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Treatment of hydrodynamic effects for toroidal containment vessels
Nuclear Engineeringand Design 53 (1979) 217-222 © North-Holland Publishing Company
TREATMENT OF HYDRODYNAMIC EFFECTS FOR TOROIDAL CONTAINMENT VESSELS Richard H. RUSH * and John E. JACKSON ** Qvil Engineering Branch, Division o f Engineering Design, Tennessee Valley Authority, Knoxville, TN, USA Received 21 December 1978
Hydrodynamic effects in liquid-shell systems may be characterized in terms of structural degrees of freedom alone if an ideal fluid is assumed. The hydrodynamic effects are modeled by means of a consistent (full) added mass matrix which is obtained via finite element methods, The procedure is demonstrated for the case of a nuclear reactor toroidal containment vessel partially filled with water. Results demonstrate the superiority of this method over diagonal added mass procedures, such as the tributary area method.
tion of the consistent mass matrix was performed to explore the reasons for the differing results of the two approaches. Solution of the system finite element equations resulting from each fluid mass matrix approach was performed. The mode shapes and frequencies from each solution were then compared. The consistent mass matrix formulation is verified by comparison with results obtained using independent public domain programs and with experimental data. These comparisons led to conclusions concerning the suitability of the various approaches in modeling hydrodynamic effects in toroidal containment vessels. A practical approach for application of the consistent added mass method to complex threedimensional systems is suggested.
1. Introduction
There are presently more than twenty nuclear power plant units operating in the United States which have toroidal-shaped primary containment vessels. These structures contain water in their normal operating configuration and are presently undergoing analysis or reanalysis for a number of transient loading conditions. Since the mass of the water comprises the major part of the total system mass, its treatment is an important consideration in the analysis procedure. This study used axisymmetric theory and finite element techniques to model a toroidal shell partially filled with water. Thus, longitudinal coupling of shell and liquid was precluded. Small motions and an ideal fluid were assumed. The investigation considers two possible approaches to modeling the fluid. One, the well known tributary area method [1 ], may be thought of as a diagonal added mass matrix procedure. Lumped masses are added at appropriate shell nodal points to represent the water mass. An alternative is a finite element fluid model in which all interior fluid degrees of freedom are eliminated. This results in a consistent (full) mass matrix in terms of the shell nodal point degrees of freedom. A solution using a systematic diagonaliza-
2. Consistent mass matrix formulation
By the use of variational methods and finite element discretization, it was shown in refs. [2] and [3] that the differential equations for fluid-structure systems may be reduced to the equations
MpqXq +KpqXq +/~prl;'r = Fp
(1)
and A rsYs - BrpXp = O.
* Control Data Corporation, Knoxville, TN, USA. ** Clemson University, Clemson, SC, USA.
The dots represent differentiation with respect to 217
(2)
11 24
24
24a) 24 b)
SOR c) SOR c)
SOR c)
15a) 15 b)
528 524
531
539a) 519 b)
19.8 20.7
22.7
NAa) 18.2 b)
19.7 18.4 18.6 17.8
811 799
807
812a) 823 b)
833 864 806 829
89.0 80.3
77.0
NAa) 82.9 b)
86.2 88.2 70.7 74.8
Gain d)
1193 1166
1179
1212 a) 1240 b)
1259 1256 1191 1258
1255
(Hz)
Frequency
Mode 3
178.3 174.5
194.1
NA a) 209.6 b)
278.7 244.1 154.3 171.0
Gain
127 555
82
521 a) 1059 b)
231 202 307 307
CPU e) (s)
33 75
6
5 a) 10 b)
4 4 4 4
Number of modes
a) MacNeil-Sehwendler version [ 6]. b) Universal Analytics version [ 8 ]. c) Surface of revolution, d) ~ / M at longitudinal station 0.1 in. e) UNIVAC 1108.
FSI
DYNASOR
NASTRAN
15 30 15 30
539 551 531 544
11 11 24 24
835
(Hz)
Frequency
Gain d)
Frequency
(Hz)
Mode 2
Mode 1
HYDRO
Sector size (deg.)
