-
Notifications
You must be signed in to change notification settings - Fork 21
/
soilproperties.py
509 lines (418 loc) · 31.7 KB
/
soilproperties.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
#!/usr/bin/env python
# -*- coding: utf-8 -*-
__author__ = 'Bruno Stuyts'
# Native Python packages
# 3rd party packages
import numpy as np
# Project imports
from groundhog.general.validation import Validator
MODULUSREDUCTION_PLASTICITY_ISHIBASHI = {
'strain': {'type': 'float', 'min_value': 0.0, 'max_value': 10.0},
'pi': {'type': 'float', 'min_value': 0.0, 'max_value': 200.0},
'sigma_m_eff': {'type': 'float', 'min_value': 0.0, 'max_value': 400.0},
'multiplier_1': {'type': 'float', 'min_value': None, 'max_value': None},
'exponent_1': {'type': 'float', 'min_value': None, 'max_value': None},
'multiplier_2': {'type': 'float', 'min_value': None, 'max_value': None},
'exponent_2': {'type': 'float', 'min_value': None, 'max_value': None},
'multiplier_3': {'type': 'float', 'min_value': None, 'max_value': None},
'exponent_3': {'type': 'float', 'min_value': None, 'max_value': None},
}
MODULUSREDUCTION_PLASTICITY_ISHIBASHI_ERRORRETURN = {
'G/Gmax [-]': np.nan,
'K [-]': np.nan,
'm [-]': np.nan,
'n [-]': np.nan,
'dampingratio [pct]': np.nan,
}
@Validator(MODULUSREDUCTION_PLASTICITY_ISHIBASHI, MODULUSREDUCTION_PLASTICITY_ISHIBASHI_ERRORRETURN)
def modulusreduction_plasticity_ishibashi(
strain, pi, sigma_m_eff,
multiplier_1=0.000102, exponent_1=0.492, multiplier_2=0.000556, exponent_2=0.4, multiplier_3=-0.0145,
exponent_3=1.3, **kwargs):
"""
Calculates the modulus reduction curve (G/Gmax) as a function of shear strain. The curve depends on the plasticity of the material (plasticity index) and the mean effective stress at the depth of interest.
The curve for cohesionless soils can be established by using a plasticity index of 0. At low plasticity, the effect of confining pressure on the modulus reduction curve is more pronounced.
Also calculates the damping ratio of plastic and non-plastic soils based on a fit to empirical data.
:param strain: Strain amplitude (:math:`\\gamma`) [:math:`pct`] - Suggested range: 0.0 <= strain <= 10.0
:param PI: Plasticity index (:math:`PI`) [:math:`pct`] - Suggested range: 0.0 <= PI <= 200.0
:param sigma_m_eff: Mean effective pressure (:math:`\\sigma_m^{\\prime}`) [:math:`kPa`] - Suggested range: 0.0 <= sigma_m_eff <= 400.0
:param multiplier_1: Multiplier in equation for K (:math:``) [:math:`-`] (optional, default= 0.000102)
:param exponent_1: Exponent in equation for K (:math:``) [:math:`-`] (optional, default= 0.492)
:param multiplier_2: First multiplier in equation for m (:math:``) [:math:`-`] (optional, default= 0.000556)
:param exponent_2: First exponent in equation for m (:math:``) [:math:`-`] (optional, default= 0.4)
:param multiplier_3: Second multiplier in equation for m (:math:``) [:math:`-`] (optional, default= -0.0145)
:param exponent_3: Second exponent in equation for m (:math:``) [:math:`-`] (optional, default= 1.3)
.. math::
\\frac{G}{G_{max}} = K \\left( \\gamma, \\text{PI} \\right) \\left( \\sigma_m^{\\prime} \\right)^{m \\left( \\gamma, \\text{PI} \\right) - m_0}
K \\left( \\gamma, \\text{PI} \\right) = 0.5 \\left[ 1 + \\tanh \\left[ \\ln \\left( \\frac{0.000102 + n ( \\text{PI} )}{\\gamma} \\right)^{0.492} \\right] \\right]
m \\left( \\gamma, \\text{PI} \\right) - m_0 = 0.272 \\left[ 1 - \\tanh \\left[ \\ln \\left( \\frac{0.000556}{\\gamma} \\right)^{0.4} \\right] \\right] \\exp \\left( -0.0145 \\text{PI}^{1.3} \\right)
n ( \\text{PI} ) = \\begin{cases}
0.0 & \\quad \\text{for PI } = 0 \\\\
3.37 \\times 10^{-6} \\text{PI}^{1.404} & \\quad \\text{for } 0 < \\text{PI} \\leq 15 \\\\
7.0 \\times 10^{-7} \\text{PI}^{1.976} & \\quad \\text{for } 15 < \\text{PI} \\leq 70 \\\\
2.7 \\times 10^{-5} \\text{PI}^{1.115} & \\quad \\text{for } \\text{PI} > 70
\\end{cases}
\\xi = 0.333 \\frac{1 + \\exp(-0.0145 PI^{1.3})}{2} \\left[ 0.586 \\left( \\frac{G}{G_{max}} \\right)^2 - 1.547 \\frac{G}{G_{max}} + 1 \\right]
:returns: Dictionary with the following keys:
- 'G/Gmax [-]': Modulus reduction ratio (:math:`G / G_{max}`) [:math:`-`]
- 'K [-]': Factor K in the equation (:math:`K ( \\gamma, \\text{PI} )`) [:math:`-`]
- 'm [-]': Exponent m in the equation (:math:`m \\left( \\gamma, \\text{PI} \\right) - m_0`) [:math:`-`]
- 'n [-]': Factor n in equations (:math:`n ( \\text{PI} )`) [:math:`-`]
- 'dampingratio [pct]': Damping ratio (:math:`\\xi`) [:math:`pct`]
Reference - Ishibashi, I., & Zhang, X. (1993). Unified dynamic shear moduli and damping ratios of sand and clay. Soils and foundations, 33(1), 182-191.
