Effects of elastoplastic strengthening of gravel soil on rockfall impact force and penetration depth

International Journal of Impact Engineering 136 (2020) 103411

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Effects of elastoplastic strengthening of gravel soil on rockfall impact force and penetration depth

T

Yusuo Wanga, , Ming Xub, Chao Yanga, Mengyuan Lua, Jie Menga, Zhilong Wanga, Mingnian Wanga ⁎

a b

School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, China Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, Chengdu 610031, China

ARTICLE INFO

ABSTRACT

Keywords: Rockfall impact Gravel soil cushion Hertz contact theory Elastoplastic strengthening model Effects of strain rate Pressure dependencies

Gravel soil backfilled on the top of a rock shed serves as a cushion and protects structures in mountainous districts that are endangered by rockfalls. Strengthening effect under rockfall impact is one of the factors that can influence the properties of the backfilled gravel soil; therefore, this effect should be considered in studies on the mechanical behaviour of the structure protected by the gravel soil. This paper introduces a bilinear enhanced constitutive model considering the strengthening coefficient and proposes a theoretical calculation method for rockfall impact force and penetration depth based on the Hertz contact theory. Back analysis and mathematical statistical analysis of the results of rockfall experiments produced a range of strengthening coefficients for the gravel soil. Based on the obtained values of the strengthening coefficient and the proposed calculating method, ranges of values of impact force and penetration depth are predicted. Based on the comparisons with previous experimental results and the results calculated using other previous methods, the calculated upper and lower limits of impact force in this study prove to be in good agreement with the previous experimental results; however, the calculated medium values are lower than the test medium values. Furthermore, the predicted medium values of the penetration depth were, on average, 20% larger than the experimental values. This study shows that the effects of the strain rate and pressure dependencies produce a certain influence on the estimated results of rockfall impact force and penetration depth. It also shows that the effects of pressure dependencies are comparatively more distinct than those of the strain rate.

Reinforced concrete sheds or arch-shaped open tunnels, with backfilled gravel soil cushions on their tops, are widely used to minimize damage due to rockfalls in mountainous districts. The falling rock, upon impact, interacts with the gravel soil cushion and generates an interactive force, called rockfall impact force. This force transmits through the cushion to the structure. Therefore, rockfall impact force has a bearing on the design load of a structure and needs consideration. Currently, the methods for calculating the rockfall impact force fall into two broad categories. One category is based on Newton's second law, where Fa•t = m(v – v0). Here m is the mass of rockfall; v0 and v are the velocities of rockfall before and after impact, respectively; t is interaction time, and Fa is the average force [1]. However, Fa is affected significantly by t in real situations, which is neither readily recorded nor estimated. The other category is based on predicting the rockfall impact force using the Hertz contact theory. For example, Refs. [2] and [3], based on regression analysis of test data, proposed formulae based on the Hertz elastic contact theory. Considering that soil is an

⁎

elastoplastic material, Ref. [4], based on the Hertz theory, deduced a calculation method for rockfall impact force considering the plastic yield of the soil by employing Mohr-Coulomb criterion. During the interaction between the falling rock and the cushion, the gravel soil first yields to the impact. Subsequently, its strength and stiffness increase with further compaction. As such, gravel soil can be seen as an elastoplastic strengthening material. Therefore, it is essential to study the influence of the strengthening characteristics of the material under impact. Drawing on the Hertz elastic contact theory, Ref. [5] proposed an elastoplastic constitutive model by incorporating linear strengthening (softening) expressed by strengthening coefficient k, and specified the relationship between the maximum contact force of two spherical particles and the relative compressive deformation. This model provided an effective way to determine the impact force on the gravel cushion and the penetration depth caused by the rockfall impact. However, the problem of how to determine k remains to be solved.

Corresponding author. E-mail address: [email protected] (Y. Wang).

https://doi.org/10.1016/j.ijimpeng.2019.103411 Received 17 February 2019; Received in revised form 4 October 2019; Accepted 6 October 2019 Available online 18 October 2019 0734-743X/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Impact Engineering 136 (2020) 103411

Y. Wang, et al.

contact load of the Hertz analytical solution as

Pe =

4 1 ER 2 3

3 2

(4)

where E is the equivalent modulus derived from Eq. (4) with δ = δ1 + δ2:

1 v 12 1 v 22 1 = + E E1 E2

(5)

where δ1, E1, μ1, δ2, E2, and μ2 are the compression deformation, elastic modulus, Poisson ratio of the half-space material and sphere, respectively. When δ is the normal contact deformation at the centre line, the normal compression deformation at any position on the contact surface can be written as

r2 , r 2R 0

(r ) =

Based on the elastoplastic solution method of the Hertz contact theory, this paper reintroduces the linearly enhanced elastoplastic model of Ref. [5] and proposes a k calculation method using the back analysis of the previous research results of rockfall impact force. This study establishes a calculation method of rockfall impact by considering the elastoplastic strengthening effect of soil. The effects of the strain rate and pressure dependencies on the estimates of the impact force and penetration depth are also discussed.

r

= z

Pm rz

1

1 z2 1+ 2 2 a

r

1

(7) (8)

=

1

z2

(9)

a2

(10)

=0

z

3

=(

1

+

3)sin

(11)

+ 2c·cos

where σ1 and σ3 are, respectively, the maximum and minimum principal stresses along the z-axis (i.e., r = 0), with σr, σθ and σz being principal stresses; hence we set σ1 = σr and σ3 = σz. The cohesive traction is denoted as c, and φ denotes the internal friction angle of the soil obtainable by geotechnical tests such as direct shear test or triaxial compression test. The integration of Eqs. (7), (9), and (11) gives

1/2

(1)

where F denotes the contact force on contact area, r is the radial distance measured from the centre of the contact surface, and a is the contact radius. The relationship between contact deformation δ and contact area is expressed as [6–8]

pm =

(2)

C v1

2c·cos C v2sin

(12)

where

Based on Eq. (1), when r = 0, the maximum contact pressure pm is given as

Pmax

+

When the half-space is an elastoplastic material, it is necessary to introduce a plastic yield criterion. Reference [9], for example, offers a contact mechanics solution based on the von-Mises criterion, which is used for the contact mechanics of metal plastic materials. Reference [4] deduced an elastoplastic model solution by introducing the Mohr–Coulomb criterion so as to explore the mechanical behaviour of the rockfall impact on the soil cushion. To specify the method for the back analysis of the strengthening coefficient (presented in the following sections), we consider that it necessary to reproduce the main derivation process and conclusions offered by Ref. [4]. In Ref. [4], the Mohr–Coulomb criterion reads as

In Fig. 1, the contact pressure distribution is assumed as

3P = 2 a2

a z

1.2. Elastoplastic model solution based on Mohr–Coulomb yield criterion

1.1. Elastic model solution

a2 = · R0

1

r

= 1+

=

z tan a

(1 + µ1) 1

where σr, σθ, and σz are the radial, circumferential, and vertical direction stresses, respectively, and τrz, τrθ, and τzθ are the corresponding shear stresses.

