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Influence of initial geometric imperfections on vibrations of thin circular cylindrical shells
Computerx & Structures ¥oi 16. NO t-.4 pp 125~130.1983
004%794918310101254)6~3.0010
Prinled in Greal Britain
Pergamon Press Lid
INFLUENCE OF INITIAL GEOMETRIC IMPERFECTIONS ON VIBRATIONS OF THIN CIRCULAR CYLINDRICAL SHELLS LaKSHMAN WATAWALA'~and WILLIAMA. NASH* Department of Civil Engineering,University of Massachusetts, Amherst, MA 01003, U.S.A. AMtl-aet--The influence of initial geometric imperfections (out-of-roundness) on vibrations of a right circular cylindrical shell is investigated on the basis on non-linear large deformation shell theory. Both free and forced motions (due to base excitation caused by seismic effects) are treated. An extension to the case of the cylindrical shell with vertical geometric axis and completelyfilled with a perfect liquid is also studied. Numerical results are presented for various initial imperfectionsand vibratory configurations.It is found that the behavior may be of the hardening type for certain types of response yet of a softening nature for other responses. BACKGROUND
Although the influence of initial geometric imperfections on buckling resistance of a cylindrical shell has been the topic of numerous investigations during the past few decades, the literature contains only a very few results associated with the influence of such imperfections of vibration characteristics of the shell. One very important problem area where imperfections may be of importance lies in the response of cylindrical liquid storage tanks to seismic excitation of the base. Since such tanks invariably have slightly non-circular cross-sections due to fabrication problems, it is of interest to examine the influence of typical imperfections on the natural frequencies of liquid filled cylindrical storage tanks. It is also of interest to ascertain the influence of realistic imperfections on response of such tanks to seismic motions of the base. Since the seismic effects on such tanks may reasonably be expected to develop displacements of the order of magnitude of the shell thickness, it is necessary to consider non-linear shell theory. Although a number of treatments of free vibrations of cylindrical shells undergoing such large displacements are to be found in the literature, several warrant particular mention. In 1%1 Chu[1] employed a single mode response form to investigate large amplitude free vibrations and concluded that the non-linear behavior of the shell was always of the hardening type, i.e. the frequency increases with the amplitude of vibration. A 1963 study of orthotropic shells by Nowinski[2] led to the same conclusion. However, experimental studies by Evensen[3] and Olson[4] indicated that the non-linearity was of the softening type, i.e. frequencies decreased with increasing amplitude. In [3] the author pointed out that the analysis in [1] implied multiple-valued circumferential displacements and advanced this as one reason for the discrepancy in results. In a later analysis, Evensen [5] employed a three term mode shape which made it possible to develop single valued circumferential displacements. The results of this work showed that the non-linearity was either softening or hardening depending on the aspect ratio (ratio of the circumferential wavelength to axil wavelength) of the mode. This work did not consider initial geometric imperfections of the shell.
A recent study of the influence of imperfections on the vibrations of thin cylindrical shells due to Rosen and Singer [6] used a linearized version of Koiter's non-linear equations with inertia terms included. This linearization was performed assuming the amplitudes of vibration to be infinitesimal so as to greatly simplify the analysis while retaining the imperfection terms. However, it appears that these authors did not satisfy the condition of single valuedness of circumferential displacement for the case when the dynamic behavior has the same wave form as that of the initial imperfections. The present investigation is concerned with the influence of initial geometric imperfections on free vibrations of cylindrical shells, as well as with the behavior of such shells filled with liquid and subject to seismic motions of the base.
ANALYSIS
Let us consider a circular cylindrical shell of mean radius R, thickness h, and length L. Further, we introduce an orthogonal coordinate system (x, y, z) where x extends along the generator, y is in the tangential, and z the inward radial direction, respectively. The middle surface displacement components are designated by u, v, and w, respectively. Here, w, denotes the total radial displacement measured from the middle surface of a shell of perfectly circular shape. The initial geometric imperfections of the shell are represented by Wo (x, y), again measured from the middle surface of a perfect shell. The dynamic response in the radial direction is represented as w: = w - wo. With these definitions, the middle surface strain-displacement relations, with a consideration of the initial geometric imperfections are:
a.
1.
1/~.,~"
.
l(a.,)'-
Ou 3v 3w 3w Y = ~ + T x ~ ax av
~Research Assistant. ±Professor of Civil Engineering.
1 (aWo~ ~"
l fOwol'-
(1)
Owo 3wo i~x ay "
These relations lead to the coupled non-linear equations 125
of motion:
patibility (3) and subsequent integration <~t ih~,. equ~tio~ yields the stress function ,, 3x: "" 3)'- ,' Oy"
",3x~y ~ ~ / ~
/Oewo
O:w,\Oe~
1 Oz~
\ ay-
oy-/a.r
~a.-Tr-~
- / -----v- z" ------r- / "-----¢ +
3"wL , ~_ -
h
(2)
~v,,a,=(o:~,t-' o:w,e:~,
l e:w, vaxoy/ - --r-ox ov-'-7-- ~ } x :
a~'~oa'-w,~ - ta'~,O'~o+.O'~,a-Wo ,----~-.-~, za..ff~va-ffa.-.;~~-_-.:..--~---.-~) \ ox- o."
ax y Ox
Ox" a y ' l
(3)
where q, represents externally applied radial load per unit surface area, p is shell density, D is shell ttexural rigidity, and • denotes a stress function of the middle surface forces. The motion equation (2) and the compatibility equation (3) govern the dynamic behavior of the shell. To gain an understanding of the influence of initial geometric imperfections on the dynamic behavior of the shell a simple model of the imperfections in considered, viz: Wo(X. y)= W,)sin
.
m~x
sm L
dO= E[A, cos 2rx + A,_ sin2sy -,- .4, ~.Os " "a"n y -.4,, cos sy sin3rx + A~ sinsy sin 3rx -- ,4,, cos sv sm :'v A7 sinsy sin rr)
where the A~ are functions of the time-dependent amplitude functions /~(t), f_.(t), and f3(t). These functions are given in [7]. With the above vatue of ~ together with the expressions for wo and w, it is possible to enforce the condition of periodicity of circumferentiai displacement which results in the relation f,(t)=~-[f,:(t)+f..:(t).-2fz(t)14,%].