495
Number of longitudinal stations
Test
Program
Table 1 Comparison of analytical and experimental results for a hemispherical-cylindrical tank partially fiUed with water
5"
c¢
R.H. Rush, J.E. Jackson /Hydronamic effects in containment vessels time. M ~ and gpq a r e the shell mass and stiffness matrices; Xp represents the nodal point displacements and rotations of the shell; and the Yr are the nodal point values of the fluid velocity potential. Fp is the shell force vector. Eqs. (1) and (2) may be combined to yield
(Mpq + J~ppA~lBsq)J~q + KpqXq
=
Fp ,
(3)
where the term BprA~lBsq has the form of an "added mass" matrix. Note that the problem has been reduced to one involving shell nodal point degrees of freedom only. The system is solved for mode shapes and frequencies using standard numerical techniques. In ref. [4] results of the computer program used in this study were compared with those obtained from other programs and with experimental data for the case of a cylindrical tank with a hemispherical bulkhead partially filled with water. Results from this report have been reproduced in table 1.
3. Tributary area method In the tributary area method water in the vicinity of a structural nodal point is lumped as added mass at that nodal point. This procedure results in a diagonal added mass matrix.
219
directional summation of the diagonal terms of the consistent mass matrix. Application of this algorithm results in a diagonal mass matrix which is lumped in proportion to the diagonal terms of the consistent mass matrix. The procedure has been shown to give good results in the solution of some structural systems [51.
5. Toroidal containment example Consider the axisymmetric system illustrated by fig. 1. The shell is modeled with conical frustum elements, and the water is discretized with linear isoparametric quadrilateral elements. Zero pressure conditions are assumed at the fluid.free surface. This system can be represented by a set of equations in the form of eq. (3) upon which modal extraction is performed. The same structural model was used for the tributary area analysis as for the consistent added mass formulation. The fluid was apportioned to the various shell nodes as indicated in fig. 2. For illustrative purposes, fig. 2 shows a coarser mesh than that actually used. Except for a different fluid mass matrix, this system is identical to the first case. Finally, the consistent added mass matrix was diagonalized using the methods described in the pre-
4. Diagonalized consistent mass matrix To aid in the comparison of the results between the consistent and tributary area added fluid mass matrix formulations, a diagonalization of the consistent mass matrix obtained in the former procedure was performed. The diagonalization procedure used was similar to that suggested by Shantaram et al. [5]. Assume that the total water mass is defined as M rThen, each diagonal term may be found from MD _ M~/MT , N
(4)
Y ---//-__-
t/
\ \
~9?~8 5d521019
I///11 ....
il
\
~ II
7
j-~l where M~/denotes the ith diagonal term of the consistent mass matrix, and N is the total number of degrees of freedom for each global directional coordinate. Thus, the denominator of eq. (4) represents the
Fig. 1. Finite element model for consistent added mass matrix formulation.
R.H. Rush, J.E. Jackson /Hydronamic effects in containment vessels
220
I I ~ P I
I
6. D i s c u s s i o n o f results
r VERTICAL
DiSTRIBUTiON
v
HORIZONTAL DISTRIBUTION
Fig. 2. Nodal point distribution of water mass by the tributary area method.
vious section, and modal extraction was again performed. Results from the above three sets of frequency calculations along with those of the NASTRAN [6] hydrodynamic analysis are compared in table 2. Mode shapes are illustrated in fig. 3. Table 2
Frequency comparison of consistent and diagonal added fluid mass methods Mode no.
NASTRAN
Consistent Diagonaladded mass (FSI)
ized consistent mass
Tributary area
(FSI) 1 2 3 4 5
1.06 7.32 8.13 9.89 -
1.07 7.69 8.66 10.69 12.59
0.79 4.02 4.23 5.89 6.61
0.78 4.05 4.24 5.76 6.39
Note from table 1 that the consistent added mass matrix formulation, termed FSI, agrees well with experimental results obtained from tests performed by Southwest Research Institute [7] on a hemispherical-cylindrical tank. The margin of difference for the first three axisymmetric bulge modes is less than 8%. The calculated results also agree favorably with those obtained using NASTRAN [6,8], DYNASOR [9], and HYDRO [10]. Thus, the accuracy of the consistent added mass matrix formulation is established. Table 2 lists the first five axisymmetric bulge modes for the consistent and diagonal mass matrix input for a toroidal containment vessel. Note that frequencies obtained from NASTRAN and FSI are in agreement. Computational costs using NASTRAN are more than an order of magnitude greater than those of FSI for calculation of a comparable number of modes. Fig. 3 is a comparison of mode shapes given by the consistent and diagonal added mass matrix input. The tributary area and diagonalized consistent added mass methods give essentially identical results and so are plotted as one set. Mode shapes from the consistent added mass matrix method are similar to those of the above two solutions but not identical. It is significant to note that the diagonalized consistent added mass and the tributary area added mass formulations result in almost identical natural frequencies even though the two methods were developed independently. For the five modes illustrated, the tributary area calculations underestimate the natural frequencies by 27-49%. The fact that the tributary area and diagonalized consistent mass matrix procedures both give essentially the same underestimated frequencies indicates that the difference is largely due to the inability of the diagonal mass matrices to account for mass coupling between nodal points in a given cross section. Since the effect of the water mass varies with each mode, it is not feasible to adjust the frequencies by application of a scaling factor. Since the effects of the fluid on the shell are manifest primarily normal to the shell surface, a practical approach to modeling hydrodynamic effects for toroidal containment vessels is suggested. First, the consistent added mass matrix due to the fluid is calculated
221
R.H. Rush, J.E. Jackson I Hydronamic effects in containment vessels MOOE 2
a
MCOE3
MOOE2
MODE 5
MODE 4
b
MODE 3
i
MOOE 4.