The contact between a sphere and half-space is a fundamental issue in contact mechanics. Hertz deduced an analytical solution to the contact stress field of two frictionless elastic spheres [6]. This solution suggested that the contact between the spherical surface and half-space can be derived from two spherical contacts when the radius of one of the spherical surfaces is assumed to be infinite. Fig. 1 depicts the stress calculation model for sphere impact on half-space. The sphere, with radius R0, is assumed to be elastic. The Hertz contact mechanics based theoretical models for halfspaces of elastic, elastoplastic, and elastoplastic strengthening materials are discussed in the following subsections.

r2 a2

=

pm

1. Calculating model for rockfall impact force based on Hertz contact theory

3F 1 2 a2

(6)

According to Refs. [6] and [9], the stress field along the z-axis (vertical to the central contact surface and r = 0) in the model of Fig. 1 is as

Fig. 1. Stress calculation model for sphere impact on half-space.

p (r ) =

a

C1 =

(3)

3 (1 + 2

2) 1

(1 + µ1)[1

tan 1 (1/ )]

and

Establishing a relation between contact force Fe (where subscript e indicates the elastic contact) and δ, Ref. [6] gave an expression for the

C2 = 2

1 (1 + 2

2) 1

(1 + µ1)[1

tan 1 (1/ )].

International Journal of Impact Engineering 136 (2020) 103411

Y. Wang, et al.

Here ξ = z/a and denotes the dimensionless penetration depth. The contact deformation resulting from the initial soil yield is termed initial yielding deformation ξ0 and the accompanying contact compressive pressure is termed initial yield contact compressive pressure py. When the initial yield contact deformation occurs, the contact compressive pressure is the smallest [8] and can be obtained by finding the partial derivative of the contact compressive pressure and setting it equal to 0:

pm

=0

(13)

Substituting Eq. (12) into Eq. (13) gives

(1 + µ1) tan

1

1

0

1+

0

0

(1

2

sin )

0

(1 +

0

2 )2

(3 + sin ) Fig. 2. Hypothesised mechanical model of pressure and deformation.

(14)

=0

where ξ0 = z0/a, z0 being the amount of displacement generated by the initial yield pressure of the soil. Setting 1-sin φ = α and 3 + sin φ = β transforms Eq. (14) into

v1 =

b a (1 +

0 2 2 1 0 ) ·tan

() 1

0

Fy =

(15)

1 + 02

3 + sin 1 sin

+ 0.4927

+ 0.9075

3 + sin 1 sin

(

3 + sin 1 sin

(

0

pm =

(16)

3 + sin 1 sin

+ 0.9075

(

0

0.2)

= 0.0274 + 1.1019(1 + µ1)

1 sin 3 + sin

(17)

C1

(18)

2c·cos C2 sin

(19)

p (r ) =

C2 =

3 (1 + 2

2 1 0)

1 (1 + 2

2 1 0)

(1 + µ1)[1 (1 + µ1)[1

0

0

=

R 0 c·cos E (C1 C2 sin )

>

(23)

y

3F 2 a2

r

( a )2

1

1/2

,

(r )

r y ),

ap r < ap

(24)

The pressure distribution on the contact surface is illustrated in Fig. 3. In Fig. 3, the integration of normal pressure along the contact area is equal to the normal load applied to the sphere, and its value is equal to the volume of the shaded area rotating around the ordinate p(r) axis; A1 and A2 are elastic stress regions, and B1 and B2 are plastic stress regions. The normal load solution formula of the contact surface can be defined as

tan 1 (1/ 0 )], tan 1 (1/ 0 )].

Combining Eqs. (2)–(4) and (19) leads to the initial yield contact deformation: y

y ),

py + k (

where

C1 =

y

where k denotes the strengthening (softening) coefficient, which is a given constant for the same material [5]; py is the yield pressure and δy is the compressive displacement when the half-space material yields (see Eqs. (19) and (20)). When the contact compressive pressure between the sphere and half-space in Fig. 1 exceeds the initial yield pressure of the half-space body, the cushion material yields immediately, with the yield radius becoming ap. With regard to the range exceeding the yield region, i.e., r > ap, it stays within the elastic contacting zone. Substituting Eqs. (1) and (6) into Eq. (23) gives the normal pressure distribution:

Inserting Eq. (18) into Eq. (12) results in the corresponding initial yield pressure:

py =

,

py + k (

The error introduced by Eq. (17) was calculated from the remainder of the first two terms in the Taylor series of Eq. (16) and proved to be less than 0.8%. Hence, Eq. (17) can be rearranged as 0

R0

p= 3 + sin 1 sin

(22)

R0

2E

Omitting the first two terms gives

(1 + µ1) = 0.1566

2E

It can be readily seen that the maximum contact compressive pressure in the elastic range is proportional to the 1/2 power of the compressive deformation. The elastoplastic linear strengthening constitutive model of materials in contact mechanics proposed in Ref. [5] is presented in Fig. 2. The corresponding constitutive relationship is defined as

0.2)

0.2) 2 + ···

0

(21)

Based on Eqs. (2)–(4), the maximum contact compressive pressure in the elastic range can be calculated as

Expanding the right-hand side of Eq. (15) to a Taylor series around ξ0 = 0.2, which corresponds to the dimensionless yielding inception depth for gravel soil with Poisson ratio varying in the range 0.2 ≤ μ1 ≤ 0.4 and internal friction angle varying in the range 30° ≤ φ ≤ 45°, yields

(1 + µ1) = 0.1566

3

1.3. Elastoplastic strengthening model solution

1

0

4R 02 c·cos 3E 2 C1 C2 sin

2

Fep = 2

(20)

Substituting Eq. (20) into Eq. (4) results in the initial yield contact force:

+2

3

ap 0

[py + k ( a

ap

3F 1 2 a2

(r ) r a

y )] rdr 2 1/2

rdr

(25)

International Journal of Impact Engineering 136 (2020) 103411

Y. Wang, et al.

Fig. 3. Normal pressure distribution on contact surface.

Substituting Eqs. (2)–(4) into Eq. (25) gives the rearranged relationship between elastoplastic contact force Fep and deformation δ:

Fep = R 0 (

y )(py

k

y)

4k R 0 3

+

3 2

+

(

2

3

y 2 )

spheres is that it increases with an increase in k, until approximating the elastic collision result. However, the k value of soil has not yet been deduced. Based on the rockfall impact test results in Refs. [3] and [10], the k value of gravel soil can be obtained by the back analysis method as follows. At first, m and v0 (calculated from free fall height), together with ξ0, py, δy, and Fy (derived by substituting c, φ, μ1, and E into Eqs. (18)–(21)), are substituted into Eq. (28). Furthermore, Fep, when set equal to Ft obtained from the experimental results in the literature, is substituted into Eq. (29), together with ξ0, py, δy, and Fy. Then, k can be calculated by the set of expressions composed of Eqs. (28) and (29). The calculation procedure for k by the back analysis method is illustrated in Fig. 4. Subsequently, Fep can be calculated by following the procedure given in Fig. 5. The parameters required for the design of the rockfall protection structure can be obtained using the flow chart in Fig. 5. First δmax is obtained, and then Fep is calculated. The calculations of k, δmax, and Fep, are illustrated in the following sections using the experimental data of Refs. [3] and [10].

