International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
A Lagrangian dynamic analysis of end effects in a generalized
shear experiment
D.Galic
!
,S.D.Glaser,R.E.Goodman
Department of Civil and Environmental Engineering,University of California,Berkeley,760 Davis Hall,Berkeley,CA 947201710,USA
Received 5 January 2007;received in revised form 9 July 2007;accepted 16 July 2007
Available online 5 September 2007
Abstract
In laboratory shear testing,a primary source of error is the surcharge force caused by relative motion between the displacement
actuator and dilating top sample.This force is referred to as end friction,and the changes it produces in experimental data are termed
end effects.The results from a laboratory shear setup always represent a superposition of natural sliding behavior and testing machine
interference;their relative proportions can be determined by externally modeling the experiment.In this paper,we construct a full
Lagrangian dynamic model for the shearing behavior of a prismatic aluminum topblock over a matching asymmetric foundation.End
friction is initially included in the analysis,whose viability is established by comparing modeled and experimental topblock sliding paths
at 12 different shear loads.The end friction force is ultimately removed from the formulation,and the end effects manifested as the
subsequent differences between modeled and experimental sliding paths.It is shown that end effects signiﬁcantly alter both the sliding
path and rotation mode of the prismatic topsample.While their impact on the trajectory of a given sample appears to decrease with
increasing shear force,it is shown that uniformsample scaling does nothing to alleviate the problem,and that end effects are functionally
scale independent.
r
2007 Elsevier Ltd.All rights reserved.
Keywords:
Contact point;End effect;End friction;In situ behavior;Laboratory behavior;Lagrangian analysis;Lateral dilation;Sample scaling;Shear
test;Sliding path
1.Introduction
The physical interaction between a testing machine and
test sample inevitably produces an additional response in
the sample that would not appear in situ.We term this
response an ‘‘end effect’’,and its origin can be traced to the
differential motion of contacting parts.In the wellknown
uniaxial test,for example,an end effect arises when the
load platens and test sample deform laterally at unequal
rates,resulting in friction along the ends of the sample.The
distorting impact of end friction on uniaxial test data has
been previously documented
[1,2]
.
Whereas a uniaxial test provides information on the bulk
properties of a medium,the direct shear test is used to
investigate the strength of existing planar discontinuities
[3]
.The shear strength of a joint is usually lower than host
rock compressive strength,and the movable halfsample
expected to slide before material failure occurs.Because
rupture or crushing are unlikely,the shear displacement
force may be imparted by a point loader instead of a load
platen.The point loader maintains contact with only a
limited portion of the test sample and its lateral deforma
tion does not adversely affect shear behavior.However,the
roughness of a joint surface virtually ensures that a sliding
sample will experience dilation,which quickly leads to
relative motion between the sample and load applicator.
Since relative motion is necessarily accompanied by
friction,this is the manner in which end effects are
introduced into a direct shear experiment.
Consider a symmetric top sample that dilates only
vertically
(
Fig.1a
).According to the Mohr–Coulomb
friction model,a joint with net friction angle
f
and normal
compression
N
requires a shear force of magnitude
N
tan
f
to displace.Assume that a force of such magnitude is
imparted by a point loader,and deﬁne
c
as the friction angle
ARTICLE IN PRESS
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r
2007 Elsevier Ltd.All rights reserved.
doi:
10.1016/j.ijrmms.2007.07.012
!
Corresponding author.Tel.:+15106429278.
Email address:
galic@ce.berkeley.edu (D.Galic).
of the load tip/sample interface.The ‘‘normal’’ force acting
on the load tip/sample interface is identical to the machine
imparted displacement force,so by Mohr–Coulomb,the
load tip friction force has magnitude
N
tan
f
tan
c
.This
force acts downward as the sliding sample dilates upward
and therefore supplements the total normal force.But since
net compressive load
N
(1+tan
f
tan
c
) is only slightly
greater than command normal load
N
,sample dilation is
unlikely to be impacted,unless we are operating near the
compressive strength of the rock.
Next consider a sample which dilates only
laterally
(
Fig.1b
).As before,the command normal load is
N
,the
shear load
N
tan
f
,and the magnitude of the load tip
friction force
N
tan
f
tan
c
.The friction force is now
directed horizontally,since relative motion between the
load tip and laterally dilating sample amounts to sideslip.
Because there are no externally applied lateral forces,the
load tip friction force comprises the
net
external force in
the horizontal direction.So whereas the vertical load of the
vertically dilating sample was increased from
N
to
N
(1+tan
f
tan
c
),the lateral load of a laterally dilating
sample increases from 0 to
N
tan
f
tan
c
.This represents a
much more serious violation of the model assumptions.
An actual joint sample will typically exhibit some
combination of vertical and lateral dilation.The vertical
dilation of a shearing top specimen provides information
on the amplitude of the joint roughness;the tendency of a
specimen to dilate laterally indicates that this roughness is
not uniformly symmetric.That a rock foundation with
strongly asymmetric topography can be expected to favor
movements involving lateral slip is of considerable interest
in dam engineering.The tendency of sliding monoliths to
dilate outward as they move forward may be key to
understanding the failure mechanics of a gravity dam
foundation
[4]
.It is therefore important,when running a
shear test designed to simulate the hydrostatic loading of a
dam monolith,that the recorded lateral slip be a property
of the foundation and not of the testing apparatus.
In this paper,we quantify the total effect of end friction
on a laterally dilating triplanar sample pair in
generalized
shear
.The lower and upper halfsamples under considera
tion are shown in
Figs.2 and 3
.The experimental
boundary conditions are referred to as
generalized shear
for the following reasons:(1) An asymmetric threeplane
sample/sample interface produces both vertical and lateral
dilation;(2) The initially matching top and bottom blocks
revert to threepoint contact when sliding begins (this
allows the block interaction to be modeled in terms of
forces rather than stresses);(3) The conﬁguration reduces
to a standard direct shear setup when the dip angles of the
interface planes are set to zero;(4) The conﬁguration
extends to a more complicated discretized surface when the
factual contact points are considered to be one entry in a
sequence of triplanar contacts.
A set of experimental data is ﬁrst generated from shear
tests performed on an aluminum sample pair.We then
model the experiment mathematically using a Lagrangian
(conﬁguration space) dynamic analysis,which captures
ARTICLE IN PRESS
Fig.1.Two limiting modes of dilation exhibited by a sample whose
nominal displacement direction is indicated by the arrow:(a) pure vertical
dilation—the sample moves upward as it is pushed forward;(b) pure
lateral dilation—the sample moves sideways as it is pushed forward.The
relative slip of the load point is indicated by a wavy line.
Fig.3.Top sample of the shearing test pair,with contact points labeled.
Fig.2.Bottom sample of the shearing test pair,with foundation planes
numbered.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
496
both the frictional and inertial qualities of rigid body
sliding.The predictions of the analytical model,presented
in the graphical form of sliding paths,are compared with
experimental results in order to establish the accuracy of
the model.Once the model has been calibrated,the forces
associated with end friction are
removed
fromthe analytical
model,and the updated model predictions compared with
the existing experimental results.The end effect is
manifested as the difference between the experimental
sliding paths (which necessarily include it) and modeled
sliding paths (fromwhich the end effect has been removed).
