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Analysis of stresses in internally loaded cylindrical shells
c~pufers & stmctures Vol. IS. No. 3, pp. 22s260.1982 Printed in Great Britain.
ANALYSIS OF STRESSES IN INTERNALLY LOADED CYLINDRICAL SHELLS J. L. URRUTIA-GALICIAand A. N. SHERBOURNE Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada
Abstract-Simply supported cylindricai shells under internal fluid and granular loading are investigated. A Fourier series solution is presented and results are shown for various shell geometries. The accuracy of the results is discussed and a comparison is made with other approximate methods available in the materials handling industry. NOMENCLATURE u, r shell radius E Young’s modulus K t/[l/r~/(tlr)l Ii, l-sincb, 1t sin I$ k t2 12a2 1 shell length m number of Fourier c~fficeints in # direction n number of Fourier coefficients in x direction Pm” pm,,,.1 double Fourier Series coefficients for loading terms
hydraulic radius shell thickness displacement in x direction Fourier coefficient for longitudinal displacement displacement in 4 direction Fourier coefficient for circumferential displacement radial displacement Fourier coeficient for radial displacement coordinates angle of repose for granular material indicator of load level density coefficient of friction Poisson’s ratio absolute maximum longitudinal stress for half-water filled pipe absolute maximum longitudinal stress for full pipe (granular) absolute maximum longitudinal stress for granular material (shell theory) absolute maximum longitudinal stress for granular material (Schorer) ~RODU~ON
The stress analysis of thin elastic shells has been one of the more challen~ng areas of structural mechanics. A wide variety of numerical methods have been applied to the governing differential equations for spherical and cylindrical structures with few results applicable to practice. This lack of information has caused a corresponding loss of confidence in designing shell-like structures. In pipeline design, for example, it is common to use safety factors of up to 10. With the growth and complexity of industry and industrial processes, large pollution control systems are needed in coal and oil fired power generation stations, in mining installations, in metallurgical plants, etc. The kind of duct work required can be of circular or rectangular cross-section. It has been estimated[l]that approximately
400,000 tons of duct work will be installed during the next decade in Canada and the U.S.A. alone, leading to investments amounting to several billion dollars. Analytical solutions for fluid filled cylindrical shells were proposed by Fltiggef2] and Timoshenko[3] using Fourier analysis. In Fliigge’s case of the horizontal pipe, h~f-~lled with water, the number of coefficients is so small that the convergence of the solution is questionable. Timoshenko’s analysis of the horizontal pipe completely filled with water is only a special case from which one cannot derive either a general treatment of this problem or any idea about the convergence of the Fourier solution. When a pipe is partially filled, a discontinuous surface load is created in the analysis. This complicates matters since the solutions for the shell differential equations are valid only in the domain in which the load is continuous and non-zero. One can observe this in the solution given by Fliigge for a half-filled shell. Timoshenko avoided it by considering a special case of loading which was continuous in the axial and ~ircumferen~al directions t~ou~out the entire domain. The analysis for twin saddle supported unstiffened cylindrical vessels is [4] conceptually similar to the flexibility method. The convergence of the solution, however, is very slow and no proof is given regarding the criteria for selecting the appropriate number of Fourier coefficients with respect to the exact solution. In the stability analysis of cylindrical shells, the use of the length parameter, Z= (/*/tr) (1 -Y*)“*, has been widespread. This parameter is intended to take into account the degree of nonlinearity of the stress distribution within the shell which is bound to affect the stability number. This paper examines the alternative parameter, K = ~~(~~~)~(~~~)Iand its relations to the stress distribution for a given shell. In the design of large penstocks for water, or any other fluid, it is generally assumed[5] that the maximum stress occurs at the bottom of the vertical diameter when the shell is half-filled. This is not necessarily true as will be shown. It will also be shown that K is a good parameter for classifying a cylinder as long or short containing, as it does, both length and sectional slenderness effects. The stress analysis is presented of a thin circular cylindrical shell, simply supported at both ends, horizontally placed, and filled with water to any desired level. The solution of the differential equations{21 includes both membrane and bending action. A new technique for calculating Fourier ~oe~cients for one- or 225
J. L. URRUTIA-GALICIAand A. N. SHERBOURNE
226
dary conditions are as follows: two-dimensional functions is given. The two-dimensional Fourier series are used to represent the load acting on au u’ o -=-= the cylinder under consideration. These double Fourier ax a series enable one to obtain analytical, continuous SO~U,I 2 tions for any of the described problems. The technique is v=$=-$=Oatx=Oandx=l 1 , (2) versatile since it is also possible to determine solutions for other loading cases of practical interest. II 2 It is shown that for K L 2.0 the shell equations revert ~~~+" to the simple beam equations regardless of load level. Typical stress profiles are computed over the range 0.6 G The solution of eqn (1) with boundary conditions (2) K s 2.0 which can be used in evaluating the elastic represents the response of a simply supported shell critical load for the internally loaded cylinder. subjected to loads applied to the middle surface of the ANALYSIS shell wall. Applying a Fourier series solution to eqns (1) This stress analysis is concerned with the solution of and using appropriate boundary conditions, leads to an infinite system of uncoupled linear equations involving the set of governing linear differential equations for a displacements as follows: circular cylindrical shell, simply supported, horizontally placed and partially filled with water or any other fluid. It will be shown that the same model can also be used for ll=g 2 U”. cos ml$ cos y the stress analysis of cylinders loaded with granular m=On=O material Fig. 1. Fl~gge[21 proposed the differential equations for cir0=2 2 omt,,sin rn+ sin? . . (3) m=, n=1 cular cylindrical shells in terms of displacements u, B and w as follows Fig. 2, w=g 2 w,. cos rnd sin 7 m=On=l (l-&+(1+ ‘. v+lJ& I;+2 2 The displacements given by eqns (3) fulfil the boun+k
(I-v)ii -[ 2
~+(I-~)‘~ ~ 2
(ltY)lu.+ii+(l-V)F+~tk 2 2
~(1+(3-G 12
gzo
+p I
“D 2
dary conditions imposed by eqns (2) for a circular cylindrical shell supported at both ends by plane diaphragms which are rigid in their own plane. Substitution of eqns (3) into eqns (1) using a Fourier representation of the loading terms of the form
I
2
P&r, cb)= m$O“$oPx_fl cos m4 cos 7
tp,;=o
P&,
4) = $,
“$, hmn sin m# sin y
-
(4)
p,(x, 4) = 2 2 prmr cos nt# sin y m=on=,
+0
allows one to drop the ~igonometric factors and arrive at the following infinite set of linear equations involving the coefficients of the displacements urn”, I),,,“,w,,,. and the coefficients of the loads pxm,,,pcbmn and pnnn
where
@g=(
)),
s=(
);k=&,D=+.
K,, Ku K,, For a simply supported shell with a rigid diaphragm at either end, length 1, radius a and thickness f, the boun-
K21
K22
rc;,,
KS,
%2
J&s
water levet 7
rigid
d~hra~
Fig. 1. Typicalgeometry.