+ Fy (26)

where py, δy, and Fy are given in Eqs. (19), (20), and (21) respectively. 1.4. Model for calculating rockfall impact force considering elastoplastic strengthening of soil The sphere in Fig. 1 is assumed to be a falling rock, an elastic body with mass m, hitting the ground with a certain impact velocity, v0, and the half-space is composed of gravel soil. When the cushion soil is considered as elastoplastic strengthening material, the impact force is to be analysed as follows. Assuming no energy loss during the rockfall impact, because the kinetic energy of the rockfall is absorbed by the elastic and plastic deformations of the contact system, the following equation can be obtained based on the law of the conservation of energy:

1 mv0 2 = 2

y

0

Fe d +

max y

Fep d

(27)

2. Prediction of rockfall impact force

When the rockfall impact is treated as a quasi-static problem, Eq. (4) still holds [4,5,7,8]. Substituting Eqs. (4), (6), and (24) into Eq. (27) gives

1 mv02 = (py 2

k

y)

8 + k R0 15

R0 (

max

y)

Using the experimentally obtained rockfall impact force in Ref. [3], the range of k of sandy gravel soil is obtained by back and statistical analyses. The calculation method of k is expounded as follows.

2

2 5 2

max

+

max

2

2

y

5 2

+

5 2

y

+ Fy (

y)

max

(28) where δmax is the maximum penetration depth. The specifications m, v0, and k render Eq. (28) a higher-order equation for δmax, thereby resulting in the calculability of the maximum penetration depth. Substitution of the calculated δmax into the Eq. (26) gives the elastoplastic impact force as

Fep = R 0 ( + Fy

max

y )(py

k

y)

+

4k R 0 3

3

max 2

(

max

+ 2

3

y 2 )

(29)

where calculating Fep requires the knowledge of the value of k. In the bilinear elastoplastic strengthening constitutive model proposed in Ref. [5] (see Fig. 2 and Eq. (23)), k is constant and is related to the nature of the material, and the influence of k on Fep between two

Fig. 4. Flow chart of back analysis for k. 4

International Journal of Impact Engineering 136 (2020) 103411

Y. Wang, et al.

Table 2 Initial yield parameters of sandy gravel soil cushion used in Ref. [3]. py [Pa]

ξ0 0.1622

1.0509

δy [m]

Fy [N] −15

4.5 × 10−15

3.4 × 10

Table 3 Strengthening coefficient k values for experimental conditions in Ref. [3] (rockfall mass m = 1000 kg).

Fig. 5. Flow chart for calculating rockfall impact force and penetration depth.

2.1. Experiment parameters To simulate the rockfall impact, two cylindrical rock models with spherical bottoms, one weighing 500 kg and the other1000 kg, made of steel shells filled with concrete, and each embedded with an accelerometer [3], were made to hit the soil cushion at specified velocities. Substituting the measured peak accelerations of the two rockfall models into Newton's second law (F = mamax) gives the maximum impact forces onto the cushion. The parameters of the falling rock and gravel soil are tabulated in Table 1. 2.2. Back analysis of strengthening coefficient According to Fig. 4, substituting the mechanical parameters of the cushion soil (given in Table 1) into formulas (18)–(21) gives the initial yield parameters of the sandy gravel soil cushion(see Table 2). Note that if c = 0, as specified in Ref. [3], is substituted into Eqs. (19)–(21), the values of py, δy, and Fy are rendered 0. This means the cushion soil goes directly into the strengthening phase without undergoing the yielding process when the impact action occurs. Therefore, c = 0 can significantly simplify the calculation process as shown in Figs. 4 and 5. Trial calculations by setting c = 0, 1, 2, and 3 Pa and substituting them into Eqs. (19)–(21), respectively, show that the values of Fep and δmax with c = 1, 2, and 3 Pa are nearly the same as those of c = 0, with the maximum difference being 0.0033%. Therefore, setting c to be either 0, 1, 2, or 3 Pa does not make significant difference to the results. Given that the actual gravel soil has some cohesive traction, we set c = 1 Pa, as given in Table1. This facilitates the illustration of the complete calculation process proposed in Figs. 4 and 5. In Ref. [3], when the rock (m = 1000 kg) falls from a specified height, the experimental maximum impact force is expressed as Ft = mamax (amax, a measured maximum acceleration of rockfall). Substituting m and v0 of the rockfall and the parameters listed in Table 2 into Eqs. (28) and (29) gives the values of k, as presented in Table 3.

R0 [m] 0.6

E2 [109 Pa] 50

μ1 0.3

E1 [106 Pa] 20

c* [Pa] 1

v0▲ [m/s]

Ft* [kN]

k [× 106 N/m5/2]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

1.0 1.0 1.0 2.0 2.0 3.0 3.0 3.0 4.0 4.0 4.0 5.0 5.0 5.0 5.0 6.0 6.0 6.0 6.0 7.0 7.0 7.0 8.0 8.0 8.0 8.0 9.0 9.0 9.0 9.5 9.5 9.5 9.5

4.43 4.43 4.43 6.26 6.26 7.67 7.67 7.67 8.86 8.86 8.86 9.90 9.90 9.90 9.90 10.85 10.85 10.85 10.85 11.72 11.72 11.72 12.53 12.53 12.53 12.53 13.29 13.29 13.29 13.65 13.65 13.65 13.65

114 120 133 196 209 260 372 390 317 395 416 418 429 462 523 402 556 584 593 481 495 522 494 561 571 580 542 570 626 565 596 623 641

0.71 0.80 1.03 0.96 1.13 1.06 2.60 2.92 1.13 1.96 2.24 1.62 1.73 2.08 2.84 1.12 2.51 2.85 2.95 1.39 1.49 1.71 1.21 1.67 1.75 1.81 1.28 1.45 1.84 1.32 1.50 1.68 1.80

2.3. Statistical analysis of strengthening coefficient Non-parametric hypothesis tests [11] of normal distribution and logarithmic (natural logarithm) normal distribution were conducted on the data samples with n = 33 (see Table 3). The test results are presented in Table 4. As can be seen from table 4, both hypotheses of normal distribution and lognormal distribution for sample data of k are acceptable. The significance level of the lognormal distribution hypothesis test reached 0.99, higher than the other one, indicating that k is more suited for lognormal distribution. Thus, k-population can be assumed to be a lognormal distribution, whose statistical histogram is illustrated in Fig. 6. Lognormal distribution is suitable for describing of the statistical characteristics of the k-population (see Table 4). Consequently, the sample of logarithms of the back-analysed values xi (i = 1,2,3…) in Eq. (30) can be considered as normally distributed. Constant k* is introduced to provide a dimensionless argument for the natural logarithm. Because k* does not influence the final results, it is set equal to an arbitrary value [12]; herein k* = 1 N/m5/2.

Gravel soil cushion μ2 0.22

H [m]

▲ Impact velocity v0 = (2 gH)0.5, where g is acceleration due to gravity, and H is the height from which the rock fall. ⁎ Measured from Fig. 12 in Ref. [3].

Table 1 Rockfall and gravel soil parameters based on Ref. [3]. Rockfall△

No.

φ [°] 41

△ Rockfall elastic modulus and Poisson ratio are specified with reference to those of concrete. ⁎ Cohesive traction c of gravel soil is set to 0 in Ref. [3].

x i = ln 5

ki , i = 1, 2, 3, ··· k*

(30)

International Journal of Impact Engineering 136 (2020) 103411

Y. Wang, et al.

Table 4 Hypotheses test results. Null Hypothesis 1 Distribution of k is normal with mean 1,701,250.759 and standard deviation 628,663.51 2 Distribution of logarithmic k is normal with mean 14.281 and standard deviation 0.37 Asymptotic significances are displayed. The significance level is 0.05.

