7.0
44 (2)
Ministerie van Verkeer en Waterstaat
Dienst Weg- en Waterbouwkunde
Directoraat-Generaal Rijkswaterstaat
EQUIVALENT LAYER THEORIES
"State of the art report"
W-DWW-94-904
NIET UITLEENBAAR
BUITEN DWW
BIDOC DWW - 015-2518363
B
I
D
O
C
(bibliotheek en documentatie)
Dienst Weg- en Waterbouwkunde
Postbus 5044, 2600 GA DELFT
Tel. 015 -2518 363/364
1 .Ü-UH
I
I
I
1
1
I
I
EQUIVALENT LAYER THEORIES
"State of the art report"
A.C. Pronk
Road and Hydraulic Engineering Division
Rijkswaterstaat
The Netherlands
(bibliotheek en documentatie)
I
I
Dienst Wfig- en Waterbouwkunde
Postbus 5CM4,2S0O GA OL1.FI
Te!. 015-2518 363/564
I
i
i
f
f
l
i
i
I
I
i
Disclaimer (Dutch)
Dit werkdocument wordt uitgegeven om geïnteresseerden de gelegenheid te bieden
om van de voortgang van het desbetreffend onderzoek, e.d. kennis te nemen.
Benadrukt wordt dat de gezichtspunten in dit werkdocument niet noodzakelijk
overeen behoeven te komen met de officiële gezichtspunten of het beleid van de
directeur-generaal van de Rijkswaterstaat Met de in dit werkdocument gegeven
informatie dient derhalve met de nodige voorzichtigheid te worden omgegaan,
aangezien de hierin vermelde conclusies in de loop van het verder onderzoek of
anderszins mogelijk herzien dienen te worden.
I
I
I
I
I
I
I
i
t
i
i
§
i
i
Abstract
An overview is given on the most known equivalent layer theories. In the equivalent
layer theory, overlaying (pavement construction) layers with different thicknesses and
moduli are combined into one layer with an equivalent thickness Heq.
Special attention is given to the applicability in case of calculations using the Hogg
and Westergaard models for concrete pavements. It is shown, that in principle the
equivalent layer thickness theories can not be used in the Westergaard model
because this model (Winkler foundation) is not comparable with the linear elastic
isotropic half space or multi layer model (Burmister model) on which all the
equivalent layer theories are based.
Only in the case of the "Supplementary Theory" model of Westergaard in which the
reaction of the foundation is 'corrected' to that of a linear elastic half space (HoggJeuffroy model) a (unique) relationship can be established between the modulus of
subgrade reaction k and the Youngs modulus Eo for both the deflection and the
stress. Therefore the relationship given by Eisenmann : k = Eo / Heq is not correct.
Contents
1. Equivalent Layerthickness Theory of Palmer and Barber
2. Odemark's Equivalent Layerthickness Theory
2.1 Deflections
2.2 Stresses and Strains
3. Ullidtz's elaboration of Odemark's Theory
3.1 General
3.2 Special application for a stress dependent subgrade modulus
3.2.1 Introduction
3.2.2 Strain calculations in a two layer system for a circular load
3.2.3 Calculations of deflections
4. Other variants on the Equivalent Layer Thickness Theory
4.1 General
4.2 Nijboer's Equivalent Layer Thickness Theory
4.3 Pronk's Equivalent Layer Thickness Theory
4.4 Comparison of the several approaches
4.5 Thenn de Barros Equivalent Layer Thickness Theory
5. Concrete Pavements
5.1 General
5.2 Two layer model
5.3 Three layer model
References
Appendix I ; Overview of formulas
1
i
I
1. Equivalent Layerthickness Theory of Palm er and Barber
In this not so well-known theory [1] the equivalency of the layers is based on a
"bending" stiffness, that is to say, each horizontal infinite layer is characterized by:
1/3
L
= H±
(1)
(1-v?)
The behaviour of a layer denoted by: (Hir Eif v) is said to be equal of that of another
layer with the parameters (Hkr Ekf vk) if the following relation is valid:
HiQi = HkQk
or
1/3
(2)
1/3
(1-v?)
If the layers are numbered starting from above (surface : 1 ; subgrade : n) the
procedure is as follows:
The equivalent layer thickness (Heq,) of all layers above layer (i+1) with
characteristics (H kv E M ,v k1 ) is calculated using equation (3).
Heqd
=
H.1
Q,
X
x-
gj-1
•H,
x•i+1
(3)
J=l
The deflection W is now written as the summation of the compressions in all
the layers above the subgrade and the deflection of the subgrade:
j=n-l
W =
+Wn =
Wn
The compression in a layer with number i (i=1,2,... n-1) is calculated using the
deflection formulas for a half space W{zrE,v} (see Appendix I) according to:
(5)
The deflection of the subgrade is calculated with:
Wn =
According to [1] this approximation of the deflection in a linear elastic multi
layer system is quite sufficient if the ratio of the Youngs moduli of layer i and
layer i+1 is bigger than 3 (E/El+1 > 3).
2. Ödemark's Equivalent Layerthickness Theory
2.1 Deflections
In the equivalent layerthickness theory of Ödemark [2, 3] two new equivalent layer
thicknesses are calculated for each pavement layer (except of course for the infinite
subgrade).
1/3
(7)
Heq1 =
Geqi = f2Hi
with f± and f2 are
constants
and E*+1 = F i EUk, Huk, \>i+k }
(8)
k = 1, 2, 3, . . up to the subgrade modulus Eo
The second equivalent layerthickness Geq, is only used in the calculation of the
(vertical) compression of.a construction layer. The first equivalent layerthickness Heq,
is known as the equivalent layerthickness.
Calculation of E-^
M and v^ ui
Odemark did not give the actual calculation 'for the replacing Poisson ratio v'M
because in his original work all the Poisson ratio's v, were taken equal.
In view of the small influence of the Poisson ratio on the calculations it can be
justified to put vM = vM.
However, the calculation of the replacing modulus E*^ is quite complex.
The calculation is based on the assumption that the deflection at the top of layer i
can be calculated on one hand as the deflection in a half space with modulus E*, at a
depth z,* = I[Heq k ] from k = 1 to k = (i-1) and on the other hand as the sum of the
compression of layer i and the deflection at the bottom of layer i.
