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TAMWAVE 7: Analysis for a Cohesive Soil

Posted on 6 January 20186 January 2018 by Don Warrington

With the analysis of the concrete pile in cohesionless soils complete, we turn to an example in cohesive soils.

The analysis procedure is exactly the same. We will first discuss the differences between the two, then consider an example.

Differences with Piles in Cohesive Soils

The unit weight is in put as a saturated unit weight, and the specific gravity of the soil particles is different (but not by much.)
Once the simulated CPT data was abandoned, the “traditional” Tomlinson formula for the unit toe resistance, namely $q_t = N_c c$ , where $N_c = 9$ , was chosen.
The ultimate resistance along the shaft is done using the formula of Kolk and van der Velde (1996). This was used as a beta method, for compatibility with the method used for cohesionless soils. Unless the ratio of the cohesion to the effective stress is constant, the whole concept of a constant lateral pressure due to cohesion needs to be discarded.
For saturated cohesive soils, an estimate of pile set-up is done using cavity expansion methods. Excess pore pressure due to cavity expansion during driving is estimated using the method described by Randolph (2003). This excess pore pressure is then added to the existing pore pressure and a new effective stress is computed at each point for the Kolk and van der Velde method, preventing negative values when the total elevated pore water pressure exceeds the total pressure of the soil. The results are within reasonable ranges.

Test Case

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The only change in basic parameters from the other case was the change to a CH soil. We opted not to perform a lateral load test this time, although the program is certainly capable of using the CLM 2 method with cohesive soils.

Pile Data
Pile Designation	12 In. Square
Pile Material	Concrete
Penetration of Pile into the Soil, ft.	100
Basic “diameter” or size of the pile, ft.	1
Cross-sectional Area of the Pile, ft²	1.000
Pile Toe Area, ft²	1.000
Perimeter of the Pile, ft.	4.000
Soil Data
Type of Soil	CH
Specific Gravity of Solids	2.7
Void Ratio	0.84
Dry Unit Weight, pcf	91.5
Saturated Unit Weight, pcf	120.0
Soil Internal Friction Angle phi, degrees
Cohesion c, psf	750
SPT N₆₀, blows/foot	6
CPT q_c, psf	12,696
Distance of Water Table from Soil Surface, ft.	50
Penetration of Pile into Water Table, ft.	50
Pile Toe Results
Effective Stress at Pile Toe, ksf	7.454
SPT (N₁)₆₀ at pile toe, blows/foot	3
Unit Toe Resistance q_p, ksf	6.8
Shear Modulus at Pile Toe, ksf	474.8
Toe Spring Constant Depth Factor	1.366
Toe Spring Constant, kips/ft	2,358.0
Pile Toe Quake, in.	0.034
Poisson’s Ratio at Pile Toe	0.500
Toe Damping, kips-sec/ft	14.0
Toe Smith-Type Damping Constant, sec/ft	2.069
Total Static Toe Resistance Q_p, kips	6.75
Pile Toe Plugged?	Yes
Final Results
Total Shaft Friction Q_s, kips	219.92
Ultimate Axial Capacity of Pile, kips	226.67
Pile Setup Factor	2.0
Total Pile Soil Resistance to Driving (SRD), kips	115.44

Shaft Segment Properties
Depth at Centre of Layer, feet	Soil Shear Modulus, ksf	Beta	Quake,inches	Maximum Load Transfer, ksf	Spring Constant for Wall Shear, ksf/in	Smith-Type Damping Constant, sec/ft	Maximum Load Transfer During Driving (SRD), ksf
0.50	34.9	2.541	0.0400	0.116	2.91	2.709	0.116
1.50	60.4	1.180	0.0322	0.162	5.03	2.559	0.162
2.50	78.0	0.827	0.0291	0.189	6.50	2.489	0.189
3.50	92.2	0.655	0.0273	0.210	7.69	2.443	0.210
4.50	104.6	0.550	0.0260	0.227	8.72	2.407	0.227
5.50	115.6	0.479	0.0250	0.241	9.64	2.378	0.241
6.50	125.7	0.427	0.0243	0.254	10.48	2.353	0.254
7.50	135.0	0.387	0.0236	0.266	11.25	2.332	0.266
8.50	143.8	0.356	0.0231	0.277	11.98	2.312	0.277
9.50	152.0	0.330	0.0226	0.287	12.66	2.294	0.287
10.50	159.8	0.308	0.0222	0.296	13.31	2.278	0.296
11.50	167.2	0.290	0.0219	0.305	13.93	2.262	0.305
12.50	174.3	0.274	0.0216	0.313	14.53	2.248	0.313
13.50	181.2	0.260	0.0213	0.321	15.10	2.234	0.321
14.50	187.8	0.248	0.0210	0.329	15.65	2.221	0.329
15.50	194.1	0.237	0.0208	0.336	16.18	2.208	0.336
16.50	200.3	0.228	0.0206	0.344	16.69	2.196	0.344
17.50	206.3	0.219	0.0204	0.351	17.19	2.184	0.351
18.50	212.1	0.211	0.0202	0.357	17.67	2.173	0.357
19.50	217.7	0.204	0.0201	0.364	18.14	2.162	0.364
20.50	223.2	0.197	0.0199	0.370	18.60	2.151	0.370
21.50	228.6	0.191	0.0198	0.377	19.05	2.141	0.377
22.50	233.9	0.186	0.0196	0.383	19.49	2.130	0.383
23.50	239.0	0.181	0.0195	0.389	19.92	2.120	0.389
24.50	244.1	0.176	0.0194	0.395	20.34	2.110	0.395
25.50	249.0	0.172	0.0193	0.401	20.75	2.100	0.401
26.50	253.8	0.168	0.0192	0.406	21.15	2.091	0.406
27.50	258.6	0.164	0.0191	0.412	21.55	2.081	0.412
28.50	263.2	0.160	0.0190	0.418	21.94	2.072	0.418
29.50	267.8	0.157	0.0190	0.423	22.32	2.062	0.423
30.50	272.3	0.154	0.0189	0.429	22.69	2.053	0.429
31.50	276.7	0.151	0.0188	0.434	23.06	2.044	0.434
32.50	281.1	0.148	0.0188	0.439	23.42	2.034	0.439
33.50	285.4	0.145	0.0187	0.445	23.78	2.025	0.445
34.50	289.6	0.143	0.0186	0.450	24.13	2.016	0.450
35.50	293.8	0.140	0.0186	0.455	24.48	2.007	0.455
36.50	297.9	0.138	0.0186	0.461	24.82	1.998	0.461
37.50	301.9	0.136	0.0185	0.466	25.16	1.989	0.466
38.50	305.9	0.134	0.0185	0.471	25.49	1.980	0.471
39.50	309.9	0.132	0.0184	0.476	25.82	1.971	0.476
40.50	313.8	0.130	0.0184	0.481	26.15	1.962	0.481
41.50	317.6	0.128	0.0184	0.487	26.47	1.953	0.487
42.50	321.4	0.126	0.0184	0.492	26.79	1.944	0.492
43.50	325.2	0.125	0.0183	0.497	27.10	1.935	0.497
44.50	328.9	0.123	0.0183	0.502	27.41	1.926	0.502
45.50	332.6	0.122	0.0183	0.507	27.72	1.917	0.507
46.50	336.2	0.120	0.0183	0.513	28.02	1.908	0.513
47.50	339.8	0.119	0.0183	0.518	28.32	1.898	0.518
48.50	343.4	0.118	0.0183	0.523	28.61	1.889	0.523
49.50	346.9	0.117	0.0183	0.528	28.91	1.880	0.528
50.50	349.7	0.116	0.0183	0.533	29.15	1.871	0.000
51.50	351.9	0.115	0.0183	0.537	29.33	1.862	0.005
52.50	354.1	0.115	0.0184	0.541	29.51	1.853	0.011
53.50	356.2	0.114	0.0184	0.546	29.69	1.844	0.018
54.50	358.4	0.114	0.0184	0.550	29.87	1.835	0.023
55.50	360.5	0.113	0.0185	0.555	30.04	1.826	0.029
56.50	362.6	0.113	0.0185	0.559	30.22	1.816	0.035
57.50	364.7	0.113	0.0185	0.564	30.39	1.807	0.041
58.50	366.8	0.112	0.0186	0.568	30.57	1.797	0.047
59.50	368.9	0.112	0.0186	0.573	30.74	1.788	0.053
60.50	371.0	0.112	0.0187	0.578	30.92	1.778	0.059
61.50	373.0	0.111	0.0187	0.583	31.09	1.768	0.064
62.50	375.1	0.111	0.0188	0.588	31.26	1.757	0.070
63.50	377.1	0.111	0.0189	0.593	31.43	1.747	0.076
64.50	379.1	0.111	0.0189	0.598	31.60	1.736	0.082
65.50	381.2	0.110	0.0190	0.603	31.76	1.726	0.088
66.50	383.2	0.110	0.0191	0.609	31.93	1.715	0.093
67.50	385.2	0.110	0.0191	0.614	32.10	1.703	0.099
68.50	387.1	0.110	0.0192	0.620	32.26	1.692	0.105
69.50	389.1	0.110	0.0193	0.626	32.43	1.680	0.111
70.50	391.1	0.110	0.0194	0.632	32.59	1.668	0.117
71.50	393.0	0.110	0.0195	0.638	32.75	1.656	0.123
72.50	395.0	0.110	0.0196	0.645	32.91	1.643	0.129
73.50	396.9	0.110	0.0197	0.652	33.07	1.630	0.135
74.50	398.8	0.110	0.0198	0.659	33.23	1.617	0.141
75.50	400.7	0.110	0.0199	0.666	33.39	1.603	0.147
76.50	402.6	0.110	0.0201	0.673	33.55	1.589	0.153
77.50	404.5	0.111	0.0202	0.681	33.71	1.575	0.159
78.50	406.4	0.111	0.0203	0.689	33.87	1.560	0.166
79.50	408.3	0.111	0.0205	0.698	34.03	1.544	0.172
80.50	410.2	0.112	0.0207	0.707	34.18	1.528	0.179
81.50	412.0	0.112	0.0209	0.716	34.34	1.512	0.186
82.50	413.9	0.113	0.0211	0.726	34.49	1.494	0.193
83.50	415.7	0.113	0.0213	0.737	34.64	1.476	0.200
84.50	417.6	0.114	0.0215	0.748	34.80	1.457	0.207
85.50	419.4	0.115	0.0217	0.760	34.95	1.437	0.215
86.50	421.2	0.116	0.0220	0.773	35.10	1.416	0.223
87.50	423.0	0.117	0.0223	0.787	35.25	1.394	0.232
88.50	424.8	0.118	0.0227	0.802	35.40	1.370	0.241
89.50	426.6	0.120	0.0230	0.819	35.55	1.345	0.250
90.50	428.4	0.121	0.0235	0.838	35.70	1.318	0.260
91.50	430.2	0.123	0.0239	0.859	35.85	1.288	0.271
92.50	432.0	0.126	0.0245	0.882	36.00	1.256	0.283
93.50	433.8	0.129	0.0252	0.910	36.15	1.220	0.297
94.50	435.5	0.132	0.0260	0.944	36.29	1.179	0.313
95.50	437.3	0.137	0.0270	0.985	36.44	1.133	0.331
96.50	439.0	0.143	0.0284	1.038	36.58	1.077	0.354
97.50	440.8	0.152	0.0303	1.113	36.73	1.006	0.385
98.50	442.5	0.168	0.0335	1.235	36.87	0.908	0.433
99.50	444.2	0.181	0.0363	1.343	37.02	0.837	0.477