The relationship between selected sample parameters
and ultimate sample trajectory is investigated.First,the
weight and dimensions of the top block are varied
independently in order to identify possible trends of end
effect intensity.Since a given combination of shear load,
sample weight,and sample size corresponds to one
particular set of ﬁeld conditions,these parameters are next
scaled uniformly.We consider the usefulness of propor
tional scaling as a means of reducing frictional end effects.
2.Experimental setup
Fig.2
shows a schematic diagram of the bottom
(foundation) sample,which can be thought of as the lower
half of a shear box.The three foundation planes are
independent,in the sense that their three normal vectors
can be used as a basis for threedimensional Euclidean
space,
R
3
.The spatial orientation of each foundation plane
is completely characterized by either its normal or dip
vector.While these quantities are variables in the analytical
formulation,speciﬁc values needed to be selected for
machining purposes.The design foundation angles pro
mote the lateral movement of the top block.Foundation
plane (FP) 1 has a dip of zero,FP2 a dip of 9.5
1
dipping
away fromthe displacement direction,and FP3 a dip of 20
1
dipping 150
1
clockwise from the displacement direction.
The top block slides freely over the foundation block,
much as the top half of a shear box moves over the
stationary bottom half.The top and bottom samples have
matching interfacial topography and ﬁt snugly together
prior to the start of sliding.Once the top block has been
displaced,it immediately reverts to threepoint contact.
The
same
three points remain in contact throughout
sliding,so long as the block is not pushed over the edge
of its foundation,and provided the center of mass
continues to project inside the triangle formed by the three
contact points.
Fig.3
shows the locations of these three
points on the top block.For the range of motion we are
interested in,each contact point is associated with exactly
one of the three foundation planes:contact point (CP) 1
slides on FP1,CP2 on FP2,and CP3 on FP3.
Fig.4
shows the test setup as it appears prior to sliding.
The foundation block is bolted to the load frame and has
plan dimensions 30 cm
!
45cm,while the free top block
has plan dimensions 15cm
!
30 cm.Also visible is the array
of laser rangeﬁnders used to track motion,as discussed in
[5]
.The shear displacement force is provided by a Copley
Controls brushless linear actuator,whose load tip (
Fig.5
)
consists of a machined steel bolt.By virtue of its design,the
actuator can move only forward or backward;the top
block,on the other hand,has the ability to dilate both
vertically and laterally as it slides.Relative motion between
the actuator load tip and the upstreamface of the top block
leads to the end effect we wish to investigate.
The experimental program consisted of approximately
50 constant force (CF) shear tests run at 12 different shear
force levels.The test range of shear force values (27.8N
through 55.6 N) was selected based on practical experience
with the aluminum blocks.It was observed that sliding
tended to be prematurely arrested by frictional rate effects
[6]
at shear loads of less than 25N,whereas shear loads of
greater than 60N resulted in more or less linear sliding
ARTICLE IN PRESS
Fig.4.Comprehensive view of the experimental setup.The top block is
displaced by a linear actuator.Note laser rangeﬁnders.
Fig.5.Loadcell mounted steel pusher shown in contact with the
aluminum top block.As the top block dilates,there is relative motion
between its aft face and the steel tip,which inscribes a curve in the moving
plane of the aft face.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
497
paths.The normal force
in all cases
amounted to the 53N
weight of the top block.The normal force was not varied,
and no additional weights placed on the block.While this
sounds like poor technique,it should be kept in mind that
we are not attempting to construct a failure envelope for
the top block.Rather,we are interested in the top block’s
sliding path as a function of driving shear force.The
‘‘output’’ of the experiment is a graphical object,not a
friction angle.
The results of the tests are compiled as contact point
traces,imaginary marks that the contact points inscribe on
their foundation planes as the top block slides.
Fig.6
shows
an overhead view of these traces for a sample shear load of
32.5 N.The forward displacement direction is indicated by
the
x
1
axis.Axis
x
3
captures the horizontal (lateral)
components of nonforward contact point movement
(dilation).The macroscale relief of the foundation most
certainly produces upward movement in the top block,but
the vertical component of dilation is lost in our two
dimensional plot.Because FP1 is entirely horizontal,the
featured trace represents CP1’s actual path;CP2 and CP3
slide on inclined foundation planes and their spiraling
threedimensional paths are seen in horizontal
projection
.
In order to account for experimental uncertainty,several
tests were run at each load level.Although it appears that
there is only one set of traces displayed,
Fig.6
in fact
contains data from all four tests run at 32.5 N.Trace sets
obtained from independent tests performed at the same
shear load were generally found to be indistinguishable,a
testament to the benevolent sliding characteristics of hard
anodized 6061 aluminum.
The experimental data shown in
Fig.6
encapsulate two
subsets of information:one set describes the interaction of
the top block with its foundation;the other describes the
interaction of the top block with the actuator load tip.Our
task is to separate the two and determine whether the
frictional interference of the load tip is negligible.
3.Analytical formulation
In the following derivation,we adopt the convention
of summation over repeated indices.For example,the
expression
a
i
b
i
is equivalent to
a
1
b
1
+a
2
b
2
+
y
a
n
b
n
,where
the value of
n
will usually be apparent from the context.In
the event that we do not wish to sumover repeated indices,
the indices will be enclosed by parentheses,for example,
F
(k)
"
P
(k)
.Coordinates are denoted by an upper index (
x
i
)
whereas vectors are denoted by a lower index (
e
i
).Boldface
is used to distinguish between points in space and their
position vectors (P vs.
P
) and between forces and their
magnitudes (
F
vs.F).
Our approach will be to examine how the energy of the
system varies among its permissible conﬁgurations,that is,
conﬁgurations which do not violate the constraints we have
placed on the system.In the ensuing sections we:establish a
series of datums by which to track motion (Section 3.1);
formulate constraint equations (Section 3.2);generalize
the constraint forces (Section 3.3);model the end effect
(Section 3.4);and formulate and numerically solve
Lagrange’s differential equations of motion (Section 3.5).
The desired output is a family of contact point traces such
as those shown in
Fig.6
.
3.1.Local and global coordinate systems
We will model the top block as a triangular frame whose
tips describe the contact points.This is not a mathematical
simpliﬁcation,but rather a graphical one;its easier to draw
only those parts of the top block that are in contact with
the foundation block.The full inertial properties of the top
block continue to be used throughout the analysis.Since
the block will be treated as a rigid body (we are not
interested in what is happening inside the block),the
dimensions of the frame are not allowed to change
during sliding.
