Empirical models for tool forces prediction of drag-typed picks based on principal component regression and ridge regression methods

Tunnelling and Underground Space Technology 62 (2017) 75–95

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Empirical models for tool forces prediction of drag-typed picks based on principal component regression and ridge regression methods Xiang Wang a,b, Yunpei Liang a,⇑, Qingfeng Wang b, Zhenyu Zhang a a b

State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, China China Coal Technology Engineering Group Chongqing Research Institute, Chongqing 400039, China

a r t i c l e

i n f o

Article history: Received 23 November 2015 Received in revised form 14 October 2016 Accepted 11 November 2016

Keywords: Drag-typed picks Tool forces Empirical models Regression analysis

a b s t r a c t The forces acting on a single drag-typed pick are important parameters for excavation machine design and selection. For better prediction of tool forces including cutting and normal forces generally, a general model of cutting forces was proposed based on theoretical models. Also, a general model of normal forces was proposed using the ratio of the normal force to cutting force. Subsequently, the effect of relevant geometrical parameters on the cutting force was discussed. The friction angle between pick and rock, the cone angle and the attack angle were employed to develop the cutting force models of conical picks. The rake angle and the friction angle between pick and rock were included in the peak cutting force model of radial picks. Finally, the peak and mean cutting forces models of conical picks and the peak cutting force model of radial picks under unrelieved cutting mode were developed using principle component regression analysis and ridge regression analysis based on the raw data from linear full-scale cutting test. The results show the proposed regression coefficients and equations are more reasonable physically. Some empirical models used for practical application were then developed by introducing relevant modified coefficients considering tool wear, relieved cutting and complex shapes of picks. The results show a good agreement between the measured and predicted cutting force of sharp picks under unrelieved cutting mode. The performance of modified models using relevant modified coefficients would be decreased to a certain extent. However, they are all statistical valid according to the results of t-test. The models of this work can be used for preliminarily estimation of tool forces acting on drag-typed picks. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Mechanical rock cutting has been applied widely in mining and similar underground excavation due to its high efficiency. Common tools used for cutting rock include conical picks (also named pointattack picks) and radial picks (also named chisel picks) which are collectively referred as drag-typed picks as shown in Fig. 1, and they are especially used in roadheaders, shearers and continuous miners. The rock cutting mechanism has been an important topic concerning the process of rock cutting. Extensive studies have been conducted regarding to different aspects, such as dust production (Fowell and Ochei, 1984; Achanti, 1998), friction and picks wear (Carbonell et al., 2013; Dewangan et al., 2014; Dogruoz and Bolukbasi, 2014), chips shape and size distribution (Rånman, 1985; Liu et al., 2009; Tuncdemir et al., 2008), rock cutting study based on fracture mechanics (Guo et al., 1992; Jiang et al., 2013), ⇑ Corresponding author. E-mail address: [email protected] (Y. Liang). http://dx.doi.org/10.1016/j.tust.2016.11.006 0886-7798/Ó 2016 Elsevier Ltd. All rights reserved.

heat production and transmission in the process of rock cutting (Loui and Rao, 2005), joint auxiliary cutting (Ciccu and Grosso, 2010; Liu et al., 2014) and specific cutting energy (Tiryaki and Dikmen, 2006; Tumac et al., 2007). The single pick is the basic element of cutting machine and the tool forces acting on it play an important role in calculating the loads of working unit of excavation machine, and in determining the depth of cut which is a fundamental aspect in determining the cutting efficiency. Also, the tool forces acting on a single pick can be used for designing the cutting head (Mustafa and Bolukbasi, 2005), studying the stability of the machine (Acaroglu and Ergin, 2006; Ergin and Acaroglu, 2007), etc. Therefore, the model development of tool forces acting on drag-typed picks is one of the main topics on the studies of cutting mechanism. So far, theoretical or semi-empirical models (Evans, 1962, 1965, 1984; Goktan, 1995, 1997; Nishimatsu, 1972; Roxborough and Liu, 1995; Goktan and Gunes, 2005; Kovrizhnykh, 2006; Bao et al., 2011), empirical models (Muro et al., 1997; Copur et al., 2003; Bilgin et al., 2006; Balci and Bilgin, 2007; Gunes et al., 2007;

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

(a) Conical pick

(b) Radial pick Fig. 1. Some parameters affecting cutting performance of drag-typed picks.

Tiryaki et al., 2010; Copur, 2010; Abu Bakar and Gertsch, 2013; Kim et al., 2012a, 2012b) and numerical models (Kou et al., 1999; Rojek et al., 2011; Su and Akcin, 2011; Van Wyk et al., 2014; Menezes et al., 2014a, 2014b) are commonly used to evaluate tool force values by engineers. These studies have led to a better understanding of rock cutting mechanism with priority of convenient usage with low cost. Widely-accepted theories include Evans’ (1962, 1965, 1984) for conical picks and radial picks, and Nishimatsu’s (1972) for radial picks. However, the cutting forces predicted by theoretical models in many cases are not in a good agreement with measured values due to heterogeneous and anisotropic nature of rocks (Bilgin et al., 2006). Numerical models may provide relatively accurate cutting force values when the applied model was reasonable and the relevant parameters were appropriately determined. In addition, some studies that are difficult to be carried out by theoretical or experimental methods can be done easily with numerical modeling

techniques. For example, Menezes et al. (2014a, 2014b) investigated the rock fragment morphology and the characteristic of fragment formation in the process of rock cutting using the explicit finite element method, and the effect of cutting speed was also involved. However, experienced engineers are required to evaluate the correction of simulation results. Although direct measurement can provide accurate values, sophisticated test facilities and intensive labour force are required, corresponding to a high volume of material consumption as well as high cost. For practical purpose, reliable empirical models based on enough test data are preferred in evaluating cutting forces, and many empirical models have been developed using single factor regression (Copur et al., 2003; Bilgin et al., 2006), multiple regression (Gunes et al., 2007; Tiryaki et al., 2010), regression tree and neural network method (Tiryaki et al., 2010). However, neither the theoretical models nor the modified ones can implement the estimation of the normal force acting on

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77

a drag-typed pick. Therefore, a reliable way to develop the normal force models for practice application is necessary. In this study, significant theoretical and empirical models for the cutting force acting on a drag-typed pick were firstly reviewed. General models of the cutting and normal forces are then developed by analyzing the structural components and the physical dimension of each parameter in theoretical models by introducing modified coefficients based on experiment data. The dominant geometrical parameters in the general model are discussed in detail, and relevant coefficients of rock strength were determined using principal component regression analysis and ridge regression analysis for the cutting force model development of conical and radial picks, respectively. Finally, the prediction performance of new models was checked by hypothesis testing and regression analysis between measured and predicted cutting force values.

Assuming that rock fracture was caused by tensile failure, Evans (1962, 1965) studied a symmetrical rock cutting mode by a wedge, and the cutting force was expressed using the rake angle by Bilgin et al. (2006) and Gunes et al. (2007) as shown in Eq. (3a). Nishimatsu (1972) attributed rock fracture to shear stress in cutting process using radial picks and proposed a model to predict the peak cutting force based on Mohr-Coulomb criterion (Eq. (3b)). It is noted that the case of negative rake angles was not involved in the rock cutting modes of Evans (1962, 1965) and Nishimatsu (1972) for radial picks. Analyzing the previous test data, Goktan (1995) proposed a semi-empirical model to predict the cutting force acting on radial picks for rocks covering the uniaxial compressive strength approximately 80–180 MPa (Eq. (3c)):

2. Previous theoretical and experimental studies on the force of drag-typed picks

(

FC r ¼

( As one of the most popular theoretical model of the cutting force acting on the conical pick, Evans (1984) considered a symmetrical V-shaped chip removed, and demonstrated theoretically that the compressive and tensile strength of the rock were dominant properties influencing on the cutting force. The calculation model of the cutting force can be expressed as Eq. (1):

16ph r2t rc cos2 ð/=2Þ 2

ð1Þ

where FC c is the cutting force acting on conical picks, rt is the tensile strength of the rock, rc is the compressive strength of the rock, h is the cutting depth and / is the cone angle of conical pick. Cutting geometry is shown in Fig. 1a. By taking the friction between pick and rock into account, Goktan (1997), Roxborough and Liu (1995) modified Evans’ (1984) model of conical picks as indicated in Eqs. (2a) and (2b), respectively. It indicates that the values of cutting force calculated by theoretical models (Eqs. (1) and (2a)) are significantly lower than the measured ones. Goktan and Gunes (2005) argued that this deviation might originate from the model assumptions where only the symmetrical cutting conditions have been considered and they introduced semi-empirical models to predict cutting forces by taking the rake angle into account. Based on the fourteen sets of test data, statistical analysis showed that there was a statistically significant relationship between measured and calculated cutting force at the 99% confidence level by setting friction angles as 10°. The models are expressed using the attack angle as shown in Eqs. (2c) and (2d):

4p h

rt sin2 ð/=2 þ uÞ cosð/=2 þ uÞ

2

FC c ¼

FC c ¼

16ph

ð2aÞ 2

rc r2t

½2rt þ ðrc cosð/=2ÞÞð1 þ tanðuÞ= tanð/=2ÞÞ2

ð2bÞ

FC cp ¼

12prt h sin ½ð/=2 þ cÞ=2 þ u cos½ð/=2 þ cÞ=2 þ u

ð2cÞ

FC cm ¼

4prt h sin ½ð/=2 þ cÞ=2 þ u cos½ð/=2 þ cÞ=2 þ u

ð2dÞ

2

2

FC r ¼ 2chd nþ1

cos w cosðubÞ 1sinðwbþuÞ

n ¼ 11:3  0:18b

2.1. Theoretical and semi-empirical models

FC c ¼

2rt hd sinðp=4  b=2Þ 1  sinðp=4  b=2Þ

2

2

where c is the attack angle, u is the friction angle between pick and rock, FC cp and FC cm are the peak and mean cutting forces acting on conical picks, respectively.

FC r ¼

ð3aÞ

ð3bÞ

Ahrc d

p=2bÞþcosðp=2bÞ A ¼ 0:80  0:01ðrc =rt Þ sinð

ð3cÞ

where FCr is the cutting force acting on radial picks, b is the rake angle, d is width of radial pick, c is the shear strength of the rock, n is stress distribution coefficient and w is internal friction angle of the rock. It is worth noting that the u in Eq. (3b) was termed the angle of friction of rock cutting by Nishimatsu (1972), which had a linear relation with the rake angle. For the convenience of calculation, the u in Eq. (3b) is usually replaced by the friction angle between pick and rock (Bilgin et al., 2006; Gunes et al., 2007). Cutting geometry is shown in Fig. 1b. In addition to the classical theories mentioned above, some researchers tried to study the cutting force from different aspects. For example, Kovrizhnykh (2006) modeled the cutting force acting on radial picks based on Mohr-Coulomb criterion by taking the cutting speed into account. Bao et al. (2011) developed a model to predict the cutting force acting on conical picks based on fracture mechanics using Evans’ (1984) rock cutting mode. 2.2. Empirical models Besides theoretical and semi-empirical models, many useful empirical models have been developed based on the detailed laboratory cutting experimental data. Bilgin et al. (2006) carried out laboratory full-scale linear cutting tests using a conical pick for 22 different rock specimens with the compressive strength values varying from 6 to 174 MPa. Regression analysis indicated that there was a strong linear relationship between the mean cutting force to cutting depth ratio and the rock compressive (tensile strength) in unrelieved cutting modes. There was also a strong power function relation between the normal force to cutting depth ratio and the rock compressive (tensile strength) in unrelieved cutting modes. Regression equations are shown as below in Eq. (4).

8 c FC m ¼ ð8:0948rc þ 213:248Þh ðR2 ¼ 0:810Þ > > > > < FC c ¼ ð123:725r þ 86:0381Þh ðR2 ¼ 0:797Þ t m c 1:014 > FN ¼ 11:9266 r h ðR2 ¼ 0:843Þ > c m > > : c FNm ¼ 154:252r0:915 h ðR2 ¼ 0:760Þ t

ð4Þ

where FNcm is the mean normal force acting on conical picks. Based on the test data of Balci and Bilgin (2007), Tiryaki et al. (2010) modeled the mean cutting force of conical picks using multiple linear regression, multiple nonlinear regression, regression

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tree and neural network methods. Regression equations are shown in Eq. (5).

