E.6 Acoustic Solid Element

E.6.1 Convective Natural Frequencies in a Rigid Rectangular Tank

For a rigid rectangular tank containing a compressible fluid with length \(\small L_{x}\), width \(\small L_{y}\), and height \(\small H\), where the top surface is subject to a zero-pressure condition and all other surfaces are subject to a zero-flux condition, the following eigenvalues can be derived analytically.

\[ \small \begin{array}{r} \omega_{lmn} = c\sqrt{\alpha_{l}^{2} + \beta_{m}^{2} + \gamma_{n}^{2}}\tag{E.6.1} \end{array} \]

where

\(\small c\) is acoustic wave velocity, 1480 m/s/s if water

\(\small \alpha_{l} = \frac{l\pi}{L_{x}},\ \beta_{m} = \frac{m\pi}{L_{y}},\gamma_{n} = \frac{(2n - 1)\pi}{2H}\)

\(\small l,m = 0,1,2,\ldots,\ \ n = 1,2,\ldots\)

For a rectangular water tank with dimensions of 40 m (length), 30 m (width), and 20 m (height), a natural frequency analysis was performed under three-dimensional conditions using three different meshes, and the results were compared with the analytical solutions.

Figure E.6.1 Contour Plot of Mode Shapes

Table E.6.1 Natural Frequencies of 3D Rigid Rectangular Tank

l	m	n	Analytic solution					Numerical solution
l	m	n	alpha	beta	gamma	wlnm	frq	8x4	12x6	16x8	16x12-P6	16x12-P8	16x12-P8+T4
0	1	0	0	0	0.07854	116.2389282	18.5000	18.6191	18.5529	18.5297	18.5242	18.5259
1	0	0	0.07854	0	0	164.3866687	26.1630	26.3314	26.2377	26.2054	26.1758	26.1755
2	0	0	0.15708	0	0	193.7315497	30.8333	31.131	30.9654	30.9076	30.9008	31.0321
0	2	0	0	0.15708	0	225.2967575	35.9631	36.2474	36.0653	35.9496	35.9369	36.3651
1	1	0	0.07854	0.07854	0	259.918145	41.3673	42.2775	41.7704	41.5937	41.5099	42.0162
0	0	1	0	0	0.07854	302.6185304	48.1632	49.0003	48.5738	48.3938	48.6972	49.1273
2	1	0	0.15708	0.07854	0	343.0194616	53.1680	54.3603	53.2028	53.2827	53.4984
1	2	0	0.07854	0.15708	0	348.7167845	55.0038	57.3957	56.7676	56.3055	56.1504	56.2974
0	1	1	0	0.07854	0.07854	350.865724	55.8415	56.9935	56.3651	56.1431	56.1939	56.1939
1	1	1	0.07854	0.07854	0.07854	357.5797658	58.5021	61.6157	59.8217	59.0764	59.5911	59.91
3	0	0	0.235619	0	0	367.5797658	58.5021	61.6157	59.8217	59.0764	59.5911	59.91
2	2	0	0.15708	0.15708	0	381.8089616	60.7346	63.6522	61.4999	61.5399	62.0894
3	1	0	0.235619	0.10472	0.07854	398.9176709	63.49972	66.4765	64.81	64.2307	64.6278	65.3594
1	3	0	0.07854	0.235619	0.10472	399.9176709	63.49972	66.4765	64.81	64.2307	64.6278	65.3594
2	1	2	0.15708	0.20944	0.07854	404.5233462	64.3819	66.7128	65.4134	64.9680	65.1404	65.5128
2	0	2	0.15708	0	0.235619	419.1054158	66.7027	69.9335	68.1335	67.5055	67.0949	69.9077

For a rectangular water tank with dimensions of 40 m (width) and 20 m (height), a natural frequency analysis was performed under two-dimensional conditions using three different meshes, and the results were compared with the analytical solutions. In the analytical solutions, the case of \(\small m = 0\) was always considered.

Table E.6.2 Natural Frequencies of 2D Rigid Rectangular Tank

l	m	n	Analytic solution					Numerical solution
l	m	n	alpha	beta	gamma	wlnm	frq	8x4	12x6	16x8	16x8-T
1	0	0	0	0	0.07854	116.2389282	18.5000	18.6191	18.5529	18.5297	18.5296
1	0	1	0.07854	0	0.07854	164.3866687	26.1630	26.3314	26.2377	26.2054	26.2881
2	0	1	0.15708	0	0.07854	259.918145	41.3673	42.2775	41.7704	41.5937	41.8000
0	2	0	0	0.235619	0	348.7167845	55.0000	58.7365	56.9352	56.3055	56.2984
3	0	0	0.235619	0	0.07854	367.5797658	58.5021	61.617	59.8217	59.2761	59.5851
2	1	0	0.07854	0.235619	0	419.1054158	66.7027	69.934	68.6325	67.6535	68.6724
4	0	0	0.314159	0	0.07854	479.2653787	76.2775	83.066	79.6017	78.1423	78.3488
3	0	2	0.235619	0	0.235619	493.1600601	78.4889	83.694	80.5185	79.6281	82.91