The model space is shown in
Fig.7
.The contact points
of
Fig.3
are now seen from above,as if the top block were
clear.Also visible are the three foundation planes,whose
three lines of intersection meet at the origin of the global
coordinate system (
x
i
).Although coordinate axes
x
i
have
been omitted from the ﬁgure,the orientation of the global
unit basis vectors
e
i
is shown in the lower left corner.The
moving frame deﬁned by contact points 1–3 is endowed
with a local,accelerating coordinate system
z
i
;the axes
ARTICLE IN PRESS
Fig.6.Experimental results are presented as contact point traces (black),
which represent the horizontal projections of the contact point sliding
paths.The displacement direction is
x
1
,with lateral dilation occurring in
the
x
3
direction.Vertical dilation in the
x
2
direction cannot be seen in this
overhead view.Foundationplane lines of intersection shown in gray.
Contact points and foundation planes as labeled.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
498
drawn inside the frame depict the orientation of local basis
vectors
g
i
.Local and global coordinate systems can be
related through the three position vectors
P
1
,
P
2
,
P
3
,which
track the locations of the contact points with respect to the
global coordinate system.For example,
g
1
is a scaled
version of the vector
P
2

P
1
.
As stated,all interaction between the top block and
foundation block is limited to the three contact points.
Focusing our attention on these three points is preferable
to tracking every point in the top block,but it is even more
convenient to shrink the block to a
single
point.This can be
done by transforming the domain of the problem from
threedimensional Euclidean space,
R
3
,to ninedimen
sional Euclidean space,
R
9
.What may sound mathemati
cally intimidating actually entails nothing more than taking
the three sets of three coordinates:
x
1
CP1
;
x
2
CP1
;
x
3
CP1
!"
;
x
1
CP2
;
x
2
CP2
;
x
3
CP2
!"
;
x
1
CP3
;
x
2
CP3
;
x
3
CP3
!"
(1)
and writing them as one set of nine coordinates:
x
1
CP1
;
x
2
CP1
;
x
3
CP1
;
x
1
CP2
;
x
2
CP2
;
x
3
CP2
;
x
1
CP3
;
x
2
CP3
;
x
3
CP3
!"
.(2)
Relabeling the coordinates in Eq.(2) as
q
instead of
x
,
we have
q
1
;
q
2
;
q
3
;
q
4
;
q
5
;
q
6
;
q
7
;
q
8
;
q
9
!"
,(3)
which are the
generalized coordinates
of the system.The
Euclidean space
R
9
whose coordinates are
q
i
is termed the
conﬁguration space
of the system,and is spanned by nine
unit basis vectors
h
i
.By virtue of the transformation from
Eq.(1) to Eq.(2),a single point in conﬁguration space
describes the locations of all three contact points and
uniquely deﬁnes the position of the block.Note that if we
wish to return to the physical model space
R
3
,we can use
the following inverse correspondence:
x
1
CP1
;
x
2
CP1
;
x
3
CP1
!"
#
q
1
;
q
2
;
q
3
!"
,
x
1
CP2
;
x
2
CP2
;
x
3
CP2
!"
#
q
4
;
q
5
;
q
6
!"
,
x
1
CP3
;
x
2
CP3
;
x
3
CP3
!"
#
q
7
;
q
8
;
q
9
!"
.
$
4
%
3.2.System constraints
The 9tuple in Eq.(3) represents the coordinates of any
point in the top block’s conﬁguration space.But the top
block’s domain of
sliding
does not encompass all of
R
9
,and
not all combinations of
q
i
are physically possible.In our
analysis,we are not interested in unstable motion or in the
elastic deformation of the block.All contact points must
therefore remain on their respective planes of sliding
throughout motion,and the dimensions of the frame
cannot be allowed to change during sliding.These
restrictions lead to six constraint equations and leave the
top block with three degrees of freedom.
The ﬁrst three constraint equations ensure that contact
points do not lift off their foundation planes or transition
to other foundation planes.In other words,the contacts
points must be elements of the foundation planes.Recall
that the equation of a plane can be written:
n
"$
P
&
P
ref
%#
0,(5)
where
n
is a normal to the plane,
P
a vector fromthe origin
to any point on the plane,and
P
ref
a vector fromthe origin
to some ﬁxed point on the plane.Because all three of our
foundation planes contain the origin of the global
coordinate system,the vector
P
ref
can always be taken as
the zero vector,and the equations that the system must
satisfy are:
n
1
"
P
1
#
0
;
n
2
"
P
2
#
0
;
n
3
"
P
3
#
0,(6)
where
P
1
,
P
2
,and
P
3
are the global position vectors of
contact points 1,2,and 3.
The next three constraint equations enforce the rigid
body constraint:they state that the distance between
contact points is constant during sliding.These equations
can be written in terms of the known frame dimensions as
P
3
&
P
2
j j
&
L
23
#
0
;
P
3
&
P
1
j j
&
L
13
#
0,
P
2
&
P
1
j j
&
L
12
#
0,
$
7
%
where
L
AB
is the length of the frame edge between contact
points
A
and
B
.Since the components of position vectors
P
1
,
P
2
,and
P
3
are simply the global coordinates of CP1,
CP2,and CP3,we can use the correspondence in Eq.(4) to
write these position vectors as functions of the
generalized
coordinates:
P
1
#
q
1
e
1
'
q
2
e
2
'
q
3
e
3
,
P
2
#
q
4
e
1
'
q
5
e
2
'
q
6
e
3
,
P
3
#
q
7
e
1
'
q
8
e
2
'
q
9
e
3
.
$
8
%
Expressions (6) and (7) then take the functional form:
f
i
q
1
...
q
9
!"
#
0,(9)
where index
i
denotes the constraint equation number and
f
i
is referred to as the
i
th constraint function.
The six equations of (9) describe a group of surfaces
S
i
in
ninedimensional conﬁguration space.All points
p
on
surface
S
i
represent valid conﬁgurations with respect to
ARTICLE IN PRESS
CP 2
CP 3
CP 1
e
1
e
2
e
3
Plane 3
dip =
!
Plane 2
dip =
"
Plane 1
dip =
"
g
3
g
2
g
1
P
1
P
2
P
3
!
Fig.7.Overhead view of the model space.The top block is represented as
a rigid frame.The contact points and foundation planes are indicated,as
well as the directions of global basis vectors
e
and local basis vectors
g
.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
499
constraint
i
.Similarly,a curve
C
on
S
i
describes a valid
sequence
of conﬁgurations.The restriction imposed by
constraint
i
prevents
C
fromever leaving surface
S
i
,and all
vectors tangent to
C
must therefore be tangent to
S
i
as well.
This fact allows us to write Eq.(9) in an equivalent form,as
a statement that the systemvelocity vector
V
through point
p
on surface
S
i
must be tangent to the constraint surface at
that point:
V
p
"
n
p
#
0
;
8
p
2
S
i
.(10)
For a point
p
with coordinates (
q
1
,
q
2
,
y
q
9
),we have
V
p
#
d
d
t
$
q
j
h
j
%#
_
q
j
h
j
,(11)
where the
h
j
are unit basis vectors and the
_
q
j
are known as
generalized velocities.The normal vector
n
to constraint
surface
S
i
at point
p
is obtained as the gradient of
the corresponding constraint function in Eq.(9),evaluated
at
p
:
n
p
#r
f
i
.(12)
We can therefore write Eq.(10) as
r
f
i
"
_
q
j
h
j
#
0.(13)
Although it is not immediately clear why (13) is a more
useful expression of the six constraint equations than (9),
this will become evident after we have formulated
Lagrange’s equations.
