Geotextiles and Geomembranes21(2003)69–84
Analysis of inextensible sheet reinforcement
subject to transverse displacement/force:
linear subgrade response
M.R.Madhav*,B.Umashankar
Department of Civil Engineering,Indian Institute of Technology,Kanpur208016,India Received9September2002;received in revised form30November2002;accepted6December2002
Abstract
Reinforced earth?lls,ground and slopes counteract the destabilizing forces by mobilising tensile forces in the reinforcement.In most studies,the pull-out resistance due to only axial pull is considered in the analysis for stability.However,the kinematics of failure clearly establishes the reinforcement being displaced obliquely.In this paper,a new approach is presented for the analysis of sheet reinforcement subjected to transverse force.Assuming a simple Winkler type response for the ground and the reinforcement to be inextensible,the resistance to transverse force is estimated.The response to the applied force is shown to depend not only on the interface shear characteristics of the reinforcement but also on the deformational response of the ground.A relation is established between pull-out resistance and transverse free end displacement.A parametric study quanti?es the contributions of depth of embedment,length and interface characteristics of the reinforcement,stiffness of the ground,etc.on the over all response.It is established that reinforcement in dense granular?lls subjected to a transverse pull generates pull-out resistances larger than purely axial pull-out capacity in reinforced earth construction.
r2003Elsevier Science Ltd.All rights reserved.
Keywords:Pull-out resistance;Reinforcement;Transverse force/displacement;Subgrade stiffness and kinematics
*Corresponding author.
E-mail address:madhav@iitk.ac.in(M.R.Madhav).
0266-1144/03/$-see front matter r2003Elsevier Science Ltd.All rights reserved.
doi:10.1016/S0266-1144(03)00002-5
1.Introduction
Reinforcement of soil and of ground have become extensive and very commonly preferred alternative to enhance the performance of the earth structures in the former case and of the in situ ground conditions in the latter.Thus reinforced earth retaining walls,embankments,slopes,foundation beds are frequently adopted
while
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70
M.R.Madhav,B.Umashankar/Geotextiles and Geomembranes21(2003)69–8471 nailing is chosen to stabilize slopes and excavations.Reinforced earth structures have a special advantage in performing better under seismic conditions
The reinforcement in all the above instances is in the form of strips,bars, grids or sheets fabricated or manufactured from metals or geosynthetics.The reinforcement is presumed to restrain tensile deformations of the soil and thus increases the over all resistance of the composite soil through interfacial bond resistance but limited by its own tensile strength.The bond resistance that operates in reinforced soil is determined either by direct shear or by pull-out tests(Jewell, 1996).Considerable literature is available(Juran et al.,1988;Farrag et al.,1993; Hayashi et al.,1994;Alfaro et al.,1995;Lopes and Ladeira,1996;Ochiai et al.,1996, Sobhi and Wu,1996,etc.)on the test procedures,analysis and interpretation of pull-out tests.
However,the kinematics of failure is usually(Fig.1)such that the failure surface intersects the reinforcement obliquely.This oblique pull can be considered as a combination of transverse and axial pulls.The reinforcement is thus subjected to both axial and transverse components of the force by the sliding mass of soil.Most available theories for the analysis and design of reinforced soil structures consider only the axial resistance of the reinforcement to pull-out and not the transverse one (Jewell,1992).The inclination of the reinforcement force(Fig.2)is considered by few researchers(Leschinsky and Reinschmidt,1985;Leschinsky and Boedeker,1989; Bergado and Long,1997,etc.)to vary between the direction of the reinforcement and the tangent to the slip surface.
Under the action of axial pull,the normal stresses on the reinforcement–soil interface remain the same as the gravity stresses.Consequently,the shear resistance mobilized at the interface is proportional to these normal stresses.However,under the action of transverse force or displacement,the soil beneath the reinforcement mobilizes additional normal stresses as the reinforcement deforms transversely.As a result,the shear resistance mobilized could be considerably different in case of reinforcement subjected to transverse force.In this paper,a method is presented for the estimation of the pull-out resistance of sheet reinforcement subjected to transverse force assuming a linear subgrade(Winkler type)and inextensible reinforcement.
Fig.1.Kinematics of reinforced embankments.
2.Problem de?nition and analysis
Fig.3a depicts a sheet reinforcement of length,L ;embedded at depth,D e ;from the surface,in a soil with a unit weight,g ;subjected to a transverse force,P ;at one of its extremes.The interface angle of shearing resistance between the reinforcement and the soil is f r :The response of the reinforcement to the transverse force is to be obtained in terms of a relation between the force,P ;and the normal displacement,w L at point B :The model proposed for the analysis is shown in Fig.3b .The reinforcement and the underlying soil responses are represented,respectively,by a
Fig.2.Oblique force in the reinforcement (Bergado and Long,1997).
