A novel finite element formulation for static bending analysis of functionally graded porous sandwich plates

doi:10.1007/s11709-022-0891-4

Frontiers of Structural and Civil Engineering

2022, Vol. 16

Issue (12): 1599-1620 https://doi.org/10.1007/s11709-022-0891-4

本期目录

A novel finite element formulation for static bending analysis of functionally graded porous sandwich plates

Van Chinh NGUYEN¹, Trung Thanh TRAN¹, Trung NGUYEN-THOI^2,³, Quoc-Hoa PHAM⁴(

)

¹. Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi, Vietnam
². Laboratory for Applied and Industrial Mathematics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City, Vietnam
³. Faculty of Mechanical-Electrical and Computer Engineering, Van Lang University, Ho Chi Minh City, Vietnam
⁴. Faculty of Engineering and Technology, Nguyen Tat Thanh University, Ho Chi Minh City, Vietnam

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Abstract：

This article aims to propose a finite element formulation based on Quasi-3D theory for the static bending analysis of functionally graded porous (FGP) sandwich plates. The FGP sandwich plates consist of three layers including the bottom skin of homogeneous metal, the top skin of fully ceramic and the FGP core layer with uneven porosity distribution. A quadrilateral (Q4) element with nine degrees of freedom (DOFs) per node is derived and employed in analyzing the static bending response of the plate under uniform and/or sinusoidally distributed loads. The accuracy of the present finite element formulation is verified by comparing the obtained numerical results with the published results in the literature. Then, some numerical examples are performed to examine the effects of the parameters including power-law index k and porosity coefficient $ξ$ on the static bending response of rectangular FGP sandwich plates. In addition, a problem with a complicated L-shape model is conducted to illustrate the superiority of the proposed finite element method.

Key words： sandwich plates functionally graded porous static bending Quasi-3D theory Q4 element

收稿日期: 2022-05-10 出版日期: 2023-01-16

Corresponding Author(s): Quoc-Hoa PHAM

引用本文:

. [J]. Frontiers of Structural and Civil Engineering, 2022, 16(12): 1599-1620.
Van Chinh NGUYEN, Trung Thanh TRAN, Trung NGUYEN-THOI, Quoc-Hoa PHAM. A novel finite element formulation for static bending analysis of functionally graded porous sandwich plates. Front. Struct. Civ. Eng., 2022, 16(12): 1599-1620.

链接本文:

https://academic.hep.com.cn/fsce/CN/10.1007/s11709-022-0891-4
https://academic.hep.com.cn/fsce/CN/Y2022/V16/I12/1599

Fig.1

material properties	Young’s moduli (GPa)	mass densities (kg/m³)	Poisson’s ratio
alumina (Al₂O₃) (ceramic)	$E c = 380$	$ρ c = 3800$	$ν c = 0.3$
aluminum (Al) (metal)	$E m = 70$	$ρ m = 2707$	$ν m = 0.3$