WI
G, = 77, = cos sy sin rx +
W)(X,
V
t) = f,(t) cos y sin rx + f2(t) sin sy sin rx + fdt) sin'- rx (5)
where the fdt) are time-dependent amplutude functions and s = n / R , r = m ~ L . The first two terms of (5) represent a harmonic distribution of response around the shell circumference and the third term corresponds to an axisymmetric response. The third term is necessary to permit satisfaction of periodicity of circumferential displacement. The vibratory configuration (5) corresponds to zero radial displacement at the ends of the shell, but does not represent a moment-free condition at those ends. Accordingly the boundary conditions represented lies intermediate between simply supported and clamped ends. Since we are interested in customary configurations of liquid storage tanks, shells whose length is small compared to the shell radius are of little interest. It is only in such relatively short shells that a precise specification of boundary conditions is of major importance. The analysis is carried out for two cases: Case 1. The circumferential wave pattern of the response is the same as that of the imperfection, i.e. = n. In this situation we have wo(x, y) = W~,sin sy sin rx w,(x.
y.
t) = fdt) cos sy sin rx + fz(t) sin sy sin rx + f3(t) sin" rx. {6)
Substitution of these relations into the equation of corn-
IS}
Consideration must now be given to the equation of motion (2). Substitution of the above values of ~v,), w,, and ~ yields a relation containing the two unknown functions [,(t) and [~.(t). Direct solution is impossible, so the Galerkin technique was employed to obtain an approximate solution. The Galerkin weighting functions employed were
(4)
where Wo is the amplitude of the imperfection. It is recognized that realistic imperfection configurations may, at times, require some, if not many, additional terms. However, the analysis and accompanying computer effort would become of almost prohibitive length were more terms employed. The characterizztion of the dynamic response is taken to be
"71
s~__~R f,_ sin'- rx
0:= ~ = sins,, sinrx + s~_~_R(.f: + Wo)sin-"rx.
19)
These functions when employed in the Galerkin approach yield two coupled equations involving/,, f=, their second derivatives with respect to time. as well as r, s. sin rx, cos sy, and Wo, together with functions representing the generalized forces acting on the shell. These forces are given by integral equations involving qAx, y, t), G~ and G~_. The external radial force q,(x, y, t) is assumed to be fixed in space and harmonic in time in the form, q,( x, y, t) = Q(x, y) cosaJt. 110) Therefore the generalized forcing functions c(t) and s(t) are given by c(t) = C... cos.Jr Ill) s(t) = S,.. cos~ot where, C... = JTOL f o 2 ~ g O(x. y) [cos sy sin rx ~-@ [t sin: rx] dx d y i12) S,.. = /)" foZ"R Q(x, y) [sin sy sin rx + ~ ( f ~ + Wo) sin-"rx] dx dy. Though the above equations were derived using the Gelerkin method, they could also have been obtained by the Rayleigh--Ritz method. Although in general these two methods are not equivalent, they can be made to coincide by proper selection of the weighting functions. The two weighting functions chosen above, G, = aw,L~f and G. = ~w~/Sfz satisfy these requirements.
Influence of initial geometric imperfections on vibrations APPLICATIONOF TIlE METItODOF AVERAGING The two coupled ordinary non-linear differential equations cannot yet be solved exactly. But an approximate solution to them can be obtained by the procedure known as the method of averaging. The unknown functions f~(t) and (.At) are taken to be in the form,
resulting in T.wo coupled ordinary non-linear differential equations to which the method of averaging may be appiied to yield - flZ(F t 4- a,F~ 3 - atFiF: 2) 4 a,Fl + 3asFi 3 + a~Fl" + a,F! F2 2 + 5a.F, ~+ a.Fi 3Ff" + a,.F, F ? = C,.,. (21)
.f,(t) = B,(t) cos tot
(13)
- fl=(F: + a,F: ~ - a~F,:F2) + a,.F: + 3a~F: ~ + a~F, ~"& + a~F~'F: + asF, eF: 3 + 5~F.- ~ = S,.,,, (22)
and then applying the method of averaging gives the approximate solution as
where the coefficients E are defined in [7]. Free vibrations of the imperfect cylindrical shell may be investigated by setting C,.,~ = S,.m = 0 in (16) and (17) for Case 1 and in (21) and (22) for Case 2. The forced excitation of the cylindrical shell having its geometric axis oriented vertically and subject to harmonic lateral excitation at its lower extremity is of importance in the case of seismically excited liquid storage tanks. For this case the support motion is taken to be,
fz(t) = B,(t) cos tot
f,(t) =/3~ coswt
(14) f~_(t) =/~: cos wt.
The resulting pair of coupled non-linear algebraic equations may conveniently be represented in terms of the dimensionless variables
0. = A~ cos tod F, =~-- F. = - ~ ; lg/o=--~--; fP = E/--7~~
(15) -
C,.m
S,,,.