L L
A
w (a) (b)(c)(d)
x Fig.3.De?nition sketch (a)reinforcement subjected to transverse force,(b)model,(c)deformed pro?le and (d)forces on an element.
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72
rough membrane and a set of Winkler springs.Fig.3c represents the deformed pro?le of the reinforcement.q t and q b and t t and t b are the normal and shear stresses acting on the top and bottom surfaces respectively of the reinforcement.
3.Characteristics of the model
(a)The reinforcement is considered to be inextensible(sheet reinforcements made of
geogrids are considered to be relatively inextensible).
(b)The front-end displacement is assumed to be small and hence the inclination of
reinforcement,y;with the horizontal is assumed to be very small and negligible.
(c)The linear normal stress–displacement relation of the soil is characterized by the
relation
q?k s w;e1Twhere k s is the modulus of subgrade reaction(Terzaghi,1955)and w the transverse displacement.Since very small transverse displacements,e.g.of the order of0.001L to0.01L,only are considered,a linear normal stress–displacement is justi?ed.
(d)Full shear resistance(rigid plastic behaviour)is developed along the interface
irrespective of its horizontal displacement(i.e.q t?t t tan f r and q b?t b tan f r) as in several axial pull-out studies(Jewell et al.,1984),wherein the full interface shear resistance is considered to have been mobilized even at in?nitesimally small displacements.
The actual case of non-linear normal stress–displacement and non-linear shear–horizontal displacement of interface responses are considered in a further study. Considering an in?nitesimal element(Fig.3d)of length,D x;unit width,the tensions and their inclinations with the horizontal at distances x and xtD x;are T and(TtD T)and y and(ytD y),respectively.The horizontal and vertical force equilibrium relations for the element are
eTtD TTcoseytD yTàT cos yàeq ttq bTtan f ráD x?0e2Tand
eTtD TTsineytD yTàT sin yàeq bàq tTáD x?0:e3TEqs.(2)and(3)on simpli?cation reduce to
cos y d T
d x
àT sin y
d y
d x
àeq ttq bTtan f r?0e4T
and
sin y d T
d x
tT cos y
d y
d x
àeq bàq tT?0:e5T
Multiplying Eq.(4)with cos y and Eq.(5)with sin y and adding the two,one gets
d T d x ?eq ttq bTcos yátan f rteq bàq tTsin y:e6TM.R.Madhav,B.Umashankar/Geotextiles and Geomembranes21(2003)69–8473
Similarly,multiplying Eq.(4)by sin y and Eq.(5)by cos y and subtracting the latter from the former,one gets
àT d y d x
àeq t tq b Ttan f r sin y teq b àq t Tcos y ?0:e7TBut tan y ?d w =d x and d y =d x ?cos 2y d 2w =d x 2and the Winkler spring response to the increase in normal stress,(q b àq t )is equal to k s áw :Substituting for these in Eqs.(6)and (7)and simplifying for small values of y (i.e.cos y ?1;sin y ?y ?0),the coupled governing equations for the reinforcement subjected to transverse force are
d T d x
?eq t tq b Ttan f r ?ek s w t2g D e Ttan f r e8Tand
àT d 2w d x 2tk s áw ?0:e9T
The original problem is to derive the response of the reinforcement in terms of w and T for a given applied transverse force,P :However,it was found simpler to obtain the force,P ;for a given free end displacement,w L :
The boundary conditions are:at x ?0;the slope,d w =d x ;of and tension in the reinforcement,T ;are zero,and at x ?L ;the displacement w ?w L :
The applied transverse load,P ,obtained from the vertical equilibrium of forces as P ?
Z L
k s w d x :e10TNon-dimensionalising Eqs.(8),(9)and (10)with X ?x =L ;W ?w =w L ;T ??T =T maxp where T maxp ?2g D e L tan f r ;the axial pull-out capacity,and P ??P =g D e L ;one gets
d T ?
d X
?f m W L W t2g =2;e11TàT ?d 2W d X tm W =e2tan f r T?0
e12Tand P ??m W L Z
10W d X ;e13T
where m ?k s L =g D e ;a relative subgrade stiffness factor and W L ?w L =L :
The parameter,m ;signi?es the product of the modulus of subgrade reaction and the length of geosynthetic relative to the overburden stress corresponding to the depth of embedment.Stiffer the subgrades,larger the reinforcement,shallower the depth of embedment,the larger will be the relative subgrade stiffness factor,m ;and vice versa.
