Practical Methods for Selecting the Parameters of a Vibratory Pile Driver

Editor’s Note: this is the original presentation of the “Savinov and Luskin Method” for sizing a vibratory driver for a specific application. It has been reproduced in many publications and in various forms. I would remind my readers that the method is presented here in Those Pesky Kilogram-Force Units.

For successful pile driving in each case it is necessary that:

the magnitude of the disturbing forces of the vibrator were sufficient to overcome the resistance of the soil to the failure of the pile at a given maximum depth of its immersion;
the amplitude of the pile vibrations sufficiently exceeded the value of the initial amplitude;
the value of the resultant of all external forces applied to the pile was sufficient to ensure the required speed of immersion (or extraction) of the pile.

Let us consider each of these conditions separately.

The first condition can be represented as a simple dependence:

$P_o = \frac{K \omega^2}{g} \geq x_o T_{kr}$ (1. 16)

Where

$K,\omega$ – respectively, as before, the moment of the vibrator eccentrics and the angular velocity (frequency) of their rotation;
$T_{kr}$ – the total calculated value of the critical resistance to pile failure at the maximum depth of its immersion;
$x_o$ – some coefficient that approximately takes into account the effect of soil elasticity.

As we have shown earlier, the effect of soil elasticity acts in different ways at different vibration frequencies. For low-frequency vibratory drivers ( $\omega = 30-60 \frac{rad}{sec}$ ) used for driving heavy reinforced concrete piles and manholes, the $x_o$ value is recommended to be = 0.6-0.8, and for high-frequency, such as, for example, vibratory pile drivers, wooden piles, etc., = 1. When designing low-frequency vibratory drivers with adjustable vibration frequency, it is allowed to reduce the value of $x_o$ to 0.4-0.5.

The value of $T_{kr}$ can be determined from the expressions:

a) for piles

$T_{kr}=s{\sum}^{n}_{i=1}\tau_{ikr}h_{i}$ (1.17)

b) for sheeting

$T_{kr}={\sum}^{n}_{i=1}\tau`_{ikr}h_{i}$ (1.18)

Here, the index i denotes the serial number of the soil layer with the thickness $h_{i}$ , passed by the pile when immersed, and n is the total number of layers.

Thus, the maximum immersion depth is $h_{max}={\sum}^{n}_{i=1}h_{i}$ . Concepts of specific critical resistance $\tau_{ikr}\left(\frac {t}{m^{2}}\right)$ and $\tau`_{ikr}\left(\frac{t}{m}\right)$ were discussed in detail in § 2 (in this case these values are also provided with the index i, which means that they are related to the soil of the i-th layer.) Finally, as before, s denotes the perimeter of the pile cross section.

In the simplest case, when piles are driven into homogeneous soil, Equations (1.17) and (1.18) take the form:

$T_{kr} = sh_{max} \tau_{kr}$ (1.17a)

And

$T_{kr} = h_{max} \tau`_{kr}$ (1.18a)

In the second paragraph, we presented some experimental data characterizing the values of $\tau_{kr}$ and $\tau`_{kr}$ . Table 8 shows the values of $\tau_{kr}$ and $\tau`_{kr}$ for various soils, recommended for practical use when selecting the parameters of pile vibratory drivers for various purposes.

\tau_{kr},\frac{t}{m^2} — Table 8 Numerical values of $\tau_{kr}$ and $\tau`_{kr}$ recommended for selecting the parameters of vibratory pile drivers

\tau`_{kr},\frac{t}{m} — Table 8 Numerical values of $\tau_{kr}$ and $\tau`_{kr}$ recommended for selecting the parameters of vibratory pile drivers

Let us move on to the second condition. Since, as we have shown above, in the case of a complete breakdown of the oscillation amplitude, the piles turn out to be very close to the $A_{\infty}$ value corresponding to the case of the absence of pile-to-soil connections, this condition can be represented in the following form:

$\xi \frac{K}{Q_o} \ge A_{(0)}$ (1. 19)

Where

$Q_o$ – weight of the pile, vibrator and other parts of the vibrator rigidly connected to the pile;
$A_{(0)}$ – required amplitude;
$\xi$ – some dimensionless coefficient taken equal to 0.8 for reinforced concrete piles, and 1 for all others.