Data for Axial Load Analysis using ALP Method
Length of the pile, in.	1,200.0
Axial stiffness EA. lbs.	720,000,000
Circumference, in.	48.000
Point resistance, lbs.	6,750
Quake of the point, in.	0.034
Number of pile elements	100
Number of loading steps	20
Maximum pile load, lbs.	226,672.5
Load Increment, lbs.	22,667.3
Failure Load, lbs.	226,672.5

Results for Loading and Unloading Test
Load Step	Force at Pile Head, kips	Pile Head Deflection, in.	Number of Plastic Shaft Springs
0	0.0	0.000	0
1	22.7	0.012	0
2	45.3	0.025	0
3	68.0	0.039	18
4	90.7	0.058	33
5	113.3	0.082	44
6	136.0	0.109	55
7	158.7	0.140	64
8	181.3	0.175	74
9	204.0	0.214	84
10	226.7	0.271	100
11	204.0	0.259	0
12	181.3	0.246	0
13	158.7	0.234	0
14	136.0	0.221	0
15	113.3	0.209	7
16	90.7	0.193	18
17	68.0	0.175	27
18	45.3	0.154	33
19	22.7	0.132	39
20	-0.0	0.108	44

alpimage — Plotted Results
x-axis = Pile Head Force
y-axis = Pile Head Deflection
Plot Limits:
x-axis from -0.000 to 226.673
y-axis from 0.000 to 0.271

Although the cohesive soils yield very different results from the cohesionless ones, the presentation is the same. Note the significant difference between the element/segment SRD for the static resistance and with the pore pressure increase included. The pile set-up factor is about 2, which is within an acceptable range. This does not apply to the toe.

Screenshot_20180106_163425

The input for the wave equation is identical, except for the hammer selected, which is much smaller than for the cohesionless soils. This is not due to set-up but to the lower capacity of the pile; the hammer selection does not account for set-up. The user will have to select a smaller hammer size to take full advantage of this, depending upon the results.

General Output for Wave Equation Analysis
2018-01-06T15:59:49-05:00
Time Step, msec	0.04024
Pile Weight, lbs.	15,000
Pile Stiffness, lb/ft	600,000
Pile Impedance, lb-sec/ft	57,937.5
L/c, msec	8.04688
Pile Toe Element Number	102
Length of Pile Segments, ft.	1
Hammer Manufacturer and Size	VULCAN 65C
Hammer Rated Striking Energy, ft-lbs	19175
Hammer Efficiency, percent	50
Length of Hammer Cushion Stack, in.	18.5
Soil Resistance to Driving (SRD) for detailed results only, kips	115.4
Percent at Toe	5.85
Toe Quake, in.	0.009
Toe Damping, sec/ft	2.07

Initial Element Output
SRD = 115.44 kips
Element	Element Weight, lbs.	Element Stiffness, kips/in	Element Cross-Sectional Area, in²	Element Soil Resistance, kips	Element Coefficient of Restitution	Element Initial Velocity, ft/sec	Element Soil Shaft Stiffness, kips/in	Element Quake, in.	Element Damping, sec/ft
Ram	6,500.0	1,880.5	99.40	0.0	0.80	9.74	0.0	1,000.000	0.00
Driving Accessory	1,100.0	711.5	144.00	0.0	0.51	0.00	0.0	1,000.000	0.00
Pile Head	150.0	60,000.0	144.00	0.5	1.00	0.00	11.6	0.040	2.71
4	150.0	60,000.0	144.00	0.6	1.00	0.00	20.1	0.032	2.56
5	150.0	60,000.0	144.00	0.8	1.00	0.00	26.0	0.029	2.49
6	150.0	60,000.0	144.00	0.8	1.00	0.00	30.7	0.027	2.44
7	150.0	60,000.0	144.00	0.9	1.00	0.00	34.9	0.026	2.41
8	150.0	60,000.0	144.00	1.0	1.00	0.00	38.5	0.025	2.38
9	150.0	60,000.0	144.00	1.0	1.00	0.00	41.9	0.024	2.35
10	150.0	60,000.0	144.00	1.1	1.00	0.00	45.0	0.024	2.33
11	150.0	60,000.0	144.00	1.1	1.00	0.00	47.9	0.023	2.31
12	150.0	60,000.0	144.00	1.1	1.00	0.00	50.7	0.023	2.29
13	150.0	60,000.0	144.00	1.2	1.00	0.00	53.3	0.022	2.28
14	150.0	60,000.0	144.00	1.2	1.00	0.00	55.7	0.022	2.26
15	150.0	60,000.0	144.00	1.3	1.00	0.00	58.1	0.022	2.25
16	150.0	60,000.0	144.00	1.3	1.00	0.00	60.4	0.021	2.23
17	150.0	60,000.0	144.00	1.3	1.00	0.00	62.6	0.021	2.22
18	150.0	60,000.0	144.00	1.3	1.00	0.00	64.7	0.021	2.21
19	150.0	60,000.0	144.00	1.4	1.00	0.00	66.8	0.021	2.20
20	150.0	60,000.0	144.00	1.4	1.00	0.00	68.8	0.020	2.18
21	150.0	60,000.0	144.00	1.4	1.00	0.00	70.7	0.020	2.17
22	150.0	60,000.0	144.00	1.5	1.00	0.00	72.6	0.020	2.16
23	150.0	60,000.0	144.00	1.5	1.00	0.00	74.4	0.020	2.15
24	150.0	60,000.0	144.00	1.5	1.00	0.00	76.2	0.020	2.14
25	150.0	60,000.0	144.00	1.5	1.00	0.00	78.0	0.020	2.13
26	150.0	60,000.0	144.00	1.6	1.00	0.00	79.7	0.020	2.12
27	150.0	60,000.0	144.00	1.6	1.00	0.00	81.4	0.019	2.11
28	150.0	60,000.0	144.00	1.6	1.00	0.00	83.0	0.019	2.10
29	150.0	60,000.0	144.00	1.6	1.00	0.00	84.6	0.019	2.09
30	150.0	60,000.0	144.00	1.6	1.00	0.00	86.2	0.019	2.08
31	150.0	60,000.0	144.00	1.7	1.00	0.00	87.7	0.019	2.07
32	150.0	60,000.0	144.00	1.7	1.00	0.00	89.3	0.019	2.06
33	150.0	60,000.0	144.00	1.7	1.00	0.00	90.8	0.019	2.05
34	150.0	60,000.0	144.00	1.7	1.00	0.00	92.2	0.019	2.04
35	150.0	60,000.0	144.00	1.8	1.00	0.00	93.7	0.019	2.03
36	150.0	60,000.0	144.00	1.8	1.00	0.00	95.1	0.019	2.03
37	150.0	60,000.0	144.00	1.8	1.00	0.00	96.5	0.019	2.02
38	150.0	60,000.0	144.00	1.8	1.00	0.00	97.9	0.019	2.01
39	150.0	60,000.0	144.00	1.8	1.00	0.00	99.3	0.019	2.00
40	150.0	60,000.0	144.00	1.9	1.00	0.00	100.6	0.019	1.99
41	150.0	60,000.0	144.00	1.9	1.00	0.00	102.0	0.018	1.98
42	150.0	60,000.0	144.00	1.9	1.00	0.00	103.3	0.018	1.97
43	150.0	60,000.0	144.00	1.9	1.00	0.00	104.6	0.018	1.96
44	150.0	60,000.0	144.00	1.9	1.00	0.00	105.9	0.018	1.95
45	150.0	60,000.0	144.00	2.0	1.00	0.00	107.1	0.018	1.94
46	150.0	60,000.0	144.00	2.0	1.00	0.00	108.4	0.018	1.93
47	150.0	60,000.0	144.00	2.0	1.00	0.00	109.6	0.018	1.93
48	150.0	60,000.0	144.00	2.0	1.00	0.00	110.9	0.018	1.92
49	150.0	60,000.0	144.00	2.1	1.00	0.00	112.1	0.018	1.91
50	150.0	60,000.0	144.00	2.1	1.00	0.00	113.3	0.018	1.90
51	150.0	60,000.0	144.00	2.1	1.00	0.00	114.5	0.018	1.89
52	150.0	60,000.0	144.00	2.1	1.00	0.00	115.6	0.018	1.88
53	150.0	60,000.0	144.00	0.0	1.00	0.00	0.0	0.018	1.87
54	150.0	60,000.0	144.00	0.0	1.00	0.00	1.2	0.018	1.86
55	150.0	60,000.0	144.00	0.0	1.00	0.00	2.5	0.018	1.85
56	150.0	60,000.0	144.00	0.1	1.00	0.00	3.8	0.018	1.84
57	150.0	60,000.0	144.00	0.1	1.00	0.00	5.1	0.018	1.84
58	150.0	60,000.0	144.00	0.1	1.00	0.00	6.4	0.018	1.83
59	150.0	60,000.0	144.00	0.1	1.00	0.00	7.6	0.018	1.82
60	150.0	60,000.0	144.00	0.2	1.00	0.00	8.9	0.019	1.81
61	150.0	60,000.0	144.00	0.2	1.00	0.00	10.1	0.019	1.80
62	150.0	60,000.0	144.00	0.2	1.00	0.00	11.3	0.019	1.79
63	150.0	60,000.0	144.00	0.2	1.00	0.00	12.6	0.019	1.78
64	150.0	60,000.0	144.00	0.3	1.00	0.00	13.8	0.019	1.77
65	150.0	60,000.0	144.00	0.3	1.00	0.00	14.9	0.019	1.76
66	150.0	60,000.0	144.00	0.3	1.00	0.00	16.1	0.019	1.75
67	150.0	60,000.0	144.00	0.3	1.00	0.00	17.3	0.019	1.74
68	150.0	60,000.0	144.00	0.4	1.00	0.00	18.4	0.019	1.73
69	150.0	60,000.0	144.00	0.4	1.00	0.00	19.6	0.019	1.71
70	150.0	60,000.0	144.00	0.4	1.00	0.00	20.7	0.019	1.70
71	150.0	60,000.0	144.00	0.4	1.00	0.00	21.8	0.019	1.69
72	150.0	60,000.0	144.00	0.4	1.00	0.00	23.0	0.019	1.68
73	150.0	60,000.0	144.00	0.5	1.00	0.00	24.1	0.019	1.67
74	150.0	60,000.0	144.00	0.5	1.00	0.00	25.2	0.019	1.66
75	150.0	60,000.0	144.00	0.5	1.00	0.00	26.2	0.020	1.64
76	150.0	60,000.0	144.00	0.5	1.00	0.00	27.3	0.020	1.63
77	150.0	60,000.0	144.00	0.6	1.00	0.00	28.4	0.020	1.62
78	150.0	60,000.0	144.00	0.6	1.00	0.00	29.4	0.020	1.60
79	150.0	60,000.0	144.00	0.6	1.00	0.00	30.5	0.020	1.59
80	150.0	60,000.0	144.00	0.6	1.00	0.00	31.5	0.020	1.57
81	150.0	60,000.0	144.00	0.7	1.00	0.00	32.6	0.020	1.56
82	150.0	60,000.0	144.00	0.7	1.00	0.00	33.6	0.021	1.54
83	150.0	60,000.0	144.00	0.7	1.00	0.00	34.6	0.021	1.53
84	150.0	60,000.0	144.00	0.7	1.00	0.00	35.6	0.021	1.51
85	150.0	60,000.0	144.00	0.8	1.00	0.00	36.6	0.021	1.49
86	150.0	60,000.0	144.00	0.8	1.00	0.00	37.6	0.021	1.48
87	150.0	60,000.0	144.00	0.8	1.00	0.00	38.6	0.021	1.46
88	150.0	60,000.0	144.00	0.9	1.00	0.00	39.6	0.022	1.44
89	150.0	60,000.0	144.00	0.9	1.00	0.00	40.6	0.022	1.42
90	150.0	60,000.0	144.00	0.9	1.00	0.00	41.5	0.022	1.39
91	150.0	60,000.0	144.00	1.0	1.00	0.00	42.5	0.023	1.37
92	150.0	60,000.0	144.00	1.0	1.00	0.00	43.4	0.023	1.34
93	150.0	60,000.0	144.00	1.0	1.00	0.00	44.4	0.023	1.32
94	150.0	60,000.0	144.00	1.1	1.00	0.00	45.3	0.024	1.29
95	150.0	60,000.0	144.00	1.1	1.00	0.00	46.2	0.025	1.26
96	150.0	60,000.0	144.00	1.2	1.00	0.00	47.2	0.025	1.22
97	150.0	60,000.0	144.00	1.3	1.00	0.00	48.1	0.026	1.18
98	150.0	60,000.0	144.00	1.3	1.00	0.00	49.0	0.027	1.13
99	150.0	60,000.0	144.00	1.4	1.00	0.00	49.9	0.028	1.08
100	150.0	60,000.0	144.00	1.5	1.00	0.00	50.8	0.030	1.01
101	150.0	60,000.0	144.00	1.7	1.00	0.00	51.7	0.034	0.91
102	150.0	786.0	144.00	1.9	1.00	0.00	52.6	0.036	0.84
Pile Toe	0.0	786.0	144.00	6.8	0.00	0.00	0.0	0.009	2.07