\
MODE 5
/
~ 6
MODE 6
Fig. 3. (a) Modes shapes - consistent added mass formulation. Co) Modes shapes - tributary area formulation.
in a manner similar to that described above. The mass matrix resulting from this two-dimensional analysis is then added to the three-dimensional structural model at appropriate cross sections. Obviously, the fluid cross sections will be uncoupled in the longitudinal direction, as is also the case with the diagonal added mass procedures, but coupling between nodal points within the plane of the cross section due to the presence of the fluid will be retained. If desired, a consistent added mass matrix coupled in three dimensions could be developed. The difference in results between the diagonal and consistent added mass approaches, illustrated herein for the case of axisymmetric toroidal fluid-shell systems, has ramifications for other liquid-shell systems. The poor performance of the diagonal added fluid mass matrix method for the toroidal system investi-
gated should not be construed to mean that the same order of error will occur for other systems, for example cylindrical or spherical liquid-shell problems. However, caution is recommeded whenever diagonal added mass matrix procedures are used in the solution of coupled fluid-structure systems and particularly fluid-structure systems using thin shell structures.
7. C o n c l u s i o n s
The treatment of hydrodynamic effects by addition of a consistent added mass matrix to structural degrees of freedom is founded upon a sound theoretical basis. The method gives results comparable with experimental data. The case of an axisymmetric toroidal containment
222
R.H. Rush, J.E. Jackson/Hydronamic effects in containment vessels
vessel partially filled with water and having dimensions comparable to those o f existing nuclear power plant structures was investigated. It was shown that there were significant discrepancies between frequencies calculated by the consistent added mass matrix formulation and those given by procedures, such as the tributary area method, which result in a diagonal added fluid mass matrix. It was postulated that these discrepancies were due to the lack of nodal coupling in the diagonal added fluid mass method and that the differences cannot be resolved by scaling the diagonal fluid mass matrix. The possible deficiency in diagonal added fluid matrix procedures may exist for liquid-shell systems of configurations other than toroidal, although the toroidal example does not indicate the magnitude o f the error for these other systems. It is suggested that a practical and economical approach to solution of certain complex three-dimensional liquid-shell systems such as toroidal containment vessels is the addition of two-dimensional consistent added mass matrices at appropriate structural cross sections.
References [1] Nuclear Reactors and Earthquakes, TID-7024, USAEC (1963). [2] R.H. Rush, Master's Thesis, University of Alabama in Huntsville, Alabama (1975). [3] T.J. Chung and R.H. Rush, Trans. ASME, J. Appl. Mech. 43, ser. E, no. 3, (1976) 507. [4] Final Report: Martin Marietta Corporation External Tank Math Model Development and Southwest Reserch Institute Tank Test Hydroelastic Vibration Analysis, Tech. Letter ESD-ED23-23367, Teledyne Brown Engineering, Huntsville, Alabama (1976). [5] D. Shantaram, D.R.J. Owen and O.C. Zienkiewicz, Earthquake Eng. Struct. Dyn. 4 (1976) 561. [6] NASTRAN Theoretical Manual, Level 16.0, NASA SP-221(03) (1976). [7] D.D. Kana and L.M. Vargas, NASA Contract NAS830270, SWRI Project 02-3851, Southwest Research Institute, San Antonio, Texas (1975). [8] UAI/NASTRAN, Universal Analytics, Incorporated, Playa Del Rey, California (1977). [9] Dynamic Analysis of Shells of Revolution - DYNASOR, Document D5-14293-1, The Boeing Company, Southeast Division (1971). [10] Hydroelastic Mathematical Model of Space Shuttle Propellant Tanks, NASA Contract NAS8-30906, Martin Marietta Corporation, Denver, Colorado (1975).