I
I
I
The compression of layer i will be calculated in this case as the difference between
the deflections in a half space with modulus £•, at the depths z(* = §[Heqk] from k = 1
to k = (i-1) and z," = zs* +
The deflection at the bottom of layer i is calculated as the deflection in a half space
with modulus E*M at a depth z,'** = §[Heqk] from k = 1 to k = i. By performing this
procedure for all layers a relation will be established between E„ E*, and E*^, for all
values of i. Because the, for the layer j just above the subgrade, modulus E*H equals
the subgrade modulus Eo all the replacing moduli E*, can be calculated.
This procedure will be explained for a three layer model with the parameters E1r vv
H v E2, v2, H2, Eo, vo. The deflection in a half space with modulus E and Poisson ratio
v at a depth z is defined as W(z,E,v).
For the first pavement layer the two equivalent layerthicknesses are defined by:
1/3
j
i'
Heqx =
E
v?)
(9)
Geq± =
The two equivalent layerthicknesses for the second pavement layer (just above the
subgrade) are given by:
Heq2 =
f±H2x
E2 ( 1 - vg)
1/3
(10)
Geq2 = f22H.
"2
The compression of the two pavement layers are calculated as follows:
AWX = W{0,E1,v1)
- W
{Geqx,E1,v1)
(ïi)
AW2 = W {Heq1, E2,v2)
- W iHeq1 + Geq2, E2, v2)
The deflection at the top of the subgrade is determined by:
Wo =
(12)
The yet still unknown replacing modulus E2* is determined by the requirement that
the deflection at the top of the second pavement layer (AW2+W0) equals the
deflection in a half space with the modulus E 2 at a depth z* = Heq, : W {Heq1rE2\v2}
AW2 + Wo =
(13)
The deflection W in a halfspace (£,v) at a depth z due a homogeneous divided
circular load P with radius a is given by:
Wiz,E, v) =
with
(1 + v) P cos(x) x 1 +
- 2v
naE
1 + s i n (x)
tg{x) = —
d.
(14)
The term in equation (14) which contaïns the factor (1-2v) disappears for v = 0,5. In
that case (1^=1^=1^3=0,5) the requirement \N(Heqv£*2,vJ = AW2+WO
will lead to equation (15):
f{a2
+ Heql) . E2
a
(15)
(Heg1 + Heq2)2)
.Eo
a
(Heq1 + Geq2)2)
y/(
.E2
1/3
with
Heq1 = f1H1x
1/3
Heq2 = fxH2
(16)
12.
'Jo
Geq2 = f2H2
Usually a value of 0,9 is taken for the constants f, and f2. In table 1 the ratio E2'/Eo is
giyen for several combinations of the ratio's E,/Eo, E2/Eo, H / a and H2/a. The ratio
E2*/Eo can be seen as the factor by which the bearing capacity of the subgrade (E*2 is
a kind of effective subgrade modulus) has been increased by the application of the
foundation (H2,E2,v2). As shown in table 1 the ratio E/E,, has limited influence on the
increase for low values of the ratio E2/Eo. The increase becomes even smaller if the
stiffness modulus of the first layer E, increases. It also appears that the influences of
the ratio's H/a and H2/a are mainly determined by the ratio H,/H2 which implies that
the increase factor E27EO nearly depends on the radius a of the load plate.
2.2 Stresses and Strains
For the calculations of the stresses and strains in the interface between two layers the
equivalent layer thickness Heqi is used. The formulas for the stresses and strains in a
halfspace (E,v) on the center line at a depth z due to a homogenous divided circular
load P are denoted by :
ar(z,v)
: Radial stress = Tangential stress (center line)
Vertical stress
Radial strain = Tangential strain (center line)
Vertical strain
Complete friction is assumed between the pavement layers -> the horizontal strain at
the bottom of layer i (erl) equals the horizontal strain at the top of layer i+1 (eM+1).
Table 1.
Combinations
Increase factor E2*/Eo
H/a
H2/H,
E,/Eo
1
2
0,5
40
1,10
1,19
1,28
1
2
0,5
20
1,12
1,24
1,35
1
1
1,0
80
1,15
1,29
1,44
1
1
1,0
40
1,18
1,35
1,53
1
1
1.0
20
1,21
1,42
1,64
2
2
1,0
80
1,16
1,31
1,46
2
2
1,0
40
1,19
1,38
1,58
2
2
1,0
20
1,23
1,47
1,73
2
1
2,0
80
1,27
1,57
1,90
2
1
2,0
40
1,32
1,67
2,09
2
1
2,0
20
1,36
1,79
2,30
H2/a
E2/Eo = 2
E2/Eo = 8
E2/Eo = 4
The horizontal strain at the bottom of layer i is calculated as :
k=i
z=
k=l
The vertical strain at the top of layer i+1 (evi+1) is calculated as:
k=i
e
v , i + i = e v ( z , £•;+!, v i + 1 } with
z = £
Heqk
(18)
k=l
The horizontal and vertical stresses at the top of layer i+1 (a-hM-, and <7vM ) are
calculated as:
r, i + 1
= oriz, v i+1 ) and
k=i
z
(19)
=
k=i
The horizontal strain at the bottom of layer i (sri) can not be calculated by:
k=i-l
ezi
* £ r ( z , ^ i / v i } with
z = J^ [Heqk] + Geq£
Jc=l
(20
>
The horizontal stress at the bottom of layer i (<ro) can be calculated with the aid of
the following relations:
for the lower layer i :
and for the upper layer:
(21)
interface : :
frietion
continuity
For a two layer system (HirE1f 1^=0,5; Eo=100 MPa,vo=0,5) the horizontal strain has
been calculated at thé bottom of the first layer due to a circular load P = 50 kN. The
En value was varied from 2000 to 8000 MPa and the ratio H/a was varied from 0,5
to 2,0. The results of the calculations are given in table 2 together with BISAR
calculations.