Final Element Output
SRD = 115.44 kips
Element	Time Step for Maximum Compressive Stress	Maximum Compressive Stress, ksi	Time Step for Maximum Tensile Stress	Maximum Tensile Stress, ksi	Maximum Deflection, in.	Final Deflection, in.	Final Velocity, ft/sec
1	183	2.90	592	0.00	0.818	0.277	-9.74
2	119	1.55	538	0.00	0.696	0.681	0.12
3	121	1.56	2	0.00	0.270	0.265	-0.02
4	123	1.56	3	0.00	0.270	0.265	-0.03
5	125	1.55	465	0.01	0.270	0.265	-0.02
6	127	1.55	467	0.05	0.270	0.265	-0.02
7	128	1.55	469	0.10	0.269	0.265	-0.02
8	130	1.55	471	0.14	0.269	0.265	-0.02
9	132	1.55	471	0.18	0.268	0.265	-0.01
10	134	1.55	473	0.22	0.268	0.265	-0.00
11	136	1.54	475	0.26	0.268	0.265	0.00
12	138	1.54	477	0.30	0.267	0.265	0.01
13	140	1.54	476	0.34	0.267	0.265	0.02
14	142	1.54	477	0.37	0.267	0.266	0.03
15	144	1.53	478	0.40	0.267	0.266	0.05
16	146	1.53	477	0.43	0.267	0.266	0.08
17	148	1.53	477	0.46	0.267	0.267	0.11
18	150	1.52	476	0.48	0.267	0.267	0.14
19	152	1.52	477	0.50	0.268	0.268	0.17
20	154	1.52	478	0.51	0.269	0.269	0.20
21	156	1.51	476	0.53	0.269	0.269	0.23
22	158	1.51	476	0.54	0.270	0.270	0.26
23	160	1.50	475	0.55	0.271	0.271	0.30
24	162	1.50	476	0.55	0.271	0.271	0.34
25	164	1.49	476	0.55	0.272	0.272	0.37
26	166	1.49	476	0.54	0.273	0.273	0.41
27	168	1.48	475	0.53	0.274	0.274	0.45
28	170	1.48	475	0.51	0.274	0.274	0.48
29	172	1.47	476	0.48	0.275	0.275	0.53
30	174	1.47	475	0.45	0.276	0.276	0.58
31	176	1.46	474	0.41	0.276	0.276	0.63
32	178	1.46	472	0.37	0.277	0.277	0.68
33	180	1.45	471	0.32	0.278	0.278	0.71
34	182	1.44	472	0.28	0.278	0.278	0.72
35	184	1.43	466	0.23	0.278	0.278	0.72
36	185	1.42	516	0.24	0.279	0.279	0.70
37	186	1.41	524	0.26	0.279	0.279	0.65
38	188	1.40	529	0.28	0.279	0.279	0.58
39	190	1.38	532	0.31	0.279	0.279	0.51
40	192	1.37	533	0.34	0.279	0.279	0.44
41	194	1.36	542	0.38	0.279	0.279	0.38
42	196	1.35	541	0.42	0.279	0.279	0.33
43	198	1.33	544	0.45	0.279	0.279	0.28
44	200	1.32	543	0.49	0.279	0.279	0.23
45	203	1.31	542	0.52	0.279	0.279	0.18
46	205	1.30	545	0.55	0.278	0.278	0.12
47	207	1.28	544	0.58	0.278	0.278	0.08
48	209	1.27	542	0.60	0.277	0.277	0.03
49	211	1.26	544	0.63	0.277	0.277	-0.01
50	213	1.24	543	0.65	0.277	0.277	-0.05
51	216	1.23	542	0.67	0.276	0.276	-0.10
52	217	1.22	540	0.69	0.276	0.276	-0.14
53	218	1.22	539	0.69	0.277	0.275	-0.18
54	220	1.22	540	0.69	0.278	0.275	-0.22
55	222	1.22	539	0.69	0.279	0.274	-0.25
56	224	1.22	538	0.69	0.281	0.274	-0.28
57	226	1.22	538	0.68	0.282	0.274	-0.32
58	228	1.22	538	0.66	0.283	0.273	-0.36
59	230	1.22	537	0.65	0.285	0.273	-0.41
60	232	1.23	536	0.63	0.286	0.273	-0.46
61	235	1.23	534	0.60	0.287	0.273	-0.52
62	237	1.23	535	0.57	0.288	0.273	-0.56
63	239	1.23	533	0.54	0.290	0.273	-0.61
64	241	1.23	532	0.50	0.291	0.273	-0.63
65	244	1.23	530	0.46	0.292	0.273	-0.66
66	246	1.23	531	0.41	0.293	0.273	-0.69
67	248	1.23	531	0.35	0.294	0.273	-0.72
68	250	1.23	530	0.29	0.294	0.274	-0.74
69	253	1.23	532	0.23	0.295	0.274	-0.75
70	255	1.23	470	0.18	0.296	0.274	-0.75
71	253	1.23	474	0.21	0.296	0.275	-0.75
72	255	1.23	473	0.24	0.296	0.275	-0.75
73	257	1.23	476	0.27	0.296	0.276	-0.74
74	260	1.23	476	0.30	0.296	0.276	-0.74
75	262	1.23	478	0.33	0.296	0.277	-0.72
76	264	1.23	478	0.35	0.295	0.277	-0.71
77	266	1.23	480	0.38	0.295	0.278	-0.70
78	268	1.22	479	0.39	0.294	0.278	-0.68
79	271	1.22	478	0.41	0.294	0.279	-0.66
80	273	1.22	480	0.43	0.293	0.279	-0.65
81	275	1.21	478	0.44	0.292	0.280	-0.64
82	277	1.21	477	0.46	0.292	0.280	-0.62
83	279	1.20	479	0.47	0.291	0.281	-0.60
84	280	1.19	477	0.48	0.290	0.282	-0.58
85	279	1.18	474	0.49	0.290	0.282	-0.55
86	280	1.17	474	0.50	0.289	0.283	-0.53
87	281	1.15	476	0.50	0.289	0.284	-0.51
88	281	1.12	469	0.51	0.288	0.284	-0.48
89	280	1.10	471	0.51	0.288	0.285	-0.45
90	281	1.06	471	0.51	0.288	0.285	-0.42
91	281	1.02	472	0.50	0.288	0.286	-0.40
92	280	0.97	473	0.49	0.288	0.287	-0.37
93	281	0.92	474	0.46	0.288	0.287	-0.34
94	282	0.87	474	0.42	0.288	0.288	-0.30
95	283	0.81	475	0.37	0.289	0.288	-0.27
96	282	0.75	476	0.31	0.289	0.289	-0.25
97	283	0.68	478	0.25	0.289	0.289	-0.23
98	289	0.62	480	0.19	0.289	0.289	-0.21
99	294	0.56	482	0.12	0.290	0.290	-0.19
100	302	0.51	485	0.07	0.290	0.290	-0.17
101	307	0.47	489	0.02	0.290	0.290	-0.15
102	316	0.46	532	0.00	0.290	0.290	-0.12

forcetime — Force-Time History, SRD = 115.44 kips
Blue Line = Pile Head Force
Red Line = Pile Head Impedance*Velocity
Vertical grid spacing from left to right is L/c, may not be complete for last spacing.
Plot Limits:
x-axis from 0.000 to 2.955
y-axis from -68,985.344 to 223,926.386

Summary of Results and Bearing Graph Data
Soil Resistance, kips	Permanent Set of Pile Toe, inches	Blows per Foot of Penetration	Maximum Compressive Stress, ksi	Element of Maximum Compressive Stress	Maximum Tensile Stress, ksi	Element of Maximum Tensile Stress	Number of Iterations
23.1 (45.3)	1.541	7.8	1.53	4	1.21	24	2000
46.2 (90.7)	0.744	16.1	1.54	4	1.05	54	1149
69.3 (136.0)	0.494	24.3	1.54	4	0.97	54	872
92.3 (181.3)	0.349	34.4	1.55	4	0.86	54	740
115.4 (226.7)	0.281	42.7	1.56	4	0.69	54	592
138.5 (272.0)	0.228	52.6	1.58	3	0.52	56	588
161.6 (317.3)	0.184	65.2	1.61	3	0.30	92	480
184.7 (362.7)	0.144	83.3	1.64	3	0.20	94	477
207.8 (408.0)	0.108	111.1	1.67	4	0.11	95	474
230.9 (453.3)	0.077	155.4	1.70	4	0.07	92	471

The bearing graph data is complete. The only difference with the cohesionless soils is the way the soil resistance is reported; the values in parentheses are ultimate resistance without set-up and those outside are the SRD with set-up. The blow count indicates that a smaller hammer may be in order.