Analysisof stresses in internally loadedcylindricalshells
221
Fig. 2. Shell element. where
defined in the interval 0~ ~9c2n: Then, by dividing interval [O,21~1into d equal parts with (R t 1) points, K
=~z*(l-M -(1tk) f#i =
(y t c)f$‘_where #a = 2
and i
x::=-Q& and K,,=_,,A-k
A3_t1-4
--i-Am'
(
>
K~~=rn2+~~'(l~3k)
From eqns (5) and the Fourier coefficients of the loads (4) one can obtain the values for the coefficients u,,, IL,,, and IV,,,,,for any loading case. With these coefficients one can then calculate the axial force N,. The stress uX can be calculated with the following equations: N, = p s
“g i-h,, 1
It follows that if c = 0 and 6 = 4, the interval [0, 2d will be divided as shown in Fig. 3, and, if c = (l/2), as in Fig. 4. The object of the constant c is to allow a shift in the points dividing the interval in such a way that one can initiate the division at any point in the domain. It should be noted that @can be even or odd. By taking very small intervals we can define, in any interval K, the value of a function f(4) equal to a constant value Wk.Furthermore, we can assume that the corresponding infinite Fourier series, which is equal to f(4) only in this interval, is[6]:
+ vmo,,
+(vtkA2)w,.]~cosm~sin~
x
cos
rndt i
cr,= -NX f
$, (1,+:, wk sin m4 d 4).sin m4 m
(7) such that:
FOURIER COEFFICIENTS IN ONE DIMENSION
The following formulation may be used to represent both continuous and discontinuous functions; it will be applied to the shell problem under consideration. Assume a periodic and piecewise continuous function
wk
‘$k-l
I
10
#k-1 ># > #k*
J. L. URRUTIA-GALICIA and A. N. SHERBOURNE
228
The integration of eqn (9) leads to the following expression: cos mf#~t 2 B, fW=Adm,&L III=,
sin rnd
(10)
where
Ao=;~$, wk
(11)
and
B,,,=$,z[cos(md[F+c]) -cos
(mf$[S$tc])]
Fig. 3. Division of shell domain.
112= 1,2,3,4, . . . m. Equation (11) can be programmed to calculate A,,, A,,, and I?,,, for the Fourier series associated with the function f(i)). DOURLEFOURIRRSERIRSFORLOADINGTERMS
Using the Fourier coefficients from eqn (1l), any of the three loading terms pr(x, d), p&x, 4), pX(x,4) of eqn (4), or any other two-dimensional functions, can be represented in the following way. Assume we are dealing with a horizontal vessel containing a certain amount of water such that, for eqn (4), p,(x, 4) # 0 and p&(x, qt) = p,.(x, 4) = 0. The water pressure in the radial direction, given by p,(4) = ya(cos 4. - cos 4) as shown in Fig. 5, can be expressed by a symmetric function
I
0~4<40
0
p,(4) =
40- cos 4)
p(cos
40 s 4 c 21T- 40 (12) 27r-&<4<2lr
or, equivalently, Fig. 4. Divisionof shell domain.
m
P,($)=
m&Am cmm4
O
(13)
If f(d) = W, in the interval K, we can establish that: f(4) = WI($) + W*(d) + * * *+ W,(d)
(8)
in the interval 0 c 0 s 27r, or f(4) =; 8, /+I;, wk d4
+ ; jitO ( k$, j-+;, wk cos m4 d4)
X cos rn4 + $ mfl ( k$, jkT, W, sin m&Q) . sin m4.
(9)
such that it is continuous in the relevant domian. Now, as the water pressure is constant in the x direction for a fixed value 4, ~~(4) = constant, and is equal to the value obtained from either eqns (12) or (13). To indicate this, the function p?(x) can be constructed in such a way that the boundary conditions given by eqn (2) are fulfilled. The product of p,(4) and p,(x) will lead to the following expression for the loading function, provided that p,(4) and pr(x) are both convergent: P,(X) . P,(d)
= P&,
4).
(14)
From eqns (14) and (12) we note that p,(x) must be
229
Analysis of stresses in internally loaded cylindrical shells
tr
Fig. 5. Schematic pressure.
expressed as p,(x) = “2, B. sin y
(IS)
such that
p,(x) =
I
1
O
0
x=-1,x=0,x=1
I-1
coefficients prmn# 0, pdmn = pX,. = 0, allows us to calculate all the coefficients u,,,., u,,,. and WC,,,for any symmetric distribution of normal pressure on the shell. The case of normal pressure due to water, as given by eqn (121, has been solved. The objectives of this research were to obtain relationships between beam and shell theories as characterized by Schorer’s formula, namely
-l
fl
(17)
01= 112 KO
Equation (15) takes the following graphical form as n +m
A -
pk--+ x- -1
*+I
lX
x-0
X-l
,-I
The loading distribution for the entire shell can therefore be represented as
&=f
_ pr(x, I$) = 2 A, cos rnb * 2 B,, sin y II=1 WI=0 =&Q, The calculation straightforward.
pnnn cos rn4 sin =. 1
where f J(>
(t6)
of coefficients pm. is then fairly
FIJJmLoADtNc Using coefficients given by eqn (16) in eqn (3, for all
g! = absolute maximum longitudinal stress for the halffilled pipe; f, = maximum beam stress for a simply supported, full pipe; r, t = radius and thickness of the shell, respectively; and to check if the modified parameter K = d[(l/r) d(t/r)] = 2 would defined the limit between these theories. The resulting solution procedure proved so effective that many exact analyses were done very rapidly. Following several trial runs on the computer it was found that eqn (17) does not relate or and fi in such
230
J. L. U~UTIA~A~CIAand A. N. SHE~BOURNE
a simple way. The parameter K, however, is effective in determining the limit between beam theory, with linear fields of stress and strain, and shell theory where the fletds of stress and strain are no~inear. From Figs. 4-8 one can see that the larger the K the smaller the nonlinearity of the stress distribution. For example, at K = 1.68 the nonlinearity is small; at K = 0.45 the stress distribution is highly nonlinear. Consider K = 1.41. This value may be obtained with two shells of different dimensions, one being long and slender, the other short and stiff. Neve~heless, they have identical stress profiies Figs. 9, 10. It is clear that the nonIin~~i~y for the first shell is due to high stresses in a very thin shell, i.e. due to large moments and deformations of the cross-section. For the second shell, less obvious, the nonlinearity is due to the strong influence of the rigid boundary diaphragm and, despite the fact that this shell is thicker, the nonlinearity will arise. The same holds true for the shells shown in Figs, II and 12 for K = 1, and in Figs, 13 and 14 for K = 0.71. In Figs. 6-14 it is also observed that, independent of K, beam and shell theory give the same stress when the cylinder is completely filled (curve 12 in all figures). One may thus conclude that the influence of K is valid only for partial loads as, for example, in Fig. 7 (K = 1.68) when the stress distribution for the half filled cylinder is sIi~tly nonlinear. For the same loading, but with a shell value K = 0.45 Fig, 8 a higher nonlinearity is observed, In general, for a given K value, the shape of the stress distributionremains the same for any shell con~guration
having this value. One can thus develop a set of profiles showing the shape and the relative magnitude of the stresses for different water levels. For a given shell, K, the stress distribution can then be scaled by a factor equal to the maximum stress calculated from beam theory for a completely filled cylinder. GRANULARLOADING
The load distribution involving granular material is essentially different from that due to water. The pressure dependsr7] mainly upon the friction angle of the material and on the friction between cylinder wall and contained particulate matter. An accepted formula for establishing the static vertical unit pressure, q, at depth, y, below the surface of a granular material and the associated static lateral pressure, p, is given by Jansenll] as
where K - l-sinqb, “-l+sin#P
and R = q
is a hydraulic radius.
Using eqn (18) and Fig. 15 it can be shown that the normal and tangential pressure components at the cylin-
S hrlll anaiysis. Fer any K valuekny stress_ distribution graph)
Fig. 6. Figs. 6-M. Stress distributions for diierent sheik.
