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International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512

A Lagrangian dynamic analysis of end effects in a generalizedshear experiment

D. Galic!, S.D. Glaser, R.E. Goodman

Department of Civil and Environmental Engineering, University of California, Berkeley, 760 Davis Hall, Berkeley, CA 94720-1710, USA

Received 5 January 2007; received in revised form 9 July 2007; accepted 16 July 2007Available online 5 September 2007

Abstract

In laboratory shear testing, a primary source of error is the surcharge force caused by relative motion between the displacementactuator and dilating top sample. This force is referred to as end friction, and the changes it produces in experimental data are termedend effects. The results from a laboratory shear setup always represent a superposition of natural sliding behavior and testing machineinterference; their relative proportions can be determined by externally modeling the experiment. In this paper, we construct a fullLagrangian dynamic model for the shearing behavior of a prismatic aluminum top-block over a matching asymmetric foundation. Endfriction is initially included in the analysis, whose viability is established by comparing modeled and experimental top-block sliding pathsat 12 different shear loads. The end friction force is ultimately removed from the formulation, and the end effects manifested as thesubsequent differences between modeled and experimental sliding paths. It is shown that end effects significantly alter both the slidingpath and rotation mode of the prismatic top-sample. While their impact on the trajectory of a given sample appears to decrease withincreasing shear force, it is shown that uniform sample scaling does nothing to alleviate the problem, and that end effects are functionallyscale 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; Sheartest; Sliding path

1. Introduction

The physical interaction between a testing machine andtest sample inevitably produces an additional response inthe sample that would not appear in situ. We term thisresponse an ‘‘end effect’’, and its origin can be traced to thedifferential motion of contacting parts. In the well-knownuniaxial test, for example, an end effect arises when theload platens and test sample deform laterally at unequalrates, resulting in friction along the ends of the sample. Thedistorting impact of end friction on uniaxial test data hasbeen previously documented [1,2].

Whereas a uniaxial test provides information on the bulkproperties of a medium, the direct shear test is used toinvestigate the strength of existing planar discontinuities[3]. The shear strength of a joint is usually lower than host

rock compressive strength, and the movable half-sampleexpected to slide before material failure occurs. Becauserupture or crushing are unlikely, the shear displacementforce may be imparted by a point loader instead of a loadplaten. The point loader maintains contact with only alimited portion of the test sample and its lateral deforma-tion does not adversely affect shear behavior. However, theroughness of a joint surface virtually ensures that a slidingsample will experience dilation, which quickly leads torelative motion between the sample and load applicator.Since relative motion is necessarily accompanied byfriction, this is the manner in which end effects areintroduced into a direct shear experiment.Consider a symmetric top sample that dilates only

vertically (Fig. 1a). According to the Mohr–Coulombfriction model, a joint with net friction angle f and normalcompression N requires a shear force of magnitude N tanfto displace. Assume that a force of such magnitude isimparted by a point loader, and define c as the friction angle

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!Corresponding author. Tel.: +1510 642 9278.E-mail address: [email protected] (D. Galic).

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of the load tip/sample interface. The ‘‘normal’’ force actingon the load tip/sample interface is identical to the machineimparted displacement force, so by Mohr–Coulomb, theload tip friction force has magnitude N tanf tanc. Thisforce acts downward as the sliding sample dilates upwardand therefore supplements the total normal force. But sincenet compressive load N(1+tanf tanc) is only slightlygreater than command normal load N, sample dilation isunlikely to be impacted, unless we are operating near thecompressive strength of the rock.

Next consider a sample which dilates only laterally(Fig. 1b). As before, the command normal load is N, theshear load N tanf, and the magnitude of the load tipfriction force N tanf tanc. The friction force is nowdirected horizontally, since relative motion between theload tip and laterally dilating sample amounts to sideslip.Because there are no externally applied lateral forces, theload tip friction force comprises the net external force inthe horizontal direction. So whereas the vertical load of thevertically dilating sample was increased from N toN(1+tanf tanc), the lateral load of a laterally dilatingsample increases from 0 to N tanf tanc. This represents amuch more serious violation of the model assumptions.

An actual joint sample will typically exhibit somecombination of vertical and lateral dilation. The verticaldilation of a shearing top specimen provides informationon the amplitude of the joint roughness; the tendency of aspecimen to dilate laterally indicates that this roughness isnot uniformly symmetric. That a rock foundation withstrongly asymmetric topography can be expected to favor

movements involving lateral slip is of considerable interestin dam engineering. The tendency of sliding monoliths todilate outward as they move forward may be key tounderstanding the failure mechanics of a gravity damfoundation [4]. It is therefore important, when running ashear test designed to simulate the hydrostatic loading of adam monolith, that the recorded lateral slip be a propertyof the foundation and not of the testing apparatus.In this paper, we quantify the total effect of end friction

on a laterally dilating tri-planar sample pair in generalizedshear. The lower and upper half-samples under considera-tion are shown in Figs. 2 and 3. The experimentalboundary conditions are referred to as generalized shearfor the following reasons: (1) An asymmetric three-planesample/sample interface produces both vertical and lateraldilation; (2) The initially matching top and bottom blocksrevert to three-point contact when sliding begins (thisallows the block interaction to be modeled in terms offorces rather than stresses); (3) The configuration reducesto a standard direct shear setup when the dip angles of theinterface planes are set to zero; (4) The configurationextends to a more complicated discretized surface when thefactual contact points are considered to be one entry in asequence of tri-planar contacts.A set of experimental data is first generated from shear

tests performed on an aluminum sample pair. We thenmodel the experiment mathematically using a Lagrangian(configuration space) dynamic analysis, which captures

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Fig. 1. Two limiting modes of dilation exhibited by a sample whosenominal displacement direction is indicated by the arrow: (a) pure verticaldilation—the sample moves upward as it is pushed forward; (b) purelateral dilation—the sample moves sideways as it is pushed forward. Therelative 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 planesnumbered.

D. Galic et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512496

both the frictional and inertial qualities of rigid bodysliding. The predictions of the analytical model, presentedin the graphical form of sliding paths, are compared withexperimental results in order to establish the accuracy ofthe model. Once the model has been calibrated, the forcesassociated with end friction are removed from the analyticalmodel, and the updated model predictions compared withthe existing experimental results. The end effect ismanifested as the difference between the experimentalsliding paths (which necessarily include it) and modeledsliding paths (from which the end effect has been removed).