TAMWAVE 6: Results of Wave Equation Analysis

Posted on 6 January 2018 by Don Warrington

With the data entered for the wave equation analysis, we can now see the results. There’s a lot of tabular data here but you need to read the notes between it to understand what the program is putting out. If you are not familiar at all with the wave equation for piles, you need to review this as well.

General Output for Wave Equation Analysis
2018-01-06T10:13:03-05:00
Time Step, msec	0.04024
Pile Weight, lbs.	15,000
Pile Stiffness, lb/ft	600,000
Pile Impedance, lb-sec/ft	57,937.5
L/c, msec	8.04688
Pile Toe Element Number	102
Length of Pile Segments, ft.	1
Hammer Manufacturer and Size	VULCAN O16
Hammer Rated Striking Energy, ft-lbs	48750
Hammer Efficiency, percent	67
Length of Hammer Cushion Stack, in.	16.5
Soil Resistance to Driving (SRD) for detailed results only, kips	572.7
Percent at Toe	35.39
Toe Quake, in.	0.220
Toe Damping, sec/ft	0.07

For those familiar with the wave equation, there are few surprises. Some explanation of the parameters can be found with the documentation for the TTI program.

Initial Element Output
SRD = 572.68 kips
Element	Element Weight, lbs.	Element Stiffness, kips/in	Element Cross-Sectional Area, in²	Element Soil Resistance, kips	Element Coefficient of Restitution	Element Initial Velocity, ft/sec	Element Soil Shaft Stiffness, kips/in	Element Quake, in.	Element Damping, sec/ft
Ram	16,250.0	4,957.5	233.71	0.0	0.80	11.37	0.0	1,000.000	0.00
Driving Accessory	3,800.0	711.5	144.00	0.0	0.51	0.00	0.0	1,000.000	0.00
Pile Head	150.0	60,000.0	144.00	0.0	1.00	0.00	16.1	0.002	45.39
4	150.0	60,000.0	144.00	0.1	1.00	0.00	28.0	0.004	19.91
5	150.0	60,000.0	144.00	0.2	1.00	0.00	36.1	0.005	13.57
6	150.0	60,000.0	144.00	0.3	1.00	0.00	42.7	0.006	10.54
7	150.0	60,000.0	144.00	0.3	1.00	0.00	48.4	0.007	8.73
8	150.0	60,000.0	144.00	0.4	1.00	0.00	53.5	0.007	7.51
9	150.0	60,000.0	144.00	0.5	1.00	0.00	58.2	0.008	6.62
10	150.0	60,000.0	144.00	0.5	1.00	0.00	62.5	0.009	5.95
11	150.0	60,000.0	144.00	0.6	1.00	0.00	66.6	0.009	5.41
12	150.0	60,000.0	144.00	0.7	1.00	0.00	70.4	0.010	4.98
13	150.0	60,000.0	144.00	0.8	1.00	0.00	74.0	0.010	4.62
14	150.0	60,000.0	144.00	0.8	1.00	0.00	77.4	0.011	4.31
15	150.0	60,000.0	144.00	0.9	1.00	0.00	80.7	0.011	4.05
16	150.0	60,000.0	144.00	1.0	1.00	0.00	83.9	0.012	3.82
17	150.0	60,000.0	144.00	1.0	1.00	0.00	87.0	0.012	3.62
18	150.0	60,000.0	144.00	1.1	1.00	0.00	89.9	0.012	3.44
19	150.0	60,000.0	144.00	1.2	1.00	0.00	92.8	0.013	3.28
20	150.0	60,000.0	144.00	1.3	1.00	0.00	95.6	0.013	3.14
21	150.0	60,000.0	144.00	1.3	1.00	0.00	98.3	0.014	3.01
22	150.0	60,000.0	144.00	1.4	1.00	0.00	100.9	0.014	2.89
23	150.0	60,000.0	144.00	1.5	1.00	0.00	103.5	0.014	2.79
24	150.0	60,000.0	144.00	1.5	1.00	0.00	106.0	0.015	2.69
25	150.0	60,000.0	144.00	1.6	1.00	0.00	108.4	0.015	2.60
26	150.0	60,000.0	144.00	1.7	1.00	0.00	110.8	0.015	2.51
27	150.0	60,000.0	144.00	1.8	1.00	0.00	113.1	0.016	2.43
28	150.0	60,000.0	144.00	1.8	1.00	0.00	115.4	0.016	2.36
29	150.0	60,000.0	144.00	1.9	1.00	0.00	117.7	0.016	2.29
30	150.0	60,000.0	144.00	2.0	1.00	0.00	119.9	0.017	2.23
31	150.0	60,000.0	144.00	2.1	1.00	0.00	122.1	0.017	2.17
32	150.0	60,000.0	144.00	2.1	1.00	0.00	124.2	0.017	2.11
33	150.0	60,000.0	144.00	2.2	1.00	0.00	126.3	0.017	2.06
34	150.0	60,000.0	144.00	2.3	1.00	0.00	128.4	0.018	2.01
35	150.0	60,000.0	144.00	2.4	1.00	0.00	130.4	0.018	1.96
36	150.0	60,000.0	144.00	2.4	1.00	0.00	132.5	0.018	1.91
37	150.0	60,000.0	144.00	2.5	1.00	0.00	134.4	0.019	1.87
38	150.0	60,000.0	144.00	2.6	1.00	0.00	136.4	0.019	1.83
39	150.0	60,000.0	144.00	2.7	1.00	0.00	138.3	0.019	1.79
40	150.0	60,000.0	144.00	2.7	1.00	0.00	140.2	0.019	1.75
41	150.0	60,000.0	144.00	2.8	1.00	0.00	142.1	0.020	1.72
42	150.0	60,000.0	144.00	2.9	1.00	0.00	144.0	0.020	1.68
43	150.0	60,000.0	144.00	3.0	1.00	0.00	145.8	0.020	1.65
44	150.0	60,000.0	144.00	3.0	1.00	0.00	147.7	0.021	1.62
45	150.0	60,000.0	144.00	3.1	1.00	0.00	149.5	0.021	1.59
46	150.0	60,000.0	144.00	3.2	1.00	0.00	151.3	0.021	1.56
47	150.0	60,000.0	144.00	3.3	1.00	0.00	153.0	0.021	1.53
48	150.0	60,000.0	144.00	3.3	1.00	0.00	154.8	0.022	1.50
49	150.0	60,000.0	144.00	3.4	1.00	0.00	156.5	0.022	1.48
50	150.0	60,000.0	144.00	3.5	1.00	0.00	158.3	0.022	1.45
51	150.0	60,000.0	144.00	3.6	1.00	0.00	160.0	0.022	1.43
52	150.0	60,000.0	144.00	3.7	1.00	0.00	161.7	0.023	1.40
53	150.0	60,000.0	144.00	3.7	1.00	0.00	163.0	0.023	1.38
54	150.0	60,000.0	144.00	3.8	1.00	0.00	164.1	0.023	1.37
55	150.0	60,000.0	144.00	3.8	1.00	0.00	165.2	0.023	1.35
56	150.0	60,000.0	144.00	3.9	1.00	0.00	166.2	0.023	1.34
57	150.0	60,000.0	144.00	4.0	1.00	0.00	167.3	0.024	1.32
58	150.0	60,000.0	144.00	4.0	1.00	0.00	168.4	0.024	1.31
59	150.0	60,000.0	144.00	4.1	1.00	0.00	169.4	0.024	1.29
60	150.0	60,000.0	144.00	4.1	1.00	0.00	170.5	0.024	1.28
61	150.0	60,000.0	144.00	4.2	1.00	0.00	171.6	0.024	1.27
62	150.0	60,000.0	144.00	4.2	1.00	0.00	172.6	0.025	1.25
63	150.0	60,000.0	144.00	4.3	1.00	0.00	173.7	0.025	1.24
64	150.0	60,000.0	144.00	4.4	1.00	0.00	174.8	0.025	1.22
65	150.0	60,000.0	144.00	4.4	1.00	0.00	175.8	0.025	1.21
66	150.0	60,000.0	144.00	4.5	1.00	0.00	176.9	0.025	1.20
67	150.0	60,000.0	144.00	4.6	1.00	0.00	178.0	0.026	1.18
68	150.0	60,000.0	144.00	4.6	1.00	0.00	179.0	0.026	1.17
69	150.0	60,000.0	144.00	4.7	1.00	0.00	180.1	0.026	1.16
70	150.0	60,000.0	144.00	4.8	1.00	0.00	181.2	0.026	1.14
71	150.0	60,000.0	144.00	4.8	1.00	0.00	182.3	0.026	1.13
72	150.0	60,000.0	144.00	4.9	1.00	0.00	183.4	0.027	1.12
73	150.0	60,000.0	144.00	5.0	1.00	0.00	184.5	0.027	1.10
74	150.0	60,000.0	144.00	5.0	1.00	0.00	185.6	0.027	1.09
75	150.0	60,000.0	144.00	5.1	1.00	0.00	186.7	0.027	1.08
76	150.0	60,000.0	144.00	5.2	1.00	0.00	187.8	0.028	1.06
77	150.0	60,000.0	144.00	5.3	1.00	0.00	189.0	0.028	1.05
78	150.0	60,000.0	144.00	5.4	1.00	0.00	190.1	0.028	1.04
79	150.0	60,000.0	144.00	5.5	1.00	0.00	191.2	0.029	1.03
80	150.0	60,000.0	144.00	5.5	1.00	0.00	192.4	0.029	1.01
81	150.0	60,000.0	144.00	5.6	1.00	0.00	193.6	0.029	1.00
82	150.0	60,000.0	144.00	5.7	1.00	0.00	194.8	0.029	0.99
83	150.0	60,000.0	144.00	5.8	1.00	0.00	196.0	0.030	0.97
84	150.0	60,000.0	144.00	5.9	1.00	0.00	197.2	0.030	0.96
85	150.0	60,000.0	144.00	6.0	1.00	0.00	198.4	0.030	0.95
86	150.0	60,000.0	144.00	6.1	1.00	0.00	199.6	0.031	0.93
87	150.0	60,000.0	144.00	6.2	1.00	0.00	200.9	0.031	0.92
88	150.0	60,000.0	144.00	6.3	1.00	0.00	202.2	0.031	0.90
89	150.0	60,000.0	144.00	6.5	1.00	0.00	203.5	0.032	0.89
90	150.0	60,000.0	144.00	6.6	1.00	0.00	204.8	0.032	0.88
91	150.0	60,000.0	144.00	6.7	1.00	0.00	206.1	0.033	0.86
92	150.0	60,000.0	144.00	6.8	1.00	0.00	207.5	0.033	0.85
93	150.0	60,000.0	144.00	7.0	1.00	0.00	208.9	0.033	0.84
94	150.0	60,000.0	144.00	7.1	1.00	0.00	210.3	0.034	0.82
95	150.0	60,000.0	144.00	7.3	1.00	0.00	211.7	0.034	0.81
96	150.0	60,000.0	144.00	7.4	1.00	0.00	213.2	0.035	0.79
97	150.0	60,000.0	144.00	7.6	1.00	0.00	214.7	0.035	0.78
98	150.0	60,000.0	144.00	7.7	1.00	0.00	216.3	0.036	0.77
99	150.0	60,000.0	144.00	7.9	1.00	0.00	217.8	0.036	0.75
100	150.0	60,000.0	144.00	8.1	1.00	0.00	219.4	0.037	0.74
101	150.0	60,000.0	144.00	8.3	1.00	0.00	221.1	0.038	0.72
102	150.0	922.6	144.00	8.5	1.00	0.00	222.8	0.038	0.71
Pile Toe	0.0	922.6	144.00	202.7	0.00	0.00	0.0	0.220	0.07

