CRREL REPORT 96-10

Structural Mechanics Solutions for

Butt Joint Seals in Cold Climates

Stephen A. Ketcham August 1996
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Cover: Nondimensionalized normal and tangential stress distributions, 

σx (y)/E and t y(y)/E, divided by the

nominal strain e, for joint seals with depth-to-width ratios 0.5, 1, 2, and 4.

Abstract: An effective, formed-in-place joint seal will

respond with elastic or viscoelastic behavior over a

reasonable design life to any large movement of the

joint without adhesive or cohesive failure. For a given

joint movement, seals with lower stiffness are most

able to deform without cohesive or adhesive failure of

the seal or of the structure to which it is bonded. It is in

recognition of this desirable response feature that

lower-modulus, rubber-based elastomeric materials

have been formulated and promoted as joint sealants.

For a seal formed from an elastomeric sealant, it

should generally be expected that the modulus of

elasticity will depend upon temperature and loading

rate, such that the modulus increases (sometimes

dramatically) with a reduction in temperature and an

increase in loading rate, and it should be expected

that the seal stiffness will depend upon the material

modulus and the shape of the seal. Measurements

from testing techniques that are routinely used to

evaluate the temperature and rate-dependent mechanical properties of rubber-like materials, together

with simple structural mechanics solutions for the

load vs. deflection behavior of rubber in the configuration of rectangular-shaped joint seals, allow these

dependencies to be modeled, and form the basis of a

practical analysis technique that could be used by

civil and mechanical engineers for sealant selection

and seal design.
CRREL Report 96-10

Structural Mechanics Solutions for

Butt Joint Seals in Cold Climates

Stephen A. Ketcham August 1996

Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

US Army Corps


of Engineers

Cold Regions Research &


Engineering Laboratory
PREFACE

This report was prepared by Dr. Stephen A. Ketcham, Research Civil Engineer,

Civil and Geotechnical Engineering Research Division, Research and Engineering

Directorate, U.S. Army Cold Regions Research and Engineering Laboratory (CRREL),

Hanover, New Hampshire. This work was funded by the U.S. Army Corps of

Engineers, Project 4A762784AT42-BS-020, Maintenance and Repair of Pavements and

Unsurfaced Areas in Cold Regions. Dr. Gregory McKenna, of the National Institute of

Standards and Technology, provided valuable suggestions for this paper.

The contents of this report are not to be used for advertising or promotional

purposes. Citation of brand names does not constitute an official endorsement or

approval of the use of such commercial products.

ii
CONTENTS

Page

Preface ................................................................................................................... ii

Introduction .......................................................................................................... 1

Extension and compression loading ................................................................. 2

Shape factor, modulus of elasticity, and apparent modulus .................... 2

Stress distributions at bonded interface ...................................................... 5

Elastic instability ............................................................................................. 6

Shrinkage or expansion stresses ................................................................... 7

Shear loading ........................................................................................................ 7

Conclusion ............................................................................................................ 8

Literature cited ..................................................................................................... 9

Appendix A: Hypothetical design example, pavement butt joint seal ........ 11

Abstract ................................................................................................................. 13

ILLUSTRATIONS

Figure

1. Real part of shear modulus G as a function of temperature T for

a natural rubber and a polysulfide sealant, from 50-Hz

harmonic loading tests..................................................................... 2

2. Original butt joint seal configuration, homogeneous deformation,

and final, inhomogeneous deformation, corresponding to

solution of Gent and Lindley for the compression of bonded

rubber blocks ..................................................................................... 3

3. Ratio of the apparent modulus to the Young’s modulus as a

function of the depth-to-width ratio .............................................. 4

4. Ratio of the nominal stress to the apparent modulus as a

function of nominal compression and extension strain for

large deformations ........................................................................... 4

5. Ratio of the nominal stress to the Young’s modulus for joint seals

with depth-to-width ratios 0.5, 1, and 2, as a function of

nominal compression and extension strain .................................. 5

6. Nondimensionalized normal and tangential stress distributions

divided by the nominal strain for joint seals with depth-towidth ratios 0.5, 1, 2, and 4 .............................................................. 6

7. Critical rupture extension and ratio of the critical rupture stress

to the Young’s modulus as functions of the depth-to-width

ratio ..................................................................................................... 6

8. The nondimensionalized average normal stress divided by the

constrained strain as a function of the depth-to-width ratio ..... 7

9. The geometry and deformation of a joint seal in shear loading ...... 8

10. Ratio of the apparent shear modulus to the shear modulus of

elasticity as a function of the depth-to-width ratio ..................... 8

iii
INTRODUCTION

An effective joint seal* that is formed in a building or pavement joint by the curing of a sealant

will respond with elastic or viscoelastic behavior

over a reasonable design life to any movement of

the joint without adhesive or cohesive failure.

Such a seal is not meant to transfer significant

forces across the joint. On the contrary, for a given

joint movement, seals with lower stiffness are

most able to deform without cohesive or adhesive failure of the seal or of the structure to which

it is bonded. It is in recognition of this desirable

response feature that lower modulus, rubberbased, elastomeric materials have been formulated and promoted as joint sealants. For a seal

formed from an elastomeric sealant, it should generally be expected that the modulus of elasticity

will depend upon temperature and loading rate,

such that the modulus increases with a reduction

in temperature and an increase in loading rate,

and it should be expected that the seal stiffness

will depend upon the material modulus and the

shape of the seal.