Test

Sig.

Decision

One-sample Kolmogorov–Smirnov test One-sample Kolmogorov–Smirnov test

0.53 0.99

Retain the null hypothesis Retain the null hypothesis

Substituting

2 32,0.05

0.4709 =

= 20.097 [13] into Eq. (35) gives (36)

upp

Step5: Calculate the 5%- and 95%- quantiles of the ln[k/k*]-population on the basis of 95% confidence intervals as

ln

ln

k k*

5%

k k*

95%

= µlow

z 0.025

= µupp + z 0.025

upp

= 13.37

upp

= 15.19

(37)

(38)

where z0.025 = 1.6449 denotes the z-value cutting an area equal to 95% of the standardised normal distribution. Fig. 6. Statistical histogram of logarithmic strengthening coefficient of sandy gravel used in Ref. [3].

Step 6: Calculate the k-values corresponding to the quantile values in accordance with the logarithmic function relation (with k* = 1 N/m5/2 ) :

Then a 6-step calculation process is deduced as follows:

k5% = 6.43 × 105N / m5/2 , k50% = 1.59 × 106N / m5/2 , k95%

Step1: Determine the mean value of the data sample as

x =

1 n

n

x i= i=1

1 n

n

ln i=1

ki = 14.28 k*

= 3.95 × 106N / m5/2 (31)

where k50% is the median, calculated from k50% = k *exp(x¯) . The probability distribution functions of the strengthening coefficient according to Eqs. (34) and (36), and estimations of the 5%- and 95%-quantiles of k according to Eqs. (37)–(39) are illustrated in Fig. 7. Therefore, the range of k∈[k5%, k95%] = [6.43 × 105, 3.95 × 106] (N/m5/2) of sandy gravel soil in Ref. [3] is obtained.

Step 2: Calculate the standard deviation of the sample data as

s=

n

1 n

1

ln i=1

2

ki k*

x

= 0.3731

(32)

2.4. Calculation of rockfall impact force considering elastoplastic strengthening

Step 3: Calculate a two-sided confidence interval for expected value μ:

µlow = x

s· tn

1,1

/2

n

µ

x +

s ·tn

1,1

n

/2

= µupp

The aforementioned k-value of gravel soil obtained by the back analysis method is based on the experimental results of 1000-kg-mass rockfall impact. To validate and justify the back analysis method, the obtained k is used to predict the impact force caused by a 500-kg-mass falling rock in Ref. [3] pursuant to Fig. 5. First, we use the mechanical parameters listed in Table 1 for calculations and k-values of gravel soil are set equal to k5%, k50%, and k95 (see Eq. (39)). Next, the values of impact forces on gravel by the 500-kg rock dropped from various heights are calculated according to the flow chart given in Fig. 5. To comparatively evaluate the calculation method for rockfall impact force proposed in this paper, two more well-accepted calculation methods are also used to predict the rockfall impact force under the same experimental conditions as described earlier. The first method for calculating the maximum impact force based on the Hertz elastic contact theory, proposed by Vincent Labiouse et al. [3], considers the geometry of the aspheric rockfall and the modulus of subgrade reaction of the ground:

(33)

where tn-1,1-α/2 denotes the t-value that cuts an area equally into 1– α/2 of the Student's t distribution with n – 1 degrees of freedom, α is the significance level (= 0.05 when the confidence interval is 95%), and n is the degree of freedom. Substituting Eqs. (31), (32), and t32,0.975 = 2.0369 into Eq. (33), gives

µlow = 14.15

µ

14.41 = µupp

(34)

Step 4: Determine the upper bound of standard deviation σ of the ln [k/k*]-population as

(n

1) s 2 2 n 1,

=

upp

(39)

(35)

Fmax = 1.765ME2/5 Rc1/5 W 3/5H 3/5

where n2 1, denotes the χ2-value that cuts an area equal to α of the chisquared distribution with n –1 degrees of freedom (see Ref. [13]).

(40)

where Fmax is the maximum impact force [kN], ME is the modulus of 6

International Journal of Impact Engineering 136 (2020) 103411

Y. Wang, et al.

Fig. 7. Probability distribution functions of the strengthening coefficient providing estimates of 5%- and 95%-quantiles of k (shaded areas equal to 5%): (a) probability density functions of normally distributed ln(k/k*)- population; (b) probability density functions of lognormal distributed k-population.

subgrade reaction [kN/m2] (whose recommended value is 3200 kN/m2 in Ref. [3]), Rc is the radius of the falling rock part in contact with the soil cushion [m] (for unavailability of Rc in Ref. [3], in this paper, we set Rc = R0), W is the weight of the falling rock [kN], and H is the falling height [m]. The second method, proposed by Japan Road Association, is based on the Hertz elastic contact theory. This method has been used for calculating the maximum impact force of a rigid sphere having a specific gravity of 2.65 hitting the sand cushion [2,14]:

Fmax = 2.108(mg )2/3

2/5H 3/5

Table 6 Predictions of rockfall (m = 500 kg) impact force and test results Ft (unit: kN) for Ref. [3].

(41)

where Fmax is the maximum impact force [kN], m is the mass of rockfall [t], g is the gravitational acceleration of 9.8 m/s2, λ is the Lame coefficient, and H is the height of a drop [m]. The generally used value of λ is 1000 kN/m2. However, its actual values range from 1000 to 10,000 kN/m2 in light of the density of the sand cushion [2]. Using the above two methods expressed by Eqs. (40) and (41), the rockfall impact force with a mass of 500 kg (as in Ref. [3]) is calculated. The calculation parameters and units used in the calculation are listed in Table 5. The experimental conditions, experimental maximum value of impact force Ft, theoretically predicted results of Fep according to Fig. 5, and predicted results of Fmax by Eqs. (40) and (41) are presented in Table 6 and Fig. 8. The results in Table 6 and Fig. 8 show that majority of the experimental data on Ft in Ref. [3] fall within the calculated Fep range between lower predicted limits with k5% and upper limits with k95%; the predicted medium Fep with k50% remarkably overlaps with the results calculated by Eq. (40). Fig. 8 shows that the values obtained from Eq. (41) are slightly smaller than those from Eq. (40). It implies that, in some sense, the Table 5 Calculation parameters based on Ref. [3] for Eqs. (40) and (41). Eq. (40)

Eq. (41)

Modulus of subgrade reaction ME = 3200 [kN/m2]; Radius of rockfall part in contact with the soil cushion Rc = 0.6 [m]; Weight of rockfall W = 5 [kN]; Height of fall H = 0.25, 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 [m]. Mass of rockfall m = 0.5 [t]; Lame coefficient λ = 1000 [kN/m2]; Height of fall H = 0.25, 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 [m].

No.

