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Vibrations of initially stressed cylinders of variable thickness
Journal of Sound and Vibration (1977) 53(2), 267-27 1
VIBRATIONS
OF INITIALLY OF VARIABLE
STRESSED
CYLINDERS
THICKNESS
S. K. RADHAMOHANAND M. MAITI Structural Engineering Division, Vikram Sarabhai Space Centre, Trivandrum-695022, India
(Received 3 November 1976, and in revisedform
1 March 1977)
Natural frequencies and buckling loads for cylindrical shells having linearly varying thickness are obtained by using a segmentation technique. The present results for free vibration of a cylinder compare very well with those obtained previously. The effect of
the thickness variation on the frequencies of a cylindrical shell is studied. Frequencies are also calculated for a cylinder of variable thickness under axial compression and a relationship between the frequency and axial compression is obtained for a particular wave number.
1. INTRODUCTION
With the advancement of rocket technology, the study of cylindrical shells of variable thickness is of great importance for the design and analysis of fuel tanks and rocket motors. Generally, shells ofvariable thickness are employed as interstages to connect shells of different thicknesses. In view of the fact that control equipment is housed in the interstages, frequency and mode shape determination is an essential part of a structural design. An investigation into the buckling phenomenon of interstages is also required because of the axial thrust experienced by the interstages. There are very few published studies on the vibrations of cylindrical shells of variable thickness under axial compression. Recently, the partition method has been used by Stoneking [l] to formulate a set of equations from which the natural frequencies and mode shapes of cylindrical shells with variable thickness may be evaluated. Further, stability of cylindrical shells of linearly varying thickness subjected to axial loads has been treated by Federhofer [2], Wagner [3] and Ross [4]. In this paper, vibration of a cylindrical shell of linearly varying thickness of finite length under axial compression is considered. The thickness is constant in the circumferential direction of the shell. The natural frequencies for different wave numbers are calculated, with a clamped-clamped boundary condition, and are compared with those of Stoneking [ 11. A relationship between the frequency and axial compression is established for a particular wave number. The present analysis is also used to evaluate the buckling loads of a cylinder of variable thickness by static and dynamic criteria. The effect of boundary conditions on the buckling behaviour is also considered in the analysis. 2. THEORY The strain displacement and equilibrium equations selected for a cylindrical shell ofvariable thickness are those of Sanders’ non-linear formulation [5], applicable to thin shells of revolu267
S. K. RADHAMOHANAND M. MAITI
268
tion. The material is assumed to be homogeneous and isotropic. All the governing equations, written in first order form, are available in reference [6]. Inertia effects have been incorporated in the equilibrium equations in the present work. The final differential equations of equilibrium are [6]
N&
-
gLpiu*, 0
where u, u and w are displacements along the $0 and normal directions, respectively, & and & are rotations of tangents along the Sand 0 directions, respectively, 4 is the rotation about the normal, N,, N, and NSOare inplane forces, MS, M, and MS0are the moment resultants, QSand Ye are the transverse shears, qs, go and q are external loads along the S, 8 and normal directions, respectively, p is the mass density of the material and L, R, to and E. are the reference length, radius, thickness and modulus, respectively. The subscript 0 refers to prebuckling quantities and the superscript * refers to buckling quantities associated with the circumferential wave number n. The other equations are the same as given in reference [6]. 3. METHOD OF COMPUTATION The above linear homogeneous differential equations with appropriate homogeneous boundary conditions were integrated by the fourth order Runge Kutta integration method. To avoid error propagation in the integration process, the shell was divided into a number of segments and integration was performed in each segment, by using a segmentation technique [7]. The integration was performed with assumed unit values for the unspecified quantities at one end and satisfying the boundary conditions at the other end. For a non-trivial solution the determinant formed by the corresponding coefficients of the homogeneous integrations shouId vanish. The corresponding load/frequency for which the determinant vanishes is evaluated by the changes of sign of the determinant between two successive load/frequency values. A computer program has been developed in FORTRAN IV language for an IBM
STRESSED CYLINDERS OF VARIABLE THICKNESS
269
360/44 computer, in which account is taken of the variation of thickness, material properties and loading along the meridian, 4. NUMERICAL
RESULTS
Natural frequencies for cylinders of variable thickness were obtained by Stoneking [ 1] by the partition method in conjunction with piecewise polynomials. This approximation can lead to inaccurate upper bound results. The results reported in this paper have been obtained by direct integration of the differential equations without assuming any displacement function. Table 1 compares the present results with those of Stoneking [l] for a particular type of thickness variation (cf. Figure 1). As the circumferential wave number decreases, the difference between numerical integration method results and partition method results increases. Since the shell (cf. Figure 1) is assumed to have a symmetric variation about its middle surface, the thickness is expressed as t = &(A, + B1 s),
O
where sis the distance from the thin edge of the shell and is non-dimensionalized with respect to the length of the shell, A, and B, are the parameters governing the thickness variation and to is the average thickness of the shell. For a particular wave number (n = 4), frequencies for shells having linear thickness variation with different values of A, and B, (cf. Table 2) have been obtained. In this analysis, the average thickness (or equivalently the weight of the structure) was kept constant. It is observed from Table 2 that the highest frequency occurs for a shell having constant thickness.