In the field of rubber technology, conventional

engineering design of rubber structures incorporates engineering mechanics-based structural

analysis techniques and corresponding material

properties. When temperature and loading rate

variations are expected, these properties are measured as a function of temperature and loading

rate so that the effect on structural response can

be evaluated. This is in contrast to the current

practice for the design of building and pavement

seals, which, being based on the “movement capability” of a model seal structure (e.g., ACI 1993,

Panek and Cook 1991), does not utilize structural

analysis and does not incorporate measurements

of the stress–strain mechanical properties of sealants. As such, the design practice is not compatible with conventional thermal analysis measures

for rubber materials, such as the modulus of elasticity vs. temperature and the coefficient of thermal expansion vs. temperature.

As indicated by the shear modulus vs. temperature data in Figure 1, measurements of the modulus of elasticity as a function of temperature can

be very revealing to the designer of a rubber structure. The data in Figure 1 were published by

Nashif and Lewis (1991) as an example of a large

database of the properties of rubbers and other

materials. The curves shown are of a natural rubber and a polysulfide sealant tested at a 50-Hz

harmonic loading frequency and at several temperatures, and were obtained using measurement

techniques that are included in standard test methods (ASTM 1991b). The shear modulus variations

of the two materials illustrate the dramatic increase in material stiffness that can occur over a

narrow, low temperature range in rubber materials, as well as a more subtle increase that is possible. By examining the data of the natural rubber, for example, a designer might suggest that

–20°C should be the lowest temperature at which

this rubber is used for loading applications at the

50-Hz frequency. Although the data shown are

from high-frequency loading tests, a designer

Structural Mechanics Solutions

for Butt Joint Seals in Cold Climates

STEPHEN A. KETCHAM

* In this paper the standard terminology for “seal” and

“sealant,” given by ASTM C717-88c (ASTM 1991a) for

buildings, is adopted. Specifically, “seal” describes a

barrier against the passage of liquids and solids, and

“sealant” describes a material that has the adhesive

and cohesive capabilities to form a seal. These definitions are used in an engineering mechanics sense to

allow distinction between the material properties of

the sealant and the load-deflection behavior of the seal.

The discussion here is limited to formed-in-place seals.
could make similar suggestions about the rubber

subjected to slower loads from modulus vs. temperature results of quasi-static loading tests, which

are also conventional thermal analysis tests.

Lewandowski et al. (1992) have demonstrated the

measurement and usefulness of such data for

pavement sealants.

It is easy to envision the potential practicality

of such information for the selection of cold climate joint sealants. For example, a designer could

use modulus vs. temperature data of the candidate sealants together with climatic temperature

data for the region of interest, and make a selection using a rationale that incorporates a severe

winter design temperature. This is in contrast to

current practice, which typically utilizes standard

bond tests of model seal structures at a given low

temperature (e.g., ASTM 1991c, d), but which does

not, in general, reveal explicitly the temperature

range at which the model seal or the sealant material stiffens. In recognition of the potential uses

of measurements of the modulus of elasticity as a

function of temperature for sealant materials, and

in recognition of the incompatibility of such data

with movement capability-based design calculations, this paper presents a review of simple engineering mechanics-based analysis techniques for

the structural design of rubber materials subjected

to tension, compression, and shear loading in long

rectangular joint seal configurations.

Solutions and techniques described by Rivlin

and Saunders (1949), Payne (1956), Gent and

Lindley (1958, 1959), Gent (1974), and Gent et al.

(1974), for rubber materials that can be considered incompressible under hydrostatic loading,

are summarized here and in some cases extended

to the plane strain configuration of a joint seal.

Previous reviews of these solutions, and of the

corresponding engineering practice in general,

have been presented by Payne and Scott (1960),

Lindley (1967), Gent and Meinecke (1970), Gent

(1978a, b), and Stanton and Roeder (1982). Recent

examinations and extensions of these solutions

have been presented by Chaloub and Kelly (1991).

A design example illustrating use of solutions

from Payne (1956) and Gent and Lindley (1959) is

shown in Appendix A.

EXTENSION AND

COMPRESSION LOADING

Gent and Lindley (1959), using small deformation, linear elasticity theory and realistic assumptions regarding structural deformations, generated solutions for the stress distribution in bonded

rubber blocks during compression of the blocks.

From these solutions they obtained expressions

for the nominal stress–strain relations of the

blocks. Payne (1956) and Gent and Lindley (1959)

also suggested approximate relations for the corresponding large deformation problem. These and

other solutions described here for a block in a

plane strain configuration are directly applicable

to the extension and compression of long, formedin-place, rectangular butt joint seals, and can be

presented as such with only slight modification

of terminology.

Shape factor, modulus

of elasticity, and

apparent modulus

The apparent modulus of a butt joint seal in

extension or compression was given by Gent and

Lindley as

Shear Modulus (Pa)

–100 –80 –60 –40 –20 0 20 40 60 80 100

a.

1010

109

108

107

106

105

1010

109

108

107

106

105

104

–100 –80 –60 –40 –20 0 20 40 60 80 100 120 140

Temperature (°C)

b.

