations based on concrete

A and a beam to slab depth ratio of b.2, The data are

however, considered applicable to the case of any T beams

with values of this ratio greater than 3*5 provided the beam

stem thickness approximates twice the slab thickness.

Each of these charts consists of a group of curves

starting near the origin and diverging slightly as scale

size increases. These individual lines correspond to the

endurance to be expected when similar structures are tested

each of which has the reinforcement located at a fixed ratio

of the maximum possible depth. An additional family of curves

has been developed in such a way that for any one of these

the product of scale ratio and percentage depth is a constant.

These curves then indicate the change in fire endurance with
-13-

scale ratio when a fixed amount of cover is provided for the

steel. It will be observed that the fire endurance is reduced

as the scale size is decreased resulting in placement of the

reinforcement near the mid plane of the beam or 100% depth

position. This effect is evidently a result of the reduction

in the amount of material backing up the reinforcement and

thus capable of soaking up heat. When increasing the scale

ratio of the beam and maintaining the reinforcement at a

fixed depth little improvement in fire endurance will result

when the size is increased beyond the point at which the

reinforcement is at less than 50 percent of maximum depth.

In fact the data indicate that for very large scale ratios and

resulting small depth ratios the endurance may again show a

very slight decrease. This probably results from reduction

of heat losses on the unexposed surface of the specimen.

Figures 8 and 9 present endurance data for T beams of

a scale ratio one as a function of variation in beam to

slab depth ratio. These curves are for critical temperatures

of 1000°F and 750°F respectively. These again are for concrete A. The charts provide the justification for the previous statement that endurance is unaffected by increasing

beam to slab depth ratio for values greater than about 3.5»

In fact for shallow depth ratios of 10 to 20 percent the

endurance does not seem to change greatly for any beam to

slab depth ratio greater than two. The figures show the

advantages to be gained by use of small beam to slab ratios.

The increased fire endurance is a direct result of the usefulness of the concrete slab in dissipating heat that would

otherwise tend to cause more rapid rise in steel temperatures.

Figure 12 presents a plot of performance of T beams

differing only in thermal properties. The data for the less

dense concretes are plotted against fire endurance

performance of specimens of concrete A but otherwise identical

in construction and failure criterion. Data plotted are based

on the limited calculations performed on the two lower density

materials and cover the following range of variables:

Critical temperature

Beam to slab ratios

Scale ratios

Depth ratios

1000° F and 750°

1.8 to 7

1/4 to 2

0.0 to 1.0 or less as limited

by endurance of 4-80 min.
This chart therefore provides a simple direct means for

estimating relative performance of specimens differing only

in thermal properties.

The good correlation shown in figure 12 between the fire

endurance of specimens having differing thermal properties was

unexpected. Because of this correlation a separate plot was

prepared in which the slope of the plots shown in figure 12,

very closely approximating the endurance ratio tx/tp, was

plotted against the reciprocal of the ratio of the corresponding thermal diffusivities ot ]_/^x , When this was

done for each of the three values of oC 5 it was found that

a straight line relationship existed between the two ratios.

While there was not a strict one to one relationship between

the two parameters, it did indicate close approach to the

theoretical relationship of;

tx/ti = oij_/dx

for the case of simple slabs and step temperature functions

on the exposed surface.

Figures 13 and l4 have been derived from the charts

of figures 8 and 9* They show fire endurance of specimens

of the indicated beam depth to slab thickness ratios relative

to that of similar specimens but having beam depth to slab

thickness ratio of 4.2 or greater. They are applicable to

beams of all thermal properties and a range of beam to slab

depth ratios. These two charts were derived only for the

scale ratio of one. However, analysis of the data available

for concrete A indicates that little error will be introduced

when they are used for the scale ratios varying from 1/4 to 2.

