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NBS TECHNICAL NOTE 1152

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

Mechanical Performance of

Built-Up Roofing Membranes

-QC

100

U5753

no, 1152

1981
NATIONAL BUREAU OF STANDARDS

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Mechanical Performance of Built-Up Roofing Membranes

M6s

f o # htCPrn- icclI n6 t c MATIONAL BUREAU

OF 8TANDAED3

UBBART

James M. Pommersheim JAN 8 198?

Q6-L clcc. _ i Bucknell University

Lewisburg, PA

Rnhprt d A/Tnthp.v

Oc/oo

no. nsr^

im

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

*< °><

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued December 1981
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a mi

National Bureau of Standards Technical Note 1 1 52

Nat. Bur. Stand. (U.S.), Tech. Note 1152, 60 pages (Dec. 1981) CODEN:NBTNAE
TABLE OF CONTENTS

Page

LIST OF TABLES iv

LIST OF FIGURES v

ABSTRACT 1

1. INTRODUCTION 1

2. BACKGROUND 8

3. THEORETICAL DEVELOPMENT 15

3.1 Determination of Stress-Strain Relations for Felt

and Bitumen 15

3.2 Mathematical Models for the Mechanical Performance of

Roofing Membranes 24

4. RESULTS AND DISCUSSION 33

5. SUMMARY AND CONCLUSIONS 50

6. REFERENCES 52

7. NOMENCLATURE 53

xn
LIST OF TABLES

Table No.

Page

Analysis Using Equation 15 of Stress-Strain Data

of Koike for Roofing Felt 19

Analysis Using Equation 18 of Stress-Strain Data

of Koike for Roofing Bitumen 23

Comparison of Test Results for Joint Movements with

Theory 43

Statistical Results for Methods used to Predict

Substrate Joint Movements needed for Felt

Rupture 47

IV
LIST OF FIGURES

Page

Figure No.

1 Schematic Representation of Roofing Membrane

Before and After Substrate Movement 9

2 Stress-Strain Diagram for Roofing Felt 11

3 Stress-Strain Diagram for Roofing Bitumen 12

4 Log (a/e) vs Z [equation 15] 18

5 Log (t/y) vs Y [equation 18] 22

6 Cross-Section of Roofing Membrane and Substrate. ... 25

Representation of Possible Solution Types

[equations 45, 46, 47] 40

MECHANICAL PERFORMANCE OF BUILT-UP ROOFING MEMBRANES

James M. Pommersheim—

Robert G. Mathey

ABSTRACT

For built-up roofing membranes with either linear or non-linear

stress-strain behavior, fully bonded to an underlying deck or substrate

which undergoes displacement, it is the equality of the complementary

strain energy of the fabric or felt layer, with the strain energy of the

bonding adhesive or bitumen layer, which governs both the conditions

under which membrane integrity is lost and the mode of failure by either

membrane splitting or adhesive disbonding. The testing criteria developed are applied to a sample case.

Key words: adhesive; bitumen; bonding; built-up roofing membrane;

complementary strain energy; felt; non-linear; roofing membrane; splitting; strain energy; stress; substrate.

1 . INTRODUCTION

A roofing membrane should be waterproof , and it should be able to

sustain building movements. Unanticipated movements can lead to failure

of the membrane. The mechanical function of a built-up roofing membrane

— Professor of Chemical Engineering, Bucknell University, on assignment to NBS under provisions of the Intergovernmental Personnel Act.
is to prevent failure by effectively transferring shear stresses from

one adjacent layer of the membrane to the next without loss in membrane

integrity.

In this report, theory is developed and tested for roofing membrane

strains and stresses which are induced by substrate movement. For

simplicity, the roofing membrane is considered to consist of a single

ply of roofing felt which is fully bonded with a bitumen (asphalt or

coal tar pitch) layer to an underlying substrate. In practice, most

bituminous built-up roofing membranes consist of asphalt saturated

organic-felts and air-blown asphalt. Other materials used in roofing

membranes are asphalt impregnated glass felts, asphalt saturated asbestos

felts, coal tar pitch saturated organic felts, and coal tar pitch. The

substrates over which membranes may be applied include insulation

materials such as fiberboard, perlite board, fiberglass or foamed plastic,

or, if no insulation is used, a roof decking material such as "concrete,

wood, or metal. In practice built-up roofing membranes consist of more

than one ply. However, the mechanical performance of a single-ply

membrane is not unlike that of a multi-ply or built-up roofing membrane

[1].

When a roof is constructed using insulation, the separate pieces or

sections (nominally, 2 or 3 x 4 ft (0.6 or 0.9 x 1.2 m)) are generally

laid down in a staggered fashion upon the roof deck. The roofing

membrane is then applied in shingle fashion over the insalation. Two to

four-ply membranes consisting of alternating layers of fully bonded

bitumen (asphalt or coal tar pitch) and felt are generally used.
Splitting of the membrane is one of the principal modes of failure

of a built-up roof [1,2]. Loss of membrane integrity and penetration of

water is the inevitable result. When splitting occurs it usually

happens over the insulation joints where the membrane is subjected to

the greatest amount of stress. In practice, staggering of the insulation boards helps to minimize the potential for lengthy splits in the

roof membrane, as does placing the membrane with its strongest direction

(the machine direction) perpendicular to the continuous joints in the

insulation. Cullen and Boone [3] claimed that their own data, as well

as that of Koike [2], supported the validity of Cullen' s [4] original

suggestions regarding the alteration of some conventional application

techniques to reduce the incidence of splitting failures resulting from

mechanically and thermally induced forces. Cullen' s recommendations

were:

(a) to place the insulation boards with the long dimension parallel

to the short dimension of the roof,

(b) to orient the roofing felt parallel to the long dimension of

the roof, and

(c) to use an adhesive (bitumen) of optimum strength to secure the

membrane to the substrate.

Although such measures help to minimize membrane failure, they

cannot prevent it when sufficiently large joint movements occur. Often

these movements arise because of sudden temperature changes either

inside or outside the building. Wetting and drying of some membranes

will also induce stress in these membranes, as will wetting and drying

of the underlying insulation or other substrate. It is the movement of

3
the roofing membrane relative to the substrate which is important in

determining if membrane failure may occur. Koike [2] found that membrane

failure occurred in single-ply membranes for joint spacings or widths

between 1.0 and 1.7 mm. Maximum movements of roof joints as high as 4

to 5 mm have been observed by Marijs and Bonafont [5] with a maximum

movement rate of 0.3 mm/h. For safety they recommend that, over the

lifetime of the membrane, a single layer of felt should not be fully

bonded across a substrate joint if movement at the joint is likely to

exceed 0.2 mm, and caution that overstressing or fatigue failure are

likely even under summer conditions whenever joint movements exceed

1 mm.

A useful calculation to perform is to estimate the temperature

difference, AT, between the substrate and the membrane needed to cause

a joint movement of 0.2 mm. For example, consider a four-ply organic

felt and asphalt roofing membrane fully bonded to an underlying substrate consisting of foam polyurethane rigid board. From 0° to 30°F

(-18° to -1°C) Mathey and Cullen [6] report a coefficient of linear

thermal expansion (a) for the membrane of 2.3 x 10 mm/mm/ C C in the

cross-machine or transverse direction and 5 x 10 mm/mm/ °C in the

machine or longitudinal direction, while in the same temperature range

the insulation has an a value of 2.2 x 10 mm/mm/°C. For a typical

roofing insulation board width, these values predict potential joint

movements of 0.2 mm when the temperature change is about 1°C. Since

temperature changes higher than this are often found in roofing insulation materials [7], this calculation points out the importance of proper

roofing design in order to minimize temperature changes in roofing systems
Mathey and Cullen [6] have proposed the thermal shock factor (TSF)

which is intended to provide a quantitative measure of the thermal-shock

resistance of bituminous built-up roofing membranes

TSF = -p- (1) Ma y

P is the tensile strength (ultimate or breaking load-force per unit

width) of the membrane, M is the elastic modulus and a is the coefficient

of linear thermal expansion.— In their definition of the thermal shock

factor, Mathey and Cullen [6] use a coefficient of linear thermal

expansion based on values measured in the temperature range to -30°F

(-18 to -34°C).