H [m]

Ft

Fep k5%

k50%

k95%

Fmax Eq. (40)

Eq. (41)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

0.25 0.25 0.25 0.5 0.5 0.5 0.5 1 1 1 1.5 2 2 2 2 3 3 4 4 4 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9

30 41 53 54 58 63 66 83 97 114 125 107 135 148 168 177 219 213 264 275 308 328 386 281 291 324 338 364 378 403 436 243 298 308 324 397 495

32 32 32 48 48 48 48 73 73 73 93 110 110 110 110 140 140 167 167 167 191 191 191 213 213 213 213 233 233 233 233 253 253 253 253 271 271

45 45 45 69 69 69 69 104 104 104 133 158 158 158 158 202 202 240 240 240 276 276 276 306 306 306 306 335 335 335 335 363 363 363 363 390 390

65 65 65 99 99 99 99 150 150 150 191 227 227 227 227 290 290 344 344 344 394 394 394 439 439 439 439 482 482 482 482 522 522 522 522 560 560

46 46 46 70 70 70 70 106 106 106 135 160 160 160 160 204 204 243 243 243 278 278 278 310 310 310 310 340 340 340 340 368 368 368 368 395 395

43 43 43 65 65 65 65 98 98 98 125 148 148 148 148 189 189 224 224 224 257 257 257 286 286 286 286 314 314 314 314 340 340 340 340 365 365

value of Rc (0.6 m, see Table 5) in Eq. (40) is set larger than its actual measured one which is unavailable in Ref. [3]. As seen in Table 6 and Fig. 8, when the gravitational potential energy decreases (say, mgH < 20 kJ), the predicted Fep with k50% and Fmax values calculated by Eq. (40) and Eq. (41) approximate the upper limit of the experimental Ft; when mgH increases (for example, mgH > 20 kJ), the three groups of results approach the lower limit of Ft. With

Note: The units against each parameter in the table are applicable exclusively to Eqs. (40) and (41). 7

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significant discreteness of experiment results; however, this trend for the 500-kg rockfall becomes relatively pronounced. Fig. 9(b) illustrates that the increasing trend of k is comparatively evident with an increase in impact force owing to increases in the bearing capacity and stiffness of the soil resulting from soil being further compacted by the rockfall and further confined by the surrounding soil [15,18,19],. Given that Ref. [3] neither offers data on the soil strain rate under the rockfall impact nor performs pressure dependencies experiments, we study the impacts of the strain rate and pressure dependencies on the rockfall impact by exploring the relationships between k and H, and between k and Ft. As can be observed from Table 7 and Fig. 9, despite significant discreteness, k does increase generally with an increase in H; in comparison, the increasing trend in k with that of Ft is more evident. Accordingly, when the impacts of the strain rate and pressure dependencies are neglected in the constitutive model proposed in Fig. 2, the calculated impact force is smaller than the test results. Because there is no relevant experimental data available, the effects of strain rate on the rock fall penetration depth cannot be verified in this section. Theoretically speaking, when effect of pressure dependencies is neglected, the rockfall penetration depth calculated with the constitutive relation of Fig. 2 is larger than that calculated by considering pressure dependencies [18, 19]; however, the calculated impact force is smaller than the experimental value at the same time.

Fig. 8. Comparison among Fep value calculated with k, Fmax by Eq. (40) with ME = 3200 [kN/m2] and Eq. (41) with λ = 1000 [kN/m2], and experimental Ft of 500 kg rockfall in Ref. [3].

the rockfall impact seen as a quasi-static problem, the constitutive model in Fig. 2, Eq. (40), and Eq. (41) are all proposed on the basis of the Hertz contact theory, considering the effect of neither of the strain rate nor pressure dependencies. Therefore, it is worthwhile to discuss the effects of these omitted parameters on the estimations of impact force and penetration depth.

3. Predictions of rockfall impact force and penetration depth The rockfall impact force on the gravel and the resulting penetration depth δ have been studied in Ref. [10], where the δ is derived from the following expression:

2.5. Effects of strain rate and pressure dependencies In geotechnical research, strain rate effects of soils are usually studied with the Split Hopkinson pressure bar [15–17]. To our knowledge, however, few strain rates of soil under rockfall impact have been measured or analysed in the literature. While the strain rate effect is related to the impact velocity when the rockfall contacts the surface of soil cushion, and the pressure dependencies are subjected to the interaction between the rockfall and soil, i.e., the impact force of the rockfall. Therefore, with k values derived from the back analysis of experimental results of rockfall impact force in Ref. [3], the relationship between k and H (corresponding impact velocity) and that between k and Ft are analysed for an indirect evaluation of the influences of strain rate effects and pressure dependencies on estimations of rockfall impact. Table 7 presents the results of the analysis of correlations [11] between k and H, Ft, and m using 70 sets of data (33 from Table 3 and 37 from Table 6). The correlations in Table 7 show that the more the value of Pearson correlation increases, the more significant the correlation is; the closer the value of Sig. (2-tailed) is to 0, the more significant the correlation is; and the closer the value is to 1, the less significant the correlation is [11]. These results indicate that the correlations between k and Ft and that between k and H are both significant, with the former being more significant than the latter; the correlation between k and m is not significant according to the results presented in Table 7. The relationships between k and H and between k and Ft are presented in Fig. 9. Fig. 9(a) shows that the increasing trend of k for the1000-kg rockfall is not noticeable with an increase in height, probably owing to the

d

k

⁎⁎

Pearson correlation Sig. (2-tailed) Sample number N

m

H

−0.040 0.744 70

0.339 0.004 70

m 3 d s BN *

N=

d

k0

(42)

(43)

where m stands for the rockfall mass, ρs is the mass density of target material, B is a dimensionless compressibility parameter of the impacted material (B = 1.2 is used for gravel [20]), and N* is the nose shape factor (N* = 0.4527 for cubic shape). The dimensionless depth of the surface crater is defined as

k 0 = 0.707 +

Hi d

(44)

where Hi is the height of the impactor nose. The impact function I is given as

I=

mv02 Rd 3

(45)

where v0 denotes the impact velocity, and R (units: N/m2) is the strength-like indentation resistance of target materials, which can be interpreted as the indentation resistance of the target material [21–23]. Using the dimensional analysis and Newton's second and third laws with the assumption that force varies linearly during contact, Ref. [10] deduced the relations of the rockfall impact force FR acting onto gravel, the penetration depth δR, and duration of impact Δti as follows

Ft ⁎⁎

for

where d is the diameter of the projectile, N is a geometry function characterizing the sharpness of the impactor nose, k0 is the dimensionless depth of the surface crater, and I is the impact function describing the intensity of the impact. Geometry function N is defined as

Table 7 Correlations. Correlations

1 + k 0 /4N 4k 0 · ·I (1 + I /N )

=

0.474⁎⁎ 0.000 70

R

=

FR =

Correlation is significant at the 0.01 level (2-tailed). 8

t 2g ti v 0 + i 2 12 2mv0 ti

(46) (47)

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Fig. 9. Two relationships of k in Ref. [3]: (a) between strengthening coefficient and height of fall; (b) between strengthening coefficient and measured experimental impact force.

where g stands for the gravitational acceleration and g is 9.81 m/s2. Based on experimental results of δR, R of the gravel has been obtained by back analysis and evaluated statistically. The estimations of penetration depths and rockfall impact forces have been performed by using Eqs. (42), (46), and (47) for Ref. [10]. In Ref. [10], granite gravel ground was hit by cubic granite blocks from specified heights [10]. The measured δt and corresponding Ft values calculated by using Eqs. (46) and (47) are listed in Table 8. In this section we aim to predict Fep and δmax under experimental conditions of Ref. [10] using k by following the flow chart in Figs. 4 and 5 and compare the predicted results with those of FR and δR with R by employing Eqs. (42), (46) and (47), as well as with those of Fmax by using Eqs. (40) and (41) in the following sections.