The relationship between selected sample parametersand ultimate sample trajectory is investigated. First, theweight and dimensions of the top block are variedindependently in order to identify possible trends of endeffect intensity. Since a given combination of shear load,sample weight, and sample size corresponds to oneparticular set of field conditions, these parameters are nextscaled 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 lowerhalf of a shear box. The three foundation planes areindependent, in the sense that their three normal vectorscan be used as a basis for three-dimensional Euclideanspace, R3. The spatial orientation of each foundation planeis completely characterized by either its normal or dipvector. While these quantities are variables in the analyticalformulation, specific values needed to be selected formachining purposes. The design foundation angles pro-mote the lateral movement of the top block. Foundationplane (FP) 1 has a dip of zero, FP2 a dip of 9.51 dippingaway from the displacement direction, and FP3 a dip of 201dipping 1501 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 thestationary bottom half. The top and bottom samples havematching interfacial topography and fit snugly togetherprior to the start of sliding. Once the top block has beendisplaced, it immediately reverts to three-point contact.The same three points remain in contact throughoutsliding, so long as the block is not pushed over the edgeof its foundation, and provided the center of masscontinues to project inside the triangle formed by the threecontact points. Fig. 3 shows the locations of these threepoints on the top block. For the range of motion we areinterested in, each contact point is associated with exactlyone of the three foundation planes: contact point (CP) 1slides 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 hasplan dimensions 30 cm! 45 cm, while the free top blockhas plan dimensions 15 cm! 30 cm. Also visible is the arrayof laser rangefinders used to track motion, as discussed in

[5]. The shear displacement force is provided by a CopleyControls brushless linear actuator, whose load tip (Fig. 5)consists of a machined steel bolt. By virtue of its design, theactuator can move only forward or backward; the topblock, on the other hand, has the ability to dilate bothvertically and laterally as it slides. Relative motion betweenthe actuator load tip and the upstream face of the top blockleads to the end effect we wish to investigate.The experimental program consisted of approximately

50 constant force (CF) shear tests run at 12 different shearforce levels. The test range of shear force values (27.8Nthrough 55.6N) was selected based on practical experiencewith the aluminum blocks. It was observed that slidingtended to be prematurely arrested by frictional rate effects[6] at shear loads of less than 25N, whereas shear loads ofgreater than 60N resulted in more or less linear sliding

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Fig. 4. Comprehensive view of the experimental setup. The top block isdisplaced by a linear actuator. Note laser rangefinders.

Fig. 5. Load-cell mounted steel pusher shown in contact with thealuminum top block. As the top block dilates, there is relative motionbetween its aft face and the steel tip, which inscribes a curve in the movingplane 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 53Nweight of the top block. The normal force was not varied,and no additional weights placed on the block. While thissounds like poor technique, it should be kept in mind thatwe are not attempting to construct a failure envelope forthe top block. Rather, we are interested in the top block’ssliding path as a function of driving shear force. The‘‘output’’ of the experiment is a graphical object, not afriction angle.

The results of the tests are compiled as contact pointtraces, imaginary marks that the contact points inscribe ontheir foundation planes as the top block slides. Fig. 6 showsan overhead view of these traces for a sample shear load of32.5N. The forward displacement direction is indicated bythe x1 axis. Axis x3 captures the horizontal (lateral)components of non-forward contact point movement(dilation). The macroscale relief of the foundation mostcertainly produces upward movement in the top block, butthe vertical component of dilation is lost in our two-dimensional plot. Because FP1 is entirely horizontal, thefeatured trace represents CP1’s actual path; CP2 and CP3slide on inclined foundation planes and their spiralingthree-dimensional paths are seen in horizontal projection.

In order to account for experimental uncertainty, severaltests were run at each load level. Although it appears thatthere is only one set of traces displayed, Fig. 6 in factcontains data from all four tests run at 32.5N. Trace sets

obtained from independent tests performed at the sameshear load were generally found to be indistinguishable, atestament 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 ofthe top block with its foundation; the other describes theinteraction of the top block with the actuator load tip. Ourtask is to separate the two and determine whether thefrictional interference of the load tip is negligible.

3. Analytical formulation

In the following derivation, we adopt the conventionof summation over repeated indices. For example, theexpression aibi is equivalent to a1b1+a

2b2+yanbn, wherethe value of n will usually be apparent from the context. Inthe event that we do not wish to sum over repeated indices,the indices will be enclosed by parentheses, for example,F(k) "P(k). Coordinates are denoted by an upper index (xi)whereas vectors are denoted by a lower index (ei). Boldfaceis used to distinguish between points in space and theirposition vectors (P vs. P) and between forces and theirmagnitudes (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 haveplaced on the system. In the ensuing sections we: establish aseries of datums by which to track motion (Section 3.1);formulate constraint equations (Section 3.2); generalizethe constraint forces (Section 3.3); model the end effect(Section 3.4); and formulate and numerically solveLagrange’s differential equations of motion (Section 3.5).The desired output is a family of contact point traces suchas those shown in Fig. 6.

3.1. Local and global coordinate systems

We will model the top block as a triangular frame whosetips describe the contact points. This is not a mathematicalsimplification, but rather a graphical one; its easier to drawonly those parts of the top block that are in contact withthe foundation block. The full inertial properties of the topblock continue to be used throughout the analysis. Sincethe block will be treated as a rigid body (we are notinterested in what is happening inside the block), thedimensions of the frame are not allowed to changeduring 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 wereclear. Also visible are the three foundation planes, whosethree lines of intersection meet at the origin of the globalcoordinate system (xi). Although coordinate axes xi havebeen omitted from the figure, the orientation of the globalunit basis vectors ei is shown in the lower left corner. Themoving frame defined by contact points 1–3 is endowedwith a local, accelerating coordinate system zi; the axes

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Fig. 6. Experimental results are presented as contact point traces (black),which represent the horizontal projections of the contact point slidingpaths. The displacement direction is x1, with lateral dilation occurring inthe x3 direction. Vertical dilation in the x2 direction cannot be seen in thisoverhead view. Foundation-plane lines of intersection shown in gray.Contact points and foundation planes as labeled.


drawn inside the frame depict the orientation of local basisvectors gi. Local and global coordinate systems can berelated through the three position vectors P1, P2, P3, whichtrack the locations of the contact points with respect to theglobal coordinate system. For example, g1 is a scaledversion of the vector P2-P1.