The larger the magnitude of the thermal shock factor of a membrane

the better able it is to withstand a combination of thermally and

mechanically induced stresses. The concept of the thermal shock factor

has proved useful as a means to assess the behavior of a wide variety of

different types of roofing membranes [3]. Nevertheless, equation 1 is

valid only for a membrane with linearly elastic (Hookian) stress-strain

behavior.

In a built-up roofing membrane, the bottom ply of felt carries most

of the load, and, in general, determines the resistance of the entire

membrane to substrate movements [5]. The amount of the total load

carried by the base felt will depend on the amount of bitumen flow

which occurs initially during the loading period. If temperature and

— A table of nomenclature is provided in Section 7 of this report.
strain are high then the amount of bitumen flow will be high, and the

base felt will carry a larger proportion of the load [5] . In addition,

the tensile strength of the membrane at higher temperatures will be

lower

.

As mentioned earlier, a single-ply system, such as that considered

in this report, has a mechanical performance not unsimilar to a multiply system. For example, for linearly elastic materials, Bonafont [1]

calculated that a two-ply membrane (consisting of a bottom bitumen

layer, felt layer, second bitumen layer, and top felt) had only 12% more

joint movement capability than a single-ply membrane (consisting of a

bottom bitumen layer and a single top felt). If the bottom ply ruptures,

the top ply has an ultimate movement capability which is then 41%

greater. However, all the stress must now be carried by the top ply,

and if it is weaker than the bottom ply, Bonafont [1] predicts that its

failure will be entrained in the failure of the base ply. For- safety he

recommends that the stronger ply be placed on top, so that even if the

base ply fails, the overall integrity of the roofing system will be

maintained. With four plies the ultimate movement capability is

increased by about a factor of two over that of a single ply system. As

a rough rule, ultimate movement capacities for an N ply membrane are

v/N times those of a single-ply system [1] .— This rule may be used to

estimate movement capacities for non-linear membranes.

— More extensive theory and discussion concerning the movement of

multi-ply membranes caused by substrate joint movements have recently

been presented in a paper by Koike e_t al. [9] . Treatment is restricted

to membranes constructed of roofing materials having linearly elastic

stress-strain characteristics.
Bonafont considers fatigue failure to be a more likely mode of

mechanical failure than splitting due to overstressing [1,8]. Slip

between fibers in the felt is irreversible so that cycling and repeated

movements can cause failure sooner. Thus, failure by fatigue occurs at

much lower values of joint spacing than does failure by overstressing.

Bonafont found that- if the felt is able to survive the first fatigue

cycle it will survive a comparatively large number of identical cycles,

since the greatest stress occurs in the first cycle. In his generalized

development, Bonafont considered that the bitumen was a linear viscoelastic material. However, he did not consider the stiffening of the

bitumen which often occurs as the membrane ages.

For proper roofing design, it is obvious that one must consider

simultaneously many factors. Materials selection, heat transfer characteristics of the roofing system and substructure, cross-sections and

thicknesses of the insulation and inter-ply bitumen layers, and the

slope, weight, and kind of surface loading are factors which can all

have a direct or indirect effect on joint movements and potential roof

failure by membrane splitting or bitumen disbonding. In some instances

a trade-off may be necessary between good insulatory properties which

are favored by the use of thicker insulation, and small joint movements

which are favored by the use of thinner insulation. Assuming that the

insulation is well adhered to the roof deck, less restraint is offered

to the change in length of the top of the thicker insulations by the

deck when the roofing surface is subjected to temperature changes.
2 . BACKGROUND

There are two basic related theories dealing with mechanical

performance of roofing membranes subjected to stresses and strains

induced by the movement of the underlying substructure; one developed

by Koike [2], and one developed by Bonafont and co-workers [1,5,8].

Koike [2] derived a relation between joint movement in a substrate

which is fully bonded to an elastic adhesive layer (generally a bitumen)

and the resultant stresses and strains induced in the adhesive and

membrane. He predicted the joint movement (or gap spacing) g needed to

rupture a fully bonded fabric or felt layer having a tensile strength P

and an elastic modulus M. The relation obtained was

g - 2 P [^]

1/2

(2)

h is the thickness of the adhesive (bitumen) film between the

substrate and the fabric and G is the shear modulus of the adhesive

(bitumen) . It is conventional in roofing system studies [1-6] to

define both P and M based on the lateral felt width. G is based on the

adhesive shear area.

Figure 1 depicts the physical situation. The system is shown both

before and after substrate movement. Each substrate element moves

horizontally a distance g away from the gap center line. Thus, the

total gap width or spacing is 2g.

The most critical assumption involved in deriving Koike's relation

is that both the fabric and the adhesive are linearly elastic or Hookian

materials. In this case, a direct proportionality exists between stress

8
?•'• . '.-'?

-o TT ^

-^ If - .M' .1 - '.!,-.' ", ...-.'•"

• •• - •

JOINT IN SUBSTRATE- J\-©-

BEFORE SUBSTRATE MOVEMENT

x- ADHESIVE

I

h [B ITUMEN) LAYER

SUBSTRATE

ELEMENTS

ADHESIVE

> (BITUMEN) LAYER

JOINT MOVEMENT

AFTER SUBSTRATE MOVEMENT

FIGURE 1. SCHEMATIC REPRESENTATION OF ROOFING MEMBRANE BEFORE

AND AFTER SUBSTRATE MOVEMENT (1)
and strain, and the moduli M and G are presumed to be strictly constant.

It is well known, however, that the moduli of roofing materials are not

constant [1,10]. Marijs and Bonafont [5], for example, have found that

a model which allows for both viscous and elastic properties, the socalled linear visco-elastic or Boltzmann model, is adequate for characterizing roofing systems. In fact, linear stress-strain diagrams for

roofing materials seem to be the exception rather than the rule. Only

at low temperatures (generally below -18°C (0°F)) do these materials

show linearly elastic behavior, and then over only narrow ranges of

stress application which are generally lower than those that cause

splitting failure in actual built-up roofing systems.

The stress-strain data obtained by Koike [2] are distinctly nonlinear. Figures 2 and 3 show stress-strain data collected by Koike [2]

at three different temperatures. Figure 2 is for asphalt-saturated

roofing felt (coated type) , while figure 3 is for bonding bitumen (a

blown-petroleum type) . Both series of figures are definitely nonlinear. In general, for the bitumen the curves bend upward, while for

the felt the curves bend downward. As shown in figure 2, the nonlinearity is more pronounced for the felt at higher temperatures, while

for the bitumen (figure 3) it is more pronounced at lower temperatures.

Figures 2 and 3 also show that at higher temperature the ultimate or

breaking strength of both bitumen and felt is lower. These facts and

the nature of the curves shown in figure 2 are also representative of

those found for multi-ply built-up roofing membranes [5,9].

