On the Validity of Flat-Plate Finite Strip Approximation of Circular Cylindrical Shells

Table 1

(a) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode 𝑠=1 (𝑅/𝑡 = 100, 𝐿/𝑅 = 10, 𝐸=1, 𝜌 = 1, and 𝜇 = 0.3). (b) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode 𝑠=2 (𝑅/𝑡 = 100, 𝐿/𝑅 = 10, 𝐸=1, 𝜌 = 1, and 𝜇 = 0.3). (c) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode 𝑠=3 (𝑅/𝑡 = 100, 𝐿/𝑅 = 10, 𝐸=1, 𝜌=1, and 𝜇=0.3). (d) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode 𝑠=1 (𝑅/𝑡 = 100, 𝐿/𝑅 = 2, 𝐸=1, 𝜌=1, and 𝜇=0.3). (e) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode 𝑠=2 (𝑅/𝑡 = 100, 𝐿/𝑅 = 2, 𝐸=1, 𝜌=1, and 𝜇=0.3). (f) Comparison of exact flat-plate finite strip results with analytical results [19] for the first 25 circumferential modes and longitudinal mode 𝑠=3 (𝑅/𝑡 = 100, 𝐿/𝑅 = 2, 𝐸=1, 𝜌=1, and 𝜇 = 0.3).

(a)  

Mode	Reference [19]			Finite strip			% Error
	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3
1	0.003838	0.477161	2.247847	0.003839	0.477154	2.247864	0.0024	−0.0014	0.0008
2	0.00053	1.60297	5.576039	0.000529	1.602897	5.57615	−0.0584	−0.0046	0.002
3	0.000644	3.513856	11.08322	0.000644	3.51365	11.08351	−0.0644	−0.0059	0.0026
4	0.001998	6.200774	18.78118	0.001997	6.200375	18.78173	−0.0205	−0.0064	0.0029
5	0.005126	9.659683	28.67419	0.005125	9.659031	28.67507	−0.0061	−0.0067	0.0031
6	0.01098	13.88911	40.76376	0.01098	13.88815	40.76505	−0.0009	−0.0069	0.0032
7	0.020762	18.88844	55.0505	0.020762	18.88711	55.05228	0.0013	−0.007	0.0032
8	0.035896	24.65741	71.53469	0.035897	24.65566	71.53703	0.0024	−0.0071	0.0033
9	0.058031	31.19588	90.21647	0.058033	31.19365	90.21944	0.0029	−0.0072	0.0033
10	0.089033	38.50377	111.0959	0.089036	38.501	111.0996	0.0033	−0.0072	0.0033
11	0.130991	46.58103	134.1731	0.130996	46.57767	134.1776	0.0035	−0.0072	0.0033
12	0.186212	55.42764	159.448	0.186218	55.42363	159.4533	0.0036	−0.0072	0.0033
13	0.257222	65.04358	186.9207	0.257232	65.03886	186.9269	0.0037	−0.0073	0.0034
14	0.34677	75.42884	216.5911	0.346783	75.42335	216.5984	0.0038	−0.0073	0.0034
15	0.457822	86.58342	248.4593	0.457839	86.5771	248.4677	0.0038	−0.0073	0.0034
16	0.593564	98.5073	282.5253	0.593586	98.5001	282.5348	0.0038	−0.0073	0.0034
17	0.757403	111.2005	318.7891	0.757432	111.1923	318.7998	0.0039	−0.0073	0.0034
18	0.952964	124.663	357.2506	0.953001	124.6538	357.2627	0.0039	−0.0073	0.0034
19	1.184095	138.8947	397.91	1.184141	138.8845	397.9235	0.0039	−0.0073	0.0034
20	1.45486	153.8958	440.7672	1.454917	153.8845	440.7821	0.0039	−0.0074	0.0034
21	1.769544	169.6662	485.8221	1.769614	169.6537	485.8386	0.0039	−0.0074	0.0034
22	2.132654	186.2059	533.0749	2.132738	186.1921	533.0929	0.004	−0.0074	0.0034
23	2.548913	203.5148	582.5254	2.549015	203.4998	582.5452	0.004	−0.0074	0.0034
24	3.023267	221.5931	634.1737	3.023388	221.5767	634.1953	0.004	−0.0074	0.0034
25	3.56088	240.4406	688.0199	3.561022	240.4228	688.0433	0.004	−0.0074	0.0034

