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Imperfection sensitivity of stiffened cylinders subjected to external pressure
004%7949lsQ$3.00+ 0.00 Q 1989 pergamott Press plc
C~rnpulers& Swxtures Vol. 34, No. I, PP. 6369, 1990 Printed in Great Britain.
IMPERFECTION SENSITIVITY OF STrFFENED CYLINDERS SUBJECTED TO EXTERNAL PRESSURE SALAM S. SELEIM and JOHN B. KENNEDY Department of Civil Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4 (Receiwd 19 llecember 1988) Abstract-The effect of initial imperfections on the elastic-buckling behaviour of ring-stiffened cylinders under external pressure is examined using the potential energy approach. Experimental results in the open
literature of four ring-stiffened cylinders tested under external pressure were used to compare and verify the th~reti~l analysis. Both the theoretical and experimental results indicate no apparent influence of initial imperfections on the buckling behaviour within the elastic range. This was so in spite of the presence of an appreciable amount of initial imperfections. The results from this work indicate that less stringent requirements in the manufacturing of ring-stiffened cylinders may be tolerated, leading to possible economy.
NOTATION
initial out-of-roundness. For a ring-stiffened cylinder, shown in Fig. 1, the amount and shape of the initial imperfections depend mainly on the manufacturing procedure as well as the relative dimensions of the shell and the stiffening rings. In this paper attention is focussed on the effect of initial imperfections on the buckling hehaviour within the elastic range of ring-stiffened cylinders subjected to external pressure. Two modes of buckling instahility are considered, nameIy shell and general instahility; each mode is dealt with exclusively assuming no interaction between the two modes. The analysis is carried out using the Rayleigh-Ritz approach in conjunction with the potential energy method.
mean radius of the shell cross-sectional area of a stiffening ring (A, = b/z) f width of a stiffening ring (see Fig. 1) b’ Ef3,‘12(1- v2) D Young’s modulus E eccentricity of centroid of a stiffening ring ; height of a stiffening ring (see Fig. 1) second moment of inertia of a stiffening ring centre-line spacing between stiffening rings : L overall length of a ring-stiffened shell m,, number of longitudinal half waves at buckling ncr number of circumferential waves at buckling number of stiffening rings N PC-r theoretical buckling pressure experimental buckling pressure P f wall thickness of shell Fmm/L ct v Poisson’s ratio
2.POTENTLU ENERGY EXPRESSIONS
Consider a circular cylindrical shell of radius a, overall length L and wall thickness t stiffened with N rings of cross-sectional area A, and spaced a distance I apart as shown in Fig. 1. Under the action of a radial pressure P, and an axial pressure P2, the total displacements U, V and W’,in the directions shown in Fig. 2, of an arbitrary point are composed of the prebuckling displacements UO, V,, and W, and the additional displacements due to buckling II, u and w. The prebuckling defo~ations under the action of the
1. INTRODUCI’ION
In the study of stiffened cylinders the discrepancy between theoretical and experimental results is usually attributed to initial imperfections. For circular cylinders the term “initial im~rf~tions” refers to
Fig. 1. A typical ring-stiffened cylinder. CM3w-E
Fig. 2. Displacements of a shell element.
63
SALAM S. SELEIM and JOHNB. KENNEDY
64
uniform external pressure can be assumed to have the following forms: U,,X= const.,
D
iJ,,0 = 0
In
w2 &2
(1)
b=z
Kw; - 2~0,~) dx d0
+
0 SH
0
+
(w.08
+
w)2
a2
J
+ 2vw,,(w,@ + w) + 2(1 - v)
I
(4)
L
U
v, = 0 W, = const.