As stated, all interaction between the top block andfoundation block is limited to the three contact points.Focusing our attention on these three points is preferableto tracking every point in the top block, but it is even moreconvenient to shrink the block to a single point. This can bedone by transforming the domain of the problem fromthree-dimensional Euclidean space, R3, to nine-dimen-sional Euclidean space, R9. What may sound mathemati-cally intimidating actually entails nothing more than takingthe three sets of three coordinates:

x1CP1;x2CP1; x

3CP1

! "; x1CP2;x

2CP2;x

3CP2

! "; x1CP3;x

2CP3;x

3CP3

! "(1)

and writing them as one set of nine coordinates:

x1CP1;x2CP1; x

3CP1; x

1CP2; x

2CP2; x

3CP2; x

1CP3; x

2CP3; x

3CP3

! ". (2)

Relabeling the coordinates in Eq. (2) as q instead of x,we have

q1; q2; q3; q4; q5; q6; q7; q8; q9! "

, (3)

which are the generalized coordinates of the system. TheEuclidean space R9 whose coordinates are qi is termed theconfiguration space of the system, and is spanned by nineunit basis vectors hi. By virtue of the transformation fromEq. (1) to Eq. (2), a single point in configuration spacedescribes the locations of all three contact points anduniquely defines the position of the block. Note that if wewish to return to the physical model space R3, we can usethe following inverse correspondence:

x1CP1;x2CP1; x

3CP1

! "# q1; q2; q3

! ",

x1CP2;x2CP2; x

3CP2

! "# q4; q5; q6

! ",

x1CP3;x2CP3; x

3CP3

! "# q7; q8; q9

! ". $4%

3.2. System constraints

The 9-tuple in Eq. (3) represents the coordinates of anypoint in the top block’s configuration space. But the topblock’s domain of sliding does not encompass all of R9, andnot all combinations of qi are physically possible. In ouranalysis, we are not interested in unstable motion or in theelastic deformation of the block. All contact points musttherefore remain on their respective planes of slidingthroughout motion, and the dimensions of the framecannot be allowed to change during sliding. Theserestrictions lead to six constraint equations and leave thetop block with three degrees of freedom.The first three constraint equations ensure that contact

points do not lift off their foundation planes or transitionto other foundation planes. In other words, the contactspoints must be elements of the foundation planes. Recallthat the equation of a plane can be written:

n " $P& Pref % # 0, (5)

where n is a normal to the plane, P a vector from the originto any point on the plane, and Pref a vector from the originto some fixed point on the plane. Because all three of ourfoundation planes contain the origin of the globalcoordinate system, the vector Pref can always be taken asthe zero vector, and the equations that the system mustsatisfy are:

n1 " P1 # 0; n2 " P2 # 0; n3 " P3 # 0, (6)

where P1, P2, and P3 are the global position vectors ofcontact points 1, 2, and 3.The next three constraint equations enforce the rigid

body constraint: they state that the distance betweencontact points is constant during sliding. These equationscan be written in terms of the known frame dimensions as

P3 & P2j j& L23 # 0; P3 & P1j j& L13 # 0,

P2 & P1j j& L12 # 0, $7%

where LAB is the length of the frame edge between contactpoints A and B. Since the components of position vectorsP1, P2, and P3 are simply the global coordinates of CP1,CP2, and CP3, we can use the correspondence in Eq. (4) towrite these position vectors as functions of the generalizedcoordinates:

P1 # q1e1 ' q2e2 ' q3e3,

P2 # q4e1 ' q5e2 ' q6e3,

P3 # q7e1 ' q8e2 ' q9e3. $8%

Expressions (6) and (7) then take the functional form:

f i q1 . . . q9

! "# 0, (9)

where index i denotes the constraint equation number andfi is referred to as the ith constraint function.The six equations of (9) describe a group of surfaces Si in

nine-dimensional configuration space. All points p onsurface Si represent valid configurations with respect to

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CP 2CP 3

CP 1

e1

e2 e3

Plane 3dip = !

Plane 2dip = "

Plane 1dip = "

g3g

2

g1

P1

P2P3

!

Fig. 7. Overhead view of the model space. The top block is represented asa rigid frame. The contact points and foundation planes are indicated, aswell as the directions of global basis vectors e and local basis vectors g.


constraint i. Similarly, a curve C on Si describes a validsequence of configurations. The restriction imposed byconstraint i prevents C from ever leaving surface Si, and allvectors tangent to Cmust therefore be tangent to Si as well.This fact allows us to write Eq. (9) in an equivalent form, asa statement that the system velocity vector V through pointp on surface Si must be tangent to the constraint surface atthat point:

Vp " np # 0; 8p 2 Si. (10)

For a point p with coordinates (q1, q2, y q9), we have

Vp #d

dt$qjhj% # _qjhj, (11)

where the hj are unit basis vectors and the _qj are known asgeneralized velocities. The normal vector n to constraintsurface Si at point p is obtained as the gradient ofthe corresponding constraint function in Eq. (9), evaluatedat p:

np # rf i. (12)

We can therefore write Eq. (10) as

rf i " _qjhj # 0. (13)

Although it is not immediately clear why (13) is a moreuseful expression of the six constraint equations than (9),this will become evident after we have formulatedLagrange’s equations.

3.3. Generalized constraint forces

Fig. 8 shows the external forces acting on the top blockas it slides. The driving force FT is imparted to the aft faceof the block by the thruster load tip. Also acting at the loadtip is the end friction force FEF, a result of relative motionbetween the load tip and test block. This force acts in thedirection opposing the relative motion. For example, if ablock-fixed point PCL begins to move to the right of theload tip, FEF will be directed toward the left. The term ‘‘endeffect’’ refers to those aspects of physical block behaviorthat result directly from the presence of FEF. Removingthis force from the analysis is equivalent to removing theend effect.