Thus, the non-linear response of the materials of construction of a

built-up roof to imposed loads may be an important consideration in

10
c/»

C/>

0.01 0.02

FELT STRAIN e, mm/mm

0.03

FIGURE 2. STRESS-STRAIN DIAGRAM FOR ROOFING FELT [2]

ii
CM

</>

u.ou 1

1

'

1

' 1

1 1

'

1

'

0.70 —

0.60 —

0.50 — -2°C

10°C -

0.40 — /

~~

0.30

20°C

0.20 —

0.10 —

/n 1 i 1 , 1I 1 I

i . 12 3 4 5

BITUMEN STRAIN 7, mm/mm

6*10 3

FIGURE 3. STRESS-STRAIN DIAGRAM FOR ROOFING BITUMEN(2)

12
their design and selection. Although Koike [2] realized that roofing

materials are not linearly-elastic, he did not account in his analysis

for the effects of the non-linearities. Koike obtained the value of M,

the elastic modulus of the felt used in equation 1, by taking the slope

of the stress-strain curve (figure 2) at the origin. Values of G, the

shear modulus of the bitumen, he took equal to the slope of a line drawn

from the origin to the ultimate or breaking stress of the bitumen. At

the lowest temperature used (-3°C), where it was expected that the base

felt would rupture, bitumen disbonding from the underlying mortar

occurred instead. Koike attributed this to the fact that the bitumen

shear tests were conducted with a stronger bond between the bitumen and

the mortar than were the actual rupture tests. Thus, the bitumen separated from the mortar surface with a smaller shearing force than

expected.

Koike [2] claimed that a rheo logical analysis of the problem of

predicting gap widths from mechanical properties of roofing materials

was impossible for non-linear or visco-elastic materials. Bonafont [1]

was able to extend Koike's work to fabric or felts with non-linear and

visco-elastic stress-strain behavior. Bonafont considered the rheological behavior of the bitumen to be that of an ideal Boltzmann-type

(linear visco-elastic) body. In addition, he presented a more general

theory for linearly elastic fabrics and adhesives, one which allows for

the finite length of the substrate elements. The equation he developed

for the joint movement is

13
IJLl (3) M[aW coth(aW)-l] K J

with

2 G

a = n\ in (4)

W is the half width of one substrate element. The bitumen is

presumed to be fully bonded across two adjacent parallel substrate

elements of total width 2W. Bonafont stated that equation 3 reduces to

Koike's relation (equation 2) under the condition that (aW) » 1.

In roofing practice this condition is the rule rather than the

exception. Because of the fact that substrate element widths are much

greater than bitumen film thicknesses, the dimensionless group (aW) will

be much larger than unity. Under this condition equation 3 becomes

2 W P . - g M(aW - 1)

K

'

Equation 5 further reduces to Koike's relation, equation 2, since

under these conditions the quantity (aW - 1) can be approximated as aW.

Koike's relation satisfactorily treats all two-ply membrane systems

composed of materials with linear stress-strain behavior.

For non-linear membranes, Marj is and Bonafont [5] derived the

following expression for the joint movement at which rupture should

occur

, r 2hU 1 l/2 f ,. g = 2 [-Q-1 (6)

14
U is the complementary strain energy of the felt, defined as the

integral under its strain-stress diagram [11], that is,

P

U = 1/2 / £ d a (7)

where £ is the strain and O is the stress in the felt.

For a Hookian or linearly elastic felt, for which the stress is

proportional to the strain, equation 7 reduces to

p

2 u =nr < 8 >

Substitution of equation 8 into equation 6 again yields Koike's relation,

equation 2.

3. THEORETICAL DEVELOPMENT

3.1 Determination of Stress-Strain Relations for Felt and Bitumen

For materials with Hookian or linearly elastic stress-strain

characteristics, the standard rate laws for the stresses in the felt and

bitumen film layers, are, respectively

a = M £ (9)

x = G y (10)

x is the stress in the bitumen (adhesive) and y is the bitumen

strain. By convention, T is based on unit bitumen area, a is based on

unit felt width.

15
Before theory can be developed and applied to calculate joint

movements for non-linear materials, it is necessary to find accurate

constitutive relations for the stress-strain behavior of these materials.

The moduli M and G are constant only for a purely elastic membrane

and film. But as was pointed out in the previous section, membranes are

usually far from elastic, so that in general

a = M(e) e (11)

and

T = G(Y) Y (12)

must be used to represent the shear stresses, in the membrane and film,

respectively. The moduli in these equations are then the ratio of

ordinate to abscissa of their respective stress-strain diagrams. They

vary continuously with the strain level.

If experimental data are available, constitutive relations can be

derived for the stresses and moduli. For the data of Koike (refer to

figure 2), as well as data from NBS roofing membrane tests [6], the

simplest formulas which were found to adequately characterize the

stress-strain behavior of roofing felts had the form

a = M £ e ae

(13)

o

16
Since M = a/e it also follows that

—otf~ M = M e (14)

o

v

M and a are empirical constants. M is the zero-strain modulus

o o for the roofing felt. Graphically, it is the slope of the stress-strain

(a,e) diagram (figure 2) at the origin. The constant a is a measure of

the deviation of the felt from linearly elastic behavior. Higher

absolute values of a give larger deviations. For linearly-elastic

fabrics or felts, a = 0, and equations 13 and 14 reduce to the Hooke's

law form, equation 9.

The constants M and a can be obtained from the linearizing plot of

the logarithm of the ratio of stress to strain vs the strain. Thus,

from equation 13

log

(f) =logK

o -^£ (15)

The intercept of a plot of log (a/e) vs e is equal to log M while the

slope is given by (-a/ 2. 303). Figure 4 shows the fit of equation 15 to

data taken from figure 2 at temperatures of -3°C, 13°C, and 20°C. For

each curve in figure 2 approximately 18 points were extracted and used

in the data analysis. The lines shown were obtained by the method of

least squares. Confidence bands (+ ts ) are drawn about each line. — r Table 1 summarizes the least squares values obtained for M and a

o at the three temperatures together with the correlation coefficients and

error band (+ ts ) of the linearizing plots, and the 95% confidence

17
2000

b| *4J

0.01 0.02 0.03

FELT STRAIN e, mm/mm

FIGURE 4. LOG oj e VERSUS e (EQUATION 15)

18
Table 1-

Analysis Using Equation 15 of Stress-Strain Data

of Koike [2] for Roofing Felt

Symbol Definition and Units

Temperature

-3°C 13°C 20°C

correlation coefficient 0.998 0.990 0.982

2/ +ts — error band on best line

(equation 15)

0.011 0.045 0.058

a felt constant mm/mm 0.325 0.336 0.332

2/ +ts — confidence band on a, mm/mm — a

0.004 0.011 0.015

M felt zero-stress modulus, N/mm 15.41 7.82 5.82

2/

+ts — confidence band on M , N/mm — M o

o 0.04 0.08 0.08

3/

m

predicted maximum stress, N/mm 17.46 8.58 6.44

4/ m

predicted strain at G , mm/mm 3.08 m2.98 3.01

1/ Based on 18 equal-spaced data points extracted from figure 2,

2/ Student's t values, 95% confidence bands.

3/ Calculated from 0.368 M /a.

4/ Calculated as 1/a.

19
bands on the values of a and M , (+ ts ) and (+ ts,, ), respectively.

o — a — M

o Also included is the predicted maximum stress and the strain at this

stress. The high values of the correlation coefficient r and the

narrowness of the confidence bands indicate that equation 15 fits the

data reasonably well.