The boundary conditions in non-dimensional form becomes:at X ?0;T ??0and d W =d X ?0and at X ?1;W ?1:0:As the coupled equations cannot be solved analytically,a ?nite difference approach is adopted.Eqs.(11),(12)and (13)in ?nite
M.R.Madhav,B.Umashankar /Geotextiles and Geomembranes 21(2003)69–84
74
difference form become respectively
T?
it1
àT?
D X ?
1
2
m W i
w L
L
t2
;e14T
àT?
i
W ià1à2W itW it1
D X
&'
t
m W i
tan f r
?0;e15T
P??m W L 1
n
W1t1
2
t
X n
i?2
W i
"#
;e16T
where D X?1=n and n the number of sub-elements in to which the reinforcement
strip is divided into,W i and T?
i are,respectively,the normalized displacement and
normalized tension at node‘i’:
Solving for normalized displacement and normalized tension,one gets
W i?
T?
i
n2W ià1tW it1
eT
2n2T itm=2tan f r
àá;e17T
T?it1?
1
2n
m W L W it2
f gtT?
i
:e18T
Arbitrary initial values are assigned for normalized displacements and tensions (W iold and T iold)at each node.Eqs.(17)and(18)are then solved along with the boundary conditions to obtain the new normalized displacements and tensions (W inew and T inew).The convergence for displacement and tension at each node is checked using the criterion e W p10à6and e T p10à6where
e W?
W inewàW iold
W inew
and e T?T inewàT iold
T inew
:
If the convergence criterion is not satis?ed,the values of W inew and T inew become W iold and T iold for the next iteration and this procedure is repeated until both the convergence criteria are satis?ed.
4.Results
The transverse force,P?is then obtained from Eq.(16)by summing the normalized displacements.A parametric study is carried out for quantifying normalized displacement,W?e?w=LT;normalized tension,T?;normalized trans-
verse force,P?;normalized maximum tension at right end,T?
max ;normalized pull-out
force,T?
max cos y L;slope or inclination of reinforcement at right end,y L;in terms of m;
f r and W L:
To check the accuracy of the solution the number,n;of elements into which the length of the geosynthetic is discretized is varied.The results did not show any further improvement for n>1000:Hence,n?1000has been adopted for further analysis.As the slope of the reinforcement,y;is considered to be small,the normalized front-end displacement,W L;is con?ned to a maximum value0.01.
M.R.Madhav,B.Umashankar/Geotextiles and Geomembranes21(2003)69–8475
Parametric studies have been carried out for w L =L ?0:001201;D e ?1210m;L ?228m;f r ?2012301(Negussey et al.,1989;O’Rourke 1990)and g ?15220kN/m 3.The values of coef?cient of subgrade reaction,k s ;considered (Scott 1981)are shown in Table 1.For the above ranges of parameters,the relative subgrade stiffness factor,m e?k s L =g D e Tranges between 50and 100,000.
4.1.Displacement and tension pro?les—effect of m
The variations of displacement pro?les,W ?with distance,X ;for front-end displacement,W L ?0:01are shown in Fig.4for different values of m and f r :The normalized displacements are negligibly small for X p 0:4to 0:9for m values of 50to 10,000,respectively,and increase sharply thereafter,i.e.for X >0:4to 0.9.For very large values of relative subgrade stiffness,m ;the displacements are localized near the right end,the rest of the reinforcement remaining undeformed.For m o 500;the displacements progress towards the farthest end.The effect of the interface angle of shear resistance on displacement pro?les is not very signi?cant being slightly larger for softer subgrades than for stiffer ones.
Tension in the reinforcement increases almost linearly with distance for 0o X o 0:9(Fig.5)as in the axial pull-out case.But for large m em X 500T;the normalized displacements become highly localized near the right end (Fig.4),tension increases
Table 1
Modulus of subgrade reaction in MN/m 3
Soil characteristics
Loose Medium dense Dense Dry or moist sand
6–1818–9090–300Submerged sand 7.52490
0.0050.01
0.51
X
W *
Fig.4.Effect of m and f r on displacement pro?les.
