drained and undrained behavior of fiber-reinforced … · drained and undrained behavior of...

Drained and Undrained Behavior of Fiber-Reinforced Sand

by Cheng-Wei Chen

Cheng-Wei Chen

University of Missouri – Columbia

Graduate Research Assistant

Department of Civil and Environmental Engineering

E2509 Lafferre Hall

Columbia, MO 65211

Telephone: (573) 882-3862

Fax: (573) 884-4784

E-mail: [email protected]

Submitted October 12, 2006

Word Count: 2290 words plus 2750 word equivalents for tables and figures (5040 total words)

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Drained and Undrained Behavior of Fiber-Reinforced Sand

by Cheng-Wei Chen

ABSTRACT

A series of CU and CD type triaxial compression tests were performed on comparable

unreinforced and fiber-reinforced specimens of Ottawa Sands to evaluate the effective stress-

strain-pore pressure-volume change behavior of fiber-reinforced soils. The results show that

fibers increase the cohesion (c’) and effective friction angle ( ’) for Ottawa Sands. The c’ and ’

determined for the reinforced specimens increases with strains. Reinforced loose specimens tend

to have a higher friction angle, but lower cohesion intercept than the reinforced medium-dense

specimens. In addition, the inclusion of fibers with loose specimens has a significant effect in

increasing effective friction angle than with medium-dense specimens. In CU tests, the loose

reinforced specimens exhibited lower pore pressures than unreinforced specimens. In CD tests,

the loose and medium-dense reinforced specimens showed more dilation than unreinforced

specimens at moderate and larger strains. These are in agreement with the response observed in

undrained tests. For Ottawa sands, the fiber resistance is mobilized at small strains in both

undrained and drained conditions. However, the mobilized shear resistances in drained tests

occur sooner than in undrained tests.

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INTRODUCTION

The behavior of fiber-reinforced soils has been studied by several investigators over the

last two decades. Fiber-reinforced soil is becoming a viable soil improvement method for

geotechnical engineering problems. Fiber-reinforced soils are currently being used or considered

include stabilization of shallow slope failures (1), construction of new embankments with

marginal soils, reduction of shrinkage cracking in compacted clay liners (2), and mechanical

stabilization of roadway subgrades (3).

PREVIOUS WORK

Fiber-reinforced soil is a mixture of soil and synthetic fibers. Synthetic fibers can be

made of different materials, shapes and lengths. Polypropylene and polyester are the most

common materials used to manufacture fibers. Fibers can be flat or round, and continuous or

discrete. Discrete fibers are manufactured in several lengths, ranging from 0.5 in to 3 in, and in

different types such as monofilament, fibrillated, tape, and mesh.

Significant fundamental research has been performed over the last few decades to

evaluate basic shear strength properties and deformation characteristics of fiber-reinforced soils.

Previous work has clearly shown that an increase in fiber content increased the shear strength of

the soils. Most investigators found that shear strength increased in direct proportion to fiber

content or area ratio (3, 5, 6, 7, 8). However, (9) observed that increase in strength was not

proportional to the reinforcement concentration.

Some of the previous research has shown that inclusion of fibers increased both the

cohesion intercept and angle of internal friction values as compare to values for unreinforced soil

(10, 11, 1). However, (4, 5, 7, 8) found that inclusion of fibers did not significantly affect the

angle of internal friction of the unreinforced soils, but rather that fiber-reinforced specimens

exhibited bi-linear failure envelopes as a result of the existence of a critical confining stress

below which the fibers tended to slip or pull-out. (12) observed an increase in the angle of

internal friction but a decrease in the cohesion intercept. (6) found that an increase in fiber

content only increased the cohesion intercept whereas the angle of internal friction remained

unchanged from that of the unreinforced soil.

Inclusion of fibers was generally found to increase the peak and post-peak strength, as

well as the strain at failure. Furthermore, inclusion of fibers has been found to not noticeably

affect the initial stiffness of the unreinforced specimens. However, some investigators have

reported an increase in the initial stiffness of specimens with increasing fiber content (11),

whereas others have shown a decrease in initial stiffness with increasing fiber content (12,13). It

is shown by many researchers that inclusion of fibers increases the shear strength under different

loading conditions.