(

FC cm ¼ 1:93 þ 0:36h þ 0:04rc þ 0:03Edyn FC cm

¼ 0:01h

1:08

r

0:32 0:29 0:52 Edyn N24 c

ðR2 ¼ 0:89Þ

2

ðR ¼ 0:95Þ

ð5Þ

where Edyn is dynamic modulus of elasticity and N24 is Schmidt hammer hardness. Gunes et al. (2007) developed a model to predict the peak cutting force acting on radial picks using multiple linear regression based on the test data of Bilgin (1977) as shown in Eq. (6).

FC rp ¼ 24:504 þ 0:513c þ 1:837h þ 0:249u þ 0:227d  0:154b  0:047rc

wpr

: FNr

wpr

h i t hd sinðp=4b=2Þ ¼ k1 k2 k3 k4 2r1sinð p=4b=2Þ ¼ k5 k6 FC rwpr

ð7Þ

where FC rwpr and FNrwpr are the cutting and normal force of worn radial picks in relieved cutting mode, k1 is a coefficient used for taking the effect of wear flat on tool forces into account, k2 is a coefficient used for taking the effect of front ridge angle of radial picks into account, k3 is a coefficient used for taking the effect of veebottom angle of radical picks into account, k4 is a coefficient used for taking the effect of relieved cutting into account, k5 is the ratio of normal to cutting force of sharp radial picks, and k6 is the ratio of peak normal force of worn radial picks to peak normal force of sharp ones. In this study, the effects of relevant parameters in classical theoretical models on the cutting force are discussed in detail later, and the best one is to be taken as a component of empirical models of this study. 3. General prediction models of tool forces 3.1. A general model of the cutting force acting on drag-typed picks The theoretical models of cutting force are very rigorous from the perspective of dimension and physical significance, which were ignored commonly in the regression equations. It is also found that theoretical models are generally made by rock strength parameters and geometric parameters. Therefore, general models of the cutting force can be expressed in Eq. (8).



FC s ¼ Rr Ra Ac þ R1 FC w ¼ Rk FC s

ðbÞ

ðaÞ

2

Ac ¼ h tan n, where n is caving semi-angle, Evans (1984) demonstrated theoretically that it equaled to 60°. 3.2. A general model of the normal force acting on drag-typed picks

ð6Þ

where FC rp is the peak cutting force acting on radial picks. Based on Evans’ (1962) cutting force model of radial picks and the previous experimental studies, Bilgin et al. (2012) developed modified cutting and normal forces models to estimate deterministically the torque and thrust requirements of the TBM equipped radial picks, confirming its validity in predicting forces of radial picks. The models are shown as in the following equation:

8 < FC r

on the test data, and he explained that it was related with ‘‘the secondary crushed zone”. It is also found that Eqs. (4) and (5) include the constants developed by using linear regression. The equivalent cutting area Ac can be expressed as Ac ¼ dh for radial picks, where the cutting forces acting on radial picks increase linearly with cutting depth and width of picks. This is consistent with experimental research conclusions of Bilgin (1977). Ac is considered as cut section area for conical picks and therefore, it can be calculated by

In order to establish the model of normal force, Bilgin et al. (2012) modeled the normal force of radial picks to estimate total thrust force requirement of a TBM equipped with radial picks successfully. Based on this way, the normal force models can be expressed in Eq. (9).



FNs ¼ knc FC s FNw ¼ kncw FC w

ð9Þ

where knc is the ratio of normal to cutting force of drag-typed picks in sharp state, kncw is the ratio of normal to cutting force of dragtyped picks in worn state. 4. Relevant parameters study in general prediction models of tool forces 4.1. Study of Ra Relevant angles include two classes in theoretical models of the cutting force. The first class is geometric angles such as cone angle (/), rake angle (b) and the attack angle (c). The second class is the angles on rock physical property such as internal friction angle of rock (w) and the friction angle between pick and rock (u). The models of Ra are shown in Eq. (10) for conical picks and in Eq. (11) for radial picks from theoretical models. 4.1.1. Ra for conical picks It can be known that the friction angle (u) was set to 10° in the semi-empirical models of Goktan and Gunes (2005), and the models could be used to predict the cutting force satisfactorily. However, the values of u of rocks are usually in a range of 26–38° averaged at 30° approximately (Bilgin et al., 2006). Therefore, the u is written as u=3 in Eq. (10c) in line with the semi-empirical models of Goktan and Gunes (2005).

REa ¼

1 cos2 ð/=2Þ

ð10aÞ

RGa ¼

sin ð/=2 þ uÞ cosð/=2 þ uÞ

ð10bÞ

2

ð8Þ

where FC s is the cutting force of sharp picks in unrelieved cutting mode, FC w is the cutting force of worn picks in relieved cutting mode, Rk is a coefficient used for taking into account the effects of tool wear, relieved cutting and shapes modified of picks on tool forces, Rr is a coefficient represented rock strength parameters

(ML1 T2 ), Ra is a coefficient represented angle parameters (nondimensional number), Ac is the equivalent cutting area (L2 ) and R1 is a constant. It can be seen that all parameters in Eq. (8a) can be found in the theoretical models (Eqs. (1)–(3)) with the exception of the constant R1 . In the work of Nishimatsu (1972), there was a constant in the model developed by using regression analysis based

sin ½ð/=2 þ cÞ=2 þ u=3 cos½ð/=2 þ cÞ=2 þ u=3 2

RGse a ¼

ð10cÞ

REa , RGa and RGse a are from Eqs. (1), (2a) and (2c), (2d) respectively developed by Evans (1984), Goktan (1997), Goktan and Gunes (2005). Obviously, there is only a cone angle (/) in Eq. (10a), the cone angle (/) and the friction angle (u) were taken into account in Eq. (10b). It is obvious that the angles in Eq. (10c) are more comprehensive compared to others. Fig. 2 shows that RGse a increases with increment of the friction angle (u), the cone angle (/) and the attack angle (c) at their common ranges, and these trends are consistent

X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

79

Fig. 2. Relation between RGse a and u; /; c.

with the actual situation. Therefore, RGse a can reflect the effects of each angle on cutting forces acting on conical picks correctly. 4.1.2. Ra for radial picks

RE2 a ¼

sinðp=4  b=2Þ 1  sinðp=4  b=2Þ 1 sinðp=2  bÞ þ cosðp=2  bÞ

ð11bÞ

1 cos w cosðu  bÞ 12:3  0:18b 1  sinðw  b þ uÞ

ð11cÞ

¼ RGse2 a RYa ¼

ð11aÞ

Gse2 RE2 and RYa are from Eqs. (3a), (3c) and (3b) respectively devela , Ra oped by Evans (1962, 1965), Goktan (1997) and Nishimatsu (1972). It can be seen that, the effect of the rake angle (b) on the cutting force was involved only in Eqs. (11a) and (11b), while all angles were fully considered in Eq. (11c). The effect trends of the rake

Gse2 Y Fig. 3. Relation between RE2 a ; Ra ; Ra and b.

Gse2 angle (b) on RE2 and RYa are shown in Fig. 3. As presented in a , Ra Gse2 decrease with the increase of rake angle (b), Fig. 3a, RE2 a and Ra which are consistent with the actual law. Fig. 3b shows that there is a singular point within common range of rake angles, which can lead to some high unrealistic cutting force values (Gunes et al., 2007). However, it should be emphasized that the case of negative rake angles was not involved in Nishimatsu’s (1972) original paper. To sum up, the existing expressions of relevant angles are very limited in theoretical cutting force models of radial picks. In order to find an appropriate model to describe the effects of relevant angles on the cutting force, a new empirical model is proposed

as below in Eq. (12) based on Eq. (11b). As presented in Fig. 4, RTa decreases with increment of the rake angle (b), and increases with increment of the friction angle (u) at their common ranges. Therefore, RTa can reflect the effects of the rake angle (b) and the friction angle (u) on cutting forces acting on radial picks correctly.

RTa ¼

sinðu  b=4Þ sinðp=2  bÞ þ cosðp=2  bÞ

ð12Þ

4.2. Study of Rr It is well known that the cutting force is also closely related to the rock strength in addition to relevant geometrical parameters. Evans (1962, 1965, 1984) demonstrated theoretically that the uniaxial compressive strength and tensile strength governed the failure in rock cutting using conical picks, and the tensile strength

Fig. 4. Relation between RTa and b; u.

governed the failure using radial picks. Nishimatsu (1972) proposed that shear failure was dominant in rock cutting process using radial picks. Goktan and Gunes (2005) suggested that the tensile strength was a major factor influenced on the rock cutting using conical picks. However, rock failure is very complex in cutting process. In practice, compressive failure, tensile failure, shear failure, and mixed failure modes may be dominant depending on rock propertied, picks and cutting geometry. Generally speaking, selecting more parameters for regression analysis, better results may be obtained, such as regression equations of Tiryaki et al. (2010) (Eq. (5)). However, it is extremely difficult to obtain such many parameters in engineering problems. In order to embody

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95 c

compressive failure, tensile failure and shear failure in rock cutting process, a model to express Rr is established by using a linear combination of the uniaxial compressive strength, the tensile strength and the shear strength of rock which are commonly used in engineering design and construction (Eq. (13)).

commercial conical pick and therefore, the factors kw is set to 1 in this work.

Rr ¼ a1 rc þ a2 rt þ a3 c

Several studies on the effect of wear on cutting performance of radial picks have been performed in the past. Among them, a systematic study was carried out by Bilgin (1977) in his Ph.D. study. Detailed laboratory cutting tests were realized on six different rocks using radial picks having a rake angle of (10°), pick width of 25.4 mm and a clearance angle of 6° with different size wear flat. Regression analysis indicated that there were strong linear relationships between the ratios of peak forces of worn picks to peak forces of sharp ones and wear flat as shown in Eq. (16) (Bilgin et al., 2012).

ð13Þ

where a1 ; a2 , and a3 are coefficients to be solved. Substituting Eqs. (10c), (12), (13) into Eq. (8a), the general model of the cutting force of sharp picks in unrelieved cutting mode can be expressed in Eq. (14).

8 < FC c ¼ ðac1 rc þ ac2 rt þ ac3 cÞ tan n sin2 ½ð/=2þcÞ=2þu=3 h2 þ Rc s 1 cos½ð/=2þcÞ=2þu=3 sinðub=4Þ : FC r ¼ ða r þ a r þ a cÞ dh þ Rr1 r1 c r2 t r3 s sinðp=2bÞþcosðp=2bÞ

ð14Þ

where FC cs and FC rs are cutting forces of sharp conical and radial picks respectively; ac1 , ac2 , ac3 , ar1 , ar2 and ar3 are coefficients, and Rc1 and Rr1 are constants. 4.3. Study of Rk

c

c

Rck ¼ kw kr Rrk

¼

r r kw kr kf kv

(

r

kwp ¼ 0:5605WF þ 0:9922 ðR2 ¼ 0:8438Þ r

kwp2 ¼ 1:5011WF þ 1:0795 ðR2 ¼ 0:7524Þ

ð16Þ

r

It should be noted that FCs in Eq. (14) is the cutting force under unrelieved cutting mode using sharp drag-typed picks of ideal shapes. In practice, picks usually have some wear or complex shapes. Furthermore, a certain number of picks are placed in a cutterhead of any mechanical miner according to certain rules. Therefore, there is always interaction between the groves generating a relieved cutting condition. Based on these two considerations, the coefficients Rk introduced to modify the cutting force models of conical and radial picks can be expressed in Eq. (15).