Analysis of stresses in internally Ioaded cyiind~c~
4513.88 Ibs/inch2
Fig. 7,
Fig. 8.
shells
231
J. L. URRUTIA-GALICIAand A. N. SHERBOURNE
232
902275 Ibs/inch2
lcnpth rkll
1000.0
Fig. 9.
,
564.23 Winch2
examined and the results are as follows:
+
(1) The shell of K = 0.8 was half-filled with coal dust of density y = 62.5 lb/ft3 (0.03611lb/in?), angle of repose c#+= 35” and friction coefficient cc= 0.30. The shell radius was 12.028in., length 180in. and thickness 0.022 in. Norma! and tangential pressures were calculated using Janssen’s Method Fig. 16. The stress analysis provided results shown dotted in Fig. 17. This figure includes the simple beam stress distributions for the half-full and full load and its maximum value is given by the following equation valid for uniformly loaded circular sections:
(inch4 Ionpth shrll radius
500.0 50.0
thicknosr
I* (20) As the cylinder is only half-filled, Q. = (r/2) and eqn (20) reduces to L*-
&(max) = 3323.85psi.
Rcfwr to $volur,
In Fig. 17 it is seen that the maximum shell stress is a, (shell) = 14663.4 psi (tension) or 4.41 times the beam stress. If the calculations are based upon eqn (17) the longitudinal bending stress becomes
Fig. 10.
der wall are given by aI = 8309.64psi. p,(4) = p sin’ a t q co2 a
(19) which is 57% of the actual shell stress. From this examp.+(4)= (q -p) sin a cos (I valid only if (I = s - 4
ple it is seen that cylinders filled with granular material and fluid, lead, in general to two different types of structural response. (2) A second analysis was carried out for a cylinder with K = 2.0 having the following dimensions: I= 180in.,
Analysisof stresses in internally loaded cylindrical shells i
ltnpth shell
36111.00Ibs/inch2
233 t
iOOO.0
2
N
Fig. 11.
2256.94 IWinch i
Fig. 12.
t = 0.85 in., r = 12.028 in., filled again with coal dust y = 62.5 lb/ft3 (0.036111 lb/in?), & = 35” (angle of repose), fiction coefficient k = 0.30. Janssen’s method for pressure dis~ibuton was again used, and the results are shown in Fig. 18. For the half-filled cylinder it is seen that she11 analysis gives approximately 10% smaller stresses than those obtained from beam theory. It is worth observing that the stress distribution for the half-filled cylinder with granular loading is almost linear. For an approximate location of the points of maximum stress, the reader is referred to Figs. 17 and 18. From the trends shown previously, with respect to the correlation between K and the degree of nonlinearity of stress distribution for a shell in the case of fluid loading, it seems plausible to infer that for K > 2.0 the stress distribution wiu be linear and beam theory will apply to within an accuracy of approximately 10%. For K < 2.0, shell and beam theory begin to diverge, the diierences
becoming marked at low K.Schorer’s formula is valid for half-filled cylinders with K values around 2 but, for small values of K, this formula is in complete disa~eement with shell theory. ACCURACYOFTHESOLUTION
Obviously the accuracy of the solution is a direct function of the accuracy of the Fourier series representation. With this in mind, Table 1 presents a set of 10 finite Fourier series for a circular shell of radius 5Oin. and over which a normal pressure is applied at dierent water levels, #o. It must be noted that these finite Fourier series, with 15 coefficients, are valid and continuous in the entire domain, 0 < Q G 27r, even in those places where the water pressure is zero. At this point the question arises regarding the number of Fourier coefficients required to obtain a good approximation of p,(4). This question is resolved by in-
J. L. U~UTIA-GALICIAand A. N. SHERBOURNE
234
902775 lbr/inch2 7(inches) length
shell
radius thickness
500.0 50.0 0.125
Fig. 13.
3
length
shell
1Ul
lbs/inch2
approaches zero if g(4) becomes an in~nite series. It can be shown that for E = 0, eqn (21) yields the well known Bessel inequality -;
lOQ0
I_~f(~)‘d~~ZAaZ+~(A,‘+E,2)
for a finite Fourier series. As m+m, becomes the Parseval equality
(22)
Bessel’s inequality
“, _; f(&’ d# = 2A,2 f 1 (A,2 t B,,,2) I
(23)
as the error E = 0. According to Fig. 5 and eqn (12) the corresponding Bessel inequality for f(4) = p,(4) is
I
+; Iyticos 6 - cos cb)l’d$
>2A,t
c (A,,,* + B,‘). m=l
(24)
Fig. 14. voking a Bessel ineq~lity and will be discussed only for the fluid loading function expressed by eqns (12) and (13). One method for measuring the degree of approximationI is by making the error E as small as possible where
In Table 2 the values for 1, are given for different water levels, 40. The error involved in the Fourier series of Table I was calculated from the Bessel inequality, modified to the following equation:
Ip - (ZAn +
E(4) =
f(d) is any arbitrary function and g(Qi) is the associated Fourier series for f(d). The error is always positive and
$, CL* + B,z)) 1,
x 100. (25)
In Table 3 it can be seen that, for lower water levels, a larger number of Fourier coefficients is required. For the lowest water level, therefore, an appropriate number of
0.575 0.361 0.197 0.0875
1.570 1.832 2.094 2.356
9.7800 8.1397 9.3429
6.4466 4.6080
0” :5
2
p,(b) is given by eqn (12).
120” 135”
105” ;:
$. (degrees)
Ip integral from equation (24)
Ip
h (degrees)
1,
-0.902 -0.6@8 -0.353 -0.164
-1.639 -1.451 -1.1%
-1.804 -1.797 -1.751
m=l
0.0735 0.2820
2.9347 0.7621 1.6300
1.364 1.099 0.8281
0.785 I.047 1.308
Table 2.
1.805 1.747 1.590
fll=O
0.00 0.261 0.523
40
0.381 0.343 0.247 0.135
0.135 0.025 0.344
0.0 0.007 0.048
m=2
0.006 0.033
4
0.00 -0.088 -0.122 -0.095
0.008
I 8
9
0.031 0.00 0.001 -0.021 -0.027 0.008 -0.004 0.013 -0.016 0.007 0.008 -0.011
0.00 0.008 -0.010 0.005
-0.003 -0.013 -0.011 -0.026 -0.008 0.00081 0.001 0.0209 0.007 -0.008
0.015
6
dQ(in radians)
0.009 -0.008 0.003 0.007
10
0.003 -o&34
I1
to.004
12
-0.002
13
-0.002
14
42.400 17.560
135”
6.450
28,220
120”
0.450 0.560 0.510 1.760
0.260 0.102
0.053
0.048
2
1.780 4.530 9.480 17.260
0.131 0.543
0.053
0.048
1
5.340
1.090
0.130 0.560 0.240 0.740
0.121 0.081
0.053
3
1.520
0.880
0.116 0.220 0.190 0.530
0.068 0.077
-
_-__-
4
1.060
0.680
__.~.
0.103 0.220 0.130 0.290
0.061 0.072
-
5
1.040
0.420
0.088 0.160 0.127 0.290
0.059 0.072
-
6
8
0.820
0.390
0.086 0.160 0.110 0.230
0.056 0.069
0.630
0.370
0084 0.150 0.110 0.220
0.067 -
Error in (%) -
7
0.560
0.590 0.600
0.320
0.320
0.320
0.082 0.140 0.105 0.200
-
-
11
0.082 0.140 0.105 0.200
-
-
10
0.082 0.150 0,110 0.210
0.066 -
-
-.-.-_--_
9
0.540
0.310
0.104 0.140 0.190
-
-
12
0.530
0.310
0.103 0.140 0.190
-
-
13
Table 3. Error in Fourier Series for eqn (12) vs number of Fourier coefficients. For different I& values independent term A0 [see eqn (IO)]t the number of Fourier coefficients A, indicated below
0.00 t0.042 0.024 -0.018
0.018 -0.026 -0.042
0.024
5
60” ; 105”
:lG
15”
0”
-0.0749 -0.040 0.024 0.053
0.094 0.053 0.123 -0.024 0.088 -0.040
0.006 0.041
3
P,(#) = $s A, cos ~C#J
Table t, Fourier Series for eqn (13) for different $Qvalues
0.528
0.304
0.103 0.138 0.190
-
-
14
$ B 45. g_ 2. fl s P G;
0, z a t z 5. c. z
B B r” 2.