C.,,. = (1-1:Eh 2L[2R) and S,.,. = (I-I2Eh : LI2R)
where to, is the frequency of the harmonic excitation. Measuring the shell deformations relative to the support. the effective radial force per unit area due to the support acceleration is. q~(x, y, t~. = p h A, cosv/R . cos wt
in the form -
I)'~(F, * a, F ~ ~ + a, F22) + a._Fi + asF, 3 _ . +-'~ 1~ a~l', - , - *-'£ 15 ~,e,- 3 -1":'+ . ~asF,~" . ~3 e~.F,1"~_"
= C..,~ (16)
(23)
(24)
which is directed radially inwards. The effective tangential force is disregarded in the analysis. The spatial distribution of the radial force Q(x, y), according to the notation given in eqn (10) is, Q(x, y) = phA~ cos y/R
(25)
and - fl:(F~ + ~, F , : F : + a , F2 ~ + 2c~, W o : F : ) + a . F : + a . & s
+3 - 2 -1"2 + ~ a ~ F I , F 2 = ~ , . , ~ i ~.a,1", 8
Then, the generalized force components C,.,~ and S.,,. are obtained by substituting the above in eqns (12)
(17) C.,., =
where the coefficients F~ are given in [7]. Case 2. The circumferential wave pattern of the response is different than that of the imperfection, i.e. j # n. In this case we take
This leads to an expression for • analogous to (7) and imposition of periodicity of circumferential displacement leads to the relation (19)
The Galerkin approach is employed, using as weighting functions
Sm~ =
(26)
phA~ cos y/R [sin sy sin rx focfo >'R +fiR.. T ( I : + Wo) sin 2 rx]
= 0 for all m and n.
(20)
(27)
Liquid filled cylinder We shall consider the case of a cylindrical shell with vertical axis and completely filled with ideal fluid. The governing equation of liquid motion is r-~rk, r-~r: +~r- 'c~O:+ ~z-" = 0
aw~ G~ = ~ = cos sv. sin rx + s~.~Rf, sin 2 rx 0wl G: = ~ = sin sv• sin rx + s~_~Rf. sin: rx
= 2RLohAs for n = 1 and odd m otherwise.
and,
w (x, v) = f,(t) cos sy sin rx + [..(t) sin sy sin rx + f, ft) sin 2 rx. (18)
[f, :(t) + f2(t)].
sZR + - - 7 - f ' sin" rx] dx dy.
m
wo(x, y) = Wo sinpy sin rx
f3(t) = ~
foL f02~rRohA~ cos y/R[cos sy sin rx
(28)
where P(r, O. z, t) is the velocity potential of the fluid. The boundary conditions of the fluid contained in the cylinder are as follows:
11) Free surface condition at the tank top.
if the frequency of vibration of the coupled iiquid-sheil system is denoted by oz then. , kL ~-=--E -tan !-E~ -
gk
:
;4~
(2) Boundary condition at the tank base. where k is an arbitrary constant yet unknown. The boundary condition at the tank wall given in (31) is the only remaining condition that has not yet been satisfied. Satisfaction of liquid-shell interface conditions yields two equations:
(3) Condition at the common boundary between the fluid and the tank wall.
D
OP
4
~-.-~[(r + 2r2s "-- s~)(f~ cos sy + f: sinsy) sin rx
Ou, ]
- ~ - -~-j,=~ =o
(3l) - 8r'f3 cos 2rx] + -~ [(f'l cos sy + f_~sin syt sin rx
-/'~
where u, is the elastic displacement of the tank wall, measured positive radially outwards. It should be noted that later in the analysis u, is replaced by - w~, so as to be compatible with the notation of the shell analysis. Working with a separable solution of (28) we obtain a solution involving harmonic functions circumferentially as well as axially and Bessel functions radially. Satisfaction of boundary conditions at the tank base lead to the velocity potential P(r, O, z, t)==~ [A~(t)cosiO-. B~(t)siniO]cos-~I~
-
2/-~ ~ f ~ i y . jcrx Eh,.~t ~ , It(k)• p i i cos ~- s , n T ÷
/ kL \
= .
.
[T,t "~,'~'/ ,.
•
iy
. j~rx ]
B,, s,n ~- sm'--L-]
= - j , cossysinrx-/=sinsysinrx-j~sinZrx.
(33)
125 h/R,I/720 L/R,2/3 v,0.272 m,5 n -25
Reference
6
Present
j ,25 Ampl.ituaes
I
05,
0.00~
2
OrOI
3
0.1
4
0.2
5
04
6
0.6
7
O8
8
I0
0 8~.
7 0
j 0 2
0 4
~6
(36)
Thus finally the set of equations governing the coupted liquid-shell behavior are eqns (34)--(36), which are to be satisfied simultaneously. Application of Galerkin's method yields a pair of equations that are of the same nature as those found previously for the empty cylin-
/~, -'-~ tan (--~-)B, = 0.
ti.5.
]
.IA,,co.'.~"s m - c +
0
=
. iv . jTrx ]
Bn s,n~,,n-L-
where the remaining terms of the equation are denoted by H~, for brevity and are given in [7], and
Imposition of the liquid free surface condition (29) now leads to the relations gk
(35)
= H (f,,f:,f~.r x s, v)
(32)
A , - - ~ tan ~ - ~ )A,
sin-"rx}
0 S
I 0
,~o Fig. 1. Frequency of vibration vs ~mplitude of iml~rfection for different values of amplitude of 'dbradon. All parameters are in non-dimensional form.