(

(2) The radial pick

ð15Þ

where Rck and Rrk are the modified coefficients (Rk ) for conical and c r radial picks, respectively. kw and kw are coefficients used for taking the effect of tool wear on the cutting force of conical and radial picks into account respectively, they are actually the ratios of cutc r ting forces of worn picks to cutting forces of sharp ones. kr and kr are coefficients accounting for the effect of relieved cutting on the cutting force of conical and radial picks respectively. kf is a coefficient used for accounting for the effect of front ridge angle of complex shapes of radial picks and kv is a coefficient accounting for the effect of v-bottom angle of complex shapes of radial picks (Fig. 1b). 4.3.1. Effect of tool wear on performance of drag-typed picks Tool wear is an important factor affecting the cutting performance of drag-typed picks. The forces of picks increased due to tool wear, usually resulting in reduced cutting depth and increased dust-making and specific cutting energy (Fowell and Ochei, 1984). (1) The conical pick There is a point that the conical pick is always rotating in its holder during rock cutting when providing a reasonable skew and title angles for picks (Fig. 1a). This rotation allows the pick to maintain their tip shape and increases pick life as well as cutting efficiency, since the rotation of the pick leads to uniform wear around the tip. However, it is worth noting that cone tips of new picks are usually made into rounded corner in order to avoid falling damage of alloy head of picks. It is also suggested that the effect of rounded corner Rc (Fig. 1a) on tool forces is necessary to further study in future. The raw data used for regression analysis in this work are taken from relative literatures of Prof. Bilgin and his group in ITU, the tool used in the experiments was a brand new

where kwp (FC rwpu =FC rspu ) is the ratio of peak cutting forces of worn radial picks to peak cutting forces of sharp ones in unrelieved cutr ting mode, kwp2 (FNrwpu =FNrspu ) is the ratio of peak normal forces of worn radial picks to peak normal forces of sharp ones in unrelieved cutting mode, and WF is width of wear flat in unit of mm (Fig. 1b).

4.3.2. Effect of relieved cutting on performance of drag-typed picks There is always an optimum ratio of cutter spacing to cutting depth (s/h) at which the specific cutting energy is minimum. Optimum ratio (s/h) is about 2 for radial picks when cutting rocks of medium strength (Bilgin et al., 2012), and 2–5 for conical picks (Bilgin et al., 2006). However, a statistical relationship between rock properties and optimum ratio (s/h) in their studies is absent. (1) The conical pick Based on the test data from Copur et al. (2003) presented in Table 1, regression analysis indicates that there are strong linear relationships between tool forces under relieved and unrelieved cutting modes as shown in Fig. 5. It can be found that tool forces acting on conical picks in relieved cutting mode (at optimum s/h) would be 20% approximately lower than the tool forces obtained in unrelieved cutting mode. Bearing in mind that the test data used for regression analysis involved peak and mean forces data, the c modified coefficient kr ¼ 0:779 can be used for modification of predication models of peak and mean cutting forces acting on conical picks later in this study. (2) The radial pick Previous study show that tool forces acting radial picks in relieved cutting mode would be 10% approximately lower than the tool forces obtained in unrelieved cutting mode (Bilgin et al., r 2012). Hence, the modified coefficient kr ¼ 0:9 can be used for modification of predication models of cutting forces acting on radial picks.

4.3.3. Effect of complex tool shape on performance of radial picks In order to minimize the wear of the tool, most of the commercially available radial picks are design with front ridge and vee-bottom angles (Fig. 1b). Their effect on tool forces has been discussed in detail in the study of Bilgin et al. (2012), the modification factors kf and kv can be selected in Table 2 according to shape parameters of radial picks.

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95 Table 1 Measured and calculated tool forces of the conical pick in different rocks (Copur et al., 2003). Rock name

rc

rt

u

(MPa) (MPa) (°)

h = 5 mm High-grade chromite Medium-grade chromite Low-grade chromite Harsburgite Serpantinite Trona Sandstone-1 h = 9 mm High-grade chromite Medium-grade chromite Low-grade chromite Harsburgite Serpantinite Trona Sandstone-1 Sandstone-2 Siltstone Limestone

Unrelieved cutting mode

Relieved cutting mode

Peak forces (kN)

Mean forces (kN)

Peak forces (kN)

Mean forces (kN)

Measure

Measured

Measure

Measured

Calculated

Calculated

Calculated

Calculated

Opt. (s/h)

Cut

Norm Cut

Norm Cut

Norm Cut

Norm Cut

Norm Cut

Norm Cut

Norm Cut

Norm

5.41

7.5

5.56

2.73

2.26

2.64

2.17

8.21

6.90

5.84

3.95

2.37

2.09

2.05

1.58

3

32

3.7

27

7.02

47

4.5

27

10.01 7.70

8.42

6.63

3.40

2.96

2.97

2.63

6.56

5.38

6.56

4.69

2.50

2.20

2.31

1.90

2

46

3.7

29

8.54

7.00

8.1

6.01

3.13

2.79

2.85

2.35

7.87

6.54

6.31

4.27

2.71

2.36

2.22

1.71

3

58 38 30 114

5.5 5.7 2.2 6.6

25 28 30 26

14.67 7.69 3.80 19.30

14.18 8.65 4.80 8.60

9.14 8.77 6.8 11.66

7.77 7.57 4.5 10.78

5.20 2.89 1.36 7.43

6.09 3.19 2.11 4.32

3.23 3.09 2.38 4.13

3.13 3.06 1.71 4.44

14.51 8.22 3.07 18.55

13.35 7.20 3.73 15.77

7.12 6.83 5.30 9.08

5.45 5.30 3.23 7.50

4.68 2.07 1.03 6.43

5.08 2.06 1.49 6.78

2.51 2.41 1.86 3.22

2.25 2.19 1.26 3.16

5 3 3 5

32

3.7

27

14.53 9.05

14.08 10.45 5.19

3.47

5.00

4.11

13.97 8.69

10.97 7.42

3.87

2.67

3.90

3.00

3

47

4.5

27

25.96 16.16 17.05 13.44 9.12

6.53

6.07

5.37

17.96 11.70 13.28 9.49

6.28

4.39

4.73

3.89

2

46

3.7

29

15.92 11.61 16.03 11.89 6.50

5.61

5.7

4.69

13.76 10.07 12.48 8.44

4.46

3.56

4.44

3.41

3

58 38 30 114 174 58 121

5.5 5.7 2.2 6.6 11.6 5.3 7.8

25 28 30 26 30 28 30

26.37 19.75 12.11 28.93 47.14 31.36 32.19

9.33 8.07 6.26 8.52 18.90 8.46 20.00

6.91 6.48 4.19 9.85 15.98 7.07 11.44

6.71 6.4 3.00 10.57 27.19 6.73 13.71

25.59 13.78 9.70 24.67 27.78 21.32 32.23

8.93 4.35 2.18 8.31 9.23 6.16 11.39

9.25 4.74 2.88 8.55 11.27 7.06 16.93

5.38 5.05 3.26 7.67 12.45 5.50 8.91

4.82 4.60 2.21 7.54 18.72 4.85 9.70

5 3 3 5 2 3 5

21.44 17.25 13.52 21.45 41.99 23.62 38.14

19.39 18.18 11.8 27.55 44.62 19.81 31.97

16.49 15.7 7.81 25.48 60.33 16.6 32.38

9.04 6.96 4.12 9.72 16.52 8.26 11.93

22.28 12.55 8.45 18.58 25.62 20.15 35.67

15.1 14.16 9.20 21.46 34.76 15.44 24.9

11.57 11.00 5.61 17.73 40.55 11.66 22.35

4.4. Study of knc and kncw The previous studies showed that the ratios knc (FNs =FC s ) and kncw (FNw =FC w ) change depending on physical, mechanical and petrographical properties of rocks, and they may also vary with the cutting depth for some samples (Bilgin, 1977; Bilgin et al., 2012). (1) The conical pick

Fig. 5. Relation between tool forces of sharp conical picks in relieved and unrelieved cutting mode (Copur et al., 2003).

Experiments (Table 3) carried out by Bilgin et al. (2006) indic cated that the knc changes from 0.583 to 1.775 for mean forces under unrelieved cutting mode, from 0.545 to 1.488 for mean forces under relieved cutting mode, from 0.557 to 1.260 for peak forces under unrelieved cutting mode, from 0.477 to 1.109 for peak forces under relieved cutting modes in different strength rocks. It is c also found that the ratio knc increases with uniaxial compressive strength of rocks in power function relation (Fig. 6a), increases exponentially with tensile strength of rocks (Fig. 6b). The regression equations are summarized in Table 4, and they will be used for estimating the normal forces acting on conical picks later.

Table 2 Summary of the effect of wear flat, front ridge and vee-bottom angles on tool forces of radial picks (Bilgin et al., 2012). WF (mm) r kwpm ðFC rwpr =FC rwmr Þ

0.5 1.27

1.0 1.55

1.5 1.83

2.0 2.11

2.5 2.39

3.0 2.67

3.5 2.95

r

1.74

2.41

3.09

3.76

4.43

5.10

5.78

kwpm ðFC rwpr =FC rwmr Þ

r

2.20

2.13

2.06

1.99

1.92

1.85

1.78

Front ridge angle Force reduction factor (kf )

– –

90° 0.65

120° 0.80

150° 0.95

180° 1.00

– –

– –

Vee bottom angle Force reduction factor (kv )

60° 0.50

90° 0.65

120° 0.80

150° 0.90

180° 1.00

– –

– –

kwp2 ðFNrwp =FN rsp Þ

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

Table 3 c The ratio knc (FN cs =FC cs ) of conical picks in different rocks (Bilgin et al., 2006).

rc (MPa)

Rock name

rt (MPa)

c

FNc

32 47 46 33 41 58 38 30 82 29 33 114 174 87 58 121 10 11 27 14 19 6

3.7 4.5 3.7 3.4 5.7 5.5 5.7 2.2 5.5 4.0 3.0 6.6 11.6 8.3 5.3 7.8 0.9 1.2 2.6 1.5 2.3 0.2

FN c

c

FN c

c

FNc

knc ( FC csmr )

knc ( FC cspu )

knc ( FC cspr )

0.759 0.790 0.870 0.698 1.069 1.106 1.130 1.541 1.133 0.767 0.639 1.027 1.292 1.045 1.257 1.775 0.667 0.676 0.690 0.682 0.645 0.583

0.675 0.731 0.784 0.65 0.951 1.040 1.102 1.320 1.018 0.758 0.652 1.011 1.231 0.852 1.143 1.488 0.571 0.667 0.708 0.667 0.609 0.545

0.705 0.692 0.769 0.679 1.004 0.890 0.994 1.196 1.066 0.715 0.585 0.859 1.117 0.882 1.055 1.260 0.590 0.618 0.663 0.682 0.571 0.557

0.613 0.680 0.719 0.65 0.951 0.874 0.916 0.934 0.848 0.685 0.652 0.786 0.928 0.712 0.941 1.109 0.497 0.593 0.635 0.556 0.528 0.477

smu

High-grade chromite Medium-grade chromite Low-grade chromite Copper ore (yellow) Copper ore (black) Harsburgite Serpantinite Trona (outlier) Anhydrite Selestite Jips Sandstone-1 Sandstone-2 Sandstone-3 Siltstone Limestone Tuff 1 Tuff 2 Tuff 3 Tuff 4 Tuff5 Tuff 6

c

knc ( FC csmu )

smr

spu

spr

FC csmu and FN csmu are mean cutting and normal forces of sharp conical picks under unrelieved cutting mode respectively; FC csmr and FNcsmr are mean cutting and normal forces of sharp conical picks under relieved cutting mode respectively; FC cspu and FNcspu are peak cutting and normal forces of sharp conical picks under unrelieved cutting mode, respectively; FC cspr and FNcspr are peak cutting and normal forces of sharp conical picks under relieved cutting mode respectively.