J. L. U~~IA-GALICIAand A. N. SH~OU~E
i
(a)
(b) Fig. 15.Variablesand loads for granularcontainment.
Fwesve distribotiin cokuloted
thickness0.022"
with Jonssen's method.
~~~~_______*__________~____
t
(a)
@$+
0.434
ji
+.J
(cl
+ 0116 & IWinclT
Fig. 16. Pressure distributions after Janssen (coal dust).
coefficients will be 14 with a resulting maximum error of 0.5% with respect to the exact solution. It is useful to notice the rapid convergence of the solution with increasing numbers of terms. From Table 3, it may also be seen that the accuracy of the solution is good and that its sensitivity to radical changes when taking more than 14 terms is negligible, being less than 0.5%. CONCLUSIONS
The paper has discussed only the solution procedure involved in the static analysis of right cylindrical, circular pipes loaded internally with a fluid or granular medium. A series of curves for various K values in the range 0.6 < K < 2.0 have been presented elsewhere [lo], as indeed has experimental evidence, and will not therefore be discussed here. The procedure for the calculation of the stress protile in bending can be automated by way of a computer pro~amme and this has been carried out
(Appendix) typically for a water filled shell. The Fourier Series sub-routine is included as part of the computational package. Several conclusions can be drawn about the genera1 analysis which are valid not only for the static, equilibrium problem but also for future stability analysis: (i) The maximum stress does not always occur for the half-filled cylinder as is commonly assumed. It must be calculated for each particular case depending upon the associated K. As an extreme example, for K = 2.0, the maximum stress will occur in the fully loaded and not the half-filled cylinder. (ii) The length parameter 2 = (/*/I?) (1 - Y*)“*which is used to indicate, in an approximate way, whether a given cylinder is long or short may be replaced by the parameter K = d[i/d(t/r)]. This parameter appears to be a better indicator of the degree of nonlinearity in stress and may therefore be more helpful.
Analysis of stresses in internally loaded cylindrical shells
237 Boom andysis
Beam omtj5is
I
I
r I
I
, .
Stressanalysis GRANULAR
MATERIAL
7 \
,
____._Half
filledcyiindcr.
_Campletely
filled
cylinder.
Shell andysis;
-.
,
\
\
I
radius 12.028” thickness 0.022” length 180.000”
#’ ,’
t K:0.8
, ,
,’
/
,’
Fig. 17. Stress distributions for coal dust (granular). Stress anatysis GRANUAR MATERIAL
-_ _ _____ Half filled cylinder
-
Completely
filled
cylinder.
mdius 12.028” thickness O.B50” length 180.000”
A I’ I’
I
I
K:2.0
,’ I’
I’
I
/’
#’
Fig. 18. Stress distributions for coal dust (granular).
(iii) If cylinders have similar cross-sectional stress distributions,
then stability analyses may be performed on entire families of shells of given K. This would simplify the number of possible cases to be analyzed. (iv) Finally, with regard to the stress analysis of cylindrical shells, horizontally placed, it is obvious from the
results for granular materials that this case and the fluid loading case are completely different. There is no reason at all to assume any similarity in the predicted results. Acknowledgements-This work was carried out with financial assistance from the National Council of Science and Technology
(CONACYT) Mexico to the primary author (J. L. UrrutiaGalicia). The authors are also grateful to the National Research Council of Canada for assistance received through grant A-1582 (to A. N. Sherbourne) which made possible the experimental and computational work presented in this paper.
I
REFERENCES
A. N. Sherboume and H. M. Haydl, Large diameter metallic ’ thin walled ducts. Metal Structures Conference. Australian Institution of Engineers, Perth (1978). 2. W. Fliigge, Stresses in SheUs. Springer-Verlag, New York (1973).
238
J. L. U~UTIA-GALICIA and A. N.
3. S. P. Timoshenko, Theoryof Plates and Shelis. McGraw-Hill, New York (1959). 4. Pressure Vessel Technology, Part I, ASME, 3rd ht. Conf. Tokyo, Japan (1977). 5. R. J. Roark, Formulas for Stress and Strain, 4th Edn. McGraw-Hill, New York (1965). 6. F. B. Hildebrand, Advanced Calculus for Applications, 2nd Edn. Prentice-Hall, Englewood Cliffs, New Jersey (1976).
SHERBOURNE
7. T. W. Lambe and R. V. Whitman, Soil Mechanics. J. Wiley, New York (1979). 8. H. M. Haydl and A. N. Sherbourne, Designof Large Circular Ducts. Proceedings Institution Civil Engineers, Part 2 (1978). 9. David L. Powers, Boundary Value Problems. Academic Press, New York (1972). 10. A. N. Sherbourne and J. L. Urrutia-Galicia, Internally loaded cylindrical shells. To be published in the March 1982issue of the Canadian Journal of Civil Engineering.
LENDS
Computerprogram-~uid filfed sheIf The following variables are used in the program. NFI, NEX the number of segments in # and x directions. MFI.MEX the maximum number of Fourier coefficients selected. Instead of =, select any m integer value to get a finite Fourier series. CFI, CEX the value of the constant “c”. GAMA the water density. ALaNG the length of the shell.
T = t the thickness of the shell. BNU, E Poisson’s ratio and Young’s modulus, respec. tively. FILEVE = $J,, the water (load) level. NP parameter used to indicate the first element in the interval [0,2r] for which p,(g)#O, and therefore starts the calculations of the Fourier coefficients. NP = 1, 2, . . . , NF1/2 (or N&X/2).
The flow diagram is as follows: Read-NFI, MFI, CFI, NEX, MEX, CEX, Y. length(Q), radius (a) thickness (t), Young modulus (E) and Poisson's ratio (v) 1 ad-Load Level I$, and NP I
-1
Write equations Prf@), for example, prt~)=ay(co~~~-cos~) and calculate all values of pr($) at each of the NFI segments for the interval [C,Zlr]. Each value is calculated at the mid-point. I
I r
I
1 IWrite equations for p,(x) and calculate values of p,(x) at each of NRX segments. Each value is again calculated at the mid-point.
I I
CALL Fourier Series Sub-routine I
Analysis ofstresses in internally loaded cylindrical shells
I
CALL Sub-routine to multiply Fourier Series for P,(X) and ~~($1
Calculate stiffness coefficients Kll, K12, K13, etc.
Calculate Us,
vmn and wmn
Calculate N, or ox, using Umnv vmn and wmn.
