To gain insight into the mechanism and structural characteristics of the traditional ramming technology, the trend of the ramming stress during the process of ramming, and the transmission law and influence rule of the stress wave in the laminated layer are monitored by measuring the ramming stress. The recovery coefficient of different ramming times and the action mechanism of the ramming stress field of the ramming hammer are discussed in detail based on dynamics and elastoplastic theory to reveal the action mechanism and scientific connotation of the traditional tamping technology.

### Obtaining numerical simulation parameters

To simulate the real ramming situation, the mechanical properties of the rammed soil layer were evaluated by changing the internal friction angle and cohesion of the rammed soil to the mechanical properties of the soil at various ramming cycles. In the numerical simulation, the No. 2 rammer was selected, the thickness of the layer was 12 cm, and the effects corresponding to eight ramming times were achieved. The mechanical properties of the soil were obtained according to triaxial tests.

The triaxial tests’s cylindrical sample (three samples in all) had radii equal to 50 mm and heights equal to 100 mm, and moisture contents of 13.8% and densities of 1.65 g/cm^{3} were achieved. The static triaxial test was conducted using a Global Digital Systems Instruments (GDS) unsaturated triaxial apparatus to measure the constitutive relationship of the rammed soil materials used in this test. The tests were conducted at the confining pressures σ_{3} of 50 kPa, 100 kPa, 150 kPa, and 200 kPa. The values of σ_{1} were measured to be 190.95 kPa, 307.71 kPa, 358.74 kPa, and 506.76 kPa, respectively. According to the requirements, the test values can be analyzed with the Mohr circle diagram (Fig. 12), and the cohesive force of the soil sample in this state was 25.14 kPa, the friction angle was 30.23°.

According to the stress–strain curve (Fig. 13) of the sample at different confining pressures σ _{3} during the test, the section with a better elastic state was eliminated from the resulting curve, and its elastic modulus after fittings were 11.05, 15.74, 17.30, and 20.66 MPa, respectively. The average value was 16.19 MPa. Table 6 shows the parameters of soil samples used in the simulation.

### Modeling and analysis of mechanism

The finite element software ABAQUS (version 2016, Dassault Systems) was used for simulations. To reduce the number of calculations, the rammer was set as a rigid body, and the rammer and covering soil model was simplified to a planar two-dimensional model. According to axisymmetric conditions, 1/2 rammer and covering soil were used for modeling. The finite element modeling adopted the linear reduction integral unit CAX4R, the total number of units of the rammer were 66 and 82 nodes (along the Y and X direction, respectively), and the total number of units that covered the soil were 2400 and 2525 nodes (along the Y and X directions respectively). Analysis of the stone pestle unearthed from the No. 3 mausoleum (Western Xia Imperial Tombs) indicated that its weight was 3.34 kg, its diameter was 14.17 cm, its height was 14.54 cm, its volume was 1632.51 cm^{3}, its density was 2.047 g/cm^{3}, and the material was sandstone. According to its curvature and diameter, a hemispherical rammer with a diameter of 12 cm and a mass of 4.03 kg was used for simulation. The height of the falling distance was 0.25 m, the thickness of the covering soil was 12 cm, while the specific simulation model is shown in Fig. 14.

#### Basic assumptions

1. The rammed soil layer was homogeneous, continuous, and isotropic, 2. ply was regarded as a semi-infinite space in the horizontal direction, 3. the rammer itself did not rotate, and 4. the rammer was in free fall Fig. 15.

#### Theoretical calculation and analysis

Hertz studied the maximum impact force of two elastic spheres with masses equal to ({m}_{1}) and ({m}_{2}) when these collided at speed (v), and expressed it with the following equation [28]:

$$F_{m} = k^{2/5} left[ {frac{5}{4}v^{2} frac{{m_{1} m_{2} }}{{m_{1} + m_{2} }}} right]^{3/5}$$

(4)

where (k) is given by the following equation:

$$k = frac{4}{3pi }sqrt {frac{{r_{1} r_{2} }}{{r_{1} + r_{2} }}} frac{1}{{C_{1} + C_{2} }}$$

(5)

$$C_{1} = frac{{1 – mu_{1}^{2} }}{{pi E_{1} }} , C_{2} = frac{{1 – mu_{2}^{2} }}{{pi E_{2} }}$$

(6)

where ({text{r}}_{{1}}) and ({text{r}}_{{2}}) are the radii of the two spheres, ({E}_{1}) and ({E}_{2}) are the elastic moduli of the two spheres, and ({mu }_{1}) and ({mu }_{2}) are the Poisson’s ratios of the two spheres. When the rammer impacted the foundation, the rammer was not necessarily a sphere and the foundation soil was not an elastomer, but some qualitative analysis can be made by using Hertz’s collision theory. The deformation modulus ({E}_{1}) of the No. 1 rammer was much larger than the deformation modulus ({E}_{2}) of the foundation soil, i.e., ({E}_{1}=infty). The foundation soil can be regarded as a half-space elastic deformation body, thus, ({r}_{2}=infty), and ({m}_{2}=infty). In addition, we set (m1=M), ({E}_{2}={E}_{s}), ({r}_{1}=R), and ({mu }_{2}=mu). Therefore, Eq. (3)becomes

$$C_{1} = 0 , C_{2} = frac{{1 – mu_{{}}^{2} }}{{pi E_{s} }}$$

(7)

In Eq. (7), ({E}_{s}) and (mu) are the deformation modulus and Poisson’s ratio of the foundation soil, respectively. Thus, Eq. (2) can be changed to,

$$k = frac{4}{3}R^{frac{1}{2}} frac{{E_{s} }}{{1 – mu^{2} }}$$

(8)