(u$
(3
>I
Considering the radial direction with initial imperfection w, which has the same shape as the buckling deformation w, the strains of the imperfect shell can be written as:
dx d9
v,~ -
w -
5 (w,@@ +
w)
*
de
>
s
Nor 2n
(ut,+Kw:,-2wv,B)de
+2a x
1
VJ -
9
0
2n
(2)
Kwt,-2wo,,)2d0
(u$+
w + 2a (u; + Kw$ - 2~0.8)
(6)
s0
s 2%
+v,
y/f
vb=g
(w,m
3
where K = 1 + 2w,/w is the imperfection parameter. The change in curvature of the stiffening ring in the 8 direction is assumed to be the same as that of the shell, i.e., wsX; this is a reasonable assumption since in the vicinity of the ring the shell bends with it as if it were an additional flange. The extensional strain at the centre of gravity of the ring, l,, is taken as:
+
w)~
where No,., No, and NoS are the membrane force resultants prior to buckling and are defined as:
P2a
Nox =- 2
(w,eo + w)
(3)
E,,, 2X L u’=2(1 __g) s o S[( o
uJ+
1 *
(A,+tl) --
Nos = where e is the eccentricity of the centre of gravity of the ring measured from the neutral axis of the shell. Using eqns (2) and (3), and ignoring those terms that will vanish during the minimization of the potential energy, the extensional and bending strain energies of the shell U, and iJ, and those of the r th stiffening ring V, and V,, respectively, can be written as follows:
PI at1
-_ NOe=
r,= to- ;
(7)
de
0
(8)
t P, aA,l (A, + tl) J
The buckling displacements u, u and w are assumed to have the following forms:
v=BsinnB.sin~
(9)
w=CcosnB.siny I
+
where n is the number of waves in the circumferential direction and m is the number of half waves in the longitudinal direction. The expressions in eqns (9) correspond to simply-supported boundary conditions in as much as they allow unrestricted end rotations. The initial imperfection w. is assumed to have the same shape as the radial buckling displacement w as follows:
vfx + Kw; 2
mxx w, = Co cos nfl * sin -. L +NOx;
(v;
+
Kw;) dx d0
(10)
The total work done by the external pressure Wr is equal to the sum of the work done in the axial
65
Imperfection sensitivity of stiffened cylinders direction W, plus that done in the radial direction W,. For the above assumed buckling shapes, Kendrick [l] showed that W, reduces to zero. The work done in the radial direction of the imperfect shell is 8iven in [Z] as follows:
+ 2nZA,(1 - v2) tL
a23 =
--n
l-f-
!.?(I
3-g~
-V)
1
[
- 2wu,, - ; GV@- O%,V,~ dx df?’ (11) >
x A,h(l+~(l--n’))+n(a-l)~I:
where 3 =w*+w. Now, the total potential energy Ur of the imperfect stiffened shell is Ur= v,+
u,+
v,+ vb+ wr.
ff,=
- &(4n)+
a33= 1 f &[&++(*2-
(12)
Substituting eqns (8X (9) and (IO) into eqn (12) yields a non-linear expression in A, B and C. For equilibrium, the total potential energy U, must be stationary, i.e.,
au,
au,
z=O,
x-0
and
au, ==O.
12vix2n) -$$$‘A,&;
+21f/* (l IL
1)2f2CL2(pl2-v)]
- G)(l - nr; I a2tL
+u -v2) -iiJ-A,(l+i(I
-n2))1
(13)
Equations (13) give a set of three non-linear equations in A, B and C, which can be solved numerically to obtain the buckling pressure and the amplitudes of the buckling deformations. The equations can also be solved ~alyti~lly if terms including quadratic and higher orders of the relatively small displacements U, , W, , I( and v are ignored in favour of similar terms of the radial displacement w. BY doing so, eqns (13) reduce to the fohowing:
a3.$=&(9u’+9nd+2n%~v}
+ 3n4A,(1 - ~3 J/45; 4a2tL aj5 = 3a2, ; as6= 3aw; b, = cd,;
I
b 2- -
0
a2, 0 aI2 412 6 a22 a23 a,3 63
a23
a33
~34
a35
0 0
-4,;
bz=(l
-*~a)&-;&;
a3t
and where:
[I
b, = Co b,
(14)
63
where:
a,,=.‘+&jyl
+g.y-nm,; a
af2 = -- “2” (1 +v) -&-v) [ aI3
=@$---
t&d 12n2
(1
--v)+&;
1
=
1
641 Et
v2)‘ ,
and
1,. =
P,a(l -v*) Et J
-?A,; The expression A is equal to unity except for m = N + 1 when it reduces to tl(A, + tl). However, the case when m = N + 1 corresponds to the shell instability mode for which the area of stiffener A,
SALAM S. SELEIM and JOHNB. KENNEDY
66
does not usually affect the buckling behaviour and can be ignored. Thus it is practically reasonable to set a equal to unity in all cases. Eliminating A and B from eqns (14), and considering only first order terms of the pressure parameters I, and 1, and up to and including third order terms of the modal amplitude C, results in the following equation: [6,0 + &,A, +
bw
+ hl+ ~*1WoC2
+ [a,,,+ 6,,1, + 42L21C3 +
[b,, 1, + &2~2lG = 0 (16)
where: 6 10 =
-&'12P,3P23 -
611=
+d3p22
+d2P33
3.
