# «PILE DRIVING ANALYSIS BY THE WAVE EQUATION For technical assistance, contact: Dr. Lee L. Lowery, Jr., P.E. Department of Civil Engineering Texas A&M ...»

If a cushion is used between the helmet and the head of the pile, it's spring constant may be placed between the helmet and the first pile weight, and the remaining pile segment springs may be moved below their corresponding weights. If no cushion is utilized, the spring rate of the first pile spring must be placed between the helmet and the head of the pile, and all following springs moved above their corresponding weights.

**Limiting Forces Between Pile Segments**

GAMMA(I) represents the minimum force that can be exerted in the I-th spring (compressive forces positive). If the parts composing the pile driver and accessories are physically separated and cannot transmit tension, then values of GAMMA(I) for the hammer assembly springs will be set equal to 0.0. In the case of diesel hammers, the minimum force in the spring under the ram is equal to the explosive force; In this case GAMMA(I) is set = to the explosive force. Thus, if any spring in the system cannot transmit tension, its value of GAMMA(I) should be set to 0.0. Any spring which can transmit tension should have its corresponding GAMMA(I) set to -1.0 to indicate that tension can be transmitted through the spring. The only exception to this is for a diesel hammer, in which case GAMMA(I) should be set equal to the explosive pressure as listed in Appendix C. This will then account for the explosive pressure force in the hammer.

**Slack in Joints**

In the case of certain mandrel driven piles, such as Raymond step-taper piles, some of the pile segments are not rigidly connected, but are connected with loose fitting pins. Thus, the ends of certain pile sections can "open up" a certain distance, known as "slack", before transmitting tension. Any such joint will thus have GAMMA(I) = -1.0 to indicate that it can transmit tension, with it's SLACK(I) set equal to the total slack in the joint, after which tension can be transmitted.

**Idealization for Soils**

The soil is idealized as a spring in parallel with a dashpot as seen in Figure 22. The soil spring can deform elastically to a limiting deformation, Q(I), after which no additional load is required to produce continued deformation (see Figure 27a). The soil resistance corresponding to a deformation Q is denoted by Ru. The dashpot is used to include dynamic loading effects in the soil resistance (see Figure 27b).

Each weight of the pile system has a soil spring associated with it. Thus, the distribution of soil resistance along the length of the pile can be specified by proper choice of the constants which describe each individual soil spring. Figure 27(a) shows the assumed static load deformation characteristic for soil springs along the side of the pile and at the tip. For the soil on the side of the pile, path OABCDEFG represents the load-deformation that occurs as the pile moves through the soil. For the soil at the point, only compressive loading can occur, since the point of the pile is free to rebound, and the load deformation path is OABCO.

It can be seen that the characteristics of the spring representing the soil stiffness are defined by the quantities Q and Ru. Q is the soil quake or maximum elastic deformation corresponding to the maximum elastic force Ru. A load-deformation diagram shown in Figure 27(a) is thus established for each soil spring.

**The stiffness of a side soil spring is given by:**

where Ru(m) = side soil resistance on segment m (kips) Q(m) = side soil quake (in).

The dynamic loading effects for the soil are included by assuming that the soil has a damper (dashpot) in parallel with the spring (see Figure 22). The dynamic resistance of the dashpot is assumed directly proportional to the velocity of the associated segment weight during displacement and the total resistance of the soil spring and dashpot during displacement is given

**by (see Figure 27b):**

R(m) = [D(m) - D'(m)] XKIM(m) [1.0 + J(m)V(m)], from O to A, and R(m) = XKIM(m)[D(m) - D'(m)] + Q(m)J(m)V(m), after A.

Where R(m) = total resistance, static plus dynamic D(m) = displacement of WAM(m) into the soil (in) D'(m) = plastic displacement of weight into soil (in) XKIM(m) = spring stiffness of soil spring (ksi) J(m) = damping constant of soil spring (sec/ft) V(m) = velocity of WAM(m) (ft/sec) Q(m) = soil quake (in) This equation will produce a dynamic load-deformation behavior as shown by path OABCDEFG in Figure 27(b) for the side soil and the path OABC for the point soil. The soil dashpot is used to include dynamic loading effects of the soil. The characteristic of a dashpot is its damping constant. Extensive data are not available for the damping characteristics of soils, however, values for sands and clays have been determined and are listed in Appendix C. Should more accurate damping values become available, they should be used instead of the approximate values.