The function G(y) can be evaluated by a similar procedure. The

curves presented in figure 3 suggest that relations of the form

t = G ye 3Y

(16)

o

and, since G = x/y

G = G e 3Y

(17)

o

are adequate for characterizing data for the bitumen shear stress and

bitumen shear moduli, respectively.

G and 8 are empirical constants, evaluable from experimental data,

G is the zero-stress modulus for the roofing bitumen. Graphically, it

is the slope of the stress-strain (x,y) diagram (figure 3) at the

origin. The parameter 2 is a measure of the deviation of the bitumen

from linearly elastic behavior. For linearly elastic bitumens, 8=0,

and equations 16 and 17 reduce to the Hooke's law form, equation lo.

The linearized form of equation 16 is

log (T/y) = log G

q - 27fo3

Y

< 18 )

20
The intercept of a plot of log (t/y) vs y is equal to log G while the

slope is given by (-3/2.303). Figure 5 shows the fit of equation 18 to

data taken from figure 3, at temperatures of 20°C, 10°C, and -2°C. The

lines shown were obtained by the method of least squares. Confidence

bands (+ ts ) are drawn about each line. Data points shown in figure 5

were extracted from figure 3. For each curve approximately 17 equallyspaced points were used. Table 2 summarizes the least squares values

obtained for G and 3 at the three temperatures together with the

correlation coefficients r of the linearizing plots, the 95% confidence

bands on 3 and G , ('+ ts ) and (+ ts^ ), respectively, and the error

o — p — G

o band on the best line (+ ts ) . The fairly high values of r and the

narrowness of the confidence bands on the slope and intercept indicate

good agreement between the data and equation 18.

Note that both equations 13 and 16 contain two empirical constants

(a, M ; and 3, G , respectively). As stated, both equations become

constants at low stress levels, reducing to the Hooke's Law forms given

by equations 9 and 10. This is in accord with both theoretical predictions and with most experimental data, where an initial linear portion

of the stress-strain diagram is observed. In addition, equation 13

permits the existence of a maximum stress at the point where £ = 1/a.

The corresponding maximum stress is [0.368 M /a]. An analogous maximum

bitumen shear stress of 0.368 G /3 at y = 1/3 exists when 3 is positive.

o Koike's data [2] yields negative 3 values for -2°C and 10°C and a

positive 3 at 20° C. In his calculations, rather than allowing for any

nonlinearities, he chose G to be the slope of a line from the origin of

the T - y plot to the point where bitumen film failure occurs. For

21
CM

0.02 0.04

BITUMEN STRAIN, mm/mm

0.06

FIGURE 5. LOG Tj 7 VERSUS 7 (EQUATION 18)

22
Table 2-

Analysis Using Equation 18 of Stress-Strain Data

of Koike [2] for Roofing Bitumen

Symbol Definition and Units

Temperature

-2°C 10°C 20°C

correlation coefficient 0.974 0.963 0.961

— r error band on best

line (equation 18)

0.054 0.035 0.018

bitumen constant,

mm/mm

-1.786 -0.882 0.493

2/ +tsR confidence band on 3,

mm/mm

0.098 0.056 0.033

bitumen zero-stress

modulus, N/mm

0.571 0.550 0.632

2/ —+ts^-' confidence band on G , G M/ 2 °

o N/mm^

0.008 0.004 0.003

1/ Based on 17 equal-spaced data points extracted from figure 3.

2/ Student's t values, 95% confidence bands.

23
bitumens having negative values of 3 this predicts joint movements for

rupture which are less than the actual movement.

3.2 Mathematical Models for the Mechanical Performance of Roofing

Membranes

In this section, theory is developed for roofing membrane strains

and stresses induced by substrate movement. The prediction of these

stresses and strains is important since they can lead to rupture or

splitting of the membrane with resultant penetration of water, and

possible subsequent slippage of the membrane. Movement of substrate

joints can also cause shear failure by disbonding of bitumen from the

underlying substrate. This begins at the location of the joint (see

figure 6) . The potential is then present for the bitumen to peel from

the substrate surface. Peeling is promoted by increased air and water

vapor pressure which develops at the disbonded joint between the membrane

and substrate due to temperature and humidity changes within the roofing

system. Peeling can lead eventually to the formation of ridge blisters

which have been observed to develop over insulation joints.

Consider the roofing section depicted in figure 6. A bitumen

coated felt is presumed to be fully bonded to an underlying substrate.

The substrate is generally some type of roofing insulation or roof deck

material. External forces such as those induced by temperature changes

or shrinkage can cause fissures in the bitumen bonding layer and gaps to

develop or widen between adjoining insulation boards or substrate

elements. Strains are then transmitted to the overlying membrane.

These in turn give rise to membrane stresses.

24
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25
Expressions are derived here for the joint movements which will

cause either splitting of the felt layer or the disbonding of the

bitumen layer from the substrate. The derivation is a generalization of

the original linear theory of Koike [2] and is also related to the nonlinear theory presented by Marijs and Bonafont [5]. However, it differs

fundamentally from the theory of these latter authors in that it applies

to both bitumens and felts having non-linear stress-strain behavior

rather than to just felts alone having non-linear behavior.

The theory is based on mechanical force balances made on the

membrane elements (bitumen and roofing felt) covering one idealized

substrate element (insulation board), located somewhere in the middle of

the roof (refer to figure 6) . The prediction of the movement of the

substrate itself is not considered in this report but is taken as known

a priori. In general, such movements are caused by temperature differences within the insulation in a direction perpendicular to the roof

deck (along the centerline in figure 6) . Such temperature differences

can be quite high, especially for today's well- insulated roofs. Calculations based on the heat-transfer model for roofing systems developed

by Rossiter and Mathey [7] showed that temperature drops as high as 72°F

(40° C) could occur across an insulation element on a hot, sunny day.

For mechanically unconstrained elements, this could cause potential

substrate movements as high as 0.4 in (10 mm).

The basic assumptions involved in the derivation of expressions for

the joint movements include:

26
1. strains and strain rates are small and equal at every adjacent

position in the bitumen and felt,

2. the membrane has low Poisson's ratio so that dominant stresses

are only in the direction of the substrate movement,

3. joint movements are produced by the translation of one substrate element relative to another, rather than by the deformation of the elements themselves,

4. joint movement is small compared to both the length of the

substrate element and to its width.

The first assumption comes from the fact that there is no relative

movement between the felt and the bitumen, while the second and third,

which are also true, imply that the problem can be treated mathematically using a single dimension. The last assumption is true in practice

and also finds use in simplifying the mathematics.

The derivation which follows also presumes that the properties of

the materials do not change with time. In practice this assumption may

not be true, especially for new roofs where rapid bitumen aging may

occur.

The shear stress in the bitumen film is given by

t = da/dx (19)

where, in general

x is the coordinate in the direction of the membrane

displacement, measured from the center of the

fissure or gap (figure 6),

t(x) is the shear stress in the bitumen or adhesive film,

a(x) is the stress in the felt or fabric.

27
This equation is readily obtained from an x-direction force balance

on a thin element of the film of volume AV = h B Ax, where Ax is the

length of the element, h is the thickness of the film between the substrate and the felt, and B is the width of the membrane (into the

diagram in figure 6)

.

Membrane strain £ and substrate strain t, are defined, respectively,

by

E - & < 20>

and

< - s < 21 >

u(x) is the membrane displacement and v(x) is the substrate displacement. Both u and v are measured relative to the center of the gap

(x = 0).