TABLE 1 Comparison of frequencies for wave numbers
Wave no. II 2 3 4 5 6
<-A-,
Frequencies (Hz)
Stoneking
Present analysis
1909 1181 866 842 988
1861.61 1139.3 825 814 966
O-3048 m (12 in) t = 0.254
Figure
1. Cylindrical
(7.367 x lo-’ lb s2/in4).
shell
mm
geometry. E= 207 GN/m’ (30 x lo6 lb/in2); v= O-3; p = 7833.4 kg/m3
270
S. K. RADHAMOHAN ANDM. MAITI TABLE2 Eflect of thickness variation on frequency (n = 4) 9.
no.
Thickness coefficients Frequency (Hz)
1
Al = 0.2 B1 = 1.6 Al = 0.4 B1 = 1.2 A, = 0.6 B, = 0.8 Al = 1.0 B1 = 0.0
2 3 4
0
1
@.I
804.73 825-O 842.4 851.97
I
I
I
I
,
I
0.2
0.3
0.4
0.5
0.6
0.7
Load
parameter,
X
Figure 2. Frequency vs. load parameter.
Frequencies of a cylinder under axial compression have been obtained for the wave number n = 6. Figure 2 shows the relationship between the frequency and axial compression. By using dynamic buckling criteria, the buckling load (corresponding to zero frequency) has been obtained and coincides well with the buckling load obtained from static analysis. It can be seen from Figure 2 that the decrease of frequency with axial load is very gradual at lower axial loads but becomes steep as the buckling load is approached. Buckling loads of a cylinder of variable thickness, subjected to axial compression are given in Table 3. The axial stress on the cylinder is taken as ocritical
=
-
t E .- avcra*e [3(1 - v~)]“’ R ”
where I is the load parameter. The lowest critical load of the shell corresponds to the wave number n = 1. The buckling load increases for higher values of n. Table 4 gives the effect of different boundary (buckling) conditions on the buckling behaviour. When the inplane displacements u and v are unconstrained, the buckling load is
STRESSEDCYLINDERSOF VARIABLETHICKNESS
271
TABLE 3 Buckling loads for direrent wave numbers
Wave no. n
Buckling load parameter
1 2 3 4 5 6
0.2255 0.2889 04071 0.5242 0.6501 0.6621
TABLE 4 Buckling loaa!sfor different boundary conditions
Boundary conditions
Buckling load parameter
w=~$=N~=N,~=O w = 4s = u = N,@= 0 w=~,=v =N, =0 w=q&=u =u =O
0.1880 0.2225 0.2257 0.2255
the lowest. However, there is no marked difference in the buckling loads for the remaining boundary conditions. 5. CONCLUSIONS
Natural frequencies of a cylinder of variable thickness, under axial compression, have been obtained for the first time. Buckling loads of shells of variable thickness have been obtained by using both static and dynamic criteria. REFERENCES 1. J. E. STONEKING1973 Nuclear Engineering and Design 24, 314-321. Free vibrations of shells of revolution with variable thickness. 1952 Zngenieur Archiv 6,277-288. Stabilitat Der Krieszylinderschale Mit Verander2. K. FEDERHOFER lither Wandstarke, Osterreichisches. 3. H. WAGNER1959 Zngenieur Archiv 13, 235-257. Die Stabilitat Der Axial Gendruckten Kreiszylinerschale Mit Vetanderlicher Wandstarke, Osterreichisches. 4. C. T. F. ROSS 1968 Quarterly Transactions of the Royal Society for Naval Architectects 110, 247-250. Elastic instability of a circular cylinder of varying shell thickness. 5. J. L. SANDERS1963 Quarterly of AppliedMathematics 21,21-36. Non-linear theories for thin shells. 6. S. K. RADHAMOHAN and A. D. SHIRODE1975 Computers and Structures 5, 155-158. Buckling of orthotropic torispherical pressure vessels. 7. J. E. GOLDBERG,A. V. SETLURand D. W. ALSPAUGH1965 Conference of the Znternational Association for Shell Structures, Budapest, Hungary. Computer analysis of non-circular cylindrical shells.