Table 9 Characteristic parameters of sandy gravel buffer layer. Resource

Mass density ρ[kg/m3]

Particle gradation[mm]

μ1*

E1* [MPa]

c△[Pa]

φ[°]

Ref. [3]

1650

0.3

20

1

41

Ref. [10]

1800

3–10 (30%); 10–32 (70%) 2–63 (60%); 63–200 (40%)

0.3

50

1

40

⁎ μ1 and E1 are set according to the conventional geotechnical reference value. △ c = 0 in Ref. [3] but is not available in Ref. [10].

A comparison of two sets of k values obtained from Ref. [3] (see Eq. (39)) and Ref. [10] (see Eq. (48)) shows that because the grain size and compactness of gravel soils in Ref. [10] are evidently larger than those in Ref. [3] (see Table 9), the k values deduced from the experimental data of Ref. [10] are larger than those of Ref. [3]. Therefore, when gravel soil is used as the cushion, k can be evaluated by factors such as the gradation and compactness of the soil, i.e., the coarser and denser the particles are, the greater the k value is. Under the experimental conditions of Ref. [10] as given in Table 8, together with the physical and mechanical parameters of gravel soil listed in Table 9, Fep values with k5% = 1.05 × 106 N/m5/2, k50% = 2.54 × 106 N/m5/2, and k95% = 6.13 × 106 N/m5/2 are predicted according to Fig. 5. Furthermore, Fmax values under the same conditions are calculated by Eq. (40) with ME = 3200 kN/m2 and Eq. (41) with λ =1000 kN/m2. The calculation parameters of Eqs. (40) and (41) for Ref. [10] are shown in Table 10. Moreover, by using the calculation method proposed in Ref. [10], i.e. using Eqs. (42), (46), and (47), FR values are calculated,

3.1. Comparative analysis of impact force Because of the unavailability of the gravel mechanical parameters such as c and φ in Ref. [10], the mechanical parameters of gravel soil are set approximately by a comparison with the mass density and gradation of the sandy gravel soil in Ref. [3]. The physical and mechanical parameters as obtained from the two references are listed in Table 9. As for the mechanical parameters of granite cube rockfall, we referred to those of common granite materials and set E2 = 50 GPa, μ2 = 0.2, and ρ = 2700 kg/m3. Similarly, the equivalent radii R0 values of the 10,160-kg and 18,260-kg falling rocks in Table 8 were set as 0.9649 and 1.1731 m, respectively. Based on Ft values listed in Table 8, the application of the backanalysis and statistic evaluations presented in Section 2 to the calculations of the 5%-, 50%-, and 95%- quintiles of k of gravel in Ref. [10] yield, respectively

k5% = 1.05 × 106N / m5/2, k50% = 2.54 × 106N / m5/2 , k95% (48)

= 6.13 × 106N / m5/2

Table 10 Calculation parameters of Eqs. (40) and (41) for Ref. [10]. Eq.(40)

Table 8 Experimental conditions and results for Ref. [10]. Test No.

Rockfall mass m [kg]

Dropping height H [m]

δt [m]

Ft [kN] (by Eqs. (46) and (47))

B.1 B.2 B.3 B.4 B.5

10,160 10,160 18,260 18,260 18,260

2.00 8.55 8.61 18.67 18.85

0.26 0.51 0.65 0.82 0.85

1566 3374 4809 8218 8005

Eq.(41)

Modulus of subgrade reaction ME = 3200 [kN/m2]; Radius of rockfall part in contact with the soil cushion Rc = 0.9649, 1.1731 [m]; Weight of rockfall W = 101.6, 182.6 [kN]; Height of fall H = 2.00, 8.55, 8.62, 18.67, 18.85 [m]. Mass of rockfall m = 10.16, 18.26 [t]; Lame coefficient λ = 1000 [kN/m2]; Height of fall H = 2.00, 8.55, 8.62, 18.67, 18.85 [m].

Note: The units of parameters in the table are specific for the calculations of Eqs. (40) and (41). 9

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Table 11 Results of different calculation methods for rockfall impact force for Ref. [10] (unit: MN). No.

Ft*

FR R5%

R50%

R95%

(FR – Ft)/Ft R = R50%

Fep k5%

k50%

k95%

(Fep – Ft)/Ft k = k50%

Fmax Eq. (40)

Eq. (41)

B.1 B.2 B.3 B.4 B.5

1.57 3.37 4.81 8.22 8.01

1.19 2.45 3.65 5.44 5.47

1.66 3.43 5.10 7.56 7.60

2.35 4.84 7.19 10.6 10.65

5.73% 1.78% 6.03% −8.03% −5.12%

0.99 2.36 3.65 5.80 5.83

1.40 3.36 5.18 8.25 8.29

2.00 4.77 7.37 11.72 11.79

−10.83% −0.30% 7.69% 0.36% 3.50%

1.07 2.57 3.66 5.83 5.86

1.10 2.64 3.91 6.23 6.26

*Ft is calculated by Eqs. (46) and (47) based on the test measured δt (see in Table 8).

in impact energy, however, Fep becomes gradually larger than FR. This is attributed to k. Fig. 10 shows that the predicted results of Fmax using Eq. (40) with ME = 3200 kN/m2 and Eq. (41) with λ = 1000 kN/m2 are notably smaller than Ft, Fep with k50%, and FR with R50%. The two values of ME and λ (3200 and 1000 kN/m2, respectively) are set relatively small. In fact, according to the trial calculation, the ME and λ of gravel soil in Ref. [10] should be set equal to 6700 and 2000 kN/m2, respectively, so as to approximate the results of Ft. We used k5%, k50%, k95%(see Eq. (48)), R5%, R50%, R95%, and Eq. (40) with ME = 6700 kN/m2 and Eq. (41) with λ = 2000 kN/m2 to predict the impact force of the 2000-kg and 20,000-kg rocks falling from different heights. The results are presented in Table 12 and Fig. 11. As can be seen from table 12 and Fig. 11(a), when m = 20,000 kg, the lower limit k5%, medium limit k50%, and upper limit k95% of Fep become larger than those of the corresponding FR with an increase in H; the medium Fep values with k50% become slightly larger than those values of Fmax calculated by Eq. (40) with ME = 6700 kN/m2 and Eq. (41) with λ = 2000 kN/m2. When m = 2000 kg (see Fig. 11(b)), the Fep values with k50% and k95% become marginally smaller than those of FR with R50% and R95% at H less than approximately 40 m; when H is larger than 40 m, the Fep values with k become increasingly larger than the FR values with R. Finally, the lower limit Fep with k5% remains slightly smaller than that of FR with R5%. Different from Fig. 11(a), majority of the Fmax results of Eq. (40) with ME = 6700 kN/m2 and Eq. (41) with λ = 2000 kN/m2 stand markedly larger than the Fep values with k50% and FR with R50%. Moreover, Fmax values deduced by Eq. (40) with ME = 6700 kN/m2 and Eq. (41) with λ = 2000 kN/m2 are almost identical. They become larger than those of Fep with k50% for relatively low impact energy (Fig. 11(b)); whereas, they become smaller for high impact energy (Fig. 11(a)). Thus, it again confirms that k reflects the strengthening characteristics of the gravel soil under impact.