Most of pervious work in this area has concentrated on the behavior of fiber-reinforced

granular (i.e. sand) and undrained behavior of clays under total stress conditions. The laboratory

work was mainly utilizing the simply direct shear tests. The data from most investigators lacks

evaluation of the load transfer mechanics in terms of effective stress. Additional tests of fiber-

reinforced sand are needed to confirm the reinforcement response with different loading

conditions in terms of effective stress measurements.

TESTING MATERIALS AND PROGRAM

The soil used in triaxial testing program was Ottawa sand (Grade F-75), which is well

known laboratory-tested sand. The particles have a mean diameter, D50 of 0.18 mm, a

uniformity coefficient, Uc of 1.7, a minimum void ratio, emin of 0.46, a maximum void ratio, emax

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of 0.77, and a specific gravity of 2.65, respectively. The soil classifies as poorly graded sands

(SP) according to the Unified Soil Classification System.

The fibers utilized in the specimens are commercially available 2-inch (50-mm) long

fibrillated polypropylene fibers of 3600 denier. The specific gravity of the fibers is 0.91 gr/cm3

(14). The ultimate tensile strength (15) and modulus of elasticity (16) of the fiber are 45 ksi, and

700 ksi, respectively.

An undercompaction process (17) was selected to produce homogeneous samples using

Ottawa sand for a parametric study in a laboratory-testing program. Unreinforced and fiber-

reinforced Ottawa sand specimens were prepared and mixed to the nominal 10 percent water

content as loose state and 3 percent water content as medium-dense state, which nominal

relatively density were equal to 10 percent (e0 = 0.74) and 55 percent (e0 = 0.60), respectively.

The soil was also allowed to hydrate overnight prior to compaction. Decided how many lifts of

compacting and calculated the desired thickness of each layers according to the undercompaction

process suggested.

All specimens were backpressure saturated at effective consolidation stresses of 2.5 psi

using the dry mounting method as specified in ASTM D4767 (18). Skempton’s pore pressure

coefficient B (18) was measured during saturation. All specimens were allowed to saturate until

measured B-values were reached at least 0.96 before consolidation and shear. Approximately 5

days were required to bring the B-value of the Ottawa sand specimens to 0.96.

The strain rate used to shear all conventional triaxial compression and extension

specimens was 10 percent per hour (deformation rate of 0.49-inch per hour) to eliminate concern

over strain rate when compared to drained and undrained test results. Most specimens were

sheared up to a maximum axial strain of 30 percent to permit evaluation of the post-peak stress-

strain behavior.

A summary of testing program undertaken to evaluate the stress-strain behavior of

unreinforced and fiber-reinforced specimens in term of effective stresses is shown in Table 1. A

total of sixteen consolidated-undrained triaxial compression tests with pore pressure

measurements (CU tests) were performed to evaluate the stress-strain-pore pressure generation

behavior of fiber-reinforced specimens under undrained loading conditions. A total of sixteen

consolidated-drained triaxial compression tests (CD tests) were also performed for specimens

compacted at loose and medium-dense state to evaluate the stress-strain-volume change behavior

of fiber-reinforced specimens under drained loading conditions. All tests were performed on

2.5-inch diameter by nominal 4.9-inch tall specimens. Specimens isotropically consolidated to

the target effective stress of 5, 20, 40, and 60-psi.

STRESS-STRAIN-PORE PRESSURE-VOLUME CHANGE RESPONSE

Typical deviatoric stress versus triaxial shear strain behavior from CU and CD tests for

unreinforced and reinforced specimens is shown in Figure 1 and 2. Loose fiber-reinforced

specimens show a strain-hardening type of behavior whereas medium-dense reinforced

specimens exhibit a noticeable peak stress at large strains of 20 percent. Figure 1 shows that

stress-strain behavior of medium-dense reinforced specimen begins to deviate at 5 percent stain

under undrained condition. However, Figure 2 shows that the considerable strength is gained by

the inclusion of fibers at 1 percent strain of medium-dense reinforced specimen under drained

condition.