VIBRATIONS
OF INITIALLY OF VARIABLE
STRESSED
CYLINDERS
THICKNESS
S. K. RADHAMOHANAND M. MAITI Structural Engineering Division, Vikram Sarabhai Space Centre, Trivandrum-695022, India
(Received 3 November 1976, and in revisedform
1 March 1977)
Natural frequencies and buckling loads for cylindrical shells having linearly varying thickness are obtained by using a segmentation technique. The present results for free vibration of a cylinder compare very well with those obtained previously. The effect of
the thickness variation on the frequencies of a cylindrical shell is studied. Frequencies are also calculated for a cylinder of variable thickness under axial compression and a relationship between the frequency and axial compression is obtained for a particular wave number.
1. INTRODUCTION
With the advancement of rocket technology, the study of cylindrical shells of variable thickness is of great importance for the design and analysis of fuel tanks and rocket motors. Generally, shells ofvariable thickness are employed as interstages to connect shells of different thicknesses. In view of the fact that control equipment is housed in the interstages, frequency and mode shape determination is an essential part of a structural design. An investigation into the buckling phenomenon of interstages is also required because of the axial thrust experienced by the interstages. There are very few published studies on the vibrations of cylindrical shells of variable thickness under axial compression. Recently, the partition method has been used by Stoneking [l] to formulate a set of equations from which the natural frequencies and mode shapes of cylindrical shells with variable thickness may be evaluated. Further, stability of cylindrical shells of linearly varying thickness subjected to axial loads has been treated by Federhofer [2], Wagner [3] and Ross [4]. In this paper, vibration of a cylindrical shell of linearly varying thickness of finite length under axial compression is considered. The thickness is constant in the circumferential direction of the shell. The natural frequencies for different wave numbers are calculated, with a clamped-clamped boundary condition, and are compared with those of Stoneking [ 11. A relationship between the frequency and axial compression is established for a particular wave number. The present analysis is also used to evaluate the buckling loads of a cylinder of variable thickness by static and dynamic criteria. The effect of boundary conditions on the buckling behaviour is also considered in the analysis. 2. THEORY The strain displacement and equilibrium equations selected for a cylindrical shell ofvariable thickness are those of Sanders’ non-linear formulation [5], applicable to thin shells of revolu267
S. K. RADHAMOHANAND M. MAITI
268
tion. The material is assumed to be homogeneous and isotropic. All the governing equations, written in first order form, are available in reference [6]. Inertia effects have been incorporated in the equilibrium equations in the present work. The final differential equations of equilibrium are [6]
N&
-
gLpiu*, 0
where u, u and w are displacements along the $0 and normal directions, respectively, & and & are rotations of tangents along the Sand 0 directions, respectively, 4 is the rotation about the normal, N,, N, and NSOare inplane forces, MS, M, and MS0are the moment resultants, QSand Ye are the transverse shears, qs, go and q are external loads along the S, 8 and normal directions, respectively, p is the mass density of the material and L, R, to and E. are the reference length, radius, thickness and modulus, respectively. The subscript 0 refers to prebuckling quantities and the superscript * refers to buckling quantities associated with the circumferential wave number n. The other equations are the same as given in reference [6]. 3. METHOD OF COMPUTATION The above linear homogeneous differential equations with appropriate homogeneous boundary conditions were integrated by the fourth order Runge Kutta integration method. To avoid error propagation in the integration process, the shell was divided into a number of segments and integration was performed in each segment, by using a segmentation technique [7]. The integration was performed with assumed unit values for the unspecified quantities at one end and satisfying the boundary conditions at the other end. For a non-trivial solution the determinant formed by the corresponding coefficients of the homogeneous integrations shouId vanish. The corresponding load/frequency for which the determinant vanishes is evaluated by the changes of sign of the determinant between two successive load/frequency values. A computer program has been developed in FORTRAN IV language for an IBM
STRESSED CYLINDERS OF VARIABLE THICKNESS
269
360/44 computer, in which account is taken of the variation of thickness, material properties and loading along the meridian, 4. NUMERICAL
RESULTS
Natural frequencies for cylinders of variable thickness were obtained by Stoneking [ 1] by the partition method in conjunction with piecewise polynomials. This approximation can lead to inaccurate upper bound results. The results reported in this paper have been obtained by direct integration of the differential equations without assuming any displacement function. Table 1 compares the present results with those of Stoneking [l] for a particular type of thickness variation (cf. Figure 1). As the circumferential wave number decreases, the difference between numerical integration method results and partition method results increases. Since the shell (cf. Figure 1) is assumed to have a symmetric variation about its middle surface, the thickness is expressed as t = &(A, + B1 s),
O
where sis the distance from the thin edge of the shell and is non-dimensionalized with respect to the length of the shell, A, and B, are the parameters governing the thickness variation and to is the average thickness of the shell. For a particular wave number (n = 4), frequencies for shells having linear thickness variation with different values of A, and B, (cf. Table 2) have been obtained. In this analysis, the average thickness (or equivalently the weight of the structure) was kept constant. It is observed from Table 2 that the highest frequency occurs for a shell having constant thickness.