9. SUMMARY

The charts presented in this paper are based on theoretical

calculations of unsteady heat transfer through homogeneous

materials. It is believed that they should provide a useful

means for estimating relative performance of simple slabs and

T-beam specimens having a stem to slab thickness ratio of two

but which differ otherwise in only one design or construction

feature such as:

1. Depth of reinforcement

2„ Scale size of specimen

3. Beam depth to slab thickness ratio

4. Thermal properties of material
-15-

The extent to which the calculated scaling relationships can

be shown to be applicable to actual performance is expected

to depend largely on the applicability of the assumptions to

the problem in hand. Experimental work is planned to obtain

data which will help evaluate the usefulness of the derived

relationships.

Any application of these relationships to actual fire

endurance problems should be made with caution and due

respect for the assumptions on which the solutions presented

have been based.
-16-

10. REFERENCES

1. "Standard Methods of Fire Tests of Building Construction

and Materials", .ASTM Designation E119-55, 19 55, ASTM

Standards Part 4-, American Society for Testing Materials,

Philadelphia, Pennsylvania.

2. "Tests of the Fire Resistance and Thermal Properties of

Solid Concrete Slabs and Their Significance", C. A.

Menzel, ASTM Proc. V 4-3, pp 1099-1153, 194-3.

3. "Lightweight Aggregate Concrete", Housing and Home

Finance Agency, 194-9, U. S. Government Printing Office.

4-. "Heat Transfer", Jakob, John Wiley & Sons, 194-9,
Schedule

of

Beam

Problems

Solved

pho to

X

Q

CD -p

CDpo

rj

OJ9 J"

XX

X

\D9

C\J

X oo

CMO CM

CO9 1 1 X

X

X

pj*o CNd

•HP

0)

PQ

CM9 X

X

X

CD P

CDPO

vDo CM -p doo

CM9 CM

009 i—

X

X

X

p(-o to

X

CD P

CDPO

CM9 PiX

XX

X

X

XXX

X

\D

CM

XX

doo

CMO CM

X

X •

GO• 1 1 X

X

X

X

X

Size

and

Shape Parameters \

Beam

to

Scale\

Slab

Ratio \

RatioV

B/S

pj- Oi—

CMOi—

ph\no 1—1 - -4-4 O.

lT\

CM \no

ph\to- CM ITN
PROTOTYPE STUDIED

0 = b= 4"

1 = 6a = 28”

BEAM TO SLAB DEPTH

RATIO -9±^-

Y

2b

FIG. I

FIG. 2
THERMAL

CONDUCTIVITY,

BTU/ft*F

FIG. 3

SURFACE TRANSFER COEFFICIENT, h BTU/ft

z °Fhr FIG. 4
FIG. 5 SEAC TEMPERATURE RISE DATA

AND ISOTHERMS INDICATED IN DEGREES

FAHRENHEIT B/S RATIO 4.2. DENSE

CONCRETE. TIME. 2 HOURS. FURNACE

TEMPERATURE I850°F OR I782°F RISE

A B
500 S3±nNM‘30NVyn0N3

SCALE

RATIO
SCALE

RATIO
ENDURANCE,

MINUTES

DURANCE,

MINUTES

FIG. 8

400

300

200

100

CONcrete"a

u (c CALE RATIO

^.= 750°F

1

m 2 CO m b

STEEL DEPTH 100%

- v ^

80

60

40

- 20

2 3 4 5

BEAM TO SLAB RATIO, B/S

FIG. 9
500 o o o o

o o o o

^ tO CM —

<

rO

in'

CM

m

O

S3±nNIW‘3DNVanCJN3

scale

ratio
500

jvl

m

(VJ

rO

K) Od S3inNIW‘3DNV8nCIN3

SCALE

RATIO
400

300

200

100

0

•ax er

• 750® F /

o a® e

c' IOOO°F /

<s& y q

x J

it . xy O Jt

/ <0 ./>

®

•

100 200 300 400

ENDURANCE OF CONCRETE A, MINUTES

FIG. 12
ENDURANCE,

%

OF

THAT

FOR

DEEP

BEAM

ENDURANCE,

%

OF

THAT

FOR

DEEP

BEAM

FIG. 13

FIG. 14

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