Change in pore pressure versus triaxial shear strain observed in tests for both

unreinforced and reinforced Ottawa sand specimens are shown in Figure 3. Loose unreinforced

specimen exhibits the lower initial increase in pore pressure than reinforced specimen at 1

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percent strain, slightly decrease in a range of 1 percent to 2 percent and increase again slowly

with addition strains. In contrast, the loose fiber-reinforced specimens show the initial increases

in pore pressure at very small strain, decrease with additional strains, and tend to level at large

strains. Both medium-dense unreinforced and reinforced specimens show the initial increases in

pore pressure followed by significant decreases and absolute value of pore pressures equal or less

than zero before 10 percent strains. It is noticed that medium-dense reinforced specimens tend to

have higher initial increases in pore pressure than unreinforced specimens and keep relative

higher pore pressures when compared to the unreinforced specimens at given strains before

absolute pore pressures equal or less than zero. When the readings of pore pressure transducer

measured less than the atmosphere pressure, cavitation is taken place. Pore pressure transducers

cannot measure negative pressure accurately. Therefore, the measure values can not represent

the real readings in the soil when the absolute value of pore pressures is equal or less than zero.

The typical volumetric strain versus triaxial shear strain response from CD tests for the

unreinforced and reinforced Ottawa sand specimens are shown in Figure 4 for the samples

compacted to loose and medium-dense state. The unreinforced specimens exhibit volumetric

strain response typical of loose and medium-dense behavior, with volume compressing at small

strains and then keeping constant at large strains for loose specimen whereas with initial volume

decreasing followed by significant dilation up to large strains. The loose fiber-reinforced

specimens show that less volume compressing than the unreinforced specimen consolidated to

20-psi or higher effective confining stresses. Conversely, medium-dense specimens exhibit less

dilation at low shear strains at the range of 2 percent to 5 percent, but more dilation at moderate

and large strains than observed on unreinforced specimens. Therefore, it indicates that the

medium-dense sand exhibit the lower pore pressure to maintain the zero volume change in

undrained loading condition at moderate to large strains, which is similar to loose sand behavior.

The volumetric strain versus triaxial shear strain behavior inspect on the fiber-reinforced

specimens under drained tests is in agreement with the pore pressure versus triaxial shear strain

response tested at undrained condition.

The causes for the unusual pore pressure responses observed in the fiber-reinforced

specimens are not clear. (20) assumed that the fibers create an “internal confining stress” (due to

tension developed in the fibers) that, when added to the applied total stresses and actual pore

pressures generated in the fiber-reinforced soil, the fibers produces additional effective stress to

prevent volume change in undrained tests. However, the results presented above show that fiber-

reinforced Ottawa sand require lower pore pressure to maintain the zero volume change in

undrained loading conditions for both loose and medium-dense state, which are different

observation with (20) for fiber-reinforced silty clay specimens. The inclusion of fibers generates

a “negative internal confining stress” and produces negative effective stress to maintain zero

volume change in undrained tests.

FAILURE ENVELOPES FOR TRIAXIAL COMPRESSION TEST

Failure envelopes were determined from the peak deviator stress (PDS) and peak

effective stress ratio (PSR) failure criteria for CU tests. Values of the Mohr-Coulomb strength

parameters, effective cohesion intercept, c', and effective internal friction angle, ', for the

unreinforced Ottawa sand specimens from CU tests and CD tests are listed in Tables 2 and 3,

respectively. The results indicate that the strength of unreinforced specimens under undrained

loading can be represented by no cohesion intercept, and an effective friction angle of 29° and

34°, for loose state and medium-dense specimens, respectively. The shear strength parameters

determined from PRS failure criteria show a slightly greater than the results from the PDS failure

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criteria. Furthermore, the shear strength parameters determined from CD tests show a greater

value than the results from the CU tests, which are presented in Table 3.

Table 2 and 3 also present the strength parameters for the fiber-reinforced Ottawa sand

specimens from CU tests and CD tests. It can be seen that inclusion of fibers has a pronounce

effect both on the effective cohesion intercept and on the measured effective friction angle.

Reinforced specimens compacted at loose state tend to have a higher friction angle, but lower

cohesion intercept than specimens compacted at medium-dense state. In addition, the inclusion

of fibers with loose specimens has a significant effect in increasing effective friction angle than

with medium-dense specimens, which is shown in Table 3.

Peak deviator stress and peak effective stress ratio for the fiber-reinforced specimens

under undrained and drained loading occurred at very large strain. Since such large strain are

seldom tolerable, the data were also analyzed for limiting strains of 5, 15, and 25 percent strain.