I
I
Table 2.
cr>1 = £r {Heq1r 100,0,5}
H/a
E1=2000
[MPa]
^=4000
[Mpa]
£r.i
E1=8000
[Mpa]
BISAR
E1=2000
[MPa]
E1=4000
[MPa]
E1=8000
[MPa]
I
0,5
1647
1320
990
780
531
343
1,0
705
482
320
421
257
152
I
2,0
209
135
86
161
93
53
1
I
i
1
i
i
i
3. Ullidtz's Elaboration of Odemark's Theory
3.1 General
Ullidtz [4] extended the theory of Odemark in one way by a more accurate
determination of the correction factor f, in the equivalent layerthickness calculation of
Odemark. Ullidtz denoted this factor as f. Instead of a constant value (= 0,9) as in
the original Odemark's theory, the correction factor f depends on the ratio of the
layerthickness H, the radius a and the obtained quantity. An overview for a two layer
systeem (HvEvvvEo,vj
is given in table 3.
Table 3
Correction cofficients f
All Hf/a values
H,/a = 2/3
H/a = 4/3
W
[um]
[Pa]
[m/m]
[m/m]
[m/m]
[m/m]
0,9
0,9
1,25
1,00
1,00
0,90
H/a > 2
Gr
[m/m]
0,90
[m/m]
0,85
A value for f of 0,8 is recommended in the case of three and more layer systems with
one exception for the calculations of the strains and stresses in the first interface
where a value of f = 1,0 is advised. However, it should be remarked that Ullidtz did
not follow the original concept of Odemark for the determination of an equivalent cq
replacing modulus E,' in the case of three and more layer systems. The implication is
that the equivalent layerthickness of layer i depends only on the material properties
of layer i+1 just below layer i and not on the layers below layer i+1.
1/3
Heqi = fxHix
(22)
Furthermore, in many cases Ullidtz neglects the influences of the Poisson ratios.
Besides the variation in the correction factor f the elaboration of Ullidtz resembles
more the equivalent layertheory of Palmer and Barber.
However, the merit of the elaboration of Odemark's theory by Ullidtz is the fact that
Ullidtz also developed an application of the equivalent layerthickness theory for multilayer systems for cases in which the moduli of the subgrade and foundation are stress
dependent [4]. Next to this important application, Ullidtz developed a methodology
for the calcultion of stresses and strains in the interfaces of multi-layer systems, due
to several circular loadings (twin wheel). In this methodology, the pointload formulas
of Boussinesq are used, but the equivalent layerthickness Heq (or actual depth z) is
replaced by a fictive depth z \ in order to 'simulate' the influence of the actual
circular load.
The second term at the right hand side of the equation above, resembles the
influence of the circular loaqV -Aa. iiffpjQyepieiQt'fer 3his transformation is
Pronk [5].
Heq
* = Heq
x
Heq+X2 a
3.2 Special application for a stress dependent subgrade modulus
3.2.1 Introduction
The equivalent layer methodology of Ullidtz can also be used if the subgrade
modulus Eo (= material modulus) is stress dependent i.c. the modulus can be
described by:
Eo = C0*x
(25)
in which <x, is the major principal stress, a0 is a reference stress of 0,1 MPa and C'o is
a constant. The procedure for a two layer system is as follows:
I
I
I
I
1
I
1
I
1
I
I
I
I
I
I
1
I
I
I
I
Surface modulus E*Oheq at the interface:
n
O.
(26)
eg
EZ* 'eg = [ 1 - 2 2 3 ] xC0*x
Equivalent layer thickness:
(27)
E,o.h,eg
Major principal stress at the interface:
3
2
2
°1 h
~
ö
'
= oT, n x 1
a
-
(28)
•
h
' e«r
1
+
2
a
The calculation is as follows:
1) Take an estimate for heq.
2) Calculate cr,Mer
3) Calculate E'oheq.
4) Calculate a new estimate for heq.
The correction coëfficiënt f is taken equal to 1 for the calculation of stresses and
deflections. For the calculations of strains the following values depending on heq [m]
should be used:
ehorizontal
^vertical
< 0.1
1,25
1,15
0,1<-<0,3
1,10
1,05
>0,3
1,00
1,00
If the construction consists out of more than one layer above the subgrade, the
equivalent layer thicknesses should be calculated as:
*^-E
ii i
•
•EQ,
1
3
(29)
3.2.2 Strain calculations in a two layer model for a circular load
The required formulas are:
ov(0,z)
= ov(0,0)
(z/a)
(z/a)
x 1 -
1+v xo v (0,0) x (z/a)
E
th(0,z)
x
(z/a)2)
1 - ( l - 2 v ) x (1+
( 1 + U / a ) 22)\
(30)
(31)
2
=
(32)
For the calculation of the vertical subgrade strain the depth z is replaced by the
equivalent layer thichness heq (calculated with the 'strain' values for the correction
coëfficiënt f) and the (material) modulus Eo should be used.
In case of the horizontal asphalt strain two procedures are possible if complete
friction exists between the layers. In both methods the equivalent layer thichkness heq
should be calculated with the f values for the horizontal strain.
method 1:
- Calculate a "vertical stress" av(0rheq).
- Calculate a "vertical asphalt strain" ev(0,heq) using the asphalt
modulus Ev
- Calculate the horizontal asphalt strain eh(0rheq) from these figures.
method 2:
- Calculate a "vertical subgrade strain" ev(0,heq) using the subgrade
material modulus Eo.
- Calculate a "horizontal subgrade strain (= horizontal asphalt strain)
eh(0,heq) using the subgrade material modulus EO.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
3.2.3 Calculations of deflections
For the calculation of the deflection the value for the correction coëfficiënt f is taken
equal to 1. The deflection at the interface with the subgrade can be calculated with
the following formula:
xaxo7(0,0)
ö(0,z) =
(33)
1+(l-2v ) x
\
For the modulus E the surface modulus E*Oheq should be used instéad of the material
modulus Eo.
For relatively large heq values (>a) it is possible to use the simple Boussinesq formulas
for a point load P on top of a half space. To simulate the circular load the depth z
should be replaced by the formula:
0, 6 a:
(34)
eq
Using the point load formulas it is possible to calculate the strains and stress due to a
twin wheel like in the Shell Pavement Design Manual. This procedure is used in the
program "I LEAP DEEP".
4. Other Variants on the Equivalent Layerthickness Theory
4.1 General
For the reduction or simplification of a multi layer system to a two layer system or
even to a half space several other theories than given here are known. An extended
overview is given by Poulos [6]. In this paragraph only the equivalent layerthickness
theories of Nijboer [7], Pronk [8] and Thenn de Barros [9] are dealed with.