(2) The radial pick The previous experimental studies indicated when using sharp r radial picks, the value of knc was around 0.5 in majority of the rocks, however it went up to 1.2 in some rock cutting tests (Bilgin et al., 2012). Practical application by Bilgin et al. (2012) showed that the upper limit solution was quite close to the measured values based on their modified model of normal force. Regression analysis using the test data of Bilgin (1977) shows that there are strong linear relationships between normal forces and cutting forces of sharp radial picks as shown in Fig. 7, and the ratio r knc changes from 0.569 to 1.397 for peak forces (Fig. 7a), from 0.760 to 1.890 for mean forces (Fig. 7b) in four different rocks having the compressive strength values varying from 112.9 to 183.6 MPa. 5. Empirical models of cutting forces using multiple linear regression analysis The purpose of multiple linear regression analysis is to explain the relations between the independent variables and a dependent variable with a straight-line fit to the data. Assuming that there are p independent variables x1 ; x2 ; . . . ; xp , the general form of the linear multiple regression equation can be expressed in Eq. (17) (Chatterjee and Hadi, 2013).

yi ¼ b0 þ b1 xi1 þ b2 xi2 þ    þ bi xip

ði ¼ 1; 2; . . . ; pÞ

ð17Þ

where yi is a dependent variable, x1 ; x2 ; . . . ; xp are independent variables, b0 is a constant, and bi ð1 6 i 6 pÞ is the regression coefficient. According to the structure of Eq. (17), Eq. (14) can be expressed as in Eq. (18) by a simple transformation.

(

FC cs ¼ ac1 r0c þ ac2 r0t þ ac3 c0 þ Rc1

FC rs ¼ ar1 r00c þ ar2 r00t þ ar3 c00 þ Rr1

c

Fig. 6. Relation between knc (FN cs =FC cs ) and

rc ðrt Þ (Bilgin et al., 2006).

ð18Þ

where r0c , r0t , c0 , r00c , r00t and c00 are new independent variables. It should be noted that there is usually a certain degree of multicollinearity in regression model due to the nature of

Table 4 Forces prediction models and relevant modification coefficients. Pick type

Conical pick

Tool force

Unrelieved cutting mode and sharp state of picks

Relieved cutting mode and worn state of picks

Cutting force

Cutting force

Normal force

Normal force

Peak force c

FC cspu

¼ ð1:0999rc þ 15:7017rt Þ

sin2 ½ð/=2þcÞ=2þu=3 2 h cos½ð/=2þcÞ=2þu=3

FN cspu ¼ kncspu FC cspu ( c kncspu ðFN cspu =FC cspu Þ c

þ 4562

ðR2 ¼ 0:819Þ

¼

kncspu ¼ 0:557  1:26 ( ðR2 ¼ 0:667Þ 0:336r0:234 c

c

c

c

c

c

FN cspr ¼ kncspr kr FC cspu ; FN cwpr ¼ kncspr kwp kr FC cspu ( c kncspr ðFN cspr =FC cspr Þ

c c c FC cspr ¼ krp FC cspu ; FC cwpr ¼ kwp krp FC cspu c c c kr ðFC sr =FC su Þ ¼ 0:779 c kwp ðFC cwp =FC csp Þ ¼ 1 ðnew picksÞ

c

¼

0:560e0:076rt ðR2 ¼ 0:704Þ

kncspr ¼ 0:447  1:109 ( ðR2 ¼ 0:688Þ 0:316r0:225 c 0:524e0:069rt ðR2 ¼ 0:653Þ

c

FC csmu

¼ ð0:3955rc þ 5:6468rt Þ

sin2 ½ð/=2þcÞ=2þu=3 2 h cos½ð/=2þcÞ=2þu=3

c

þ 1581

2

ðR ¼ 0:852Þ

Radial pick

FN c ¼ kncsmu FC csmu ( smuc kncsmu ðFNcsmu =FC csmu Þ

¼

kncsmu ¼ 0:583  1:775 ( ðR2 ¼ 0:687Þ 0:313r0:285 c

c

c

c

c

c

c

c

FN c ¼ kncsmr kr FC csmu ; FNcwmr ¼ kncsmr kwm kr FC csmu ( smr c kncsmr ðFNcsmr =FC csmr Þ

c

FC csmr ¼ kr FC csmu ; FC cwmr ¼ kwm kr FC csmu c kwm ðFC cwm =FC csm Þ ¼ 1 ðnew picksÞ

c

¼

0:585e0:092rt ðR2 ¼ 0:711Þ

kncsmr ¼ 0:545  1:488 ( ðR2 ¼ 0:698Þ 0:311r0:268 c 0:561e0:085rt ðR2 ¼ 0:705Þ

Peak force

FC rspu ¼ ð0:059rc þ 3:93cÞ sinðub=4Þdh sinðp=2bÞþcosðp=2bÞ

þ 2172

¼

r knc FC rspu

ðR2 ¼ 0:816Þ

r

r

FC rspr ¼ krp FC rspu ; FC rwpr ¼ kf kv kwp kr FC rspu

FN rspr ¼ knc kr FC rspu

r kr ðFC rsr =FC rsu Þ ¼ 0:9; kf ; kv ðTable r kwp ðFC rwpu =FC rspu Þ ¼ 0:5605WF þ

r r r r FN rwpr ¼ kf kv kwp kwp2 knc kr FC rspu r r r kwp2 ðFN wp =FN sp Þ ¼ 1:5011WF

r

FNrspu

r

r

2Þ 0:9922

þ 1:0795

Mean force r

r

r

r

r

r

r

r r

FC rsmu ¼ kspm FC rspu

FN rsmu ¼ knc kspm FC rspu

FC rsmr ¼ kspm kr FC rspu ; FC rwmr ¼ kwpm kf kv kwp kr FC rspu

FN rsmr ¼ kspm kr knc FC rspu

r kspm ðFC rsmu =FC rspu Þ

r knc ðFN rsmu =FC rsmu Þ

r kwpm ðFC rwpr =FC rwmr Þ

FN rwmr

r

¼ 0:5

r

¼ 0:5  1:2

ðTable 2Þ

¼

r r r r r kwpm kf kv kwp kwp2 kr knc FC rspu

Note: FC cspu and FC rspu are peak cutting forces acting on sharp concial and radial picks under unrelieved cutting mode respectively (in unit of N); FN cspu and FN rspu are peak normal forces acting on sharp concial and radial picks under unrelieved cutting mode respectively (in unit of N); FC csmu and FC rsmu are mean cutting forces acting on sharp concial and radial picks under unrelieved cutting mode respectively (in unit of N); FNcsmu and FN rsmu are mean normal forces acting on sharp concial and radial picks under unrelieved cutting mode respectively (in unit of N); FC cspr and FC rspr are peak cutting forces acting on sharp conical and radial picks under relieved cutting mode respectively (in unit of N); FC cwpr and FC rwpr are peak cutting forces acting on conical and radial picks in worn state under relieved cutting mode respectively (in unit of N); FNcspr and FNrspr are peak normal cutting forces acting on sharp conical and radial picks under relieved cutting mode respectively (in unit of N); FN cwpr and FN rwpr are peak normal forces acting on conical and radial picks in worn state under relieved cutting mode respectively (in unit of N); FC csmr and FC rsmr are mean cutting forces acting on sharp conical and radial picks under relieved cutting mode respectively (in unit of N); FC cwmr and FC rwmr are mean cutting forces acting on conical and radial picks in worn state under relieved cutting mode respectively; FN csmr and FNrsmr are mean normal cutting forces acting on sharp conical and radial picks under relieved cutting mode respectively (in unit of N); FNcwmr and FN rwmr are mean normal cutting forces acting on conical and radial picks in worn state under relieved cutting mode respectively (in unit of N). rc , rt and c are compressive, tensile and shear strength of rocks respectively (in unit of MPa); / and c are the cone and attack angle of conical c picks respectively; u is the friction angle between the pick and rock; b is the rake angle; h is depth of cut (in unit of mm); d is the width of radial pick (in unit of mm). kncspu is the ratio of the peak normal to cutting force of sharp c c concial picks under unrelieved cutting mode; kncsmu is the ratio of the mean normal to cutting force of sharp concial picks under unrelieved cutting mode; kncspr is the ratio of the peak normal to cutting force of sharp concial picks c r under relieved cutting mode; kncsmr is the ratio of the mean normal to cutting force of sharp concial picks under relieved cutting mode; knc is the ratio of the mean normal to cutting force of sharp radial picks under unrelieved r r c r cutting mode; kspm is the ratio of the mean to peak cutting force of sharp radial picks under unrelieved cutting mode; kwpm is the ratio of the mean to peak cutting force of worn radial pick under unrelieved cutting mode; kr and kr c r are the ratio of cutting force under relieved cutting mode to cutting force under unrelieved cutting mode for conical and radial picks respectively; kwp and kwp are the ratio of peak cutting force of worn picks to peak cutting force of c sharp ones under unrelieved cutting mode for conical and radial picks respectively; kwm is the ratio of mean cutting force of worn concial picks to mean cutting force of sharp ones under unrelieved cutting mode; kf ; kv are r coefficients used for taking into account the effect of complex-shapes of radial pick; kwp2 is the ratio of peak normal force of worn radial picks to peak normal force of sharp ones.

X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

Mean force

83

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

instead of the original variables. Ridge regression, an improved method of least squares, is a biased estimation of regression method. Although some information is lost, and prediction accuracy is relatively reduced using the principal component and ridge regression methods, many reasonable coefficients and equations can be obtained. The principal component and ridge regression all can be realized in SPSS software. 5.1. The cutting force models of sharp conical picks in unrelieved cutting mode using the principal component regression

Fig. 7. Relation between normal and cutting forces of sharp radial picks in unrelieved cutting mode (Bilgin, 1977).

observational data. Multicollinearity occurs in regression models when independent variables are highly correlated and it can lead to the unstable prediction model and the unrealistic regression coefficients. The principal component regression and ridge regression are powerful means to avoid the serious multicollinearity (Chatterjee and Hadi, 2013). The essence of the principal component regression is to reduce the dimension of samples, and regression analysis is then carried out using the principal components

The raw data used for regression analysis are from literatures published by Copur et al. (2003), Goktan and Gunes (2005) and Bilgin et al. (2006) for conical picks with a total of 55 samples under unrelieved cutting modes in Appendix A (u is set to 30° when it is unknown.). It deserves to note that all test data of above literatures are from the same research team in ITU using the same test platform. In this study, repeated data of Copur et al. (2003), Goktan and Gunes (2005) and Bilgin et al. (2006) was removed in regression. The compressive and tensile strengths of rocks were used for regression analysis. First of all, multiple linear regression analysis was carried out directly with the test data in SPSS software, and the variances inflation factor (VIF) of the variables are respectively: r0c (11.300), r0t (11.300). When VIF is equal to 1 it indicates that there is not multicollinearity, and value of VIF is more than 10 indicates that multicollinearity is very serious in regression analysis model (Gunes et al., 2007). Therefore, the result indicates that multicollinearity is a serious problem that must be solved. As seen in Fig. 8, the correlation coefficient (r) and determination coefficient (R2) are all more than 0.9, indicating that there is a strong linear relationship between r0c and r0t . The gap between long axis and short axis of the ellipse where all data fall in is very big, thus it is reasonable to reduce the dimension of variables (Zhang and Dong, 2004). In practice, only one principal component of r0c and r0t was extracted as new variable used for linear regression analysis. The characteristic value and the accumulation contribution rate of the principal component are respectively 1.955 and 97.736%, indicating that the principal component contains most characteristics of original variables. Based on the above analysis and calculation, the cutting force models of conical picks are developed in Eq. (19) using the principal component regression.

FC cspu ¼ ð1:0999rc þ 15:7017rt Þ

sin ½ð/=2 þ cÞ=2 þ u=3 2 h þ 4562 cos½ð/=2 þ cÞ=2 þ u=3 2

ðR2 ¼ 0:819Þ FC csmu ¼ ð0:3955rc þ 5:6468rt Þ ðR2 ¼ 0:852Þ FC cspu

ð19aÞ sin ½ð/=2 þ cÞ=2 þ u=3 2 h þ 1581 cos½ð/=2 þ cÞ=2 þ u=3 2

ð19bÞ

FC csmu

where and are peak and mean cutting forces of sharp conical picks under unrelieved cutting mode respectively in unit of N. The uniaxial compressive strength of rock (rc ) is in unit of MPa, the tensile strength of rock (rt ) is in unit of MPa, and the cutting depth (h) is in unit of mm. 5.2. The cutting force model of sharp radial picks in unrelieved cutting mode using ridge regression analysis

Fig. 8. Scatter diagram of

r0c and r0t .