For these analyses, the PDS and PSR failure criteria were taken to be the maximum values

measured at strains close or equal to the limiting strains. Figure 5 and 6 show the Cambridge

stress path diagrams and failure envelopes for the chosen limiting strains from CU tests and CD

tests on unreinforced and reinforced loose specimens. The failure envelopes of unreinforced

specimens do not show a significant difference at established limiting strains for both loose and

medium-dense state. In contrast, the shear strength parameters calculated from reinforced

specimens increase with chosen limiting strains in terms of effective cohesion intercept and

effective friction angle.

The results of Mohr-Coulomb strength parameters for the unreinforced and reinforced

specimens at the limiting strains from CU and CD tests are summarized in Table 4 and 5,

respectively. In general, the strength parameters measured from CU tests are greater than those

from CD tests for loose reinforced specimens, whereas the effective friction measured from CU tests are less than those from CD tests. Furthermore, reinforced specimens compacted at loose

state show a significant increase in the effective friction angle from CU and CD tests. The

opposite was observed for specimens compacted at medium-dense state, a significant increase in

the effective cohesion intercept.

CONCLUSIONS

The results of the triaxial compression tests performed on loose and medium-dense

Ottawa sand show that inclusion of fibers can improve the strength of soils under undrained and

drained loading conditions. Shear strength parameters of effective cohesion intercept and

effective friction angle increase significantly in the CU and CD tests, due to the addition of

fibers. The angle of internal friction under drained loading was slightly greater than for those

under undrained loading for both unreinforced and reinforced specimens. It is noted the

reinforcing fibers alter the pore pressure response of specimens tested under undrained loading

conditions and the volume change response of specimens tested under drained loading condition.

However, the response in Ottawa sand is totally different from the response in silty clay (20).

It was shown that fiber reinforced specimens must deform before developing and increase

in shear strength due to the inclusion of fibers. Under undrained conditions, deformations can be

high for most structures. However, drained specimens mobilized the shear fiber resistance at

very low strain, which can be tolerable for must structures. As a result, more strains are needed

to mobilize the fiber shear strain for specimens consolidated at high effective stresses.

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REFERENCES

1. Gregory, G.H., and D.S. Chil. Stabilization of Earth Slopes with Fiber Reinforcement.

Proceedings of the Sixth International Conference on Geosynthetics, March 25-29, Atlanta,

Georgia, 1998, pp. 1073-1078.

2. Rifai, S.M. Impact of Polypropylene Fibers on Dessication Cracking and Hydraulic

Conductivity of Compacted Clay Liners. Dissertation submitted in partial fulfillment for the

requirements of the Doctoral Degree, Wayne State University, Detroit, Michigan, 2000.

3. Santoni, R.L., J.S. Tingle, and S.L. Webster. Engineering Properties of Sand-Fiber Mixtures

for Road Construction, Journal of Geotechnical and Environmental Engineering, ASCE,

Vol. 127, No. 3, 2001, pp. 258-268.

4. Gray, D.H., and H. Ohashi. Mechanics of Fiber Reinforcement in Sand. Journal of the

Geotechnical Engineering Division, ASCE, Vol. 109, No. 3, 1983, pp. 335-353.

5. Ranjan G., R.M. Vasan, and H.D. Charan. Probability Analysis of Randomly Distributed

Fiber-reinforced Soil. Journal of the Geotechnical Engineering Division, ASCE, Vol. 122,

No. 6, 1996, pp. 419-426.

6. Bauer, G., and A. Oancea. Soils Reinforced with Discrete Synthetic Fibers. Geosynthetics’99

– Specifying Geosynthetics and Developing Design Detail, IFAI, Boston, Massachusetts,

1999, pp. 465-475.

7. Maher, M.H., and D.H. Gray. Static Response of Sands Reinforced with Randomly

Distributed Fibers. Journal of the Geotechnical Engineering Division, ASCE, Vol. 116, No.

11, 1990, pp. 1661-1677.

8. Gray, D.H., and T. Al-Refeai. Behavior of Fabric- versus Fiber-reinforced Sand. Journal of

the Geotechnical Engineering Division, ASCE, Vol. 112, No. 8, 1986, pp. 804-820.