3.3.Generalized constraint forces
Fig.8
shows the external forces acting on the top block
as it slides.The driving force
F
T
is imparted to the aft face
of the block by the thruster load tip.Also acting at the load
tip is the end friction force
F
EF
,a result of relative motion
between the load tip and test block.This force acts in the
direction opposing the relative motion.For example,if a
blockﬁxed point P
CL
begins to move to the right of the
load tip,
F
EF
will be directed toward the left.The term‘‘end
effect’’ refers to those aspects of physical block behavior
that result directly from the presence of
F
EF
.
Removing
this force from the analysis is equivalent to removing the
end effect.
The driving force
F
T
,end friction
F
EF
,and block weight
W
are resisted by reactions at the three contact points.
These include shear and normal reactions.The normal
reactions are constraint forces,since they ensure that the
contact points will not sink beneath the foundation planes.
In fact,they are
the
constraint forces that maintain
constraint conditions (6).On the other hand,the constraint
forces related to constraint conditions (7) are not shown in
Fig.8
.These forces promote rigid body motion,and it is
not entirely obvious where to display them in the ﬁgure.
Apart from this graphical omission,the rigid body
constraint forces will be treated in the same way as the
more tangible forces shown in
Fig.8
.
We wish to examine how the work done by each of these
forces varies from point to point.Because the ultimate
trajectory of the block is still unknown,this can only be
done locally,measuring the differences in energy between
slightly different sliding paths.The current conﬁguration of
the block and an alternative inﬁnitesimally distant conﬁg
uration are separated by the incremental position vector
d
P
.The differential amount of work that must be
performed by force
F
in order for the system to reach its
alternate conﬁguration is:
d
W
#
d
P
"
F
#
q
P
q
q
j
"
F
#$
d
q
j
,(14)
where
d
q
j
is a small change in generalized coordinate
q
j
(see
e.g.
[7]
).The bracketed quantity in Eq.(14) is known as the
j
th generalized force,
Q
j
:
Q
j
#
q
P
q
q
j
"
F
.(15)
Generalizing forces is a crucial step in the formulation of
Lagrange’s equations,and Eq.(15) must be applied to each
of the forces acting on the block.Often,the most
challenging aspect of this process is obtaining an expres
sion for
P
,the position vector of the point through which
the force in question acts.
It is remarkably easy to formulate
P
for the con
straint forces.This is because the constraint forces in
habit conﬁguration space,and their ‘‘location’’ is
simply the point describing the current conﬁguration of
the system:
P
constraints
#
q
1
h
1
'
q
2
h
2
'
...
q
9
h
9
#
q
j
h
j
.(16)
Obtaining coordinate expressions for the constraint forces
themselves is straightforward,but requires an appreciation
of their geometric meaning.Constraint force
i
is always
normal to
S
i
because it enforces the requirement that
permissible motions remain embedded in the constraint
surface.By Eq.(12),constraint force
i
must therefore be
parallel to the gradient of
f
i
.So while the magnitude of
constraint force
i
is still unknown,its vector can be
conveniently written as a scalar multiple of
r
f
i
.The vector
sum of all six constraint forces is then:
F
constraints
#
l
1
r
f
1
'
l
2
r
f
2
'
...
l
6
r
f
6
#
l
i
r
f
i
,(17)
ARTICLE IN PRESS
CM
P
L
CP 2
CP 3
W
F
T
N
1
S
1
N
2
S
2
S
3
N
3
F
EF
CP 1
Fig.8.External forces acting on the top block during sliding.The contact
points,center of mass,and instantaneous load point are indicated.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
500
where the unknown magnitudes
l
are termed Lagrange
multipliers.The
l
are independent of
q
j
and ignored by the
delta operator in Eq.(14).Using Eq.(15) we have
Q
constraints
j
#
q
q
k
h
k
!"
q
q
j
"
l
i
r
f
#
l
i
r
f
i
"
h
j
.(18)
Eq.(18) accounts for contributions to
Q
j
from the
internal rigid body forces and from the foundation normal
reactions.The foundation
shear
reactions are not con
straint forces (since they resist rather than restrict system
motion) but can be related to the normal reactions through
the Mohr–Coulomb failure criterion.For a sliding contact
point
a
,
S
a
j j
#
N
$
a
%
%
%
%
%
tan
f
$
a
%
,(19)
where
f
a
is the dynamic (residual) angle of friction between
contact point
a
and foundation plane
a
.In terms of the
Lagrange multipliers,(19) can be written as
S
a
j j
#
l
$
a
%
tan
f
$
a
%
.(20)
Eq.(20) gives only the magnitude of the shear force at
contact point
a
.The complete shear force vector
S
a
is
antiparallel to
_
P
a
,the instantaneous velocity of contact
point
a
:
S
a
#&
_
P
$
a
%
_
P
$
a
%
%
%
%
%
l
$
a
%
tan
f
$
a
%
.(21)
We now deﬁne
S
a
#&
_
P
$
a
%
_
P
$
a
%
%
%
%
%
tan
f
$
a
%
,(22)
after which the generalized shear reactions can be written
as
Q
shears
j
#
l
a
S
a
"
h
j
.(23)
The remaining forces in
Fig.8
are not directly associated
with the contact points and therefore cannot be written
without using local coordinates.For example,weight
W
points downward through the mass center (CM) of the top
block.Referenced to contact point 1,the local coordinates
of CM are (
C
1
,
C
2
,
C
3
),as shown in
Fig.9
.The global
position vector to CM is then:
P
CM
#
P
1
'
C
b
g
b
.(24)
The timeﬁxed differential of this vector is
d
P
CM
#
q
P
1
q
q
j
d
q
j
'
C
b
q
g
b
q
q
j
d
q
j
.(25)
By replacing
W
with
&
W
e
2
,we get
Q
W
j
#&
W
e
2
"
q
P
1
q
q
j
'
C
b
q
g
b
q
q
j
#$
.(26)
Note that the magnitude of the weight,W,is a known
quantity.
3.4.Modeling the end effect
The laboratory top block is sheardisplaced by a
mechanical thruster.At the
start
of the experiment,the
thruster load tip (see
Fig.5
) is in contact with the aft face
centerline,at a point labeled P
CL
in
Fig.9
.Unfortunately,
the machine imparted drive force
F
T
can only remain at
P
CL
over small displacements.This is because the linear
thruster has a single degree of freedom (i.e.,
F
T
#
F
T
e
1
),
whereas the sliding top block can dilate both vertically and
laterally.As a result,the blockﬁxed point P
CL
inevitably
slips away from the load tip,and the
instantaneous
load
point P
L
is distinct from P
CL
.These two points together
with the spaceﬁxed initial load point P
0
CL
(P
CL
frozen at
its presliding position) form a triangle,as illustrated in
Fig.10
.