I
G+T$ 1 and calculate another loading case
239
_w.._-- For part Number N
MI~M~~~rn~rn number of Fourier Coeificients c___--
ANALYSIS OF STRESSES IN INTERNALLY LOADED CYLINDRICAL SHELLS J. L. URRUTIA-GALICIAand A. N. SHERBOURNE Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada
Abstract-Simply supported cylindricai shells under internal fluid and granular loading are investigated. A Fourier series solution is presented and results are shown for various shell geometries. The accuracy of the results is discussed and a comparison is made with other approximate methods available in the materials handling industry. NOMENCLATURE u, r shell radius E Young’s modulus K t/[l/r~/(tlr)l Ii, l-sincb, 1t sin I$ k t2 12a2 1 shell length m number of Fourier c~fficeints in # direction n number of Fourier coefficients in x direction Pm” pm,,,.1 double Fourier Series coefficients for loading terms
hydraulic radius shell thickness displacement in x direction Fourier coefficient for longitudinal displacement displacement in 4 direction Fourier coefficient for circumferential displacement radial displacement Fourier coeficient for radial displacement coordinates angle of repose for granular material indicator of load level density coefficient of friction Poisson’s ratio absolute maximum longitudinal stress for half-water filled pipe absolute maximum longitudinal stress for full pipe (granular) absolute maximum longitudinal stress for granular material (shell theory) absolute maximum longitudinal stress for granular material (Schorer) ~RODU~ON
The stress analysis of thin elastic shells has been one of the more challen~ng areas of structural mechanics. A wide variety of numerical methods have been applied to the governing differential equations for spherical and cylindrical structures with few results applicable to practice. This lack of information has caused a corresponding loss of confidence in designing shell-like structures. In pipeline design, for example, it is common to use safety factors of up to 10. With the growth and complexity of industry and industrial processes, large pollution control systems are needed in coal and oil fired power generation stations, in mining installations, in metallurgical plants, etc. The kind of duct work required can be of circular or rectangular cross-section. It has been estimated[l]that approximately
400,000 tons of duct work will be installed during the next decade in Canada and the U.S.A. alone, leading to investments amounting to several billion dollars. Analytical solutions for fluid filled cylindrical shells were proposed by Fltiggef2] and Timoshenko[3] using Fourier analysis. In Fliigge’s case of the horizontal pipe, h~f-~lled with water, the number of coefficients is so small that the convergence of the solution is questionable. Timoshenko’s analysis of the horizontal pipe completely filled with water is only a special case from which one cannot derive either a general treatment of this problem or any idea about the convergence of the Fourier solution. When a pipe is partially filled, a discontinuous surface load is created in the analysis. This complicates matters since the solutions for the shell differential equations are valid only in the domain in which the load is continuous and non-zero. One can observe this in the solution given by Fliigge for a half-filled shell. Timoshenko avoided it by considering a special case of loading which was continuous in the axial and ~ircumferen~al directions t~ou~out the entire domain. The analysis for twin saddle supported unstiffened cylindrical vessels is [4] conceptually similar to the flexibility method. The convergence of the solution, however, is very slow and no proof is given regarding the criteria for selecting the appropriate number of Fourier coefficients with respect to the exact solution. In the stability analysis of cylindrical shells, the use of the length parameter, Z= (/*/tr) (1 -Y*)“*, has been widespread. This parameter is intended to take into account the degree of nonlinearity of the stress distribution within the shell which is bound to affect the stability number. This paper examines the alternative parameter, K = ~~(~~~)~(~~~)Iand its relations to the stress distribution for a given shell. In the design of large penstocks for water, or any other fluid, it is generally assumed[5] that the maximum stress occurs at the bottom of the vertical diameter when the shell is half-filled. This is not necessarily true as will be shown. It will also be shown that K is a good parameter for classifying a cylinder as long or short containing, as it does, both length and sectional slenderness effects. The stress analysis is presented of a thin circular cylindrical shell, simply supported at both ends, horizontally placed, and filled with water to any desired level. The solution of the differential equations{21 includes both membrane and bending action. A new technique for calculating Fourier ~oe~cients for one- or 225
J. L. URRUTIA-GALICIAand A. N. SHERBOURNE
226
dary conditions are as follows: two-dimensional functions is given. The two-dimensional Fourier series are used to represent the load acting on au u’ o -=-= the cylinder under consideration. These double Fourier ax a series enable one to obtain analytical, continuous SO~U,I 2 tions for any of the described problems. The technique is v=$=-$=Oatx=Oandx=l 1 , (2) versatile since it is also possible to determine solutions for other loading cases of practical interest. II 2 It is shown that for K L 2.0 the shell equations revert ~~~+" to the simple beam equations regardless of load level. Typical stress profiles are computed over the range 0.6 G The solution of eqn (1) with boundary conditions (2) K s 2.0 which can be used in evaluating the elastic represents the response of a simply supported shell critical load for the internally loaded cylinder. subjected to loads applied to the middle surface of the ANALYSIS shell wall. Applying a Fourier series solution to eqns (1) This stress analysis is concerned with the solution of and using appropriate boundary conditions, leads to an infinite system of uncoupled linear equations involving the set of governing linear differential equations for a displacements as follows: circular cylindrical shell, simply supported, horizontally placed and partially filled with water or any other fluid. It will be shown that the same model can also be used for ll=g 2 U”. cos ml$ cos y the stress analysis of cylinders loaded with granular m=On=O material Fig. 1. Fl~gge[21 proposed the differential equations for cir0=2 2 omt,,sin rn+ sin? . . (3) m=, n=1 cular cylindrical shells in terms of displacements u, B and w as follows Fig. 2, w=g 2 w,. cos rnd sin 7 m=On=l (l-&+(1+ ‘. v+lJ& I;+2 2 The displacements given by eqns (3) fulfil the boun+k
(I-v)ii -[ 2
~+(I-~)‘~ ~ 2
(ltY)lu.+ii+(l-V)F+~tk 2 2
~(1+(3-G 12
gzo
+p I
“D 2
dary conditions imposed by eqns (2) for a circular cylindrical shell supported at both ends by plane diaphragms which are rigid in their own plane. Substitution of eqns (3) into eqns (1) using a Fourier representation of the loading terms of the form
I
2
P&r, cb)= m$O“$oPx_fl cos m4 cos 7
tp,;=o
P&,
4) = $,
“$, hmn sin m# sin y
-
(4)
p,(x, 4) = 2 2 prmr cos nt# sin y m=on=,
+0
allows one to drop the ~igonometric factors and arrive at the following infinite set of linear equations involving the coefficients of the displacements urn”, I),,,“,w,,,. and the coefficients of the loads pxm,,,pcbmn and pnnn
where
@g=(
)),
s=(
);k=&,D=+.
K,, Ku K,, For a simply supported shell with a rigid diaphragm at either end, length 1, radius a and thickness f, the boun-
K21
K22
rc;,,
KS,
%2
J&s
water levet 7
rigid
d~hra~
Fig. 1. Typicalgeometry.
