STRESS GRADIENT ON TENSILE

STRENGTH FROM THREE-POINT BENDING

TEST OF SANDSTONE

MR.NARAWIT KATHANCHAROEN

D6110307

MISS.PRATABJAI PRASUJAN

M6110833

A PRAJECT OF ADVANCED ROCK MECHANICS

INSTITUTE OF CIVIL, TRANSPORTATION AND

GEO-RESOURCES ENGINEERING

SURANAREE UNIVERSITY OF TECHNOLOGY

ACADEMIC YEAR 2018

I

TABLE OF CONTEVTS

Page

TABLE OF CONTENT........................................................................................................I

LIST OF TABLE ............................................................................................................... III

LIST OF FIGURES ........................................................................................................... IV

CHAPTERT

I INTRODUCTION ................................................................................... IV

1.1 Background and rationale ................................................................... 1

1.2 Research objectives ............................................................................ 1

1.3 Scope and limitations.......................................................................... 1

1.4 Research methodology ....................................................................... 1

1.4.1 Literature review .................................................................. 2

1.4.2 Sample preparation .............................................................. 2

1.4.3 Laboratory testing ................................................................ 3

II LITERATURE REVIEW ......................................................................... 4

2.1 Introduction ........................................................................................ 4

2.2 Three-point bending test ..................................................................... 4

2.3 Researcher on bending test ................................................................. 5

2.4 Stress gradient effect .......................................................................... 9

2.5 Tensile strength of rock ...................................................................... 9

III SAMPLE PREPARATION .................................................................... 11

3.1 Introduction ...................................................................................... 11

3.2 Sample preparation ........................................................................... 11

IV LABORATORY TESTING AND TEST RESULTS ............................ 14

4.1 Three-point bending test ................................................................... 14

4.2 Test result ......................................................................................... 16

VI DISCUSSIONS AND CONCLUSIONS................................................. 17

5.1 Discussions and conclusions ............................................................ 17

5.2 Recommendation for future studies .................................................. 17

REFERENCES .................................................................................................... 20

II

LIST OF TABLES

Table Page 3.1 Physical properties of specimen for bending tests ................................................. 13 4.1 Results of three-point bending test ........................................................................ 19

III

LIST OF TABLES

Figure Page 1.1 Research methodology ............................................................................................ 1

2.1 Schematic of a suitable apparatus for flexure test by center-point loading method

(ASTM (C293–02)) ................................................................................................. 4

2.2 Difference specimen size utilized (Cardani and Meda,2004) .................................. 5

2.3 Three (a) and four (b) point bending configuration (Cardani and Meda,2004) ...... 7

2.4 Elastic beam with (a) fixed ends and (b) simple (pin) supports (Diederichs and

Kaiser,1998). ........................................................................................................... 7

2.5 Variation of the stresses along the height of a beam (Exadaktylos et al.,2001). ..... 8

3.1 a) Rectangular specimen with dimensions 50x30x200 mm3 ................................ 12

b) Rectangular specimen with dimensions 50x50x200 mm3 ................................ 12

c) Rectangular specimen with dimensions 50x70x200 mm3 ................................ 12

d) Rectangular specimen with dimensions 50x90x200 mm3 ................................ 13

e) Rectangular specimen with dimensions 50x110x200 mm3 .............................. 13

4.1 Bending test equipment ......................................................................................... 14

4.2 Loading diagram for three-point bending test ....................................................... 15

4.3 Pre and post-test some PWSS specimens of three-point bending test ................... 15

4.4 Relation between tensile stress ( t) versus deflection of PWSS specimens.......... 16

4.5 Relation between tensile strength ( t) versus thickness of PWSS specimens ....... 17

4.6 Stress gradient from tensile strength of the various thickness PWSS specimens . 18

CHAPTER I

INTRODUCTION

1.1 Background and rationale Bend testing (e.g.,ASTM C293-02,ASTM D790-17) has been widely used to

determine the tensile strength. The tensile strength can be obtained from the laboratory by

various methods but the bending test is more preferable than other for design and stability

analysis of underground structure such as the tunnel roof. This is primarily due to the

stresses variation along the height of a roof that can find from tensile strength.