The raw data are from doctoral dissertation of Bilgin (1977) and Gunes et al. (2007) for radial picks with a total of 55 samples. When multiple linear regression analysis was carried out directly with the tests data, the VIF of the variables are respectively: r00c (9.842), r00t (8.335), c00 (4.484), indicating that multicollinearity is

X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

85

where FC rsmu is the mean cutting force of sharp radial picks in unrelieved cutting mode. 6. The performance check of new models Based on the tool forces prediction models of drag-typed picks in ideal state developed in the previous section, some modified models were also developed by introducing into relevant modification coefficients. The models and relevant coefficients are summarized in Table 4. 6.1. The performance check of the force models of conical picks

Fig. 9. Ridge trace.

not a serious problem. However, the negative coefficient of r00c is unrealistic. Ridge regression method can be used to solve this problem. The ridge trace under the first trial is shown in Fig. 9. As seen that the standardized coefficient of r00t is not sensitive with increase of ridge parameter k, and therefore it is omitted from the model due to lack of statistical significance. After many trials, when ridge parameter k is equal to 0.1, a reasonable result can be obtained as below in Eq. (20a).

FC rsp ¼ ð0:059rc þ 3:93cÞ ðR2 ¼ 0:816Þ

sinðu  b=4Þ dh þ 2172 sinðp=2  bÞ þ cosðp=2  bÞ

6.1.1. The performance check of the cutting force models Goktan and Gunes (2005) stated that the theoretical values of cutting forces calculated by Eqs. (1) and (2a) are significantly smaller than measured ones, while the cutting force values calculate by Eqs. (2c) and (2d) fitted statistically well with measured ones based on 14 sets measured values (The correlation coefficients r were more than 0.9 between measured and calculated cutting force statistically significant at the 99% confidence level). A statistical analysis carried out by Bilgin et al. (2006) using t-test showed that the measured mean cutting force values were not significantly different from the theoretical values calculated by Eq. (2a) at 5 mm depth of cut, but it is significantly different between the measured and theoretical values calculated by Eqs. (1) and (2b).

ð20aÞ

where the uniaxial compressive strength of rock (rc ) is in unit of MPa, the shear strength of rock (c) is in unit of MPa, the width of pick (d) is in unit of mm, the cutting depth (h) is in unit of mm, and the peak cutting force FC rsp is in unit of N. In order to estimate mean cutting forces of sharp radial picks in r unrelieved cutting mode, the ratio kspm ðFC rsmu =FC rspu Þ of mean cutting forces to peak cutting forces can be used for modeling mean r forces as shown in Eq. (20b). The ratio kspm ðFC rsmu =FC rspu Þ is equal to 0.5 approximately for all rocks and for each level of experimental variables in the experiments of Bilgin (1977) as presented in Fig. 10. However, when picks are in worn state, the ratio r kwpm ðFC rwp =FC rwm Þ decreases with increasing of width of wear flat (WF) (Bilgin et al., 2012). r

FC rsmu ¼ kspm FC rsp

ð20bÞ

Fig. 10. Relation between mean and peak cutting forces of sharp radial picks in unrelieved cutting mode.

Fig. 11. Relation between measured and calculated cutting forces in unrelieved cutting mode by different models of sharp conical picks.

86

X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

better the model performs (Gunes et al., 2007). In this regard, the performance of this work models is also very superior.

In order to check and compare the prediction performance of the cutting force models of this work (Eq. (19)) with Goktan and Gunes’ (2005) models (Eqs. (2c) and (2d)), regression curves were plotted to show the relations between calculated and measured cutting force values for each model in Fig. 11 based on the data used for regression analysis, and relevant regression equations and statistical parameters are presented in Table 5. As seen that, on the one hand, data points calculated by the models of this work (Eqs. (19a) and (19b)) are all evenly distributed over, above and below the line y = x without any outlying data points. On the other hand, the determination coefficients R2 statistics indicate that the models as fitted explains respectively 81.9% and 85.2% of the variability in calculated peak and mean cutting forces. The correlation coefficients r yielded by the peak and mean cutting force models are respectively 0.905 and 0.923, indicating strong relationships between variables. Since the p-values are all less than 0.05, indicating that the relationships between measured and calculated cutting force are all statistically significant at the 95% confidence level. A statistical analysis by t-test at the 95% confidence level carried out to check whether the measured force values are significantly different from estimated ones. As can be seen from Table 5, the p-values of t are all greater than 0.05, indicating that there is no significant difference between estimated force values by the model given in Eq. (19) and measured ones. Therefore, it is concluded that the models of this work Eqs. (19a) and (19b) are statistically valid. The statistical parameters of Goktan and Gunes’ (2005) semiempirical models presented in Table 5 show that they can be also used to predict the cutting force under unrelieved cutting mode satisfactorily. It is worth noting that the compressive strength of rocks plays an important role in the models of this work, but it is ignored in Goktan and Gunes’ (2005) semi-empirical models. For further compared the performance of the models, the variance account for (VAF) and the root mean square error (RMSE) were calculated by Eq. (21) summarized in Table 5. In general, the higher the VAF, the better the model performs, the lower the RMSE, the

 VAF ¼

RMSE ¼

1

 ^i Þ varðyi  y  100% varðyi Þ

ð21aÞ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN ^ i Þ2 ðyi  y i N

ð21bÞ

^i is the where var symbolize the variance, yi is the measured value, y calculated value, and N is the number of samples. It is well known that the relation between the cutting force and cutting depth is also very important since cutting depth is changing over the time in rock cutting process by using roadheaders or shearers. The cutting force acting on conical picks is proportional to the square of cutting depth in most of theoretical models such as Eqs. (1) and (2). However, Bilgin et al. (2006) suggested that the cutting force acting on conical picks was proportional to the cutting depth supported by experimental data and previous studies (Eq. (4)). Authors of this paper suggested that this fact might originate from that the range of cutting depth is too narrow in general rock cutting tests (3–10 mm), all the test points can fit very well with line or parabola in this range. The curves of relation between the cutting force and cutting depth using Eqs. (19), (2c), and (2d) are plotted in Fig. 12, where relevant parameters are from literature of Bilgin et al. (2006). As can be seen that the intersections of curves generally fall at the range of cutting depth from 5 to 10 mm with different input parameters, and the reason is that the most test points of the cutting force were measured at this range of the cutting depth. It has been known that relieved cutting is a normal state using c drag-typed picks in mechanical excavation. The factor kr was used to modify Eq. (19) to calculate the tool forces under relieved cutting mode in this work as presented in Table 4. Based on the test data of Copur et al. (2003) (Table 1), the relations between measured and calculated cutting forces in relieved cutting mode by the model of this work were plotted in Fig. 13, and the statistical

Table 5 Relation between measured and calculated forces by different models and related statistical parameters at a ¼ 0:05 level. Picks Conical pick

Radial pick

Figure

Force

Regression equation

R2

r

F-value

p-value

t-value

p-value

VAF (%)

Fig. 11a

FC cspuT FC cspuG FNcspuT

y = x  0.001

0.819

0.905

239.429

0.000

0.000

1.000

81.88

4.77

y = 0.812x + 3.905

0.795

0.892

206.051

0.000

0.706

0.482

75.26

5.80

y = 0.716x + 4.985

0.837

0.915

77.225

0.000

0.155

0.878

70.62

5.70

Fig. 11b

FC csmuT FC csmuG FNcsmuT

y = 1.020x  0.099 y = 0.890x + 1.286 y = 0.741x + 2.285

0.852 0.822 0.775

0.923 0.907 0.869

305.029 244.515 51.574

0.000 0.000 0.000

0.001 1.071 0.325

0.999 0.287 0.747

85.16 80.94 68.02

1.55 1.95 2.95

Fig. 13a

FC csprT

y = 0.898x + 4.190

0.731

0.855

40.633

0.000

1.034

0.309

72.11

5.14

FNcsprT

y = 0.702x + 6.240

0.587

0.766

21.347

0.000

1.014

0.318

48.18

6.67

Fig. 13b

FC csmrT FNcsmrT

y = 0.876x + 1.103 y = 0.725x + 2.215

0.696 0.567

0.834 0.753

34.363 19.664

0.000 0.000

0.534 0.676

0.597 0.504

68.21 48.60

1.72 3.03

Fig. 14

FC rspuT

y = 1.092x  1.035

0.825

0.923

255.592

0.000

0.02

0.998

90.75

1.43

FC rspuN

y = 0.999x + 0.004

0.900

0.949

476.563

0.000

0.005

0.996

89.99

1.49

FC rspuT

y = 0.860x + 0.954

0.755

0.869

302.019

0.000

0.963

0.528

73.49

3.78

FC rspuN

y = 1.064x  0.934

0.708

0.841

237.141

0.000

0.251

0.802

70.50

3.94

FNrspuT

y = 0.711x + 1.942

0.684

0.827

212.198

0.000

0.859

0.391

57.14

3.55

FC rsmuT FNrsmuT

y = 0.925x + 0.083 y = 0.745x + 1.106

0.789 0.618

0.888 0.786

365.391 158.306

0.000 0.000

0.656 0.830

0.512 0.408

78.32 54.60

1.81 2.72

Fig. 16a

Fig. 16b

RMSE

FC cspuT and FC cspuG are peak cutting forces of sharp conical picks under unrelieved cutting mode calculated by the models of this work and Goktan and Gunes (2005) respectively; FC csmuT and FC csmuG are mean cutting forces of sharp conical picks under unrelieved cutting mode calculated by the model of this work and Goktan and Gunes (2005) respectively; FN cspuT and FNcsmuT are peak and mean normal forces of sharp conical picks under unrelieved cutting mode calculated by the models of this work; FC csprT and FC csmrT are peak and mean cutting forces of sharp conical picks under relieved cutting mode calculated by the models of this work; FNcsprT and FNcsmrT peak and mean normal forces of sharp conical picks under relieved cutting mode calculated by the models of this work; FC rspuT and FC rspuN are peak cutting forces of sharp radial picks under unrelieved cutting mode calculated by the models of this work and Gunes et al. (2007) respectively; FNrspuT is the normal force of sharp radial picks under unrelieved cutting mode calculated by the model of this work; FC rsmuT and FNrsmuT are mean cutting and normal forces of sharp radial picks under unrelieved cutting mode calculated by the models of this work.

X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

87

Fig. 13. Relation between measured and calculated tool forces of sharp conical picks in relieved cutting modes by the models of this work.

Fig. 12. Relation between the cutting force and cutting depth in unrelieved cutting mode by different models of sharp conical picks.

parameters were presented in Table 5. F and p-values (F = 40.633, 34.395; p = 0.000, 0.000 for peak and mean cutting forces respectively) indicate that relationships between measured and calculated cutting forces are all statistically significant at the 95% confidence level. The performance of cutting force models under relieved cutting mode is slightly weaker than that under unrelieved cutting mode judging from values of R2 and r (R2 = 0.731, 0.696; r = 0.855, 0.834 for peak and mean cutting forces respectively). The results of t-test (Table 5) show that there is no significance between calculated cutting force values by the models of this work and measured ones under relieved cutting mode. Therefore, the modified cutting force models are also statistical valid. 6.1.2. The performance check of the normal force models In order to check the performance of the normal force models of conical picks, relations between measured and calculated normal force values were plotted in Fig. 11 for unrelieved cutting mode, and in Fig. 13 for relieved cutting mode based on the test data of Copur et al. (2003) (Table 1). All statistical parameters are prec sented in Table 5. Note that the modified factor knc was predicated based on tensile strength of rocks as presented in Table 4. The Fvalues are all more than 19, and p-values are all equal to 0.000,

indicating that relationships between measured and calculated normal forces are all statistically significant at the 95% confidence level. The values of R2 and r (R2 = 0.837, 0.775; r = 0.915, 0.869 for peak and mean normal forces, respectively, under unrelieved cutting mode) indicate that the normal force models under unrelieved cutting model can be used to predict the forces values well, and their performance is better than that of normal force models under relieved cutting mode according to the values of R2 and r (R2 = 0.587, 0.567; r = 0.766, 0.753 for peak and mean normal forces respectively under relieved cutting mode). The analysis above also shows that the performance of modified models after introducing some factors declines from a statistical sense. In other words, the result is dramatically sensitive to the modified factors. This is the obvious shortcoming by using indirect method to predict force values, thus it should be very careful to select factor values in practical application. However, the results of t-test indicate that there is no significance difference between calculated and measured forces, thus the models of this work are accepted as statistically meaningful and reliable. It is suggested that they can be used for preliminarily estimation tool forces of conical picks in practical application. 6.2. The performance check of the force models of radial picks 6.2.1. The performance check of the cutting force models Gunes et al. (2007) have compared the prediction performance of Eqs. (3a), (3c) and (6), and the result indicated that the