The driving force FT, end friction FEF, and block weightW are resisted by reactions at the three contact points.These include shear and normal reactions. The normalreactions are constraint forces, since they ensure that thecontact points will not sink beneath the foundation planes.In fact, they are the constraint forces that maintainconstraint conditions (6). On the other hand, the constraintforces related to constraint conditions (7) are not shown inFig. 8. These forces promote rigid body motion, and it isnot entirely obvious where to display them in the figure.Apart from this graphical omission, the rigid bodyconstraint forces will be treated in the same way as themore 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 ultimatetrajectory of the block is still unknown, this can only bedone locally, measuring the differences in energy betweenslightly different sliding paths. The current configuration ofthe block and an alternative infinitesimally distant config-uration are separated by the incremental position vectordP. The differential amount of work that must beperformed by force F in order for the system to reach itsalternate configuration is:

dW # dP " F #qPqqj

" F# $

dqj , (14)

where dqj is a small change in generalized coordinate qj (seee.g. [7]). The bracketed quantity in Eq. (14) is known as thejth generalized force, Qj:

Qj #qPqqj

" F. (15)

Generalizing forces is a crucial step in the formulation ofLagrange’s equations, and Eq. (15) must be applied to eachof the forces acting on the block. Often, the mostchallenging aspect of this process is obtaining an expres-sion for P, the position vector of the point through whichthe 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’’ issimply the point describing the current configuration ofthe system:

Pconstraints # q1h1 ' q2h2 ' . . . q9h9 # qjhj. (16)

Obtaining coordinate expressions for the constraint forcesthemselves is straightforward, but requires an appreciationof their geometric meaning. Constraint force i is alwaysnormal to Si because it enforces the requirement thatpermissible motions remain embedded in the constraintsurface. By Eq. (12), constraint force i must therefore beparallel to the gradient of fi. So while the magnitude ofconstraint force i is still unknown, its vector can beconveniently written as a scalar multiple of rfi. The vectorsum of all six constraint forces is then:

Fconstraints # l1rf 1 ' l2rf 2 ' . . . l6rf 6 # lirf i, (17)

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CM

PL

CP 2

CP 3

W

FTN1

S 1N2

S2

S3

N3

FEF

CP 1

Fig. 8. External forces acting on the top block during sliding. The contactpoints, center of mass, and instantaneous load point are indicated.


where the unknown magnitudes l are termed Lagrangemultipliers. The l are independent of qj and ignored by thedelta operator in Eq. (14). Using Eq. (15) we have

Qconstraintsj #

q qkhk! "

qqj" lirf # lirf i " hj. (18)

Eq. (18) accounts for contributions to Qj from theinternal rigid body forces and from the foundation normalreactions. The foundation shear reactions are not con-straint forces (since they resist rather than restrict systemmotion) but can be related to the normal reactions throughthe Mohr–Coulomb failure criterion. For a sliding contactpoint a,

Saj j # N$a%%% %%tanf$a%, (19)

where fa is the dynamic (residual) angle of friction betweencontact point a and foundation plane a. In terms of theLagrange multipliers, (19) can be written as

Saj j # l$a%tanf$a%. (20)

Eq. (20) gives only the magnitude of the shear force atcontact point a. The complete shear force vector Sa isantiparallel to _Pa, the instantaneous velocity of contactpoint a:

Sa # &_P$a%_P$a%%% %% l

$a%tanf$a%. (21)

We now define

Sa # &_P$a%_P$a%%% %% tanf$a%, (22)

after which the generalized shear reactions can be writtenas

Qshearsj # laSa " hj. (23)

The remaining forces in Fig. 8 are not directly associatedwith the contact points and therefore cannot be writtenwithout using local coordinates. For example, weight Wpoints downward through the mass center (CM) of the top

block. Referenced to contact point 1, the local coordinatesof CM are (C1, C2, C3), as shown in Fig. 9. The globalposition vector to CM is then:

PCM # P1 ' Cbgb. (24)

The time-fixed differential of this vector is

dPCM #qP1

qqjdqj ' Cb qgb

qqjdqj. (25)

By replacing W with &We2, we get

QWj # &We2 "

qP1

qqj' Cb qgb

qqj

# $. (26)

Note that the magnitude of the weight, W, is a knownquantity.

3.4. Modeling the end effect

The laboratory top block is shear-displaced by amechanical thruster. At the start of the experiment, thethruster load tip (see Fig. 5) is in contact with the aft facecenterline, at a point labeled PCL in Fig. 9. Unfortunately,the machine imparted drive force FT can only remain atPCL over small displacements. This is because the linearthruster has a single degree of freedom (i.e., FT # FTe1),whereas the sliding top block can dilate both vertically andlaterally. As a result, the block-fixed point PCL inevitablyslips away from the load tip, and the instantaneous loadpoint PL is distinct from PCL. These two points togetherwith the space-fixed initial load point P0

CL (PCL frozen atits pre-sliding position) form a triangle, as illustrated inFig. 10.A position vector to the instantaneous load point PL can

be assembled from position vectors PCL and P0CL. Using

Fig. 9 and the dimensions of the top block:

PCL # P1 ' Ag2 & Bg3,

P0CL # PCL$t # 0%

# & L12e1 ' Ae2 ' L23 & B$ %e3. $27%

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g1

g2

g3

C1C3

C2

AB

CP1

PCL

CM

Fig. 9. The locations of the original load point PCL and mass center CM,shown with respect to contact point CP1.

FT

PL

P˚CL

PCLk2g2+k3g3

Global Origin

Current Configuration

Starting Configuration

Fig. 10. The instantaneous location PL of the load tip can be expressed interms of its original location on the block (PCL) and its original location inspace (P0

CL).


Write PL # PCL+k2g2+k3g3, where the quantities

k2 # P0CL & PCL '

PCL & P0CL

! "" g1

e1 " g1e1

!

" g2,

k3 # P0CL & PCL '

PCL & P0CL

! "" g1

e1 " g1e1

!