1.2 Research objectives The objectives of this study are to determine the stress gradient of specimens that

are sandstone in Phra Wihan formations. The tensile strength is made in the laboratory by

three-point bending test. Mathematical relationship between the thickness of specimen,

deflection and tensile strength.

1.3 Scope and limitations The scope and limitations of the research include as follows.

1) Laboratory experiments are conducted on specimens from Phra Wihan.

2) Laboratory testing made under constant rate loading and static loading.

3) Different specimen thickness are 30 mm, 50 mm, 70 mm, 90 mm and 110 mm.

4) All test rocks are prepared in the laboratory.

5) Up to 1 samples 5 size are tested for sandstone.

6) No field testing is conducted.

1.4 Research methodology

As shown in Figure 1.1, the research methodology comprises 5 steps’ including

literature review, sample collection and preparation, laboratory testing (bending test),

discussions and conclusions and report writing.

2

Literature review

Sample Collection and

Preparation

Bending test

Discussions and Conclusions

Report writing

Figure 1.1 Research methodology

1.4.1 Literature review

Literature review has been carried out to improve an understanding of

tensile strength, bend testing and stresses variation. The sources of information are from

text books, journals, technical reports and conference papers. A summary of the literature

review is given in chapter two.

1.4.2 Sample preparation

The specimens used for bending test are the Phra Wihan sandstone. They

are prepared to obtain rectangular shape with dimension of same width 50mm, same length

200mm but vary thickness 30 50 70 90 and 110mm

3

1.4.3 Laboratory testing

Laboratory testing made under constant rate loading and static loading. The

test procedure is in accordance with the ASTM (C293-16, D6272-17) standard practice.

1.5 Report contents

This report is devices into five chapters. The first chapter includes background and

rationale, scope and limitations, and research methodology. Chapter II present results of

the literature review to improve an understanding of tensile strength, bend testing and

stresses variation. Chapter III describes sample collection and preparation. Chapter IV

describes the laboratory testing. Chapter V is discussions, conclusions and future studies.

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

The topic reviewed here include three point bending tests, research bending test,

stress gradient effect and tensile strength of rock.

2.2 Three-point bending test The American Society for Testing and Materials (ASTM (C293–16)) specified the

methods and sample preparation for the three-point bending test. All forces should be

applied perpendicular to the face of the specimen continuously without eccentricity. A

diagram of an apparatus that accomplishes this purpose is shown in Figure 2.1.

The modulus of rupture is calculated as follows:

23PLR =2bd

(2.1)

where R is modulus of rupture, MPa (psi), P is maximum applied load indicated by the

testing machine, N (lbf), L is span length, mm (in.), b is average width of specimen, at the

fracture, mm (in.) and d is average depth of specimen, at the fracture, mm (in.). A bar of

rectangular cross section rests on two supports and is loaded at two points or two loading

noses. The distance between the loading noses (the load span) is either one third or one half

of the support span.

Figure 2.1 Schematic of a suitable apparatus for flexure test by center-point loading

method (ASTM (C293–16)).

5

2.3 Research on bending test Cardani and Meda (2004) study The tensile behavior of marble at failure is analysed.

A series of monotonic and cyclic bending tests were carried out in the laboratory on

geometrically similar specimens of different size. The strain distribution was monitored with

strain gauges and laser interferometry techniques. Three and four-point bending

configuration was considered. It is shown that cyclic loads in a three-point bending

configuration induce permanent tensile deformation.

A series of bending tests on geometrically similar specimens of four different sizes

(Figure 2.2) were performed with a four-point bending configuration. In order to obtain the

softening branch a crack mouth opening displacement control was used. For this reason a

notch was made in the middle of the span to localise the crack propagation. In addition three

and four point bending tests were performed to compare the two different configurations

(Figure 2.3). Experimental evidence shows that strength may be a function of the structure

dimensions and a decrease in strength with increasing size is observed.