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

performance of Eq. (3c) was slight better compared to Eq. (3a). However, Eq. (6) was highly superior to others. The model of Nishimatsu (1972) (Eq. (3b)) was omitted due to highly unrealistic force values prediction. The relations between measured and calculated cutting force values by the models of this work (Eq. (20a)) and Gunes et al. (2007) (Eq. (6)) were shown in Fig. 14 based on the data in regression analysis. The values of variance account for (VAF) and root mean square error (RMSE) were summarized in Table 5. Table 5 indicates that there is not obvious difference between the models performance of Eqs. (6) and (20a) in term of numerical magnitude. As seen in Fig. 14 that data points calculated by the model of this work (Eq. (20a)) are all evenly distributed around the regression line without any outlying data points. The correlation coefficient (r) of 0.910 indicates that there is a strong relationship between variables. The determination coefficient (R2) shows that the model as fitted explains 82.5% of the variability in calculated cutting force values. Since the p-value is less than 0.05, indicating that the relationship between measured and calculated cutting force by Eq. (20a) is statistically significant of 95% confidence level. Therefore, it is concluded that the model of this work (Eq. (20a)) are statistically valid. The statistical parameters of Eq. (6) (R2 = 0.900, r = 0.949, and p = 0.000) show that it also can be used to predict the cutting force satisfactorily for existing test data, and its performance is slight superior to the model of this work in this regard. However, the shortcomings of Eq. (6) are also very obvious. On the one hand, it is not reasonable that the regression coefficient of rock compressive strength is negative in Eq. (6). On the other hand, the cutting force values calculated by Eq. (6) are negative when cutting depth and rock strength are relatively small as shown in Fig. 15a due to several negative coefficients and the negative constant in the model, but when rock strengths are relatively big, the negative cutting force values are not appear as shown in Fig. 15b. This is the advantage of ridge regression, namely, more reasonable coefficients can be obtained using ridge regression, while the prediction accuracy decreased relatively within the acceptable range. This result also indicates that using the form of trigonometric functions to describe the effects of relevant angels on the cutting force is much better than that by a simple linear combination of angles. In order to further check the performance of models of tool forces acting on radial picks, based on the test data of Bilgin (1977) carried on four rocks having different strength with a total of 100 sets under unrelieved cutting mode, the relations between measured and calculated cutting forces were plotted in Fig. 16, and the statistical parameters were summarized in Table 5.

Fig. 15. Relation between the cutting force and cutting depth in unrelieved cutting mode by different models of sharp radial picks.

Fig. 16a shows that the cutting force values calculated by Eq. (6) are negative at a relatively smaller depth of cut (h = 1.5 mm). The model of this work (Eq. (20a)) can predict that satisfactorily according to the values of statistical parameters (R2 = 0.755, 0.789; r = 0.869, 0.888; p = 0.000, 0.000 for peak and mean cutting forces respectively) as shown in Fig. 16. 6.2.2. The performance check of the normal force models Fig. 16 and Table 5 show that the normal forces model of this work (Eq. (20b)) is statistically valid (R2 = 0.684, 0.827; r = 0.618, 0.789; p = 0.000, 0.000) for peak and mean normal forces respecr tively). It is noted that the factors knc used for Eq. (20b) are presented in Fig. 7. Also, it is worth noting that its performance is slightly worse than that of cutting force models. The results of ttest indicate that there is no significance difference between that calculated by the model of this work and measured. Therefore, the normal force model under unrelieved cutting mode is accepted as statistically meaningful and reliable.

Fig. 14. Relation between measured and calculated peak cutting forces in unrelieved cutting mode by different models of sharp radial picks using the data of regression analysis.

6.2.3. Validation of the modified models for practical application In practical application, tool wear and relieved cutting should be also considered in prediction models. Bilgin et al. (2012) substituted Eq. (7) into Eq. (22) to estimate deterministically the torque and thrust requirements of the TBM equipped radial picks used for excavating the Beykoz tunnel in Istanbul.

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

TBM (3.175 m for the TBM used in Beykoz tunnel). T F is the torque of TBM to overcome the friction and to turn the TBM cutterhead in idle position (26 kN m for the TBM used in Beykoz tunnel), and FT F is the thrust requirement of TBM to overcome the friction (46 kN for the TBM used in Beykoz tunnel). The thrust and torque of the TBM used in Beykoz tunnel were estimated in rock formations of different strength presented in Table 6 using the modified models of this work (Table 4) by taking r r r kwp = 2.67, kwp2 = 5.1, kwpm = 1.85 for 3 mm of wear flat, kf = 0.95 for 150° of front ridge angle, kv = 0.90 for 150° of vee-bottom angle, r r krp = 0.9, b = 10°, u = 30°, knc = 0.5 and 1.2 for lower and upper limit thrust force requirement of the TBM respectively. The shearer strength of rock c is evaluated by Eq. (23) (Nishimatsu, 1972). It is worth noting that the values of all modified factors are as the same as those of Bilgin et al. (2012). As can be seen from Table 6, the mean predicted torque value is about 2.9% lower than the realized mean torque value of the TBM by the model of this work. The result shows that the estimated mean torque requirement of the TBM is quite close to the realized mean torque values of the TBM. A hypothesis testing carried out by t-test at the 95% confidence level shows that there is no significant difference between the torques predicted by the model of this work and measured ones.

rc rt

c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rt ðrc  3rt Þ

Fig. 16. Relation between measured and calculated tool forces of sharp radial picks under unrelieved cutting mode by the models of this work using the data of Bilgin (1977).

(

T ¼ Nc FC rwmr D4 þ T F

ð22Þ

FT T ¼ Nc FNrwmr þ FT F

where T and FT T are the torque and thrust requirement of TBM, respectively. N c is the number of tools on the cutterhead of TBM (23 for the TBM used in Beykoz tunnel). D is the diameter of

ð23Þ

The results presented in Table 6 also indicate that the predicted r mean thrust force values for the lower limit value of knc (0.5) is about 41% lower than the measured ones. The predicted mean r thrust force values for upper limit value of knc (1.2) is about 36% higher than the measured ones. However, the mean value of predicted lower and upper limit thrust force values by the model of this work is quite close to the measured ones (only 2.6% lower than measured mean values). In this regard, further studies are needed to precisely predict the ratio mean normal force to mean cutting force. The results of t-test indicate that the lower and upper limit thrust estimations by the models of this work are not statistically reliable. However, the result of hypothesis testing indicates that there is no significant difference between mean values of lower and upper limit thrust estimations and measured ones. It’s also worth noting that the estimated values of data of No. 6, 7 and 8 are dramatically lower than measured ones by the modified model based on Evans (1962). This is due to that the tensile strength of rocks of No. 6, 7 and 8 is relatively lower, and the model of Evans (1962) only includes one rock strength parameter which

Table 6 Performance of the TBM equipped with radial picks in different strength rock formations and comparison of the measured and estimated thrust and torque values (Bilgin et al., 2012). No. 1 2 3 4 5 6 7 8 9 10 Mean Std

rc

rt

(MPa)

(MPa)

c (MPa)

T MEAS (kN m)

T ESTB (kN m)

T ESTT (kN m)

FT TMEAS (kN m)

FT TESTLB (kN m)

FT TESTLT (kN m)

FT TESTUB (kN m)

FT TESTUT (kN m)

FT TESTMT (kN m)

73.5 54.8 61.6 50.7 33.3 21.3 55.9 52 52.7 73.1 52.9 16:0

7.5 4.9 6.3 4.7 2.9 2 2.5 2.5 4.9 7.3 4.6 2:0

14.1 9.6 11.8 9.08 5.7 3.9 6.4 6.2 9.5 13.8 9.0 3:5

242.0 208.4 216.7 228.7 221.0 207.7 207.7 210.0 204.9 194.9 219 27

300 205 256 197 132 99 117 117 205 292 192 74

311 235 273 226 169 136 184 180 233 307 225.4 59

1318 1245 1262 1318 870 1043 1101 1233 1051 1221 1166 145

925 620 784 597 386 280 339 339 620 902 579 238

963 718 838 690 505 400 554 542 711 948 687 189

2156 1424 1818 1368 862 609 749 749 1424 2099 1326 572

2246 1659 1947 1591 1147 896 1266 1236 1641 2210 1584 454

1605 1189 1393 1141 826 648 910 889 1176 1579 1136 322

T MEAS : measured torque requirement of the TBM, T ESTB : estimated torque requirement of the TBM by Bilgin et al. (2012), T ESTT : estimated torque requirement of the TBM in this work, FT TMEAS : measured thrust force requirement of the TBM, FT TESTLB : estimated lower limit thrust force requirement of the TBM by Bilgin et al. (2012), FT TESTLT : estimated lower limit thrust force requirement of the TBM in this work, FT TESTUB : estimated upper limit thrust force requirement of the TBM by Bilgin et al. (2012), FT TESTUT : estimated upper limit thrust force requirement of the TBM in this work, and FT TESTMT : mean value of FT TESTLT and FT TESTUT ; std is standard deviation.

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

is the tensile strength. The model of this work includes compressive and shearer strength of rock, thus the prediction performance of the model in this work is improved to a certain extent compared to the modified model based on Evans (1962). On the other hand, the modified model is developed based on Eq. (20), which is a regression equation using the test data in high strength rocks with compressive strength from 84.9 to 183.6 MPa. However, practical application indicated they are also reliable for rocks of medium strength. 7. Conclusions The tool forces acting on a single drag-typed pick are basic parameters in designing working units of the excavation machines and evaluating their performance. The tool forces prediction using empirical models is preferred in this study with the advantage of convenient use. According to the structure of some theoretical models of the cutting force, general models were proposed considering the dimension and physical significance of each parameter. Relevant modified factors were discussed in detail for developed modified force models to estimate tool forces considering tool wear, complex shapes, and relieved cutting. The peak and mean cutting force models of conical picks and the peak cutting force model of radial picks under unrelieved cutting mode were developed by conducting principle component and ridge regression analyses over the raw data obtained from linear

full-scale cutting test. With the new model, serious multicollinearity in multiple linear regression models can be avoided, and the induced regression coefficients and equations are more reasonable physically. The performance of proposed models was evaluated by using statistical correlation and hypothesis testing, and good agreement exists between the measured and predicted cutting force values under relieved cutting mode for sharp picks. The performance of models after introducing relevant modified factors would decrease to some extent. However, they are also statistical valid. Besides, it should be noted that the rock and ore samples used for regression analysis have the uniaxial compressive strength varying from 6 to 174 MPa, and therefore the models for conical picks have a very wide range of applications. Although the models of radial picks were developed based on the test data in rock formations with the uniaxial compressive strength varying from 84.9 to 183.6 MPa, practical application shows that they can also be used for estimating tool forces in medium rocks well. In addition, it is suggested that the models developed in this study can be used for preliminary estimation of torque requirements of roadheader cutting heads, shearer drums and other excavation machines equipped with drag-typed picks. Due to the limited number of samples, especially fewer samples of conical picks, the general applicability of the developed models need to be checked and updated in further studies. However, the general model and model development framework have a good guiding significance for future studies.

