Simplified Boundary Element Method for Kinematic Response of Single Piles in Two-Layer Soil

Table 1

Comparison of FE, BDWF, and the present simplified approach for dimensionless maximum kinematic pile bending 𝑀_max(𝜔₁)/𝜌_𝑑𝑑⁴𝑈̈_𝑔 during the harmonic excitation.

Case	𝐿/𝑑	𝐸𝑝/𝐸𝑎	𝐻𝑎/𝐻𝑏	𝑉𝑎/𝑉𝑏	Previous 𝜔1𝑑/𝑉𝑎	Computed 𝜔1𝑑/𝑉𝑎	FE method	δ comp	BDWF method	Error1: %	Simplified results	Error2: %
1	10	500	1	0.58	0.0964	0.0968	709	2.21	760	7.2	752	6.0
2	20				0.0482	0.0484	1253	2.41	1282	2.3	1326	5.8
3	40				0.0241	0.0242	2088	2.63	2196	5.2	2313	10.8
4	10	5000	1		0.0964	0.0968	1854	1.66	1623	−12.5	1921	3.6
5	20				0.0482	0.0484	7718	1.81	8019	3.9	7735	0.2
6	40				0.0241	0.0242	13120	1.97	13320	1.5	13406	2.2
7	20	1000	3	1.5	0.0916	0.0910	508	2.27	515	1.4	555	9.2
8	20	10000	1/3		0.1164	0.1159	1113	1.42	1097	−1.4	1119	0.6
9	20		1		0.1058	0.1056	931	1.55	1070	14.9	663	−28.8
10	20		3		0.0913	0.0910	1482	1.70	1433	−3.3	1547	4.4
11	10	500	1	1.73	0.2310	0.2293	86	2.05	79	−8.1	96	11.6
12	20				0.1155	0.1147	269	2.24	253	−5.9	315	17.2
13	40				0.0578	0.0573	656	2.44	577	−12.0	778	18.7
14	10	5000	1		0.2310	0.2293	273	1.54	266	−2.6	265	−2.9
15	20				0.1155	0.1147	998	1.68	860	−13.8	963	−3.5
16	40				0.0578	0.0573	3054	1.83	2825	−7.5	3144	3.0
17	20	1000	3	3	0.1017	0.1009	1046	2.17	1075	2.8	1331	27.3
18	20	10000	1/3		0.2117	0.2094	875	1.35	896	2.4	1217	39.1
19	20		3		0.1415	0.1403	2949	1.48	3042	3.2	3470	17.7
20	20		3		0.1019	0.1009	4867	1.63	4911	0.9	5608	15.2
21	20	1000	3	6	0.1042	0.1037	1486	2.07	1480	−0.4	2077	39.7
22	20	10000	1/3		0.2897	0.2849	1128	1.29	1274	12.9	1827	62.0
23	20		1		0.1535	0.1526	3809	1.42	4193	10.1	5073	33.2
24	20		3		0.1045	0.1037	6730	1.55	6831	1.5	8675	28.9

(1) Values of previous 𝜔₁𝑑/𝑉_𝑎, FE, δ _comp, and BDWF computed form a proposed simple analytical expression [19].
(2) Error1: percentage error in the BDWF moment compared with the FE moment. Error2: percentage error in the elastic moment compared with the FE moment.
(3) The values of computed 𝜔₁𝑑/𝑉_𝑎 are associated with the computed time interval.