4.2 Nijboer's Equivalent Layerthickness Theory
A pavement construction can almost be simulated by or modelled into a three layer
system (H1,Evv1,H2rE2,v2,Eo,vo). When Nijboer developed his theory, the multi-layer
model of Burmister was already known, but no computer programs like BISTRO or
BISAR were evailable at that time. For the calculations of strains, stresses and
deflections graphs were used. These graphs were developed by Odemark and were
only valid for two layer systems. Nijboer developed a methodology for the
combination of the two upper pavement layers to one layer in order to be able to
use the Odemark's graphs. Nijboer used his theory (in which v1~v2=ve,} especially for
the stiffness measurements which were carried out with ther Road Vibration Machine
(RVM).
The point of departure in all most each equivalent layerthickness theorie is the
requirement that the stress distribution in the interface will be the same. Odemark
tried to fulfill this major requirement by the requirement that the 'bending moments'
(expressed by the quantity E.h3) of the original layers and the equivalent layer are the
same. Nijboer used the same approach in the replacement of two layers by one
equivalent layer but he used also a second requirement for the place of the neutral
axis. His second requirement was that the actual depth of the neutral axis should also
be the same. However, at this point it should be noticed that in contrast with
ordinary beams, it is not possible to talk about a neutral axis in the case of a multilayer system. In that case several neutral "surfaces" can occur.
The properties of the new equivalent layer (H',E\v'=v,=v2) are determined
by the following two equations:
= (A + N)2x A* + 4A3N + 6A2N + AAN + N2 x
(A2 + 2A + N)3
x
(A + N) . (N + 1 )
with
A = —-
and
2
N =
H
Hl
E
2
Nijboer (I)
Besides this replacement of the two pavement layers by one different layer (Nijboer I:
H*,E*)r Nijboer also gave another equivalent one layer substitution (Nijboer II). In his
second method, Nijboer dropped the second requirement for the place of the neutral
axis and replaced it by the requirement, that the equivalent layerthickness H* should
equal the sum of the layerthicknesses of the two original layers.
His second approach is given by the following two equations:
E* (II) =
A
'
+
6A N+
4 M
*A3N+
"
3
(A+l) (A + N)
H* (JJ)
+ j y 2
xK
O7)
- K + H9
(38)
Nijboer(II)
4.3 Pronk's Equivalent Layer Thickness Theory
Pronk [8] gives a supplement on the methodology of Nijboer. For the second
requirement, Pronk suggested that the modulus of the equivalent layer, should equal
the modulus of the lower original layer. This requirement leads to the following two
equations for the calculation of the equivalent layer (H*,E*):
E*
H* =
(39)
=
+ 6A2N + 4AN + N2
(A + l ) 3 x (A + N)
1/3
x (Hr + H2)
(40)
Pronk
The last equation resembles more the equivalency as used in the theory of Odemark.
Although these three methods (Nijboer (I), Nijboer (II) and Pronk) have been
developed for the simplification of a three layer model into a equivalent two layer
model, the methodology can easily be extended to more than three layers. The
procedure is as follows:
1)
The first (H^E,) and second (H2,E2) pavement layers are transformed into one
new equivalent layer (H2\E2*).
2)
This new layer and the next layer
new equivalent layer (H3*,E3*).
3)
The steps above are repeated until all layers above the subgrade (E,,) are
transformed into one new equivalent layer (H*,, £').
are transformed into the following
After this procedure the solution techniques for a two layer model can be used.
4.4 Compar/son of the several approaches
A comparison has been made for the several approaches, using a four layer model of
which the material properties and dimensions are given in Table 4a and 4b.
Table 4a: Construction A "Full depth asphalt"
Load P = 2 x 25 kN ; SPDM
Complete friction between layers
Layer number
E, [MPa]
Asphalt 1
2000
0,35
0,04
Asphalt 2
4000
0,35
0,04
Asphalt 3
6000
0,35
0,10
Subgrade o
100
0,35
v,
[m/m]
H, [m]
For these two four layer models the horizontal strain (er) at the bottom of the lowest
pavement layer (i=3) and the vertical strain (ev) at the top of the subgrade have been
calculated for the several equivalent layerthickness approaches. The results are given
in Table 5a and 5b
Table 4b: Construction B "Asphalt on top of an unbound base"
Complete friction between layers
Load P = 2 x 25 kN ; SPDM
H,
[m]
Layer number
E, [MPa]
Asphalt 1
2000
0,35
0,04
Asphalt 2
4000
0,35
0,04
Asphalt 3
6000
0,35
0,10
400
0,35
0,20
100
0,35
Base
4
Subgrade o
Vi
[m/m]
oo
Table 5a : Results for the full depth asphalt construction (A)
e
BISAR
Nijboer
(I)
Nijboer
(II)
Pronk
Ullidtz
er
168
127
196
171
221 (185)
ev
530
320
561
525
535 (455)
Table 5b : Results for the asphalt construction on top of an unbound base (B)
e
BISAR
er
122
85
138
126
112 (96)
ev
385
212
377
373
280 (233)
Nijboer
(I)
Nijboer
(II)
Pronk
Ullidtz
For the correction coëfficiënt f in the approach of Ullidtz a value of 0,9 has been
used. The strain values (between brackets) are calculated if a value of f = 1,0 is
adopted. Better comparison is reached if for construction A the values 1,065 (er) and
0,905 (ev) are used and the values 0,85 (er) and 0,745 (ev) for construction B. base.
I
I
I
I
I
I
I
I
I
I
1
I
I
I
I
I
I
I
I
I
The replacing properties for the new equivalent layer in the approaches of Nijboer,
Pronk and Ullidtz are given below:
BlSAR Calculations for two layer system with Eo = 100 MPa and v0 = 0,35
Nijboer (I)
Nijboer (II)
Pronk
E* = 1374 Mpa;
E* = 3951 Mpa;
E* = 6000 Mpa (=E3);
H* = 0,405 m
H* = 0,180 m (= H1+H2+H3)
H* = 0,1591 m
Calculations with "I LEAP DEEP" : Boussinesq formulas (point load) with a correction
for the equivalent depth (circular load).
Ullidtz 4a
Heq= 0,6369 m (er asphalt and ev subgrade)
Eeq= 100 MPa
Ullidtz 4b
Heq= 0,4012 m (er asphalt)
Eeq= 100 MPa
Ullidtz 4b
Heq= 0,9543 m (ev subgrade)
Eeq= 100 MPa
In case of the "Nijboer" and "Pronk" formulas, the first step was to take the upper
two asphalt layers together into an equivalent layer. The second step was to combine
this equivalent upper layer with the lowest asphalt layer into a new equivalent layer.