TABLE 1 Comparison of frequencies for wave numbers
Wave no. II 2 3 4 5 6
<-A-,
Frequencies (Hz)
Stoneking
Present analysis
1909 1181 866 842 988
1861.61 1139.3 825 814 966
O-3048 m (12 in) t = 0.254
Figure
1. Cylindrical
(7.367 x lo-’ lb s2/in4).
shell
mm
geometry. E= 207 GN/m’ (30 x lo6 lb/in2); v= O-3; p = 7833.4 kg/m3
270
S. K. RADHAMOHAN ANDM. MAITI TABLE2 Eflect of thickness variation on frequency (n = 4) 9.
no.
Thickness coefficients Frequency (Hz)
1
Al = 0.2 B1 = 1.6 Al = 0.4 B1 = 1.2 A, = 0.6 B, = 0.8 Al = 1.0 B1 = 0.0
2 3 4
0
1
@.I
804.73 825-O 842.4 851.97
I
I
I
I
,
I
0.2
0.3
0.4
0.5
0.6
0.7
Load
parameter,
X
Figure 2. Frequency vs. load parameter.
Frequencies of a cylinder under axial compression have been obtained for the wave number n = 6. Figure 2 shows the relationship between the frequency and axial compression. By using dynamic buckling criteria, the buckling load (corresponding to zero frequency) has been obtained and coincides well with the buckling load obtained from static analysis. It can be seen from Figure 2 that the decrease of frequency with axial load is very gradual at lower axial loads but becomes steep as the buckling load is approached. Buckling loads of a cylinder of variable thickness, subjected to axial compression are given in Table 3. The axial stress on the cylinder is taken as ocritical
=
-
t E .- avcra*e [3(1 - v~)]“’ R ”
where I is the load parameter. The lowest critical load of the shell corresponds to the wave number n = 1. The buckling load increases for higher values of n. Table 4 gives the effect of different boundary (buckling) conditions on the buckling behaviour. When the inplane displacements u and v are unconstrained, the buckling load is
STRESSEDCYLINDERSOF VARIABLETHICKNESS
271
TABLE 3 Buckling loads for direrent wave numbers
Wave no. n
Buckling load parameter
1 2 3 4 5 6
0.2255 0.2889 04071 0.5242 0.6501 0.6621
TABLE 4 Buckling loaa!sfor different boundary conditions
Boundary conditions
Buckling load parameter
w=~$=N~=N,~=O w = 4s = u = N,@= 0 w=~,=v =N, =0 w=q&=u =u =O
0.1880 0.2225 0.2257 0.2255
the lowest. However, there is no marked difference in the buckling loads for the remaining boundary conditions. 5. CONCLUSIONS
Natural frequencies of a cylinder of variable thickness, under axial compression, have been obtained for the first time. Buckling loads of shells of variable thickness have been obtained by using both static and dynamic criteria. REFERENCES 1. J. E. STONEKING1973 Nuclear Engineering and Design 24, 314-321. Free vibrations of shells of revolution with variable thickness. 1952 Zngenieur Archiv 6,277-288. Stabilitat Der Krieszylinderschale Mit Verander2. K. FEDERHOFER lither Wandstarke, Osterreichisches. 3. H. WAGNER1959 Zngenieur Archiv 13, 235-257. Die Stabilitat Der Axial Gendruckten Kreiszylinerschale Mit Vetanderlicher Wandstarke, Osterreichisches. 4. C. T. F. ROSS 1968 Quarterly Transactions of the Royal Society for Naval Architectects 110, 247-250. Elastic instability of a circular cylinder of varying shell thickness. 5. J. L. SANDERS1963 Quarterly of AppliedMathematics 21,21-36. Non-linear theories for thin shells. 6. S. K. RADHAMOHAN and A. D. SHIRODE1975 Computers and Structures 5, 155-158. Buckling of orthotropic torispherical pressure vessels. 7. J. E. GOLDBERG,A. V. SETLURand D. W. ALSPAUGH1965 Conference of the Znternational Association for Shell Structures, Budapest, Hungary. Computer analysis of non-circular cylindrical shells.