Figure 1. Real part of shear modulus G as a function

of temperature T for (a) a natural rubber and (b) a

polysulfide sealant, from 50-Hz harmonic loading

tests (Nashif and Lewis 1991).

2
 

E d

w a = + E 



 



 















4

3

1

3

2

(1)

where E is the Young’s modulus of the sealant,

and d and w are the depth and width of the seal

within the joint, respectively, as illustrated in Figure 2a. In the literature of building and pavement

seals, d/w is often called the shape factor of a seal.

The term “apparent” was used by Gent and

Lindley to distinguish the bonded extension or

compression deformation as an inhomogeneous

structural deformation and to refer to the nominal stress and strain of the structural response in

explicit contrast to the homogeneous stress and

strain of a material property test. The two terms

in eq 1 originate, respectively, from solutions to

(a) a plane strain deformation in which the material is free to slip on the bonded interface and to

deform homogeneously in the section, while remaining constrained in the long direction, and (b)

a subsequent inhomogeneous shear deformation

that restores the material of the bonded interface

to the bonded position. These deformations are

depicted schematically in Figure 2, in parts b and

c, respectively. The material is assumed to be incompressible, i.e., it is assumed that there is no

volume change during deformation. For the inhomogeneous shear deformation, it is assumed

that planes parallel to the bonded surface remain

plane, and that planes normal to these distort to

form parabolas in the cross section. The solution

to the homogeneous deformation problem (a) is

the uniform normal stress, σx1, i.e.,

 

σx1

4

3 = Ee (2)

where e = ∆w/w is the joint extension or compression, i.e., the nominal strain of the seal, ∆w is the

total joint movement in the x direction, and x

refers to the coordinate axis of Figure 2. The solution to the shear deformation problem (b) is a

hydrostatic pressure distribution, p(y), that varies

with the joint extension or compression, and along

the interface of the seal in the y direction, according to

 

p y

d

w

y

w ( ) = Ee 



 



 − 



 





















1

2

2

2 2

 . (3)

(See Chaloub and Kelly [1991] for an illustrative

derivation of the governing equation that p(y)

satisfies.) A normal stress distribution, σx2 (y), is

in equilibrium with this pressure. The total normal stress distribution is found by the superposition of the stresses σx1 and σx2 (y), i.e.,

 

σx y

d

w

y

w ( ) = + Ee 



 



 − 



 





















4

3

1

2

2

2 2

(4)

where σx is the total normal stress. The average of

this stress, i.e., the nominal stress 

σx , is found by

integrating σx (y) over the area d × 1 of the interface and dividing by this area. The nominal stress–

strain relation that follows is

 

σx

a

= + 



 





















=

4

3

1

3

2 d

w

Ee

E e .

(5)

Figure 2. (a) Original butt joint seal configuration,

(b) homogeneous deformation, and (c) final, inhomogeneous deformation, corresponding to solution of

Gent and Lindley (1959) for the compression of bonded

rubber blocks.

d

w

y

x

Substrate Seal

a

w

w + ∆w

Undeformed

d Configuration

b

w

w + ∆w

Undeformed

d Configuration

c

3
The apparent modulus is thus greater than the

Young’s modulus of the associated homogeneous

plane strain deformation by the value of the second term in eq 1, which models the additional

structural stiffness that arises from the constraint

of the bond. Gent and Lindley showed that eq 1

and 5 correctly represented results of load and

deflection experiments of rubber structures with

length-to-width ratios of three. The joint compressions in these experiments were less than 5%

and the material modulus of elasticity was approximately 1800 kPa. Lindley (1967) suggested

that eq 1 is applicable up to nominal strains of

about 10%.

The plane strain structural stiffness, F/∆w,

which corresponds to the apparent modulus Ea,

is given by

 

F

w

d

w

E

∆ = a . (6)

In this equation F is the resultant compressive or

extensive force at the interface per unit length

along the seal.

The relationships of eq 1 and 6 describe

the structural stiffness of a butt joint seal in

extension or compression and its dependence on shape factor and elastic modulus.

The relationship of eq 1 is depicted in Figure 3 in the form of the ratio of the apparent modulus to the Young’s modulus, for

depth-to-width ratios from 0.25 to 5. As

indicated in the figure, the modulus ratio

increases from 4/3 to nearly 10 as the depthto-width ratio increases from 0.25 to 5, revealing that the average normal stress in

the seal can increase by a factor of 7 over

this small range of d/w. It is readily observed from Figure 3 that in order to keep

the seal stresses at reasonably low levels,

e.g., below failure stress levels, both the

depth-to-width ratio of the seal and the

modulus of elasticity of the sealant should

be kept optimally small. It is in this context

that measurements of the shear modulus or

the Young’s modulus as a function of temperature appear practical for seal design.

Recognizing the limitations of the above

relations beyond small deformations, Payne

(1956) and Gent and Lindley (1959) suggested

the following expression for a large deformation, nominal stress–strain relationship of

a rubber block.