So that soil elasticity does not have a significant adverse effect on the pile driving process, the value of $A_{(0)}$ should be at least 4-5 times greater than the value of the initial amplitude. This requirement is met by the data placed in Table 9. The table shows that the higher the vibration frequency, the lower the recommended value of $A_{(0)}$ . In a rough approximation, we can assume that the amplitude of the oscillation velocity for successful pile driving should be approximately equal to 0.5-0.8 m/s. If this circumstance is taken into account, then Equation (1.16) can be represented as:

$\omega \ge \frac{x_o g T_{kr}}{Q_o v_{(0)}}$ (1.16a)

where $v_{(0)}$ is the required oscillation velocity amplitude.*

*From the formula, by the way, it is clearly seen that the greater the weight of $Q_o$ , the lower the vibration frequency should be, and vice versa.

The third and last condition can be considered practically fulfilled when:

$Q \ge p_{(0)}F$ (1.20)

$v_1 < \frac{Q}{P_o} < v_2$ (1.21)

Where

$Q$ – weight of the pile, vibrator and additional weights (if any);
$F$ – cross-sectional area of the pile;
$p_{(0)}$ – required specific pressure on the pile;
$P_o$ – the amplitude of the dynamic force of the vibrator;
– values of coefficients:
- for steel sheet pile – $v_1 = 0.15, v_2 = 0.5$ ;
- for light (wooden, tubular steel) piles – $v_1 = 0.30,v_2 = 0.6$
- for heavy (reinforced concrete) piles and wells – $v_1 = 0.40,v_2 = 1.0$

For successful pile driving, it is practically sufficient that the value of $p_{(0)}$ be 1.2-1.5 times higher than the value of the initial pressure (see § 2.)

Recommended $p_o$ values are given in Table. 10.

A_{(0)},mm — Table 9 Recommended values of vibration amplitudes required for efficient driving of sheet piles and piles

p_{(0)},\frac{kg}{cm^2} — Table 10 Recommended values of the required pressure (for piles immersed in water-saturated sandy and weak clay soils)

The selection of the parameters of the pile vibratory driver must be carried out in the following order:

a) based on the initial data characterizing the soil conditions, using Equation (1.17) or (1.18) and Table 8 to determine the total calculated value $T_{kr}$ of the critical resistance to pile failure at a given maximum depth $h_{max}$ of its insertion;

b) outline the approximate value of the weight $Q_o$ of the pile and parts of the vibratory driver rigidly connected to it, select according to Table 9 values $A_{(0)}$ of the required amplitude and, taking the corresponding value of the coefficient $\xi$ included in Equation (1.19), calculate the approximate value K of the moment of vibrator unbalances according to the formula:

$K = \frac {A_{(0)}Q_o}{\xi}$ (1. 22)

and then according to the formula

$\omega = \sqrt{\frac{gT_{kr}}{K}}$ (1.23)

calculate the required vibration frequency $\omega$ (angular speed of rotation of vibrator eccentrics.)

In cases where at first it is not clear in what range the frequency $\omega$ will be, first you should determine this frequency using Equation (1.16a), and then find the eccentric moment from the expression:

$K = \frac{Q_o v_{(0)}}{\xi \omega}$ (1.22a)

c) using Table 10, determine the required minimum weight of the vibrator according to the formula (1. 20) and check whether this weight satisfies the condition expressed by the formula (1.21.) If it turns out that the weight obtained by formula (1. 20) $Q < v_1P_o$ then increase it to $Q = v_1P_o$ ; if it turns out that $Q > v_2P_o$ then increase the magnitude of the perturbing force to such a limit that condition (1. 21) is satisfied;

d) finally set the moment of the eccentrics and the number of revolutions of the vibrator, as well as the weight of the vibrator and, if necessary, additional surcharge, and then check the finally adopted parameters using Equations (1.16), (1.19), (1.20) and (1.21.)