"""
strain = 0.01 * strain
if pi == 0:
_n = 0
elif 0 < pi <= 15:
_n = 3.37e-6 * (pi ** 1.404)
elif 15 < pi <= 70:
_n = 7e-7 * (pi ** 1.976)
else:
_n = 2.7e-5 * (pi ** 1.115)
_m = 0.272 * (1 - np.tanh(np.log((multiplier_2 / strain) ** exponent_2))) * \
np.exp(multiplier_3 * (pi ** exponent_3))
_K = 0.5 * (1 + np.tanh(np.log(((multiplier_1 + _n) / strain) ** exponent_1)))
_G_over_Gmax = min(_K * (sigma_m_eff ** _m), 1)
_damping = 100 * 0.333 * 0.5 * (1 + np.exp(-0.0145 * (pi ** 1.3))) * \
(0.586 * (_G_over_Gmax ** 2) - 1.547 * _G_over_Gmax + 1)
return {
'G/Gmax [-]': _G_over_Gmax,
'K [-]': _K,
'm [-]': _m,
'n [-]': _n,
'dampingratio [pct]': _damping,
}
GMAX_SHEARWAVEVELOCITY = {
'Vs': {'type': 'float', 'min_value': 0.0, 'max_value': 600.0},
'gamma': {'type': 'float', 'min_value': 12.0, 'max_value': 22.0},
'g': {'type': 'float', 'min_value': 9.7, 'max_value': 10.2},
}
GMAX_SHEARWAVEVELOCITY_ERRORRETURN = {
'rho [kg/m3]': np.nan,
'Gmax [kPa]': np.nan,
}
@Validator(GMAX_SHEARWAVEVELOCITY, GMAX_SHEARWAVEVELOCITY_ERRORRETURN)
def gmax_shearwavevelocity(
Vs, gamma,
g=9.81, **kwargs):
"""
Calculates the small-strain shear modulus (shear strain < 1e-4%) from the shear wave velocity and the bulk unit weight if the soil based on elastic theory.
Often, the result of an in-situ or laboratory test will provide the shear wave velocity, which is then converted to the small-strain shear modulus using this function.
:param Vs: Shear wave velocity (:math:`V_s`) [:math:`m/s`] - Suggested range: 0.0 <= Vs <= 600.0
:param gamma: Bulk unit weight (:math:`\\gamma`) [:math:`kN/m3`] - Suggested range: 12.0 <= gamma <= 22.0
:param g: Acceleration due to gravity (:math:`g`) [:math:`m/s2`] - Suggested range: 9.7 <= g <= 10.2 (optional, default= 9.81)
.. math::
G_{max} = \\rho \\cdot V_s^2
\\rho = \\gamma / g
:returns: Dictionary with the following keys:
- 'rho [kg/m3]': Density of the material (:math:`\\rho`) [:math:`kg/m3`]
- 'Gmax [kPa]': Small-strain shear modulus (:math:`G_{max}`) [:math:`kPa`]
Reference - Robertson, P.K. and Cabal, K.L. (2015). Guide to Cone Penetration Testing for Geotechnical Engineering. 6th edition. Gregg Drilling & Testing, Inc.