PllPzzP33;
-2P,3h*q23 -
+Pd3
Equation (16) can be used to study the effect of initial imperfection on the buckling behaviour of ring-stiffened cylinders under any combination of axial and lateral pressure effects. The case of external lateral pressure (1, = 0) was studied by Seleim and Roorda [2], who found that there is no significant effect of initial imperfections on the buckling behaviour. The case of external axial pressure (1, = 0) is not a practical one, since stringer-stiffened (i.e. using axial stiffeners) rather than ring-stiffened cylinders are commonly used for such a case of loading. The case of external uniform pressure (a, = 1,) is a common case for marine structures and some nuclear reactor components and will now be examined further.
+p23q12
2Pl,P23%3
+p:2433
To study the elastic buckling behaviour of ringstiffened cylinders under a uniform pressure, one may rewrite eqn (16) as follows:
-P22413)
-P:3qll
+ 2P,,P*,cl23
+ 2P,,P33ql*
BUCICLING BEHAVIOUR(I, = 1, = A)
-PllP**q33 1 =
_
-P22P23%G 62 = 422(P:3
-PIIP33)
630 = PII (P23P24 - P2lP34 + P23P35)
631
=P24h1(1*3
-PL3P24
+P23c?lI
+P34(2PL*q12 +P3dPIIQ23 632 =
-P12413
-P134,2)
-P**%*) -P12%3
-P,3412);
-PllP34!722;
6O, = b,k+3P22 + (1 -
602 =
-PLSPSd;
+P23qll
-Pl2P23) ~‘HP:,
-;(P:*
+ b2(P,,P23
-P,2P,3)
-P,,P**h
-PllP22);
and where pij, qii and & are defined using the coefficient matrix [a,] of eqns (14), as follows:
,
,
,
(a)
(17)
,
,
I
a,,=min
6
( > -2
6I1
,
m = 1 or m = N + 1 and n = 2,3,. . . (18)
au =pij + q$, + C&I.,; i = l-3, j = l-6.
L,,
+ SOCO
where 6, =6,, +a,, and 60=60, +ao2. The terms . . contaming the coefficients &,,, 6*,, 6,, and 6,, were omitted since they were found to have insignificant effect (approximately f 1%) on the buckling pressure parameter a. For a perfect ring-stiffened cylinder, the buckling pressure can be obtained from eqn (17) by setting Co = 0 and minimizing the linear part with respect to m and n. To simplify the process of minimization, Kendrick [1] suggested that one may consider the case when m = 1 which corresponds to the general instability mode (Fig. 3a) and m = N + 1 which corresponds to the shell instability mode (Fig. 3b). This is acceptable since it was assumed that these two modes do not interact. The critical buckling pressure parameter 1, and the corresponding critical values mcr and ner can be derived from the following expression obtained by Kendrick [ 11:
2PL*P%q12;
+PL2(P12P34
+ d30C3
6,C --P,#22);
+ q33(P:2
6 20 =d2P36; 621 =
6lOC
Lcl
(b)
Fig. 3. Basic buckling modes of ring-stiffened cylinders under external pressure. (a) General instability mode (G.I.M.). (b) Shell instability mode (WM.).