 

′ = − 



 

 σ λ  λ x a

1

3

1

2 E (7)

where λ = 1 + e is the ratio of the deformed seal

width to the original width. The expression is a

modification of the large, homogeneous deformation relationship for the simple extension or

uniaxial compression of an elastic material (Treloar

1975). The Payne/Gent and Lindley expression

accounts for the inhomogeneous deformation in

an approximate manner by the use of the apparent modulus of the structure at small strains Ea

rather than the Young’s modulus E. Gent and

Lindley presented experimental results which suggest that, for accuracy consistent with that required for design of building and pavement seals,

the relationship of eq 7 is valid for nominal compression strains up to 30%.

The expression of eq 7 is illustrated by nondimensional relationships in Figures 4 and 5. Figure 4 shows the ratio of the nominal stress to the

apparent modulus as a function of nominal compression and extension strain, and Figure 5 shows

similar relationships for the nominal stress di10

8

6

4

2

012 345

d/w

Ea/E

Figure 3. Ratio of the apparent modulus to the Young’s modulus, Ea/E, as a function of the depth-to-width ratio d/w.

0.3

0.2

0.1

0

– 0.1

– 0.2

– 0.3

– 0.4

– 0.20 0 0.10 0.20

e

Compression

Extension

– 0.10

σx /Ea Figure 4. Ratio of the nominal stress to the apparent modulus,

 

σx /Ea, as a function of nominal compression and extension

strain, e, for large deformations.

4
vided by the elastic modulus E, which is a function of the depth-to-width ratio. Predictions for

compressions and extensions up to 0.25 are depicted. In the latter figure, predictions are shown

for three depth-to-width ratios: 2, 1, and 0.5. The

curves in both figures demonstrate the nonlinear

stress–strain response predicted by the Payne/

Gent and Lindley expression, and the curves in

Figure 5 further demonstrate the effect of increasing d/w values on the average bond stress acting

on a seal.

Figure 5 also includes results from large deformation numerical analyses of butt joint seals.

These analyses were finite element analyses, using the commercially available software ABAQUS

(Hibbitt, Karlsson, and Sorenson, Inc. 1993), that

incorporated strain energy constitutive models of

two silicone sealants at 0°C for 1 s and 1000 s

relaxation times (Ketcham et al. 1996). The sealants are designated “sealant A” and “sealant B”

in Figure 5. The measured Young’s moduli of the

sealants at 0°C were E = 540 kPa and E = 350 kPa

for the respective t = 1 s and t = 1000 s responses

of sealant A, and E = 330 kPa and E = 170 kPa for

the 1 s and 1000 s responses of sealant B. The

analyses were conducted to allow comparison of

predictions of eq 7 with the more realistic numerical results. The general nonlinearity of the

finite element predictions and the effect that the

depth-to-width ratio has on the behavior are indeed captured by the eq 7 approximation. For

compressive strains the differences between the

predictions in Figure 5 are typically less than 10%.

For extensive strains, however, the eq 7 relation

predicts considerably stiffer responses than the

finite element analyses. As indicated in Figure 5,

the eq 7 predictions are closest to the numerical

results for the d/w = 0.5 and d/w = 1 seals. In this d/

w range, and for extensions nearing 25%, the comparisons indicate that the Payne/Gent and Lindley

equation would provide conservative estimates

of the average bond stress that are roughly 20–

30% high.

The small and large deformation relations of

eq 5 and 7 indicate that butt joint seals with large

depth-to-width ratios should be avoided. It should

be noted that, for asphalt pavement crack seals,

these relations provide a structural analysis-based

argument for preparing a joint at a crack rather

than simply filling the crack. A hypothetical example for an asphalt pavement crack seal design,

illustrating the use of eq 5 and 7, is presented in

Appendix A.

Stress distributions

at bonded interface

Equation 4 gives the normal stress distribution

at the interfaces between the seal and the joint for

small deformations. The corresponding tangential stress distribution ty (y) is found, as suggested

by Gent et al. (1974), from the pressure distribution p(y) according to

 

t y w p

y

y

w

Ee

y ( ) = ∂

∂

= −

2

2 .

(8)

Expressions like eq 4 and 8 for bonded rubber

cylinders have been derived and validated by

experiment by Gent et al. (1974).

The normal and tangential stress distributions

are illustrated in Figure 6 for the d/w values 4, 2, 1,

and 0.5. The stresses, per unit nominal strain, are

shown in a nondimensional form divided by the

elastic modulus E as a function of the position y/d

Sealant A

d/w = 0.5

Sealant B

d/w = 0.5

Sealant A

d/w = 1

Sealant B

d/w = 1

Sealant A

d/w = 2

Sealant B

d/w = 2

–0.2 0 0.2

e

–0.1 0.1

0

0.2

–0.2

–0.4

0

0.2

0.4

–0.2

–0.4

–0.6

0.4

0

–0.4

–0.8

σx/E σx/E σx/E

 1 s Response

1000 s Response

–0.2 0 0.2

e

–0.1 0.1

Figure 5. Ratio of the nominal stress to the Young’s

modulus, 

σx /E, for joint seals with depth-to-width ratios 0.5, 1, and 2, as a function of nominal compression

and extension strain e (continuous curves). Finite element data for silicone sealants A and B at 0°C and

relaxation times 1 s and 1000 s (discrete points).

5
along the interface. As indicated, the peak of the

normal stress σx (y) occurs at the mid-depth of the

seal, and the maximums of the tangential stress ty

(y) occur at the upper and lower edges of the seal.