129
Influence of initial geometric imperfections on vibrations drical shell, but with p replaced by a new variable p, defined as
l.(t) 2_ L / L r
"
Thus. the application of the method of averaging may be carried out on the liquid-filled cylindrical shell in the same manner as for the empty shell. NUMERICAL RESULTS
As a necessary check on the numerical accuracy of the present approach, the case of free vibrations of an initially perfect cylindrical shell was considered. The first four circumferential modes as well as the first four axial modes of a cylinder having a thickness/radius ratio of 0.01 and a length/radius ratio of 1.0 were determined by the present approach and compared with the tabulated values given in [8] which were obtained using a threedimensional wave propagation theory. In all cases the natural frequencies differed by no more than two percent, thus lending confidence to the current approach. Next, free vibrations of an imperfect cylindrical shell were considered. The case of a shell having a thickness/radius ratio of 1/720, a length/radius ratio of 2/3 and Poisson's ratio of 0.272 was examined in detail. These parameters were selected because they are representative of contemporary prototype liquid storage tanks. To gain some indication of the approximate number of axial and circumferential waves that should reasonabl~ be considered in the free vibration problem the corresponding number that would form during asymmetric buckling of a perfect shell was determined from [9]. Each of these numbers was then varied to ascertain minimum natural frequencies for each mode. For example, Fig. 1 indicates the results of Case 1 with n = j = 25 and m = 5. The amplitude of imperfection is increased
from zero (perfect shell) to unity and the amplituae of vibration from 0.001 to 1.0. It is evident that the noniinearitx in this case is of the softening type. Also included in that figure is the curve corresponding to the analysis of [6]. Analogous plots for n ¢ j were developed for the circumferential wave number (n) varying from 15 to 30 with m = 5 and the circumferential wave number of imper|ection j = 25 but are not shown for the sake of brevity. In each of these the non-linearity was of the softening type. Figure 2 indicates the relationship between frequency of vibration and the circumferential wave number (n) for varying values of amplitude of initial imperfection at a very small amplitude of vibration. Results of the present analysis and those of [6] are shown by solid and broken lines, respectively. It is evident that the results are in good agreement except when n = j. At this value of n (25 for this example) [6] predicts an increase of frequency whereas the present study predicts a significant decrease. Analogous plots are given in [7] for other values of j. Additional information pertinent to the behavior of the shell as n and ] vary (but m = constant) is indicated in Figs. 3 and 4. Here. the interesting parameter is the aspect ratio of the vibrational configuration, characterized as (rrRIn)l(LIm). It is to be observed that at higher values of the aspect ratio the non-linearity is of the hardening type, whereas at lower values it is of a softening nature. Response of an empty cylindrical shell with axis vertical and lower extremity clamped to a rigid slab undergoing horizontal seismic motion represented by the standard artificial earthquake [10] with peak acceleration of 0.5 g is represented in Fig. 5 for the case n = j = 1. In this case, the upper extremity of the shell was considered to be unsupported and the forced response indicated represents peak radial displacement of a generator lying in a diametral plane about which the base excitation is symmetric.
I 40,
h l R = I/720 LIR,2/3 m,,5 j.25 I 30
•
I 20
-
. . . .
Reference 6 Present
/
', \
/
/
',
t .00
0 90-
o8o I
i
i
i
n
Fig. 2. Frequency of free vibration vs circumferential wave number for different values of amplitude of imperfection. Amplitude of vibration = 0.00l.
130
'37
A~mp(i t,u,~.es 000! 2 90~ 3 ol 402 '504 606 7 98 8 o
] h/L:t/720 L / R =2./5 i p : 0272 m=5
n=
j =~
~ = 15/2 54 ¸
;2 ./.>>2
5i
3,
1,
Natural frequencies of the ~ame cylindr~cai sneii completely filled with water were determined for a range of values of amplitude of initial imperfection ,Jp ~o md including a value equal to the shell thickness. For very small dimensionless amplitudes of vibration (0.OOL~it was found that the decrease in natural frequency is very nearb the same as for the empty shell, namely slightly more than ten percent. Maximum decrease was found to occur at an imperfection amplitude of approximately half ~he shell thickness, with less severe decreases with increasing values of imperfection. Acknowledgement--The authors would like to express their gratitude to the National Science Foundation for their support of this study under grant CEE-76-t483L
46.
, .45 t-,
02
04.
08
06
~0
REFERENCES
,g,o Fig. 3. Frequency of free vibration vs amplitude of imperfection for different values of amplitude of vibration. Case 1. n = ].
24
h/R
•
I 1720
I 0.0Ol 2 0.0t 3 0,1
t.../ R =2/3 t, =0.272. I:6
m=5
4
n= j,20 ( • 3 rr/8 08
0.2
0.4 6 0.,6 7 0.8 80I 5
1 2 ~ ,-, " ~
O0 C92 084
)
~; 2
04
06
08
0
#o Fig. 4. Frequency o[ free vibration vs amplitude of imperfection for different values of amplitude of vibration. Case 1, n =/. 15
1. H. N. Chu, Influence of large amplitudes on flexural vibrations of a thin circular cylindrical shell. J. Aerospace Sci. 28, 602-..609 (1961). 2. J. Nowinski, Non-linear transverse vibrations of orthotropic cylindrical shells. AIAA ]. 1,617-.620 (1963). 3. D. A. Evensen, Some observations on the non-linear vibrations of thin cylindrical shells. AIAA J, 1, 2857-2858 (1%3t. 4. M. D. Olson, Some experimental observations on the nonlinear vibrations of thin cylindrical shells. A[AA 7. 3, 17751777 (1%5). 5. D. A. Evensen. Non-linear flexurat vibrations of thin walled circular cylinders. NASA TN D-4090, p. 29 (1%7). 6, A. Rosen and J. Singer, Influence of asymmetric imperfections on the vibrations of axially compressed cylindrical shells. TAE Rep. 212, Technion-Israel Institute of Technology, Haifa, Israel, 1975. 7. L. Watawala, Influence of initial geometric imperfections on vibrations of thin circular cylindrical shells. PhlD. Dis~ertation, University of Massachusetts, 1981. 8. A. E. Armenakas, D. C. Gazis and G. Herrman, Free Vibrations of Circular Cylindricai Shells. Pergamon Press. London (1%9). 9. W. T. Koiter, On the stability of elastic equilibrium. Thesis, Delft, The Netherlands, 1941. Available in English as NASA TTF-10833, 1967. IO. P. Ruiz and J. Penzien, Artificial generation of earthquake accelerograms. Programs available as PSEQGN from the National Information Service, Earthquake Engineering, University of California, Berkeley, California.
y
/~,~./
GOg 2. "
0
h/R. 1/720 L/R • 213 m .5 n~ : 0j 272 :1 C5.~ 0.00t S5,~ O0
/
09-
F, o,
O3
© I 43
; 45
I 47
I 4.9
I ,51
53
,G Fig. 5. Amplitude of forced vibration vs frequency of excitation for different values of amplitude of imperfection.