**Soil Quake and Damping**

The values of Q (commonly called "quake") and J (the damping constant) for various soils are still being studied. However, most soils have a Q value on the order of 0.1 in, its value is usually considered to be constant for all soils and equal to 0.1 in. More accurate values for Q should be used if available. Until more accurate values are available, the authors recommend use of the values listed in Appendix C. These values have been determined by full scale pile tests wherein the values were varied and those which gave the most accurate correlation with load tests were selected. Additional information regarding these tests can be found in Reference 31.

Introduction The following section lists the required information necessary to perform an analysis using the wave equation computer program. A detailed discussion is included that explains the required input parameters. Any number of data sets may be loaded in sequence.

Program Input Data

1) Card 1 (Required) NCARDS = Total number of identification cards to be read, including Card 1 (Maximum of 8 cards).

= 1 Read Card 1 only (Read "NCARDS" plus read and print 68 columns of Alphanumeric problem identification).

2) Card 101 (Required) l/DELTEE = 1/Time interval. If left blank, Delta Tcritical/2 will be used. (l/sec) (Normally left blank).

NSTOP = Maximum number of time intervals the program is to run. (See Chapter 4, page 6) IPRINT = Print frequency. For example, if a print of the solution at every 5th time interval is wanted, set IPRINT = 5.

NSl-NS6 = The element numbers for which solutions vs. time interval will be printed.

NOP(I) Value NOP(I) Functions

NOP(l) Used to specify long or short print-out of solution.

=l Print out information needed for checking problem solution, and all final answers.

(Normally used).

=2 Print out all variables needed to check program operations using fixed formats. (Dump print).

=3 Print out all variables needed to check program operations using floating formats.

(Dump print).

NOP(2) Used to specify the input method for the segment weights WAM(I) =1 Read one weight for each segment (card series 200). Note that this number is automatically inserted in the data set for you since no other value is possible.

NOP(3) Used to specify the input method for the internal spring stiffness. XKAM(I) =l Read one stiffness for each internal spring from card series 300.

NOP(4) Used to specify what soil resistance distribution acts along the pile. NOTE: RUP is the total point resistance and RUT is the total resistance on the pile. Both are read in on card 103.

=1 Read RUM(I) for each element from card series 400, including the point bearing soil resistance RUM(MP+l). (MP is defined on card 102).

=2 Set all side resistances equal to zero, and set RUM(MP+l)= RUP.

=3 Distribute RUT-RUP uniformly along the side of the pile from segment MO thru MP, and set RUM(MP+l) = RUP. (MO is defined on card 103).

=4 Distribute RUT-RUP triangularly along the pile between segments MO and MP, and set RUM(MP+l) = RUP.

NOP(5) Used to specify the input method for GAMMA(I).

NOTE: The significance of GAMMA(I) is discussed in the 500 card series.

=1 Read GAMMAl, GAMMA2, and GAMMA3 from card 102 and assign GAMMAl to internal spring number 1, GAMMA2 to spring number 2, and GAMMA3 to spring number 3.

Then set GAMMA(I) of the remaining springs to -1.0. (Normally used).

=2 Read GAMMA(I) for each spring from card series 500.

NOP(6) Used to specify the input method for EEM(l). (Coefficients of restitution for springs) =1 Read EEMl, EEM2, and EEM3 from card 102, set EEM(l)= EEMl, EEM(2) = EEM2, and EEM(3) = EEM3. Then set EEM(I) for all other springs equal to 1.0 (perfectly elastic).

(Normally used).

=2 Read EEM(I) for each spring from card series 600.

NOP(7) Used to specify whether program is to run for a full NSTOP iterations, or if program should run only until maximum permanent set of pile has been reached.

=l Run until permanent set of pile is found. (Normally used).

=2 Run full NSTOP iterations.

NOP(8) Used to specify input method for VEL(I).

=1 Read VELMI from card 102 and set VEL(l,t=O) equal to VELMI. Set all other VEL (I,t=0)=0.0. (Normally used).

=2 Read VEL(I) for each segment from card series 800.

NOP(9) Used to specify input method for Q(I).

=1 Read QSIDE and QPOINT from card 103 and set all Q(I)along side of the pile equal to QSIDE. Set Q(MP + 1)under pile tip equal to QPOINT. (Normally used).

=2 Read Q(I) for each element including Q(MP + 1) from card series 900.

NOP(10) Used to specify input method for soil damping SJ(I).