In Eq. (5), R is the radius of the rammer. If Eq. (5) is substituted in Eq. (1) and ({text{v}}^{{2}} {text{ = 2gh}}) is considered, then Eq. (1) can be simplified to

$$F_{m} = 1.944R^{frac{1}{2}} m^{frac{3}{4}} left[ {frac{{E_{s} }}{{1 – mu^{2} }}} right]^{frac{2}{5}} left( {gh} right)^{frac{3}{5}}$$

(9)

In the experiment, the contact area between the rammer and the paving layer was equivalent to the projected area S of the surface where the diameter of the shaped rammed nest was located. Thus, the maximum impact stress of the rammer was

$$sigma_{{text{m}}} { = }frac{{F_{m} }}{s}$$

(10)

Based on measurements, we found that the projected area of the ramming pit was mainly affected by the weight of the rammer, while the influence of the soil-paving thickness on the projected area could be ignored. This is owing to the different potential energies generated by different rammers subject to the same condition, that is to say, the impact energies on the paving soil were different. The specific expression is as follows,

$$s = pi R^{2} left{ begin{gathered} left( {0.13 + 0.2sin left( {0.5pi dN} right) + 0.9e^{{ – 0.16N^{2} }} } right)left( {frac{{m_{0} }}{m}} right)^{frac{1}{2}} left( {frac{d}{0.12}} right)^{frac{3}{10}} hfill \ 0.24 + 6sin left( { – 0.0004pi dN} right) + 0.6e^{{ – 0.09N^{2} }} left( {frac{d}{0.2}} right)^{frac{3}{10}} hfill \ end{gathered} right.$$

(11)

The problem of collision between the rammer and foundation soil is different from that of the general elastomer and elastoplastic body because there are not only elastic and plastic deformations, but also viscosity, hardening, and friction energy dissipation in the collision process between the rammer and the foundation soil [29,30,31]. The maximum impact stress of the rammer must be reduced owing to the collision characteristics described above. According to the analysis of the calculation method of impact stress Eqs. (3–11), it is concluded that different rammer diameters (weight, cross-sectional area) and layer thicknesses mainly affect the internal friction angle and cohesion of the buffer layer. Therefore, the maximum impact stress must be corrected, and a parameter influence coefficient λ is derived. The actual maximum impact stress value is obtained by multiplying the calculated result of the elastoplastic Hertz contact theory by a parameter λ (parameter influence coefficient). The calculation formula is as follows

$$sigma = lambda sigma_{m}$$

(12)

The parameter influence coefficient λ was calculated according to the experimental data subject to the working condition of rammer No. 1 and a paving soil thickness equal to 12 cm. Polynomial fitting was conducted with the use of the calculated parameter influence coefficient as follows,

$$lambda { = }left{ begin{gathered} frac{{text{a}}}{{3 + e^{0.1N} }}{text{ 2R > d }} cap {text{m}} le {text{3m}}_{0} hfill \ frac{a}{{1 + e^{0.2N} }}{text{ others}} hfill \ end{gathered} right.$$

(13)

and

$${text{a}} = left{ begin{gathered} 0.3left( {frac{mH}{{m_{0} d}}} right)^{frac{4}{5}} {text{N}}^{{frac{13}{{10}}}} left( {0.1 + 0.2sin left( {frac{dNpi }{2}} right) + e^{{ – 0.16N^{2} }} } right) , 2R le d cap m le 3m_{0} hfill \ , 0.25frac{m}{{m_{0} }}N^{frac{1}{2}} frac{H}{d} , 2R > d cap m < 3m_{0} hfill \ , 0.2{(}frac{m}{{m_{0} }}{)}^{frac{2}{5}} N^{frac{4}{5}} {(}frac{H}{d}{)}^{frac{1}{2}} , others , hfill \ end{gathered} right.$$

(14)

where *m*_{0} is the minimum weight of the rammer subject to this working condition and is equal to 3.32 kg, H is the drop height of the rammer, d is the thickness of the paving soil, and m is the weight of the rammer. Additionally, a is the dimensionless coefficient, and the correction factor λ is the dimensionless parameter. Fig. 16 s shows the effect of soil thickness and rammer quality on parameter a. In the figure, X1 represents the paving thickness, X2 represents the rammer mass, and Y represents parameter a. When the rammer weight increases, a increases; when the paving thickness increases, a decreases.

Comparison of the theoretical calculation value and the actual value obtained from Eqs. (13) and (14), and the fitting curve in Fig. 17, it can be concluded that the overall trend of different working conditions is that the error decreases gradually as a function of the ramming times. This is attributed to the fact that as the number of ramming times increases, the layer density and elastic modulus increase, while as the transmission energy consumption of the impact wave decreases, the detection accuracy of the sensor increases, and the theoretical calculation value becomes closer to the actual contact force. Based on the test results, we found that when the thickness of the paving soil was 20 cm, the consolidation and stratification of the paving soil layer would occur after ramming with the No. 1 and No. 2 rammers. This means that there is no uniform compaction within the range of the paving soil thickness. It has been proved that the ramming energies of the No. 1 and No. 2 rammers were not sufficient to be transmitted to the bottom part of the rammed layer, or the energy transmitted to the bottom part was too small to be rammed. To achieve the ideal rammed state, the ramming energy of the rammer should be increased and analyzed according to actual operation and measured data. When the No. 1 and No. 2 rammers rammed the layer that had a thickness of 20 cm, the measured impact stress was less than 300 kPa. Therefore, when the thickness of the rammed layer is fixed, the actual impact stress value can be calculated with Eqs. 13 and 14. The deviation of the data of some paving thickness was large that was mainly owing to the influences of the soil particles, ramming technology, and other aspects in the test process, showing that the adaptability of the rammer used was not high and the error was large.

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