Figure 2.1 Difference specimen size utilized (Cardani and Meda,2004)

Diederichs and Kaiser (1999) proposed a more reasonable stability limit for this

failure mode in rockmasses and particularly for data limited cases. Design charts, based on

this linearity limit for unsupported stability of jointed rock beams, are presented here

summarizing critical span±thickness±modulus relationships.

Consider a laminated rock beam above an exca-vation with a horizontal span given

by S. The normal thickness of the single layer under analysis is T. For an elastic beam, with

no joints and with constant cross section, a distribution of compression and tension,

symmetrical about the horizontal centreline of the beam, is found across all plane sections

within the beam (Figure 2.4a). The solution for the maximum stress values, at the abutments,

6

for compression (bottom of beam) or tension (top of beam), max, as well as the maximum

beam deection, , can be easily calculated using closed form beam equartion as follows:

2

maxγSσ =2T

(2.2)

4

2γSδ =

32ET (2.3)

where E is the Young's modulus of the rock and g is the speci®c weight. The maximum

stress at the mid-span is one half of the maximum stress at the abutments. Therefore, for

such a beam with fixed ends and distributed loading, yield is assumed when the maximum

tensile stress, in the upper part of the beam at the abutments, exceeds the tensile strength of

the rock. Vertical tensile fractures form at the abutments and the beam becomes simply

supported (assuming no slip at the abutments), as shown in Fig. 4(b), with a maximum

tensile stress at the midspan given by

2

max2γSσ =3T

(2.4)

This stress is higher than the previous abutment stress, and therefore higher than

the rock tensile strength. This leads to subsequent fracturing centered about the mid span as

shown by Stimpson and Ahmed (1992). This process of progressive cracking at the

abutments, followed by cracking at the midspan and other parts of the beam can be

responsible for a flurry of low-level seismic emissions (rock noise), often encountered in

newly developed underground spans at low to moderate depth. This initial elastic phase fol-

lowed by progressive fracture and deformation of laminated hangingwalls has been observed

and is described in detail by Milne (1996)

7

Figure 2.3 Three (a) and four (b) point bending configuration (Cardani and Meda,2004).

Figure 2.4 Elastic beam with (a) fixed ends and (b) simple (pin) supports (Diederichs and

Kaiser,1999).

8

Exadaktylos et al. (2001) study elastic theory and Voussoir beam analogy. The strain

distribution in each section is assumed to be triangular (Figure 2.5(a)), with a symmetrical

distribution of compression and tensile stresses about the horizontal midline. But there are

studies that refute the hypothesis of a symmetrical distribution of compression and tensile

stresses. In this case, the distribution of stresses adopts a form similar to that depicted in

Figure 2.5(b), where the tensile strength of the rock is approximately 10% of its compressive

strength. In this work, a non-linear failure criterion for rock mass is proposed, based on the

double theory of elasticity, and from it the determination of the maximum span that can

support the tensile stresses through the roof. Besides, the behavior of the roof when the

tensile stresses increase the tensile strength is analyzed. The method is applied on two real

examples: Kampanzar Quarry situated in the municipality of Arrasate in Gipuzkoa (Spain)

and Calzada Quarry situated in Villamartı´n de Vadeorras in Orense (Spain). The numerical

results have been compared with those ones obtained from the Stephansson formulation, and

the behavior after failure is compared with the compression arch assumed by the Voussoir

beam analogy.

Figure 2.5 Variation of the stresses along the height of a beam (Exadaktylos et al.,

2001).

9

2.4 Stress gradient effect Claesson and Bohloli (2002) state that the tensile strength of rock is among the most

important parameters influencing rock deformability, rock crushing and blasting results. To

calculate the tensile strength from the indirect tensile (Brazilian) test, one must know the

principal tensile stress, in particular at the rock disc center, where a crack initiates. This

stress can be assessed by an analytical solution. A study of this solution for anisotropic

(transversely isotropic) rock is presented.

Liao et al. (1997) study the tensile behavior of a transversely isotropic rock by a

series of direct tensile tests on cylindrical argillite specimens. To study the deformability of

argillite under tension, two components of an electrically resistant type of strain gage with

a parallel arrangement, or a semiconductor strain gage, are adopted for measuring the small

transverse strain observed on specimens during testing. The curves of axial stress and axial

strain and average volumetric strain are presented for argillite specimens with differently

inclined angles of foliation. Experimental results indicate that the stress-strain behavior

depends on the foliation inclination of specimens with respect to the loading direction. The

five elastic constants of argillite are 13 calculated by measuring two cylindrical specimens.