M.R.Madhav,B.Umashankar /Geotextiles and Geomembranes 21(2003)69–84
76
sharply in that portion of the reinforcement length due to large normal stresses
acting on the reinforcement.Thus while for m ?50;the maximum tension,T ?max ;is
only 1.03,it increases to 1.55for m ?10;000:
4.2.Variations of P ?;T ?max ;y L and T ?max ;cos y L with W L F effect of m
The variation of normalized transverse force,P ?;evaluated from Eq.(18)with normalized front-end displacement,W L ;is depicted in Fig.6for f r ?301:For low values of m e?k s L =g D e To 1000;implying soft subgrade,short reinforcement or large depths of embedment,the transverse force increases linearly with the normalized end displacement,W L :The curves tend to become concave upward for m >1;000indicating that larger forces are required to mobilize larger displacements.Longer 0
0.81.6
00.5
1
X
T *
Fig.5.Effect of m on tension pro?les.0
0.71.4
00.005
0.01
w L /L P *
μ=10,000
5,000
2,000
50050
φ r = 30
o 1,000
Fig.6.Normalized transverse force versus W L =L for j r ?301:
M.R.Madhav,B.Umashankar /Geotextiles and Geomembranes 21(2003)69–8477
reinforcement or reinforcement placed at shallow depth tend to deform signi?cantly at larger displacements requiring greater forces to be mobilized.At a normalized frontend displacement,W L ;of 0.01,the normalized transverse force increase from a low value of 0.07for m ?50to a high value of 1.15for m ?10;000:
The variations of maximum normalized tension,T ?max and the inclination or slope
of the reinforcement,y L ;at the right end,with normalized free end displacement,W L ;are presented in Figs.7and 8,respectively.The normalized maximum tension,T ?max increases gradually (Fig.7)and nominally with W L for relatively compressible
subgrades (m ?50).In contrast,the maximum normalized tension,T ?max increases
rapidly with increasing relative subgrade stiffness,m (stronger/stiffer subgrades).T ?max values range from 1.00to 1.03at W L equal to 0.001(very small transverse displacement)to 1.02to 1.55at W L equal to 0.01for m increasing from 50to 100,000.
0.0010.0055
0.01
w L /L T *m a x Fig.7.Normalized maximum tension versus W L =L for j r ?301:
0.0010.0055
0.01
w L /L Fig.8.Inclination of reinforcement at X ?1versus W L =L :
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78
As has been shown in Fig.4,the transverse displacements extend over a wide range in case of relatively compressible subgrades (m ?50)and over a very narrow range in case of stiffer subgrades (m ?10;000).Hence the maximum inclinations,y L ;are relatively small for relatively compressible subgrades than those for stiffer subgrades (Fig.8).At a normalized front-end displacement,W L ;equal to 0.01,the maximum inclination of the reinforcement increases from about 41for m ?502391for m ?10;000because of localization of displacement at the front end for very stiff subgrades.
The axial pull-out resistance,T ?max cos y L ;of sheet reinforcement subject to
transverse pull is thus effected by the relative stiffness,m (Fig.9).At a normalized front-end displacement of 0.01,the axial pull-out resistance of sheet reinforcement increases from 1.03for m ?50(compressible subgrade)to a value of 1.21for m ?10;000(stiffer subgrade)for f r ?301:It could thus be reported that at larger displacements,the pull-out force could be considerably more at failure where the displacements are very large.
4.3.Variations of P ?;T ?max ;y L and T ?max ;cos y L with m F effect of W L
The transverse force,P ?;required to cause small front-end displacements,W L e?0:001Tis almost independent of the relative stiffness of soil,m (Fig.10).The effect of m on the required transverse force,P ?;becomes signi?cant for large front-end displacements W L eX 0:005T:P ?increases non-linearly with m and the rate of increase increases with the front-end displacement,W L :P ?values range from 0.03to 0.07for m equal to 50(relatively compressible subgrades)to 0.55to 1.15at m ?10;000for f r ?301and front-end displacement,W L ;increasing from 0.005to 0.01.Similar trends of variations are observed for the variation of maximum tension,T ?max with W L and relative subgrade stiffness,m (Fig.11).
The inclination or slope of the reinforcement at right end,y L ;increases with increase in relative stiffness,m ;(Fig.12)because of the localization of displacements 11.121.240.0010.0055
0.01
w L /L
T *m a x c o s (θ)L φ r = 30o 50
500
2,0005,000
μ=10,000
Fig.9.Pull-out force versus W L =L M.R.Madhav,B.Umashankar /Geotextiles and Geomembranes 21(2003)69–8479
near the right end for relatively stiff soils.The rate of increase of y L with m increases with front-end displacement.y L ranges from 0.41to 41for m ?50while the values vary from 51to 391for m ?10;000for W L increasing from 0.001to 0.01and f r ?301:
The variation of axial pull-out force,T ?max cos y L ;with m is depicted in Fig.13.