In the case of the "Pronk" formulas, this approach gives satisfying results for a
general modelling of different asphalt layers (differences in E moduli due to differencesin temperature) in a common pavement type (asphalt, unbound foundation and
subgrade) into one new equivalent asphalt layer [8].
4.5 Thenn de Barros's Equivalent Layer Thickness Theory
Next to the approaches developed by Nijboer and Pronk, another equivalent
approach is developed by Thenn de Barros [9]. In his approach Thenn de Barros uses
the
requirement for the equivalent layerthickness H" as Nijboer (II), but the formula for
the replacing stiffness modulus E* is different. Nevertheless, the answers for E* do not
differ much, as can be seen in Table 6.
n.1/3
v2
E* =
H2
N1/3 + A
1 +A
H* =
13
-
(41)
3
v
<E2
H2
Thenn de Barros
(42)
Table 6 Ratios E*/E2 for the Nijboer II and the Thenn de Barros model.
Ratios E/E2 ; Y\2/H,
E/E2
Hj/H,
Ratio
E*/E2
Thenn de Barros
Nijboer (II)
0,5
0,5
0,64
0,69
0,5
1,0
0,72
0,69
0,5
2,0
0,81
0,71
2,0
0,5
1,62
1,43
2,0
1,0
1,44
1,38
2,0
2,0
1,28
1,37
5. Application of the Equivalent Layer Theory in Concrete Pavement Design
5.1 General
The equivalent layerthickness approaches, as described in the former paragraphs, are
sometimes used in calculations for concrete pavements. Often, a relation is made
with regard to the Herz/Westergaard model, in which the behaviour of the subgrade
is decribed by one modulus of subgrade reaction (k) [10, 11] and the Hogg/Jeuffroy
model [12, 13]. However, one forgets that the equivalent layer approaches are
fundamentally based on a linear elastic multi-layer model (Boussinesq/Burmister), in
which complete friction between the layers is assumed. In the Westergaard and Hogg
models, the pavement is modelled as a thin (or thick) plate frictionless resting on the
subgrade. Hogg also gives the formulas for a two layer model, in which complete
friction between plate and subgrade is assumed, but this model is only used by
Wiseman [14].
The subgrade in the (ordinary) Westergaard models, is modelled by a spring constant
k (modulus of subgrade reaction). The deflection W is linear proportional to the
vertical stress av. In the case of a point load, both the stress and the deflection will
go to infinity at the surface in a halfspace. So, the modellation of the subgrade by a
system of uncoupled/linked springs is mainly used for pavement layers (plates) resting
on the subgrade. The subgrade is considered to be a dense liquid (Winkler foundation) and there is no horizontal stress transfer in the subgrade, in contrast with a
linear elastic subgrade, in which horizontal load transfer occurs (Poisson ratio). In a
linear elastic layer a deflection/deformation can occur, without the presence of a
vertical stress.
In spite of these differences the linear elastic theory of Burmister has been misused
for the 'determination' of relationships between the modulus of subgrade reaction k
and the Youngs' modulus Eo as will be shown.
The starting point will be a halfspace loaded by a homogenous divided load P
(constant contact stress cr0) on a circle with radius a.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
•
The definition for k is given by:
0
Ir
= JS
(43)
I
W
Therefore k can be calculated from the equation belovv (assuming that the deflection
below the loading plate is constant).
k =
Z— 01 by
na2N0 .
(44)
I
•
I
M
P = 2 7t fovir}rdi
o
=
2nk(w{r)idr
o
Outside the area r=a the vertical stress at the surface is zero. So, in case of a Winkler
foundation, no deflection may occur at the surface for r > a (in the case of a halfspace). However, in a linear elastic halfspace the surface deflection for r > a is not
zero. In spite of the non-compatibility of these two types of foundations,
relationships between k and Eo are often made. The k values based on these (wrong)
relationships will be denoted by k*.
Homogeneous Isotropic Linear Elastic Halfspace :
I
-
Flexible loading (constant contact stress cr0 over the plate)
-> The deflection under the loading plate is not constant!
na2wc
8
•
-
Rigid loading (constant deflection Wo under the plate)
"
•j
P
=
na2Wo
1
_
•
•
(45)
In this formula the deflection Wc is the central deflection at r=0.
ic# =
•
- v2)
2a(1
2E
na (1 - v2)
Wiseman uses a variant on the integral definition for the calculation of k given by:
.ie»
^ P
a
r , ^
27i / w{i)rdx
l
»
r , ,
2%
(47
>
\wii)xdr
(outside the plate (r>a) the deflection is zero for a halfspace).
The two first relationships depend on the radius a and so the calculated parameter kwill not be a constant, but will be inversely proportional to the applied radius of the
I
I
I
loading plate. Therefore, the two mechanical models (Burmister-Westergaard) are
principally not compatible. The third definition has its advantage when it is used for
two or more layer systems (see paragraph 5.2), in which the compressions of the
pavement layers are negligible (plates). This definition gives a right k value, if the
subgrade reacts as a Winkler foundation and the volume of the whole deflection
bowl is measured. However, it should be noticed that for large distances from the
center, the surface deflection will become negative [15] in the Westergaardmodel.
I
™
Westergaard also presented a model in which a correction term was introduced in
order to get a better match between theory and practice. In this so called Supplementary Theory, he presented an extreme case in which the subgrade reaction
resembles that of a linear elastic subgrade (Hogg model; no friction). Westergaard
himself stated that in that case the quantity k = k x L became a constant which
implied that the original concept of k = constant has been leaved.
—
I
Therefore the use of the linear elastic multi-layer model for the establishment of
relationships k - Eo, can only be justified if the extreme case is used of the
Supplementary Theory. It is recommended to use in that case a relationship between
k x L = K - Eo rather than a relationship between k and Eo and also to calculate the
stresses and strains using the Westergaard formulas, which are expressed in this
quantity K.