TREATMENT OF HYDRODYNAMIC EFFECTS FOR TOROIDAL CONTAINMENT VESSELS Richard H. RUSH * and John E. JACKSON ** Qvil Engineering Branch, Division o f Engineering Design, Tennessee Valley Authority, Knoxville, TN, USA Received 21 December 1978
Hydrodynamic effects in liquid-shell systems may be characterized in terms of structural degrees of freedom alone if an ideal fluid is assumed. The hydrodynamic effects are modeled by means of a consistent (full) added mass matrix which is obtained via finite element methods, The procedure is demonstrated for the case of a nuclear reactor toroidal containment vessel partially filled with water. Results demonstrate the superiority of this method over diagonal added mass procedures, such as the tributary area method.
tion of the consistent mass matrix was performed to explore the reasons for the differing results of the two approaches. Solution of the system finite element equations resulting from each fluid mass matrix approach was performed. The mode shapes and frequencies from each solution were then compared. The consistent mass matrix formulation is verified by comparison with results obtained using independent public domain programs and with experimental data. These comparisons led to conclusions concerning the suitability of the various approaches in modeling hydrodynamic effects in toroidal containment vessels. A practical approach for application of the consistent added mass method to complex threedimensional systems is suggested.
1. Introduction
There are presently more than twenty nuclear power plant units operating in the United States which have toroidal-shaped primary containment vessels. These structures contain water in their normal operating configuration and are presently undergoing analysis or reanalysis for a number of transient loading conditions. Since the mass of the water comprises the major part of the total system mass, its treatment is an important consideration in the analysis procedure. This study used axisymmetric theory and finite element techniques to model a toroidal shell partially filled with water. Thus, longitudinal coupling of shell and liquid was precluded. Small motions and an ideal fluid were assumed. The investigation considers two possible approaches to modeling the fluid. One, the well known tributary area method [1 ], may be thought of as a diagonal added mass matrix procedure. Lumped masses are added at appropriate shell nodal points to represent the water mass. An alternative is a finite element fluid model in which all interior fluid degrees of freedom are eliminated. This results in a consistent (full) mass matrix in terms of the shell nodal point degrees of freedom. A solution using a systematic diagonaliza-
2. Consistent mass matrix formulation
By the use of variational methods and finite element discretization, it was shown in refs. [2] and [3] that the differential equations for fluid-structure systems may be reduced to the equations
MpqXq +KpqXq +/~prl;'r = Fp
(1)
and A rsYs - BrpXp = O.
* Control Data Corporation, Knoxville, TN, USA. ** Clemson University, Clemson, SC, USA.
The dots represent differentiation with respect to 217
(2)
11 24
24
24a) 24 b)
SOR c) SOR c)
SOR c)
15a) 15 b)
528 524
531
539a) 519 b)
19.8 20.7
22.7
NAa) 18.2 b)
19.7 18.4 18.6 17.8
811 799
807
812a) 823 b)
833 864 806 829
89.0 80.3
77.0
NAa) 82.9 b)
86.2 88.2 70.7 74.8
Gain d)
1193 1166
1179
1212 a) 1240 b)
1259 1256 1191 1258
1255
(Hz)
Frequency
Mode 3
178.3 174.5
194.1
NA a) 209.6 b)
278.7 244.1 154.3 171.0
Gain
127 555
82
521 a) 1059 b)
231 202 307 307
CPU e) (s)
33 75
6
5 a) 10 b)
4 4 4 4
Number of modes
a) MacNeil-Sehwendler version [ 6]. b) Universal Analytics version [ 8 ]. c) Surface of revolution, d) ~ / M at longitudinal station 0.1 in. e) UNIVAC 1108.
FSI
DYNASOR
NASTRAN
15 30 15 30
539 551 531 544
11 11 24 24
835
(Hz)
Frequency
Gain d)
Frequency
(Hz)
Mode 2
Mode 1
HYDRO
Sector size (deg.)