The movement of the film is defined in terms of the separate

movements of the membrane and substrate by y(x), the shear deformation

of the bitumen film

y(x) = u(x) - v <*> (22) h

Combining equations 12 and 19

ff = G(Y) y (23)

28
Differentiating equation 23, using equation 22

or

dx

h 4 = f[£ -^

<25)

dx

Equation 25 follows from the strain definitions given by equations 20

and 21, and from equation 12.

Equation 25 is a non-linear equation of the second order. As such

it needs two boundary conditions. These are given by

da _ G(0) g. , . ( dx} x=0 " h 2 (26)

and

€>x-W - C27)

where G(0) is the value of G at x = 0. The first condition, equation

26, follows since there is no membrane displacement g at the center of

the joint. The second condition, equation 27, is the result of the fact

that x = W is a plane of system symmetry, and thus a line of zero

bitumen shear stress.

29
The complementary strain energy U is defined by [11]

f

s

*

U does not have the obvious physical meaning that the regular strain

energy does, but is a useful mathematical construct for solving stressstrain problems involving non-linear materials or structures situations,

Note that equation 7 is the integral of equation 28.

Introducing equation 28 into equation 25 and rearranging

2

d a .. dT dU dx . . h ^2-a^-d7 C (29)

Substituting equation 19 into equation 29, using the fact that

gives

d

2 a dT dx

TT = ta

= T cw

(30)

dx

, dT dT dU dT ...

hT d^

= dT a^ - d7 C (31)

30
Dividing through by di/dy and using the fact that the displacement £ is

gives

c = m (32)

dy _ du g , hT

do" -

da" +2W~ (33)

Multiplying by da gives an exact equation which on integration becomes

Y

h / TdY = U+|r7 a + C 1 (34) 2W 1

Equation 34 is expressed in terms of the strain energy for the bitumen

film, Jt dy, the complementary strain energy for the felt U and the

membrane stress a. It must be true at each and every position x within

the structure. In particular, at positions far from the joint the

strain energies and stress levels will be small. This implies that the

constant C may be taken equal to zero so that equation 34 becomes

h / Tdy = U+f?? a (35)

When the gap spacing g is much less than the membrane width W, the last

term in equation 35 can be dropped. As discussed previously this is a

reasonable assumption for all practical situations of interest. Thus

31
Y(0) = fr- o(0)

h / t dy = / £ da (36)

Equation 36 is a major result. It shows the equality of the

product of the bitumen film thickness times the strain energy for

bitumen shear with the complementary strain energy for felt or fabric

shear. Since the greatest stresses occur above the joints (at x = 0)

this equation is of greatest applicability there. This is shown in

equation 36 by the use of the integration limits y(0) and a(0) which

represent the bitumen strain and membrane stress, respectively, at the

joint. The corresponding maximum bitumen shear stress at the joint is

t(0) - ^2LA (37)

Equation 36 can be used whenever separate stress-strain data for

the bitumen and the felt are available. The gap spacing g that will

cause rupture of the felt is the one which would first give rise to the

ultimate stress. Therefore, equation 36 can be solved graphically, or

analytically if equations are available, to find the value of g which

will first cause stress o(0) corresponding to P. Since equation 36 can

be used to predict the incidence of membrane failure, it can be considered to be a general criterion for membrane failure.

In some cases the bitumen itself will not be able to transfer

stress t(0) corresponding to P. An ultimate shear strength T is reached

which destroys the ability of the bitumen film to transfer shear

stresses. T will generally be the lesser of (a) the shear required to

32
break the bond at the interfaces between dissimilar materials, or (b)

the shear required to rupture the adhesive itself.

In both cases the integrity of the fabric will be maintained at the

expense of the adhesive. Bonafont [1] maintains that an artificial slip

layer will be formed which will be able to subsequently absorb large

substrate movements. This may be true, but it also appears to be a

potential source of formation of ridge blisters in built-up roofs. This

occurs because in the evening warm moisture laden air from the sides or

below enters the peeled joint. Later the moisture condenses out as the

roof cools. During the daytime when the roof is warm this moisture

vaporizes and the pressure within the peeled joint rises, causing more

peeling. Over time this day-to-day cycling action causes a ridge

blister to develop as the peeling joint propagates the weakest direction,

which is down the joint. Eventually the blister may enlarge to the

point where the overlying fabric is stretched to its yield point and

splitting occurs.

4. RESULTS AND DISCUSSION

In this section the theory derived for the mechanical performance

of roofing membranes is applied to the constitutive stress-strain

relations developed in the previous section. Results are discussed in

terms of the mechanical conditions causing membrane failure and the mode

of failure. Methods are presented for analyzing data to determine the

incidence and mode of membrane failure. Comparisons are made with the

results of Koike [2] for linearly elastic roofing materials.

33
It is easy to understand whether the strength of the felt (P) or

the strength of the bitumen (t) controls failure. The stronger material

is whichever one has the greater strain energy as determined by the

integrals in equation 36.

Quantitative predictions of the incidence and mode of failure can

be obtained. Substituting equations 13 and 16 into equation 36, and

integrating

- [1- (I&+1) expU] -

3

2 ^ v 2h *' TMv

2hM ? 7 -| [e

a£

(a e + ae + 1) -1] (38) a

Equation 38 is the criterion for membrane failure. It can be considered

to be the working form of equation 36. The term before the equals sign

in the equation represents the bitumen strain energy-film thickness

product, while the term after the equals sign of equation 38 represents

the complementary strain energy for the felt (U) . Of course the

validity of the equation is conditional on the applicability of the

constitutive stress-strain laws (equations 13 and 16) to the data being

considered.

All terms in equation 38 are evaluated at x = 0, where the greatest

stresses act. If felt tensile strength P controls failure, equation 38

can be solved for the critical spacing g needed for rupture. The

strain £ is replaced by the ultimate strain E corresponding to P on

34
the stress-strain diagram. If the bitumen film yield strength T

controls failure the g calculated from

g = 2h Y (39)

will be less than that calculated from equation 38. y is the bitumen

shear strain corresponding to the stress T. Use of this value of g will

allow calculation of the felt strain from equation 38 and the felt

stress from equation 21. This calculation is of importance for determining the residual stress still present in the felt. At higher stress

levels felts having positive values of a can show two different strains

(e) at the same stress. This is because they may not rupture at the

point of greatest stress.

For materials with linear stress-strain diagrams the solution is

similar. Substituting equations 9 and 10 into equation 36, integrating

and solving for the gap spacing g needed for rupture, yields Koike's

relation, equation 2, which is valid for linearly elastic materials

only. As mentioned in the previous section, Bonafont [1] developed a

more general theory for membranes with linear stress-strain behavior, a

theory which reduces to Koike's relation for small h/W. Since h/W is

small in all roofing applications of interest, Koike's relation is valid

as long as the membrane materials are linearly elastic.

In some cases the felt may be linearly-elastic while the bitumen

remains non-linear. For this case a = 0, M = M, and equation 38

o

n reduces to (in the limit)

35
P

2 = 2hMG

Q

[1 - (|| + 1) exp |g] (40)

It is also possible that the felt has non-linear stress-strain

behavior, but the bitumen is a linearly elastic material. For this case

3 = 0, G = G, and equation 38 reduces to (in the limit)

„ 8hM -ae „ _ g = 1 [e p

(a e + a e + 1) -1] (41)

G a

2 P P

For given mechanical properties of the bitumen and felt and a

specified bitumen film thickness, equations 40 and 41 may be used to

calculate the gap spacing g which governs failure of the roofing membrane by rupture of the felt. If the g calculated from equation 39 is

less than that calculated from equation 40 or 41, the bitumen film

strength controls failure.