A detailed output of the parameters for each segment/element. TAMWAVE no longer uses the simplifications used in the past for resistance distribution along the shaft, i.e., uniform, triangular, etc., but constructs one based on the soil properties. Much of this data is repeated from the static analysis.

Final Element Output
SRD = 572.68 kips
Element	Time Step for Maximum Compressive Stress	Maximum Compressive Stress, ksi	Time Step for Maximum Tensile Stress	Maximum Tensile Stress, ksi	Maximum Deflection, in.	Final Deflection, in.	Final Velocity, ft/sec
1	50	3.70	164	0.00	1.299	1.299	-0.11
2	176	2.64	1	0.00	1.300	1.261	-2.56
3	178	2.64	2	0.00	0.650	0.646	-1.01
4	180	2.65	3	0.00	0.646	0.643	-0.93
5	182	2.66	4	0.00	0.641	0.639	-0.85
6	184	2.66	5	0.00	0.637	0.635	-0.78
7	186	2.67	6	0.00	0.632	0.631	-0.70
8	187	2.67	7	0.00	0.628	0.627	-0.62
9	190	2.68	8	0.00	0.623	0.622	-0.53
10	192	2.69	9	0.00	0.619	0.618	-0.45
11	194	2.69	10	0.00	0.614	0.613	-0.37
12	196	2.69	11	0.00	0.609	0.609	-0.30
13	198	2.70	12	0.00	0.604	0.604	-0.22
14	359	2.71	13	0.00	0.599	0.599	-0.14
15	361	2.72	14	0.00	0.594	0.594	-0.06
16	363	2.73	15	0.00	0.588	0.588	0.01
17	365	2.74	16	0.00	0.583	0.583	0.07
18	367	2.75	17	0.00	0.578	0.578	0.13
19	369	2.75	18	0.00	0.572	0.572	0.19
20	372	2.76	19	0.00	0.567	0.567	0.24
21	374	2.77	20	0.00	0.561	0.561	0.27
22	376	2.78	21	0.00	0.556	0.556	0.29
23	378	2.79	22	0.00	0.550	0.550	0.30
24	379	2.80	23	0.00	0.544	0.544	0.29
25	381	2.80	24	0.00	0.539	0.539	0.28
26	384	2.81	25	0.00	0.533	0.533	0.26
27	386	2.82	26	0.00	0.527	0.527	0.23
28	388	2.82	27	0.00	0.522	0.522	0.19
29	390	2.83	28	0.00	0.516	0.516	0.15
30	392	2.83	29	0.00	0.511	0.511	0.11
31	393	2.84	30	0.00	0.505	0.505	0.07
32	395	2.84	31	0.00	0.500	0.500	0.03
33	397	2.84	32	0.00	0.496	0.494	-0.01
34	399	2.84	33	0.00	0.491	0.489	-0.05
35	399	2.84	34	0.00	0.487	0.483	-0.08
36	400	2.84	35	0.00	0.483	0.478	-0.11
37	401	2.83	36	0.00	0.479	0.473	-0.14
38	400	2.82	37	0.00	0.474	0.468	-0.17
39	401	2.81	38	0.00	0.470	0.463	-0.19
40	400	2.80	39	0.00	0.466	0.457	-0.21
41	401	2.78	40	0.00	0.462	0.452	-0.24
42	399	2.76	41	0.00	0.458	0.447	-0.26
43	400	2.74	42	0.00	0.454	0.442	-0.27
44	399	2.71	43	0.00	0.449	0.437	-0.29
45	398	2.68	44	0.00	0.445	0.432	-0.30
46	397	2.65	45	0.00	0.441	0.427	-0.31
47	267	2.64	46	0.00	0.437	0.422	-0.32
48	270	2.64	47	0.00	0.433	0.417	-0.33
49	272	2.63	48	0.00	0.429	0.412	-0.33
50	275	2.62	49	0.00	0.425	0.407	-0.34
51	277	2.61	50	0.00	0.420	0.402	-0.34
52	279	2.60	51	0.00	0.416	0.397	-0.35
53	282	2.59	52	0.00	0.412	0.393	-0.35
54	284	2.58	53	0.00	0.407	0.388	-0.36
55	283	2.57	54	0.00	0.403	0.383	-0.36
56	286	2.56	55	0.00	0.398	0.378	-0.36
57	288	2.55	56	0.00	0.393	0.373	-0.36
58	290	2.54	57	0.00	0.389	0.368	-0.36
59	293	2.53	58	0.00	0.384	0.363	-0.36
60	295	2.52	59	0.00	0.379	0.358	-0.35
61	298	2.51	60	0.00	0.374	0.353	-0.35
62	300	2.50	61	0.00	0.368	0.349	-0.35
63	303	2.49	62	0.00	0.363	0.344	-0.35
64	301	2.47	63	0.00	0.358	0.339	-0.34
65	304	2.46	64	0.00	0.352	0.334	-0.34
66	306	2.45	65	0.00	0.347	0.329	-0.33
67	309	2.44	66	0.00	0.341	0.324	-0.32
68	311	2.43	67	0.00	0.336	0.319	-0.32
69	478	2.42	68	0.00	0.330	0.315	-0.31
70	480	2.43	69	0.00	0.324	0.310	-0.31
71	479	2.44	70	0.00	0.319	0.305	-0.30
72	481	2.44	71	0.00	0.313	0.300	-0.29
73	482	2.44	72	0.00	0.307	0.296	-0.29
74	481	2.43	73	0.00	0.302	0.291	-0.28
75	482	2.42	74	0.00	0.296	0.286	-0.28
76	480	2.40	75	0.00	0.290	0.282	-0.27
77	482	2.38	76	0.00	0.285	0.277	-0.26
78	479	2.35	77	0.00	0.280	0.273	-0.26
79	482	2.32	78	0.00	0.274	0.269	-0.25
80	483	2.28	79	0.00	0.269	0.264	-0.25
81	481	2.25	80	0.00	0.264	0.260	-0.24
82	483	2.21	81	0.00	0.259	0.256	-0.24
83	485	2.17	82	0.00	0.255	0.252	-0.23
84	483	2.13	83	0.00	0.250	0.248	-0.22
85	485	2.09	84	0.00	0.246	0.244	-0.21
86	487	2.05	85	0.00	0.241	0.240	-0.20
87	490	2.00	86	0.00	0.237	0.236	-0.19
88	487	1.95	87	0.00	0.233	0.232	-0.18
89	489	1.91	88	0.00	0.229	0.229	-0.18
90	492	1.86	89	0.00	0.226	0.225	-0.17
91	489	1.80	90	0.00	0.222	0.221	-0.16
92	492	1.75	91	0.00	0.218	0.218	-0.15
93	495	1.69	92	0.00	0.215	0.215	-0.15
94	497	1.63	93	0.00	0.212	0.211	-0.14
95	494	1.57	94	0.00	0.208	0.208	-0.15
96	497	1.51	95	0.00	0.205	0.205	-0.14
97	506	1.45	96	0.00	0.202	0.202	-0.15
98	508	1.39	97	0.00	0.199	0.199	-0.13
99	517	1.33	98	0.00	0.196	0.196	-0.16
100	521	1.28	99	0.00	0.193	0.193	-0.14
101	529	1.23	100	0.00	0.190	0.190	-0.15
102	532	1.24	101	0.00	0.188	0.187	-0.12

This table shows the end results of the run for the “target” SRD of the pile. “SRD” is “soil resistance to driving,” and in TAMWAVE for cohesionless soils, SRD and the ultimate capacity are the same. That’s not the case with cohesive soils, as we will see. In any case TAMWAVE always does a “bearing graph” analysis, which proportionally varies the SRD and obtains different results for the blow count, maximum tensile and compressive stresses. The bearing graph method isn’t perfect but it’s probably the best way we have to account for varying site conditions and to make judgments about the effect of those on our hammer selection.

The adoption of “Smith-type” damping was originally done for comparison purposes but for bearing graph analysis has one important advantages: it varies the soil radiation damping with the SRD, which is more realistic than just assuming fixed damping.

The table above only appears if the target SRD is actually achieved during bearing graph analysis. If it doesn’t come up, the bearing graph analysis could not achieve net pile penetration at the target SRD, which means you need to revisit your hammer selection.

Here we see the second graphical output: the force-time history at the target SRD. There are actually two histories: the actual pile head force (blue) and the pile head velocity multiplied by the impedance (red.) For semi-infinite piles, the two should be the same; they will differ for actual finite piles, as is easily seen. Although a “semi-infinite pile” may seem a very theoretical concept, the relationship of the two plots is very important in the field application of pile dynamics.

Summary of Results and Bearing Graph Data
Soil Resistance, kips	Permanent Set of Pile Toe, inches	Blows per Foot of Penetration	Maximum Compressive Stress, ksi	Element of Maximum Compressive Stress	Maximum Tensile Stress, ksi	Element of Maximum Tensile Stress	Number of Iterations
114.5	1.707	7.0	2.61	30	0.67	43	1590
229.1	0.754	15.9	2.64	29	0.20	25	1124
343.6	0.355	33.8	2.67	28	0.00	102	719
458.1	0.111	108.2	2.71	32	0.00	102	567
572.7	0.000	0.0	2.84	34	0.00	102	549

The final results are shown here. In this case, at the target SRD, no permanent set of the pile is recorded. It will be necessary to vary the size of the hammer, being mindful of the stresses (whose allowable values are described here.)