495
Number of longitudinal stations
Test
Program
Table 1 Comparison of analytical and experimental results for a hemispherical-cylindrical tank partially fiUed with water
5"
c¢
R.H. Rush, J.E. Jackson /Hydronamic effects in containment vessels time. M ~ and gpq a r e the shell mass and stiffness matrices; Xp represents the nodal point displacements and rotations of the shell; and the Yr are the nodal point values of the fluid velocity potential. Fp is the shell force vector. Eqs. (1) and (2) may be combined to yield
(Mpq + J~ppA~lBsq)J~q + KpqXq
=
Fp ,
(3)
where the term BprA~lBsq has the form of an "added mass" matrix. Note that the problem has been reduced to one involving shell nodal point degrees of freedom only. The system is solved for mode shapes and frequencies using standard numerical techniques. In ref. [4] results of the computer program used in this study were compared with those obtained from other programs and with experimental data for the case of a cylindrical tank with a hemispherical bulkhead partially filled with water. Results from this report have been reproduced in table 1.
3. Tributary area method In the tributary area method water in the vicinity of a structural nodal point is lumped as added mass at that nodal point. This procedure results in a diagonal added mass matrix.
219
directional summation of the diagonal terms of the consistent mass matrix. Application of this algorithm results in a diagonal mass matrix which is lumped in proportion to the diagonal terms of the consistent mass matrix. The procedure has been shown to give good results in the solution of some structural systems [51.
5. Toroidal containment example Consider the axisymmetric system illustrated by fig. 1. The shell is modeled with conical frustum elements, and the water is discretized with linear isoparametric quadrilateral elements. Zero pressure conditions are assumed at the fluid.free surface. This system can be represented by a set of equations in the form of eq. (3) upon which modal extraction is performed. The same structural model was used for the tributary area analysis as for the consistent added mass formulation. The fluid was apportioned to the various shell nodes as indicated in fig. 2. For illustrative purposes, fig. 2 shows a coarser mesh than that actually used. Except for a different fluid mass matrix, this system is identical to the first case. Finally, the consistent added mass matrix was diagonalized using the methods described in the pre-
4. Diagonalized consistent mass matrix To aid in the comparison of the results between the consistent and tributary area added fluid mass matrix formulations, a diagonalization of the consistent mass matrix obtained in the former procedure was performed. The diagonalization procedure used was similar to that suggested by Shantaram et al. [5]. Assume that the total water mass is defined as M rThen, each diagonal term may be found from MD _ M~/MT , N
(4)
Y ---//-__-
t/
\ \
~9?~8 5d521019
I///11 ....
il
\
~ II
7
j-~l where M~/denotes the ith diagonal term of the consistent mass matrix, and N is the total number of degrees of freedom for each global directional coordinate. Thus, the denominator of eq. (4) represents the
Fig. 1. Finite element model for consistent added mass matrix formulation.
R.H. Rush, J.E. Jackson /Hydronamic effects in containment vessels
220
I I ~ P I
I
6. D i s c u s s i o n o f results
r VERTICAL
DiSTRIBUTiON
v
HORIZONTAL DISTRIBUTION
Fig. 2. Nodal point distribution of water mass by the tributary area method.
vious section, and modal extraction was again performed. Results from the above three sets of frequency calculations along with those of the NASTRAN [6] hydrodynamic analysis are compared in table 2. Mode shapes are illustrated in fig. 3. Table 2
Frequency comparison of consistent and diagonal added fluid mass methods Mode no.
NASTRAN
Consistent Diagonaladded mass (FSI)
ized consistent mass
Tributary area
(FSI) 1 2 3 4 5
1.06 7.32 8.13 9.89 -
1.07 7.69 8.66 10.69 12.59
0.79 4.02 4.23 5.89 6.61
0.78 4.05 4.24 5.76 6.39
Note from table 1 that the consistent added mass matrix formulation, termed FSI, agrees well with experimental results obtained from tests performed by Southwest Research Institute [7] on a hemispherical-cylindrical tank. The margin of difference for the first three axisymmetric bulge modes is less than 8%. The calculated results also agree favorably with those obtained using NASTRAN [6,8], DYNASOR [9], and HYDRO [10]. Thus, the accuracy of the consistent added mass matrix formulation is established. Table 2 lists the first five axisymmetric bulge modes for the consistent and diagonal mass matrix input for a toroidal containment vessel. Note that frequencies obtained from NASTRAN and FSI are in agreement. Computational costs using NASTRAN are more than an order of magnitude greater than those of FSI for calculation of a comparable number of modes. Fig. 3 is a comparison of mode shapes given by the consistent and diagonal added mass matrix input. The tributary area and diagonalized consistent added mass methods give essentially identical results and so are plotted as one set. Mode shapes from the consistent added mass matrix method are similar to those of the above two solutions but not identical. It is significant to note that the diagonalized consistent added mass and the tributary area added mass formulations result in almost identical natural frequencies even though the two methods were developed independently. For the five modes illustrated, the tributary area calculations underestimate the natural frequencies by 27-49%. The fact that the tributary area and diagonalized consistent mass matrix procedures both give essentially the same underestimated frequencies indicates that the difference is largely due to the inability of the diagonal mass matrices to account for mass coupling between nodal points in a given cross section. Since the effect of the water mass varies with each mode, it is not feasible to adjust the frequencies by application of a scaling factor. Since the effects of the fluid on the shell are manifest primarily normal to the shell surface, a practical approach to modeling hydrodynamic effects for toroidal containment vessels is suggested. First, the consistent added mass matrix due to the fluid is calculated
221
R.H. Rush, J.E. Jackson I Hydronamic effects in containment vessels MOOE 2
a
MCOE3
MOOE2
MODE 5
MODE 4
b
MODE 3
i
MOOE 4.
