Methods for Approximate Calculation of the Parameters of Vibratory Drivers and Impact-Vibration Hammers

Calculation of the parameters of longitudinal-action vibrational pile drivers.

Note: this part of the chapter is basically the “Savinov and Luskin” method for determining the drivability of piles by vibration which goes back to the late 1950’s. The rationale for this is discussed in my paper Reconstructing a Soviet-Era Plastic Model to Predict Vibratory Pile Driving Performance. Another presentation of it is in my post Vibratory Machines for Sinking Piles: Vibratory Pile Drivers.

The initial data for calculation are: the mass of the pile to be driven mo, in kg; the geometric dimensions of the pile to be driven;the depth of sinking l, in m; and the soil conditions.

1. The resistance of the soil is determined. The calculated value FCR of the critical separation resistance at a given maximum sinking depth (kN) is determined on the basis of the original data characterizing the soil conditions:

where i is the ordinal number of the soil layer of thickness li passed through during the sinking, k is the total number of layers and Z is the perimeter of the cross section. The values of the specific separation resistance σ are assumed according to the data of Table 5.

Type of soilFor a pile, kPaFor a sheet pile, kN/m
Steel Pipe PilesReinforced concrete pilesTubular piles open on the bottomLight sectionsHeavy sections
Water-soaked sandy and fluid-plastic clay soils6751214
The same, soils with interlayers of compact clay or gravelly soils81071720
Stiff plastic clay soils1518102025
Semihard and hard clay soils2530204050
Table 5. Dependence of the specific separation resistance of the soil type.

2. After determination of the tentative mass value mo, in kg, of the pile to be driven and the parts of the vibratory pile driver rigidly connected with it, the approximate value of the static moment of the mass of the eccentrics, in kg • m, is calculated:

where ψ = 0.8 for reinforced concrete piles and ψ = 1 for the other piles sunk.

The recommended vibrational amplitude Ao required for an effective sinking is determined with the data of Table 6.

Types of piles to be drivenAo, mm
Sandy soilsClayey soils
Vibration frequency, Hz5-1213-1718-255-1213-1718-25
Steel sheet pile, steel tubes with open end, and other piles with cross-sectional area up to 150 cm28-104-610-126-8
Tubular piles (with closed end) with cross-sectional area up to 800 cm210-126-812-158-10
Reinforced concrete, square or rectangular cross section with area up to 2,000 cm212-1515-20
Reinforced concrete tubular piles with large diameter, inserted with excavation of soil from tube cavity6-104-68-126-10
Table 6. Dependence of the vibration amplitude Ao, in mm, on the types of piles to be driven

3. The frequency of the vibrations of the vibratory pile driver, Hz, is calculated as follows:

When the set of parameters of the vibratory pile driver is derived with a previously undetermined interval of change in the θ value, it must be determined from the condition

The amplitude of the vibration velocity vo for a successful sinking should be within the interval of 0.5-0.8 m/s ; is a coefficient that takes the resilience of the soil into account: = 0.6-0.8 for low-frequency vibratory pile drivers (5-10 Hz) and = 1 for the others. If the value θ is determined by this method, the static moment of mass of the eccentrics is calculated with the formula, in kg • m:

4. The required minimum mass of the vibratory pile drive and driven piles, in kg, is determined as follows:

where Uc is the cross sectional surface, in cm2; po are the recommended pressure values required. The dependence of the pressure po, in MPa, on the type and dimensions of piles driven into saturated sandy and loose clayey soils is given below:

Steel tubes of small diameter and other piles with a cross sectional surface up to 150 cm20.15 – 0.3
tubular steel (with closed end) piles with a cross sectional surface up to 800 cm20.4 – 0.5
reinforced concrete piles of square and rectangular section with an area up to 2000 cm20.6 – 0.8

5. The value of the ratio of the force of gravity to the amplitude of the compelling force Pθ is verified:

v1v2
for a steel sheet pile0.150.5
for light piles0.30.6
for heavy piles and tubular ones0.41.0

In performing the calculations with respect to this point, either the mass mo or the amplitude of the compelling force (due to an increase in K or θ) is increased if necessary.

6. Finally, the values K, θ and mo are precisely defined, after which these parameters are verified with the formulas:

P_{o}\geq\aleph F_{cr};\,A_{o}\leq\frac{1000\psi K}{m_{0}}

In addition, the precisely defined parameters are verified with the formulas of paragraphs 4 and 5.

7. The power of the driving motor is determined by:

where D is the diameter of the journals of the shafts of a vibration exciter, in cm.

The efficiency of transfer from the motor to vibration exciter (equal to 0.9), the coefficient of rolling friction in the bearings of the vibration exciter (equal to 10-3), and the additional consumption of power in the vibration of the soil mass, assumed to be 15% of the power expended to overcome the resistance of the soil, were studied here.

Calculation of the parameters of longitudinal-action impact-vibration hammers.

The original data for the calculation are the same parameters as for the vibratory pile drivers.

1. On the basis of the original data on the pile driven, the mass of the impact part of the vibratory hammer is determined, in kg:

m_1 = (0.7-1.2)m_2

2. The amplitude value of the compelling force, kN, is determined:

P_o = dm_1 \times 10^{-2}

The lower limit of the parameter d (d = 2-6) is designated for impact-vibration hammers that drive piles with a relatively small cross sectional surface (up to 50 cm2). The parameter d also increases with an increase in cross sectional surface.