A position vector to the instantaneous load point P
L
can
be assembled from position vectors
P
CL
and
P
0
CL
.Using
Fig.9
and the dimensions of the top block:
P
CL
#
P
1
'
A
g
2
&
B
g
3
,
P
0
CL
#
P
CL
$
t
#
0
%
#&
L
12
e
1
'
A
e
2
'
L
23
&
B
$ %
e
3
.
$
27
%
ARTICLE IN PRESS
g
1
g
2
g
3
C1
C3
C2
A
B
CP1
P
CL
CM
Fig.9.The locations of the original load point P
CL
and mass center CM,
shown with respect to contact point CP1.
F
T
P
L
P˚
CL
P
CL
k
2
g
2
+
k
3
g
3
Global Origin
Current Configuration
Starting Configuration
Fig.10.The instantaneous location P
L
of the load tip can be expressed in
terms of its original location on the block (P
CL
) and its original location in
space (P
0
CL
).
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
501
Write
P
L
#
P
CL
+
k
2
g
2
+
k
3
g
3
,where the quantities
k
2
#
P
0
CL
&
P
CL
'
P
CL
&
P
0
CL
!"
"
g
1
e
1
"
g
1
e
1
!
"
g
2
,
k
3
#
P
0
CL
&
P
CL
'
P
CL
&
P
0
CL
!"
"
g
1
e
1
"
g
1
e
1
!
"
g
3
$
28
%
represent the orthographic projections of vector
(
P
0
CL
–
P
CL
) onto the aft face of the block,as shown in
Fig.10
.The terms
k
2
and
k
3
are functions of the
generalized coordinates
q
i
,and are therefore timedepen
dent.However,treating them as constant within the small
duration of a solution increment leads to a simpliﬁed
expression for
d
P
L
.In that case
d
P
L
#
d
P
1
'
k
2
'
A
$ %
d
g
2
'
k
3
&
B
$ %
d
g
3
.(29)
The generalized driving forces are
Q
T
j
#
F
T
q
P
1
q
q
j
'$
k
2
'
A
%
q
g
2
q
q
j
'$
k
3
&
B
%
q
g
3
q
q
j
#$
"
e
1
.(30)
The instantaneous load point P
L
is also the locus of end
friction force
F
EF
.This force is a result of load tip drag as
the original load point P
CL
slips away from P
L
.
F
EF
always
resists the relative motion;it is a surface force on the aft
face of the top block and tangent to the blockﬁxed curve
describing the migration of P
CL
.In the context of an
incremental solution,the unscaled direction
D
EF
of end
friction force
F
EF
is given by
D
EF
current increment
$ %#
P
L
current increment
$ %
&
P
L
previous increment
$ %
,
$
31
%
The magnitude of
F
EF
depends on the magnitude of
displacement force
F
T
,according to the Mohr–Coulomb
formulation.Deﬁning
f
EF
as the dynamic friction angle
between top block and load tip,and introducing a
correction factor to account for top block rotation,the
magnitude of
F
EF
can be written as
F
EF
#
F
T
e
1
"
g
1
!"
tan
f
EF
.(32)
The complete end friction force vector is
F
EF
#
D
EF
D
EF
j j
F
T
$
e
1
"
g
1
%
tan
f
EF
,(33)
and the generalized end friction forces are
Q
EF
j
#
F
T
tan
f
EF
e
1
"
g
1
D
EF
j j
q
P
1
q
q
j
'$
k
2
'
A
%
q
g
2
q
q
j
#
'$
k
3
&
B
%
q
g
3
q
q
j
$
"
D
EF
.
$
34
%
Expression (34) is only meaningful in the context of an
incremental solution.
To summarize,the generalized forces associated with the
entire system are:
Q
j
#
l
i
h
j
"r
f
i
'
l
i
h
j
"
R
i
'
Q
W
j
'
Q
T
j
'
Q
EF
j
,
#
l
i
Q
Lambda
ji
'
Q
W
j
'
Q
T
j
'
Q
EF
j
,
$
35
%
where the terms connected with lambda have been
separated for convenience.If we wish to investigate how
the system behaves in the absence of end effects,the
Q
j
EF
term is simply omitted from Eq.(35).This is equivalent to
removing force
F
EF
from
Fig.8
.
3.5.Equations of motion
Lagrange’s equations provide a relationship between the
generalized forces
Q
j
and system kinetic energy
T
.There is
one equation for each generalized coordinate
q
j
:
q
T
q
q
j
&
d
d
t
q
T
q
_
q
j
#$
'
Q
j
#
0.(36)
The equations of motion for our top block consist of
nine Lagrange equations (36) and six constraint equations
(13).The 15 unknowns (
q
1
–
q
9
and
l
1
–
l
6
) are thus balanced
by 15 differential equations.
Equation sets (36) and (13) can be written in a
computationally efﬁcient format by isolating the coefﬁ
cients of generalized accelerations
!
q
j
and Lagrange multi
pliers
l
i
(see
[8]
).This is accomplished by expanding the
kinetic energies in Eq.(36) and differentiating both sides
of Eq.(13):
M
jl
!
q
l
&
l
i
Q
Lambda
ji
#
1
2
q
M
kl
q
q
j
_
q
k
_
q
l
&
q
M
jl
q
q
k
_
q
k
_
q
l
'
Q
Weight
j
'
Q
Thruster
j
'
Q
End Friction
j
,
$
37
%
&r
f
i
"
!
q
j
h
j
#
_
r
f
i
"
_
q
j
h
j
.(38)
The 81 coefﬁcients
M
jl
in Eq.(37) are functions of the
generalized coordinates.Their calculation is largely me
chanical,and coordinate expressions accounting for
both translation and block rotation are given in
[9]
.
Without going into further detail here,we note that the
inertial properties of the top block enter the formulation
through
M
jl
.
At the instant before sliding begins,generalized coordi
nates
q
j
retain their known initial values,while the
velocities
_
q
j
are identically zero.In fact,all quantities on
the right hand sides of Eqs.(37) and (38) are known in the
ﬁrst increment of motion,and the system can be solved for
!
q
j
and
l
i
.Denoting the right hand sides of Eqs.(37) and
(38) by the column vectors
V
1
and
V
2
,the two equation sets
are written as a single matrix equation:
!
q
1
:
!
q
9
l
1
:
l
6
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
9
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
;
#
M
( )
&
Q
( )
T
& r
f
( )
0
( )
"#
&
1
V
1
V
2
( )
,(39)
where [M] is the matrix of
M
jl
(9
!
9),[Q] the matrix of
Q
ji
Lambda
(9
!
6),[
r
f] the matrix of
r
f
i
"
h
j
(6
!
9),and [0] a
matrix of zeros (6
!
6).The accelerations
!
q
j
are then used
ARTICLE IN PRESS
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
502
to calculate the starting positions and velocities for the next
increment:
q
1
.
.
.
q
9
_
q
1
.
.
.
_
q
9
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
n
#
q
1
.
.
.
q
9
_
q
1
.
.
.
_
q
9
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
n
&
1
'
D
t
_
q
1
.
.
.
_
q
9
!
q
1
.
.
.