At this point the analysis of this pile is complete. The program gives you the choice of simply trying another hammer or starting over. The latter is what we will do next with a sample case for cohesive soils.

TAMWAVE

TAMWAVE 5: Wave Equation Analysis, Overview and Initial Entry

Posted on 5 January 2018 by Don Warrington

With the static analysis complete, we turn to the wave equation analysis. TAMWAVE (as with the previous version) was based indirectly on the TTI wave equation program. Although the numerical method was not changed, many other aspects of the program were, and so we need to consider these.

Shaft and Toe Resistance

Most wave equation programs in commercial use still use the Smith model for shaft and toe resistance during impact. Referencing specifically their use in inverse methods, Randolph (2003) makes the following comment:

Dynamic pile tests are arguably the most cost-effective of all pile-testing methods, although they rely on relatively sophisticated numerical modelling for back-analysis. Theoretical advances in modelling the dynamic pile-soil interaction have been available since the mid-1980s, but have been slow to be implemented by commercial codes, most of which still use the empirical parameters of the Smith (1960) model. In order to allow an appropriate level of confidence in the interpretation of dynamic pile tests, and estimation of the static response, it is high time that appropriate scientific models were used for pile-soil interaction, including explicit modelling of the soil plug for open-ended piles.

And that was in 2003…and the use of the Smith model in inverse methods was proceeded by its use in forward methods such as this one. The model he is referring to from the mid-1980’s is, of course, the Randolph and Simons (1986) model, which was used in the ZWAVE program in the late 1980’s. The details of this model were discussed in Warrington (1997).

The Randolph and Simons model is the one which is being used for the wave equation portion of this routine, as the static component was used for the ALP static axial pile analysis. In converting the code from the Smith model to this one, there are some things that need to be understood. We have discussed some of these earlier but others are as follows:

Randolph and Simons (1985) used a visco-elastic-plastic model for both shaft and toe, the major difference being the location of the plastic slider for the shaft resistance (as is evident in the ZWAVE poster.) Some contemporary “experimental” codes (such as Salgado, Loukidis, Abou-Jaoude and Zhang (2015)) add a series of springs and masses to replicate the soil mass that surrounds the piles. While these doubtless enhance the performance of the models, we stuck with the simple visco-elastic-plastic model in TAMWAVE because these are better replicated in true 3D continuum models like STADYN. 1D code is good because of its simplicity, especially with an online routine like TAMWAVE.
The 1′ segment/element lengths are carried over to the wave equation. This is shorter than is customarily used even in commercial work but it saves interpolation of the properties along the shaft.
The “Smith-type” damping constants are simply the damping of the element computed divided by its ultimate/plastic resistance. Unlike the Smith model, however, the damping force does not vary with the instantaneous static resistance, but is simply the velocity multiplied by the damping constant and the ultimate resistance of that element, be it shaft or toe. Thus different Smith type constants should be expected from the model being used. Additionally, with the shaft resistance, the resistance of a shaft segment is limited to its ultimate static resistance. This means that all additional damping forces must take place during elastic shearing of the soil surface. Implicit in the Randolph and Simons model is that, once plasticity is achieved, the soil closest to the pile is effectively decoupled from the soil mass, and thus the pile movement can no longer radiate additional energy into the soil. The result of this is that, as seen here, the Smith-type damping constants are much higher than one would normally assign. Corte and Lepert (1985), in a direct comparison of the two models, note that the two give nearly the same result if the original Smith damping constants are multiplied by 7.5 for the new model. Dividing the new result by this brings the damping constants much closer, especially in the lower reaches of the pile where most of the shaft resistance is found, although the ratio of 7.5 should be regarded as study-specific. Bringing some rationality to the issue of damping constants would go a long way to improve the results of pile dynamics, forward and inverse, since variations of these have a significant impact on the results.
We mentioned earlier that the toe quakes that resulted seemed high for this size of pile. This may be due to the fact that “significant residual pressures are locked in at the pile base during installation (equilibriated by negative shear stresses along the pile shaft, as if the pile were loaded in tension.) This will lead to a stiffer overall pile response in compression, and significantly higher end-bearing stresses mobilised at small displacements.” (Randolph, 2003) He goes on to state that “(f)or driven closed ended-piles the residual stress will be lower, but may still be as high as 75% of the base capacity…” There are two ways to deal with this. The first is to run the ALP program first and preload the base and shaft before using the resulting prestressed deflections to run the wave equation analysis. This would be in effect a residual stress analysis (RSA,) which has been used in this field for many years. The second is to use a “quick and dirty” method, i.e., to reduce the toe quake and thus simulate the higher toe stiffness and lower quake. The latter was adopted in TAMWAVE, although one motivation from switching from P4XC3 to ALP was to make an RSA easier. This is a possible point of future modification of the code.
A change not related to the pile-soil interaction is the elimination of slack computation, as the pile is uniform and continuous (the hammer-cap and cap-pile interface is obviously inextensible.

Initial Wave Equation Input

For our example the initial input of the wave equation is shown below.

Most of the data required has been carried over from the static analysis. The hammer database was added in 2010; however, it was reordered in ascending rated striking energy order and a hammer was suggested using the “initial guess” criterion in the Soils and Foundations Handbook, which essentially suggests to set the initial hammer energy in ft-lbs at 8% of the ultimate capacity in pounds. This is a “rule of thumb” designed to help students who, faced with a wave equation program for the first time, will have some idea of where to start, although there is no guarantee that the hammer will be either too large or small. Since the energies are sorted, the user can move up or down the list to try another hammer.

The cushion material properties of the hammer, and the coefficient of restitution used to model cushion plasticity, are discussed (with sample properties) in the WEAP87 documentation. No attempt was done to either convert coefficients of restitution to viscous damping or alter the rebound curve as was done in ZWAVE. Pile cushion thickness is only input for concrete piles; the input is not shown for others.

References

In addition to those already cited, the following is included:

Corte, J.-F., and Lepert, P. (1986) “Lateral resistance during driving and dynamic pile testing.” Proceedings of the Third International Conference on Numerical Methods in Offshore Piling, Nantes, France, 21-22 May. Paris: Éditions Technip, pp. 19-34.

TAMWAVE

TAMWAVE 4: Shaft Resistance Profile, ALP and CLM2

Posted on 4 January 2018 by Don Warrington

With the basic parameters established, we can turn to the static analysis of the pile, both axial and lateral.

Shaft Resistance Profile

Shaft Segment Properties
Depth at Centre of Layer, feet	Soil Shear Modulus, ksf	Beta	Quake,inches	Maximum Load Transfer, ksf	Spring Constant for Wall Shear, ksf/in	Smith-Type Damping Constant, sec/ft	Maximum Load Transfer During Driving (SRD), ksf
0.50	48.4	0.163	0.0022	0.009	4.03	45.394	0.009
1.50	83.9	0.163	0.0038	0.027	6.99	19.911	0.027
2.50	108.3	0.163	0.0050	0.045	9.02	13.572	0.045
3.50	128.1	0.163	0.0059	0.063	10.68	10.543	0.063
4.50	145.3	0.163	0.0067	0.081	12.11	8.730	0.081
5.50	160.6	0.164	0.0074	0.098	13.38	7.509	0.098
6.50	174.6	0.164	0.0080	0.116	14.55	6.623	0.116
7.50	187.6	0.164	0.0086	0.134	15.63	5.948	0.134
8.50	199.7	0.164	0.0091	0.152	16.64	5.414	0.152
9.50	211.1	0.164	0.0097	0.170	17.59	4.980	0.170
10.50	222.0	0.164	0.0102	0.188	18.50	4.618	0.188
11.50	232.3	0.164	0.0106	0.206	19.36	4.313	0.206
12.50	242.2	0.164	0.0111	0.224	20.18	4.050	0.224
13.50	251.7	0.164	0.0115	0.242	20.98	3.822	0.242
14.50	260.9	0.164	0.0120	0.260	21.74	3.621	0.260
15.50	269.8	0.164	0.0124	0.278	22.48	3.444	0.278
16.50	278.4	0.164	0.0128	0.296	23.20	3.285	0.296
17.50	286.7	0.164	0.0132	0.314	23.89	3.142	0.314
18.50	294.8	0.164	0.0135	0.332	24.57	3.013	0.332
19.50	302.7	0.164	0.0139	0.351	25.22	2.895	0.351
20.50	310.4	0.164	0.0143	0.369	25.86	2.787	0.369
21.50	317.9	0.164	0.0146	0.387	26.49	2.688	0.387
22.50	325.2	0.164	0.0149	0.405	27.10	2.597	0.405
23.50	332.4	0.165	0.0153	0.423	27.70	2.512	0.423
24.50	339.4	0.165	0.0156	0.441	28.29	2.434	0.441
25.50	346.3	0.165	0.0159	0.460	28.86	2.361	0.460
26.50	353.1	0.165	0.0162	0.478	29.42	2.292	0.478
27.50	359.7	0.165	0.0166	0.496	29.98	2.228	0.496
28.50	366.3	0.165	0.0169	0.515	30.52	2.168	0.515
29.50	372.7	0.165	0.0172	0.533	31.06	2.112	0.533
30.50	379.0	0.165	0.0175	0.552	31.58	2.058	0.552
31.50	385.2	0.165	0.0178	0.570	32.10	2.007	0.570
32.50	391.3	0.166	0.0181	0.589	32.61	1.960	0.589
33.50	397.4	0.166	0.0183	0.607	33.11	1.914	0.607
34.50	403.3	0.166	0.0186	0.626	33.61	1.871	0.626
35.50	409.2	0.166	0.0189	0.645	34.10	1.830	0.645
36.50	415.0	0.166	0.0192	0.664	34.58	1.790	0.664
37.50	420.7	0.166	0.0195	0.683	35.06	1.753	0.683
38.50	426.4	0.166	0.0197	0.702	35.53	1.717	0.702
39.50	432.0	0.167	0.0200	0.721	36.00	1.682	0.721
40.50	437.5	0.167	0.0203	0.740	36.46	1.649	0.740
41.50	443.0	0.167	0.0206	0.759	36.92	1.618	0.759
42.50	448.4	0.167	0.0208	0.778	37.37	1.587	0.778
43.50	453.8	0.168	0.0211	0.798	37.82	1.558	0.798
44.50	459.1	0.168	0.0214	0.817	38.26	1.530	0.817
45.50	464.4	0.168	0.0216	0.837	38.70	1.502	0.837
46.50	469.6	0.168	0.0219	0.856	39.13	1.476	0.856
47.50	474.8	0.169	0.0221	0.876	39.56	1.450	0.876
48.50	479.9	0.169	0.0224	0.896	39.99	1.426	0.896
49.50	485.0	0.169	0.0227	0.916	40.42	1.402	0.916
50.50	489.1	0.169	0.0229	0.933	40.76	1.382	0.933
51.50	492.3	0.170	0.0231	0.947	41.03	1.367	0.947
52.50	495.5	0.170	0.0233	0.960	41.30	1.352	0.960
53.50	498.7	0.171	0.0234	0.974	41.56	1.337	0.974
54.50	501.9	0.171	0.0236	0.988	41.83	1.323	0.988
55.50	505.1	0.171	0.0238	1.002	42.09	1.308	1.002
56.50	508.3	0.172	0.0240	1.016	42.36	1.294	1.016
57.50	511.5	0.172	0.0242	1.031	42.63	1.280	1.031
58.50	514.7	0.173	0.0244	1.045	42.89	1.266	1.045
59.50	517.9	0.173	0.0246	1.060	43.16	1.252	1.060
60.50	521.1	0.174	0.0248	1.075	43.42	1.238	1.075
61.50	524.3	0.174	0.0250	1.091	43.69	1.224	1.091
62.50	527.5	0.175	0.0252	1.106	43.96	1.211	1.106
63.50	530.7	0.176	0.0254	1.122	44.22	1.197	1.122
64.50	533.9	0.176	0.0256	1.139	44.49	1.184	1.139
65.50	537.1	0.177	0.0258	1.155	44.76	1.170	1.155
66.50	540.4	0.178	0.0260	1.172	45.03	1.157	1.172
67.50	543.6	0.178	0.0262	1.189	45.30	1.144	1.189
68.50	546.9	0.179	0.0265	1.207	45.57	1.130	1.207
69.50	550.2	0.180	0.0267	1.224	45.85	1.117	1.224
70.50	553.5	0.181	0.0269	1.243	46.12	1.104	1.243
71.50	556.8	0.182	0.0272	1.262	46.40	1.091	1.262
72.50	560.1	0.183	0.0274	1.281	46.68	1.078	1.281
73.50	563.5	0.184	0.0277	1.300	46.96	1.065	1.300
74.50	566.9	0.185	0.0280	1.321	47.24	1.051	1.321
75.50	570.3	0.186	0.0282	1.341	47.52	1.038	1.341
76.50	573.7	0.187	0.0285	1.363	47.81	1.025	1.363
77.50	577.2	0.188	0.0288	1.385	48.10	1.012	1.385
78.50	580.7	0.190	0.0291	1.407	48.39	0.999	1.407
79.50	584.3	0.191	0.0294	1.431	48.69	0.985	1.431
80.50	587.9	0.193	0.0297	1.455	48.99	0.972	1.455
81.50	591.5	0.194	0.0300	1.479	49.29	0.959	1.479
82.50	595.2	0.196	0.0303	1.505	49.60	0.945	1.505
83.50	598.9	0.197	0.0307	1.532	49.91	0.932	1.532
84.50	602.7	0.199	0.0310	1.559	50.22	0.919	1.559
85.50	606.5	0.201	0.0314	1.587	50.54	0.905	1.587
86.50	610.4	0.203	0.0318	1.617	50.87	0.891	1.617
87.50	614.4	0.205	0.0322	1.647	51.20	0.878	1.647
88.50	618.4	0.207	0.0326	1.678	51.53	0.864	1.678
89.50	622.5	0.210	0.0330	1.711	51.87	0.850	1.711
90.50	626.7	0.212	0.0334	1.745	52.22	0.837	1.745
91.50	630.9	0.215	0.0339	1.781	52.58	0.823	1.781
92.50	635.2	0.217	0.0343	1.817	52.94	0.809	1.817
93.50	639.7	0.220	0.0348	1.856	53.30	0.795	1.856
94.50	644.2	0.223	0.0353	1.896	53.68	0.781	1.896
95.50	648.8	0.226	0.0358	1.937	54.07	0.767	1.937
96.50	653.5	0.229	0.0364	1.981	54.46	0.753	1.981
97.50	658.3	0.233	0.0369	2.026	54.86	0.739	2.026
98.50	663.3	0.236	0.0375	2.073	55.27	0.725	2.073
99.50	668.3	0.240	0.0381	2.122	55.69	0.710	2.122

