A mechanical device for the measurement of combined shear and tension in rocks

Int. J. Rock Mech. Min. Sci. Vol. 34, No. 1, pp. 147-151, 1997

Pergamon PII: S0148-9062(96)00046-0

© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/97 $17.00 + 0.00

Technical Note A Mechanical Device for the Measurement of Combined Shear and Tension in Rocks C. T. AIMONE-MARTINf K. I. ORAVECZt T. K. NYTRA:~

INTRODUCTION A mechanical rock testing device has been designed and used in laboratory experiments to measure deformations in rock salt under combined shear and tension loading. The device was designed for use with an MTS Systems servo-controlled hydraulic testing machine to test 304.8 mm (12.0 in.) long cylindrical rock cores 101.6 mm (4.0 in.) in diameter. The experimental program involved the correlation of measured rock deformations to the response of coaxial cables embedded axially within cores using time domain reflectometry (TDR) techniques. In previous work, rock deformations in pure shear and pure tension were correlated to embedded cable response in an effort to demonstrate the application of TDR to predict rock displacements [1]. In the work reported herein, combined loading tests were conducted using two different ratios of shear to tension deformation. The application of a testing device that combines both shear and extension may be useful in general rock mechanics testing. Induced tensile stresses combined with shear stresses do occur around underground openings. Further, the device can be readily adapted for combined compressive stress and shear stress experiments. No laboratory method is currently in use to study the response of rock to such states of stress. The device controls deformations in the two directions. The shear and tensile forces (stresses) develop spontaneously depending on the shear and tensile stiffness of the sample tested. This paper describes the design elements and operation of the apparatus. The mechanics of the device are analyzed to illustrate load calculations from measured displacements.

fDepartment of Mineral and Environmental Engineering, New Mexico Institute of Mining and Technology, Socorro, NM 87801, U.S.A. ++Golder Construction Services, East Lansing, MI 48823, U.S.A.

D E S C R I P T I O N O F C O M B I N E D S H E A R AND T E N S I O N DEVICE

A diagram of the device is shown in Fig. 1. The device is a symmetrical four-link mechanism that develops extensional displacements along the axis of cylindrical rock samples. The mechanism is centered within the loading platens of the testing machine by loading plates fixed to the platens. Two 381.0mm (15.0in.) long structural tubes are designed to carry and transfer the total load, P, to the rock. Three holes, spaced 95.3 mm (3.8 in.) apart, are drilled in the tubes to accommodate 12.7 mm (0.5 in.) diameter pin connections. The holes link tensile load carrying members (outer linkage members) from the pin locations at "a" and "d", shown in Fig. 2, to the pins located at "b" and "c", and are designed to vary the ratio of transverse shear to longitudinal extension displacement depending upon the hole locations used. The shear to extension ratios for the top and bottom pins are approximately 0.75 and 2.0, respectively. In preparation for testing, the rock sample is epoxied within two circular sample jackets. A 12.7 mm (0.5 in.) gap is left between the two steel jackets to accommodate a short length of rock in the proximity of the anticipated shear region. The jackets are then placed inside the shear sleeves. The sleeves are designed to slide past one another in a vertical plane, thereby allowing transverse shear to develop on the rock sample. Reinforcing bars are designed to prevent buckling of the device. Outer linkage members are pinned at "b" and "c" to tension bolts instrumented with electrical resistance strain gages that are calibrated for the measurement of tensile loads. Extensional and shear displacements are measured by means of four linear variable differential transformers (LVTDs). The extensional deformations exerted at pins "b" and "c" are transferred to the sample by means of chain links connected to end caps bolted to the sample jackets. The device is fabricated from structural steel and designed for a maximum compressive load of 222.0 kN (50,000 lb). The maximum extensional load is limited by the tensile strength of the chains links. The measurable range of tensile force is 44.5 kN (10,000 lb). 147

148

AIMONE-MARTIN et

al.:

TECHNICAL NOTE ~'-"

--'-

p'a'e I r n

-'