!
q
9
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
n
&
1
,(40)
where
n
is the increment number and
D
t
the time length of
an increment.Repeatedly applying Eqs.(39) and (40)
generates a solution sequence describing the system
trajectory in conﬁguration space.Inverse relations (4) are
then used to recover the individual paths of the contact
points.
Expression (40) is a variant of Euler’s method,a simple
numerical solution procedure for ordinary differential
equations (see e.g.
[10]
).The use of small time steps
D
t
often ensures convergence,but can lead to excessively
sluggish runtimes.In the event that the solution does not
converge,a higher order numerical method may be used in
place of Eq.(40) with no additional modiﬁcation to the
analysis.
4.Results
The parameters used in the analysis need to match actual
experimental values in order for direct comparison to be
possible.The dimensions,mass,and inertial properties of
the top block are either known or calculable.Residual
friction angles for the foundation plane/contact point
interfaces may be obtained by running direct shear tests
over representative surfaces.The values used here are 17
1
(FP1/CP1),14
1
(FP2/CP2) and 11
1
(FP3/CP3);they differ
because the foundation roughness unequally affects con
tact points of varying convexity.The friction angle between
the steel load tip and aluminum top block aft face has a
measured value of 8
1
.
As stated,the experimental results consist of contact
point traces from sliding motion at 12 different shear load
levels.They are compared below with the predictions of the
analytical model where:the predicted results include a
modeled end effect (Section 4.1);the predicted results do
not include an end effect (Section 4.2);and the predicted
results do not include an end effect and the displacement
force is modeled as ideally hydrostatic (Section 4.3).
In each
of these comparisons,the experimental results necessarily
contain the end effect
.We next use the analytical model to
investigate the effects of end friction on laboratory samples
different from our test block,including its effects on:
samples with different size but equal weight (normal load)
as the test block (Section 4.4);samples with equal size
but different weight (normal load) than the test block
(Section 4.5);samples with different size and different
weight,but equal density as the test block (Section 4.6).
The latter case is used to investigate whether laboratory
end effects can be reduced through uniformsample scaling.
4.1.Comparison with end friction included in the modeled
results (
Fig.11
)
The predictions of the analytical model with end friction
included
are compared with experimental results in
Fig.11
.
The individual panels of the ﬁgure represent overhead
views of the foundation block,with the vertical axis
suppressed and the horizontal axes equally scaled.The
graphical correspondence between contact point number
CP
n
and its associated trace is as given in
Fig.6
.
Each panel of
Fig.11
corresponds to one shear force
level (value noted above the panel);the normal force in all
cases is 53 N.Mathematically predicted contact point
traces are shown in gray;observed experimental traces
are in black.The two sets of traces show a good quality of
match at all 12 shear levels,with the experimental traces
superimposed directly onto their predicted paths.
4.2.Comparison with end friction excluded from modeled
results (
Fig.12
)
By removing the term
Q
j
EF
from Eq.(35),we effectively
remove all contact friction between the top block and
actuator load tip.Their interaction is limited to an
imparted shear displacement force,and the analytical
model now describes how the block would behave in the
absence of an end effect.The experimental data,on the
other hand,continue to describe how the block
actually
behaves when pushed by a steel load tip.The two sets
of results are compared in
Fig.12
with black de
noting experimental traces and gray the calculated traces.
In contrast to
Fig.11
,the sliding paths no longer align
and there is signiﬁcant trace mismatch.This is especially
true for the lower shear loads,and at the ﬁrst contact
point.
4.3.Comparison with end friction and load point slip
excluded from modeled results (
Fig.13
)
As in Case B,end friction force
F
EF
is excluded from the
analytical results.Additionally,driving force
F
T
is modeled
as ﬁxed throughout sliding to its starting position on the
centerline,at 1/3 of the top block’s height.
F
T
is further
required to remain orthogonal to the aft face of the
block,simulating the force imparted by static ﬂuid
pressure.
Fig.13
compares the predicted traces of this
frictionless hydrostatic loading to the observed experi
mental traces seen previously.Experimental traces are
shown in black,predicted traces in gray.There are serious
disparities between the observed and calculated results.The
existing mismatch of
Fig.12
is compounded by the newly
leftward concavity of the predicted CP2 traces.
ARTICLE IN PRESS
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
503
ARTICLE IN PRESS
Fig.11.Case A:Comparison of sliding paths with end friction included in the derived results (shown in gray).The 12 panels correspond to 12 different
shear loads.Experimental results shown in black.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
504
ARTICLE IN PRESS
Fig.12.Case B:Comparison of sliding paths with end friction excluded fromthe derived results (shown in gray).The 12 panels correspond to 12 differe
nt
shear loads.Experimental results shown in black.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
505
ARTICLE IN PRESS
Fig.13.Case C:Comparison of sliding paths with end friction excluded fromthe derived results (shown in gray) and an orthogonal shear force applied a
t
a ﬁxed point of the top block.Experimental results shown in black.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
506
4.4.Samples of different size (
Fig.14
)
Consider samples half the size and twice the size of the
laboratory sample.The mass remains 5.4 kg.
Fig.14
compares two sets of modeled results (we are no longer
comparing with laboratory data).Black traces indicate
predictions which include an end effect.Gray traces do
not include an end effect.The left,middle,and right
columns depict sliding at 26.7,40,and 53.4 N shear force,
respectively.
The ﬁrst row of
Fig.14
compares end effectfree and end
effectincluded traces for a halfscale sample.The second
row of
Fig.14
makes the comparison for a doublescale
sample.There seems to be less mismatch between end
effectincluded/free traces at the higher two shear loads
than at the lowest shear load.Scaling of axes aside,the top
and bottom rows appear to be identical.
4.5.Samples of different weight (
Fig.15
)
Consider samples with half the mass and twice the mass
of the 5.4 kg laboratory top block.External dimensions are
held ﬁxed.The ﬁrst row of
Fig.15
compares end effectfree
traces (gray) with end effectincluded traces (black) for a
2.7 kg top block under shear loads of 13.3,20,and 26.7 N.
The second row makes this comparison for a 10.9 kg block
under shear loads of 53.4,80.1,and 106.8 N.The disparity
between end effectfree/included traces appears to decrease
with increasing shear load.The top and bottomrows of the
ﬁgure are identical.
4.6.Uniformly scaled samples (
Fig.16
)
Fig.16
compares end effectfree traces with end effect
included traces at three sample scales:0.5:1,1:1 (lab
sample),and 2:1.In each case,the density remains equal to
the density of the lab sample.Block weight is varied to
maintain constant density.The shear force (value given
above each panel) is held ﬁxed at 1/2 block weight,that is,
at 50%of the normal load.All of the traces are calculated
rather than experimental.
The ﬁrst column of
Fig.16
shows sliding at 1/2 scale,the
second column at full scale,and the ﬁnal column at double
scale.Traces in the ﬁrst row do not include end effects,
while traces in the second row do include them.The entries
in each row are identical to one another,but the entries in
ARTICLE IN PRESS
Fig.14.Case D:Samples of different size than the experimental topblock.The upper row features halfsize samples tested at three different shear l
oads.