Fig.10. Comparisons of predicted values of rockfall impact force among Fep, FR, and Ft in Ref. [10] and results of Fmax by Eq. (40) with ME = 3200 kN/m2 and Eq. (41) with λ =1000 kN/m2.

respectively, with R5% = 4.58 × 106 Pa, R50% = 9.22 × 106 Pa and R95% = 18.58 × 106 Pa deduced in Ref. [10]. The calculated results are listed in Table 11 and Fig. 10. Table 11 and Fig. 10 illustrate that the medium values of Fep with k50% are in good agreement with the calculated Ft values based on δt. The Fep values with k are slightly smaller than FR with R when gravitational potential energy or impact energy is lower than approximately 1000–1500 kJ. With an increase in the impact energy, the calculated Fep becomes increasingly larger than FR. Fig. 10 shows that the results of Fep would approximate FR when the impact energy (evaluated by mgH) is relatively small; with an increase Table 12 Predictions of rockfall impact force (unit: MN) for Ref. [10]. H [m]

5 20 40 60 80 100 H [m] 5 20 40 60 80 100

Rockfall m = 20,000 kg Fep k5% k50% 2.81 4.00 6.46 9.19 9.79 13.92 12.49 17.76 14.84 21.11 16.97 24.13 Rockfall m = 2000 kg Fep k5% k50% 0.52 0.74 1.19 1.70 1.81 2.57 2.31 3.28 2.74 3.90 3.14 4.46

k95%

FR R5%

R50%

R95%

Fmax Eq. (40) (ME = 6700 kN/m2)

Eq. (41) (λ = 2000 kN/m2)

5.69 13.06 19.80 25.25 30.01 34.31

2.94 6.00 8.78 11.11 13.24 15.25

4.13 8.32 11.97 14.93 17.54 19.95

5.82 11.67 16.64 20.57 23.97 27.04

3.88 8.91 13.50 17.22 20.46 23.39

3.91 8.98 13.60 17.35 20.62 23.58

k95% 1.05 2.41 3.66 4.67 5.55 6.34

FR R5% 0.63 1.29 1.88 2.39 2.85 3.28

R50% 0.88 1.79 2.57 3.21 3.77 4.29

R95% 1.25 2.51 3.58 4.42 5.16 5.82

Fmax Eq. (40) (ME = 6700 kN/m2) 0.84 1.92 2.91 3.71 4.41 5.04

Eq. (41) (λ = 2000 kN/m2) 0.84 1.93 2.93 3.74 4.44 5.08

10

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Fig. 11. Estimates of impact forces for various rock falling heights for Ref. [10]: (a) m = 20,000 kg, (b) m = 2000 kg.

It must be pointed out that the results presented in Table 11 and Fig. 10 are inadequate for the evaluation of influences of the strain rate and pressure dependencies on the impact force values predicted in Section 2.5 owing to two factors: The Ft value in Table 8 is calculated by inputting the measured penetration depth from Ref. [10] into Eqs. (46) and (47); whereas, the Ft value in Table 3 is derived from the measured maximum acceleration of rockfall, namely, Ft = mamax. In addition, Ref. [10] provides less data.

and 6% larger than those with R5%, R50%, and R95%, respectively. By analogy, for cases with m = 2000 kg in Fig. 13(b), the upper, medium, and lower limit penetration depth values, respectively, with k5%, k50%, and k95% are 23–42%, 14–37%, and 8–35% larger than those with R5%, R50%, and R95%, respectively. On average, these values are 27%, 21%, and 18% larger than those with R5%, R50%, and R95%, respectively. Table 14 and Fig. 13 show that as the impact energy increases, the predictions for δR with R gradually approximate those δmax with k. Based on robustness considerations, the designed thickness of the gravel cushion on the top of the shed structure should be 2 m for m = 2000 kg and H = 100 m, and 3 m for m = 20,000 kg and H = 100 m when the evaluated upper limits of δmax with k5% are taken into account. It is to be noted that the penetration depth calculation method in this paper is based on the Hertz theory, which is appropriate for small indentations only. The calculated medium values and test results for the penetration depth are less than the half of the side length or equivalent radius of rockfall in Ref. [10], which implies that the rockfall does not penetrate the gravel soil completely. Therefore, one should apply the method proposed herein for very large penetrations with significant caution. Some studies (Refs. [24] and [25], for example) on large indentation elastoplastic models may provide alternative analyses and methods.

3.2. Comparative analysis of impact penetration depth Using the methods given in Fig. 5 and Eq. (42), respectively, give δmax and δR with 5%-, 50%-, and 95%-quantiles of k and R corresponding to the test conditions in Ref. [10]. These results are listed in Table 13 and Fig. 12, together with the experimentally measured values of δt in Ref. [10]. As is clear from Table 13 and Fig. 12, all experimentally obtained values of δt fall within the predicted ranges of δmax with k and δR with R. The δmax values are evidently larger than their corresponding δR values. The δmax values with k50% are, on average, 20% larger than the experimental values of δt; whereas, the difference between δR values with R50% and their corresponding δt is within 10%. The penetration depth predicted by k50% is noticeably larger than the measured value, which is in agreement with the evaluation in Section 2.5; i.e., the predicted penetration depth is larger than the actual value when the constitutive model in Fig. 2 does not consider pressure dependencies. With k5%, k50%, and k95% of Eq. (48), the predicted penetration depths of 2000-kg and 20,000-kg rockfalls from 5 to 100 m height are in contrast with those with R5%, R50%, and R95% in Ref. [10], as shown in Table 14 and Fig. 13. For cases with m = 20,000 kg in Fig. 13(a), the upper, medium, and lower limit penetration depth values, respectively, with k5%, k50%, and k95% are 11–28%, 3–27%, and −1–26% larger than those with R5%, R50%, and R95%, respectively. On average, these value are 15%, 10%, Table 13 Results of penetration depth (unit: m) for Ref. [10]. No.

B.1 B.2 B.3 B.4 B.5

δt

0.26 0.51 0.65 0.82 0.85

δmax k95%

k50%

k5%

(δmax – δt)/δt [%] k50%

0.25 0.45 0.52 0.71 0.72

0.35 0.63 0.74 1.01 1.02

0.50 0.90 1.06 1.44 1.45

34.62 23.53 13.85 23.17 20.00

δR R95%

R50%

R5%

(δR – δt)/δt [%] R50%

0.17 0.35 0.43 0.63 0.64

0.24 0.50 0.61 0.89 0.90

0.35 0.71 0.86 1.24 1.25

−7.69 −1.96 −6.15 8.54 5.88

Fig. 12. Penetration depth versus rockfall impact energy for Ref. [10]. 11

International Journal of Impact Engineering 136 (2020) 103411

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Table 14 Estimation of penetration depths (unit: m) for Ref. [10]. H [m]

Rockfall mass m = 20000kg k5% k50% k95%

R5%

R50%

R95%

Rockfall mass m = 2000kg k5% k50% k95%

R5%

R50%

R95%

5 20 40 60 80 100

0.87 1.52 2.00 2.36 2.64 2.89

0.68 1.32 1.80 2.13 2.38 2.58

0.48 0.95 1.32 1.58 1.80 1.97

0.34 0.68 0.95 1.15 1.31 1.45

0.47 0.82 1.08 1.28 1.43 1.56

0.32 0.61 0.84 0.99 1.11 1.20

0.22 0.44 0.61 0.74 0.83 0.92

0.16 0.31 0.44 0.53 0.61 0.68

0.61 1.07 1.41 1.66 1.86 2.03

0.43 0.75 0.99 1.17 1.31 1.43

0.33 0.58 0.76 0.90 1.01 1.10

0.23 0.41 0.54 0.63 0.71 0.77

Fig. 13. Penetration depth predictions for Ref. [10]: (a) m = 20,000 kg, (b) m = 2000 kg.