Having the data obtained as a result of the selection of the main parameters of the vibratory driver, it remains to calculate the required engine power and complete the calculation.

The formula for calculating the required engine power is usually given as follows:

$W=\frac{\sum^k_{j=1}W_{j}+W_{o}}{\eta_{per}}$ (1.24)

Where

$\sum^k_{j=1}W_{j}$ is the total power required to overcome the resistances in the vibrator mechanism;

$W_{o}$ – power required to overcome ground resistance;

‘ $\eta_{per}$ – coefficient taking into account energy losses in transmission from the engine to the vibrator shafts.

The value l: $\sum^k_{j=1}W_{j}$ takes into account the energy losses: on the one hand, the friction in the bearings of the shafts carrying the eccentrics, and on the other hand, processes accompanying the operation of the vibrator: churning of oil in the vibrator housing, vibrations of machine parts, overcoming air resistance, etc. The power required to overcome friction in bearings can be determined by the formula:

$W_n = 5P_odn_of\times10^{-6}\,kW$ (1. 25)

Where

$P_o$ – amplitude of the perturbing force in kg:
$d$ – diameter of vibrator shaft pins in cm;
$n_0$ – vibrator revolutions per minute;
$f$ – coefficient of friction of rolling bearings, reduced to the diameter of the shaft journal; the value can be taken equal to 0.01.

As for other resistances that arise in the vibrator mechanism, the power required to overcome them cannot be accurately determined. It is obvious, however, that these resistances depend on the same factors as the resistances taken into account by Equation (1. 25,) the magnitude of the disturbing force and the number of revolutions of the vibrator. Measurements made by M. G. Tseitlin [61] showed that the power required to overcome friction in bearings and the power required to overcome other resistances in the vibrator mechanism are approximately the same. Thus, in practical calculations, it can be assumed that

$\sum^k_{j=1}W_{j} = P_odn_of\times10^{-5}\,kW$ (1.26)

The calculation of the coefficient $\eta_{ner}$ can be made according to the instructions that are available in mechanical engineering reference books; in practical calculations, its value can be taken equal to 0.9.

E. M. Sinelnikov [55] proposed the following approximate formula for determining the power $W_o$ required to overcome the soil resistance:

$W_{0max} = \frac{K^2\omega^3}{4Q_o}\times 10^{-7} kW$ (1.27)

This formula can be used when selecting the parameters of pile vibratory drivers. Taking into account that during its derivation, energy losses due to vibrations of the soil mass were not taken into account, it is advisable in each case to increase the value of $W_{0max}$ obtained by Equation (1. 27) by 10-20%.

In conclusion, we give several examples of the selection of parameters for pile vibratory pile drivers for various purposes.

Example 1. Select the parameters of a metal sheet pile vibrator based on the following data: sheet pile type -SHP-I; the maximum depth of immersion in water-filled sandy and weak clay soils with clay or gravel layers is 15 m; the largest weight of one sheet pile (with a length of 20 m) is 1400 kg.

According to tables 8 and 9, we accept:

$\tau`_{kr} = 1.7 \frac{t}{m},17 kg-cm, A_{(0)} = 5\,mm$ .

We accept the weight of the vibrator approximately 700 kg. We find:

a) the value of the total critical resistance to stall:

$T_{kr} = 1500 \times 17 = 25500\,kg$

b) moment of vibrator eccentrics:

$K = 0.5(1400+700) =1050\,kg-cm$ ;

c) oscillation frequency:

$\omega = \sqrt{\frac{981 \times 25500}{1050}} =153\frac{rad}{sec}$

$n_o = 153 \times 9.55 = 1460\,RPM$

$K = 1000\,kg-cm;\,n_o = 1500\,RPM$

d) determine the required weight of the vibrator.