Structural ste~ r holes

Oufer linkage

sleeves Sample jacket Chains

End caps

sample Gap between j

g bars | plate

Bolts

3latens Fig. 1. Schematicof combined shear and tension testing device. DEVICE OPERATION Shear deformation, regulated by the testing machine, is the only controlling parameter throughout combined deformations experiments. The free body diagram in Fig. 2 illustrates the forces induced during loading of the mechanism. Rock sample testing for TDR experiments on rock salt consisted of a series of incremental deformations or machine stroke steps. During each step, the total compressive load developed on the device, P, was measured and recorded along with machine stroke. A portion of this load, transferred as a shear load, E~o (equal to Fdo) to the sample jacket, induced a shear force on the sample. Simultaneously, the outer linkage members induced a tensile load in the sample, forcing a horizontal separation of the sample jacket. Extensional strains generated in the rock sample eventually resulted in a tensile failure. Shear deformation in the vertical direction and extensional deformation in the horizontal direction were measured and recorded using four LVDTs mounted in opposing directions as shown in Fig. 3. The average values of both extensional and shear displacements were reported. The tensile loads that developed in the chain links (Tb and To) were also measured and recorded.

chain links, are assumed to remain a constant length between the four outer reference boundary points ("a"-"d"). The four inner members, lying at right angles to one another (oa, ob, oc and od) vary in length as the vertical members slide past each other and the horizonal members lengthen as the rock sample is pulled apart. In order to determine the vertical shear force, two sets of equations describing the equilibrium of forces applied to pins "a" and "b" (or, similarly, pins "d" and "c") P

Fh a

~°I

i:i:i:i:i:i:i:i:i:i:i:i:!:i:i:i:i:i:i:i:!:i:i:i:i:i:i:i:i:i:i Fd°

DEVICE MECHANICS During deformation of the combined shear and extension testing mechanism, the total load, P, and tensile loads, Tb and To, are measured. The shear load, E,o, is not known and must be derived using equations of static equilibrium and considering the relative positions of the load-carrying members during testing. The four outer linkages, transmitting tensile loads to the

Fh

P Fig. 2. Free body diagram showing forces within members and at pinned connections.

AIMONE-MARTIN et al.:

/~

Shear

TECHNICAL NOTE

149

80

LVDT1

TEST Average Average Shear Extension 1 • ©

LVDT1

E

so

4

tl:: I=

4.

÷

•

[]

SHEAR DEFORMATION = 0.970761 * MACHINE S T R O K E

.2

~ 4o "0 "0

E W W 0

Sheor

LVDT2 -/

=E

J ° l/

20

.... '

Fig. 3. Location of LVDTs mounted vertically (shear) and horizontally (extension). Extension LVDT2 is mounted behind the device. =~

o

must be solved simultaneously. The solutions to these equations are given in the Appendix. It should be noted that horizontal forces, Fh, result at the contact points between the platens and the device. The horizontal forces counteract the couple (moment) generated by the movement of the steel sleeves. Hence, rotational forces are balanced and equilibrium is maintained. RESULTS OF COMBINED SHEAR AND EXTENSION TESTING

Figures 4 and 5 show typical displacement responses of extension and shear deformation for tests conducted using the top and bottom pins, respectively. The data represent the average shear and extension displacement computed for two pairs of LVDTs during tests conducted on TDR coaxial cables embedded in rock salt. Hence, the upper limit to the data is controlled by the tensile and shear strength of the cables. The shear displacements closely track the machine stroke with a slight lag in measured rock shear deformation, as shear movement cannot occur until a tensile failure plane

TEST Average Average Shear Extension

EXTENSION

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DEFORMATION =

1

•

1.184 X ~ " + 1 . 1 t 2 X2 + 1.0 8 - -4- X

2

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A

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4

O

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20

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40

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stroke,

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Fig. 5. Measured deformation vs machine stroke for tests conducted using the bottom pin connection.