The lower row features double size samples tested at three different shear loads.Block weight is held ﬁxed.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
507
each column are dissimilar.Column entries do not begin to
resemble one another with increasing shear force.
5.Discussion
5.1.Case A (
Fig.11
)
Each of the 12 experimental (black) sliding paths in
Fig.11
comprise the superimposed results of four
independent tests at the speciﬁed shear load.Because the
shear tests were manually terminated,their traces tend to
be shorter in length than the modeled traces (gray),which
were automatically cut off at
D
q
1
#
8cm.The portions of
the traces which do overlap exhibit an excellent quality of
match.A slight misalignment of CP2 and CP3 traces at the
higher shear loads may be due to the breakdown of the
Mohr–Coulomb model at high shearing rates.The peak
sliding velocity of the top block was measured close to 1m/s,
suggesting the presence of frictional rate effects (see
[6]
).
The block appears to rotate counterclockwise as it
slides forward and right.The rotation is more pronounced
at low shear loads,as the traces of all three contact
points appear to straighten with increasing shear force.
Part of this phenomenon is certainly attributable to the
greater linear momentum of the top block under large
displacement forces.We cannot yet speculate on the
contribution of end friction force
F
EF
to the physical
sliding behavior,because the force remains embedded
in the analytical formulation.However,the fact that
observed sliding paths are correctly predicted suggests that
the dynamics of the top block have been accurately
modeled.
5.2.Case B (
Fig.12
)
With end friction force
F
EF
removed from the formula
tion,the modeled sliding behavior is quite different from
the observed behavior.The most obvious disparity occurs
at CP1,where the experimental traces (black) feature very
little lateral dilation and the calculated traces (gray) predict
signiﬁcant rightward dilation.A more subtle distinction is
indicated by connecting the endpoints of the calculated
traces.Doing so reveals that in the process of moving
forward and rightward,the modeled block rotates clock
wise.In contrast,the experimental traces show the block
rotating counterclockwise.
ARTICLE IN PRESS
Fig.15.Case E:Samples of different weight than the experimental topblock.The upper row features halfweight samples tested at three different sh
ear
loads.The lower row features double weight samples tested at three different shear loads.The external dimensions of the block are held ﬁxed.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
508
This modal difference is fully explained by the end effect.
The ‘‘natural’’ tendency of the top block to rotate counter
clockwise over the threeplane foundation results in
differential motion between its aft face and the load tip.
The relative motion is quickly resisted by end friction force
F
EF
,which is tangent to the line of load tip drift and
directed leftward in response to clockwise rotation.The
magnitude of
F
EF
is proportional to shear load and
apparently great enough linearize the path of CP1.In the
ﬁnal two panels of
Fig.12
(
j
F
T
j#
51.6 N,55.6 N),the
experimental dilation of CP1 (black) is entirely suppressed.
End friction thus changes the concavity of trace CP1 and
imparts the top block with an observed tendency to rotate
clockwise.The relative contribution of linear momentum
may be discerned in the calculated CP1 trace (gray),which
straightens with increasing shear force,but remains
diagonal to the
e
1
direction.Higher shearing velocities
induce a more linear sliding path in the sense that the block
tends to dilate laterally without rotating.In contrast,the
end effect suppresses uniform dilation at high shear loads,
but does not prevent the block from rotating.
It is important to note that the relative severity of the
end effect appears to lessen with increasing shear load,as
indicated by the improved convergence of predicted and
observed contact point traces.This is especially true for
CP2 and CP3,whose black and gray traces line up nicely at
the highest shear loads despite being skew at the lower
ones.The CP1 traces remain incongruent at a shear force
of 55.6 N but appear to be slowly converging.We may
therefore conjecture that for a given sample,sliding path
distortion always decreases with increasing shear load,a
claim further investigated below.
5.3.Case C (
Fig.13
)
Test sample dilation can involve two types of motion,
translation and rotation.Lateral translation causes the
original load point (the bodyﬁxed point on the aft face
centerline) to migrate away from the forward moving load
tip.Rotation decreases the nominally right angle between
aft face and load tip by cos
&
1
(
e
1
"
g
1
)
1
.While end friction is
a result of sample translation speciﬁcally,both dilation
modes change the assumed boundary conditions by
introducing loading asymmetry.The resulting moments
lead to ‘‘directional effects’’,which occur independently of
the end friction.
Consider a shear experiment designed to simulate the
failure of a gravity dam monolith under upstream ﬂuid
pressure.Whereas the net force of water always acts along
the centerline of a conﬁning surface,orthogonal to the
surface,the mechanical thruster can only produce such
loading until the test sample begins to dilate.In order to
ARTICLE IN PRESS
Fig.16.Case F:The sliding paths,at three different shear loads,of scaled samples with the same density as the original block.The results in the uppe
r row
do not include end effects.The results in the lower row do include end effects.The ratio of shear load to block weight is held ﬁxed.
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
509
maintain hydrostatic conditions thereafter,the thruster
would have to be equipped with a servocontrolled robotic
arm.We now investigate whether our nonrobotic shear
setup provides a reasonable approximation of hydrostatic
loading.
The calculated traces in
Fig.13
(gray) depict the sliding
behavior of the top block under a frictionless bodyﬁxed
shear force.The experimental traces are the same ones
previously seen in
Figs.10 and 11
,but may now be said to
contain directional effects in addition to frictional end
effects.It is evident that the two sets of traces are even
more disparate than above.The calculated traces of all
three contact points exhibit a leftward concavity indicative
of vigorous clockwise rotation.At low shear loads,
rotation is in fact the primary dilation mechanism.At
high shear loads,path distortion appears to lessen,but the
traces of CP2 and CP3 never align neatly as in
Fig.12
.
The increased curvature of the calculated traces in
Fig.13
compared with those of
Fig.12
is due to the fact
that the bodyﬁxed shear force ‘‘follows’’ the block along
its spiraling trajectory.Because the driving force vector
continues to project near the center of mass,it produces no
signiﬁcant moment,allowing topographical effects to
prevail.In contrast,the nonattached modeled load tip of
Case B drifts left over the block’s aft face,always
increasing its distance from the center of mass.The
resulting moment opposes clockwise rotation and reduces
the sliding path’s curvature.
It is clear from
Fig.13
that our experimental setup does
a poor job of modeling hydrostatic loading conditions.The
existence of directional and frictional effects in general
should call into question the propriety of relying on
experimental data in a problem where failure
route
is
important.For example,had our experimental sliding
paths been generated during a suite of tests involving a trial
dam monolith,their careless identiﬁcation with hydrauli
cally induced sliding paths would lead to a ﬂawed
understanding of how best to design a monolith footprint
or reinforce some critical section of the dam.
5.4.Cases D and E (
Figs.14 and 15
)
It was noted above that calculated contact point traces
tend to be more closely resembled by experimental traces at
high shear loads than at low.Since lateral dilation is
ultimately both the source and victim of end friction,it is
not surprising that those tests which feature less dilation
(due to large block inertia or high shearing velocity) are the
ones that yield the most accurate
experimental
data.