004%794918310101254)6~3.0010
Prinled in Greal Britain
Pergamon Press Lid
INFLUENCE OF INITIAL GEOMETRIC IMPERFECTIONS ON VIBRATIONS OF THIN CIRCULAR CYLINDRICAL SHELLS LaKSHMAN WATAWALA'~and WILLIAMA. NASH* Department of Civil Engineering,University of Massachusetts, Amherst, MA 01003, U.S.A. AMtl-aet--The influence of initial geometric imperfections (out-of-roundness) on vibrations of a right circular cylindrical shell is investigated on the basis on non-linear large deformation shell theory. Both free and forced motions (due to base excitation caused by seismic effects) are treated. An extension to the case of the cylindrical shell with vertical geometric axis and completelyfilled with a perfect liquid is also studied. Numerical results are presented for various initial imperfectionsand vibratory configurations.It is found that the behavior may be of the hardening type for certain types of response yet of a softening nature for other responses. BACKGROUND
Although the influence of initial geometric imperfections on buckling resistance of a cylindrical shell has been the topic of numerous investigations during the past few decades, the literature contains only a very few results associated with the influence of such imperfections of vibration characteristics of the shell. One very important problem area where imperfections may be of importance lies in the response of cylindrical liquid storage tanks to seismic excitation of the base. Since such tanks invariably have slightly non-circular cross-sections due to fabrication problems, it is of interest to examine the influence of typical imperfections on the natural frequencies of liquid filled cylindrical storage tanks. It is also of interest to ascertain the influence of realistic imperfections on response of such tanks to seismic motions of the base. Since the seismic effects on such tanks may reasonably be expected to develop displacements of the order of magnitude of the shell thickness, it is necessary to consider non-linear shell theory. Although a number of treatments of free vibrations of cylindrical shells undergoing such large displacements are to be found in the literature, several warrant particular mention. In 1%1 Chu[1] employed a single mode response form to investigate large amplitude free vibrations and concluded that the non-linear behavior of the shell was always of the hardening type, i.e. the frequency increases with the amplitude of vibration. A 1963 study of orthotropic shells by Nowinski[2] led to the same conclusion. However, experimental studies by Evensen[3] and Olson[4] indicated that the non-linearity was of the softening type, i.e. frequencies decreased with increasing amplitude. In [3] the author pointed out that the analysis in [1] implied multiple-valued circumferential displacements and advanced this as one reason for the discrepancy in results. In a later analysis, Evensen [5] employed a three term mode shape which made it possible to develop single valued circumferential displacements. The results of this work showed that the non-linearity was either softening or hardening depending on the aspect ratio (ratio of the circumferential wavelength to axil wavelength) of the mode. This work did not consider initial geometric imperfections of the shell.
A recent study of the influence of imperfections on the vibrations of thin cylindrical shells due to Rosen and Singer [6] used a linearized version of Koiter's non-linear equations with inertia terms included. This linearization was performed assuming the amplitudes of vibration to be infinitesimal so as to greatly simplify the analysis while retaining the imperfection terms. However, it appears that these authors did not satisfy the condition of single valuedness of circumferential displacement for the case when the dynamic behavior has the same wave form as that of the initial imperfections. The present investigation is concerned with the influence of initial geometric imperfections on free vibrations of cylindrical shells, as well as with the behavior of such shells filled with liquid and subject to seismic motions of the base.
ANALYSIS
Let us consider a circular cylindrical shell of mean radius R, thickness h, and length L. Further, we introduce an orthogonal coordinate system (x, y, z) where x extends along the generator, y is in the tangential, and z the inward radial direction, respectively. The middle surface displacement components are designated by u, v, and w, respectively. Here, w, denotes the total radial displacement measured from the middle surface of a shell of perfectly circular shape. The initial geometric imperfections of the shell are represented by Wo (x, y), again measured from the middle surface of a perfect shell. The dynamic response in the radial direction is represented as w: = w - wo. With these definitions, the middle surface strain-displacement relations, with a consideration of the initial geometric imperfections are:
a.
1.
1/~.,~"
.
l(a.,)'-
Ou 3v 3w 3w Y = ~ + T x ~ ax av
~Research Assistant. ±Professor of Civil Engineering.
1 (aWo~ ~"
l fOwol'-
(1)
Owo 3wo i~x ay "
These relations lead to the coupled non-linear equations 125
of motion:
patibility (3) and subsequent integration <~t ih~,. equ~tio~ yields the stress function ,, 3x: "" 3)'- ,' Oy"
",3x~y ~ ~ / ~
/Oewo
O:w,\Oe~
1 Oz~
\ ay-
oy-/a.r
~a.-Tr-~
- / -----v- z" ------r- / "-----¢ +
3"wL , ~_ -
h
(2)
~v,,a,=(o:~,t-' o:w,e:~,
l e:w, vaxoy/ - --r-ox ov-'-7-- ~ } x :
a~'~oa'-w,~ - ta'~,O'~o+.O'~,a-Wo ,----~-.-~, za..ff~va-ffa.-.;~~-_-.:..--~---.-~) \ ox- o."
ax y Ox
Ox" a y ' l
(3)
where q, represents externally applied radial load per unit surface area, p is shell density, D is shell ttexural rigidity, and • denotes a stress function of the middle surface forces. The motion equation (2) and the compatibility equation (3) govern the dynamic behavior of the shell. To gain an understanding of the influence of initial geometric imperfections on the dynamic behavior of the shell a simple model of the imperfections in considered, viz: Wo(X. y)= W,)sin
.
m~x
sm L
dO= E[A, cos 2rx + A,_ sin2sy -,- .4, ~.Os " "a"n y -.4,, cos sy sin3rx + A~ sinsy sin 3rx -- ,4,, cos sv sm :'v A7 sinsy sin rr)
where the A~ are functions of the time-dependent amplitude functions /~(t), f_.(t), and f3(t). These functions are given in [7]. With the above vatue of ~ together with the expressions for wo and w, it is possible to enforce the condition of periodicity of circumferentiai displacement which results in the relation f,(t)=~-[f,:(t)+f..:(t).-2fz(t)14,%].