The rate of increase of T ?max cos y L for a given front-end displacement,W L ;
decreases with increase in the stiffness of the soil.The pull-out force at W L =0.01is 3%more compared to the value at W L ?0:001for m ?50whereas the
corresponding increase is 15%for m ?10;000:While T ?max increases signi?cantly
with m ;so does the value of y L :Consequently the product of T ?max and cos y L
increases only marginally.
0.6
1.2
05000
10000
μP *
φ r = 30o w L /L=0.01
0.0050.001
Fig.10.Normalized transverse force versus m :1
1.3
1.6
505025
10000
μ
T
*m a x Fig.11.Normalized maximum tension versus m :M.R.Madhav,B.Umashankar /Geotextiles and Geomembranes 21(2003)69–84
80
4.4.Variation of P *with f r —effect of m and W L
The in?uence of m on the variation of normalized transverse force,P ?;with interface friction angle,f r ;is presented in Fig.14.P ?varies almost linearly with f r :An equation of the form P ??a tb f r can be ?tted to the above curves.The constants ‘a ’and ‘b ’are tabulated in Table 2for f r ?301and for various m values for W L ?0:01:
The normalized transverse force,P ?;increases linearly (Fig.15)with f r for low values of W L =L ;typically in the range 0.001–0.01.The rate of increase of P ?with f r increases with increase in normalized front-end displacement,W L =L :An equation of the type P ??c td f r can once again be ?tted with c and d varying with W L :The constants ‘c ’and ‘d ’are tabulated in Table 3for f r ?301for various W L values for m ?500:
2040
50502510000
μ(θ )L i n d e g .φ r =30o w L /L=0.01
0.005
0.001
Fig.12.Maximum inclination at X ?1versus m :
1
1.11
1.22
505025
10000
μ
T *m a x c o s (θ)L φ r =30o w L /L=0.01
0.001
0.005Fig.13.Pull-out force versus m :M.R.Madhav,B.Umashankar /Geotextiles and Geomembranes 21(2003)69–8481
0.81.6
2030
40
φr (in deg.)P *w L /L=0.01
μ=10,000
5,000
500
Fig.14.Normalized transverse force versus f r —Effect of m 0
0.15
0.3
20
3040
φr (in deg.)P *w L /L=0.01
0.0075
0.005
0.00250.001
μ=500
φr =30o Fig.15.Normalized transverse force versus f r —Effect of W L =L :
Table 2
Values of a and b for various m
m
a b 500
0.1880.0055000
0.6260.01710,0000.900.025
M.R.Madhav,B.Umashankar /Geotextiles and Geomembranes 21(2003)69–84
82
M.R.Madhav,B.Umashankar/Geotextiles and Geomembranes21(2003)69–8483 Table3
Values of c and d for various W L
W L a b
0.0010.1860.0023 0.0050.09340.0117 0.010.18820.0238
N
5.Conclusions
Reinforcement in reinforced earth constructions and nails in nailed soil structures are rarely subjected to pure axial pull-out force.The kinematics of the problem often dictates a non-axial movement of the sliding soil and an imposition of an oblique force on the reinforcement.Conventionally reinforcement is considered to be subjected to an axial pull-out force only,though in few stability analyses non-axial reinforcement forces are included.
In this paper,an analysis of a reinforcement sheet embedded in soil at depth to a transverse force is proposed modeling the soil response by a set of linear Winkler springs.The governing differential equation is normalized and solved numerically to obtain normalized force versus normalized tip displacement relationships and normal displacement pro?les for a range of parameters considered.Results of a parametric study are presented.The kinematics of downward transverse pull increases the normal stresses and consequently the shear resistance mobilized on the bottom surface of the reinforcement.The axial component of the pull-out force due to a transverse pull is somewhat greater than the purely axial pull-out capacity of the reinforcement.The increase is more than10%for m greater than2000and front-end displacement,W L;greater than0.005L:For example,the value of relative subgarde stiffness factor,m;works out to be5000for a reinforcement of length6m embedded at a depth2m in medium dense subgardes/?lls(unit weight,g;equal to18kN/m3) with modulus of subgrade reaction,k s;equal to30MN/m3.Thus dense granular?lls would have m>5000:It is therefore established that reinforcement in dense granular ?lls subjected to a transverse pull generates pull-out resistances larger than purely axial pull-out capacity in reinforced earth construction.This axial component of pull-out force increases with the relative subgrade stiffness factor and the front-end displacement of reinforcement.
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