9. Shewbridge, S.E., and N. Sitar. Deformation Characteristics of Reinforced Soil in Direct

Shear. Journal of the Geotechnical Engineering Division, ASCE, Vol. 115, No. 8, 1989, pp.

1134-1147.

10. Kumar, R., V.K. Kanaujia, and D. Chandra. Engineering Behaviour of Fibre-Reinforced

Pond Ash and Silty Sand,” Geosynthetics International, Vol. 6, No. 6, 1999, pp. 509-518.

11. Nataraj, M.S., and K.L. McManis. Strength and Deformation Properties of Soils Reinforced

with Fibrillated Fibers," Geosynthetics International, Vol. 4, No. 1, 1997, pp. 65-79.

12. Consoli, N.C., P.D.M. Prietto, and L.A. Ulbrich. Influence of Fiber and Cement Addition on

Behavior of Sandy Soil. Journal of the Geotechnical Engineering Division, ASCE, Vol. 124,

No.12, 1998, pp. 1211-1214.

13. Michalowski R.L., and J. Cermak. Triaxial Compression of Sand Reinforced With fibers,

Journal of the Geotechnical Engineering Division, ASCE, Vol. 192, No. 2, 2003, pp. 125-

136.

14. ASTM. D792. “Standard test methods for density and specific gravity (relative density) of

plastics by displacement.” Annual Book of ASTM Standards, Vol. 08.01, Philadelphia.

15. ASTM. D2256. “Standard test method for tensile properties of yarns by the single-strand

method.” Annual Book of ASTM Standards, Vol. 07.01, Philadelphia.

16. ASTM. D2101. “Standard test methods for tensile properties of single man-made textile

fibers taken from yarns and tows.” Annual Book of ASTM Standards, Vol. 07.01,

Philadelphia.

17. Ladd, R. S. Preparing Test Specimens Using Undercompaction Geotechnical Testing

Journal, Vol. 1, No. 1,1978, pp. 16-23.

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18. ASTM D4767, “Standard test method for consolidated-undrained triaxial compression test on

cohesive soils,” Annual Book of ASTM Standards, Vol. 04.08, Philadelphia.

19. Skempton, A.W. The Pore Pressure Coefficient A and B,” Geotechnique, Vol. 4, 1954, pp.

143-147.

20. Romero, R.J. Development of a Constitutive Model for Fiber-Reinforced Soils. Dissertation

submitted in partial fulfillment for the requirements of the Doctoral Degree, University of

Missouri-Columbia, 2003.

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List of Tables

TABLE 1 Summary of Triaxial Tests Performed to Evaluate the Stress-Strain Behavior of

Unreinforced and Reinforced Ottawa Sand Specimens

TABLE 2 Mohr-Coulomb Strength Parameters, c' and ', Measured for Unreinforced and

Reinforced Ottawa Sand Specimens from CU Tests


Reinforced Ottawa Sand Specimens from CD Tests

TABLE 4 Mohr-Coulomb Strength Parameters, c' and ', from CU Tests on Unreinforced

and Reinforced Ottawa Sand Specimens When Strength Is Taken as Peak Stress

Experienced at Limiting Strains of 5, 15, 25 Percent

TABLE 5 Mohr-Coulomb Strength Parameters, c' and ', from CD Tests on Unreinforced



List of Figures

FIGURE 1 Deviatoric stress (q) versus triaxial shear strain ( q) curves from CU tests for

Ottawa sand specimens consolidated to 20-psi effective stress and compacted at: a) loose

state (e0 = 0.74), and b) medium-dense state (e0 = 0.60).

FIGURE 2 Deviatoric stress ( q) versus triaxial shear strain ( q) curves from CD tests for



FIGURE 3 Change in pore pressure ( u ) versus triaxial shear strain ( q) curves from CU

tests for Ottawa sand specimens consolidated to 20-psi effective stress and compacted at: a)

loose state (e0 = 0.74), and b) medium-dense state (e0 = 0.60).




FIGURE 5 Cambridge stress paths and failure envelopes for limiting strains of 5, 15, and

25 percent strain from CU tests on Ottawa sand specimens compacted at loose state

(e0 = 0.74): a) 0.0 percent fiber content, and b) 0.4 percent fiber content.