Besides the Westergaard model (Winkler foundation), the Pastemak model [16] can
also be used. This model can be considered as a general Winkler foundation with a
kind of horizontal coupling between the vertical springs k, denoted by a damping
constant k*. The Westergaard model is a special case of the Pastemak model (k' = 0).
|
•
*
•
«
J
m
|
•
8
•
I
5.2 Two Layer Systems
I
In order to get still a relationship between k and Eo, one might calculate the vertical
stress and deflection at the interface between the pavement layer and the subgrade
in a linear elastic two layer system. The parameter k will be defined as the ratio of
the calculated stress and deflection. However, the deflection in a point is not solely
determined by the vertical stress, but this fact is completely disregarded.
In a linear elastic two layer model (HvEvvvEo,v() the deflection W can be written as
the sum of the deflection in the subgrade (Wo) and the compression of the pavement
layer (AW,) -> W = Wo + AW, . If the equivalent layer theory of Odemark will be
used, these two terms can be calculated with the aid of the formulas for a halfspace.
AW1 = wiO,E1,v1} - W{Geq1,E1,v1)
Wo =
wiHeqlfEo,vJ
wi th
Geqx = 0 , 9 / ^
I
"
m
•
I
(48)
1/3
and
I
*
•
•
Heq1 = 0, 9 Hx x
I
I
I
I
•
•
•
Because on one hand there will be p.d. no compression of the pavement layer in the
plate theory and on the other hand the compression for stiff pavement layers like
concrete will be very small in the linear elastic theory it is justified to neglect the
compression term AW V Therefore a definition for the analog parameter k* might be:
# =
|
I
f
.
I
1
1
1
•
°
[1 + s i n ( P ) 3 x [l - sin 3 ((3) 3
a [1 + vo3 xcos(P) x [2 - 2v o + s i n ( p ) 3 (5°>
tan<P, -
The layerthickness H, will be nearly always bigger than the radius a :: (H, > a) and
the same is even more true for the ratio of the moduli E, and Eo :: (E, » Ë^. These
conditions imply that :
COS(P)
°
. .._.
sin(P) =
a
HeQl
(51)
Heq,
+ a2
Before these approximations are used in the relationship above, the term
(1 - sin3(/?)) has to be rewritten:
[ l - s i n 3 ( P ) 3 = l - s i n ( p ) x [ l - c o s 2 ( P ) ] *cos 2 (P)
•
M9)
Using the formulas for a halfspace with a circular load P (constant contact stress a
as given in the appendix, thè following relationship will be found:
,:# _ E
I
oviHeqi,Eo,vo)
M H E J
(52)
Using the approximations above the following relationship will be obtained:
(1 + v o ) x (3 - 2v o ) xHeqx
(53)
'2 (1 - v2o) xHeqi
»
(3 - 2 v o )
If the Poisson ratio vo for the subgrade is taken equal to 0.5, the relationship
established in this way will be:
I
I
A wrong relationship between fc*(denoted as £**) and Eor is given by Eisenmann [17J:
E
=
°
(55)
Heq1
I
Eisenmann deduced this relationship by combining the equivalent layerthickness
theory of Odemark, together with the Boussinesq formulas for a point load P on a
halfspace [18]. For positions on the centerline the vertical stress <7v(z) and deflection
W(z) are given by:
for
v
°
= 0 f 5 ifc follows
_
o iz)
•
_
E
—ïj—r- = —WXZJ
z
For the depth z Eisenmann uses the equivalent layerthickness Heq, for which in
reference [18] the Poisson ratio v, is taken equal to 0,5. In later reference [17]
Eisenmann uses a more comon value of 1/6 « 0,16 for the Poisson ratio v, of the
(concrete) pavement layer, which leads to:
1/3
*1
Heqx = 0 , 825
(57)
The large difference between the two relationships (^-E o and fc*#-E„) can be traced
back to the use of a pointload P instead of a circular load P.
Pointload
P
Circular
3P
_
2
load P (z>a)
P_
71 Z2
2 71 Z
vo) x ( 3 - 2 v o )
2 P ( 1 - v2o)
(58)
2nzEo
2 n zE
o
Afterwards calculated fc*# value is used in the Westergaard model. In the Westergaard
model a circular load is applied. So, only for this reason alone the procedure as
suggested by Eisenmann ought to be dissuaded.
Another objection against the procedures above is the fact that, in the Westergaard
model, the pavement layer (plate) rests frictionless on the foundation i'.c. the
subgrade which is not the case in the linear elastic models used in the equivalent
layer theory.
|
•
•
I
I
I
If as a starting point the Hogg model without friction between plate and the linear
elastic subgrade (E^vJ is chosen, an impression of the influence of this ignoration
can be obtained. According to Eisenmann [18] at the interface due to a point load
the vertical stress av {H,} is given by:
x
6EO(1 - y\)
Ex(l
1/3
(59)
-" v|)
Using Odemark's expression for the equivalent layerthickness Heq, this equation can
be rewritten as:
ov = 1,078 x
(60)
2 it Heql
In the same way the deflection W {0} = W {Hl} can be obtained:
W{0}
= ^& x
SD
= 3,955x
2 D ( 1 - vl)
2/3
PU - v|)
(61)
2 7i EoHeqx
+ v o ) x (3 - 2v o )
2nEoHeqx
3 , 9 5 5 x (1 - v o )
(3 - 2v o )
The relationship for /c*#f is given by:
3,955
'o
x
= Cx
(62)
The value of the coëfficiënt C in the relationship above is given in Table 7 as a
function of the Poisson ratio vn.
Table 7
C =
Coeffiient C as a function of the Poisson ratio
0,15
0,25
0,35
0,45
0,50
I
0,836
0,872
0,932
1,025
1,090
f
It should be remarked that both the stress av as well as the deflection W in the Hogg
model (no-friction), do not differ much from the stress av and deflection W in a
linear elastic two layer model (Burmister) with complete friction between the two
layers and using a point load. The ratio WHogg/WBurmister = [3,955 x(1-iO3'/K3-2«O]
varies from 1,245 at vo = 0,15 to 0,989 at vo = 0,50.
The influence of friction or no-friction between the layers in the Hogg model can be
obtained from the two following expressions for the deflection:
Hogg;no-friction
*Hogg; friction
o
= O, 1 9 2 5 X
(63)
2 (1 - Vl)
W(no-friction)
W {friction)
and
( 1 + v j x (3 - 4 v J
_ (1 + vo) x (3 - 4 v
4X (1 - vl)
The ratio of the defiections Kv,) are presented in table 8 as a function of the Poisson
ratio vo.