For a given material and joint extension, the effect

of an increasing depth-to-width ratio on the peak

normal or tangential stress is dramatic. Also, for

higher d/w values, the contribution of the shear

deformation to the total, normal stress is much

greater than the contribution of the homogeneous

deformation, which is constant at the σx (y)/Ee

ratio of 4/3.

tion elasticity analysis, this pressure was found to

be approximately (5/6)E. Gent and Lindley validated this solution with experimental results, and

used the solution with normal stress predictions

from small deformation relations like eq 5 to predict the nominal stress and extension values of a

rubber structure at failure.

Applying Gent and Lindley’s analysis technique to the plane strain structure of a butt joint

seal, the critical average stress 

σ ′

x at which an

internal rupture occurs can be found as a function

of the material elastic modulus E and the depthto-width ratio of the seal. This relation is

 

′ = 



 



 +

















σx

5

3

4

3

1

3

2

w

d

E . (9)

The critical extension, e′ = 

σ ′

x /Ea, is

 

′ = 



 

 e  w

d

5

3

2

 . (10)

These relations are illustrated in Figure 7 for

depth-to-width ratios of a seal from 4 to 8. Below

d/w = 4, the relations should not be applied since

the resulting critical extensions are large and vio0.4

0.2

0

–0.2

–0.4

y/d

0 2 4 6 8 10 0 –4 2 4 –2

d/w = 4

2 1

0.5

σx(y)/Ee ty(y)/Ee

Figure 6. Nondimensionalized normal and tangential

stress distributions, 

σx (y)/E and ty(y)/E, divided by

the nominal strain e, for joint seals with depth-towidth ratios 0.5, 1, 2, and 4.

Figure 7. Critical rupture extension, e′, and ratio of the critical

rupture stress to the Young’s modulus, 

σ ′

x /E, as functions of the

depth-to-width ratio d/w.

Elastic instability

Gent and Lindley (1958) observed in

experiments of rubber cylinders that an

internal rupture was possible at a comparatively small tensile load when the diameter-to-thickness ratio of the cylinder

was high. The rupture was described as

consisting of the sudden appearance of

internal cracks at a repeatable, small tensile load. The experiments and analysis

described by Gent and Lindley showed

the internal rupture to be governed by an

elastic instability and the failure stress to

depend upon the elastic modulus. The

cracking stress was found to be independent of the extensibility and strength of

the rubber material.

The elastic instability was shown by

Gent and Lindley to occur when a small

cavity or imperfection within the rubber is

subjected to a tensile and primarily hydrostatic stress, such as the maximum normal

stress of the curve for d/w = 4 in Figure 6.

At a critical pressure the cavity expands,

forming a crack. Using a large deforma1.05

0.95

0.85

0.75

0.6545678

d/w

σ’ /E x 0.10

0.08

0.06

0.04

0.02

e’

6
late the small deformation assumption of the

derivation. At d/w = 4, the critical extension

is about 10%, which occurs at a critical stress

approximately equal to the modulus E. At

higher d/w values, the critical values become

lower. Using this information, it is possible

to attribute some seal failures, e.g., some of

the “cohesive” failures observed in butt joint

seals with high d/w values, to the elastic instability mechanism.

Shrinkage or

expansion stresses

Using a superposition of solutions, Gent

(1974) demonstrated the use of the nominal

stress–strain relations of bonded rubber structures to evaluate interface stresses generated

by the shrinkage of the rubber. Gent suggested

that prediction of the interface stresses caused by

thermally induced volume changes would be an

application of the technique. For example, Gent

suggested that interface stresses generated after a

rubber structure is formed and bonded at an elevated temperature could be predicted. Using the

small deformation, nominal stress–strain relation

for a rubber block in a plane strain configuration,

the prediction of stresses generated during the

cooling of a hot-applied seal would be a specific

example of this suggested application.

Following Gent’s analysis procedure, when

there is no joint movement, and when the seal, if

unconstrained, would otherwise extend or compress homogeneously by e1 = ∆w/w due to shrinkage or expansion, the following expression for

the average, normal interface stress applies.

 

σx =− + 



 



 















2

3

2

2

1

d

w

Ee . (11)

e1 must be known in order to evaluate the stress.

For volume changes due to temperature changes,

e1 can be related to the coefficient of linear thermal expansion of the material, α, by e1 = α∆T,

where ∆T is the temperature change. When α

should be treated as a function of temperature, an

integration must be performed to calculate e1.

The normal stress of eq 11, divided by Ee1, is

illustrated in Figure 8 as a function of the depthto-width ratio of the seal. The figure indicates

that, as expected, the average normal stress increases dramatically with d/w for a given material

and constrained strain.

Expressions for the stress distributions σx (y)

and ty (y) for a constrained strain e1 could also be

found by following Gent’s procedure. Gent suggested that these stresses could be superimposed

with stresses generated by the joint movement in

order to evaluate their significance.

SHEAR LOADING

Rivlin and Saunders (1949) studied the effect

of shape on the shear behavior of rubber cylinders bonded at their ends. In particular they analyzed the problem in which one end of a cylinder

is displaced parallel to the other. They demonstrated by experiment and theory that, when the

diameter-to-height ratio of the cylinder is relatively small, the actual deformation can be considered to be made up of a component due to

bending in addition to a component due to shear.