" g3 $28%

represent the orthographic projections of vector(P0

CL–PCL) onto the aft face of the block, as shown inFig. 10. The terms k2 and k3 are functions of thegeneralized coordinates qi, and are therefore time-depen-dent. However, treating them as constant within the smallduration of a solution increment leads to a simplifiedexpression for dPL. In that case

dPL # dP1 ' k2 ' A$ %dg2 ' k3 & B$ %dg3. (29)

The generalized driving forces are

QTj # FT

qP1

qqj' $k2 ' A%

qg2qqj

' $k3 & B%qg3qqj

# $" e1. (30)

The instantaneous load point PL is also the locus of endfriction force FEF. This force is a result of load tip drag asthe original load point PCL slips away from PL. FEF alwaysresists the relative motion; it is a surface force on the aftface of the top block and tangent to the block-fixed curvedescribing the migration of PCL. In the context of anincremental solution, the unscaled direction DEF of endfriction force FEF is given by

DEF current increment$ % # PL current increment$ %& PL previous increment$ %,

$31%

The magnitude of FEF depends on the magnitude ofdisplacement force FT, according to the Mohr–Coulombformulation. Defining fEF as the dynamic friction anglebetween top block and load tip, and introducing acorrection factor to account for top block rotation, themagnitude of FEF can be written as

FEF # FT e1 " g1! "

tanfEF. (32)

The complete end friction force vector is

FEF #DEF

DEFj jFT$e1 " g1%tanfEF, (33)

and the generalized end friction forces are

QEFj #

FTtanfEFe1 " g1DEFj j

qP1

qqj' $k2 ' A%

qg2qqj

#

'$k3 & B%qg3qqj

$"DEF. $34%

Expression (34) is only meaningful in the context of anincremental solution.

To summarize, the generalized forces associated with theentire system are:

Qj # lihj " rf i ' lihj " Ri 'QWj 'QT

j 'QEFj ,

# liQLambdaji 'QW

j 'QTj 'QEF

j , $35%

where the terms connected with lambda have beenseparated for convenience. If we wish to investigate howthe system behaves in the absence of end effects, the Qj

EF

term is simply omitted from Eq. (35). This is equivalent toremoving force FEF from Fig. 8.

3.5. Equations of motion

Lagrange’s equations provide a relationship between thegeneralized forces Qj and system kinetic energy T. There isone equation for each generalized coordinate qj:

qTqqj

&d

dt

qTq _qj

# $'Qj # 0. (36)

The equations of motion for our top block consist ofnine Lagrange equations (36) and six constraint equations(13). The 15 unknowns (q1–q9 and l1–l6) are thus balancedby 15 differential equations.Equation sets (36) and (13) can be written in a

computationally efficient format by isolating the coeffi-cients of generalized accelerations !qj and Lagrange multi-pliers li (see [8]). This is accomplished by expanding thekinetic energies in Eq. (36) and differentiating both sidesof Eq. (13):

Mjl !ql & liQLambda

ji #1

2

qMkl

qqj_qk _ql &

qMjl

qqk_qk _ql 'QWeight

j

'QThrusterj 'QEnd Friction

j , $37%

&rf i " !qjhj # _rf i " _q

jhj. (38)

The 81 coefficients Mjl in Eq. (37) are functions of thegeneralized coordinates. Their calculation is largely me-chanical, and coordinate expressions accounting forboth translation and block rotation are given in [9].Without going into further detail here, we note that theinertial properties of the top block enter the formulationthrough Mjl.At the instant before sliding begins, generalized coordi-

nates qj retain their known initial values, while thevelocities _qj are identically zero. In fact, all quantities onthe right hand sides of Eqs. (37) and (38) are known in thefirst increment of motion, and the system can be solved for!qj and li. Denoting the right hand sides of Eqs. (37) and(38) by the column vectors V1 and V2, the two equation setsare written as a single matrix equation:

!q1

:

!q9

l1

:

l6

8>>>>>>>>><

>>>>>>>>>:

9>>>>>>>>>=

>>>>>>>>>;

#M( ) & Q( )T

& rf( ) 0( )

" #&1V1

V2

( )

, (39)

where [M] is the matrix of Mjl (9! 9), [Q] the matrix ofQji

Lambda (9! 6), [rf] the matrix of rfi " hj (6! 9), and [0] amatrix of zeros (6! 6). The accelerations !qj are then used

ARTICLE IN PRESSD. Galic et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 495–512502

to calculate the starting positions and velocities for the nextincrement:

q1

..

.

q9

_q1

..

.

_q9

2

666666666664

3

777777777775

n

#

q1

..

.

q9

_q1

..

.

_q9

2

666666666664

3

777777777775

n&1

' Dt

_q1

..

.

_q9

!q1

..

.

!q9

2

666666666664

3

777777777775

n&1

, (40)

where n is the increment number and Dt the time length ofan increment. Repeatedly applying Eqs. (39) and (40)generates a solution sequence describing the systemtrajectory in configuration space. Inverse relations (4) arethen used to recover the individual paths of the contactpoints.

Expression (40) is a variant of Euler’s method, a simplenumerical solution procedure for ordinary differentialequations (see e.g. [10]). The use of small time steps Dtoften ensures convergence, but can lead to excessivelysluggish runtimes. In the event that the solution does notconverge, a higher order numerical method may be used inplace of Eq. (40) with no additional modification to theanalysis.

4. Results

The parameters used in the analysis need to match actualexperimental values in order for direct comparison to bepossible. The dimensions, mass, and inertial properties ofthe top block are either known or calculable. Residualfriction angles for the foundation plane/contact pointinterfaces may be obtained by running direct shear testsover representative surfaces. The values used here are 171(FP1/CP1), 141 (FP2/CP2) and 111 (FP3/CP3); they differbecause the foundation roughness unequally affects con-tact points of varying convexity. The friction angle betweenthe steel load tip and aluminum top block aft face has ameasured value of 81.

As stated, the experimental results consist of contactpoint traces from sliding motion at 12 different shear loadlevels. They are compared below with the predictions of theanalytical model where: the predicted results include amodeled end effect (Section 4.1); the predicted results donot include an end effect (Section 4.2); and the predictedresults do not include an end effect and the displacementforce is modeled as ideally hydrostatic (Section 4.3). In eachof these comparisons, the experimental results necessarilycontain the end effect. We next use the analytical model toinvestigate the effects of end friction on laboratory samplesdifferent 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 sizebut 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 laboratoryend effects can be reduced through uniform sample scaling.