Input File

tank2d-8x4.inp : 2 dimensional rectangular tank, 8x4 elements, AC2D4
tank2d-12x6.inp : 2 dimensional rectangular tank, 12x6 elements, AC2D4
tank2d-16x8.inp : 2 dimensional rectangular tank, 16x8 elements, AC2D4
tank2d-16x8-T3.inp : 2 dimensional rectangular tank, 16x8*2 elements, AC2D3
tank3d-8x6x4.inp : 3 dimensional rectangular tank, 8x6x4 elements, AC3D8
tank3d-12x9x6.inp : 3 dimensional rectangular tank, 12x9x6 elements, AC3D8
tank3d-16x12x8.inp : 3 dimensional rectangular tank, 16*12*8 elements, AC3D8
tank3d-16x12x8-T4.inp : 3 dimensional rectangular tank, 16*12*8*6 elements, AC3D4
tank3d-16x12x8-P6.inp : 3 dimensional rectangular tank, 16*12*8*2 elements, AC3D6

E.6.2 Sloshing Natural Frequencies in a Rigid Rectangular Tank

For an incompressible fluid, the natural frequencies of free-surface sloshing in a rectangular tank can be derived analytically (Lamb, 1945).

\[ \small \begin{array}{r} \omega_{mn} = \sqrt{gk_{mn}\tanh\left( k_{mn}H \right)}\tag{E.6.2} \end{array} \]

where

\[ \small \begin{array}{r} \ k_{mn} = \sqrt{\alpha_{m}^{2} + \beta_{n}^{2}},\ \ \alpha_{m} = \frac{m\pi}{L_{x}},\ \ \beta_{n} = \frac{n\pi}{L_{y}},\ where\ \ l,m = 0,1,2,\ldots,\ \ n = 1,2,\ldots\tag{E.6.3} \end{array} \]

Housner (1957) proposed the following approximate formula for the fundamental frequency.

\[ \small \begin{array}{r} \omega^{2} = \frac{g}{L/2}\sqrt{\frac{5}{2}}\tanh\left( \sqrt{\frac{5}{2}}\frac{H}{L/2}\ \right)\tag{E.6.4} \end{array} \]

Here, \(\small L\) is the length in the direction of excitation.

Table E.6.3 Natural Frequencies of Sloshing Modes in 3D Rigid Rectangular Tank

m	n	kmn	f	Numerical solution
m	n	kmn	f	10x30x6	20x60x11	20x60x11-P6	20x60x11	20x60x11-T4
0	0	0.000	0.000	0	0	0	0
0	1	0.053	0.084	0.084416	0.084368	0.084368	0.084372
0	2	0.107	0.149	0.149068	0.148796	0.148783	0.148874
1	0	0.160	0.194	0.195056	0.19439	0.194329	0.194702
0	3	0.160	0.194	0.195056	0.19439	0.194343	0.195071
1	1	0.169	0.200	0.201226	0.200551	0.200536	0.201349
1	2	0.193	0.216	0.216947	0.216181	0.216238	0.217327
0	4	0.214	0.229	0.230264	0.229004	0.228874	0.229771
1	3	0.227	0.236	0.237313	0.236266	0.236372	0.237929
0	5	0.267	0.257	0.259196	0.257612	0.257536	0.258952
1	4	0.267	0.257	0.259909	0.257787	0.257734	0.260181
1	5	0.312	0.278	0.281321	0.278901	0.278968	0.281839
0	6	0.321	0.282	0.286557	0.283247	0.282861	0.285658
2	0	0.321	0.282	0.286557	0.283247	0.282862	0.287921
2	1	0.325	0.284	0.28854	0.285216	0.284879	0.290092
2	2	0.338	0.290	0.294289	0.290893	0.290692	0.296245

Table E.6.4 Natural Frequencies of Sloshing Modes in 2D Rigid Rectangular Tank

m	km	f	Numerical solution
m	km	f	30x6	60x11	60x11-T
0	0.000	0.000	0	0	0
1	0.053	0.084	0.084416	0.084368	0.084372
2	0.107	0.149	0.149068	0.148796	0.148875
3	0.160	0.194	0.195056	0.19439	0.194722
4	0.214	0.229	0.230264	0.229004	0.229799
5	0.267	0.257	0.259909	0.257787	0.259258
6	0.321	0.282	0.286557	0.283247	0.285611
7	0.374	0.305	0.311438	0.306572	0.310062
8	0.427	0.326	0.335203	0.323875	0.333244
9	0.481	0.346	0.358247	0.349017	0.355533
10	0.534	0.364	0.380845	0.368738	0.377183
11	0.588	0.382	0.403204	0.387712	0.398377

Input File

sloshing2d-30x6.inp : 2 dimensional rectangluar tank, 30*6 elements, AC2D4
sloshing2d-60x11.inp : 2 dimensional rectangluar tank, 60*11 elements, AC2D4
sloshing2d-60x11-T3.inp : 2 dimensional rectangluar tank, 60*11*2 elements, AC2D3
sloshing3d-10x30x6.inp : 3 dimensional rectangluar tank, 10*30*6 elements, AC3D8
sloshing3d-20x60x11.inp : 3 dimensional rectangluar tank, 20*60*11 elements, AC3D8
sloshing3d-20x60x11-T4.inp : 3 dimensional rectangluar tank, 20*60*11*6 elements, AC3D4
sloshing3d-20x60x11-P6.inp : 3 dimensional rectangluar tank, 20*60*11*6 elements, AC3D6

E.6.3 Natual Frequencies in a Rigid Cylinderical Tank

For a cylindrical tank, various natural frequencies were calculated. The exact solution for sloshing is given as follows.

