# «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 ...»

d) If NOP(4) = 4, distribute (RUT-RUP) triangularly between MO and MP, and set RUM(MP+1) = RUP.

8) 500 CARD SERIES (Required when NOP(5) = 2).

I = element number GAMMA(I) = The minimum force possible in spring I after the peak compressive force has passed, except that any negative GAMMA(I) is construed to mean that spring can transmit a tensile force of any magnitude (kip). Total number of GAMMA(I) values = MP-1 (one for each internal spring.) 9) 600 CARD SERIES (Required when NOP(6) = 2) I = Element number EEM(I) = The coefficient of restitution for MP-1 internal springs. This determines the slope of the unloading curve (dimensionless).

10) 700 CARD SERIES (Not Used) 11) 800 CARD SERIES (Required when NOP(8) = 2) I = Element number VEL(I) = The initial velocities of each of the MP weights (ft/sec).

12) 900 CARD SERIES (Required when NOP(9) = 2) I = Element number Q(I) = The soil "quake" for MP+1 soil springs (in.).

13) 1000 CARD SERIES (Required when NOP(10) = 2) I = Element number SJ(I)= The soil damping factor for MP+1 soil springs (sec/ft).

14) 1100 CARD SERIES (Required when NOP(ll) = 2) I = Element number A(I)= The cross-sectional area of the MP-l internal springs (in^2).

15) 1200 CARD SERIES (Required when NOP(12) = 2) I = Element number SLACK(I)= The slack or "looseness" in the MP-l internal springs (in), input for segments which have slack present. For segments which do not have slack, input 0, such that the element will not have slack or looseness.

Coding Sheets - Since EDITWAVE handles the forming of the data sheets, coding sheets are no longer required. The data is entered in its proper position in ASCII format, and can be viewed with an ASCII editor if desired.

The following is a summary for properties of drop and steam hammers, diesel hammers, cushions/capblocks, and soil properties. Figure 28 shows the idealization for commonly used steam and drop hammers, while Table Cl lists their specific values. Figure 29 shows the idealization for diesel hammers, and Table C2 lists their properties for use in the wave equation.

Table C3 gives a summary of various cushion and capblock materials commonly used, and recommended wave equation input values. Table C4 gives a summary of common soil properties.

*Maximum stroke. For actual stroke use field observations - may vary from 4.0 to 8.0 ft.

**Maximum stroke. Determine true value in field from bounce chamber pressure (he = E/Ram weight where E = Indicated Energy).

C - distance from exhaust parts to anvil.

*After reference 31

## APPENDIX D - SAMPLE PROBLEMS

Case I Assume that a 12"x12" prestressed concrete pile 60-feet long is to be driven to a penetration of 30 feet below the mudline with clay at the side and point of the pile, using a standard Vulcan 08 steam hammer.Hammer

**From Appendix C, the properties of a Vulcan 08 hammer are found to be:**

Hammer Stroke = 3.25 feet Ram Weight (WAM(1)) = 8.0 kips Efficiency = 66% Helmet Weight (WAM(2)) = 1.0 kip (assumed) Note that helmet weights and cushion dimensions are not listed in Appendix C, since they vary from job to job, and with each contractor. They must be individually determined. Values used in all following example cases were selected from previous cases solved by the author.

**The ram velocity at impact (VELMI) is computed by:**

VELMI = sqrt[2g(Hammer Stroke)(Efficiency)] VELMI = sqrt[(64.4)(3.25)(0.66)] VELMI = 11.75 ft/sec Capblock

**The properties for this capblock are:**

Capblock: OAK (from contractor) Diameter (from contractor)= 14.0" Thickness (from contractor)= 1.0" Modulus of Elasticity (E)= 45 kips/in (Table C3) Coefficient of Restitution (AIM)= 0.5 (Table C3) Since this spring is between the ram and the helmet it cannot transmit tension, so GAMMA1 = 0.

Spring rate for CAPBLOCK (XKAM(1)) = AE/L where A = Area of Capblock E = Modulus of Elasticity L = Length or Thickness XKAM(1) = ( pi)(Diameter)^2(45 ksi)/(4*1.0") = 6927 kips/inch Cushion (from contractor) = OAK (see Appendix C) Size (from contractor)= 12"x12" square (same size as pile) Thickness (from contractor)= 1.0" Modulus of Elasticity (E)= 45 ksi Coefficient of Restitution = (EEM2)= 0.5 GAMMA2 = 0.0

**Spring rate for cushion XKAM(2):**

XKAM(2) = AE/L = (12")(12")(45 ksi)/1.0" = 6480 kips/inch Pile The pile to be driven is a 12"x12" prestressed concrete pile, 60 feet long. Given information is as

**follows:**

Area = 144 in^2 Length = 60 feet Modulus of Elasticity = 3000 ksi Prestress = 2000 psi Maximum allowable compressive stress = 2000 psi above prestress Maximum allowable tensile stress = 500 psi above prestress A series of 10-foot pile segments will be used to idealize the pile in this case. Although a slight increase in accuracy would be possible by using shorter segment lengths, this is probably not justified because of the increased solution time required. The idealization for this case is shown in Figure 30.