(b)  

Mode	Reference [19]			Finite strip			% Error
	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3
1	0.038162	0.714701	2.415242	0.038162	0.714689	2.415267	0.0025	−0.0016	0.001
2	0.00641	1.786224	5.826179	0.006409	1.786145	5.8263	−0.0173	−0.0044	0.0021
3	0.002154	3.667229	11.36764	0.002152	3.667019	11.36794	−0.0788	−0.0057	0.0026
4	0.002579	6.339256	19.08146	0.002577	6.338854	19.08202	−0.072	−0.0063	0.0029
5	0.005463	9.790132	28.98279	0.005461	9.789479	28.98368	−0.0335	−0.0067	0.0031
6	0.011266	14.01483	41.07719	0.011265	14.01387	41.07849	−0.0147	−0.0069	0.0032
7	0.02107	19.01119	55.36697	0.021068	19.00986	55.36875	−0.0062	−0.007	0.0032
8	0.036261	24.77816	71.85317	0.03626	24.77641	71.85552	−0.002	−0.0071	0.0033
9	0.058475	31.31524	90.53636	0.058475	31.31301	90.53934	0.0002	−0.0071	0.0033
10	0.089573	38.62211	111.4168	0.089574	38.61935	111.4205	0.0015	−0.0072	0.0033
11	0.13164	46.69862	134.4948	0.131643	46.69526	134.4992	0.0023	−0.0072	0.0033
12	0.186982	55.54465	159.7702	0.186987	55.54064	159.7756	0.0027	−0.0072	0.0033
13	0.258126	65.16014	187.2434	0.258134	65.15542	187.2496	0.0031	−0.0073	0.0033
14	0.347819	75.54504	216.9142	0.347831	75.53955	216.9214	0.0033	−0.0073	0.0034
15	0.459027	86.69932	248.7827	0.459043	86.69301	248.791	0.0035	−0.0073	0.0034
16	0.594936	98.62296	282.8489	0.594958	98.61576	282.8584	0.0036	−0.0073	0.0034
17	0.758954	111.3159	319.1129	0.758982	111.3078	319.1236	0.0037	−0.0073	0.0034
18	0.954705	124.7783	357.5746	0.95474	124.7691	357.5867	0.0037	−0.0073	0.0034
19	1.186036	139.0099	398.2341	1.186081	138.9997	398.2476	0.0038	−0.0073	0.0034
20	1.457012	154.0109	441.0914	1.457068	153.9995	441.1063	0.0038	−0.0074	0.0034
21	1.771919	169.7811	486.1464	1.771987	169.7686	486.1629	0.0039	−0.0074	0.0034
22	2.135261	186.3207	533.3993	2.135344	186.307	533.4174	0.0039	−0.0074	0.0034
23	2.551765	203.6296	582.8499	2.551864	203.6146	582.8697	0.0039	−0.0074	0.0034
24	3.026373	221.7078	634.4983	3.026492	221.6914	634.5199	0.0039	−0.0074	0.0034
25	3.564252	240.5553	688.3445	3.564392	240.5375	688.3679	0.0039	−0.0074	0.0034

(c)  