(1) Veletos (1984) proposed the following equation.

\[ \small \begin{array}{r} \omega_{j}^{2} = \frac{\lambda_{j}g}{R}\tanh\left( \lambda_{j}\frac{H}{R}\ \right)\tag{E.6.5} \end{array} \]

여기에서 \(\small \lambda_{j}\)는 \(\small \frac{dJ_{1}(x)}{dx} = 0\)의 해이며, 저차 3개 값은 \(\small \lambda_{1} = 1.8411837813406593\), \(\small \lambda_{2} = 5.331442773525033\), \(\small \lambda_{3} = 8.536316366346286\) 이다.

(2) Housner (1957) proposed the following approximate formula for the fundamental frequency of sloshing.

\[ \small \begin{array}{r} \omega^{2} = \frac{g}{R}\sqrt{\frac{27}{8}}\ \tanh\left( \sqrt{\frac{27}{8}}\frac{H}{R}\ \right)\tag{E.6.6} \end{array} \]

All sloshing frequencies are based on the assumption of incompressible water. When compressibility is considered, however, the exact solution for the impulsive component is the same as that for a rectangular tank with \(L_{x} = L_{y} = 2R\).

Solutions were compared for both a wide cylinder and a tall cylinder. The wide cylinder has \(R = 18.3\ \text{m}\) and \(H = 12.2\ \text{m}\), while the tall cylinder has \(R = 10\ \text{m}\) and \(H = 30\ \text{m}\). Two mesh types were prepared for each case, and the fundamental frequencies were compared.

Table E.6.5 Sloshing Freqencies of wideCylinder Model

Numerical Solution		Analytic Solution
Model	Natural Frequency	Analytic Solution
wideCylinderIncomp-15x10	0.145258 Hz	0.145075 Hz(Veletsos), 0.144846 Hz((Housner)
wideCylinder-15x10	0.145256 Hz
wideCylinderIncomp-30x15	0.14516 Hz
wideCylinder-30x15	0.145158 Hz

Table E.6.6 Impulsive Frequencies of wideCylinder Model

Numerical Solution		Analytic Solution
Model	Natural Frequency	Analytic Solution
wideCylinder-15x10	30.3591 Hz	30.3279 Hz
wideCylinder-30x15	30.3417 Hz	30.3279 Hz

Figure E.6.2 Sloshing Mode of wideCylinder-20.x10

Figure E.6.3 Impulsive Mode of wideCylinder-20.x10

Table E.6.7 Solshing Freuencies of tallCylinder Model

Numerical Solution		Analytic Solution
Model	Natural Frequency	Analytic Solution
tallCylinderIncomp-10x10	0.21567 Hz	0.21389 Hz(Veletsos), 0.213656 Hz((Housner)
tallCylinder-10x10	0.215668 Hz
tallCylinderIncomp-15x15	0.21567 Hz
tallCylinder-15x15	0.215668 Hz

Table E.6.8 Impulsive Frequencies of tallCylinder Model

Numerical Solution		Analytic Solution
Model	Natural Frequency	Analytic Solution
tallCylinder-10x10	12.346 Hz	12.3333 Hz
tallCylinder-15x15	12.346 Hz	12.3333 Hz

Figure E.6.4 Sloshing Mode of tallCylinder-10.x10

Figure E.6.5 Impulsive Mode of tallCylinder-10.x10

Input File

wideCylinder-20x10.inp : Wide Cylinder with 20*10 elements
wideCylinderIncomp-20x10.inp : Wide Cylinder with 20*10 elements, Incompressible Water
wideCylinder-30x15.inp : Wide Cylinder with 30*15 elements
wideCylinderIncomp-30x15.inp : Wide Cylinder with 30*15 elements, Incompressible Water
tallCylinder-10x10.inp : Tall Cylinder with 10*10 elements
tallCylinderIncomp-10x10.inp : Tall Cylinder with 10*10 elements, Incompressible Water
tallCylinder-15x15.inp : Tall Cylinder with 15*15 elements
tallCylinderIncomp-15x15.inp : Tall Cylinder with 15*15 elements, Incompressible Water

References

Lamb, H. (1945) Hydrodynamics, 6^th Edition, Dover Publications.
Veletos, A.S.(1984), Seismic reesponse and design of liquid storage tanks, Guidelines for the seismic design of oil and gas pipeline systems, ASCE
Housner, G. W. (1957). Dynamic pressures on accelerated fluid containers. Bulletin of the seismological society of America, 47(1), 15-35.