3. The preliminary depression force of the working springs is calculated, in kN:

Q_{V,P} = \left ( \sin \alpha - \frac{1}{d} \right ) P_o

The parameter sin α = 0.3 – 0.5. An attempt must be made to assure the value sin α = 0.4 in designing the hammers.

4. The static moment of mass of the eccentrics is determined, in kg • m:

K = 25.3\frac{P_o}{\theta^2}

The frequency of the compelling force of the hammer θ lies within the interval of 6-10 Hz.

5. The optimal stiffness of the working springs of the vibratory hammer is, in N/cm:

c_1 = (3.5 - 10) \times 10^{-2} m_1 \theta^2

In designing the hammers it is desirable to obtain the minimum c1, value from the recommended optimal range.

6. The ratio γ1 between the frequency of the natural vibrations of the impact part and the frequency of the compelling force:

7. The dimensionless resistance of the soil f and γ is calculated with the F and R values (9). The existence of an impact-vibrational mode is verified in terms of the parameters, sin α, γ1, and f + γ (see Figure 22).

8. The dimensionless impact velocity is calculated (correlation formula):

9. The velocity of the impact part at the moment of impact is determined, in m/s:

x_{imp} = 6.28 \times K \theta / m_1 \dot y_1

With respect to the durability of the hammer, the impact velocity should not exceed 2 m/s. If the value ximp » 2 m/s with the parameters chosen, it is necessary to select other parameters that assure a value ximp ≤ 2 m/s

10. The dimensionless sinking per impact (correlation formula)

11. The possibility of driving under the given soil conditions is checked:

The value ΔL is adopted according to the data (Yu. R. Perkov, V. N. Shaevich, 1974) given below:

water-soaked sands of medium coarseness and compactness0.16
low-moisture sands of medium coarseness and compactness0.46
wet sands of medium coarseness and compactness0.22
loams of stiff-plastic consistency0.32
macroporous sandy loams of a hard consistency0.28

If it turns out that xPL < ΔL, it is necessary to reduce the parameter sin α to 0.2-0.3 and repeat the calculation. When xPL again proves to be less than ΔL, it can be assumed that the impact vibrational driving is ineffective under the given soil conditions.

12. The dimensionless rising height of the impact part is calculated:

13. The maximum rise value of the impact part is determined, in cm:

14. The maximum reaction of the vibratory hammer springs is calculated, in kN:

15. The drive engine power required is calculated, in kW:

The transfer efficiency from the vibration exciter to the motor (equal to 0.9) and the friction coefficient in the supports of the shafts (equal to 10-3) are taken into account in this formula.

The selection of the parameters of the the impact-vibrational machines for extracting the piles from the soil is done by the same method, but in this case, in kN,

Calculation of the parameters of longitudinal-rotatory-action vibratory pile drivers.

The initial data for the calculation are the same characteristics as for longitudinal-action vibratory pile drivers.

1. The basic calculation characteristics of the driving system are determined:

where σP,V is the specific resistance on the lateral surface of the shell being driven with longitudinal-rotational vibrations; the σP,V values are given in Table 7, compiled according to the results of experimental studies:

where Ut is the area of the end surface of the shell, in m2 ; R is the calculated resistance of the soil under the end of the shell, in kPa (undertaken according to the data of SNip 11-02 — 03.85).

soilstubular piles with closed lower endshells driven with removal of the soil
water-soaked and soft -plastic clayey soils4.02.5
the same, with interlayers of compact or gravelly soils6.03.5
clayey stiff -plastic soils10.05.0
Table 7- Specific resistance of the soil σP,V, in kPa on the lateral
surface of the driven shell under the effect of longitudinal-rotational
vibrations.

2. The required dimensionless velocity values of the vibrations are determined:

where vn is the projected driving velocity in m/min (recommended value: vn = 0.2 -1.0 m/min) ; vn = vo/a2 ; θ is the frequency (recommended value: θ = 6-7 Hz).

3. The amplitudes of the compelling force and the static moment of mass of the eccentrics are determined, in kg – m:

4. The separation conditions of the shell relative to the adjacent soil are verified:

The Amin values are given in Table 8.

Pile drivensandy soilsclayey soils
6-9 Hz13-16 Hz6-9 Hz13-16 Hz
tubular pile with closed lower end3.01.04.01.5
shells, driven with removal of the soil2.41.23.02.0
Table 8. Minimum separation amplitudes Amin, in mm, under the effect
of longitudinal-rotational vibrations.

5. The magnitude of eccentricity of application of the compelling force is determined:

r_1=\delta r

The δ value is selected as a function of the ratio of vibrator mass m1 and the mass of the driven shell m2;

6. The drive power of the vibratory longitudinal-rotational action pile driver is calculated, in kW:

where D is the diameter of the support of the eccentric shaft, in mm; ηver is the coefficient that takes into account the losses in transfer from the engine to the eccentric shafts.

7. It is recommended that the effort in extracting the tubular pile from the soil under the action of longitudinal-rotational vibrations be determined with the formula

S = 10m_0 + F_{IZ}

where FIZ is the mean force required to overcome the lateral resistance of the soil during vibratory extraction: FIZ = ζFe; Fe is the separation resistance along the lateral surface of a tubular pile; and ζ is the coefficient of decrease in the lateral resistance as a function of the vibratory mode.

The ζ value is determined by the graphs plotted with Equation (35) and given in Figure 19, as a function of the ratio of the extraction velocity (vIZ) to the amplitude of the velocity of the longitudinal component of the vibrations (). The rate of rise in the tube should not exceed 0.5-1.0 m/min in the first extraction stage.

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