The results should be self explanatory; however, some observations are in order.

A 1′ increment was used for the analysis. This will be carried over to both the static and dynamic axial analyses. For this routine it’s probably overkill, but for a real system with multiple soil layers this eliminates a great deal of interpolation and adjustment.
Both the shear modulus and the maximum shear stress on the shaft surface vary with effective stress. This tends to homogenise the quake to some degree. The increase of shear modulus with depth also increases the shaft element stiffness as well.
Beta values are about 50% higher at the pile toe than at the pile head. This is mostly due to the depth effect of the $K$ value computed by the method used.
The resulting quakes are lower than the “traditional values.” This varies from run to run.
The Smith-type damping constants are considerably higher than is usually expected. This will be discussed with the wave equation analysis itself.
There is no difference between ultimate capacity and SRD with this run because of the cohesionless soils. This will change with cohesive ones.

ALP Program

The original routine used the PX4C3 routine to construct the axial load-deflection curve. For this routine it was replaced by the ALP program, which is described in Verruijt. The Turbo Pascal code in the text was converted to php and modified for the online application. The ALP99 program, which allows for layered soils, has been used in a classroom setting, is a good program but has three serious weaknesses:

There is no guidance on what values of quake to use for either shaft or toe, and for beginners this is very confusing.
The guidance on entering shaft resistance properties is primitive, to say the least.
The program simply crashes if a resistance in excess of the ultimate resistance is entered, even though the latter is easily computed.

This online version of ALP addresses all of these by limiting the highest resistance during the “load test” and furnishing quake and resistance values all along the shaft and toe.

The basic parameters of ALP returned by TAMWAVE are shown below.

Data for Axial Load Analysis using ALP Method
Length of the pile, in.	1,200.0
Axial stiffness EA. lbs.	720,000,000
Circumference, in.	48.000
Point resistance, lbs.	202,673
Quake of the point, in.	0.879
Number of pile elements	100
Number of loading steps	20
Maximum pile load, lbs.	572,676.9
Load Increment, lbs.	57,267.7
Failure Load, lbs.	572,676.9

Some of these are repetitious from earlier data output. The results of the actual “load test” are shown below.

Results for Loading and Unloading Test
Load Step	Force at Pile Head, kips	Pile Head Deflection, in.	Number of Plastic Shaft Springs
0	0.0	0.000	0
1	57.3	0.033	22
2	114.5	0.082	39
3	171.8	0.144	52
4	229.1	0.216	64
5	286.3	0.300	74
6	343.6	0.395	85
7	400.9	0.601	100
8	343.6	0.571	10
9	286.3	0.534	22
10	229.1	0.489	31
11	171.8	0.437	39
12	114.5	0.378	45
13	57.3	0.314	52
14	0.0	0.244	58

The program ceases to load the pile and begins to unload when all of the shaft friction is mobilised or the ultimate load is achieved, whichever comes first. This is intended to prevent the routine from going unstable with the applied load too near the maximum capacity of the pile, thus violating static equilibrium.

ALP solves the system by constructing a tridiagonal matrix and then solving the non-linear problem. In some cases it will achieve a result before coming to actual convergence according to the convergence criterion. In such cases ALP will report that no convergence was achieved.

One new feature with the current version of TAMWAVE is the inclusion of two basic graphs of the results. This is one of them. Contrary to American practice, the deflection (y) axis is upward even though the actual deflection is downward. For serious plotting purposes it is probably best for the student to copy and paste the results into a spreadsheet or other plotting program and then make the results look more presentable.

CLM 2 Routine for Lateral Loads

To analyse lateral pile loading, the CLM2 Method is employed. Details on this method can be found with the CLM 2 spreadsheet here. Some notes about this are as follows:

The analyser is for single piles only, no group or bent analysis.
The following cases can be considered:
- Free (Pinned) Head, Lateral Force Only
- Free Head, Moment Only
- Free Head, Combined Force and Moment
- Fixed Head, Lateral Force Only
Any lateral load or pile head moment is entered when the soil properties are confirmed. If zero load or moment is entered, the results are expanded or truncated accordingly.

For this example the results of the CLM 2 analysis are here.

Data for Lateral Load Analysis using CLM2 Method
Nominal Soil Unit Weight, lb/in³	0.06944
Pile Moment of Inertia, in⁴	1,728.00
Pile Section Modulus, in³	288.00
Pile Solid Circle Moment of Inertia, in⁴	1,017.88
Moment of Inertia Ratio R_i	1.698
Pile Moment of Inertia Ratio Product, ksi	8,488.3
Pile-Soil Interaction Variable	97,803
Pile L/D Ratio	100.0
Characteristic Load, lbs.	2,745,232.8
Characteristic Moment, in-lbs.	196,821,533.6
Pile Head Fixity	Free
Pile Head Lateral Load, lbs.	5,000.0
P_t/P_c	0.00182
Y_t/D	0.00800
Pile Head Deflection due to Load, inches	0.096
Maximum Moment Due to Pile Head Lateral Load, in-lbs	136,112.3
Maximum Bending Stress Due to Pile Head Lateral Load, in-lbs	472.6

The results are explained in the CLM 2 documentation. The bending stresses are not really meaningful in concrete piles, as flexure is generally transmitted through the reinforcement. Parametric studies must be run manually, i.e., one load at a time.

CLM 2 is a quick way to obtain estimates of lateral loads, shears and moments for groundline piles and simple soil profiles, and both of these are present in TAMWAVE. Since all of the soil input is already done, this source of error is eliminated.

Once these results are complete, the user can proceed to run a wave equation analysis.

TAMWAVE

TAMWAVE 3: Basic Results of Pile Capacity Analysis

Posted on 4 January 20188 April 2018 by Don Warrington

With the soil properties and lateral loads finalised, we can proceed to look at the program’s static results. These are shown below. We will concentrate on cohesionless soils in this post; a sample case with cohesive results will come later.

Pile Data
Pile Designation	12 In. Square
Pile Material	Concrete
Penetration of Pile into the Soil, ft.	100
Basic “diameter” or size of the pile, ft.	1
Cross-sectional Area of the Pile, ft²	1.000
Pile Toe Area, ft²	1.000
Perimeter of the Pile, ft.	4.000
Soil Data
Type of Soil	SW
Specific Gravity of Solids	2.65
Void Ratio	0.51
Dry Unit Weight, pcf	109.5
Saturated Unit Weight, pcf	130.5
Soil Internal Friction Angle phi, degrees	32
Cohesion c, psf	0
SPT N₆₀, blows/foot	20
CPT q_c, psf	211,600
Distance of Water Table from Soil Surface, ft.	50
Penetration of Pile into Water Table, ft.	50
Active Earth Pressure Coefficient (K_min)	0.453
Frictional Angle Between Pile and Soil delta, degrees	27.9
Minimum Value for Beta	0.240
Pile Toe Results
Effective Stress at Pile Toe, ksf	8.880
N_q	22.8
Relative Density at Pile Toe, Percent	40
SPT (N₁)₆₀ at pile toe, blows/foot	10
Unit Toe Resistance q_p, ksf	202.7
Shear Modulus at Pile Toe, ksf	675.7
Toe Spring Constant Depth Factor	1.410
Toe Spring Constant, kips/ft	2,767.9
Pile Toe Quake, in.	0.879
Poisson’s Ratio at Pile Toe	0.310
Toe Damping, kips-sec/ft	13.2
Toe Smith-Type Damping Constant, sec/ft	0.065
Total Static Toe Resistance Q_p, kips	202.67
Pile Toe Plugged?	No
Final Results
Total Shaft Friction Q_s, kips	370.00
Ultimate Axial Capacity of Pile, kips	572.68
Pile Setup Factor	1.0
Total Pile Soil Resistance to Driving (SRD), kips	572.68

Pile Data

The pile data is pretty straightforward. Reproducing it here is an opportunity for you to confirm you’ve selected the correct pile.