\
MODE 5
/
~ 6
MODE 6
Fig. 3. (a) Modes shapes - consistent added mass formulation. Co) Modes shapes - tributary area formulation.
in a manner similar to that described above. The mass matrix resulting from this two-dimensional analysis is then added to the three-dimensional structural model at appropriate cross sections. Obviously, the fluid cross sections will be uncoupled in the longitudinal direction, as is also the case with the diagonal added mass procedures, but coupling between nodal points within the plane of the cross section due to the presence of the fluid will be retained. If desired, a consistent added mass matrix coupled in three dimensions could be developed. The difference in results between the diagonal and consistent added mass approaches, illustrated herein for the case of axisymmetric toroidal fluid-shell systems, has ramifications for other liquid-shell systems. The poor performance of the diagonal added fluid mass matrix method for the toroidal system investi-
gated should not be construed to mean that the same order of error will occur for other systems, for example cylindrical or spherical liquid-shell problems. However, caution is recommeded whenever diagonal added mass matrix procedures are used in the solution of coupled fluid-structure systems and particularly fluid-structure systems using thin shell structures.
7. C o n c l u s i o n s
The treatment of hydrodynamic effects by addition of a consistent added mass matrix to structural degrees of freedom is founded upon a sound theoretical basis. The method gives results comparable with experimental data. The case of an axisymmetric toroidal containment
222
R.H. Rush, J.E. Jackson/Hydronamic effects in containment vessels
vessel partially filled with water and having dimensions comparable to those o f existing nuclear power plant structures was investigated. It was shown that there were significant discrepancies between frequencies calculated by the consistent added mass matrix formulation and those given by procedures, such as the tributary area method, which result in a diagonal added fluid mass matrix. It was postulated that these discrepancies were due to the lack of nodal coupling in the diagonal added fluid mass method and that the differences cannot be resolved by scaling the diagonal fluid mass matrix. The possible deficiency in diagonal added fluid matrix procedures may exist for liquid-shell systems of configurations other than toroidal, although the toroidal example does not indicate the magnitude o f the error for these other systems. It is suggested that a practical and economical approach to solution of certain complex three-dimensional liquid-shell systems such as toroidal containment vessels is the addition of two-dimensional consistent added mass matrices at appropriate structural cross sections.
References [1] Nuclear Reactors and Earthquakes, TID-7024, USAEC (1963). [2] R.H. Rush, Master's Thesis, University of Alabama in Huntsville, Alabama (1975). [3] T.J. Chung and R.H. Rush, Trans. ASME, J. Appl. Mech. 43, ser. E, no. 3, (1976) 507. [4] Final Report: Martin Marietta Corporation External Tank Math Model Development and Southwest Reserch Institute Tank Test Hydroelastic Vibration Analysis, Tech. Letter ESD-ED23-23367, Teledyne Brown Engineering, Huntsville, Alabama (1976). [5] D. Shantaram, D.R.J. Owen and O.C. Zienkiewicz, Earthquake Eng. Struct. Dyn. 4 (1976) 561. [6] NASTRAN Theoretical Manual, Level 16.0, NASA SP-221(03) (1976). [7] D.D. Kana and L.M. Vargas, NASA Contract NAS830270, SWRI Project 02-3851, Southwest Research Institute, San Antonio, Texas (1975). [8] UAI/NASTRAN, Universal Analytics, Incorporated, Playa Del Rey, California (1977). [9] Dynamic Analysis of Shells of Revolution - DYNASOR, Document D5-14293-1, The Boeing Company, Southeast Division (1971). [10] Hydroelastic Mathematical Model of Space Shuttle Propellant Tanks, NASA Contract NAS8-30906, Martin Marietta Corporation, Denver, Colorado (1975).