Analysisof stresses in internally loadedcylindricalshells
221
Fig. 2. Shell element. where
defined in the interval 0~ ~9c2n: Then, by dividing interval [O,21~1into d equal parts with (R t 1) points, K
=~z*(l-M -(1tk) f#i =
(y t c)f$‘_where #a = 2
and i
x::=-Q& and K,,=_,,A-k
A3_t1-4
--i-Am'
(
>
K~~=rn2+~~'(l~3k)
From eqns (5) and the Fourier coefficients of the loads (4) one can obtain the values for the coefficients u,,, IL,,, and IV,,,,,for any loading case. With these coefficients one can then calculate the axial force N,. The stress uX can be calculated with the following equations: N, = p s
“g i-h,, 1
It follows that if c = 0 and 6 = 4, the interval [0, 2d will be divided as shown in Fig. 3, and, if c = (l/2), as in Fig. 4. The object of the constant c is to allow a shift in the points dividing the interval in such a way that one can initiate the division at any point in the domain. It should be noted that @can be even or odd. By taking very small intervals we can define, in any interval K, the value of a function f(4) equal to a constant value Wk.Furthermore, we can assume that the corresponding infinite Fourier series, which is equal to f(4) only in this interval, is[6]:
+ vmo,,
+(vtkA2)w,.]~cosm~sin~
x
cos
rndt i
cr,= -NX f
$, (1,+:, wk sin m4 d 4).sin m4 m
(7) such that:
FOURIER COEFFICIENTS IN ONE DIMENSION
The following formulation may be used to represent both continuous and discontinuous functions; it will be applied to the shell problem under consideration. Assume a periodic and piecewise continuous function
wk
‘$k-l
I
10
#k-1 ># > #k*
J. L. URRUTIA-GALICIA and A. N. SHERBOURNE
228
The integration of eqn (9) leads to the following expression: cos mf#~t 2 B, fW=Adm,&L III=,
sin rnd
(10)
where
Ao=;~$, wk
(11)
and
B,,,=$,z[cos(md[F+c]) -cos
(mf$[S$tc])]
Fig. 3. Division of shell domain.
112= 1,2,3,4, . . . m. Equation (11) can be programmed to calculate A,,, A,,, and I?,,, for the Fourier series associated with the function f(i)). DOURLEFOURIRRSERIRSFORLOADINGTERMS
Using the Fourier coefficients from eqn (1l), any of the three loading terms pr(x, d), p&x, 4), pX(x,4) of eqn (4), or any other two-dimensional functions, can be represented in the following way. Assume we are dealing with a horizontal vessel containing a certain amount of water such that, for eqn (4), p,(x, 4) # 0 and p&(x, qt) = p,.(x, 4) = 0. The water pressure in the radial direction, given by p,(4) = ya(cos 4. - cos 4) as shown in Fig. 5, can be expressed by a symmetric function
I
0~4<40
0
p,(4) =
40- cos 4)
p(cos
40 s 4 c 21T- 40 (12) 27r-&<4<2lr
or, equivalently, Fig. 4. Divisionof shell domain.
m
P,($)=
m&Am cmm4
O
(13)
If f(d) = W, in the interval K, we can establish that: f(4) = WI($) + W*(d) + * * *+ W,(d)
(8)
in the interval 0 c 0 s 27r, or f(4) =; 8, /+I;, wk d4
+ ; jitO ( k$, j-+;, wk cos m4 d4)
X cos rn4 + $ mfl ( k$, jkT, W, sin m&Q) . sin m4.
(9)
such that it is continuous in the relevant domian. Now, as the water pressure is constant in the x direction for a fixed value 4, ~~(4) = constant, and is equal to the value obtained from either eqns (12) or (13). To indicate this, the function p?(x) can be constructed in such a way that the boundary conditions given by eqn (2) are fulfilled. The product of p,(4) and p,(x) will lead to the following expression for the loading function, provided that p,(4) and pr(x) are both convergent: P,(X) . P,(d)
= P&,
4).
(14)
From eqns (14) and (12) we note that p,(x) must be
229
Analysis of stresses in internally loaded cylindrical shells
tr
Fig. 5. Schematic pressure.
expressed as p,(x) = “2, B. sin y
(IS)
such that
p,(x) =
I
1
O
0
x=-1,x=0,x=1
I-1
coefficients prmn# 0, pdmn = pX,. = 0, allows us to calculate all the coefficients u,,,., u,,,. and WC,,,for any symmetric distribution of normal pressure on the shell. The case of normal pressure due to water, as given by eqn (121, has been solved. The objectives of this research were to obtain relationships between beam and shell theories as characterized by Schorer’s formula, namely
-l
fl
(17)
01= 112 KO
Equation (15) takes the following graphical form as n +m
A -
pk--+ x- -1
*+I
lX
x-0
X-l
,-I
The loading distribution for the entire shell can therefore be represented as
&=f
_ pr(x, I$) = 2 A, cos rnb * 2 B,, sin y II=1 WI=0 =&Q, The calculation straightforward.
pnnn cos rn4 sin =. 1
where f J(>
(t6)
of coefficients pm. is then fairly
FIJJmLoADtNc Using coefficients given by eqn (16) in eqn (3, for all
g! = absolute maximum longitudinal stress for the halffilled pipe; f, = maximum beam stress for a simply supported, full pipe; r, t = radius and thickness of the shell, respectively; and to check if the modified parameter K = d[(l/r) d(t/r)] = 2 would defined the limit between these theories. The resulting solution procedure proved so effective that many exact analyses were done very rapidly. Following several trial runs on the computer it was found that eqn (17) does not relate or and fi in such
230
J. L. U~UTIA~A~CIAand A. N. SHE~BOURNE
a simple way. The parameter K, however, is effective in determining the limit between beam theory, with linear fields of stress and strain, and shell theory where the fletds of stress and strain are no~inear. From Figs. 4-8 one can see that the larger the K the smaller the nonlinearity of the stress distribution. For example, at K = 1.68 the nonlinearity is small; at K = 0.45 the stress distribution is highly nonlinear. Consider K = 1.41. This value may be obtained with two shells of different dimensions, one being long and slender, the other short and stiff. Neve~heless, they have identical stress profiies Figs. 9, 10. It is clear that the nonIin~~i~y for the first shell is due to high stresses in a very thin shell, i.e. due to large moments and deformations of the cross-section. For the second shell, less obvious, the nonlinearity is due to the strong influence of the rigid boundary diaphragm and, despite the fact that this shell is thicker, the nonlinearity will arise. The same holds true for the shells shown in Figs, II and 12 for K = 1, and in Figs, 13 and 14 for K = 0.71. In Figs. 6-14 it is also observed that, independent of K, beam and shell theory give the same stress when the cylinder is completely filled (curve 12 in all figures). One may thus conclude that the influence of K is valid only for partial loads as, for example, in Fig. 7 (K = 1.68) when the stress distribution for the half filled cylinder is sIi~tly nonlinear. For the same loading, but with a shell value K = 0.45 Fig, 8 a higher nonlinearity is observed, In general, for a given K value, the shape of the stress distributionremains the same for any shell con~guration
having this value. One can thus develop a set of profiles showing the shape and the relative magnitude of the stresses for different water levels. For a given shell, K, the stress distribution can then be scaled by a factor equal to the maximum stress calculated from beam theory for a completely filled cylinder. GRANULARLOADING
The load distribution involving granular material is essentially different from that due to water. The pressure dependsr7] mainly upon the friction angle of the material and on the friction between cylinder wall and contained particulate matter. An accepted formula for establishing the static vertical unit pressure, q, at depth, y, below the surface of a granular material and the associated static lateral pressure, p, is given by Jansenll] as
where K - l-sinqb, “-l+sin#P
and R = q
is a hydraulic radius.
Using eqn (18) and Fig. 15 it can be shown that the normal and tangential pressure components at the cylin-
S hrlll anaiysis. Fer any K valuekny stress_ distribution graph)
Fig. 6. Figs. 6-M. Stress distributions for diierent sheik.