Table 8
1/
"0
The ratio of the defiections in the two Hogg models (friction versus nofriction) as a function of the Poisson ratio
=
f(O =
0,15
0,25
0,35
0,45
0,50
0,793
0,763
0,723
0,668
0,630
As already stated above, it is quite curious that a remarkable resemblance exists
between the stresses, strains and defiections in the Hogg model, with no-friction
between the layers and a linear elastic two layer model with friction between the
layers, when in both models a pointload P is applied.
From the exercise above, it can be concluded that the (original) Westergaard model is
not compatible with a Hogg model or a linear elastic Burmister model. Only in the
extreme case of the Supplementary Theory of Westergaard [11], which resembles the
Hogg model (no-friction) [12], a relationship between k and Eo might be established.
However, in that case, the parameter k has no real physical meaning and may only
be considered as a mathematical quantity.
By equalization of the defiections in both models the following relationship can be
obtained:
ie = 0 , 1 2 8 x
Heq1
for
v o = 0 , 5 i t follows:
(64)
k = 0,17x
Heg1
If the relationship above is used the extreme case of the Supplementary Theory of
Westergaard has to be used.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
5.3 Th ree Layer Models
A three layer model can be simplified into a two layer model by either:
A:
Combination of the two pavement layers into one equivalent layer
(Concrete layer on a foundation of lean concrete)
or by:
To consider the second pavement layer as an increase of the bearing
B:
capacity of the subgrade (Concrete on a foundation of unbound material)
Possibility A:
If the two pavement layers are taken together in order to calculate the stresses
and strains at the bottom of the second pavement layer the following
procedures can be used:
A1)
Equivalent layer theory of Barber and Palmer
(chapter 1)
A2)
Equivalent layer theory of Odemark
(chapter 2)
A3)
Equivalent layer theory of Ullidtz
(chapter 3)
A4)
Equivalent layer theory of Nijboer and Pronk
(chapter 4)
Possibility B:
If the second pavement layer is considered as an increase in the bearing
capacity of the subgrade the increase can be obtained using:
B1)
I
I
I
I
I
I
I
I
I
Equivalent layer theory of Odemark
(chapter 2)
Needless to say that, in all the theories mentioned above complete frictions between
all layers is assumed. For systems where no-friction between the first and second
pavement layer is assumed (Jeuffroy/Bachelez model), no equivalent layer theories
are found in the literature.
If complete friction between all layers is assumed, calculations at the top of the
subgrade can be performed, using the (total) equivalent layerthickness for layer 1 and
layer 2 which is given in the case yo-v^=v2 by:
Heq2
= 0 , 9 x H, x
+ H2x\
(65)
1/3
1/3
= 0,9x
X
H2
Si •
E
o
E2 • E2
The term H1.{(E1.EO)/(E2.E*2)}1/3 can be considered as an effective increase of the
second pavement layer. The expression above can also be written as:
.1/3
Heq2
= 0,9x
Er
(66)
1/3
H2 xE2
+ H±xE-
Besides the factor Eo/E*2 this procedure can be seen as the summation of the terms
H,E1/3
If there is no friction (complete slip) between layer 1 and 2 in analogy with the
expression above one might define an equivalent layer as:
1/3
(67)
Also in the case of three layer systems, the equivalent layer theory is misused to
obtain a /f* - Eo relationship based on the ratio <7V/W.
Eisenmann [17] gives the following relationship, if the compression of the second
pavement layer, may not be neglected.
Heq1 x
Heqt
Heq2
x(
' Heqt
Heq±
(68)
with
:
Heqt = Heq1 + Heq2
1/3
Heq1 =
1/3
; Heq2 = 0,9
xH2x
E,
The starting point for this relationship was a pointload and the second pavement
layer was considered as an increase of the subgrade properties. Furthermore, the
influences of the Poissonratios were ignored by taking vo - v, = v2 = 0,5.
In view of all these neglections the following more 'fundamentaP approach is advised
if one would establish a relationship for /f*:
v
Wo
For the calculations of HeqvHeq2, and Geq2 the Odemark's formulas are used.
(69)
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Palmer, L.A. and Barber, E.S.,
"Soil displacement under a circular loaded area"
Proceedings Highway Research Board, Vol, 20, 1940
Odemark, N.,
"Investigations to the Elastic Properties of the Soils and the Design of
Pavements to the Theory of Elasticity", Stockholm, 1949
Ullidtz, P.,
"En Studie af to Dybdeasfaltbeasestelser",
Ph.D. dissertion, The Technical University of Denmark, Lyngby, 1973
Ullidtz, P.,
"Some Simple Methods Determining the Critical Strains in Road Structures",
Technical University of Denmark, Department for Road construction,
Transportation and Town Planning, Lyngby, 1974
l
Pronk, A.C.,
"Berekening van Deflecties, Rekken en Spanningen in een Half Oneindig
Medium en in 2- of Meerlagen Systemen",
Report DIM-R-80-75, Road and Hydraulic Engineering Division,
The Netherlands, 1980.
Poulos, H.G. and Davis, E.H.,
"Elastic Solutions for Soil and Rock Mechanics",
John Wiley & Sons, Inc., New York-London-Sydney-Toronto, 1974
Nijboer, L.W.,
"Dynamic Investigations of Road Constructions",
Shell Bitumen Monograph no. 2.
Pronk, A.C.,
"Equivalente laagdikte theorieën",
Internal Report, TW-N-86-42, Road and Hydraulic Engineering Division,
The Netherlands, 1986.
Thenn de Barros, S.r
"Deflection Factor Charts for Two- and Three-layer Elastic Systems",
Highway Research Record, No.145, 1966
Westergaard, H.M.,
"Stresses in Concrete Pavements Computed by Theoretical Analysis",
Public Roads7, no. 2, 1926
Westergaard, H.M.H.,
"Analytical Tools for Judging Results of Structural Tests of Concrete
Pavements", Public Roads 14, no. 10, 1933
Hogg, A.H.A.,
"Equilibrium of a Thin Plate, symetrically loaded, resting on an Elastic
Foundation of Infinite Depth",
London, Edinberg and Dublin, Philosophical Magazine and Journal of Science
Vol. 25, (168), 1938.