"""
_rho = 1000 * gamma / g
_Gmax = 1e-3 * _rho * (Vs ** 2)
return {
'rho [kg/m3]': _rho,
'Gmax [kPa]': _Gmax,
}
DAMPINGRATIO_SANDGRAVEL_SEED = {
'cyclic_shear_strain': {'type': 'float', 'min_value': 0.0001, 'max_value': 1.0},
}
DAMPINGRATIO_SANDGRAVEL_SEED_ERRORRETURN = {
'D LE [pct]': np.nan,
'D BE [pct]': np.nan,
'D HE [pct]': np.nan,
}
@Validator(DAMPINGRATIO_SANDGRAVEL_SEED, DAMPINGRATIO_SANDGRAVEL_SEED_ERRORRETURN)
def dampingratio_sandgravel_seed(
cyclic_shear_strain,
**kwargs):
"""
Damping ratios for sand are compiled from a dataset comprising several sands and gravels. Average values and upper and lower bounds are provided. The comparison of the trends proposed for sand with the datapoints measured on gravel suggests that the trend is applicable for gravels too.
:param cyclic_shear_strain: Cyclic shear strain (:math:`\\gamma_{cyc}`) [:math:`pct`] - Suggested range: 0.0001 <= cyclic_shear_strain <= 1.0
:returns: Dictionary with the following keys:
- 'D LE [pct]': Low estimate damping ratio (:math:`D_{LE}`) [:math:`pct`]
- 'D BE [pct]': Average or best estimate damping ratio (:math:`D_{BE}`) [:math:`pct`]
- 'D HE [pct]': High estimate damping ratio (:math:`D_{HE}`) [:math:`pct`]
.. figure:: images/dampingratio_sandgravel_seed_1.png
:figwidth: 500.0
:width: 450.0
:align: center
Proposed trends and measurement data on gravels
Reference - Seed, H. B., Wong, R. T., Idriss, I. M., & Tokimatsu, K. (1986). Moduli and damping factors for dynamic analyses of cohesionless soils. Journal of geotechnical engineering, 112(11), 1016-1032.
"""
gamma_cyc_le = np.array(
[0.0001, 0.00010736704570827216, 0.00012251883726624774, 0.0001456046695997414, 0.0001730404914240103,
0.00020774487550410804, 0.00021704006888420929, 0.00023242841823261729, 0.00026473594476528894,
0.00028545466903303663, 0.0002994757536988713, 0.00033877427235285735, 0.0003652873729599878,
0.00038851611499061847, 0.00043055932487400233, 0.00046425569330965606, 0.0004988776087409925,
0.0005780361825306242, 0.0006232743623828213, 0.0006674651133236992, 0.0007654682597921579,
0.0008253752203841818, 0.0008808729401071887, 0.0009888154107768829, 0.001063466628084799,
0.0011466954916785786, 0.0011948068117258286, 0.0013423677618519564, 0.0014474239435819526,
0.001544747835016312, 0.0018021607982596848, 0.0019432012327136867, 0.002031677764622028, 0.002322020118922107,
0.002503745704508012, 0.0026996934702496465, 0.0028967682011055013, 0.003160369520630482, 0.003407705879658681,
0.003587352758504253, 0.0039619645139493676, 0.004272035178497375, 0.004497247627352388, 0.005104850422187586,
0.005504365449960532, 0.005794543683655353, 0.006399642854956974, 0.006900490731286996, 0.007364475463440597,
0.00859167346223717, 0.009200831362914412, 0.009989097232837956, 0.010994494638962928, 0.011534526934504477,
0.012782733940858966, 0.013689042054936413, 0.015274674020277513, 0.016470098236609144, 0.0182524073101425,
0.020089502548822056, 0.02467272437991229, 0.02771985351909183, 0.029845419300759967, 0.033033086887161785,
0.03535094067449866, 0.03734224475575858, 0.04096019231635378, 0.04416581266121802, 0.0492816362557288,
0.052236254697168326, 0.0569060794502507, 0.060317811781970765, 0.06437354416877875, 0.07158447562120226,
0.07718680897099323, 0.08943429734581976, 0.09577527492826368, 0.11021469138190157, 0.11843396317394378,
0.12510530005291384, 0.13864355029454234, 0.14446054728207172, 0.15845677830834998, 0.16567152579104538,
0.17986515724109198, 0.18805465663823706, 0.19594477030621169, 0.2127320114510672, 0.22471511895922835,
0.2456441316685955, 0.2577097853132104, 0.2648686953618417, 0.2836481402663653, 0.3320498963900685,
0.34361965946695683, 0.3730587002567181, 0.3927255475416774, 0.4220130950950124, 0.4442607121262613,
0.4856372698471063, 0.5112390156726162, 0.5607710125231429, 0.5943913324010771, 0.6740809134497746,
0.7250124447179741, 0.8007223631227072, 0.8604363145928147, 0.9502879628574636, 1.0])
damping_le = np.array(
[0.4534883720930267, 0.4529499243609152, 0.4519499500012856, 0.4506422912233034, 0.4493346324453249,
0.4828337734929846, 0.4963895688177154, 0.502407173529626, 0.519485771203005, 0.5027180916858249,