4. Summary and comments

properties of gravel soil. Therefore, ME and λ should be described as ranges of values.

A method for calculating rockfall impact force and penetration depth, based on the Hertz contact theory and considering elastoplastic strengthening of sand soil, is proposed. Using the experimental values of rockfall impact force, the range of strengthening coefficient of gravel soil was obtained by back analysis and statistical evaluation. Then, by a comparative analysis, the recommended strengthening coefficient was verified. The verification of the value confirms that the impact force prediction proposed in this study is reliable, as well as the range of predicted values is in agreement with the real situation. The following conclusions can be drawn from this study.

4.2. Rockfall impact force This study, considering the strengthening of cushion soil, proposes a theoretical analytical expression for rockfall impact force based on the Hertz contact theory. In addition, the consideration of the discreteness of soil enlarges the estimated range of rockfall impact force, which approximates the tested values measured in a related reference. However, with an increase in height or impact velocity of the rockfall, the predicted median value is smaller than the measured one because of the influences of pressure dependencies and strain rate. Because the lower, medium, and upper limits of rockfall impact force can be calculated from their corresponding quantile k values, the calculation of impact load is possible after the relationship between the impact force acting on the soil cushion and impact load transferred from the soil cushion acting on the structure is established. This can render a robust reliability design of the protective structure feasible. However, the establishment of this relationship needs to be further explored in the future.

4.1. Strengthening coefficient of gravel soil When a rockfall occurs, the interacting force between the falling rock and the cushion layer increases with an increase in strengthening coefficient k; however, the depth of the rock fall penetrating the soil becomes smaller. In addition, k is closely related to the gradation and compactness of the soil. The analysis of the data in the study shows that the effect of the strain rate on k, though less noticeable, exhibits an increasing trend with an increase in impact velocity. In contrast, under the dynamic impact of the falling rock, the trend displayed by the pressure dependencies is comparatively more pronounced, i.e., the k value of gravel soil increases with an increase in impact force. Based on the analyses of two kinds of gravel soil, the proper range of k can be identified as k ∈ [5 × 105 N/m5/2, 1 × 107 N/m5/2]. The influences of pressure dependencies and strain rate on k require a consideration in the evaluation of the rockfall impact. Moreover, similar to the strength-like indentation resistance R and k, the modulus of subgrade reaction ME of Eq. (40) and the Lame coefficient λ of Eq. (41) are both related to the physical and mechanical

4.3. Penetration depth by rockfall impact The penetration depth is a key issue to be considered in the rock protection structure design. When the penetration depth reaches or exceeds the thickness of the cushion layer, it can damage the structure. The impact force is negatively correlated with the penetration depth; soil being soft, the impact force and impact load decrease, but the penetration depth increases, and vice versa. Therefore, in the design of cushion on a structure, compactness and gradation of soil materials should be well balanced. 12

International Journal of Impact Engineering 136 (2020) 103411

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The theoretical formula established in this study can be used to evaluate penetration depth. In fact, compared with the existing experimental values, the medium predicted results by using the proposed formula are, on average, 20% larger than the experimental values, which is applicable in designs considering uncertainties and robustness. The fact that the penetration depths predicted herein are larger than the measured ones also proves that the mechanical behaviour of the rockfall impact on gravel soil is indeed subjected to effects of pressure dependencies.

interest that represents a conflict of interest in connection with the paper submitted. Acknowledgements Wang YS is indebted to Professor XZ Hu of the University of Western Australia (UWA) for guidance as a supervisor at UWA. He also acknowledges the support from the Science and Technology plan project in Sichuan Province, China (no. 2013GZ0047). Wang MN would like to acknowledge the National Natural Science Foundation of China (no. 51878567, 51578458 and 51878568.) for its support.

Declaration of Competing Interest We declare that we do not have any commercial or associative Appendix. Detailed derivation of Eq. (29)

Fe =

4 E R0 3

3 2

p (r ) =

r2 a2

3Fe 1 2 a2

1 2

a y2 a =

ap = a2

R0

Calculation:

Fep = Fe

ap

2 ·

p (r )

0

py

r2 2R 0

k

rdr

y

Substitute Fe into the above integration equation

Fep =

4 E R0 3

3 2

2 ·

ap 0

ap

[p (r )] rdr + 2 ·

0

(py ) rdr + 2 ·

ap 0

r2 rdr 2R 0

k

ap

2 ·

0

k

y rdr

Substitute p(r) in the above equation, and integrate the constant terms of the integral formula:

Fep =

4 E R0 3

4 E R0 3

3 2

4 = E R0 3

3 2

=

3 2

2 ·

3Fe 2 a2

ap 0

a2 2

3Fe · a2

ap 0

r2 a2

1 3 ap 2

r2 a2

3F 2 + e· · 1 2 3

r2 a2

1

+

1 2

1 2

ap

rdr + 2 ·

0

r2 a2

d 1

(py ) rdr + 2 k·

0

+ ·py · r 2 ap + 2 k·( R 0 ) 0

2 2 kR0· 3

·py · ap2

r2 rdr 2R 0

ap

0

3 ap 2

r2 2R 0

k·

2 k·

ap 0

r2 d 2R

ap 0

y rdr

r2 2R 0

k·

2 ap y ·r 0

2 y · ap

0

Substituting ap2 = a2 – ay2 into above equation, we obtain

Fep =

4 E R0 3

3 2

+ Fe ·

ay

3

a

1 + · py ·(a2

a y2

a2

4 kR0 · 3

a y2)

3 2

3 2

2R

k·

2 y ·(a

a y2 )

Next, substitute a2 = R0∙ δ into the above equation:

Fep =

=

4 E R0 3

4 E R0 3

3 2·

= Fy + R0 (

3 2

+ Fe ·

y

3 2

y

+ R0 (

y )(py

k

3 2

1 + ·py · R0 ·(

y )(py

y) +

k

4 kR0 · 3

y)

3 2

+

4 kR 0 · 3 +

+

4 kR0 · 3

y)

y

3 2

y

3 2

2

+

y

3 2

k·

y · R 0 ·(

y)

3 2

2

3 2

2

[3] Montani S, Descoeudres FO, Labiouse V. Experimental Study of Rock Sheds Impacted by Rock Blocks. Structural Engineering International 1996;6(, 3):171–6. [4] Siming H, Xinpo L, Yong W. Research on yield property of soil under rock-fall impact. Yanshilixue Yu Gongcheng Xuebao/Chinese Journal of Rock Mechanics and Engineering 2008;27:2973–7. [5] Siming H, Xinpo L, Yong W. Theoretical model on elastic-plastic granule impact. Gongcheng Lixue/Engineering Mechanics 2008;25:19–24. [6] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1985.

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