develops (between 2% and 4% strain for rock salt). Shown in Fig. 6, a sudden "jump" in shear deformation occurred upon rock tensile failure, resulting in abnormally high initial ratios of shear to extension deformation, particularly for the bottom pin connection (i.e. at high ratios of shear displacement to extensional displacement). Shear and extension deformations fall within the same data range up to a machine stroke of about 8.0 mm (0.3 in.). For rock salt, this is approximately the size of the mineral grains. Up to this limit, the asperities along the shear failure surface constrained shear movement. This produced higher than design extensional displacements for the bottom pin test data, resulting in a low shear to extension deformation ratio shown in Figs 5 and 6. Beyond this value, the top pin test data suggest that extension progressed rapidly as asperities were quickly overridden, while in the case of the bottom pin connection (Fig. 5), shear development was further constrained until the asperities were, in part, sheared through. Figure 6 shows that the design shear to extension ratio of 2.0 of the bottom pin connection was not fully attained until a machine stroke of 20.0 mm (0.8 in.) was reached. The behavior of the testing device f"

8

=.O

.2

DEFORMATION= 0.088 ) ( ' - 0.077 X ' * 5.63 X + 8.6E

mr_

[]

PiN CONNECTION

.o []

i

l

~ 20

[]

4-

9

Ze-

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W ~--

o

Bottom

"I" 0

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S DEFORMATION HEAR =, 0.992323 * M A C H I N E

I 20

'

I

'

40

I 60

C?nU~Urn []

_o '

' 80

• achine stroke, mm

Fig. 4. Measured deformation vs machine stroke for tests conducted using the top pin connection.

• •

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0

I 10

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I 20

'

I 30

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I 40

50

M a c h i n e stroke, mm

Fig. 6. Ratio of shear to extension deformation plotted against machine stroke for both the top and bottom pin connections.

A I M O N E - M A R T I N et al.: TECHNICAL NOTE

150

40

shear component, the high-angled asperities that developed along the failure surface contributed to high shear loads, inhibiting tensile load development. Sample dilation resulted from the movement of the two rock surfaces advancing past each other, sliding around asperities contained on the surfaces, and is apparent in the successive, undulating peaks in the loads. The compressive load indicated that a horizontal compressive component of force, greater than the tensile loads developing on pins "b" and "c", was generated along the sample axis. Tensile loads developed after 20.0 mm (0.8 in.) of machine stroke once asperities were sheared through. This observation is consistent with that found in Fig. 6 for the bottom pin.

Top Pin Connection

30

~.~ 2O o,

10

/ v

ShearLoad

I

I

10

I

20

40

30

Machine stroke, mm

Fig. 7. Load plots for a typical top pin connection test.

beyond this point transmitted a smooth, nearly constant ratio of deformations as the loads were fully transferred from the rock to the cable instrumentation embedded within the axis of the rock. The effects of rock salt asperities on the performance of the combined testing device are demonstrated in the load plots of Figs 7 and 8. The loads transmitted to the sample for the device pinned at the top connection (Fig. 7) indicate that the shear load tracked closely with total load once rock failure occurred (as indicated by the strong total load peak). The average extension load at pins "b" and "c" quickly developed, dominating over the shear load. Load transfer was smooth through the test. In contrast, the response shown in Fig. 8 for the bottom pin connection indicated that, up to 20.0 mm (0.8 in.) machine stroke, nearly all of the total load was transferred to the sample as shear load; very little extension load developed and, in some cases, compressive loads were recorded. In this case of a dominant

CONCLUSIONS A combined shear and tension rock testing device was fabricated and used to load cylindrical samples of rock salt in which TDR cables were embedded. The device, designed to vary the shear to tension deformation ratio between 0.75 and 2.0, was instrumented to measure and record both shear and tension deformation, as well as total and tensile loads. The shear load is not known and must be computed using equations of static equilibrium and the device geometry. The use of the combined testing mechanism has been demonstrated and the relationships between shear and tension deformations and loads explained during testing with rock salt using two different pin connections. The testing device may be applicable in general rock mechanics testing to measure small displacements when a combined stress state is used. Acknowledgements--The authors wish to thank Mr Irl Downes for his time and dedication to this project. Mr Floyd Hewitt was responsible for the details of the design and machining of the testing device.

Accepted for publication 12 July 1996.