Tab.1

Fig.2

Fig.3

k	scheme	displacement and stress	mesh size					Zenkour [40]	Thai et al. [50]
k	scheme	displacement and stress	$12 × 12$	$14 × 14$	$16 × 16$	$18 × 18$	$20 × 20$	Zenkour [40]	Thai et al. [50]
0	2-1-2	$w ∗$	0.1983	0.1979	0.1976	0.1976	0.1976	0.1961	0.1961
	2-1-2	$σ x ∗ (h / 2)$	2.0204	2.0136	2.0092	2.0092	2.0092	2.0499	1.9758
	1-1-1	$w ∗$	0.1983	0.1979	0.1976	0.1976	0.1976	0.1961	0.1961
	1-1-1	$σ x ∗ (h / 2)$	2.0204	2.0136	2.0092	2.0092	2.0092	2.0499	1.9758
	2-2-1	$w ∗$	0.1983	0.1979	0.1976	0.1976	0.1976	0.1961	0.1961
	2-2-1	$σ x ∗ (h / 2)$	2.0204	2.0136	2.0092	2.0092	2.0092	2.0499	1.9758
	1-2-1	$w ∗$	0.1983	0.1979	0.1976	0.1976	0.1976	0.1961	0.1961
	1-2-1	$σ x ∗ (h / 2)$	2.0204	2.0136	2.0092	2.0092	2.0092	2.0499	1.9758
1	2-1-2	$w ∗$	0.3097	0.3091	0.3087	0.3087	0.3087	0.3063	0.3064
	2-1-2	$σ x ∗ (h / 2)$	1.4808	1.4758	1.4726	1.4726	1.4726	1.4959	1.4517
	1-1-1	$w ∗$	0.2952	0.2946	0.2943	0.2943	0.2943	0.2920	0.2920
	1-1-1	$σ x ∗ (h / 2)$	1.4111	1.4063	1.4033	1.4033	1.4033	1.4262	1.3830
	2-2-1	$w ∗$	0.2836	0.283	0.2827	0.2827	0.2827	0.2809	0.2809
	2-2-1	$σ x ∗ (h / 2)$	1.3027	1.2983	1.2954	1.2954	1.2954	1.3206	1.2775
	1-2-1	$w ∗$	0.2739	0.2734	0.2730	0.2730	0.2730	0.2709	0.2710
	1-2-1	$σ x ∗ (h / 2)$	1.3076	1.3033	1.3004	1.3004	1.3004	1.3231	1.2810
2	2-1-2	$w ∗$	0.3562	0.3555	0.3551	0.3551	0.3551	0.3523	0.3526
	2-1-2	$σ x ∗ (h / 2)$	1.7070	1.7013	1.6976	1.6976	1.6976	1.7214	1.6750
	1-1-1	$w ∗$	0.3365	0.3359	0.3355	0.3355	0.3355	0.3329	0.3330
	1-1-1	$σ x ∗ (h / 2)$	1.6130	1.6076	1.6041	1.6041	1.6041	1.6275	1.5824
	2-2-1	$w ∗$	0.3189	0.3183	0.3179	0.3179	0.3179	0.3162	0.3163
	2-2-1	$σ x ∗ (h / 2)$	1.4521	1.4472	1.4439	1.4439	1.4439	1.4710	1.4253
	1-2-1	$w ∗$	0.3059	0.3053	0.3049	0.3049	0.3049	0.3026	0.3027
	1-2-1	$σ x ∗ (h / 2)$	1.4646	1.4597	1.4565	1.4565	1.4565	1.4799	1.4358
10	2-1-2	$w ∗$	0.4086	0.4078	0.4073	0.4073	0.4073	0.4041	0.3894
	2-1-2	$σ x ∗ (h / 2)$	1.9573	1.9508	1.9465	1.9465	1.9465	1.9713	1.9216
	1-1-1	$w ∗$	0.3898	0.3890	0.3886	0.3886	0.3886	0.3855	0.3724
	1-1-1	$σ x ∗ (h / 2)$	1.8710	1.8647	1.8606	1.8606	1.8606	1.8838	1.8375
	2-2-1	$w ∗$	0.3648	0.3641	0.3637	0.3637	0.3637	0.3622	0.3492
	2-2-1	$σ x ∗ (h / 2)$	1.6450	1.6393	1.6357	1.6357	1.6357	1.6666	1.6160
	1-2-1	$w ∗$	0.3520	0.3514	0.3509	0.3509	0.3509	0.3482	0.3361
	1-2-1	$σ x ∗ (h / 2)$	1.6899	1.6843	1.6806	1.6806	1.6806	1.7042	1.6587

Tab.2

Fig.4

power law-index	$w ¯ = E c h 3 12 (1 − ν 2) q 0 a 4 w (a 2, b 2)$		$σ ¯ x = h q 0 a σ x (a 2, b 2, h 2)$		$τ ¯ x z = h q 0 a τ x z (0, b 2, 0)$
power law-index	present	Vasiraja Nagaraj [70]	present	Vasiraja Nagaraj [70]	present	Vasiraja Nagaraj [70]
ceramic	0.001244	0.001237	28.3122	28.308	0.639017	0.638044
0.5	0.001883	0.001949	36.6621	36.652	0.665653	0.665576
1	0.002426	0.002540	42.8435	42.822	0.693451	0.692580
2	0.003065	0.003264	48.16327	49.154	0.666412	0.665902
5	0.003644	0.003870	54.4678	54.402	0.603652	0.603208
metal	0.006455	0.006883	28.3114	28.308	0.639017	0.638044