For cylinders where the bending deformation is

significant, Rivlin and Saunders suggested the

use of an apparent shear modulus Ga to describe

the combined deformation behavior. Their analysis can be applied to the problem of the shear

loading of a long, rectangular joint seal, as indicated by the analysis of long, bonded rubber blocks

by Lindley (1967) and Gent and Meinecke (1970).

Neglecting inertial effects, the problem corresponds to the transverse shear displacement of a

pavement joint seal by traffic loading. The geometry of the seal, and the shear and bending deformations, are illustrated schematically in Figure 9.

For the joint seal geometry, the apparent shear

modulus derived by Rivlin and Saunders has the

form

 

G G

w

d

a =

+ 



 

 1  1

3

2 (12)

10

8

6

4

2

0 12 345 d/w

12

14

16

18

σ /Ee x 1

Figure 8. The nondimensionalized average normal stress 

σx /

E divided by the constrained strain e1, as a function of the

depth-to-width ratio d/w.

7
where G is the shear modulus of elasticity. Ga can

be used in the nominal stress–strain relation

 

t G

G

w

y aa

a

=

=

γ

∆ (13)

where 

ty denotes the nominal or average interface shear stress that is generated by the transverse joint displacement ∆, and γa = ∆/w is the

corresponding apparent shear strain. The expression for the apparent shear modulus can be derived by assuming small shear and bending displacements, ∆s and ∆b, respectively, as the

superimposed components of the total displacement. For the shear component, ∆s = γw, where γ

is the shear strain, and thus

 

∆s

y = t

G

w . (14)

For the bending component, beam theory

yields

 

∆b

y = 



 



 1

3

2 t

G

w

d

w (15)

for a volume-incompressible material. Ga can

be found by substituting the displacement

∆ = ∆s + ∆b into eq 13. The plane strain shear

stiffness F/∆ that corresponds to Ga is

 

F d

w

G

∆ = a (16)

where, for shear loading, F is the resultant

shear force at the interface per unit length

along the seal.

A significant bending contribution has the

effect of reducing the apparent shear modulus of

the seal relative to the actual shear modulus of

the sealant. This is illustrated in Figure 10, which

depicts the relation of eq 12 in a nondimensional

form. For example, at d/w = 1, the apparent shear

modulus is 75% of the shear modulus. Like the

expressions of eq 1 and 5 for extension and compression loading, the shear loading relations of eq

12 and 13 allow the use of the modulus of elasticity, measured as a function of temperature, in

design calculations.

CONCLUSION

The relations reviewed here constitute the basis of a practical analysis technique for evaluating

load vs. deflection responses of rectangular joint

seals subjected to tension, compression, and shear.

The nominal stress–strain relations presented have

y

d x

w

a

y

x

∆s

b

y

x

c d

y

x

∆ ∆ b

Figure 9. The geometry and deformation of a joint seal in shear loading. (a) original

configuration, (b) shear component of total deformation, (c) bending component of

total deformation, and (d) total deformation, corresponding to solution of Rivlin and

Saunders (1949) for the shear behavior of rubber cylinders bonded at their ends.

10

8

6

4

2

0 12345

d/w

G /Ga Figure 10. Ratio of the apparent shear modulus to the shear

modulus of elasticity, Ga/G, as a function of the depth-towidth ratio d/w.

8
been shown by their developers to be valid for

experimental structures formed with rubber materials. The relations provide a simple and rational mechanics-based approach to the selection of

seal shape factor, and allow use of the modulus of

elasticity of the sealant as a design variable. In

this way the effect of temperature on the modulus of elasticity, which is routinely measured for

elastomeric materials that are used in cold climates, can be directly incorporated into the sealant selection and seal design process, as can the

effect of loading rate or time. Future work in this

area should focus on (1) incorporating temperature and rate-dependent mechanical properties in

the sealant selection and seal design process, and

(2) establishing by field evaluation how results

from standard tests of model seals, in which displacements and loads are measured, can be used

in conjunction with the relations described here

for a practical mechanics-based seal design.

LITERATURE CITED

ACI (1993) Guide to sealing joints in concrete

structures, ACI 504R-90. In ACI Manual of Concrete Practice. Part 5. Detroit, Michigan: American

Concrete Institute.

ASTM (1991a) Standard terminology of building

seals and sealants. C 717-88c. Annual Book of ASTM

Standards. Vol. 4.07. Philadelphia, Pennsylvania:

American Society for Testing and Materials.

ASTM (1991b) Standard practice for determining

and reporting dynamic mechanical properties of

plastics. D 4065-82. Annual Book of ASTM Standards. Vol. 8.03. Philadelphia, Pennsylvania:

American Society for Testing and Materials.

ASTM (1991c) Standard methods of testing joint

sealant, hot-applied, elastomeric-type, for portland cement concrete pavements, or joint sealant,

hot-applied, elastomeric, jet-fuel-resistant-type, for

portland cement concrete pavements. D 3583-85.

Annual Book of ASTM Standards. Vol. 4.03. Philadelphia, Pennsylvania: American Society for Testing and Materials.