Soil Data

Soil data affords the same opportunity for verification; however, it also shows the way the soil data is interpreted to generate the necessary parameters for shaft and toe resistance to load, both static and dynamic.

The first thing that is shown is assumed specific gravity and void ratio. TAMWAVE assumes cohesionless soils have a particle specific gravity of 2.65 and for cohesive soils 2.7. The void ratio is then computed using basic soil mechanics formulae. To do this it is necessary to know the unit weight. The typical properties tables show this in two ways. For cohesionless soils, the “moist” unit weight is shown, and for cohesive soils the saturated unit weight is shown. In both cases this is reduced to dry and saturated unit weights by assuming that S=50% for the cohesionless soils and S=100% for the cohesive ones. Thus, for cohesionless soils neither value will be the same as given in the typical properties.

The internal friction angle, cohesion and $N_{60}$ values are taken from the typical properties as modified (or not) by the user. The equivalent $q_c$ is also reported here, based on the Robertson and Campanella research as reported by Fellenius. As noted earlier, neither the $N_{60}$ values nor the $q_c$ values are actually used in the analysis.

Finally we get to the data necessary to compute the shaft friction. The methods used in TAMWAVE for ultimate shaft resistance are as follows:

Cohesionless soils: Randolph, Dolwin and Beck (1994), which are described here
Cohesive soils: Kolk and van der Velde (1996), which are described here

For cohesionless soils, it is necessary to compute the minimum/active earth pressure coefficient, which of course is strictly a function of $\phi$ . Discussion of $K_{act}$ brings us to the issue of computing $\beta$ . In this post $\beta$ was initially computed using the following formula

$\beta = K tan \phi$

However, as pointed out in the same place, both retaining wall practice and empirical pile capacity formulae show that the friction angle between the wall/pile shaft and the soil is not equal to the internal friction angle of the soil, and so this formula should really be written as

$\beta = K tan \delta$

This actually has a theoretical basis, and in fact is one of the knottiest problems in theoretical soil mechanics. We can consider this by considering the failure along the pile surface as a “direct shear” type of failure, where failure is induced along a predetermined surface. For the case where the principal stresses are normal and tangential to the surface (which is generally the case with driven piles) the failure surface predicted by Mohr’s circle and Mohr-Coulomb theory is not the same as the “predetermined” surface. The most acrimonious manifestation of this problem was with the shear failure of cellular cofferdams, which led to the dispute between Karl Terzaghi and Dmitri Krynine.

Although various studies have been made to determine friction on an empirical basis, probably the simplest solution, suggested by Šuklje (1969), is to compute the apparent friction angle by the formula

$\delta = tan^{-1} (sin \phi)$

Using this result and the active earth pressure coefficient, the minimum value for $\beta$ is readily computed.

Pile Toe Results

Now we get to the application of these parameters. The decision to not use equivalent CPT values has two immediate results. The first is that the unit toe resistance is most easily computed (for cohesionless soils) by the equation

$q_t = N_q \sigma'_{vo}$

Use of bearing capacity factors for toe resistance is both well embedded in literature and practice and well criticised in the same place. Additionally it is necessitated by the fact that the shaft friction is dependent upon $N_q$ , as discussed here.

So what value of $N_q$ should we adopt? As is all too common in geotechnical engineering, there has been a proliferation of values for this parameter. We experimented with several, including that of Vesic (1977). In addition to the usual theortical vs. empirical (and all the variations in between) divide, another factor is whether provision for soil elasticity is taken into consideration. Although methods such as Vesic (1977) do this, their main weakness is their tendency to be stiff, thus returning unrealistically high values of $N_q$ . Taking into account both theoretical methods and empirical ones such as Dennis and Olson, the first method tried for TAMWAVE was from Verruijt), namely:

$N_{\sigma} = K_p e^{\pi tan \phi}$

Note that we’re not at $N_q$ quite yet. For reasons explained by Vesic (1977), the pile toe unit resistance should be a function of $\frac {I_1}{3}$ . (An explanation of this quantity can be found here.) Thus,

$N_q = K_p e^{\pi tan \phi} \frac {I_1}{3 \sigma'_{vo}}$

If we use Jaky’s Equation for normally consolidated soils for the pile toe condition (we will definitely change this for the shaft,)

$\frac {I_1}{3 \sigma'_{vo}} = 1 - \frac {2}{3} sin \phi$

and so

$q_t = K_p e^{\pi tan \phi} \left( 1 - \frac {2}{3} sin \phi \right) \sigma'_{vo}$

This method yields conservative values of $N_q$ . It only takes into consideration the plasticity of the soil, and is additionally subject to the criticism of the entire concept of “bearing capacity” at the pile toe. Given this, a different (and more empirical) approach based on Randolph, Dolwin and Beck (1994) was chosen, which estimates the bearing capacity coefficient as follows:

$N_q=\frac{0.065S+25}{\left( \frac{\sigma'_{vo}}{p_{atm}} \right)^{0.25}}$

$S$ is the coefficient from the Hardin and Black (1968) method of estimating the shear modulus of the soil (see below.) Although this coefficient varies with silt content, for TAMWAVE we will assume $S = 315$ , thus this reduces to

$N_q=\frac{45.475}{\left( \frac{\sigma'_{vo}}{p_{atm}} \right)^{0.25}}$

If static capacity were our sole interest, we would be done with toe. But what about its response to movement? For both toe and shaft resistance, in both static and dynamic cases, we intend to use an elastic-purely plastic model. Assuming no preloading of the system, there are only two parameters we need to know: the ultimate/purely plastic resistance of the soil, and the deflection at which we reach that resistance. The spring constant can be computed by dividing the ultimate resistance by that deflection, or conversely we can determine that deflection by dividing the resistance by a known spring constant. It is the latter operation we will use in TAMWAVE, which leaves us to determine the spring constant of the toe and eventually along the shaft.

We will have occasion to return to this topic, but to determine spring constants we will use the model of Randolph and Simons (1985). For the toe this in turn is dependent upon Lysmer’s Analogue; both of these are discussed in detail in Warrington (1997). They are dependent upon determining values for the soil shear modulus $G$ . (They are also dependent upon the dry unit weight $\gamma$ and Poisson’s Ratio $\nu$ , but both of these parameters are known from basic soil properties and, indirectly, through Jaky’s Equation.) That in turn brings us to another “sticky wicket,” namely determining the shear (or for that matter the elastic) modulus of the soil. An interesting discussion of this topic can be found in Salgado, Loukidis, Abou-Jaoude and Zhang (2015). Assuming a hyperbolic type of soil deformation, there are two basic extremes to this parameter:

The small-strain (or tangent) value, the highest possible value.
The large-strain (or secant) value, the lowest possible value.

Based on their review of the literature, they conclude that the value for (2) can be 10-50% of (1). Although this problem is frought with uncertainties, it is hard to avoid the conclusion that this is a substantial spread and, for our purposes, raises as many questions as it answers. The “solution” to this problem is found in this post, where one attempts to define a ratio between (1) and (2) based on some consideration of anticipated deflections under load for a given application.

Based on some experimentation with the code and earlier considerations, we decided to use a ratio between the two of 0.15, i.e., the secant modulus used in elastic-purely plastic models is 15% of the tangent modulus from the hyperbolic model. We should emphasise that this is not “set in stone” but subject to variation. One of the advantages of a project such as TAMWAVE is the ability to alter parameters and see the results without affecting results on actual projects.

“Fixing” this ratio allows us to determine the shear modulus based on the tangent or small-strain value, and this can be computed by the method proposed in Hardin and Black (1968). There is little difference between the correlation for cohesionless and cohesive soils. There are many ways of expressing this; the one we used (for values of $G$ in psf) is as follows:

$G=S p_{atm}\frac{\left(3-e\right)^{2}}{1+e}\sqrt{\frac{\sigma'_{vo}}{p_{atm}}\frac{1+2K}{3}}$

As before, for TAMWAVE (and many applications) $S = 315$ . The same formula is used for the shaft friction, the main difference is that the lateral earth pressure coefficient is different thus the lateral/confining stresses are different. At the toe we use the result from Jaky’s Equation, which was explained in Verruijt’s method above.

Once this is computed, the pile toe stiffness is computed. The stiffness is increased by multiplying it by a depth factor (Salgado, Loukidis, Abou-Jaoude and Zhang (2015)

$D_f = 1+\left( 0.27- 0.12 ln \nu \right)\left\{ 1-e^{\left[ -0.83\left( \frac{D}{B} \right)^{0.83} \right]} \right\}$

Even at this, when compared to “conventional” toe quakes in dynamic analysis, the toe quake shown above seems rather large. We will leave this as it is for the static analysis and will return to this topic with the dynamic analysis.

Since we are computing stiffnesses for shaft and toe here, we will also do the same for damping. Traditionally wave equation programs have used “Smith damping,” but as we will see this will be modified for the wave equation analysis. To start let us redefine the “Smith type damping constant” as

$j = \frac {\mu}{R_u}$

In this case $\mu$ is the damping constant for the toe or shaft element in question, computed using the formulae given in Warrington (1997). $R_u$ is the ultimate resistance of the toe or shaft element in question. The toe damping constant that results in this case is somewhat lower than “standard” values; this will be discussed later.

Final Results

The final results are at the end of the table. The shaft friction computation will be discussed in the next post. The cohesive calculations have a provision for pile set-up using cavity expansion theory and this will be discussed later.

References

In addition to works already cited in this and the STADYN study, the following should be noted:

Hardin, B.O., and Black, W.L. (1968). “Vibration modulus of normally consolidated clay.” J. Soil Mech. Found. Div. 94, No. 2, 353-370.
Salgado, R., Loukidis, D., Abou-Jaoude, G., and Zhang, Y. (2015) “The role of soil stiffness non-linearity in 1D pile driving simulations.” Geotechnique 65, No. 3, 169-187. http://dx.doi.org/10.1680/geot.13.P.124
Vesic, A.S. (1977) Design of Pile Foundations. NCHRP Synthesis 42. Washington, DC: Transportation Research Board.