Table A1 Physical and mechanical properties of the rocks tested, cutting parameters and cutting forces of conical picks (Copur et al., 2003) (/ ¼ 80 ; c ¼ 55 ). Rock name

rc (MPa)

rt (MPa)

h (mm)

u (°)

FC cspuM (kN)

FC cspuG (kN)

FC cspuT (kN)

FC csmuM (kN)

FC csmuG (kN)

FC csmuT (kN)

Limestone Siltstone Sandston-1 Sandston-2 Claystone Claystone Claystone Claystone

121 58 114 174 58 58 58 58

7.8 5.3 6.6 11.6 5.6 5.6 5.6 5.6

3 3 3 3 3 5 7 9

30 28 26 30 30 30 30 30

11.60 7.33 8.95 9.02 3.74 8.79 10.77 16.58

3.50 2.30 2.78 5.21 2.52 6.99 13.69 22.64

7.58 6.26 7.09 8.99 6.34 9.56 14.39 20.82

3.87 3.07 3.83 4.02 1.17 2.95 3.17 5.24

1.17 0.77 0.93 1.74 0.84 2.33 4.56 7.55

2.6 2.19 2.42 3.12 2.14 3.33 5.1 7.47

FC cspuM is measured peak cutting force of sharp conical picks under unrelieved cutting mode, FC cspuG is the peak cutting force of sharp conical picks calculated by the model of Goktan and Gunes (2005), FC cspuT is the peak cutting force of sharp conical picks calculated by the model of this work, FC csmuM is measured mean cutting force under unrelieved cutting mode, FC csmuM is the mean cutting force calculated by the model of Goktan and Gunes (2005), FC csmuT is the mean cutting force of sharp conical picks calculated by the model of this work.

Table A2 Physical and mechanical properties of the rocks tested and forces of conical picks (Bilgin et al., 2006) (/ ¼ 80 ; c ¼ 55 ; h ¼ 9 mm). Rock name

rc (MPa)

rt (MPa)

u (°)

FC cspuM (kN)

FC cspuG (kN)

FC cspuT (kN)

FC csmuM (kN)

FC csmuG (kN)

FC csmuT (kN)

High-grade chromite Medium-grade chromite Low-grade chromite Copper ore (yellow) Copper ore (black) Harsburgite Serpantinite Trona Anhydrite Selestite Jips Sandstone-1 Sandstone-2 Sandstone-3 Siltstone Limestone Tuff 1 Tuff 2 Tuff 3 Tuff 4 Tuff 5 Tuff 6

32 47 46 33 41 58 38 30 82 29 33 114 174 87 58 121 10 11 27 14 19 6

3.7 4.5 3.7 3.4 5.7 5.5 5.7 2.2 5.5 4.0 3.0 6.6 11.6 8.3 5.3 7.8 0.9 1.2 2.6 1.5 2.3 0.2

27 27 29 38 32 25 28 30 30 30 30 26 30 30 28 30 32 29 30 27 30 32

14.53 25.96 15.92 14.77 25.30 26.37 19.75 12.11 15.97 8.89 6.34 28.93 47.14 15.60 31.36 32.19 3.94 11.60 7.71 7.154 7.203 2.136

14.23 17.31 14.71 15.7 23.82 20.47 22.29 8.89 22.23 16.17 12.13 24.98 46.89 33.55 20.73 31.53 3.76 4.77 10.51 5.77 9.3 0.84

14.06 17.03 16.01 15.53 19.47 19.37 18.17 11.78 23.48 14.7 13.49 27.54 44.62 28.79 19.8 31.96 7.32 7.8 12.1 8.51 10.65 5.61

5.19 9.12 6.50 4.99 8.90 9.04 6.96 4.12 5.09 3.36 3.31 9.72 16.52 6.42 8.26 11.93 1.58 3.70 2.69 2.43 2.93 1.02

4.74 5.77 4.9 5.24 7.94 6.82 7.43 2.96 7.41 5.39 4.04 8.33 15.63 11.18 6.91 10.51 1.25 1.59 3.5 1.92 3.1 0.28

4.98 6.08 5.7 5.52 6.97 6.94 6.5 4.14 8.45 5.22 4.77 9.94 16.23 10.41 7.1 11.57 2.5 2.68 4.26 2.94 3.73 1.87

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95 Table A3 Physical and mechanical properties of the rocks tested and cutting forces of conical picks (Bilgin et al., 2006) (/ ¼ 80 ; c ¼ 55 ; h ¼ 5 mm). Rock name

rc (MPa)

rt (MPa)

u (°)

FC cspuM (kN)

FC cspuG (kN)

FC cspuT (kN)

FC csmuM (kN)

FC csmuG (kN)

FC csmuT (kN)

High-grade chromite Medium-grade chromite Low-grade chromite Copper ore (yellow) Copper ore (black) Harsburgite Serpantinite Trona Anhydrite Selestite Jips Sandstone-1 Sandstone-2 Sandstone-3 Siltstone Limestone Tuff 1 Tuff 2 Tuff 3 Tuff 4 Tuff 5 Tuff 6

32 47 46 33 41 58 38 30 82 29 33 114 174 87 58 121 10 11 27 14 19 6

3.7 4.5 3.7 3.4 5.7 5.5 5.7 2.2 5.5 4.0 3.0 6.6 11.6 8.3 5.3 7.8 0.9 1.2 2.6 1.5 2.3 0.2

27 27 29 38 32 25 28 30 30 30 30 26 30 30 28 30 32 29 30 27 30 32

7.02 10.01 8.54 4.31 7.18 14.67 7.69 3.80 12.27 4.65 8.55 19.30 22.79 8.91 22.58 21.08 2.01 6.73 3.70 2.77 3.37 1.30

4.39 5.34 4.54 4.85 7.35 6.32 6.88 2.75 6.86 4.99 3.74 7.71 14.47 10.36 6.4 9.73 1.16 1.47 3.24 1.78 2.87 0.26

7.47 8.39 8.07 7.93 9.14 9.11 8.74 6.77 10.38 7.67 7.3 11.63 16.91 12.02 9.25 13 5.39 5.54 6.87 5.76 6.42 4.87

2.73 3.40 3.13 1.67 2.65 5.20 2.89 1.36 3.31 1.47 3.93 7.43 8.04 3.71 7.26 7.31 0.73 1.92 1.23 0.91 1.34 0.46

1.46 1.78 1.51 1.62 2.45 2.11 2.29 0.92 2.29 1.66 1.25 2.57 4.82 3.45 2.13 3.24 0.39 0.49 1.08 0.59 0.96 0.09

2.56 2.9 2.78 2.73 3.17 3.16 3.03 2.3 3.63 2.63 2.49 4.09 6.03 4.23 3.21 4.59 1.79 1.85 2.34 1.93 2.17 1.6

Table A4 Physical and mechanical properties of the rocks tested, cutting parameters and cutting forces of conical picks (cited in Goktan and Gunes, 2005). Rock name

rc (MPa)

rt (MPa)

u (°)

c (°)

/ (°)

h (mm)

FC cspuM (kN)

FC cspuG (kN)

FC CTP spuT (kN)

FC csmuM (kN)

FC csmuG (kN)

FC csmuT (kN)

Sandstone-2 Copper-1 Copper-2

174 33 41

11.6 3.4 5.7

30 38 32

57 55 55

105 80 80

9 10 10

60.50 15.07 25.82

67.93 16.97 28.45

62.61 18.11 22.97

23.50 5.09 9.08

22.64 5.66 9.48

22.85 6.47 8.26

Table A5 Physical and mechanical properties of the rocks tested, cutting parameters and cutting forces of radial picks used for regression analysis (Bilgin, 1977; cited in Gunes et al., 2007). Rock name and mechanical properties

b (°)

d (mm)

h (mm)

FC rspuM (kN)

FC rspuN (kN)

FC rspuT (kN)

Limestone 1

10 10 10 10 5 5 5 10 5 0 5 10

20 20 20 20 10 20 30 20 20 20 20 20

3.0 6.0 9.0 12.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

4.40 8.70 13.30 17.70 6.10 7.20 8.30 6.70 6.80 7.00 7.20 7.40

4.00 9.60 15.20 20.90 4.70 7.00 9.20 4.65 5.40 6.20 7.00 7.70

6.18 10.19 14.2 18.21 5.03 7.89 10.75 6.07 6.55 7.14 7.89 8.85

10 10 10 10 5 5 5 0 5 10

20 20 20 20 10 20 30 20 20 20

3.0 4.5 6.0 7.5 5.0 5.0 5.0 5.0 5.0 5.0

8.40 14.00 19.60 25.20 11.30 14.50 17.80 13.30 14.50 15.90

11.90 14.70 17.50 20.32 12.60 14.90 17.15 14.10 14.90 15.65

9.79 13.6 17.41 21.21 7.66 13.14 18.63 11.81 13.14 14.87

10 10 10 5 5 5

20 20 20 10 20 30

3.0 4.5 6.0 5.0 5.0 5.0

4.50 6.70 7.80 4.70 6.50 8.40

3.80 6.60 9.40 4.50 6.80 9.00

7.03 9.47 11.9 5.69 9.21 12.72

rc ¼ 84:9 MPa rt ¼ 6:2 MPa c ¼ 28 MPa

u ¼ 25:6 w ¼ 18:6

Limestone 2

rc ¼ 127:3 MPa rt ¼ 7:5 MPa c ¼ 44 MPa

u ¼ 32:3 w ¼ 37

Anhydrite

(continued on next page)

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Table A5 (continued) Rock name and mechanical properties

rc ¼ 112:9 MPa rt ¼ 5:5 MPa c ¼ 25 MPa

u ¼ 36:3 w ¼ 43

Granite

rc ¼ 179:1 MPa rt ¼ 10:7 MPa c ¼ 50 MPa

u ¼ 21:5 w ¼ 42

Greywacke

rc ¼ 183:6 MPa rt ¼ 16:45 MPa c ¼ 52 MPa

u ¼ 23:6 w ¼ 33

b (°)

d (mm)

h (mm)

FC rspuM (kN)

FC rspuN (kN)

FC rspuT (kN)

10 5 0 5 10

20 20 20 20 20

5.0 5.0 5.0 5.0 5.0

6.10 6.20 6.40 6.50 6.70

4.50 5.25 6.00 6.80 7.60

7.21 7.73 8.38 9.21 10.28

10 10 10 5 5 5 0 5 10 15 20

20 20 20 10 20 30 20 20 20 20 20

2.0 3.0 4.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

6.50 9.40 12.00 9.00 12.00 15.00 12.50 13.70 15.00 16.50 18.10

8.00 9.90 11.75 10.60 12.90 15.10 12.10 12.90 13.60 14.40 15.20

8.57 8.4 10.48 6.58 10.98 15.39 9.76 10.98 12.55 14.66 17.63

10 10 10 5 5 5 0 5 10 15 20

20 20 20 10 20 30 20 20 20 20 20

3.0 4.5 6.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

8.20 13.30 18.30 11.20 13.30 15.50 11.90 13.40 14.90 16.80 18.80

11.20 14.00 16.80 11.90 14.20 16.45 13.40 14.20 15.00 15.70 16.50

8.65 12.68 16.18 7.15 12.12 17.09 10.79 12.12 13.84 16.15 19.41

FC rspuM is measured peak cutting force of sharp radial picks under unrelieved cutting mode, FC rspuN is the peak cutting force of sharp radial picks calculated by the model of Gunes et al. (2007), and FC rspuT is the peak cutting force of sharp radial picks calculated by the model of this work.