0.5253259597629274, 0.5497060051741833, 0.5726665903384252, 0.5760196208360711, 0.5949526272379515,
0.6157739981501145, 0.6371561167893276, 0.6654547137022, 0.6873456917404006, 0.6957402011768394,
0.7467566480829397, 0.7707868403732192, 0.7902153339791624, 0.8540509495685562, 0.8587139708224107,
0.8891618058689197, 0.8952681186568192, 0.9574928375161492, 0.9858014583105792, 1.0168619294121868,
1.0836144945343875, 1.127967222219393, 1.150091697733842, 1.2164651242771818, 1.255469816331999, 1.295544115512854,
1.3322938553890538, 1.3852672896808509, 1.436037660122082, 1.4546340241488522, 1.5354391867524733,
1.5958360213280578, 1.6352897243125746, 1.7410231010658812, 1.8078375783976883, 1.8569177455165509,
1.9543018185737682, 2.024325117283695, 2.0845323825698467, 2.257978258514367, 2.3479006000021663,
2.4816681331904107, 2.6144286659272207, 2.6900075535795587, 2.8924545826060135, 3.018149340197983,
3.246215051921304, 3.4167814204788307, 3.6677886130274224, 3.9455641030077793, 4.404651588779963,
4.781935028875684, 4.965347801873111, 5.310490057916855, 5.540583659842966, 5.726458461003531, 6.069993195304786,
6.343241847961984, 6.759558413868043, 6.996570145060081, 7.360657570938454, 7.577881570298783, 7.923406536455651,
8.345275350010269, 8.726696936680806, 9.420148551706026, 9.702362485408274, 10.355898335818587, 10.74231359752742,
11.022220447379462, 11.529420617872384, 11.806823440339912, 12.249383421043241, 12.467271467459032,
12.917106208602505, 13.12659019902799, 13.435688197102575, 13.798260823546753, 14.07580986880152,
14.566352393169891, 14.828265755570815, 14.967285396497914, 15.384448095935323, 16.285290007349026,
16.483086151880176, 16.917793592241107, 17.153894565347766, 17.552206235202867, 17.79133266846603,
18.22766903169697, 18.456710597771767, 18.911603318417228, 19.16150975773984, 19.725081461367047,
19.977719644082526, 20.363309357250767, 20.60470966171605, 20.9318988258026, 21.104574241686112])
gamma_cyc_be = np.array(
[0.0001, 0.00011235607753106597, 0.00012004861395727114, 0.00013551207945889716, 0.000188648345808323,
0.00020234025531060845, 0.00021853190388122838, 0.00023563461452460284, 0.0002540758149086333,
0.00027474330647545226, 0.0002954008937595277, 0.0003185195136042041, 0.00034344743936099386,
0.00037032627065416437, 0.0003993086889562561, 0.00043055932487400233, 0.00046425569330965606,
0.0005005892018097682, 0.0005397662378292022, 0.000582009341086441, 0.0006275584676695892,
0.0006766723530736442, 0.0007296299819115822, 0.0007867321726477517, 0.0008483032863554371,
0.0009146930692049297, 0.000982647472538318, 0.001063466628084799, 0.0011466954916785786, 0.00123643799994364,
0.0013332039227491347, 0.001437542925496226, 0.0015500476914161474, 0.001671357288225084,
0.0018021607982596848, 0.0019432012327136867, 0.0020952797522099218, 0.0022592602176821127,
0.0024360740974170613, 0.00262672575813104, 0.0030331108307883967, 0.003547820971844297, 0.004128194645775764,
0.00445127478763278, 0.004799639778441663, 0.005175268457207639, 0.005580294530533349, 0.006477673843176275,
0.008143353805556489, 0.008785091473186155, 0.009303207942506421, 0.010994494638962928, 0.011854944107167406,
0.012782733940858966, 0.013689042054936413, 0.014760371783422216, 0.016583306222028756, 0.01763784402092674,
0.02027382798670632, 0.024504292983635444, 0.025973417011258847, 0.027530620519893242, 0.03081321999038474,
0.033123722073301484, 0.03535094067449866, 0.03759891840998422, 0.04012704644711407, 0.05081101695004028,
0.057297226031172224, 0.06052475542257685, 0.06415344091522583, 0.06823297894393368, 0.0714008281326666,
0.07587623271087471, 0.087017054005354, 0.09297411230021717, 0.11128944438907216, 0.12442159494433225,
0.13125027575125686, 0.13807489209485932, 0.1435972726387898, 0.15176399757025846, 0.1600933225387565,
0.1673825840529291, 0.17728991405192868, 0.18562297266826935, 0.20059360979178895, 0.21358375759392,
0.22625970876951595, 0.2364369961823236, 0.2657774301199742, 0.2807485644436065, 0.3331891223067882,
0.3543767262109732, 0.3769116566908065, 0.4022549635490181, 0.4293023386977912, 0.459740286851477,
0.4923363147645632, 0.5272434323648003, 0.5685064777243272, 0.6025905676771414, 1.0])