4.1. Comparison with end friction included in the modeledresults (Fig. 11)

The predictions of the analytical model with end frictionincluded are compared with experimental results in Fig. 11.The individual panels of the figure represent overheadviews of the foundation block, with the vertical axissuppressed and the horizontal axes equally scaled. Thegraphical correspondence between contact point numberCPn 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 allcases is 53N. Mathematically predicted contact pointtraces are shown in gray; observed experimental tracesare in black. The two sets of traces show a good quality ofmatch at all 12 shear levels, with the experimental tracessuperimposed directly onto their predicted paths.

4.2. Comparison with end friction excluded from modeledresults (Fig. 12)

By removing the term QjEF from Eq. (35), we effectively

remove all contact friction between the top block andactuator load tip. Their interaction is limited to animparted shear displacement force, and the analyticalmodel now describes how the block would behave in theabsence of an end effect. The experimental data, on theother hand, continue to describe how the block actuallybehaves when pushed by a steel load tip. The two setsof 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 alignand there is significant trace mismatch. This is especiallytrue for the lower shear loads, and at the first contactpoint.

4.3. Comparison with end friction and load point slipexcluded from modeled results (Fig. 13)

As in Case B, end friction force FEF is excluded from theanalytical results. Additionally, driving force FT is modeledas fixed throughout sliding to its starting position on thecenterline, at 1/3 of the top block’s height. FT is furtherrequired to remain orthogonal to the aft face of theblock, simulating the force imparted by static fluidpressure. Fig. 13 compares the predicted traces of thisfrictionless hydrostatic loading to the observed experi-mental traces seen previously. Experimental traces areshown in black, predicted traces in gray. There are seriousdisparities between the observed and calculated results. Theexisting mismatch of Fig. 12 is compounded by the newlyleftward concavity of the predicted CP2 traces.

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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 differentshear loads. Experimental results shown in black.


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Fig. 12. Case B: Comparison of sliding paths with end friction excluded from the derived results (shown in gray). The 12 panels correspond to 12 differentshear loads. Experimental results shown in black.


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Fig. 13. Case C: Comparison of sliding paths with end friction excluded from the derived results (shown in gray) and an orthogonal shear force applied ata fixed point of the top block. Experimental results shown in black.


4.4. Samples of different size (Fig. 14)

Consider samples half the size and twice the size of thelaboratory sample. The mass remains 5.4 kg. Fig. 14compares two sets of modeled results (we are no longercomparing with laboratory data). Black traces indicatepredictions which include an end effect. Gray traces donot include an end effect. The left, middle, and rightcolumns depict sliding at 26.7, 40, and 53.4N shear force,respectively.

The first row of Fig. 14 compares end effect-free and endeffect-included traces for a half-scale sample. The secondrow of Fig. 14 makes the comparison for a double-scalesample. There seems to be less mismatch between endeffect-included/free traces at the higher two shear loadsthan at the lowest shear load. Scaling of axes aside, the topand bottom rows appear to be identical.

4.5. Samples of different weight (Fig. 15)

Consider samples with half the mass and twice the massof the 5.4 kg laboratory top block. External dimensions areheld fixed. The first row of Fig. 15 compares end effect-free

traces (gray) with end effect-included traces (black) for a2.7 kg top block under shear loads of 13.3, 20, and 26.7N.The second row makes this comparison for a 10.9 kg blockunder shear loads of 53.4, 80.1, and 106.8N. The disparitybetween end effect-free/included traces appears to decreasewith increasing shear load. The top and bottom rows of thefigure are identical.

4.6. Uniformly scaled samples (Fig. 16)

Fig. 16 compares end effect-free traces with end effect-included traces at three sample scales: 0.5:1, 1:1 (labsample), and 2:1. In each case, the density remains equal tothe density of the lab sample. Block weight is varied tomaintain constant density. The shear force (value givenabove each panel) is held fixed at 1/2 block weight, that is,at 50% of the normal load. All of the traces are calculatedrather 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 doublescale. Traces in the first row do not include end effects,while traces in the second row do include them. The entriesin each row are identical to one another, but the entries in

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Fig. 14. Case D: Samples of different size than the experimental top-block. The upper row features half-size samples tested at three different shear loads.The lower row features double size samples tested at three different shear loads. Block weight is held fixed.


each column are dissimilar. Column entries do not begin toresemble one another with increasing shear force.

5. Discussion

5.1. Case A (Fig. 11)

Each of the 12 experimental (black) sliding paths inFig. 11 comprise the superimposed results of fourindependent tests at the specified shear load. Because theshear tests were manually terminated, their traces tend tobe shorter in length than the modeled traces (gray), whichwere automatically cut off at Dq1 # 8 cm. The portions ofthe traces which do overlap exhibit an excellent quality ofmatch. A slight misalignment of CP2 and CP3 traces at thehigher shear loads may be due to the breakdown of theMohr–Coulomb model at high shearing rates. The peaksliding 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 itslides forward and right. The rotation is more pronouncedat low shear loads, as the traces of all three contactpoints appear to straighten with increasing shear force.

Part of this phenomenon is certainly attributable to thegreater linear momentum of the top block under largedisplacement forces. We cannot yet speculate on thecontribution of end friction force FEF to the physicalsliding behavior, because the force remains embeddedin the analytical formulation. However, the fact thatobserved sliding paths are correctly predicted suggests thatthe dynamics of the top block have been accuratelymodeled.

5.2. Case B (Fig. 12)

With end friction force FEF removed from the formula-tion, the modeled sliding behavior is quite different fromthe observed behavior. The most obvious disparity occursat CP1, where the experimental traces (black) feature verylittle lateral dilation and the calculated traces (gray) predictsignificant rightward dilation. A more subtle distinction isindicated by connecting the endpoints of the calculatedtraces. Doing so reveals that in the process of movingforward and rightward, the modeled block rotates clock-wise. In contrast, the experimental traces show the blockrotating counterclockwise.

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Fig. 15. Case E: Samples of different weight than the experimental top-block. The upper row features half-weight samples tested at three different shearloads. The lower row features double weight samples tested at three different shear loads. The external dimensions of the block are held fixed.