Based on theoretical analysis results, the range of the foliation inclination of the specimens

tested is investigated for feasibility obtaining the five elastic moduli. A dipping angle of the

foliations (φ) of 30-60° with respect to the plane normal to the loading direction is

recommended. The final failure modes of the specimens are investigated in detail. A saw

toothed failure plane occurs for the specimens with a high inclination of foliation with

respect to the plane perpendicular to the loading direction. On the other hand, a smooth plane

occurs along the foliation for specimens with low inclination of foliation with respect to the

plane normal to the loading direction. A conceptual failure criterion of tensile strength is

proposed for specimens with a high inclination of foliation.

2.5 Tensile strength of rock

The tensile strength is the maximum amount of tensile stress that can be applied

before the rock fails, and is a characteristic property of the rock. Anisotropic strength

characteristics very common for rocks that have pronounced directional features such as

flow structure, are ation and lamination. The tensile strength parallel to the bedding is

usually higher than the foli ensile strength perpendicular to the bedding. There is however

no guarantee that the tensile strength at an angle to the bedding has a value that is between

10

the strength parallel and perpendicular to the bedding. because all fractures across the

bedding planes are caused by tension. Tensile strength of rock is an important parameter

used in the design and stability analysis of underground structures Rock tensile strength

dictates the maximum roof span of underground openings, the stability of boreholes, and the

minimum borehole pressures for hydraulic fracturing process (Klanphumeesri, 2010).

CHAPTER III

SAMPLE PREPARATION

3.1 Introduction

This chapter describes the sample preparation for the three-point bending test. the

sample rocks using are Phra Wihan sandstones (PWSS) formation. The specimens are

choose at the perfect, homogeneous, no crack and fresh rocks.

3.2 Sample preparation

The specimens used for the three-point bending test were prepared from the Phra

Wihan sandstones by saw-cutting. The specimens were prepared to obtain rectangular

shape that various thickness but same width and length with dimension 50x30x200,

50x50x200, 50x70x200, 50x90x200 and 50x110x200 mm3 (Figures 3.1).

12

Figure 3.1 a) Rectangular specimen with dimensions 50x30x200 mm3

Figure 3.1 b) Rectangular specimen with dimensions 50x50x200 mm3

Figure 3.1 c) Rectangular specimen with dimensions 50x70x200 mm3

13

Figure 3.1 d) Rectangular specimen with dimensions 50x90x200 mm3

Figure 3.1 e) Rectangular specimen with dimensions 50x110x200 mm3

Table 3.1 Physical properties of the specimen for three-point bending test.

specimen no. Dimension (mm3) Weight (g) Density (g/cc)

PW-01 52.1x31.5x200.8 739.69 2.25

PW-02 50.9x51.7x200.7 1181.09 2.24

PW-03 51.5x71.4x200.1 1642.33 2.23

PW-04 51.5x90.4x200.2 2174.06 2.33

PW-05 52.0x111.8x200.0 2641.34 2.27

Mean ± SD 2.24 0.04

CHAPTER IV

LABORATORY TESTING

4.1 Three-point bending test

The objective of the three-point bending test is determine the tensile strength of the

specimen that vary thickness. The test procedures of three-point bending test follow the

American Society for Testing and Materials (ASTM D790). Figure 4.1 shown the

laboratory arrangement of bending test. Figure 4.2 shown a diagram of rectangular

specimen cross section rests on two bottom supports and on top loaded at mid span. The

load for the rate-controlled testing is under stress rates 1 MPa/min at the center of

specimen. The specimen deformations are monitored and recorded every 0.03 MPa of load

increment until failure. Figure 4.3 shown the pre and post-test of the specimens for three-

point bending test

Figure 4.1 Bending test equipment

15

Figure 4.2 Loading diagram for three-point bending test

Figure 4.3 Pre and post-test of PWSS specimens for three-point bending test

16

4.2 Test results

The standard formula for tensile strength of a beam at flexure point is as follows

t 2

3PLσ =2bd

Where t is tensile strengths, P is the applied load, L is support span, b is specimen width

and d is specimen thickness.