damping_be = np.array(
[0.6986440349973506, 0.7362373606667489, 0.7407480828812112, 0.7729599329917427, 0.8186545698812111,
0.8516093206127415, 0.8610737104687196, 0.8904519383891873, 0.9166213449315456, 0.9428434096098536,
0.9892826934109812, 1.0165217070793735, 1.053387184882112, 1.0881134484327717, 1.1271181404875892,
1.172540475298632, 1.2158235958576, 1.259106716416568, 1.3055986583536523, 1.3456729575345037,
1.3953737208496977, 1.4397264485347063, 1.487287997597825, 1.5466152250473648, 1.6091512738750249,
1.665269679946455, 1.7610737670214751, 1.795689813231956, 1.8667827190679205, 1.941084446281998,
2.015386173496076, 2.093966329214308, 2.1704072706804642, 2.245778605020579, 2.3307764034950367,
2.421122237599693, 2.510398464578305, 2.60609233431315, 2.704995025426108, 2.8060369307911444,
3.0338432008741916, 3.208051218844247, 3.485021647551964, 3.632056659336644, 3.7753480461801945,
3.9421707897965814, 4.038934266657467, 4.410357087416075, 4.872586516988363, 5.041283899108514,
5.224062563648147, 5.627511939977103, 5.848349843458428, 6.076674996822021, 6.257632789259301,
6.5034281923481885, 6.986959273630376, 7.242247227037264, 7.836749412144936, 8.58936308987851,
8.863781359172831, 9.136238682069413, 9.653075256343936, 9.936211125087073, 10.206851707897634,
10.4817015871874, 10.766793750723105, 11.8840614853599, 12.520696060655148, 12.764558848148788,
13.042572185838962, 13.330364311353787, 13.578570507093573, 13.89235165841942, 14.624351308903234,
14.939264978677109, 15.851867343610868, 16.45195445225551, 16.719807190507087, 16.998269794637665,
17.226521036615786, 17.504654474165264, 17.77079584101518, 18.037191843938626, 18.375354933730755,
18.583218233193925, 18.93216185095902, 19.245739648268426, 19.5441511525224, 19.81200207088116,
20.356593802114066, 20.628656866375167, 21.420329030058628, 21.683936150001365, 21.946235972345615,
22.21883750155392, 22.470783728706056, 22.73934319366042, 22.997894191935426, 23.2402949715018,
23.499638731520506, 23.675504588674016, 23.915041650104733])
gamma_cyc_he = np.array(
[0.0001, 0.00011015945244167936, 0.00011878073476376348, 0.00012962130074026585, 0.00013810025650511809,
0.0001489082377876431, 0.00015837736072440495, 0.00017916033092457522, 0.00019318174950993465,
0.0002083005102252808, 0.0002200339930754089, 0.00024720861642004354, 0.00026655561958109636,
0.00033877427235285735, 0.0003652873729599878, 0.0003911865985572175, 0.0004486240836775954,
0.00048373423352078283, 0.0005205213923692329, 0.0005472119560568889, 0.0006105967251195091,
0.0006583831532063964, 0.0006978556860562095, 0.0007867321726477517, 0.0008483032863554371,
0.000905342675064425, 0.001063466628084799, 0.0011466954916785786, 0.0012217046166771036,
0.0013332039227491347, 0.001437542925496226, 0.0015185191326326875, 0.0016486157247629332,
0.0017776394380147169, 0.0021096817549321007, 0.0022747893477521397, 0.0024832491772025485,
0.002644780679665561, 0.003225987328182171, 0.003419397335963503, 0.0038024279755937366, 0.0041000130158274795,
0.004638034661967479, 0.004916102189457339, 0.005392404566036912, 0.005774731229112175, 0.006201841151930438,
0.006924165540298022, 0.007595021510246519, 0.008050371460496924, 0.010011931835549694, 0.011455785063641071,
0.012142602230882892, 0.013456612009179428, 0.014410698725553015, 0.015743860333017508, 0.016812061962493838,
0.01850418719610731, 0.019680874436571974, 0.02181063714833311, 0.02327717091085021, 0.025358107025649997,
0.026878420398955924, 0.02829539255074718, 0.029940489927209605, 0.03233902143328173, 0.03427786673884472,
0.03696051593511605, 0.03998984577574319, 0.04224245749081527, 0.04649413871882389, 0.04962037587212595,
0.054241807140663435, 0.057493806381175336, 0.059497088795807535, 0.06393409022925085, 0.06776717809686499,
0.07510059254738255, 0.0925504965751776, 0.10082434014077037, 0.10871503932295457, 0.11843396317394378,
0.12510530005291384, 0.13722627505333002, 0.14446054728207172, 0.15900042572547404, 0.17085789860592426,
0.1780264966357266, 0.19130284068466888, 1.0])
damping_he = np.array(
[0.8935734930337595, 0.8897939000470387, 0.914893699463356, 0.9297414266082099, 0.955466834161644,