ASTM (1991d) Standard test method for adhesion and cohesion of elastomeric joint sealants

under cyclic movement (Hockman cycle). C 719-

86. Annual Book of ASTM Standards. Vol. 4.07. Philadelphia, Pennsylvania: American Society for Testing and Materials.

Chaloub, M.S. and J.M. Kelly (1991) Analysis of

infinite-strip-shaped base isolator with elastomer

bulk compression. Journal of Engineering Mechanics, ASCE, 117(8): 1791–1805.

Gent, A.N. (1974) Fracture mechanics of adhesive

bonds. Rubber Chemistry and Technology, 47: 202–

212.

Gent, A.N. (1978a) Rubber elasticity: Basic concepts and behavior. Chapter 1 in Science and Technology of Rubber (F.R. Eirich, Ed.). New York: Academic Press.

Gent, A.N. (1978b) Strength of elastomers. Chapter 10 in Science and Technology of Rubber (F.R.

Eirich, Ed.). New York: Academic Press.

Gent, A.N. and P.B. Lindley (1958) Internal rupture of bonded rubber cylinders in tension. Proceedings of the Royal Society, 249(Ser. A): 195–205.

Gent, A.N. and P.B. Lindley (1959) The compression of bonded rubber blocks. Proceedings of the

Institute of Mechanical Engineers, 173(3): 111–117.

Gent, A.N. and A.A. Meinecke (1970) Compression, bending, and shear of bonded rubber blocks.

Polymer Engineering and Science, 10(1): 48–53.

Gent, A.N., R.L. Henry, and M.L. Roxbury (1974)

Interfacial stresses for bonded rubber blocks in

compression and shear. Journal of Applied Mechanics, 41: 855–859.

Hibbitt, Karlsson, and Sorenson, Inc. (1993)

ABAQUS User’s Manual. Version 5.2. Providence,

Rhode Island.

Ketcham, S.A., J.M. Niemiec, and G.B. McKenna

(In press) Extension and compression of elastomeric butt joint seals. Journal of Engineering Mechanics.

Lewandowski, L.H., L. Lynch, C. Dorman, and

H. McKnight (1992) The application of dynamic

mechanical analysis to the characterization of

field-molded joint sealants. Proceedings of the Pavement Systems Division 1992 Workshop—Crack and

Joint Sealing for Rigid and Flexible Pavements (L.

Lynch, Ed.). U.S. Army Corps of Engineers Waterways Experiment Station, p. 176–196.

Lindley, P.B. (1967) The stiffness of rubber springs.

Chapter 1 in Use of Rubber in Engineering (P.W

Allen, P.B. Lindley, and A.R. Payne, Eds.). London: MacLaren and Sons.

Nashif, A.D. and T.M. Lewis (1991) Data base of

the dynamic properties of materials. Sound and

Vibration, 25(7): 14–25.

Panek, J.R. and J.P. Cook (1991) Construction Sealants and Adhesives. 3rd Edition. New York: John

Wiley.

Payne, A.R. (1956) Effect of shape on the static

and dynamic stress-strain relationship of bonded

rubber in compression. Nature, 177: 1174–1175.

9
Payne, A.R. and J.R. Scott (1960) Engineering Design with Rubber. London: McLaren and Sons.

Rivlin, R.S. and D.W. Saunders (1949) Cylindrical shear mountings. Transactions of the Institute of

Rubber Industry, 24(6): 296-306.

Stanton, J.F. and C.W. Roeder (1982) Elastomeric

Bearings Design, Construction, and Materials. National Cooperative Highway Research Program

Report 248. Transportation Research Board. National Research Council, Washington, D.C.

Treloar, L.R.G. (1975) The Physics of Rubber Elasticity. 3rd Edition. Oxford: Clarendon Press.

10
A maintenance engineer at a northern airfield

has received two contract bids for sealing cracks

in asphalt concrete pavement. Both contracts

specify good installation techniques, e.g., both

contractors have specified that cracks will be

routed, cleaned, and sealed with a “low-modulus” sealant over a rectangular backer material,

without bonding of the sealant to the backer material. Contractor 1 has specified that low-modulus sealant 1 will be used to form seals of 20-mm

width and a depth-to-width ratio of 1. Contractor

2 has specified that low-modulus sealant 2 will be

used to form seals of 15-mm width, but with a

depth-to-width ratio of 3 in order to have “better

adhesion.”

The engineer has material property data given

in Figure A-1 for the two sealants. (Data of this

form could be generated using the techniques of

Lewandowski et al. [1992]). He/she also has results of standard bond tests conducted at –29°C.

From these tests, average normal stresses across

the bond interface at failure are calculated. (These

data could be generated by extending the standard bond tests to include load measurements.)

Failure stress for the model seal formed with sealant 1 for an asphalt concrete substrate was 2 × 106

Pa, and for the seal formed with sealant 2 was 8 ×

105 Pa.

The engineer has noticed in the past that seals

made with sealant 1 have debonded during cold

winter periods, and knows that every five years

or so it gets down to –40°C. Although the engineer expects routed joint openings as large as 5

mm resulting from the thermal contraction of the

pavement, the exact opening is not known. To

evaluate the combined effects of temperature and

depth-to-width ratio on the adhesive bond

APPENDIX A: HYPOTHETICAL DESIGN EXAMPLE, PAVEMENT BUTT JOINT SEAL

stresses, the engineer does some simple calculations using eq 5 and 7.