25 percent strain from CD tests on Ottawa sand specimens compacted at loose state

(e0 = 0.74): a) 0.0 percent fiber content, and b) 0.4 percent fiber content.

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TABLE 1 Summary of Triaxial Tests Performed to Evaluate the Stress-Strain Behavior of

Unreinforced and Reinforced Ottawa Sand Specimens

Effective Confining Stress

Loose Specimens (e0=0.74) Dense Specimens (e0=0.60)

Type of

Triaxial

Testing

Fiber

Content

(%) 5 psi 20 psi 40 psi 60 psi 5 psi 20 psi 40 psi 60 psi

0.0 1 1 1 1 1 1 1 1 CU

0.4 1 1 1 1 1 1 1 1

0.0 1 1 1 1 1 1 1 1 CD

0.4 1 1 1 1 1 1 1 1

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Reinforced Ottawa Sand Specimens from CU Tests

0.0% Fiber Content 0.4% Fiber Content

Peak PSD Peak PSR Peak PSD/PSR

Initial

Void

Ratio c' (psi) ' (deg) c' (psi) ' (deg) c' (psi) ' (deg)

0.74 0.0 28.8 0.0 30.4 12.4 43.1

0.60 0.0 33.5 0.0 33.7 - -a - -

a

a Data not available.

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Reinforced Ottawa Sand Specimens from CD Tests

0.0% Fiber Content 0.4% Fiber Content

Peak PSD Peak PSD

Initial

Void

Ratio c' (psi) ' (deg) c' (psi) ' (deg)

0.74 0.0 29.8 4.2 44.2

0.60 0.0 34.2 10.9 41.8

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TABLE 4 Mohr-Coulomb Strength Parameters, c' and ', from CU Tests on Unreinforced



0.0% Fiber content 0.4% Fiber content

5% Strain 15% Strain 25% Strain 5% Strain 15% Strain 25% Strain Initial

Void

Ratio c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

0.74 0.0 29.9 0.0 30.2 0.0 29.6 2.3 36.7 4.0 42.0 7.0 44.9

0.60 0.0 33.6 - -a

- -a

- -a

- -a

8.1 35.6 - -a

- -a

- -a

- -a

a Data not available.

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TABLE 5 Mohr-Coulomb Strength Parameters, c' and ', from CD Tests on Unreinforced



0.0% Fiber content 0.4% Fiber content

5% Strain 15% Strain 25% Strain 5% Strain 15% Strain 25% Strain Initial

Void

Ratio c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

c'

(psi)

'

(deg)

0.74 0.0 28.3 0.0 29.7 0.0 29.4 1.2 33.8 2.2 40.2 3.0 44.2

0.60 0.0 33.9 0.0 33.1 0.0 31.8 5.2 37.4 6.6 41.5 9.8 41.5

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FIGURE 1 Deviatoric stress (q) versus triaxial shear strain ( q) curves from CU tests for

Ottawa sand specimens consolidated to 20-psi effective stress and compacted at:

a) loose state (e0 = 0.74), and b) medium-dense state (e0 = 0.60).

*: start point of suspicious measurement.

a) Loose state (e0 = 0.74)

b) Medium-dense state (e0 = 0.60)

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FIGURE 2 Deviatoric stress (q) versus triaxial shear strain ( q) curves from CD tests for





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FIGURE 3 Change in pore pressure ( u ) versus triaxial shear strain ( q) curves from CU

tests for Ottawa sand specimens consolidated to 20-psi effective stress and

compacted at: a) loose state (e0 = 0.74), and b) medium-dense state (e0 = 0.60).

*: start point of suspicious measurement.



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25 percent strain from CU tests on Ottawa sand specimens compacted at loose

state (e0 = 0.74): a) 0.0 percent fiber content, and b) 0.4 percent fiber content.

a) 0.0 percent fiber content

b) 0.4 percent fiber content

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25 percent strain from CD tests on Ottawa sand specimens compacted at loose

state (e0 = 0.74): a) 0.0 percent fiber content, and b) 0.4 percent fiber content.

a) 0.0 percent fiber content

b) 0.4 percent fiber content

drained and undrained behavior of fiber-reinforced … · drained and undrained behavior of...

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