0.9827058478300366, 1.0164662809107696, 1.0668734330556229, 1.0994604823542071, 1.1384651744090206,
1.164576324510335, 1.2345022146771392, 1.2927598350006413, 1.4642200979369235, 1.526756146764587,
1.5861352625421787, 1.7262856366181405, 1.8048657923363685, 1.8804666146001097, 1.9365630860601328,
2.0758514063222755, 2.170475668931079, 2.2367067956656683, 2.404440140130081, 2.5204565452596555,
2.6210414795537496, 2.9027331886815912, 3.0529770218443915, 3.183523927749576, 3.3727176164386736,
3.5507712348784644, 3.6795646690794532, 3.9309414480374194, 4.026360275939851, 4.4869805451690254,
4.694983163137883, 4.934766674472193, 5.093874685059003, 5.7191596994038605, 5.905365092109072,
6.248439514045536, 6.491739167173666, 6.9336225274485335, 7.1497769196828225, 7.51434726079616,
7.789416615047254, 8.025755323863304, 8.488469839332105, 8.902242108243207, 9.148167232152225,
10.040707011077169, 10.641518816876305, 10.886374333659287, 11.341319951572515, 11.609058535502374,
12.004760943127547, 12.280887851824437, 12.768306338508301, 13.047601029632936, 13.472169805166676,
13.728248875408589, 14.146425739424538, 14.427217084511506, 14.68420213675406, 14.910263366986214,
15.259708555291633, 15.592068121719082, 15.909017865525847, 16.356720236454017, 16.60779650533933,
17.097110573541013, 17.39258981072291, 17.849469421930138, 18.12029262282861, 18.27298700021181,
18.64175374434101, 18.922054852828538, 19.44190779650686, 20.52512074972923, 20.933917753476006,
21.280434494266828, 21.704003960876555, 21.929004311591964, 22.369348677534212, 22.57368231899753,
22.978764826023692, 23.259526672727194, 23.38275496581629, 23.694000615611877, 23.95869241126077])
_D_LE = np.interp(np.log10(cyclic_shear_strain), np.log10(gamma_cyc_le), damping_le)
_D_BE = np.interp(np.log10(cyclic_shear_strain), np.log10(gamma_cyc_be), damping_be)
_D_HE = np.interp(np.log10(cyclic_shear_strain), np.log10(gamma_cyc_he), damping_he)
return {
'D LE [pct]': _D_LE,
'D BE [pct]': _D_BE,
'D HE [pct]': _D_HE,
}
MODULUSREDUCTION_DARENDELI = {
'mean_effective_stress': {'type': 'float', 'min_value': 0.0, 'max_value': 1600.0},
'pi': {'type': 'float', 'min_value': 0.0, 'max_value': 60.0},
'ocr': {'type': 'float', 'min_value': 1.0, 'max_value': 20.0},
'N': {'type': 'float', 'min_value': 1.0, 'max_value': None},
'frequency': {'type': 'float', 'min_value': 0.05, 'max_value': 100.0},
'soiltype': {'type': 'string', 'options': ('sand', 'fine sand', 'silt', 'clay', 'all'), 'regex': None},
'min_strain': {'type': 'float', 'min_value': None, 'max_value': None},
'max_strain': {'type': 'float', 'min_value': None, 'max_value': None},
'no_points': {'type': 'int', 'min_value': 10.0, 'max_value': None},
}
MODULUSREDUCTION_DARENDELI_ERRORRETURN = {
'strains [pct]': np.nan,
'G/Gmax [-]': np.nan,
'D [pct]': np.nan,
'sigma_ND [-]': np.nan,
'sigma_D [pct]': np.nan,
}
@Validator(MODULUSREDUCTION_DARENDELI, MODULUSREDUCTION_DARENDELI_ERRORRETURN)
def modulusreduction_darendeli(
mean_effective_stress, pi, ocr,N,frequency,soiltype,
min_strain=0.0001,max_strain=1.0,no_points=250,custom_coefficients=None, **kwargs):
"""
Darendeli (2001) proposed a comprehensive framework for estimating the modulus reduction curve and damping curve for sand, fine sand, silt and clay based on extensive laboratory testing. The framework is initially based on the work by Hardin and Drnevich but extends the formulation to include the effect of soil type, plasticity, overconsolidation ratio, stress ratio, loading frequency and number of cycles applied.
The author used a Bayesian approach to calibrate the parameters of the parametric equations and also formulated expressions to estimate the standard deviation on the estimates. Parameters for individual soil types can be used as well as parameters calibrated to the entire credible dataset (using ``'all'`` for ``soiltype``).
The formulation is based on Masing damping but takes into account the damping at small strains, which is not zero but difficult to estimate from resonant column or cyclic DSS tests. The Masing damping at large strains is also adjusted to be in line with experimental observations (Masing damping overestimates the damping ratio at large strains).