WI
G, = 77, = cos sy sin rx +
W)(X,
V
t) = f,(t) cos y sin rx + f2(t) sin sy sin rx + fdt) sin'- rx (5)
where the fdt) are time-dependent amplutude functions and s = n / R , r = m ~ L . The first two terms of (5) represent a harmonic distribution of response around the shell circumference and the third term corresponds to an axisymmetric response. The third term is necessary to permit satisfaction of periodicity of circumferential displacement. The vibratory configuration (5) corresponds to zero radial displacement at the ends of the shell, but does not represent a moment-free condition at those ends. Accordingly the boundary conditions represented lies intermediate between simply supported and clamped ends. Since we are interested in customary configurations of liquid storage tanks, shells whose length is small compared to the shell radius are of little interest. It is only in such relatively short shells that a precise specification of boundary conditions is of major importance. The analysis is carried out for two cases: Case 1. The circumferential wave pattern of the response is the same as that of the imperfection, i.e. = n. In this situation we have wo(x, y) = W~,sin sy sin rx w,(x.
y.
t) = fdt) cos sy sin rx + fz(t) sin sy sin rx + f3(t) sin" rx. {6)
Substitution of these relations into the equation of corn-
IS}
Consideration must now be given to the equation of motion (2). Substitution of the above values of ~v,), w,, and ~ yields a relation containing the two unknown functions [,(t) and [~.(t). Direct solution is impossible, so the Galerkin technique was employed to obtain an approximate solution. The Galerkin weighting functions employed were
(4)
where Wo is the amplitude of the imperfection. It is recognized that realistic imperfection configurations may, at times, require some, if not many, additional terms. However, the analysis and accompanying computer effort would become of almost prohibitive length were more terms employed. The characterizztion of the dynamic response is taken to be
"71
s~__~R f,_ sin'- rx
0:= ~ = sins,, sinrx + s~_~_R(.f: + Wo)sin-"rx.
19)
These functions when employed in the Galerkin approach yield two coupled equations involving/,, f=, their second derivatives with respect to time. as well as r, s. sin rx, cos sy, and Wo, together with functions representing the generalized forces acting on the shell. These forces are given by integral equations involving qAx, y, t), G~ and G~_. The external radial force q,(x, y, t) is assumed to be fixed in space and harmonic in time in the form, q,( x, y, t) = Q(x, y) cosaJt. 110) Therefore the generalized forcing functions c(t) and s(t) are given by c(t) = C... cos.Jr Ill) s(t) = S,.. cos~ot where, C... = JTOL f o 2 ~ g O(x. y) [cos sy sin rx ~-@ [t sin: rx] dx d y i12) S,.. = /)" foZ"R Q(x, y) [sin sy sin rx + ~ ( f ~ + Wo) sin-"rx] dx dy. Though the above equations were derived using the Gelerkin method, they could also have been obtained by the Rayleigh--Ritz method. Although in general these two methods are not equivalent, they can be made to coincide by proper selection of the weighting functions. The two weighting functions chosen above, G, = aw,L~f and G. = ~w~/Sfz satisfy these requirements.
Influence of initial geometric imperfections on vibrations APPLICATIONOF TIlE METItODOF AVERAGING The two coupled ordinary non-linear differential equations cannot yet be solved exactly. But an approximate solution to them can be obtained by the procedure known as the method of averaging. The unknown functions f~(t) and (.At) are taken to be in the form,
resulting in T.wo coupled ordinary non-linear differential equations to which the method of averaging may be appiied to yield - flZ(F t 4- a,F~ 3 - atFiF: 2) 4 a,Fl + 3asFi 3 + a~Fl" + a,F! F2 2 + 5a.F, ~+ a.Fi 3Ff" + a,.F, F ? = C,.,. (21)
.f,(t) = B,(t) cos tot
(13)
- fl=(F: + a,F: ~ - a~F,:F2) + a,.F: + 3a~F: ~ + a~F, ~"& + a~F~'F: + asF, eF: 3 + 5~F.- ~ = S,.,,, (22)
and then applying the method of averaging gives the approximate solution as
where the coefficients E are defined in [7]. Free vibrations of the imperfect cylindrical shell may be investigated by setting C,.,~ = S,.m = 0 in (16) and (17) for Case 1 and in (21) and (22) for Case 2. The forced excitation of the cylindrical shell having its geometric axis oriented vertically and subject to harmonic lateral excitation at its lower extremity is of importance in the case of seismically excited liquid storage tanks. For this case the support motion is taken to be,
fz(t) = B,(t) cos tot
f,(t) =/3~ coswt
(14) f~_(t) =/~: cos wt.
The resulting pair of coupled non-linear algebraic equations may conveniently be represented in terms of the dimensionless variables
0. = A~ cos tod F, =~-- F. = - ~ ; lg/o=--~--; fP = E/--7~~
(15) -
C,.m
S,,,.