REFERENCES 1. Aimone-Martin C. T., Oravecz K. I. and Nytra T. K. TDR calibration for quantifying rock mass deformation at the WIPP site, Carlsbad, NM. Proc. Time Domain Reflectometry in Environmental, Infrastructure, and Mining Applications. pp. 507517. T D R Specialty Conference, Northwestern University (1994).

40 Bottom Pin Connection

Total Load/ /

//

30

APPENDIX

Z 20

/A

~ V

~ /

/ Sh:ar / / Load / j/

-I 10 ExtensionLoads~

~ I

10

i /

~

--

XTc

I

20

I

30

40

Machine stroke, mm

Fig. 8. Load plots for a typical bottom pin connection test.

Relative positions of the structural members of the combined shear and extensional testing device are defined by distances between set boundaries at the pin connections and the contact shear plane of the device sleeve jackets containing the sample. With respect to the four outer linkage members, the distances between the reference boundaries remain constant throughout the experiment. The load intensity on each device member varies for each stroke step and is a function of the changing angles ~ and 0. The angles ~ and 0 are a function of the shear displacement corresponding to the distance between set boundaries of member ao. At the beginning of an experiment using the top pin connection, the distances between reference boundaries of members ao and bo are at their maximum and shortest length, 564 mm (22.2 in.) and 391 mm (15.4 in.), respectively. The angle ~ between members ao and bo is initially 391 tan-Jet = ~ = 34.7 °.

(A1)

A I M O N E - M A R T I N et al.: The lengths of the outer linkage members remain constant and are 686 m m (27.02 in.). Angle 0 is zero degrees at the beginning of an experiment, as the sample within the sleeves is initially parallel to the x-axis. During shear deformation the distances between reference boundaries of the vertical members ao and do decrease, while the distances between reference boundaries of the horizontal members bo and co increase. This occurs at some ratio dependent upon the location of the pin connection positioned along members ao and bo. The m a x i m u m vertical (shear) deformation of the device is 69.9 m m (2.75 in.). At this position, the relative distances of members ao and bo are calculated to be 529 m m (20.8in.) and 4 3 7 m m (17.2in.), respectively. Hence, c~is 39.5 °, corresponding to a 0 angle of 25.4 °. The m a x i m u m ratio of shear to tension displacements for the top pin connection is 0.76. For the bottom pin connection, a varies from 57 ° to 60.5 ° as 0 varies from 0 ° to 7.6 °. The ratio of shear to tension displacements for the bottom pin connection is 1.66. The device geometry, c~ and 0 angle relationships developed above, combined with measured total and tensile load data are used to calculate forces across members. Consider pin " a " shown in Fig. 2. Three u n k n o w n forces, F,b, Fac and F~o, act at the connection in response to the total applied load, P. Summing the forces in the vertical and horizontal directions, respectively, - P + E~o + F.b cos c~+ F.c cos c~ = 0

(A2)

TECHNICAL NOTE

151

F~b sin ~ -- E,~ sin ~ - Fh = O.

(A3)

The above equations can be solved by considering the forces acting at connections " b " and "c". However, analyzing the free body diagrams of pins " b " and "c", the equilibrium equations readily yield u n k n o w n forces F~b, Fdb, F,~ and Fa~ from measured tensile forces Th and T~. The equilibrium equations incorporate variable angles a and 0 which are readily calculated from the shear displacement and initial device geometry. Consider point "b", shown in Fig. 2. Using the law of sines, an expression can be written to find F~b in terms of Tb, ~ and 0 F~b -- Tb sin(90 ° -- ~ -- 0) sin(2ct)

(A4)

where (90 ° - ct - 0) is the angle formed between F~ and Tb. Similarly for F~c substituting equations (4) and (5) into equation (2) yields Fdc --

Tc sin(90 ° - ct + 0) sin(2ct)

F~o = P -- Tb sin(90 -- c~ + 0) cos ~ -- Tc sin(90 -- ~ + 0) cos ~ sin(2cQ sin(2~) "

(A5)

(A6)

This equation is easily computed in parts within a spreadsheet for known values of P, machine stroke, Tb and T¢. Depending on the total vertical (shear) deformation, the angle c~ is known and 0 is easily computed, and the u n k n o w n shear force, E,o is determined.