It is possible to design a single-ply system such that the bitumen

and felt layers have equal strength according to the criterion for

failure developed in this report. In such a case the bitumen and felt

would be predicted to fail at the same joint substrate movement g*. The

design thickness of bitumen to use such that bitumen and felt have equal

strengths is h*.

For linearly elastic membranes equation 2 can be rearranged to give

g* - 4 (42>

M T

36
or, equivalently

T T Y

The last form (equation 43) is more useful when only ultimate

stress data are available. Equation 43 was first presented in a slightly

different form by Koike [2] who used it to predict modes of roofing

failure. For values of bitumen film thickness less than h*, failure is

predicted to occur by bitumen disbonding, while for values greater than

h* it is predicted to occur by membrane rupture. Based on Koike's data

and methods of obtaining moduli, equation 43 predicts that failure

should occur by membrane rupture. Koike found that this was true for

all six samples at the higher temperatures (13 and 18°C) , but at the

lowest temperature (-3°C) he found that bitumen disbonding occurred

instead.

Jones [12] applied Koike's theory to some of his own data obtained

for a two-ply asphalt-asbestos membrane bonded with asphalt between two

adjacent aluminum plates. In all cases the mode of failure agreed with

that predicted by equation 43. Thus, with an asphalt thickness of

0.020 in (0.5 mm) Jones found that failure occurred by bitumen disbonding, while with a thickness of 0.042 in (1.1 mm) it occurred by membrane

rupture. For each thickness three separate experiments were conducted.

Jones obtained moduli M and G in the same way as Koike. The stressstrain curves he obtained for the membrane were distinctly non-linear

even at the lowest temperature. No stress-strain curves were given for

the bitumen.

37
For the same system, Jones [12] presents a plot (equation 43) of

the critical thickness h* as a function of temperature for two and fourply membranes in both the machine and cross direction, h* was obtained

from the right-hand side of equation 43 using the experimentally determined mechanical properties found at each temperature. Jones found that

h* always decreased with increasing temperature, the rate of drop with

temperature being greatest for a four-ply membrane in the machine

direction and least for a two-ply membrane in the cross direction.

For non-linear membranes, the value of h* can be obtained by substituting equation 39 into equation 38

M 3

2 [e p

(a

2

£

2 + ae +1) -1]

h* = -2 E E _ (44)

G a

2 [l - (3 7+1) e~ 3y

] o

The corresponding value of g* can be found from equation 39.

Using equation 38 and equation 39, data can be analyzed to determine the incidence and mode of membrane failure for non-linear materials

A convenient way to analyze data is to first write equation 38 in the

form

1 - (1 + 0)e

9

= k (45)

where

e = % (46) 2h

38
and

2 M 3 -ae k = ^-t? [e p

(a e + ae + 1) -1] (47)

g g a

P P

is the dimensionless group defined by equation 46 while k is the

constant defined by equation 47.

To evaluate the bitumen film thickness h which is needed to prevent

a rupture of the felt for a given gap spacing g, a plot of the lefthand side of equation 45 vs. 9 is made. A line of slope k is drawn on

this plot and where the line intersects the curve a solution for 9 is

obtained. Equation 46 can then be solved for the h corresponding to

this value of 9.

Figure 7 illustrates the method. Depending on the numerical value

of the constant k and the sign of the bitumen strain rate 3 three kinds

of solutions are possible. If 3 is negative (region A in figure 7) k

and will be negative, and a single solution is present, denoted by 9

in figure 7. If 3 is positive, k and 9 will be positive and either two

solutions or no solution is present. No solution is present (region B

in figure 7) when k is greater than 0.2984, while two solutions are

present (region C) when k is less than 0.2984. The value of 9 at this

value of k is 1.793. The two solutions in region C are denoted as 9

CI

and 9 „ in figure 7.

When no solution is present, this indicates that failure by membrane

splitting can not occur at the gap spacing regardless of the value of h.

The membrane is strong enough to sustain appreciable substrate movements

in region B of figure 7. Further increases in gap spacing g will lower

39
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.2984

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40
the value of k until k = 0.2984, at which failure is predicted if it

happens that h = g3/3.586.

When two solutions are present (region C) , this means physically

that two different bitumen film thicknesses are capable of sustaining

the same substrate movement. Treating this as a design problem, one

where h is to be specified, the solution with the smallest value of the

film thickness is preferred provided bitumen disbonding does not occur.

This will be the one with the largest value of 9 (indicated as 9 „ on

figure 7)

.

This procedure may also be used to treat the problem of determining

at what gap spacing failure by felt rupture will occur for an existing

membrane, one whose value of h is fixed. In this case only one solution

will be present, one corresponding to either 9 . or 9 _ on figure 7, but

not both.

Bitumen disbonding will occur if

£-f > T (48)

Viewing this as a design problem, the larger value of h should be chosen

in hopes that the yield strain y of the bitumen will not be exceeded, so

that the bitumen is capable of sustaining more lateral movement before

disbonding.

Figure 7 shows how the selection of the mode of failure can be

quantified. A vertical line is drawn at an illustrating value of

9 = By, where 9 is the value corresponding to incipient failure by

bitumen disbonding. If the roofing membrane is designed for equal

41
strength felt and bitumen layers, then 9=0. The value of the bitumen

d

film thickness to use in this case is h* as given by equation 44. Since

these correspond to small values of 0, the smallest value of is preferred (0 .. in figure 7) . For an existing membrane, the value of

calculated from equation 46 will either be larger or smaller than 0..

d

If < , then failure by felt rupture is predicted, whereas if > 9,, d d

then failure by bitumen disbonding should occur.

Table 3 presents the major results of the present study. At

temperatures of -3°C, 13°C, and 18°C, and three different bitumen thicknesses, the experimental results of Koike [2] for the substrate joint

movement needed to cause felt rupture are compared with the theoretical

predictions for both linear and non-linear membrane materials. Experimental results at -3°C were not available for comparison with the

theoretical predictions since bitumen disbonding occurred [2]

.

Also given in table 3 are the joint movements which are needed to

cause bitumen disbonding (third column from the right in table 3) . The

last two columns in table 3 present the joint movements g* and film

thicknesses h* at which the bitumen and felt would have equal strengths,

that is, the condition under which failure by bitumen disbonding and

felt rupture would be equally probable.

For linearly elastic membranes the theoretical predictions of

substrate joint movement g (fourth column from the left in table 3) were

obtained in the same manner as that employed by Koike [2], that is, by

an application of equation 2. His value for the modulus M was equal to

the physically measured initial slope of the stress-strain diagram of

the roofing felt, while he took G equal to the slope of a line drawn

42
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43
from the origin of the stress-strain diagram of the bitumen out to the

ultimate or breaking stress of the bitumen. In this report the corresponding value of M is M in equation 13, and G is the ratio of the

bitumen breaking stress to the strain at that point. The values for g

calculated by this procedure do not differ substantially from those

reported by Koike [2],

Koike's method of obtaining the moduli M and G is somewhat arbitrary since the procedure could be reversed to obtain G from an initial

slope, and M as the ratio of coordinates at the yield point. Employing

this procedure gives a significantly different set of results for predicted joint movements than if M is obtained from an initial slope and G

as the ratio of coordinates at the ultimate stress. For consistency it

would have been better to define both moduli at the same point, either

at zero stress or at the ultimate stress or at some well-defined halfway point. Koike [2] did not give a reason for his choice of method.