Tab.3

power law-index	$w ¯ = 10 E c h 3 q 0 a 4 w (a 2, b 2)$		$σ ¯ x = h q 0 a σ x (a 2, b 2, h 2)$		$τ ¯ x z = h q 0 a τ x z (0, b 2, 0)$
power law-index	present	Vasiraja Nagaraj [70]	present	Vasiraja Nagaraj [70]	present	Vasiraja Nagaraj [70]
ceramic	0.4702	0.466	2.9075	2.872	0.5020	0.491
0.5	0.7048	0.712	3.7887	3.719	0.5119	0.501
1	0.9009	0.927	4.4909	4.345	0.5028	0.491
2	1.1344	1.193	5.2529	4.988	0.4657	0.466
5	1.3751	1.444	6.1746	5.521	0.4083	0.427
metal	2.4443	2.530	2.9075	2.872	0.4992	0.491

Tab.4

Fig.5

parameters	$k = 0.5$	$k = 1$	$k = 2$	$k = 4$	$k = 10$
max ( $w ∗$ )	1.5769	1.7310	1.8768	1.9648	2.0067
$σ x ∗ (h / 2)$	4.7358	4.9465	5.0721	5.0709	5.0223
$σ z ∗ (h / 2)$	–0.1026	–0.0448	–0.0221	–0.0698	–0.1711
$τ x z ∗ (0)$	0.5491	0.5105	0.4040	0.2722	0.2378

Tab.5

Fig.6

parameters	$k = 0.5$	$k = 1$	$k = 2$	$k = 4$	$k = 10$
max ( $w ∗$ )	1.1924	1.2681	1.3406	1.3885	1.4152
$σ x ∗ (h / 2)$	2.6989	2.7840	2.8436	2.8592	2.8451
$σ z ∗ (h / 2)$	0.0083	0.0353	0.0483	0.0385	0.0103
$τ x z ∗ (0)$	0.5763	0.5054	0.3598	0.206	0.1549

Tab.6

Fig.7

parameters	$ξ = 0.1$	$ξ = 0.2$	$ξ = 0.3$	$ξ = 0.4$	$ξ = 0.5$
$w ∗$	2.3609	2.5479	2.7840	3.0943	3.5251
$σ x ∗ (h / 2)$	2.4028	2.5142	2.6486	2.8175	3.0417
$σ z ∗ (h / 2)$	–0.0106	0.0174	0.0573	0.1155	0.2046
$τ x z ∗ (0)$	–0.0851	–0.0716	–0.0575	–0.0425	–0.0266

Tab.7

Fig.8

parameters	$ξ = 0.1$	$ξ = 0.2$	$ξ = 0.3$	$ξ = 0.4$	$ξ = 0.5$
$w ∗$	2.2870	2.4194	2.5749	2.7611	2.9897
$σ x ∗ (h / 2)$	12.1422	12.5523	13.0116	13.5344	14.1424
$σ z ∗ (h / 2)$	–0.2010	–0.1692	–0.1247	–0.0624	0.0257
$τ x z ∗ (0)$	0.6928	0.6543	0.6099	0.5580	0.4970

Tab.8

Fig.9

parameters	value
parameters	0-1-0	1-1-1	1-2-1	1-8-1	2-2-1
$w ∗$	6.1509	5.5924	5.6977	5.8287	6.0131
$σ x ∗ (h / 2)$	10.7513	9.4641	9.2555	9.4142	8.9882
$σ z ∗ (h / 2)$	–0.6205	0.4326	0.2359	–0.2589	–0.072
$τ x z ∗ (0)$	–0.0074	–0.0035	–0.0044	–0.0064	–0.2087

Tab.9

Fig.10

parameters	scheme
parameters	0-1-0	1-1-1	1-2-1	1-8-1	2-2-1
$w ∗$	5.0841	4.4759	4.5641	4.7329	4.8117
$σ x ∗ (h / 2)$	4.7835	4.2647	4.1825	4.2377	4.0886
$σ z ∗ (h / 2)$	–0.3376	0.2420	0.1333	–0.1398	–0.0373
$τ x z ∗ (0)$	–0.0065	–0.0036	–0.0044	–0.0058	–0.1977