In this case, condition (1.21) is decisive.

The total weight of the vibratory driver with the pile according to this condition should be equal to: $Q = 0.15 \times 25000 = 3500 kg$ .

Therefore, the weight of the vibrator itself must be at least:

$Q_B = 3500-1400=2100\,kg$

Given that the required weight of the vibrator is 700 kg, the design of the vibrator should include a static load equal to;

$2100 - 700 = 1400\,kg$

Example 2. Check whether it is possible to use a vibratory driver, the parameters of which were selected in the first example, to plunge a steel sheet pile of the Larsen V type (weight $102 \frac{kg}{m}$ ) 12 m long to a depth of 10 m into the following soils:

0.0 – 4.0 – water-saturated medium-grained sand;
4.0 – 7.5 – fine-grained sand with separate layers of weak clay soil;
7.5 – 10 – hard plastic clay soil.

According to the table 8 we accept the values $\tau`_{kr}$ for the first layer – 1.4 t/m, for the second – 2 t/m and for the third – 2.5 t/m.

We determine the total value of the critical stall resistance:

$T_{kr} = 1.4 \times 4.0 + 2.0 \times 3.5 + 2.5 \times 2.5 = 18.8 t < 25.0 t$ .

The amplitude of fluctuations of the tongue will be equal to:

$A = \frac{1000 \times 10}{700+12 \times 102} = 5.2 > 5\,mm$ .

Thus, a vibratory hammer with the specified parameters for the conditions under consideration is suitable.

Example 3. Select the parameters of a vibratory driver for reinforced concrete shell piles. Pile dimensions: outer diameter = 1.5 m, wall thickness = 0.1 m, maximum length = 30 m. Weight = 34 t; maximum immersion depth = 25 m, including: in weak clay soils = 10 m, in hard-plastic clays = 15 m. The weight of the vibrator is assumed to be 10 tons.

According to the table 8 we set the values of $\tau`_{kr}$ for the upper soil layer = 0.5 t/m and for the lower one = 1 t/m.

We define:

a) the total value of the critical shear resistance, assuming that the height of the unexcavated soil column inside the pile cavity does not exceed 3 m, and assuming $x_o = 0.8$ ,

$T_{kr} = 0.8 \times [(15 + 3) \times 3.14 \times 1.5 \times 1.0 + 10 \times 3.14 \times 1.5 \times 0.5] = 87\,t$ ;

b) the moment of the eccentrics of the vibrator, taking according to Table 9 $A_{(0)} = 6\,mm$ :

$K = \frac{0.6}{0.8} (34000 + 10000) = 33000 kg-cm$ ;

c) vibration frequency:

$\omega = \sqrt{\frac{981 \times 87000}{33000}}=51\frac{rad}{sec}$

d) the required weight (at the required pressure $p_o = 6 \frac{kg}{cm^2}$ :

$Q = \frac{\pi}{4} (1.5^2-1.3^2) \times 60 == 26.6\,t$ .

In fact, the weight of the pile and the vibrator is 44 tons; we make an additional check according to the formula (1.21):

$0.4\times87 = 34.8 t < 44 < 0.9\times 87 = 78.3\,t$ .

The test results are satisfactory.

However, the results obtained should be regarded as indicative. To work in various ground conditions, it is necessary to be able to change the vibration frequency in the range from $30 \frac{rad}{sec}$ to $60 \frac{rad}{sec}$ .

The eccentric moment must be adjustable, and for operation at low frequencies it is desirable to increase it by at least 20%, i.e. bring it up to 40,000 kg-cm,

Example 4. Select the parameters of a vibrator for wooden piles with the following initial data: the maximum depth of the pile = 12 m; the largest diameter of the average cross section = 26 cm; estimated weight = 0.8 tons.

Soils up to a depth of 8 m are water-saturated sandy and weakly plastic clay soils with dense interlayers ( $\tau_{kr} = 0.8 \frac{t}{m^2}$ ), below – hard-plastic clays ( $\tau_{kr} = 1.5 \frac{t}{m^2}$ ) .