C.,,. = (1-1:Eh 2L[2R) and S,.,. = (I-I2Eh : LI2R)
where to, is the frequency of the harmonic excitation. Measuring the shell deformations relative to the support. the effective radial force per unit area due to the support acceleration is. q~(x, y, t~. = p h A, cosv/R . cos wt
in the form -
I)'~(F, * a, F ~ ~ + a, F22) + a._Fi + asF, 3 _ . +-'~ 1~ a~l', - , - *-'£ 15 ~,e,- 3 -1":'+ . ~asF,~" . ~3 e~.F,1"~_"
= C..,~ (16)
(23)
(24)
which is directed radially inwards. The effective tangential force is disregarded in the analysis. The spatial distribution of the radial force Q(x, y), according to the notation given in eqn (10) is, Q(x, y) = phA~ cos y/R
(25)
and - fl:(F~ + ~, F , : F : + a , F2 ~ + 2c~, W o : F : ) + a . F : + a . & s
+3 - 2 -1"2 + ~ a ~ F I , F 2 = ~ , . , ~ i ~.a,1", 8
Then, the generalized force components C,.,~ and S.,,. are obtained by substituting the above in eqns (12)
(17) C.,., =
where the coefficients F~ are given in [7]. Case 2. The circumferential wave pattern of the response is different than that of the imperfection, i.e. j # n. In this case we take
This leads to an expression for • analogous to (7) and imposition of periodicity of circumferential displacement leads to the relation (19)
The Galerkin approach is employed, using as weighting functions
Sm~ =
(26)
phA~ cos y/R [sin sy sin rx focfo >'R +fiR.. T ( I : + Wo) sin 2 rx]
= 0 for all m and n.
(20)
(27)
Liquid filled cylinder We shall consider the case of a cylindrical shell with vertical axis and completely filled with ideal fluid. The governing equation of liquid motion is r-~rk, r-~r: +~r- 'c~O:+ ~z-" = 0
aw~ G~ = ~ = cos sv. sin rx + s~.~Rf, sin 2 rx 0wl G: = ~ = sin sv• sin rx + s~_~Rf. sin: rx
= 2RLohAs for n = 1 and odd m otherwise.
and,
w (x, v) = f,(t) cos sy sin rx + [..(t) sin sy sin rx + f, ft) sin 2 rx. (18)
[f, :(t) + f2(t)].
sZR + - - 7 - f ' sin" rx] dx dy.
m
wo(x, y) = Wo sinpy sin rx
f3(t) = ~
foL f02~rRohA~ cos y/R[cos sy sin rx
(28)
where P(r, O. z, t) is the velocity potential of the fluid. The boundary conditions of the fluid contained in the cylinder are as follows:
11) Free surface condition at the tank top.
if the frequency of vibration of the coupled iiquid-sheil system is denoted by oz then. , kL ~-=--E -tan !-E~ -
gk
:
;4~
(2) Boundary condition at the tank base. where k is an arbitrary constant yet unknown. The boundary condition at the tank wall given in (31) is the only remaining condition that has not yet been satisfied. Satisfaction of liquid-shell interface conditions yields two equations:
(3) Condition at the common boundary between the fluid and the tank wall.
D
OP
4
~-.-~[(r + 2r2s "-- s~)(f~ cos sy + f: sinsy) sin rx
Ou, ]
- ~ - -~-j,=~ =o
(3l) - 8r'f3 cos 2rx] + -~ [(f'l cos sy + f_~sin syt sin rx
-/'~
where u, is the elastic displacement of the tank wall, measured positive radially outwards. It should be noted that later in the analysis u, is replaced by - w~, so as to be compatible with the notation of the shell analysis. Working with a separable solution of (28) we obtain a solution involving harmonic functions circumferentially as well as axially and Bessel functions radially. Satisfaction of boundary conditions at the tank base lead to the velocity potential P(r, O, z, t)==~ [A~(t)cosiO-. B~(t)siniO]cos-~I~
-
2/-~ ~ f ~ i y . jcrx Eh,.~t ~ , It(k)• p i i cos ~- s , n T ÷
/ kL \
= .
.
[T,t "~,'~'/ ,.
•
iy
. j~rx ]
B,, s,n ~- sm'--L-]
= - j , cossysinrx-/=sinsysinrx-j~sinZrx.
(33)
125 h/R,I/720 L/R,2/3 v,0.272 m,5 n -25
Reference
6
Present
j ,25 Ampl.ituaes
I
05,
0.00~
2
OrOI
3
0.1
4
0.2
5
04
6
0.6
7
O8
8
I0
0 8~.
7 0
j 0 2
0 4
~6
(36)
Thus finally the set of equations governing the coupted liquid-shell behavior are eqns (34)--(36), which are to be satisfied simultaneously. Application of Galerkin's method yields a pair of equations that are of the same nature as those found previously for the empty cylin-
/~, -'-~ tan (--~-)B, = 0.
ti.5.
]
.IA,,co.'.~"s m - c +
0
=
. iv . jTrx ]
Bn s,n~,,n-L-
where the remaining terms of the equation are denoted by H~, for brevity and are given in [7], and
Imposition of the liquid free surface condition (29) now leads to the relations gk
(35)
= H (f,,f:,f~.r x s, v)
(32)
A , - - ~ tan ~ - ~ )A,
sin-"rx}
0 S
I 0
,~o Fig. 1. Frequency of vibration vs ~mplitude of iml~rfection for different values of amplitude of 'dbradon. All parameters are in non-dimensional form.