The segment weights are computed from WAM(I) = AL(Density) where WAM(I) = Weight of element I(kips), A = Cross-sectional area of the pile at element I(in^2), L = Length of element I, (inches), and the density of the material at element I is 0.150 kips/ft^3.

Thus, WAM(3)=(144 in^2)(10ft x 12 in/ft)(0.150 kips/ft^3)/(1728 in^3/ft^3) = 1.5 kips. Thus, WAM(3) through WAM(8) = 1.5 kips.

Note that although there are 8 element weights in the system, there are only 7 internal springs (one between each two adjacent weights). There are also 9 soil springs associated with the 8 weights, since the last weight has a "side" soil spring (side friction) and also a "point" soil spring (point bearing). The top 5 soil springs are not shown on weights 1 through 5 as they are absent for this case.

**To compute the spring rates for each of the 10-foot pile segments:**

XKAM(3) through XKAM(7) = AE/L = (12"x12")(3000 ksi)/(10ftx12"/ft) = 3600 kips/inch Coefficients of restitution for the concrete pile should be set to 1.0, as the damping in the pile is negligible. Also, since each of the pile springs 3 through 7 can transmit tension, GAMMA(3) through GAMMA(7) = -1.0.

Soil

**The soil properties, as determined from soil borings and tests are assumed as follows:**

To obtain a starting value of resistance for use in the wave equation, some value around 5 to 10 times the pile weight is normally selected. This is then increased in increments as desired. Thus for Case I, since the pile weighs 9 kips, the initial total resistance will be assumed as 50 kips. Of this, 95% (47.5 kips) will be distributed uniformly along the side of the pile, and 5% (2.5 kips) will be placed under the pile point. Thus, input data for the program will be: total soil resistance RUT = 50 kips, total point resistance RUP = 2.5 kips.

Card Input: Case I Using the values from Case I, the following information is input in EDITWAVE

1) Card 1 NCARDS = Total number of identification cards to be read (maximum of 8 cards). For this case, NCARDS = 2, and two lines of identification are entered.

2) Card 101 1/Deltee - 1/Time interval. If left blank, Deltee critical/2 will be used. (1/sec). 1/Deltee will be left blank for this case, and the computer will calculate the time interval from the properties of the weights and springs involved in the idealization.

NSTOP = Maximum number of intervals the program is to run. Set at 200 for this case.

IPRINT = Print frequency. For this case, set IPRINT equal to 5, to print every 5th iteration.

NS1-NS6 = The element number for which solution vs. time interval will be printed. For this case, NS1 = 1 (Ram), NS2 = 2 (Helmet), NS3 = 3 (First Pile Segment), NS4 = 5, NS5 = 7, NS6 = 8 (Last Segment of the Pile).

NOP(1) = 1, to print out normal information needed for checking problem solution, and a summary of all final answers.

NOP(2) = 1, to read the WAM(I) (weights) values for each weight used in the hammer-pile system from card series 200.

NOP(3) = 1, read values of XKAM(I) (spring rates) for each internal spring from card series 300.

NOP(4) = 3, distribute Total Resistance (RUT) minus Point Resistance (RUP) uniformly along the side of the pile from segment MO through MP, and set the point bearing soil resistance = RUP. (MO is defined on card 103).

**NOP(5) = 1, read GAMMA(1), GAMMA(2) and GAMMA(3) from card 102. For this case:**

GAMMA(1) = 0.0, GAMMA(2) = 0.0, and GAMMA(3) = -1.0, meaning that the pile springs can transmit tensile forces. All remaining GAMMA(I) values of the system will then be set to -1.0, indicating that the remaining pile springs can transmit tension.

NOP(6) = 1, read EEM(1), EEM(2), and EEM(3) from card 102. (EEM(1) is discussed on the 102 card.) Thus all remaining EEM(I) will be set equal to 1.0 (no damping).

NOP(7) = 2, run the program for full NSTOP iterations as specified on the 101 card. This is used when you want to make sure that the maximum stresses have been observed in the pile. If you let the computer shut the run down, can only check to see if the pile has reached its maximum penetration and shuts down. The maximum tensile waves may not have yet been recorded.

NOP(8) = 1, read VELMI from card 102. This is the velocity of the ram at impact. All element velocities following the ram are then set equal to 0.0.

NOP(9) = 1, read QSIDE and QPOINT from card 103 and set all Q(I) along the side of the pile equal to QSIDE. Set Q at the pile tip equal to QPOINT. For this case, QSIDE = 0.1 and QPOINT = 0.1.