Tab.10

Fig.11

$a h$	scheme	$w ∗$				$σ x ∗ (h / 2)$
$a h$	scheme	$k = 0.5$	$k = 1.5$	$k = 4.5$	$k = 10.5$	$k = 0.5$	$k = 1.5$	$k = 4.5$	$k = 10.5$
15	2-1-2	0.8784	0.9727	1.0567	1.0909	6.3092	6.5531	6.6895	6.7067
	1-1-1	0.8068	0.9597	1.1057	1.1628	5.9845	6.4243	6.615	6.5761
	2-2-1	0.9338	1.1028	1.2584	1.3349	6.3485	6.4606	6.194	6.0005
	1-8-1	0.5541	0.8466	1.2147	1.4443	4.6512	5.8802	6.3579	6.2968
25	2-1-2	2.4161	2.6738	2.9015	2.9934	6.3056	6.5505	6.6869	6.7034
	1-1-1	2.2182	2.6359	3.0298	3.1793	5.9786	6.4198	6.609	6.5663
	2-2-1	2.5643	3.0147	3.3936	3.5356	6.343	6.4494	6.1549	5.9230
	1-8-1	1.5180	2.3169	3.2765	3.689	4.6408	5.8674	6.3219	6.1527
45	2-1-2	7.7980	8.6274	9.3581	9.652	6.305	6.5509	6.6881	6.7047
	1-1-1	7.1580	8.5027	9.7641	10.2371	5.9768	6.4193	6.6094	6.5661
	2-2-1	8.2710	9.7063	10.8665	11.2379	6.3419	6.4476	6.1447	5.8988
	1-8-1	4.8916	7.4630	10.4927	11.5449	4.6368	5.8629	6.3109	6.1010
65	2-1-2	16.2553	17.983	19.5043	20.1155	6.3049	6.5511	6.6886	6.7053
	1-1-1	14.9205	17.7219	20.3464	21.3277	5.9764	6.4193	6.6098	6.5665
	2-2-1	17.2388	20.2215	22.6094	23.3412	6.3418	6.4475	6.1431	5.8942
	1-8-1	10.1931	15.5496	21.8323	23.8898	4.6358	5.862	6.3088	6.0899

Tab.11

$a h$	scheme	$w ∗$				$σ x ∗ (h / 2)$
$a h$	scheme	$k = 0.5$	$k = 2.5$	$k = 5.5$	$k = 9.5$	$k = 0.5$	$k = 2.5$	$k = 5.5$	$k = 9.5$
5	1-1-2	0.0305	0.0325	0.0347	0.0374	2.4846	2.5974	2.7286	2.8848
	1-1-1	0.0373	0.0406	0.0448	0.0504	2.6051	2.7543	2.9404	3.1836
	2-2-1	0.0444	0.0498	0.0571	0.0678	2.6459	2.8007	3.0011	3.2800
	1-6-1	0.0405	0.0445	0.0496	0.0564	2.6255	2.7572	2.9185	3.1251
30	1-1-2	0.8036	0.8552	0.9168	0.9921	2.4375	2.5461	2.6727	2.8238
	1-1-1	0.9119	0.9948	1.1029	1.2510	2.4950	2.6311	2.8017	3.0271
	2-2-1	0.9554	1.0553	1.1933	1.3994	2.4546	2.5721	2.7213	2.9263
	1-6-1	0.9096	0.9885	1.0912	1.2321	2.4831	2.5921	2.724	2.8919
70	1-1-2	4.3360	4.6143	4.947	5.3542	2.4358	2.5443	2.6707	2.8215
	1-1-1	4.9072	5.3533	5.9353	6.7345	2.4914	2.6270	2.7970	3.0214
	2-2-1	5.1152	5.6463	6.3798	7.4766	2.4492	2.5654	2.7130	2.9153
	1-6-1	4.8788	5.2999	5.8479	6.6002	2.479	2.5873	2.7183	2.8847
95	1-1-2	7.9786	8.4909	9.1032	9.8526	2.4357	2.5441	2.6705	2.8212
	1-1-1	9.0272	9.848	10.9188	12.3892	2.4911	2.6266	2.7965	3.0208
	2-2-1	9.405	10.3807	11.7283	13.7437	2.4486	2.5647	2.7121	2.9141
	1-6-1	8.9720	9.7459	10.7531	12.1361	2.4786	2.5868	2.7177	2.8840