Analysis of stresses in internally Ioaded cyiind~c~
4513.88 Ibs/inch2
Fig. 7,
Fig. 8.
shells
231
J. L. URRUTIA-GALICIAand A. N. SHERBOURNE
232
902275 Ibs/inch2
lcnpth rkll
1000.0
Fig. 9.
,
564.23 Winch2
examined and the results are as follows:
+
(1) The shell of K = 0.8 was half-filled with coal dust of density y = 62.5 lb/ft3 (0.03611lb/in?), angle of repose c#+= 35” and friction coefficient cc= 0.30. The shell radius was 12.028in., length 180in. and thickness 0.022 in. Norma! and tangential pressures were calculated using Janssen’s Method Fig. 16. The stress analysis provided results shown dotted in Fig. 17. This figure includes the simple beam stress distributions for the half-full and full load and its maximum value is given by the following equation valid for uniformly loaded circular sections:
(inch4 Ionpth shrll radius
500.0 50.0
thicknosr
I* (20) As the cylinder is only half-filled, Q. = (r/2) and eqn (20) reduces to L*-
&(max) = 3323.85psi.
Rcfwr to $volur,
In Fig. 17 it is seen that the maximum shell stress is a, (shell) = 14663.4 psi (tension) or 4.41 times the beam stress. If the calculations are based upon eqn (17) the longitudinal bending stress becomes
Fig. 10.
der wall are given by aI = 8309.64psi. p,(4) = p sin’ a t q co2 a
(19) which is 57% of the actual shell stress. From this examp.+(4)= (q -p) sin a cos (I valid only if (I = s - 4
ple it is seen that cylinders filled with granular material and fluid, lead, in general to two different types of structural response. (2) A second analysis was carried out for a cylinder with K = 2.0 having the following dimensions: I= 180in.,
Analysisof stresses in internally loaded cylindrical shells i
ltnpth shell
36111.00Ibs/inch2
233 t
iOOO.0
2
N
Fig. 11.
2256.94 IWinch i
Fig. 12.
t = 0.85 in., r = 12.028 in., filled again with coal dust y = 62.5 lb/ft3 (0.036111 lb/in?), & = 35” (angle of repose), fiction coefficient k = 0.30. Janssen’s method for pressure dis~ibuton was again used, and the results are shown in Fig. 18. For the half-filled cylinder it is seen that she11 analysis gives approximately 10% smaller stresses than those obtained from beam theory. It is worth observing that the stress distribution for the half-filled cylinder with granular loading is almost linear. For an approximate location of the points of maximum stress, the reader is referred to Figs. 17 and 18. From the trends shown previously, with respect to the correlation between K and the degree of nonlinearity of stress distribution for a shell in the case of fluid loading, it seems plausible to infer that for K > 2.0 the stress distribution wiu be linear and beam theory will apply to within an accuracy of approximately 10%. For K < 2.0, shell and beam theory begin to diverge, the diierences
becoming marked at low K.Schorer’s formula is valid for half-filled cylinders with K values around 2 but, for small values of K, this formula is in complete disa~eement with shell theory. ACCURACYOFTHESOLUTION
Obviously the accuracy of the solution is a direct function of the accuracy of the Fourier series representation. With this in mind, Table 1 presents a set of 10 finite Fourier series for a circular shell of radius 5Oin. and over which a normal pressure is applied at dierent water levels, #o. It must be noted that these finite Fourier series, with 15 coefficients, are valid and continuous in the entire domain, 0 < Q G 27r, even in those places where the water pressure is zero. At this point the question arises regarding the number of Fourier coefficients required to obtain a good approximation of p,(4). This question is resolved by in-
J. L. U~UTIA-GALICIAand A. N. SHERBOURNE
234
902775 lbr/inch2 7(inches) length
shell
radius thickness
500.0 50.0 0.125
Fig. 13.
3
length
shell
1Ul
lbs/inch2
approaches zero if g(4) becomes an in~nite series. It can be shown that for E = 0, eqn (21) yields the well known Bessel inequality -;
lOQ0
I_~f(~)‘d~~ZAaZ+~(A,‘+E,2)
for a finite Fourier series. As m+m, becomes the Parseval equality
(22)
Bessel’s inequality
“, _; f(&’ d# = 2A,2 f 1 (A,2 t B,,,2) I
(23)
as the error E = 0. According to Fig. 5 and eqn (12) the corresponding Bessel inequality for f(4) = p,(4) is
I
+; Iyticos 6 - cos cb)l’d$
>2A,t
c (A,,,* + B,‘). m=l
(24)
Fig. 14. voking a Bessel ineq~lity and will be discussed only for the fluid loading function expressed by eqns (12) and (13). One method for measuring the degree of approximationI is by making the error E as small as possible where
In Table 2 the values for 1, are given for different water levels, 40. The error involved in the Fourier series of Table I was calculated from the Bessel inequality, modified to the following equation:
Ip - (ZAn +
E(4) =
f(d) is any arbitrary function and g(Qi) is the associated Fourier series for f(d). The error is always positive and
$, CL* + B,z)) 1,
x 100. (25)
In Table 3 it can be seen that, for lower water levels, a larger number of Fourier coefficients is required. For the lowest water level, therefore, an appropriate number of
0.575 0.361 0.197 0.0875
1.570 1.832 2.094 2.356
9.7800 8.1397 9.3429
6.4466 4.6080
0” :5
2
p,(b) is given by eqn (12).
120” 135”
105” ;:
$. (degrees)
Ip integral from equation (24)
Ip
h (degrees)
1,
-0.902 -0.6@8 -0.353 -0.164
-1.639 -1.451 -1.1%
-1.804 -1.797 -1.751
m=l
0.0735 0.2820
2.9347 0.7621 1.6300
1.364 1.099 0.8281
0.785 I.047 1.308
Table 2.
1.805 1.747 1.590
fll=O
0.00 0.261 0.523
40
0.381 0.343 0.247 0.135
0.135 0.025 0.344
0.0 0.007 0.048
m=2
0.006 0.033
4
0.00 -0.088 -0.122 -0.095
0.008
I 8
9
0.031 0.00 0.001 -0.021 -0.027 0.008 -0.004 0.013 -0.016 0.007 0.008 -0.011
0.00 0.008 -0.010 0.005
-0.003 -0.013 -0.011 -0.026 -0.008 0.00081 0.001 0.0209 0.007 -0.008
0.015
6
dQ(in radians)
0.009 -0.008 0.003 0.007
10
0.003 -o&34
I1
to.004
12
-0.002
13
-0.002
14
42.400 17.560
135”
6.450
28,220
120”
0.450 0.560 0.510 1.760
0.260 0.102
0.053
0.048
2
1.780 4.530 9.480 17.260
0.131 0.543
0.053
0.048
1
5.340
1.090
0.130 0.560 0.240 0.740
0.121 0.081
0.053
3
1.520
0.880
0.116 0.220 0.190 0.530
0.068 0.077
-
_-__-
4
1.060
0.680
__.~.
0.103 0.220 0.130 0.290
0.061 0.072
-
5
1.040
0.420
0.088 0.160 0.127 0.290
0.059 0.072
-
6
8
0.820
0.390
0.086 0.160 0.110 0.230
0.056 0.069
0.630
0.370
0084 0.150 0.110 0.220
0.067 -
Error in (%) -
7
0.560
0.590 0.600
0.320
0.320
0.320
0.082 0.140 0.105 0.200
-
-
11
0.082 0.140 0.105 0.200
-
-
10
0.082 0.150 0,110 0.210
0.066 -
-
-.-.-_--_
9
0.540
0.310
0.104 0.140 0.190
-
-
12
0.530
0.310
0.103 0.140 0.190
-
-
13
Table 3. Error in Fourier Series for eqn (12) vs number of Fourier coefficients. For different I& values independent term A0 [see eqn (IO)]t the number of Fourier coefficients A, indicated below
0.00 t0.042 0.024 -0.018
0.018 -0.026 -0.042
0.024
5
60” ; 105”
:lG
15”
0”
-0.0749 -0.040 0.024 0.053
0.094 0.053 0.123 -0.024 0.088 -0.040
0.006 0.041
3
P,(#) = $s A, cos ~C#J
Table t, Fourier Series for eqn (13) for different $Qvalues
0.528
0.304
0.103 0.138 0.190
-
-
14
$ B 45. g_ 2. fl s P G;
0, z a t z 5. c. z
B B r” 2.