Table 3 shows a wide variation in predicted joint movements depending on what model is used to calculate them. For the non-linear model

of the present study the predicted value of the joint movement g rises

with the value of the bitumen film thickness h. According to the

linear theory (equation 2), the joint spacing for a given linearly

elastic membrane should rise in proportion to the square root of the

film thickness. For membrane materials with non-linear stress-strain

behavior this was found at 13°C, but at -3°C the predicted gap spacing

increased more rapidly than that predicted by the linear theory while at

18°C it increased more slowly.

44
Table 3 shows that as the temperature falls the substrate is able

to sustain more movement. This is a consequence of the greater complementary strain energies of the felt at the lower temperatures. Predictions of the non-linear theory gave permissible substrate movements

which were 5% higher at 13°C than at 18°C and over 20% higher at -3°C

than at 18° C.

Koike [2] found that bitumen disbonding was the mode of membrane

failure at -3°C. This result is not consistent with prediction.

Table 3 presents a column (third from the right) which gives the predicted substrate joint movement needed to cause failure by bitumen

disbonding. These values were calculated from equation 39. In all

cases the joint movements calculated exceeded the experimental values

and those predicted from the linear and non-linear theories. This

indicates that the bitumen film should be stronger than for the roofing

felt, but such was evidently not the case at -3°C. Koike [2] attributes

this anomalous result to a difference in experimental technique.

Note from table 3 that failure by bitumen disbonding is more

likely at lower temperatures where g values for bitumen and felt are

closer. At some temperature even lower than -3°C the g values for

bitumen and felt might become equal in which case failure by either

bitumen disbonding or felt rupture would be equally likely. In this

event the felt and bitumen can be considered to have equal strengths.

The last two columns in table 3 summarize for each temperature the

predicted values of joint spacing Cg*) and bitumen film thickness (h*)

which would be needed to give equal strengths of bitumen and felt

layers. These were calculated using equations 39 and 44, respectively.

45
Also included in parenthesis below each value are the corresponding

values calculated using equations 42 and 43 for membrane materials with

linear stress-strain response. At -3°C values of g* and h* are equivalent (1 mm) regardless of whether the membrane had linear or non-linear

stress-strain characteristics. At higher temperatures the values of g*

and h* are about twice as high for membranes with linear behavior as for

membrane materials with non-linear behavior. As pointed out in the last

section, g* and h* are useful for membrane design. At the higher

temperatures, gap widths or spacings are greater than predicted for

membranes with linear stress-strain behavior. In practice an actual

membrane would not sustain nearly as much substrate movement before

failure.

To quantify the differences between the different methods or

models for predicting substrate joint movements equation 49 is used

9 n (g -g .) 2 m j. ti ei

(49)

i=l

n" 1

2

s represents the sample variance of the difference between the theoretical (g .) and experimental (g .) values of the joint width, n denotes

the number of data points at each temperature (three here) . Relatively

low values of the sample variance indicate good agreement between

experimental and predicted values.

Table 4 summarizes the results of the calculations. Each of the

linear models are characterized in terms of the assumptions made to

obtain their moduli from the stress-strain diagrams for the roofing

felts and roofing bitumens (second column of table 4) . In the third

46
Table 4. Statistical Results for Methods used to Predict Substrate

Joint Movements needed for Felt Rupture (data from [2])

Model

Assumptions Made to

Obtain Moduli from

Variance of Joint

2/ 2 Movement— _j_ mm

1/

Variance

3/ Linearity Stress-Strain Diagrams (Theoretical-Experimental)— Ratio—1

13°C 18°C 13°C 18°C

none 0.19 0.05 1.0 1.0 XT T • 3/ Non-Linear—

Linear— M zero-stress slope,—

G from ultimate stress

coordinate ratio

0.02 0.06 0.1 1.2

Linear— G from zero-stress slope,

M from ultimate stress

coordinate ratio

1.91 0.30 9.5 6.0

Linear— M and G from zero-stress

slopes

0.12 0.14 0.6 2.

Linear— M and G from ultimate

stress coordinate ratios

0.64 0.72 3.3 14.4

Linear— M and G from coordinate

ratio of point at half

the ultimate stress

(half-height method)

0.10 0.05 0.5 1.0

1/ Defined and calculated from equation 49.

If Ratio of variances of joint movements relative to the non-linear

model at the same temperature.

3/ Results for present theory for non-linear materials (equation 38)

4-/ Results for previous theory for linear materials (equation 2).

5/ Method used by Koike [2] in his data analysis.

47
column of the table variances between the theoretical and experimental

joint widths calculated using equation 49 are presented. Examining

table 4 shows a wide range in variances calculated from equation 49

depending upon the convention adopted for choosing the moduli. For

roofing materials with non-linear stress-strain characteristics, no

assumptions are involved so that this forms a convenient basis for

comparison. The last column gives the ratio of the variance of the

linear model to that of the non-linear model. Variance ratios are

generally greater than unity, indicating that the non-linear model

usually gives a closer fit to the experimental results, especially at

the higher temperature (18°C)

Comparing the variances of the linear to the non-linear model, only

the linear model of Koike [2] (second row in table 4) and that based on

the values of M and G obtained from the coordinate ratio of the point at

half the ultimate stress (last row in table 4) , called the half-ultimate

stress point,— gave better results than the non-linear model. This

latter model works well because it partially accounts for the nonlinearities in the stress-strain curves. The method of drawing a line

from the origin through a point on the stress-strain curve located at

half the ultimate stress (last row in table 4) tends to give strain

energies equal to the non-linear values. This is because, for the

stress-strain data used in this study, the line subtends and excludes

about equal areas when drawn with the stress-strain curve, and by the

mean value theorem of calculus, approximately equal strain energies

— Denoted as the half-height method.

48
result. In systems where non-linearities are more pronounced this may

not be the case.

All of the models examined (refer to table 4) involve assumptions

about the choice of moduli except for the non-linear model. The nonlinear model is more general since it reduces to the linear models for

roofing materials which have linear stress-strain characteristics. As

discussed, the convention adopted for choosing the moduli for the

linear models is arbitrary, and, in fact, there is no theoretical basis

for adopting one linear model compared to another. In addition, it

would seem to be more consistent to define the moduli M and G in a

similar fashion either as the zero-stress slopes of their respective

stress-strain diagrams (4th row in table 4) or as the ultimate stress

coordinate ratios (5th row in table 4) . However, these models generally

give larger variance ratios when compared to the non-linear model,

especially at the higher temperature.

In general, the non-linear model is preferred because it is exact

and does not depend on any assumptions about the moduli. However, in

some specific cases, it may be preferable for estimation purposes to use

a simpler procedure such as the half-height method (last row in table 4)

which may account satisfactorily for non-linearities. If such a procedure is used it is recommended that it should be checked, where stressstrain data for roofing materials are available, against the criterion

for failure (equation 36 or 38) developed in this report.

49
5. SUMMARY AND CONCLUSIONS

Rupture of built-up roofing membranes resulting from movements of

the underlying deck or substrate is an important cause for water leakages

in roofs. Similarly, bitumen or adhesive disbonding without membrane

rupture is a potential cause of the formation_^if roof blisters, which

themselves often lead to water penetration and subsequent rupture.

This report develops and applies visco-elastic theory to the calculation of membrane and adhesive stresses and strains in a single-ply

fabric or felt, which, except at the joints between adjacent substrate

sections, is fully bonded to the substrate with a layer of adhesive or

bitumen. The theory applies to fabric and adhesive layers with both

linear and non-linear stress-strain behavior. For materials with

linear stress-strain characteristics it is shown that the theory reduces

to previously published work.