Tab.12

Fig.12

power law-index	displacement and stress	value
power law-index	displacement and stress	$12 × 12$	$14 × 14$	$16 × 16$	$18 × 18$	$20 × 20$
$k = 0.5$	$w ∗$	0.1563	0.1559	0.1556	0.1556	0.1556
	$σ x ∗ (h / 2)$	1.5970	1.5965	1.5962	1.5962	1.5962
	$σ z ∗ (h / 2)$	–0.0161	–0.0157	–0.0154	–0.0154	–0.0154
	$τ x z ∗ (0)$	–0.3376	–0.3371	–0.3367	–0.3367	–0.3367
$k = 1$	$w ∗$	0.1769	0.1765	0.1763	0.1763	0.1763
	$σ x ∗ (h / 2)$	1.6900	1.6906	1.6901	1.6900	1.6900
	$σ z ∗ (h / 2)$	0.0183	0.0180	0.0178	0.0177	0.0177
	$τ x z ∗ (0)$	–0.2857	–0.2853	–0.2850	–0.2849	–0.2849
$k = 2$	$w ∗$	0.1990	0.1990	0.1991	0.1990	0.1990
	$σ x ∗ (h / 2)$	1.7621	1.7617	1.7614	1.7613	1.7613
	$σ z ∗ (h / 2)$	0.0432	0.0428	0.0424	0.0424	0.0424
	$τ x z ∗ (0)$	–0.1859	–0.1855	–0.1851	–0.1850	–0.1850
$k = 5$	$w ∗$	0.2201	0.2199	0.2197	0.2196	0.2196
	$σ x ∗ (h / 2)$	1.7743	1.7739	1.7735	1.7735	1.7735
	$σ z ∗ (h / 2)$	0.0289	0.0285	0.0281	0.0280	0.0280
	$τ x z ∗ (0)$	–0.0420	–0.0416	–0.0412	–0.0411	–0.0411
$k = 10$	$w ∗$	0.2275	0.2270	0.2268	0.2267	0.2267
	$σ x ∗ (h / 2)$	1.7586	1.7581	1.7576	1.7576	1.7576
	$σ z ∗ (h / 2)$	–0.0021	–0.0015	–0.0011	–0.0010	–0.0010
	$τ x z ∗ (0)$	–0.0128	–0.0123	–0.0120	–0.0119	–0.0119

Tab.13

Fig.13

porosity coefficient	displacement and stress	value
porosity coefficient	displacement and stress	$12 × 12$	$14 × 14$	$16 × 16$	$18 × 18$	$20 × 20$
$ξ = 0.05$	$w ∗$	0.2876	0.2870	0.2865	0.2864	0.2864
	$σ x ∗ (h / 2)$	0.7686	0.7680	0.7675	0.7674	0.7674
	$σ z ∗ (h / 2)$	–0.0397	–0.0392	–0.0389	–0.0388	–0.0388
	$τ x z ∗ (0)$	0.0128	0.0122	0.0118	0.0117	0.0117
$ξ = 0.15$	$w ∗$	0.3130	0.3126	0.3121	0.3121	0.3121
	$σ x ∗ (h / 2)$	0.7991	0.7987	0.7983	0.7982	0.7982
	$σ z ∗ (h / 2)$	–0.0350	–0.0345	–0.0342	–0.0341	–0.0341
	$τ x z ∗ (0)$	0.0084	0.0079	0.0076	0.0075	0.0075
$ξ = 0.25$	$w ∗$	0.3470	0.3466	0.3461	0.3460	0.3460
	$σ x ∗ (h / 2)$	0.8365	0.8360	0.8355	0.8354	0.8354
	$σ z ∗ (h / 2)$	–0.0272	–0.0267	–0.0263	–0.0263	–0.0263
	$τ x z ∗ (0)$	0.3471	0.3465	0.3461	0.3460	0.3460
$ξ = 0.35$	$w ∗$	0.3945	0.3940	0.3936	0.3935	0.3935
	$σ x ∗ (h / 2)$	0.8837	0.8832	0.8827	0.8827	0.8827
	$σ z ∗ (h / 2)$	–0.0141	–0.0137	–0.0133	–0.0132	–0.0132
	$τ x z ∗ (0)$	–0.0029	–0.0024	–0.0019	–0.0018	–0.0018
$ξ = 0.45$	$w ∗$	0.4673	0.4668	0.4664	0.4663	0.4663
	$σ x ∗ (h / 2)$	0.9493	0.9489	0.9485	0.9485	0.9485
	$σ z ∗ (h / 2)$	0.0114	0.0107	0.0101	0.0100	0.0100
	$τ x z ∗ (0)$	–0.0080	–0.0073	–0.0069	–0.0068	–0.0068