=1 Read SIDEJ and POINTJ from card 103. Set all SJ(I) along side of pile equal to SIDEJ and set SJ(MP + 1)under pile tip equal to POINTJ. (Normally used).

=2 Read SJ(I) for each element including SJ(MP + 1) from card series 1000.

NOP(ll) Used to specify input method for cross sectional area of pile A(I).

=1 Read AREA from card 102 and set all A(I) equal to AREA. (Set A(I) = 1000.0 for conversion to kips). (Normally used).

=2 Read A(I) for each internal spring from card series 1100.

NOP(12) Used to read in "slack" present in any of the segments.

=l No slack present in any of the joints. (Normally used).

=2 Read in joint slacks for each spring from card series 1200.

3) Card 102 (Required) MP = Total number of segments in the system to be analyzed.

MH = Element number of the first pile segment.

VELMI = Initial velocity of the ram. (ft/sec) AREA = A constant used to convert the output forces into stresses or other more convenient values if desired. Note that you can change pounds, which is the normal computer output unit, to kips by setting AREA = 1000.0. Or you can change pounds to ksi stress by setting AREA = 1000*area of pile.

EEMl = Coefficient of Restitution of spring number 1, directly under the ram.

EEM2 = Coefficient of Restitution of spring number 2.

EEM3 = Coefficient of Restitution of spring number 3.

GAMMAl = The minimum force in the spring between the ram and the anvil, once that force has reached a maximum. (kip) For example, if the diesel hammer explosive pressure causes a 158.7 kip minimum force in this spring, set GAMMAl = 158.7 kip. If the minimum force the spring can transmit is zero (for example, when no tensile force can exist between the ram and the anvil, and no explosive pressure force is acting, set the corresponding GAMMA(I) = 0.0. If the spring represents a continuous body such as the spring between any two pile segments, that spring can transmit tensile forces between the elements. This is signified by setting GAMMA(I) equal to any negative value, usually -1.0 kip.

GAMMA2 = Same as above, but for spring number 2. Note that GAMMA2 will only be -1 or 0, never positive, since only GAMMA1 will have a diesel explosive pressure.

GAMMA3 = Same as GAMMA2, but for spring number 3.

4) Card 103 (Required) RUT = The total static soil resistance acting on the pile (kip).

RUP = The total static soil resistance acting beneath the pile point (kip).

MO = Number of the first element upon which soil resistance acts.

QSIDE = Soil quake along the side of the pile (inches). If the soil quake varies along the pile, use input on card series 900, and QSIDE will be ignored.

QPOINT = Soil quake on the point segment (inches).

SIDEJ = Soil damping factor in shear along the side of the pile.(sec/ft) POINTJ = Soil damping factor in compression beneath the pile point (sec/ft) DRl = Increment by which RUT and RUP are to be increased when the problem is re-run.

DR2 = Second increment by which RUT and RUP are to be increased and the problem re-run.

DR3 = Etc. For example, if three levels of resistance were to be run with RUT = 100, 250, 400, 800 kips, with corresponding point resistances of 10, 25, 40 and 80 kips, set RUT = 100, RUP = 10, and set DRl = 2.5, DR2 = 4.0, and DR3 = 8.0.

5) 200 CARD SERIES (Required) I = Element number (from top down) (Last value must be MP) WAM(I) = The weight of element number I (kip). Note that only the last weight of a string of identical weights must be input, if desired. For example, if WAM(l) = 10.0, WAM(2-8) = 5.0, and

**WAM(9-53) = 2.0, it is sufficient to input:**

1 10.0 8 5.0 53 2.0 The program will understand that WAM(2) thru WAM(8) = 5.0, WAM(9-53) = 2.0. All input data utilizes this method.

6) 300 CARD SERIES (Required) I = Element number XKAM(I) = The internal spring rate of spring I (kip/in.) Similar to card series 200, only the last value of a string of identical values must be input. Values must be input from the top of the system down. Last value must be MP-l.

7) 400 CARD SERIES (Required if NOP(4) = 1) I = Element number RUM(I) = The ultimate static soil resistance acting on pile segment I (kip).

a) If NOP (4) = 1, read MP + 1 ultimate soil resistances, from card series 400.

b) If NOP(4) = 2, set all side friction = 0.0 and set RUM(MP+1) = RUP.

c) If NOP(4) = 3, distribute (RUT-RUP) uniformly along the pile starting from segment number MO to number MP, and set RUM(MP+1) = RUP.