J. L. U~~IA-GALICIAand A. N. SH~OU~E
i
(a)
(b) Fig. 15.Variablesand loads for granularcontainment.
Fwesve distribotiin cokuloted
thickness0.022"
with Jonssen's method.
~~~~_______*__________~____
t
(a)
@$+
0.434
ji
+.J
(cl
+ 0116 & IWinclT
Fig. 16. Pressure distributions after Janssen (coal dust).
coefficients will be 14 with a resulting maximum error of 0.5% with respect to the exact solution. It is useful to notice the rapid convergence of the solution with increasing numbers of terms. From Table 3, it may also be seen that the accuracy of the solution is good and that its sensitivity to radical changes when taking more than 14 terms is negligible, being less than 0.5%. CONCLUSIONS
The paper has discussed only the solution procedure involved in the static analysis of right cylindrical, circular pipes loaded internally with a fluid or granular medium. A series of curves for various K values in the range 0.6 < K < 2.0 have been presented elsewhere [lo], as indeed has experimental evidence, and will not therefore be discussed here. The procedure for the calculation of the stress protile in bending can be automated by way of a computer pro~amme and this has been carried out
(Appendix) typically for a water filled shell. The Fourier Series sub-routine is included as part of the computational package. Several conclusions can be drawn about the genera1 analysis which are valid not only for the static, equilibrium problem but also for future stability analysis: (i) The maximum stress does not always occur for the half-filled cylinder as is commonly assumed. It must be calculated for each particular case depending upon the associated K. As an extreme example, for K = 2.0, the maximum stress will occur in the fully loaded and not the half-filled cylinder. (ii) The length parameter 2 = (/*/I?) (1 - Y*)“*which is used to indicate, in an approximate way, whether a given cylinder is long or short may be replaced by the parameter K = d[i/d(t/r)]. This parameter appears to be a better indicator of the degree of nonlinearity in stress and may therefore be more helpful.
Analysis of stresses in internally loaded cylindrical shells
237 Boom andysis
Beam omtj5is
I
I
r I
I
, .
Stressanalysis GRANULAR
MATERIAL
7 \
,
____._Half
filledcyiindcr.
_Campletely
filled
cylinder.
Shell andysis;
-.
,
\
\
I
radius 12.028” thickness 0.022” length 180.000”
#’ ,’
t K:0.8
, ,
,’
/
,’
Fig. 17. Stress distributions for coal dust (granular). Stress anatysis GRANUAR MATERIAL
-_ _ _____ Half filled cylinder
-
Completely
filled
cylinder.
mdius 12.028” thickness O.B50” length 180.000”
A I’ I’
I
I
K:2.0
,’ I’
I’
I
/’
#’
Fig. 18. Stress distributions for coal dust (granular).
(iii) If cylinders have similar cross-sectional stress distributions,
then stability analyses may be performed on entire families of shells of given K. This would simplify the number of possible cases to be analyzed. (iv) Finally, with regard to the stress analysis of cylindrical shells, horizontally placed, it is obvious from the
results for granular materials that this case and the fluid loading case are completely different. There is no reason at all to assume any similarity in the predicted results. Acknowledgements-This work was carried out with financial assistance from the National Council of Science and Technology
(CONACYT) Mexico to the primary author (J. L. UrrutiaGalicia). The authors are also grateful to the National Research Council of Canada for assistance received through grant A-1582 (to A. N. Sherbourne) which made possible the experimental and computational work presented in this paper.
I
REFERENCES
A. N. Sherboume and H. M. Haydl, Large diameter metallic ’ thin walled ducts. Metal Structures Conference. Australian Institution of Engineers, Perth (1978). 2. W. Fliigge, Stresses in SheUs. Springer-Verlag, New York (1973).
238
J. L. U~UTIA-GALICIA and A. N.
3. S. P. Timoshenko, Theoryof Plates and Shelis. McGraw-Hill, New York (1959). 4. Pressure Vessel Technology, Part I, ASME, 3rd ht. Conf. Tokyo, Japan (1977). 5. R. J. Roark, Formulas for Stress and Strain, 4th Edn. McGraw-Hill, New York (1965). 6. F. B. Hildebrand, Advanced Calculus for Applications, 2nd Edn. Prentice-Hall, Englewood Cliffs, New Jersey (1976).
SHERBOURNE
7. T. W. Lambe and R. V. Whitman, Soil Mechanics. J. Wiley, New York (1979). 8. H. M. Haydl and A. N. Sherbourne, Designof Large Circular Ducts. Proceedings Institution Civil Engineers, Part 2 (1978). 9. David L. Powers, Boundary Value Problems. Academic Press, New York (1972). 10. A. N. Sherbourne and J. L. Urrutia-Galicia, Internally loaded cylindrical shells. To be published in the March 1982issue of the Canadian Journal of Civil Engineering.
LENDS
Computerprogram-~uid filfed sheIf The following variables are used in the program. NFI, NEX the number of segments in # and x directions. MFI.MEX the maximum number of Fourier coefficients selected. Instead of =, select any m integer value to get a finite Fourier series. CFI, CEX the value of the constant “c”. GAMA the water density. ALaNG the length of the shell.
T = t the thickness of the shell. BNU, E Poisson’s ratio and Young’s modulus, respec. tively. FILEVE = $J,, the water (load) level. NP parameter used to indicate the first element in the interval [0,2r] for which p,(g)#O, and therefore starts the calculations of the Fourier coefficients. NP = 1, 2, . . . , NF1/2 (or N&X/2).
The flow diagram is as follows: Read-NFI, MFI, CFI, NEX, MEX, CEX, Y. length(Q), radius (a) thickness (t), Young modulus (E) and Poisson's ratio (v) 1 ad-Load Level I$, and NP I
-1
Write equations Prf@), for example, prt~)=ay(co~~~-cos~) and calculate all values of pr($) at each of the NFI segments for the interval [C,Zlr]. Each value is calculated at the mid-point. I
I r
I
1 IWrite equations for p,(x) and calculate values of p,(x) at each of NRX segments. Each value is again calculated at the mid-point.
I I
CALL Fourier Series Sub-routine I
Analysis ofstresses in internally loaded cylindrical shells
I
CALL Sub-routine to multiply Fourier Series for P,(X) and ~~($1
Calculate stiffness coefficients Kll, K12, K13, etc.
Calculate Us,
vmn and wmn
Calculate N, or ox, using Umnv vmn and wmn.
I
G+T$ 1 and calculate another loading case
239
_w.._-- For part Number N
MI~M~~~rn~rn number of Fourier Coeificients c___--