Mode	Reference [19]			Finite strip			% Error
	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3
1	0.117569	1.030781	2.751855	0.117572	1.030763	2.751891	0.0026	−0.0018	0.0013
2	0.0259	2.065125	6.259916	0.025898	2.06504	6.260051	−0.0083	−0.0041	0.0022
3	0.007829	3.914095	11.84728	0.007825	3.91388	11.84759	−0.0483	−0.0055	0.0026
4	0.004751	6.566632	19.58415	0.004747	6.566226	19.58472	−0.0902	−0.0062	0.0029
5	0.006564	10.00599	29.49815	0.006559	10.00533	29.49906	−0.0669	−0.0066	0.0031
6	0.012017	14.22358	41.60012	0.012013	14.22261	41.60143	−0.0356	−0.0068	0.0032
7	0.021736	19.21531	55.89471	0.021732	19.21398	55.8965	−0.0182	−0.0069	0.0032
8	0.036963	24.97915	72.38416	0.036959	24.97739	72.38651	−0.0092	−0.007	0.0033
9	0.059275	31.514	91.06963	0.059273	31.51176	91.07262	−0.0043	−0.0071	0.0033
10	0.090513	38.81925	111.9518	0.090512	38.81648	111.9555	−0.0015	−0.0071	0.0033
11	0.13275	46.89453	135.0309	0.132751	46.89116	135.0354	0.0002	−0.0072	0.0033
12	0.188288	55.73962	160.3074	0.188291	55.7356	160.3127	0.0013	−0.0072	0.0033
13	0.25965	65.35437	187.7812	0.259655	65.34964	187.7875	0.002	−0.0072	0.0033
14	0.349581	75.73868	217.4526	0.34959	75.73318	217.4599	0.0025	−0.0073	0.0034
15	0.461047	86.89248	249.3216	0.46106	86.88616	249.33	0.0029	−0.0073	0.0034
16	0.597234	98.81572	283.3882	0.597253	98.80852	283.3978	0.0031	−0.0073	0.0034
17	0.761547	111.5084	319.6525	0.761572	111.5002	319.6633	0.0033	−0.0073	0.0034
18	0.957613	124.9704	358.1146	0.957646	124.9613	358.1267	0.0034	−0.0073	0.0034
19	1.189277	139.2018	398.7743	1.189319	139.1916	398.7878	0.0035	−0.0073	0.0034
20	1.460605	154.2026	441.6318	1.460658	154.1913	441.6467	0.0036	−0.0073	0.0034
21	1.775882	169.9727	486.687	1.775947	169.9602	486.7035	0.0037	−0.0074	0.0034
22	2.139612	186.5121	533.94	2.139692	186.4984	533.9581	0.0037	−0.0074	0.0034
23	2.556522	203.8209	583.3908	2.556619	203.8058	583.4106	0.0038	−0.0074	0.0034
24	3.031555	221.8989	635.0393	3.031671	221.8825	635.0609	0.0038	−0.0074	0.0034
25	3.569876	240.7463	688.8856	3.570014	240.7285	688.909	0.0039	−0.0074	0.0034

(d)  