:param mean_effective_stress: Mean effective stress at the depth under consideration (:math:`\\sigma_0^{\\prime}`) [:math:`kPa`] - Suggested range: 0.0 <= mean_effective_stress <= 1000.0
:param pi: Plasticity index (difference between liquid limit and plastic limit) (:math:`PI`) [:math:`pct`] - Suggested range: 0.0 <= plasticity_index <= 60.0
:param ocr: Overconsolidation ratio of the soil (:math:`OCR`) [:math:`-`] - Suggested range: 1.0 <= OCR <= 20.0
:param N: Number of cycles (:math:`N`) [:math:`-`] - Suggested range: N >= 1.0
:param frequency: Loading frequency (:math:`f`) [:math:`Hz`] - Suggested range: 0.05 <= frequency <= 20.0
:param soiltype: Soil type used for calculating modulus reduction and damping curves - Options: ('sand', 'fine sand', 'silt', 'clay', 'all')
:param min_strain: Minimum value for the strain (:math:`\\gamma_{min}`) [:math:`pct`] (optional, default= 0.0001)
:param max_strain: Maximum value for the strain (:math:`\\gamma_{max}`) [:math:`pct`] (optional, default= 1.0)
:param no_points: Number of points used for the strain curve calculation (:math:``) [:math:`-`] - Suggested range: no_points >= 10.0 (optional, default= 250)
:param custom_coefficients: Dictionary with custom calibration coefficients (:math:``) [:math:`-`] (optional, default= None)- Elementtype: float, order: ascending, unique: True, empty entries allowed: False
.. math::
\\frac{G}{G_{max}} = \\frac{1}{1 + \\left( \\frac{\\gamma}{\\gamma_r} \\right)^a}
\\gamma_r = \\left( \\phi_1 + \\phi_2 \\cdot PI \\cdot OCR^{\\phi_3} \\right) \\cdot \\sigma_0^{\\prime \\phi_4}
a = \\phi_5
D_{\\text{adjusted}} = b \\cdot \\left( \\frac{G}{G_{max}} \\right)^{0.1} \\cdot D_{\\text{Masing}} + D_{\\text{min}}
D_{\\text{min}} = \\left( \\phi_6 + \\phi_7 \\cdot PI \\cdot OCR^{\\phi_8} \\right) \\cdot \\sigma_0^{\\prime \\phi_9} \\cdot \\left[1 + \\phi_{10} \\cdot \\ln(f) \\right]
b = \\phi_{11} + \\phi_{12} \\cdot \\ln(N)
\\sigma_{\\text{NG}} = \\exp(\\phi_{13}) + \\sqrt{\\frac{0.25}{\\exp(\\phi_{14})} - \\frac{\\left(G / G_{max} - 0.5 \\right)^2}{\\exp(\\phi_{14})}}
\\sigma_D = \\exp (\\phi_{15} ) + \\exp (\\phi_{16} ) \\cdot \\sqrt{D}
\\rho_{i,j} = \\exp \\left( \\frac{-1}{\\exp ( \\phi_{17} )} \\right) \\cdot \\exp \\left( \\frac{- | \\ln \\gamma_i - \\ln \\gamma_j |}{\\exp ( \\phi_{18} )} \\right)
:returns: Dictionary with the following keys:
- 'strains [pct]': List of strains for the modulus reduction curve (:math:`\\gamma`) [:math:`pct`]
- 'G/Gmax [-]': Modulus ratio for given strains (:math:`G / G_{max}`) [:math:`-`]
- 'D [pct]': Damping ratios for given strains (:math:`D`) [:math:`pct`]
- 'sigma_ND [-]': Standard deviation for the modulus reduction curve (:math:`\\sigma_{ND}`) [:math:`-`]
- 'sigma_D [pct]': Standard deviation for the damping curve (:math:`\\sigma_D`) [:math:`pct`]
Reference - Darendeli, M. B. (2001). Development of a new family of normalized modulus reduction and material damping curves. The university of Texas at Austin.