The observations in question were made for one
particular test specimen;we now vary the dimensions and
mass of that original sample.This is done mathematically,
since the model has been shown to be viable,and since the
actual test block comes in only one size.The required
modiﬁcations to the solution routine are minimal.Apart
from explicitly changing the block weight/length,the mass
moment tensor
I
must be scaled (
I
enters the formulation
through coefﬁcients
M
ij
).Since
I
has dimensions
ML
2
,a
length change
cL
0
requires that its components be multi
plied by
c
2
;a weight change
dM
0
requires that the
components be multiplied by
d
.
Fig.14
shows the consequences of shear force increase in
half size and double size samples.The samples react
identically because their weight is the same (the larger
sample is allowed more time to slide and covers more
ground).The distortive impact of the end effects appears to
lessen with increasing shear load,as end effectfree/
included traces converge at CP2 and CP3.The convergence
of CP1 traces improves between 26.7 and 40N shear load,
then changes little between 40 and 53.4 N,where it appears
that the gray trace continues to linearize and the black
trace is already linear.This behavior is consistent with the
explanation given in Section 5.2.
Fig.15
portrays the sliding motion of half weight and
double weight samples.The ﬁrst and second rows are
identical because the blocks in a given column slide with
the same acceleration (weight and shear load have been
increased proportionally).The geometric similarity be
tween Case D/E sliding paths is not a coincidence,but
rather a consequence of the binary manner in which we
have scaled the original specimen,and of the distance
(rather than time)dependent slide parametrization.As in
Case D,there is a lessening of end effectfree/included trace
disparity with increased shear level.
5.5.Case F (
Fig.16
)
The beneﬁts of running the experiment at higher shear
loads were ﬁrst suggested in
Fig.12
,where the decreasing
severity of the end effect was indicated by an improved
similarity between experimental and calculated paths.In
the preceding section,this trend was seen to hold for
proportional samples of different size or weight,not being
restricted apparently to the particular laboratory specimen
used.Because the pivotal drop in lateral dilation that
accompanies high shearing velocities is responsible for the
end effect decrease,we term the phenomenon ‘‘velocity
shielding’’.
Cases D and E considered two methods of scaling a
laboratory sample,neither of which are appropriate for the
purposes of physical modeling.The weight of an actual
dam monolith cannot be preserved in a smaller sized
sample,nor can a lighter model with the same dimensions
be cast.More importantly,a combination of load and
sample parameters corresponds to one speciﬁc set of ﬁeld
scale conditions;unilaterally increasing the experimental
shear load while holding sample parameters constant
changes the physical situation being modeled.
In situ shearing conditions can be replicated by casting a
laboratory specimen with the same density as the ﬁeld
monolith,then applying a shear force whose magnitude
preserves the ratio of monolith weight to hydrostatic
displacement force.This type of scaling makes it possible
to retain the ﬁeld monolith’s frictional properties,since a
ARTICLE IN PRESS
D.Galic et al./International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512
510
model can be cast from the same material.In our case,we
are simulating the test numerically and only need to adjust
the components of mass moment tensor
I
.The dimensions
ML
2
of the components can be written (
M
/
L
3
)
L
5
or
r
L
5
,
so for a length change
cL
0
,density is preserved by
multiplying them by
c
5
.
Fig.16
shows modeled test results under uniform
scaling.The columns correspond to equal density top
blocks of scale 0.5:1,1:1 (laboratory specimen),and 2:1.
The left to right shear force increase preserves a 0.5:1 shear
load to weight ratio.Sliding paths in the top row do not
include a modeled end effect and simulate test results from
an ideal machine;bottom row sliding paths do include an
end effect and depict the shearing of an aluminum top
block pushed by a steel load tip.
The sliding paths in a given row are geometrically
similar,which conﬁrms that we have chosen an appropriate
means of scaling,but at once reveals the total absence of
velocity shielding.That the column entries do not begin to
assimilate with increasing shear load indicates there is no
advantage to uniformly varying the scale of a laboratory
shear specimen.The end effects themselves appear to be
scale independent,in the sense that proportionally scaled
samples exhibit geometrically similar path aberrations.
This follows from our acceptance of the Mohr–Coulomb
criterion,which linearly relates the magnitude of end
friction force
F
EF
to the magnitude of the driving shear
force.The consequences of choosing a velocitydependent
friction model (see e.g.
[11]
) are beyond the scope of
this paper.
6.Conclusion
An experimental setup can provide only indirect
information about its own limitations,for example,
through observed inconsistencies between tests performed
on specimens with different properties.Analysis,on the
other hand,allows one to isolate speciﬁc parameters and
investigate their individual contributions to the measured
data.The objective of this paper has been to predict the
laboratory sliding path of a triplanar monolith by
accounting for both planned and incidental contact
interactions.Removing the end friction term
F
EF
from
the sum of external forces,we were able to envision how
the monolith might slide when displaced by a low viscosity
hydrostatic force or frictionless mechanical load tip.Such
modeled sliding behavior was seen to be vastly different
from laboratory sliding behavior,which always involves a
superposition of ‘‘natural’’ shear characteristics and testing
machine interference.
The results of our comparison indicate that in the
context of a generalized shear experiment,end effects are
nontrivial and cannot be ignored without compromising
model assumptions.Although the results we have pre
sented are speciﬁc to an aluminum sample pair with the
stated foundation angles,the conclusion holds for any
shear experiment involving signiﬁcant lateral dilation.End
friction results whenever there is relative movement
between a sliding sample and displacement actuator.
However,when a sample is free to dilate laterally,the
absence of side constraint allows the moments induced by
F
EF
to rotate the block unopposed.The experimental data
from a laterally unconstrained shear experiment is tainted
with a loss of distinction between this spurious rotation
and topographically induced lateral dilation.
A number of methods could be used to physically reduce
end effects in a direct sheartype experiment.Although
lubricant coatings are usually thwarted by the size and
convexity of the actuator/sample interface,the load tip
itself can be manufactured from a lowfriction material
such as graphite or teﬂon.Another approach would be to
change the manner in which displacement force is applied.
Replacing the mechanical actuator with a ﬂuid jet would
obviate entirely the threat of end friction,but also present
new challenges such as the management of spent ﬂuid.
Manually tilting the sample pair is a simple means of
applying frictionless displacement force
[4]
.However,it
offers very little control over model parameters and is only
practical for small samples.
Until an inexpensive frictionless method of applying
displacement force becomes widely available,end friction
will continue to be an insidious factor in laboratory shear
experiments.End effects are inevitable,but it is possible to
understand and anticipate them.The engineer should be
wary of equating laboratory results with ﬁeld behavior,and
understand that certain kinds of shear tests are conspicu
ously susceptible to end effects.In situations where
signiﬁcant lateral dilation is expected or where the path
of the block is important,it may be wise to dispense
altogether with physical testing and instead focus one’s
resources on numerical simulation.Commercially available
DDA codes
[12,13]
can be useful for of modeling a variety
of sliding problems.
Acknowledgments
This research was funded by the National Science
Foundation under Grant CMS0408389.The authors
would like to Pete Thuesen for his help in preparing the
aluminum sample pair.
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