129
Influence of initial geometric imperfections on vibrations drical shell, but with p replaced by a new variable p, defined as
l.(t) 2_ L / L r
"
Thus. the application of the method of averaging may be carried out on the liquid-filled cylindrical shell in the same manner as for the empty shell. NUMERICAL RESULTS
As a necessary check on the numerical accuracy of the present approach, the case of free vibrations of an initially perfect cylindrical shell was considered. The first four circumferential modes as well as the first four axial modes of a cylinder having a thickness/radius ratio of 0.01 and a length/radius ratio of 1.0 were determined by the present approach and compared with the tabulated values given in [8] which were obtained using a threedimensional wave propagation theory. In all cases the natural frequencies differed by no more than two percent, thus lending confidence to the current approach. Next, free vibrations of an imperfect cylindrical shell were considered. The case of a shell having a thickness/radius ratio of 1/720, a length/radius ratio of 2/3 and Poisson's ratio of 0.272 was examined in detail. These parameters were selected because they are representative of contemporary prototype liquid storage tanks. To gain some indication of the approximate number of axial and circumferential waves that should reasonabl~ be considered in the free vibration problem the corresponding number that would form during asymmetric buckling of a perfect shell was determined from [9]. Each of these numbers was then varied to ascertain minimum natural frequencies for each mode. For example, Fig. 1 indicates the results of Case 1 with n = j = 25 and m = 5. The amplitude of imperfection is increased
from zero (perfect shell) to unity and the amplituae of vibration from 0.001 to 1.0. It is evident that the noniinearitx in this case is of the softening type. Also included in that figure is the curve corresponding to the analysis of [6]. Analogous plots for n ¢ j were developed for the circumferential wave number (n) varying from 15 to 30 with m = 5 and the circumferential wave number of imper|ection j = 25 but are not shown for the sake of brevity. In each of these the non-linearity was of the softening type. Figure 2 indicates the relationship between frequency of vibration and the circumferential wave number (n) for varying values of amplitude of initial imperfection at a very small amplitude of vibration. Results of the present analysis and those of [6] are shown by solid and broken lines, respectively. It is evident that the results are in good agreement except when n = j. At this value of n (25 for this example) [6] predicts an increase of frequency whereas the present study predicts a significant decrease. Analogous plots are given in [7] for other values of j. Additional information pertinent to the behavior of the shell as n and ] vary (but m = constant) is indicated in Figs. 3 and 4. Here. the interesting parameter is the aspect ratio of the vibrational configuration, characterized as (rrRIn)l(LIm). It is to be observed that at higher values of the aspect ratio the non-linearity is of the hardening type, whereas at lower values it is of a softening nature. Response of an empty cylindrical shell with axis vertical and lower extremity clamped to a rigid slab undergoing horizontal seismic motion represented by the standard artificial earthquake [10] with peak acceleration of 0.5 g is represented in Fig. 5 for the case n = j = 1. In this case, the upper extremity of the shell was considered to be unsupported and the forced response indicated represents peak radial displacement of a generator lying in a diametral plane about which the base excitation is symmetric.
I 40,
h l R = I/720 LIR,2/3 m,,5 j.25 I 30
•
I 20
-
. . . .
Reference 6 Present
/
', \
/
/
',
t .00
0 90-
o8o I
i
i
i
n
Fig. 2. Frequency of free vibration vs circumferential wave number for different values of amplitude of imperfection. Amplitude of vibration = 0.00l.
130
'37
A~mp(i t,u,~.es 000! 2 90~ 3 ol 402 '504 606 7 98 8 o
] h/L:t/720 L / R =2./5 i p : 0272 m=5
n=
j =~
~ = 15/2 54 ¸
;2 ./.>>2
5i
3,
1,
Natural frequencies of the ~ame cylindr~cai sneii completely filled with water were determined for a range of values of amplitude of initial imperfection ,Jp ~o md including a value equal to the shell thickness. For very small dimensionless amplitudes of vibration (0.OOL~it was found that the decrease in natural frequency is very nearb the same as for the empty shell, namely slightly more than ten percent. Maximum decrease was found to occur at an imperfection amplitude of approximately half ~he shell thickness, with less severe decreases with increasing values of imperfection. Acknowledgement--The authors would like to express their gratitude to the National Science Foundation for their support of this study under grant CEE-76-t483L
46.
, .45 t-,
02
04.
08
06
~0
REFERENCES
,g,o Fig. 3. Frequency of free vibration vs amplitude of imperfection for different values of amplitude of vibration. Case 1. n = ].
24
h/R
•
I 1720
I 0.0Ol 2 0.0t 3 0,1
t.../ R =2/3 t, =0.272. I:6
m=5
4
n= j,20 ( • 3 rr/8 08
0.2
0.4 6 0.,6 7 0.8 80I 5
1 2 ~ ,-, " ~
O0 C92 084
)
~; 2
04
06
08
0
#o Fig. 4. Frequency o[ free vibration vs amplitude of imperfection for different values of amplitude of vibration. Case 1, n =/. 15
1. H. N. Chu, Influence of large amplitudes on flexural vibrations of a thin circular cylindrical shell. J. Aerospace Sci. 28, 602-..609 (1961). 2. J. Nowinski, Non-linear transverse vibrations of orthotropic cylindrical shells. AIAA ]. 1,617-.620 (1963). 3. D. A. Evensen, Some observations on the non-linear vibrations of thin cylindrical shells. AIAA J, 1, 2857-2858 (1%3t. 4. M. D. Olson, Some experimental observations on the nonlinear vibrations of thin cylindrical shells. A[AA 7. 3, 17751777 (1%5). 5. D. A. Evensen. Non-linear flexurat vibrations of thin walled circular cylinders. NASA TN D-4090, p. 29 (1%7). 6, A. Rosen and J. Singer, Influence of asymmetric imperfections on the vibrations of axially compressed cylindrical shells. TAE Rep. 212, Technion-Israel Institute of Technology, Haifa, Israel, 1975. 7. L. Watawala, Influence of initial geometric imperfections on vibrations of thin circular cylindrical shells. PhlD. Dis~ertation, University of Massachusetts, 1981. 8. A. E. Armenakas, D. C. Gazis and G. Herrman, Free Vibrations of Circular Cylindricai Shells. Pergamon Press. London (1%9). 9. W. T. Koiter, On the stability of elastic equilibrium. Thesis, Delft, The Netherlands, 1941. Available in English as NASA TTF-10833, 1967. IO. P. Ruiz and J. Penzien, Artificial generation of earthquake accelerograms. Programs available as PSEQGN from the National Information Service, Earthquake Engineering, University of California, Berkeley, California.
y
/~,~./
GOg 2. "
0
h/R. 1/720 L/R • 213 m .5 n~ : 0j 272 :1 C5.~ 0.00t S5,~ O0
/
09-
F, o,
O3
© I 43
; 45
I 47
I 4.9
I ,51
53
,G Fig. 5. Amplitude of forced vibration vs frequency of excitation for different values of amplitude of imperfection.