"""
if custom_coefficients is None:
parameters = {
'clay': {
'phi1': 2.58e-2, 'phi2': 1.95e-3, 'phi3': 9.92e-2,
'phi4': 2.26e-1, 'phi5': 9.75e-1, 'phi6': 9.58e-1,
'phi7': 5.65e-3, 'phi8': -1.0e-1, 'phi9': -1.96e-1,
'phi10': 3.68e-1, 'phi11': 4.66e-1, 'phi12': 2.23e-2,
'phi13': -5.65e0, 'phi14': 4.0e0, 'phi15': -5.0e0,
'phi16': -7.25e-1, 'phi17': 7.67e0, 'phi18': 2.16e0},
'silt': {
'phi1': 4.16e-2, 'phi2': 6.89e-4, 'phi3': 3.21e-1,
'phi4': 2.80e-1, 'phi5': 1.0e0, 'phi6': 7.12e-1,
'phi7': 3.03e-3, 'phi8': -1.0e-1, 'phi9': -1.89e-1,
'phi10': 2.34e-1, 'phi11': 5.92e-1, 'phi12': -7.67e-4,
'phi13': -5.02e0, 'phi14': 3.93e0, 'phi15': -5.2e0,
'phi16': -6.42e-1, 'phi17': 4.06e0, 'phi18': 1.94e0},
'fine sand': {
'phi1': 3.34e-2, 'phi2': -5.79e-5, 'phi3': 2.49e-1,
'phi4': 4.82e-1, 'phi5': 8.45e-1, 'phi6': 8.89e-1,
'phi7': 2.02e-2, 'phi8': -1.0e-1, 'phi9': -3.72e-1,
'phi10': 2.33e-1, 'phi11': 7.76e-1, 'phi12': -2.94e-2,
'phi13': -3.98e0, 'phi14': 4.32e0, 'phi15': -5.34e0,
'phi16': -2.66e-1, 'phi17': 4.92e0, 'phi18': 2.68e0},
'sand': {
'phi1': 4.74e-2, 'phi2': -2.34e-3, 'phi3': 2.5e-1,
'phi4': 2.34e-1, 'phi5': 8.95e-1, 'phi6': 6.88e-1,
'phi7': 1.22e-2, 'phi8': -1.0e-1, 'phi9': -1.27e-1,
'phi10': 2.88e-1, 'phi11': 7.67e-1, 'phi12': -2.83e-2,
'phi13': -4.14e0, 'phi14': 3.61e0, 'phi15': -5.15e0,
'phi16': -2.32e-1, 'phi17': 5.15e0, 'phi18': 3.12e0},
'all': {
'phi1': 3.52e-2, 'phi2': 1.01e-3, 'phi3': 3.25e-1,
'phi4': 3.48e-1, 'phi5': 9.19e-1, 'phi6': 8.01e-1,
'phi7': 1.29e-2, 'phi8': -1.07e-1, 'phi9': -2.89e-1,
'phi10': 2.92e-1, 'phi11': 6.33e-1, 'phi12': -5.66e-3,
'phi13': -4.23e0, 'phi14': 3.62e0, 'phi15': -5.0e0,
'phi16': -2.5e-1, 'phi17': 5.62e0, 'phi18': 2.78e0
}
}
params = parameters[soiltype]
else:
params = custom_coefficients
_gamma = np.logspace(np.log10(min_strain), np.log10(max_strain), no_points)
sigma_0_eff = mean_effective_stress / 100 # Stresses in formulae are in atm
_gamma_r = \
(params['phi1'] + params['phi2'] * pi * (ocr ** params['phi3'])) * (sigma_0_eff ** params['phi4'])
_a = params['phi5']
_Dmin = \
(params['phi6'] + params['phi7'] * pi * (ocr ** params['phi8'])) * \
(sigma_0_eff ** params['phi9']) * \
(1 + params['phi10'] * np.log(frequency))
_b = params['phi11'] + params['phi12'] * np.log(N)
_G_Gmax = 1 / (1 + ((_gamma / _gamma_r) ** _a))
_D_masing_a1 = (100 / np.pi) * (4 * (
(_gamma - _gamma_r * np.log((_gamma + _gamma_r) / _gamma_r)) /
((_gamma ** 2) / (_gamma + _gamma_r))
) - 2)
c_1 = -1.1143 * (_a ** 2) + 1.8618 * _a + 0.2523
c_2 = 0.0805 * (_a ** 2) - 0.0710 * _a - 0.0095
c_3 = -0.0005 * (_a ** 2) + 0.0002 * _a + 0.0003
_D_masing = c_1 * _D_masing_a1 + c_2 * (_D_masing_a1 ** 2) + c_3 * (_D_masing_a1 ** 3)
_F = _b * (_G_Gmax ** 0.1)
_D = _F * _D_masing + _Dmin
_sigma_ND = np.exp(params['phi13']) + np.sqrt(
(0.25 / np.exp(params['phi14'])) -
(((_G_Gmax - 0.5) ** 2) / np.exp(params['phi14']))
)
_sigma_D = np.exp(params['phi15']) + np.exp(params['phi16']) * np.sqrt(_D)
return {
'strains [pct]': _gamma,
'G/Gmax [-]': _G_Gmax,
'D [pct]': _D,
'sigma_ND [-]': _sigma_ND,
'sigma_D [pct]': _sigma_D,
'gamma_r [pct]': _gamma_r,
'a [-]': _a,
'Dmin [pct]': _Dmin,
'b [-]': _b,
'Dmasing,a=1 [pct]': _D_masing_a1,
'Dmasing [pct]': _D_masing
}