From eq 5 the engineer plots the ratio of the

apparent seal modulus to the sealant Young’s

modulus for a given sealant temperature, i.e., Figure 3. The plot and eq 5 show the engineer that

the average normal stress across the bond interface for a d/w = 3 seal made of sealant 2 should be

about 2.5 times that of a d/w = 1 seal also made of

sealant 2. In order to keep the bond stresses as far

below failure stresses as reasonable, he/she decides that the depth-to-width ratio of 3 suggested

by contractor 2 should be abandoned and that a

depth-to-width ratio of 1 should be considered

for both sealants. Thus the engineer’s calculations will compare the performance of seals

formed with sealant 1 and sealant 2 at 20-mm

width and depth-to-width ratio of 1 for a 5-mm

joint opening.

The engineer calculates the average bond stress

of a 20-mm2 seal cross section for a 5-mm (25%)

joint opening when the temperature of the sealant is –40°C, which is the design condition he/

she feels is appropriate. He/she conservatively

selects the rapid-loading shear modulus curve for

the material property because it is suspected that

the wintertime movement of the joint is a rapid

stick–slip movement. The engineer assumes the

material to be volume-incompressible and uses

the corresponding relation between Young’s

modulus of elasticity E and the shear modulus of

elasticity G in the calculations, i.e., E = 3G.

Sealant 1: G ≈ 3 × 106 Pa, and so E ≈ 9 × 106 Pa;

d/w = 1; e = 25%. Thus the linear result from eq 5 is

 

σx = 3.75 × 106 Pa and the nonlinear result from

eq 7 is 

σx = 3.05 × 106 Pa.

Sealant 2: G ≈ 1.5 × 105 Pa, and so E ≈ 4.5 ×

105 Pa; d/w = 1; e = 25%. Thus the linear result

from eq 5 is 

σx = 1.875 × 105 Pa and the nonlinear result from eq 7 is 

σx = 1.525 × 105 Pa.

Based on these calculations, the engineer

realizes that sealant 2 results in much less bond

stress at large extensions and cold temperatures than does sealant 1, which suggests that

sealant 2 should be the selected material. However, the engineer must also compare these

stresses with the bond failure stresses. He/she

notes that the calculated average normal

stresses in the seal formed with sealant 2 at

–40°C and 25% extension is less than the failure stress of the standard bond test (8 × 105 Pa),

107

106

105

104

G(Pa)

–40 –20 0 20 40

Temperature (°C)

Sealant Loading

1

2

1

2

Rapid

Rapid

Slow

Slow

Figure A-1. Hypothetical sealant shear modulus vs. temperature data for design example.

11
whereas the average normal stress in the sealant 1

seal under these conditions is greater than the

bond failure stress of the sealant 1 standard test (2

× 106 Pa). As a result, the engineer no longer

considers sealant 1.

The engineer performs further calculations to

evaluate the additional factor of safety provided

by using a wider joint, and what effect a joint

opening larger than 5 mm would have. The engineer recognizes that the constructed seals will not

be exactly square or rectangular in shape, that the

sealant has a more complicated viscous behavior

than modeled in the analysis, that other effects

such as material aging have not been accounted

for, and that his/her calculations are theoretical

approximations of the actual response of the seal.

However, he/she is satisfied that the 20-mm-wide

joint, as designed above to include the effects of

both temperature and shape on the bond stress, is

adequate.

12
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August 1996

Structural Mechanics Solutions for Butt Joint Seals in Cold Climates PE: 6.27.84A

PR: 4A762784AT42

TA: BS

WU: 020

Stephen A. Ketcham

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road CRREL Report 96-10

Hanover, New Hampshire 03755-1290

Office of the Chief of Engineers

Washington, D.C. 20314-1000

For conversion of SI units to non-SI units of measurement consult ASTM Standard E380-93, Standard Practice for Use of the

International System of Units, published by the American Society for Testing and Materials, 1916 Race St., Philadelphia, Pa.

19103.

Approved for public release; distribution is unlimited.

Available from NTIS, Springfield, Virginia 22161

20

Elastomeric butt joint seal Extension and compression loading

Shear loading

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

An effective, formed-in-place joint seal will respond with elastic or viscoelastic behavior over a reasonable

design life to any large movement of the joint without adhesive or cohesive failure. For a given joint movement, seals with lower stiffness are most able to deform without cohesive or adhesive failure of the seal or of

the structure to which it is bonded. It is in recognition of this desirable response feature that lower-modulus,

rubber-based elastomeric materials have been formulated and promoted as joint sealants. For a seal formed

from an elastomeric sealant, it should generally be expected that the modulus of elasticity will depend upon

temperature and loading rate, such that the modulus increases (sometimes dramatically) with a reduction in

temperature and an increase in loading rate, and it should be expected that the seal stiffness will depend upon

the material modulus and the shape of the seal. Measurements from testing techniques that are routinely used

to evaluate the temperature and rate-dependent mechanical properties of rubber-like materials, together with

simple structural mechanics solutions for the load vs. deflection behavior of rubber in the configuration of

rectangular-shaped joint seals, allow these dependencies to be modeled, and form the basis of a practical

analysis technique that could be used by civil and mechanical engineers for sealant selection and seal design.