NOP(10) = 1, read SIDEJ and POINTJ from card 103. Set all SJ(I) along the side of the pile equal to SIDEJ and SJ(MP+1) under pile tip equal to POINTJ.

NOP(11) = 1, read AREA from card 102 and set all A(I) equal to AREA. Note that this is of course incorrect for the hammer area, but since we are not interested in the true stresses in the hammer, we do not care. The stresses in the pile will be accurate. Note also that this in no way affects the solution, even though the areas of some elements are incorrect. The areas are used only after the solution for FORCES in the springs are determined, and only to change that output from pounds to psi or ksi.

**NOP(12) = 1, NO SLACK present in any of the joints.**

3) Card 102 MP = 8 for this case because the ram and helmet are represented as one weight each, and the pile is 60 feet long, and it is divided into 10-foot segments. (see Figure 30).

MH = 3, the first pile segment (see Figure 30).

VELMI = 11.75 ft/sec as calculated above. AREA = 144 in^2 of concrete for a 12"x12" pile.

EEM1 = 0.5 for an oak capblock (see Appendix C). EEM2 = 0.5 for an oak cushion (see Appendix C). EEM3 = 1.0 for the pile coefficient of restitution.

GAMMA1 = 0.0, minimum spring force between the pile and the helmet (no tension in spring 1).

GAMMA2 = 0.0 (no tension in spring 2).

GAMMA3 = -1.0, the continuous body (pile) can transmit tensile forces (springs 3 through 7).

4) Card 103 RUT = 50 kips = starting value of total static soil resistance acting on the pile.

RUP = 2.5, starting value of static soil resistance acting beneath the pile.

MO = 6, (see Figure 30) first element of the pile upon which soil resistance acts.

QSIDE = 0.1, soil quake along the side of the pile (inches).

QPOINT = 0.1, soil quake at the point segment of the pile (inches).

SIDEJ = 0.2, for clay - soil damping factor in shear along the side of the pile (sec/ft). (See Appendix C).

POINTJ = 0.01, for clay - soil damping factor in compression beneath the pile point (sec/ft). (See Appendix C).

DR1-DR7 = 2.0, 3.0, 4.0, 6.0, etc. These values increase RUT and RUP proportionally to develop points to plot the RUT vs. Blow Count curve (see Figure 22). For example, RUT = 50 and RUP = 2.5, initially, these resistances will then be increased to 100 and 5 kips, 150 and 7.5 kips, etc., respectively.

5) 200 Card Series WAM(1) = 8.0 kips, ram weight, see Appendix C.

WAM(2) = 1.0 kips, helmet weight, from contractor.

WAM(8) = 1.5 kips, pile weight per 10-foot segment (see Figure 30). Entered on card 200 as WAM(8) only because the internal program sets WAM(3) through WAM(8) = 1.5 kips, as calculated above.

6) 300 Card Series XKAM(1) = 6927 kips/inch, spring rate for the capblock as calculated above.

XKAM(2) = 6480 kips/inch, spring rate for the cushion, as calculated above.

XKAM(7) = 3600 kips/inch, spring rate for pile elements as calculated above.

Case II Assume that a 12"x12" prestressed concrete pile 60 feet long is to be driven to a penetration of 30 feet below the mudline with clay at the side and at the point of the pile, using a standard Vulcan 010 steam hammer. (Same as Case I, except for change in hammer).

Hammer

**From the contractor and Appendix C, the properties of a Vulcan 010 hammer are found to be:**

Hammer Stroke= 3.25 feet Ram Weight (WAM(1))= 10.0 kips Efficiency= 66% Helmet Weight (WAM(2))= 1.0 kips

**The ram velocity at impact (VELMI) is computed by the same equation as in Case I:**

VELMI = sqrt[(64.4)(3.25)(0.66)] VELMI = 11.75 ft/sec (Same as Case I) Capblock and Cushion From Appendix C, the properties for the capblock and cushion are found to be the same as used in Case I. The dimensions are also assumed to be the same. Thus, the spring rate and all other input parameters for this case are the same as in Case I.

The pile used in this case is the same pile driven in Case I.

Soil Identical to Case I.

Card Input: Case II The same input for Case I is used for Case II except the following: 0001 and 0002 cards for problem identification are changed. WAM(1) = 10.0 kips (Ram weight for the Vulcan 010 hammer). All other input data entry for Case II will thus be identical to that used in Case I.

Case III Assume that a 12"x12" prestressed concrete pile 60 feet long is to be driven to a penetration of 30 feet below the mudline with clay at the side and at the point of the pile, using a standard Delmag D-15 open end diesel hammer. Case III is the same as Case I, except for the hammer and capblock utilized to drive the pile.