Mode	Reference [19]			Finite strip			% Error
	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3
1	0.360519	1.711392	4.171047	0.36053	1.711367	4.171102	0.0032	−0.0015	0.0013
2	0.117813	2.821872	7.754094	0.117809	2.821778	7.754261	−0.0035	−0.0033	0.0021
3	0.042117	4.649301	13.42077	0.042108	4.649077	13.42111	−0.0236	−0.0048	0.0026
4	0.019279	7.270339	21.2091	0.019267	7.269925	21.2097	−0.0624	−0.0057	0.0028
5	0.01385	10.68516	31.15515	0.013838	10.68449	31.15608	−0.092	−0.0062	0.003
6	0.016449	14.88542	43.27761	0.016436	14.88445	43.27896	−0.0782	−0.0065	0.0031
7	0.025049	19.86499	57.58586	0.025036	19.86365	57.58769	−0.0504	−0.0067	0.0032
8	0.039939	25.62014	74.08476	0.039927	25.61838	74.08715	−0.0303	−0.0069	0.0032
9	0.062317	32.14864	92.777	0.062306	32.14641	92.78003	−0.0181	−0.007	0.0033
10	0.093854	39.44914	113.6641	0.093844	39.44637	113.6679	−0.0107	−0.007	0.0033
11	0.136544	47.5208	136.7471	0.136536	47.51743	136.7516	−0.0062	−0.0071	0.0033
12	0.192647	56.36306	162.0265	0.192641	56.35903	162.0318	−0.0032	−0.0071	0.0033
13	0.264666	65.97556	189.5026	0.264663	65.97083	189.5089	−0.0013	−0.0072	0.0033
14	0.355332	76.35806	219.1759	0.355333	76.35256	219.1832	4𝐸−05	−0.0072	0.0033
15	0.467605	87.51038	251.0464	0.46761	87.50406	251.0548	0.001	−0.0072	0.0034
16	0.604665	99.43241	285.1143	0.604675	99.4252	285.1238	0.0016	−0.0072	0.0034
17	0.769916	112.124	321.3796	0.769932	112.1159	321.3904	0.0021	−0.0073	0.0034
18	0.966981	125.5852	359.8425	0.967005	125.5761	359.8546	0.0025	−0.0073	0.0034
19	1.199705	139.8159	400.503	1.199739	139.8057	400.5165	0.0028	−0.0073	0.0034
20	1.472153	154.816	443.3611	1.472198	154.8047	443.3761	0.003	−0.0073	0.0034
21	1.78861	170.5856	488.4169	1.788667	170.5731	488.4334	0.0032	−0.0073	0.0034
22	2.153579	187.1245	535.6703	2.153651	187.1108	535.6885	0.0033	−0.0073	0.0034
23	2.571786	204.4329	585.1215	2.571875	204.4178	585.1413	0.0034	−0.0074	0.0034
24	3.048175	222.5106	636.7704	3.048283	222.4942	636.792	0.0035	−0.0074	0.0034
25	3.587911	241.3577	690.617	3.58804	241.3398	690.6405	0.0036	−0.0074	0.0034

(e)  

Mode	Reference [19]			Finite strip			% Error
	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3
1	0.785423	4.335855	12.10394	0.785472	4.335863	12.10402	0.0062	0.0002	0.0006
2	0.46872	5.6709	15.53683	0.468725	5.670845	15.53704	0.001	−0.001	0.0014
3	0.256867	7.642798	21.19587	0.256843	7.642607	21.19629	−0.0096	−0.0025	0.002
4	0.142906	10.30723	29.03288	0.142865	10.30684	29.03358	−0.0287	−0.0038	0.0024
5	0.086917	13.71786	39.03489	0.086867	13.71722	39.03594	−0.0566	−0.0047	0.0027
6	0.062743	17.89803	51.20571	0.06269	17.89707	51.20718	−0.0844	−0.0054	0.0029
7	0.057383	22.85481	65.55247	0.057328	22.85348	65.55443	−0.0948	−0.0058	0.003
8	0.065245	28.58905	82.08135	0.065191	28.58729	82.08388	−0.0836	−0.0061	0.0031
9	0.084499	35.09969	100.7969	0.084445	35.09745	100.8	−0.0638	−0.0064	0.0031
10	0.115233	42.38534	121.7022	0.11518	42.38257	121.7061	−0.0457	−0.0066	0.0032
11	0.158594	50.44476	144.7996	0.158543	50.44139	144.8043	−0.0321	−0.0067	0.0032
12	0.21637	59.27693	170.0905	0.216321	59.27291	170.096	−0.0224	−0.0068	0.0032
13	0.29079	68.88109	197.576	0.290744	68.87635	197.5825	−0.0157	−0.0069	0.0033
14	0.384424	79.25664	227.2569	0.384382	79.25113	227.2644	−0.0109	−0.0069	0.0033
15	0.500133	90.40314	259.1337	0.500095	90.3968	259.1423	−0.0074	−0.007	0.0033
16	0.641033	102.3202	293.2069	0.641001	102.313	293.2166	−0.0049	−0.0071	0.0033
17	0.810488	115.0077	329.4767	0.810463	114.9995	329.4876	−0.0031	−0.0071	0.0033
18	1.012094	128.4653	367.9434	1.012077	128.4561	367.9556	−0.0017	−0.0071	0.0033
19	1.249678	142.6929	408.6071	1.249671	142.6826	408.6207	−0.0006	−0.0072	0.0033
20	1.527292	157.6903	451.468	1.527295	157.679	451.4831	0.0002	−0.0072	0.0034
21	1.849211	173.4575	496.5262	1.849227	173.445	496.5429	0.0009	−0.0072	0.0034
22	2.219934	189.9945	543.7818	2.219965	189.9807	543.8001	0.0014	−0.0072	0.0034
23	2.64418	207.301	593.2348	2.644228	207.2859	593.2548	0.0018	−0.0073	0.0034
24	3.126891	225.3771	644.8854	3.126958	225.3607	644.9071	0.0022	−0.0073	0.0034
25	3.673229	244.2228	698.7335	3.673318	244.2049	698.7571	0.0024	−0.0073	0.0034