From standard the three-point bending test. According to the results of PWSS

specimens. Figure 4.4 shown relation between tensile stress ( t) versus deflection and

figure 4.5 shown relation between tensile strength ( t) versus thickness of PWSS

specimens.

Figure 4.4 Relation between tensile stress ( t) versus deflection of PWSS specimens.

17

Figure 4.5 Relation between tensile strength ( t) versus thickness of PWSS specimens. Stress gradient or variation of the stresses that along the height of a beam form the

three-point bending test can plot from the maximum stress values shown in table 4.1, for

compression (top of beam) and tension (bottom of beam), as shown in figure 4.6 with the

maximum tensile stress at the mid span of the beam.

18

Figure 4.6 Stress gradient from tensile strength of the various thickness PWSS specimens.

t max = 2.95 MPa

t max = 8.03 MPa

t max = 10.90 MPa

t max = 11.53 MPa

t max = 11.77 MPa

19

Table 4.1 Results of three-point bending test.

No. width (mm) thickness (mm) length (mm) P (kN) t (MPa)

1 50 30 200 0.49 2.95

2 50 50 200 3.72 8.03

3 50 70 200 10.30 10.90

4 50 90 200 11.30 11.53

5 50 110 200 26.36 11.77

CHAPTER V

DISCUSSIONS, CONCLUSION AND RECOMMENDATIONS FOR FUTURE STUDIES

5.1 Discussions and conclusions

The post-test specimens obtained from the three-point bending test show that the

fractures occur at the center of specimen for all testing. This is because this point is under

maximum tensile stresses. The specimen with high thickness (110 mm) show higher

tensile strengths at failure than those with lower thickness (30 mm) indicate that the

thickness of specimen affect on tensile strength. The thickness increases, the tensile

strength is increased but stresses variation is decreased due to the effect of thickness is

greater than tensile strength.

5.2 Recommendations for future studies

More testing is required on a various rock type and specimen size. May be change

width or length only or both for test about the stress gradient and more type of laboratory

testing should be performed.

21

REFERENCES

ASTM C293–16. Standard test method for flexural strength of concrete (Using Simple Beam

With Center-Point Loading). Annual Book of ASTM Standards, American Society

for Testing and Materials, West Conshohocken, P.A., Vol.04.02.

Cardani, G. and Meda, A. (2004). Marble behaviour under monotonic and cyclic loading in

tension. Construction and Building Materials 18: 419-424.

Claesson, J. and Bohloli, B. (2002). Brazilian test: stress field and tensile strength of

anisotropic rocks using an analytical solution. International Journal of Rock

Mechanics and Mining Sciences, 39: 991–1004.

Chanpen, S. (2017). Effect of thinly-stratified sandstone on tunnel roof stability.

Suranaree University of Technology, Nakhonratchasima.

Diederichs, M.S. and Kaiser, P.K. (1999) Stability of large excavations in laminated hard

rock masses: the voussoir analogue revisited. International Journal of Rock

Mechanics and Mining Sciences, 36: 97–117.

Exadaktylos, G.E., Vardoulakis, I. and Kourkoulis, S.K. (2001). Influence of nonlinearity

and double elasticity on flexure of rock beams — II. Characterization of Dionysos

marble. International Journal of Solids and Structures, 38: 4119–4145.

Klanphumeesri, S. (2010). Direct tension testing of rock specimens. Suranaree University

of Technology, Nakhonratchasima.

Liao, J.J., Yang, M.T. and Hsieh, H.Y. (1997). Direct tensile behavior of a transversely

isotropic rock. International Journal of Rock Mechanics and Mining Sciences,

34(5): 831–849.

Snyder, V.W. (1983). Analysis of beam building using fully grouted roof bolts. Proceedings

of the International Symposium on Rock Bolting, Abisko, Sweden, pp. 187–194.