We find the value of the total critical resistance to the failure of the pile:

$T_{kr} = \pi \times 0.26 \times (8 \times0.8+4 \times 1.5)= 10\,t$ .

In this case, it is not yet clear in what frequency range the vibrodriver should operate; so Let us start with the definition of $\omega$ using formula (1.16a.) Since the weight of the immersed element is relatively small, it is obvious that the weight of the vibrator will also be small. We initially accept $Q_B = 0.7t$ .

We calculate:

$\omega = \sqrt{\frac{1 \times 981 \times 10000}{1 \times 1500 \times 50}} = 133 \frac{rad}{sec}$ .

We accept $\omega = 156 \frac{rad}{sec}$ ( $n_o = 1500\,RPM$ ) and define:

$K = \frac{1500 \times 50}{1 \times 156}= 480\,kg-cm \approx 500\,kg-cm$ .

The required total weight of the machine (for $p_{(0)} = 5\frac{kg}{cm^2} = 50\frac{t}{m^2}$ ) is:

$Q_P = \frac{\pi \times 0.26^2}{4} \times 50 = 2.7\,t$ .

Since Equation (1.21) is not satisfied for such a weight $\frac{Q}{P_o} = \frac{2.7}{10} = 0.27$ , it is necessary to slightly increase the weight of the vibrodriver. Let us take it equal to 3.2 tons. Thus, the value of the static load should be equal to:

$Q_P = 3.2-1.5 = 1.7 t$ .

Checking the correctness of the selected parameters ( $\omega$ , K and Q) gives positive results, which allows you to keep them unchanged.

The practical method of selecting the parameters of a pile vibratory driver described in this chapter is approximate, but at the same time it gives a fairly correct orientation and allows you to avoid gross errors when choosing the type and parameters of vibratory drivers in relation to certain specific conditions.

Type of Pile	$A_{(0)},mm$
	Sandy Soils			Clay Soils
	300-700 RPM	800-1000 RPM	1200-1500 RPM	400-700 RPM	800-1200 RPM	1200-1500 RPM
Steel sheet pile,steel tubes with open end, and other elements with cross-sectional area up to 150 cm²	–	8-10	4-6	–	10-12	6-8
Wooden and tubular steel(with closed end) with cross-sectional area up to 800 cm²	–	10-12	6-8	–	12-15	8-10
Reinforced concrete,square or rectangular cross section with area up to 2,000 cm²	12-15	–	–	15-20	–	–
Reinforced concrete tubular piles with large diameter, inserted with excavation of soil from tube cavity	6-10	4-6	–	8-12	6-10	–

Types and Dimensions of Piles	$p_{(0)},\frac{kg}{cm^2}$
Steel sheet pile,steel tubes with open end, and other elements with cross-sectional area up to 150 cm²	1.5-3.0
Wooden and tubular steel(with closed end) with cross-sectional area up to 800 cm²	4.0-5.0
Reinforced concrete,square or rectangular cross section with area up to 2,000 cm²	6.0-8.0

Practical Methods for Selecting the Parameters of a Vibratory Pile Driver

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Type of Soil	Forces of friction exerted on inserted element.
	For Piles ( $\tau_{kr},\frac{t}{m^2}$ )			For Sheeting ( $\tau`_{kr},\frac{t}{m}$ )
	Wooden piles, steel tubes	Reinforced concrete piles	Reinforced concrete tubular piles, open at the bottom, inserted with excavation of soil	Light profile sheet pile	Heavy profile sheet pile
Water-saturated sandy and slightly plastic clayey soils	0.6	0.7	0.5	1.2	1.4
Same, but with layers of thick clay or gravely soils	0.8	1.0	0.7	1.7	2.0
Only slightly plastic clayey soils	1.5	1.8	1.0	2.0	2.5
Same, semi-hard and hard	2.5	3.0	2.0	4.0	5.0

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