(f)  

Mode	Reference [19]			Finite strip			% Error
	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3	Root 1	Root 2	Root 3
1	0.908799	8.995999	25.62642	0.908914	8.996064	25.62654	0.0126	0.0007	0.0004
2	0.704499	10.29172	28.98691	0.704552	10.29174	28.98716	0.0075	0.0002	0.0009
3	0.497109	12.32846	34.57777	0.4971	12.32836	34.57824	−0.0018	−0.0008	0.0014
4	0.338551	15.07368	42.38017	0.338496	15.07339	42.38095	−0.0163	−0.0019	0.0018
5	0.233578	18.5403	52.37722	0.233492	18.53974	52.37837	−0.0368	−0.003	0.0022
6	0.171066	22.74951	64.55981	0.170961	22.74864	64.56141	−0.0613	−0.0038	0.0025
7	0.139392	27.71679	78.92533	0.139277	27.71553	78.92744	−0.0831	−0.0045	0.0027
8	0.130515	33.45085	95.47451	0.130393	33.44916	95.4772	−0.0932	−0.0051	0.0028
9	0.13989	39.9559	114.2094	0.139765	39.95372	114.2127	−0.0888	−0.0055	0.0029
10	0.165521	47.23361	135.1321	0.165396	47.23087	135.1361	−0.0753	−0.0058	0.003
11	0.207115	55.28438	158.2446	0.206991	55.28104	158.2494	−0.0597	−0.006	0.0031
12	0.265506	64.10807	183.5485	0.265385	64.10407	183.5543	−0.0458	−0.0062	0.0031
13	0.342301	73.70429	211.0453	0.342182	73.69957	211.0519	−0.0347	−0.0064	0.0032
14	0.439659	84.07261	240.7358	0.439544	84.06712	240.7434	−0.0261	−0.0065	0.0032
15	0.560166	95.21262	272.6208	0.560057	95.2063	272.6296	−0.0196	−0.0066	0.0032
16	0.706759	107.1239	306.7011	0.706655	107.1167	306.711	−0.0147	−0.0067	0.0032
17	0.882675	119.8063	342.977	0.882578	119.7981	342.9882	−0.011	−0.0068	0.0033
18	1.091425	133.2593	381.449	1.091336	133.2501	381.4615	−0.0081	−0.0069	0.0033
19	1.336774	147.4828	422.1174	1.336695	147.4726	422.1313	−0.0059	−0.0069	0.0033
20	1.622731	162.4767	464.9824	1.622664	162.4653	464.9977	−0.0041	−0.007	0.0033
21	1.953542	178.2407	510.0441	1.953488	178.2282	510.061	−0.0028	−0.007	0.0033
22	2.333682	194.7748	557.3029	2.333643	194.761	557.3214	−0.0017	−0.0071	0.0033
23	2.767854	212.0787	606.7587	2.767833	212.0637	606.7789	−0.0008	−0.0071	0.0033
24	3.260987	230.1526	658.4117	3.260985	230.1361	658.4337	−5𝐸−05	−0.0071	0.0033
25	3.818234	248.9961	712.2621	3.818254	248.9783	712.2859	0.0005	−0.0072	0.0033