Tab.14

Fig.14

$a h$	scheme	$w ∗$				$σ x ∗ (h / 2)$
$a h$	scheme	$k = 0.5$	$k = 1.5$	$k = 5.5$	$k = 8.5$	$k = 0.5$	$k = 1.5$	$k = 5.5$	$k = 8.5$
15	2-1-2	0.1198	0.1323	0.1448	0.1472	1.9343	2.0125	2.0664	2.0719
	1-1-1	0.1104	0.1308	0.1528	0.1568	1.8389	1.9771	2.0566	2.0576
	2-2-1	0.1275	0.1507	0.179	0.1867	1.9545	2.0177	1.9894	1.9789
	1-8-1	0.0774	0.1172	0.1789	0.2003	1.4551	1.8269	2.0487	2.0761
25	2-1-2	0.3216	0.3542	0.3860	0.3918	1.9076	1.9744	2.0157	2.0187
	1-1-1	0.2962	0.3495	0.4040	0.4133	1.8193	1.9404	1.9979	1.9947
	2-2-1	0.3404	0.3965	0.4501	0.4608	1.9218	1.9632	1.9058	1.8871
	1-8-1	0.2054	0.3097	0.4466	0.4772	1.4480	1.8032	1.9732	1.9803
55	2-1-2	1.5322	1.6856	1.8328	1.8589	1.8937	1.9538	1.9872	1.9886
	1-1-1	1.4107	1.6613	1.9112	1.9516	1.8093	1.9206	1.9642	1.9581
	2-2-1	1.6179	1.8705	2.076	2.104	1.9044	1.9325	1.8578	1.8351
	1-8-1	0.9731	1.4646	2.0519	2.1366	1.4450	1.7915	1.9327	1.9309
85	2-1-2	3.6507	4.0154	4.3645	4.4264	1.8915	1.9504	1.9825	1.9836
	1-1-1	3.3611	3.9569	4.5485	4.6434	1.8077	1.9174	1.9585	1.9518
	2-2-1	3.8534	4.4501	4.921	4.9794	1.9016	1.9274	1.8495	1.8261
	1-8-1	2.3166	3.4856	4.861	5.0403	1.4445	1.7896	1.9259	1.9227

Tab.15

$a h$	scheme	$w ∗$				$σ x ∗ (h / 2)$
$a h$	scheme	$ξ = 0.1$	$ξ = 0.2$	$ξ = 0.3$	$ξ = 0.4$	$ξ = 0.1$	$ξ = 0.2$	$ξ = 0.3$	$ξ = 0.4$
45	$2 - 1 - 2$	0.8525	0.9221	1.0107	1.1283	1.6144	1.6928	1.7883	1.9100
	$1 - 1 - 1$	0.8607	0.9339	1.0283	1.1562	1.6015	1.6778	1.7713	1.8918
	$2 - 2 - 1$	0.8907	0.9741	1.0868	1.2504	1.5934	1.6561	1.7322	1.8312
	$1 - 8 - 1$	0.8891	0.9636	1.0607	1.1951	1.6648	1.7271	1.7999	1.8891
65	$2 - 1 - 2$	1.7718	1.9165	2.1006	2.3452	1.6121	1.6899	1.7847	1.9052
	$1 - 1 - 1$	1.7881	1.9400	2.1361	2.4017	1.5991	1.6747	1.7673	1.8864
	$2 - 2 - 1$	1.8469	2.0190	2.2516	2.5891	1.5908	1.6527	1.7274	1.8238
	$1 - 8 - 1$	1.8442	1.9980	2.1987	2.4760	1.6628	1.7244	1.7963	1.8839
80	$2 - 1 - 2$	2.6806	2.8995	3.1781	3.5482	1.6114	1.689	1.7835	1.9036
	$1 - 1 - 1$	2.7049	2.9347	3.2313	3.6332	1.5983	1.6737	1.766	1.8846
	$2 - 2 - 1$	2.7922	3.0521	3.4032	3.9126	1.5900	1.6516	1.7259	1.8214
	$1 - 8 - 1$	2.7885	3.0208	3.3237	3.7424	1.6621	1.7236	1.7951	1.8822
100	$2 - 1 - 2$	4.1849	4.5266	4.9615	5.5394	1.6108	1.6883	1.7827	1.9025
	$1 - 1 - 1$	4.2225	4.5811	5.0441	5.6714	1.5978	1.673	1.7651	1.8834
	$2 - 2 - 1$	4.3569	4.762	5.3093	6.1032	1.5894	1.6508	1.7248	1.8197
	$1 - 8 - 1$	4.3514	4.7135	5.1858	5.8385	1.6617	1.7230	1.7943	1.8810

Tab.16

Fig.15

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