Jeuffroy, G.,
"Note sur Ie Comportement des Chaussées", Annales des Ponts et Chaussées,
no. 3, 1959
14.
15.
16.
17.
18.
Wiseman, G.,
"Flexible Pavement Evaluation Using Herz Theory",
Transportation Engineering Journal, ASCE, Vol. 99, No.TE3, Proceeding
Paper 9921, 1973.
Pronk, A.C.,
"A Further Investigation into the Westergaard Formulas for Interior Loading
and the (Mis)use of the Modulus of the Subgrade Reaction k",
2nd International Workshop on Theoretical Design of Concrete Pavements,
1990, Siguenzq, Spain.
Pronk, A.C.,
"The Pastemak Foundation - An Attractive Altemative for the Winkler
Foundation",
Proceedings of the 5th International Conference on Concrete Pavement Design
and Rehabilitation, Purdue University, West Lafayette, Indiana, 1993.
Eisenmann, J.,
"Betonfahrbanen", Verlag von Wilhelm Ernst & Sohn, Berlin-MunchenDusseldorf, 1979.
Meier, H., Eisenmann, J. and Koroneos, E.,
"Beanspruchung der Strasse unter Verkehrslast",
Forschunsarbeiten aus dem Strassenwesen, Neue Folge, Heft 76, Kirschbaum
Verlag, Bad Godesberg.
APPENDIX:
1)
The
The
The
The
Overview of Formulas
Formulas for the deflections, stresses and strains in a halfspace due to a point
load P (Boussinesq).
horizontal distance to the centerline is denoted by: r [m]
depth below the point load is denoted by:
z [m]
distance to the point load is denoted by:
R [m]
angle between the centerline and the position r,z: a [°]
(70)
R = Jz2 +
sin(a) =
cos(a) =
R
(71)
_z
~R
2
W = P ( 1 + v) x 2 ( 1 - v ) + c o s ( « )
2nRE
(72)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
a = 3xPxcos3(g)
2uR2
I
(
3xcos(a) xsin2(a) |
I
px
(73)
1
2 ]
+ C O S (Ot)
2
2nR
ev = ^ x ( l + v) x 3cos 3 (g) - 2vxcos(g)
„ 6)
I
e
•
r
- - x ( l + v ) x
'
•
•
3
- 3 x c o s ( a ) + (3 - 2 v ) x c o s ( a ) _
• (77)
1
2v
~ , ..
1 + c o s (a)
- cos(a) + - —
For point on the centerline these formulas can be simplified to:
(79)
I
I
E
27i z
or{z) = oc(z) = - Px
°
I
I
(1
'
2 v )
4 11 Z 2
(H)
(1 - v) xor(z} -
er{zl = zt\
2)
\xoviz}
(83)
Formulas for the deflection in a halfspace due to a homogenous loaded
circular plate.
xPx
ƒ
_1_
2x(l-v)x[T 2 -r i ]-C 2 x
T
dy
X
\
\
1- [
a
i
y]
2 _
i-[^y] 2
\
\
a
i
(84)
v/l-- y 2
•
r2
z2
2
y
+
+ -JT2
r
On the surface (z=0) this equation reduces to a Hypergeometric serie.
(85)
(2n) ! T a r
(n\)2
3)
n +1
Formulas for the deflections, stresses and strains in a halfspace
on the centerline under a circular homogenous divided load P.
tan(p) = —
a
(86)
- 2v)
1 +
Wiz)
=P x ( l
+
v)xcos(P)x
n aE
w{0)
-
2P
(88)
- sin3(p)]
n
(89)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
P
ariz} = o t{z) =
na2
x
(90)
±-2ï. - (1 + v) xsin(p) + -ixsin 3 (p)l
u
£i
J
_ -PU + V) w
na2E
[ l - s i n
3
( P )
oi)
+2xvx[sin(P) - 1 ] ]
{zY=
x
na2E
(92)
I
I
f
z{z)
B
AMz) =
4)
•
I
I
= — L 2^ x s i n ( p ) x c o s 2 ( p )
2na E
3 P
2
4na E
i f v = 0 , 5 os)
xsin(p) xcos2(p)
; (v-0.5)(94)
The requirement that there will be no volume change (ev + 2 er = 0 ) for v = 0,5 is
fulfilled. The shear stress r2 equals in that case - e r x E
If the depth z is smaller than two times the radius of the loading plate (z < 2.a) the
compression of the material between 0 and z can be very well estimated by:
"
I
3
- (1 - v) xsin(P) + - | x s i n ( p ) |
er{z) = et{z}=
•
—
"2
2XV
lt should be noticed explicitely that in these equations compression (negative strain)
has a positive sign (because the vertical loading is taken positive).
If the Poisson ratio v equals 0,5 the formulas for the strains can even more simplified:
I
I
1
2
2na E
x[2x(l
-v
2
) - 0 , 7 ]
os)
Formulas for the deflections, stresses and strains in a halfspace
on the centerline under a circular rigid load P.
If the loading plate is stiff (rigid) instead of a flexible plate (constant contact stress
cr=P//7.a2) the following formulas have to be applied.
p
Contactstress
oo = — —
•
for r < a (96)
2
2
2naja
-r
1 + 3x
(97)
= Px
W{Z)
.
(98)
(1 - v) x7C - 2 x a r c t a n ( —) +
ax 1 +
*M =
1 -
2 (1 - v)
1 +
5)
1 +
z2
a2
z2'
a2
2
(99)
Formulas for the deflection in a stress dependent halfspace due to a homogenous loaded plate
If the modulus of the halfspace is stress dependent according to:
Material modulus E = Cx
with o± = major principal
n
stress
(100)
os = referencestress
n = coëfficiënt
C = constant
than the surface modulus calculated from deflection measurements at r = 0 is given
by:
n
= (1-272) xCx
(101)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
For r = O the following equation yields:
n
2 a
= (1-2x2) xCx
ofl-D2)
(102)
The major principal stress can be approximated by:
aAzY=
1
3 a2
*
r n
2
The surface modulus for r o 0 is given by:
n
I
I
I
I
I
I
I
I
I
I
I
It should be remarked that by application of these equations in the equivalent layer
theory the surface modulus Es must be used for the calculation of the defiections and
that the modulus-E-must be used for the calculation of strain and stresses.

Download Report