Table A6 Results of rock cutting experiment in anhydrite (rc ¼ 112:9 MPa;rt ¼ 5:5 MPa;c ¼ 25 MPa;u ¼ 36:3 ; w ¼ 43 ) (Bilgin, 1977). h (mm)

d (mm)

b (°)

FC rspuM (kN)

FC rspuN (kN)

FC rspuT (kN)

FC rsmuM (kN)

FC rsmuT (kN)

FNrspuM (kN)

FN rspuT (kN)

FN rsmuM (kN)

FNrsmuT (kN)

1.5 4.5 7.5 3 6 3 6 1.5 4.5 7.5 4.5 7.5 3 6 1.5 6 1.5 4.5 7.5 3 7.5 3 6 1.5 4.5

10 20 30 40 50 20 30 40 50 10 30 40 50 10 20 40 50 10 20 30 50 10 20 30 40

10 10 10 10 10 0 0 0 0 0 10 10 10 10 10 20 20 20 20 20 30 30 30 30 30

1.46 6.75 13.25 6.24 15.94 3.68 8.67 5.46 10.04 6.21 8.50 13.44 7.57 5.80 1.74 10.79 5.05 4.20 9.82 6.22 13.03 2.99 5.20 3.40 4.43

1.28 6.61 14.5 8.34 16.23 2.26 10.15 3.99 11.88 8.42 5.8 13.69 7.53 4.07 2.09 9.34 3.18 0.28 7.61 1.45 12.88 4.63 3.26 2.9 4.99

3.39 9.5 20.49 11.94 26.59 5.92 13.4 5.92 16.21 6.85 9.01 17.36 9.77 5.21 3.69 12.44 5.38 4.1 8.59 6.02 16.13 3.29 6.64 3.85 8.87

0.7 3.08 6.00 3.02 9.72 1.40 4.57 2.35 5.90 3.17 3.22 8.53 4.24 1.86 0.82 6.50 2.77 1.25 3.66 2.51 6.02 1.34 2.70 1.98 1.57

1.7 4.75 10.24 5.97 13.29 2.96 6.7 2.96 8.11 3.43 4.5 8.68 4.88 2.61 1.85 6.22 2.69 2.05 4.29 3.01 8.07 1.64 3.32 1.92 4.44

1.56 5.41 17.45 6.14 21.19 3.09 9.42 6.91 13.8 6.02 10.48 24.6 9.25 3.96 1.60 21.51 8.42 5.00 13.62 7.77 20.82 5.03 8.48 4.80 3.76

4.68 13.1 28.25 16.46 36.67 8.16 18.49 8.16 22.36 9.45 12.42 23.95 13.47 7.19 5.09 17.15 7.42 5.65 11.84 8.3 22.25 4.54 9.16 5.31 12.24

0.90 3.55 11.7 3.81 15.61 1.61 5.32 4.65 10.22 2.73 5.53 22.73 5.77 2.34 0.96 13.39 5.70 2.32 8.11 4.86 13.88 2.75 5.43 3.76 1.71

3.21 8.98 19.36 11.28 25.13 5.59 12.67 5.59 15.32 6.48 8.51 16.41 9.23 4.92 3.49 11.75 5.08 3.87 8.11 5.69 15.25 3.11 6.27 3.64 8.39

where FC rsmuM is measured mean cutting force of sharp radial picks under unrelieved cutting mode, FC rsmuT is the mean cutting force of sharp radial picks under unrelieved cutting mode calculated by the model of this work, FNrspuM and FN rsmuM are measured peak and mean normal forces of sharp radial picks under unrelieved cutting mode, FN rspuT and FNrsmuT are the peak and mean normal forces of sharp radial picks under unrelieved cutting mode calculated by the models of this work.

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95 Table A7 Results of rock cutting experiment in limestone (rc ¼ 127:3 MPa;rt ¼ 7:5 MPa;c ¼ 44 MPa;u ¼ 32:3 ; w ¼ 37 ) (Bilgin, 1977). h (mm)

d (mm)

b (°)

FC rspuM (kN)

FC rspuN (kN)

FC rspuT (kN)

FC rsmuM (kN)

FC rsmuT (kN)

FN rspuM (kN)

FN rspuT (kN)

FN rsmuM (kN)

FNrsmuT (kN)

1.5 4.5 7.5 3 6 3 6 1.5 4.5 7.5 4.5 7.5 3 6 1.5 6 1.5 4.5 7.5 3 7.5 3 6 1.5 4.5

10 15 20 25 30 15 20 25 30 10 20 25 30 10 15 25 30 10 15 20 30 10 15 20 25

0 0 0 0 0 5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20

2.715 9.683 18.707 7.442 15.254 5.674 20.469 3.868 16.738 18.08 12.479 21.314 8.696 11.902 3.085 20.021 4.378 10.734 28.95 8.207 40.155 5.625 21.286 4.304 19.853

5.21 11.96 18.72 11.42 18.18 9.92 16.68 9.38 16.14 17.22 14.64 21.39 14.1 15.18 7.88 19.35 12.06 13.14 19.89 12.6 24.07 11.1 17.85 10.56 17.31

3.62 8.68 16.63 9.4 19.53 7.11 15.34 6.29 16.98 10.4 13.6 25.97 13.6 9.79 5.03 24.7 8.93 8.93 19.07 11.18 43.33 7.66 18.64 7.66 22.75

1.594 3.548 7.406 3.622 7.114 2.612 9.941 2.247 8.325 9.199 5.110 9.450 3.895 4.121 1.734 11.356 2.676 4.348 12.835 3.912 22.49 2.717 10.41 2.402 10.603

1.81 4.34 8.32 4.7 9.76 3.55 7.67 3.14 8.49 5.2 6.8 12.99 6.8 4.89 2.51 12.35 4.46 4.46 9.53 5.59 21.67 3.83 9.32 3.83 11.38

2.378 4.234 12.472 5.07 7.118 3.616 9.992 3.13 8.657 8.312 6.116 9.521 5.126 5.367 2.214 14.23 2.991 5.239 8.756 4.294 28.499 3.428 14.735 2.971 13.564

2.06 4.94 9.46 5.35 11.11 4.04 8.73 3.58 9.66 5.92 7.74 14.78 7.74 5.57 2.86 14.05 5.08 5.08 10.85 6.36 24.66 4.36 10.6 4.36 12.95

1.85 2.478 7.597 3.437 4.007 2.273 5.416 2.334 5.153 4.500 3.366 5.502 3.060 2.888 1.632 11.356 2.227 3.268 5.150 2.696 20.553 2.156 8.770 2.095 8.500

1.37 3.30 6.32 3.57 7.42 2.70 5.83 2.39 6.45 3.95 5.17 9.87 5.17 3.72 1.91 9.38 3.39 3.39 7.24 4.25 16.47 2.91 7.08 2.91 8.65

Table A8 Results of rock cutting experiment in granite (rc ¼ 179:1 MPa;rt ¼ 10:7 MPa;c ¼ 50 MPa;u ¼ 21:5 ; w ¼ 42 ) (Bilgin, 1977). h (mm)

d (mm)

b (°)

FC rspuM (kN)

FC rspuN (kN)

FC rspuT (kN)

FC rsmuM (kN)

FC rsmuT (kN)

FN rspuM (kN)

FN rspuT (kN)

FN rsmuM (kN)

FNrsmuT (kN)

1 3 5 2 4 2 4 1 3 5 3 5 2 4 1 4 1 3 5 2 5 2 4 1 3

10 15 20 25 30 15 20 25 30 10 20 25 30 10 15 25 30 10 15 20 30 10 15 20 25

0 0 0 0 0 5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20

2.064 7.448 12.936 7.346 11.995 4.808 10.413 3.800 10.473 10.488 10.526 17.51 8.082 8.300 3.102 15.712 4.500 6.099 14.511 7.387 21.428 6.692 13.582 4.762 13.185

2.22 7.11 11.99 7.5 12.38 6.00 10.88 6.40 11.28 10.49 9.78 14.66 10.18 9.38 4.90 13.56 9.07 8.28 13.16 8.68 17.34 7.18 12.06 7.57 12.46

2.93 5.59 9.76 5.97 11.28 4.81 9.22 4.37 10.1 6.58 8.4 15.15 8.4 6.33 3.73 14.66 5.92 5.92 11.54 7.17 25.36 5.26 11.45 5.26 13.77

1.284 3.659 6.034 4.859 6.987 2.349 5.752 2.850 6.584 4.592 5.618 7.798 4.81 3.500 1.764 9.043 3.400 2.672 6.658 4.127 1.887 3.475 6.976 2.92 7.058

1.47 2.79 4.88 2.98 5.64 2.41 4.61 2.19 5.05 3.29 4.2 7.58 4.20 3.16 1.86 7.33 2.96 2.96 5.77 3.58 12.68 2.63 5.72 2.63 6.88

2.792 6.225 9.675 9.849 11.399 4.351 8.856 5.400 11.034 4.500 8.665 10.031 8.706 5.700 3.217 11.944 6.200 4.172 10.776 6.325 13.103 4.787 9.650 4.525 11.800

2.24 4.26 7.45 4.55 8.61 3.67 7.03 3.34 7.71 5.02 6.41 11.56 6.41 4.83 2.85 11.19 4.52 4.52 8.81 5.47 19.35 4.02 8.73 4.02 10.5

2.072 4.692 7.529 8.409 8.581 3.871 7.012 4.100 7.732 4.172 6.351 6.475 7.423 4.300 2.676 9.16 5.200 3.457 8.015 4.527 9.018 3.665 7.34 3.962 10.00

1.63 3.10 5.41 3.31 6.25 2.67 5.11 2.43 5.60 3.65 4.66 8.40 4.66 3.51 2.07 8.13 3.28 3.28 6.40 3.98 14.06 2.92 6.35 2.92 7.63

Table A9 Results of rock cutting experiment in greywacke (rc ¼ 183:6 MPa;rt ¼ 16:45 MPa;c ¼ 52 MPa;u ¼ 23:6 ; w ¼ 33 ) (Bilgin, 1977). h (mm)

d (mm)

b (°)

FC rspuM (kN)

FC rspuN (kN)

FC rspuT (kN)

FC rsmuM (kN)

FC rsmuT (kN)

FN rspuM (kN)

FN rspuT (kN)

FN rsmuM (kN)

FNrsmuT (kN)

1.5 4.5 7.5 3.0 6.0 3.0 6.0 1.5 4.5 7.5

10 15 20 25 30 15 20 25 30 10

0 0 0 0 0 5 5 5 5 5

3.03 8.01 15.29 7.44 13.35 5.97 16.15 3.89 11.11 20.79

4.5 11.25 18.01 10.71 17.47 9.21 15.97 8.67 15.43 16.51

3.46 7.99 15.09 8.63 17.68 6.65 14.11 5.9 15.6 9.63

1.03 3.18 9.55 3.25 6.13 2.83 7.39 2.30 5.60 10.11

1.73 3.99 7.55 4.32 8.84 3.32 7.05 2.95 7.8 4.82

2.57 5.17 6.13 5.9 9.15 5.04 9.93 4.40 8.59 11.53

2.26 5.21 9.84 5.63 11.53 4.34 9.2 3.85 10.17 6.28

1.26 3.00 5.00 3.61 4.69 3.99 5.74 3.39 6.48 7.46

1.46 3.36 6.35 3.63 7.43 2.80 5.93 2.48 6.56 4.05 (continued on next page)

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X. Wang et al. / Tunnelling and Underground Space Technology 62 (2017) 75–95

Table A9 (continued) h (mm)

d (mm)

b (°)

FC rspuM (kN)

FC rspuN (kN)

FC rspuT (kN)

FC rsmuM (kN)

FC rsmuT (kN)

FNrspuM (kN)

FN rspuT (kN)

FN rsmuM (kN)

FNrsmuT (kN)

4.5 7.5 3.0 6.0 1.5 6.0 1.5 4.5 7.5 3.0 7.5 3.0 6.0 1.5 4.5

20 25 30 10 15 25 30 10 15 20 30 10 15 20 25

10 10 10 10 10 15 15 15 15 15 20 20 20 20 20

10.56 20.72 9.22 10.84 3.79 26.08 5.96 12.30 33.84 10.05 30.62 6.38 18.20 4.69 15.46

13.93 20.68 13.39 14.47 7.17 18.64 11.35 12.43 19.18 11.89 23.36 10.39 17.14 9.85 16.6

12.68 24.06 12.68 9.17 4.8 23.14 8.46 8.46 17.9 10.56 40.95 7.34 17.68 7.34 21.56

4.15 8.37 4.01 4.07 1.81 14.54 2.54 5.53 15.55 5.14 16.80 2.94 10.34 2.37 9.32

6.34 12.03 6.34 4.59 2.4 11.57 4.23 4.23 8.95 5.28 20.48 3.67 8.84 3.67 10.78

6.93 10.59 7.24 5.68 4.14 18.52 4.78 8.44 16.86 7.34 24.53 5.54 14.38 4.57 12.66

8.26 15.68 8.26 5.98 3.13 15.09 5.52 5.52 11.67 6.89 26.7 4.79 11.53 4.79 14.06

3.05 6.63 3.95 2.83 3.16 11.79 2.82 5.00 6.87 4.52 17.28 4.39 11.83 3.80 10.88

5.33 10.12 5.33 3.86 2.02 9.73 3.56 3.56 7.53 4.44 17.22 3.09 7.44 3.09 9.07

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