A method was developed for accurately fitting non-linear stressstrain data for roofing materials to a constitutive equation having only

two adjustable constants. The method was applied to stress-strain

roofing felt and roofing bitumen data from the literature [2] obtained

at three different temperatures. The constitutive equation was shown to

reduce to Hooke's law for linearly elastic materials at low values of

stress.

The joint width between substrate sections, the bitumen layer

thickness, and the mechanical properties of the felt and bitumen, were

important parameters in the mathematical models developed, while the

dimensions of the substrate section were not.

50
A criterion for failure indicating either rupture of the felt or

disbonding of the bitumen was established. Specifically, it was found

that failure was predicted when the product of the strain energy of the

bitumen with the bitumen thickness was equal to the complementary

strain energy of the felt. The highest strain energy that either

material can accommodate was one corresponding to its ultimate stress.

In accord with the criterion, whenever the lower of these two energies

was reached, failure was predicted in that mode.

The theory developed for the mechanical performance of roofing

membranes was applied at three different temperatures to non-linear

stress-strain data obtained for roofing felts and bitumens bonded to a

concrete deck [2], Predicted results agreed well with the theory.

Also, as predicted by the failure criterion, at temperatures of 18°C and

13°C failure occurred by membrane rupture. At -3°C it occurred by

bitumen disbonding, although the theory also predicted failure by membrane rupture.

Based on the criterion for failure, an equation was developed which

predicted the bitumen interply thickness at which the adhesive and

fabric (bitumen and felt) were equally likely to fail. Substrate

movements necessary to produce failure were presented as a function of

system parameters.

For the best mechanical strength, it is recommended that, within

other constraints, roofing membranes be selected or designed which have

strain energies and interply thicknesses compatible with one another

according to the criterion for failure.

51
6. REFERENCES

1. Bonafont, R. L., "Research into Mechanical Behaviour of Waterproofing Membrane Systems," International Symposium on Roofs and

Roofing, (Paper No. 9), Brighton, England, September 1974.

2. Koike, M. , "Elastic Analysis of Ruptures of Bitumen-Felt Roof

Coverings Caused by Cracks in Substructures," BRI Occasional

Report (No. 15), Building Research Institute of Japan, December

1963.

3. Cullen, W. C. and Boone, T. H., "Thermal-shock resistance for

built-up membranes," National Bureau of Standards, Building

Science Series 9, August 1967.

4. Cullen, W. C. , "Solar Heating, Radiative Cooling, and Thermal

Movement-Their Effects on Built-up Roofing," National Bureau of

Standards (U.S.) Technical Note 231, December 1963.

5. Marijs, P. D. and Bonafont, R. L. , "Mechanical Behaviour of

Bituminous Waterproofing Membrane Systems," J. Appl. Chem. , 20 , pp. 11-20, 1970.

6. Mathey, R. G. and Cullen, W. C. , "Preliminary Performance Criteria

for Bituminous Membrane Roofing," National Bureau of Standards,

Building Science Series 55, November 1974.

7. Rossiter, W. J., Jr. and Mathey, R. G. , "Effect of Insulation on

the Surface Temperature of Roof Membranes," National Bureau of

Standards (U.S.) NBSIR 76-987, February 1976.

8. Bonafont, R. L. , "Application of Performance Concept in Evaluation,

Specification, and Selection of Roofing Materials," Proceedings of

the Symposium on Roofing Technology, (Paper No. 11) pp. 70-85,

National Bureau of Standards, Washington, D.C., September 1977.

9. Koike, M. , Tanaka, K. and Tomiita, T. , "Strains in Multi-ply

Membranes Caused by Joint Movement in Substrate," Report of the

Research Laboratory of Engineering Materials (Number 3) , Tokyo

Institute of Technology, 1978.

10. Heukelom, W. , "Observations on the Rheology and Fracture of Bitumens

and Asphalt Mixes," Shell Bitumen Reprint No. 19, Shell International Petroleum Company Limited, London, 1966.

11. Timoshenko, S. P. and Gere, J. M. , Mechanics of Materials , Chapter

11, Van No strand-Re inho Id Co., New York, 1972.

12. Jones, P. M. , "Load-Strain Properties and Splitting of Roof

Membranes," Engineering Properties of Roofing Systems , ASTM

STP409, Am. Soc. Testing Mats. , 1967.

52
7 . NOMENCLATURE

a parameter in equation 3, defined by equation 4, ram

B width of the membrane, mm

C, integration constant, equation 34

g joint substrate movement (half the gap spacing), mm

g* joint substrate movement for equal strengths of bitumen and felt,

mm

2 G shear modulus of bitumen (adhesive) film, N/mm

2 G shear modulus of bitumen film at zero stress, N/mm

o h thickness of bitumen (adhesive) film, mm

h* bitumen film thickness for equal strengths of bitumen and felt, mm

i index of summation, equation 49

g . experimental value of g, mm

g . theoretical (predicted) value of g, mm

k constant defined by equation 47

M elastic modulus of membrane or felt (fabric) , N/mm

M elastic modulus of felt at zero stress, N/mm

n number of data points, equation 49

N number of plies in a built-up roofing membrane

P tensile strength, ultimate or breaking load (force/unit width) of

membrane or roofing felt, N/mm

r correlation coefficient

s sample standard deviation

s sample standard deviation on a

s

fi sample standard deviation on 3

s sample residual standard deviation

s_, sample standard deviation on G

(j o53
s„ sample standard deviation on M

M o o

2

s sample variance, equation 49

t student's t

TSF thermal shock factor, defined by equation 1

u membrane displacement, mm

U complementary strain energy of the felt, equation 28

v substrate displacement, mm

3

Y volume of membrane system, mm

W half-width of one substrate element, mm

x membrane displacement measured from the center of the gap, mm

ot felt constant, equation 13; coefficient of linear thermal expansion,

equation 1

3 bitumen or adhesive constant, equation 16

Y bitumen or adhesive strain, equation 22

Y bitumen strain corresponding to T

£ felt or fabric strain, equation 20

£ felt strain at maximum stress

m

£ felt strain at ultimate stress P

P

C substrate strain, equation 21

9 dimensionless group, defined by equation 46

2 felt or fabric stress (N/mm )

2 maximum stress, calculated from 0.368 M /a, N/mm

m o

2 T bitumen or adhesive stress, N/mm — 2 T bitumen film yield strength, N/mm

54
NBS-114A (REV. 2-8CI U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions,)

1. PUBLICATION OR

REPORT NO. TN 1152

2. Performing Organ. Report No 3. Publication Date

December 1981

4. TITLE AND SUBTITLE

Mechanical Performance of Built-Up Roofing Membranes

5. AUTHOR(S)

James M. Pommersheim and Robert G. Mathey

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as no. 6

10. SUPPLEMENTARY NOTES

J Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

For built-up roofing membranes with either linear or non-linear stress-strain

behavior, fully bonded to an underlying deck or substrate which undergoes

displacement, it is the equality of the complementary strain energy of the

fabric or felt layer with the strain energy of the bonding adhesive or bitumen

layer, which governs both the conditions under which membrane integrity is

lost and the mode of failure by either membrane splitting or adhesive bonding.

The testing criteria developed are applied to a sample case.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

adhesive; bitumen; bonding; built-up roofing membrane; complementary strain energy;

felt; non-linear; roofing membrane; splitting; strain energy; stress; substrate.

13. AVAILABILITY

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20402.

[X^ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

60

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