This modal difference is fully explained by the end effect.The ‘‘natural’’ tendency of the top block to rotate counter-clockwise over the three-plane foundation results indifferential motion between its aft face and the load tip.The relative motion is quickly resisted by end friction forceFEF, which is tangent to the line of load tip drift anddirected leftward in response to clockwise rotation. Themagnitude of FEF is proportional to shear load andapparently great enough linearize the path of CP1. In thefinal two panels of Fig. 12 (jFTj # 51.6N, 55.6N), theexperimental dilation of CP1 (black) is entirely suppressed.

End friction thus changes the concavity of trace CP1 andimparts the top block with an observed tendency to rotateclockwise. The relative contribution of linear momentummay be discerned in the calculated CP1 trace (gray), whichstraightens with increasing shear force, but remainsdiagonal to the e1 direction. Higher shearing velocitiesinduce a more linear sliding path in the sense that the blocktends to dilate laterally without rotating. In contrast, theend 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 theend effect appears to lessen with increasing shear load, asindicated by the improved convergence of predicted andobserved contact point traces. This is especially true forCP2 and CP3, whose black and gray traces line up nicely at

the highest shear loads despite being skew at the lowerones. The CP1 traces remain incongruent at a shear forceof 55.6N but appear to be slowly converging. We maytherefore conjecture that for a given sample, sliding pathdistortion always decreases with increasing shear load, aclaim further investigated below.

5.3. Case C (Fig. 13)

Test sample dilation can involve two types of motion,translation and rotation. Lateral translation causes theoriginal load point (the body-fixed point on the aft facecenterline) to migrate away from the forward moving loadtip. Rotation decreases the nominally right angle betweenaft face and load tip by cos&1(e1 " g1)1. While end friction isa result of sample translation specifically, both dilationmodes change the assumed boundary conditions byintroducing loading asymmetry. The resulting momentslead to ‘‘directional effects’’, which occur independently ofthe end friction.Consider a shear experiment designed to simulate the

failure of a gravity dam monolith under upstream fluidpressure. Whereas the net force of water always acts alongthe centerline of a confining surface, orthogonal to thesurface, the mechanical thruster can only produce suchloading until the test sample begins to dilate. In order to

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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 upper rowdo 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.


maintain hydrostatic conditions thereafter, the thrusterwould have to be equipped with a servo-controlled roboticarm. We now investigate whether our non-robotic shearsetup provides a reasonable approximation of hydrostaticloading.

The calculated traces in Fig. 13 (gray) depict the slidingbehavior of the top block under a frictionless body-fixedshear force. The experimental traces are the same onespreviously seen in Figs. 10 and 11, but may now be said tocontain directional effects in addition to frictional endeffects. It is evident that the two sets of traces are evenmore disparate than above. The calculated traces of allthree contact points exhibit a leftward concavity indicativeof vigorous clockwise rotation. At low shear loads,rotation is in fact the primary dilation mechanism. Athigh shear loads, path distortion appears to lessen, but thetraces of CP2 and CP3 never align neatly as in Fig. 12.

The increased curvature of the calculated traces inFig. 13 compared with those of Fig. 12 is due to the factthat the body-fixed shear force ‘‘follows’’ the block alongits spiraling trajectory. Because the driving force vectorcontinues to project near the center of mass, it produces nosignificant moment, allowing topographical effects toprevail. In contrast, the non-attached modeled load tip ofCase B drifts left over the block’s aft face, alwaysincreasing its distance from the center of mass. Theresulting moment opposes clockwise rotation and reducesthe sliding path’s curvature.

It is clear from Fig. 13 that our experimental setup doesa poor job of modeling hydrostatic loading conditions. Theexistence of directional and frictional effects in generalshould call into question the propriety of relying onexperimental data in a problem where failure route isimportant. For example, had our experimental slidingpaths been generated during a suite of tests involving a trialdam monolith, their careless identification with hydrauli-cally induced sliding paths would lead to a flawedunderstanding of how best to design a monolith footprintor 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 tracestend to be more closely resembled by experimental traces athigh shear loads than at low. Since lateral dilation isultimately both the source and victim of end friction, it isnot surprising that those tests which feature less dilation(due to large block inertia or high shearing velocity) are theones that yield the most accurate experimental data.

The observations in question were made for oneparticular test specimen; we now vary the dimensions andmass of that original sample. This is done mathematically,since the model has been shown to be viable, and since theactual test block comes in only one size. The requiredmodifications to the solution routine are minimal. Apartfrom explicitly changing the block weight/length, the massmoment tensor I must be scaled (I enters the formulation

through coefficients Mij). Since I has dimensions ML2, alength change cL0 requires that its components be multi-plied by c2; a weight change dM0 requires that thecomponents be multiplied by d.Fig. 14 shows the consequences of shear force increase in

half size and double size samples. The samples reactidentically because their weight is the same (the largersample is allowed more time to slide and covers moreground). The distortive impact of the end effects appears tolessen with increasing shear load, as end effect-free/included traces converge at CP2 and CP3. The convergenceof CP1 traces improves between 26.7 and 40N shear load,then changes little between 40 and 53.4N, where it appearsthat the gray trace continues to linearize and the blacktrace is already linear. This behavior is consistent with theexplanation given in Section 5.2.Fig. 15 portrays the sliding motion of half weight and

double weight samples. The first and second rows areidentical because the blocks in a given column slide withthe same acceleration (weight and shear load have beenincreased proportionally). The geometric similarity be-tween Case D/E sliding paths is not a coincidence, butrather a consequence of the binary manner in which wehave scaled the original specimen, and of the distance(rather than time)-dependent slide parametrization. As inCase D, there is a lessening of end effect-free/included tracedisparity with increased shear level.

5.5. Case F (Fig. 16)

The benefits of running the experiment at higher shearloads were first suggested in Fig. 12, where the decreasingseverity of the end effect was indicated by an improvedsimilarity between experimental and calculated paths. Inthe preceding section, this trend was seen to hold forproportional samples of different size or weight, not beingrestricted apparently to the particular laboratory specimenused. Because the pivotal drop in lateral dilation thataccompanies high shearing velocities is responsible for theend effect decrease, we term the phenomenon ‘‘velocityshielding’’.Cases D and E considered two methods of scaling a

laboratory sample, neither of which are appropriate for thepurposes of physical modeling. The weight of an actualdam monolith cannot be preserved in a smaller sizedsample, nor can a lighter model with the same dimensionsbe cast. More importantly, a combination of load andsample parameters corresponds to one specific set of fieldscale conditions; unilaterally increasing the experimentalshear load while holding sample parameters constantchanges the physical situation being modeled.In situ shearing conditions can be replicated by casting a

laboratory specimen with the same density as the fieldmonolith, then applying a shear force whose magnitudepreserves the ratio of monolith weight to hydrostaticdisplacement force. This type of scaling makes it possibleto retain the field monolith’s frictional properties, since a


model can be cast from the same material. In our case, weare simulating the test numerically and only need to adjustthe components of mass moment tensor I. The dimensionsML2 of the components can be written (M/L3) L5 or rL5,so for a length change cL0, density is preserved bymultiplying them by c5.

Fig. 16 shows modeled test results under uniformscaling. 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 shearload to weight ratio. Sliding paths in the top row do notinclude a modeled end effect and simulate test results froman ideal machine; bottom row sliding paths do include anend effect and depict the shearing of an aluminum topblock pushed by a steel load tip.

The sliding paths in a given row are geometricallysimilar, which confirms that we have chosen an appropriatemeans of scaling, but at once reveals the total absence ofvelocity shielding. That the column entries do not begin toassimilate with increasing shear load indicates there is noadvantage to uniformly varying the scale of a laboratoryshear specimen. The end effects themselves appear to bescale independent, in the sense that proportionally scaledsamples exhibit geometrically similar path aberrations.This follows from our acceptance of the Mohr–Coulombcriterion, which linearly relates the magnitude of endfriction force FEF to the magnitude of the driving shearforce. The consequences of choosing a velocity-dependentfriction model (see e.g. [11]) are beyond the scope ofthis paper.

6. Conclusion

An experimental setup can provide only indirectinformation about its own limitations, for example,through observed inconsistencies between tests performedon specimens with different properties. Analysis, on theother hand, allows one to isolate specific parameters andinvestigate their individual contributions to the measureddata. The objective of this paper has been to predict thelaboratory sliding path of a tri-planar monolith byaccounting for both planned and incidental contactinteractions. Removing the end friction term FEF fromthe sum of external forces, we were able to envision howthe monolith might slide when displaced by a low viscosityhydrostatic force or frictionless mechanical load tip. Suchmodeled sliding behavior was seen to be vastly differentfrom laboratory sliding behavior, which always involves asuperposition of ‘‘natural’’ shear characteristics and testingmachine interference.

The results of our comparison indicate that in thecontext of a generalized shear experiment, end effects arenon-trivial and cannot be ignored without compromisingmodel assumptions. Although the results we have pre-sented are specific to an aluminum sample pair with thestated foundation angles, the conclusion holds for anyshear experiment involving significant lateral dilation. End

friction results whenever there is relative movementbetween a sliding sample and displacement actuator.However, when a sample is free to dilate laterally, theabsence of side constraint allows the moments induced byFEF to rotate the block unopposed. The experimental datafrom a laterally unconstrained shear experiment is taintedwith a loss of distinction between this spurious rotationand topographically induced lateral dilation.A number of methods could be used to physically reduce

end effects in a direct shear-type experiment. Althoughlubricant coatings are usually thwarted by the size andconvexity of the actuator/sample interface, the load tipitself can be manufactured from a low-friction materialsuch as graphite or teflon. Another approach would be tochange the manner in which displacement force is applied.Replacing the mechanical actuator with a fluid jet wouldobviate entirely the threat of end friction, but also presentnew challenges such as the management of spent fluid.Manually tilting the sample pair is a simple means ofapplying frictionless displacement force [4]. However, itoffers very little control over model parameters and is onlypractical for small samples.Until an inexpensive frictionless method of applying

displacement force becomes widely available, end frictionwill continue to be an insidious factor in laboratory shearexperiments. End effects are inevitable, but it is possible tounderstand and anticipate them. The engineer should bewary of equating laboratory results with field behavior, andunderstand that certain kinds of shear tests are conspicu-ously susceptible to end effects. In situations wheresignificant lateral dilation is expected or where the pathof the block is important, it may be wise to dispensealtogether with physical testing and instead focus one’sresources on numerical simulation. Commercially availableDDA codes [12,13] can be useful for of modeling a varietyof sliding problems.

Acknowledgments

This research was funded by the National ScienceFoundation under Grant CMS-0408389. The authorswould like to Pete Thuesen for his help in preparing thealuminum sample pair.

References

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[2] Szczepanik Z, Milne D, Hawkes C. The confining effect of endroughness on unconfined compressive strength. In: Eberhardt E,Stead D, Morrison T, editors. Proceedings of the first annals ofCanada—US rock mechanics symposium. London: Taylor andFrancis; 2007. p. 793–7.

[3] Franklin JA. International society for rock mechanics commissionon standardization of laboratory and field tests—suggested methodsfor determining shear strength. In: Brown ET, editor. Rockcharacterization testing and monitoring. Oxford: Pergamon; 1981.p. 129–40.

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[6] Scholz CH. The mechanics of earthquakes and faulting. Cambridge:Cambridge University Press; 1990.

[7] Ardema MD. Analytical dynamics. New York: Kluwer; 2005.[8] Ginsberg JH. Advanced engineering dynamics. 2nd ed. Cambridge:

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[9] Galic D, Glaser SD, Goodman RE. Calculating the shear strength ofa sliding asymmetric block under varying degrees of lateralconstraint. Int J Rock Mech Min Sci, submitted for publication.

[10] Faires JD, Burden RL. Numerical methods. Boston: PWS-Kent;1993.

[11] Mohammadi S. Discontinuum mechanics. Southampton: WIT Press;2003.

[12] Shi GH. Block system modeling by discontinuous deformationanalysis. Southampton: Comput Mech Pub; 1993.

[13] Shi GH. Cutting joint blocks and finding removable blocks forgeneral free surfaces using 3-D DDA. In: Yale DP, Holz SC, BreedsC, Ozbay U, editors. Proceedings of the 41st US rock mechanicssymposium. Golden: ARMA; 2006 on CD-ROM.


a lagrangian dynamic analysis of end effects in a ...glaser.berkeley.edu/glaserdrupal/pdf/ijrm...

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