Imperfection sensitivity of stiffened cylinders
67
Table 1. Mechanical properties of test material Static yield Specimen
ASTM
Heat
Thickness
Dynamic yield stress
stress
Modulus of elasticity
specification number t (in) (ksi) x lo-’ F, $ (ksi) F,§ (ksi) 35.2 0.517 A36 85886 39.3 28.2 1 35.2 0.517 A36 85886 39.3 28.2 2 40.3 0.260 A36 71554 3 45.3 28.2 39.1 0.261 A36 71554 4 44.6 29.0 t These specimens corresponds to specimen nos 3, 4, 17 and 18, respectively, in the original text [4]. $ Crosshead speed zero. 0 Crosshead speed 0.5 in/min with 8 in gauge length. Note: 1 in = 25.4 mm, 1ksi = 6.895 MPa. ll0.t
Table 2. Geometry of test specimen Specimen no. 1 2 3 4
Mean diameter
Thickness
Overall length
D (in)
Bay length
1 (in)
L (in)
I (in)
15.50 15.51 15.73 15.71
0.517 0.517 0.260 0.261
192.0 192.0 192.0 192.0
44.82 60.76 31.69 48.67
Ring size b * h (in) 0.513 x 0.513 x 0.385 x 0.272 x
3.04 3.06 1.30 1.45
Note: 1 in = 25.4 mm.
Using m = m,, and n = nor in eqn (17) yields
(19)
A computer solution program was developed based on the above theoretical analysis. Equation (19) will now be used to study the effect of initial imperfections on the buckling behaviour of four ring-stiffened cylinders tested under a uniform external pressure. 4. EFFECT
OF IMPERFECTIONS
In order to compare and examine the analysis presented herein, the results of four ring-stiffened cylinders tested under external pressure will be used. The results are part of a test program carried out by Miller and Kinra [3] on 20 ring-stiffened fabricated steel cylinders. The four selected specimens were
made from typical offshore platform steel A36 following routine platform fabrication procedure. The mechanical properties and the geometries of the four specimens are shown in Tables 1 and 2, respectively. Using eqn (18), the buckling pressure and the corresponding critical values ma and n,, were computed for the four selected specimens. A comparison between the experimental and theoretical results is shown in Table 3. Specimens 1 and 2 failed by the shell instability mode (Fig. 3b), while specimens 3 and 4 failed by the general instability mode (Fig. 3a). All the four specimens failed at pressures higher than those predicted by eqn (18) as shown in Table 3. The post-buckling behaviour of the four specimens is shown in Fig. 4a-d as derived from eqn (19). The ratios between the stresses at failure to the dynamic yield stress were 0.94, 0.80, 0.20 and 0.16 for specimens 14, respectively [4]. The behaviour of specimens 3 and 4 was clearly elastic; however, specimens 1 and 2 exhibited inelastic behaviour.
Table 3. Experimental and theoretical buckling results
Specimen no.
Experimental buckling pressure -P, (psi)
Measured initial imperfection 2, t (in)
1 2
2120 1790
0.084 0.052
0.163 0.101
PJPC, 1.07 1.31
&I 4 3
3 4
545 402
0.102 0.094
0.392 0.360
1.62 1.64
2 2
tc,= Lx
- 4, 4
.
Note: 1 psi = 0.006895 MPa, 1 in = 25.4 mm.
W
zs 478.3 878.38 > 468.5 1102.2
Failure mode Shell instability General instability
SALAM
S. SELEIM and JOHNB. KENNEDY
2.o1(a)
c,h
2’o(b)
0.00
1.5-
0.05
1.5 -
0.00
0.10
0.05
0.15
0.10
0.20
J 0.5
0.15 0.20
0.0
0.1
0.2
0.3
0.4
(x5
1
2’o------a00
0.05
L”L 1.0
0.10 0.15 0.20
z
0.0
0.2
0.4
0.6
aa
Fig. 4. (a) Buckling behaviour of specimen no. 1 @A.M.). (b) Buckling behaviour of specimen no. 2 (S.I.M.). (c) Buckling behaviour of specimen no. 3 (G.I.M.). (d) Buckling behaviour of specimen no. 4 (G.I.M.).
All specimens had an appreciable amount of initial imperfections ranging from 10 to 39% of the shell wall thickness. The initial out-of-roundness factor y f&,)/0.01&,,) was 2.11, 1.29, 2.54 (r =(LXand 2.36 for specimens l-4, respectively [3]; the values of y far exceeded the limit value “y < 1.0” set by the current specifications [5,6]. In spite of such excessive initial imperfections, all specimens failed at pressures ranging from 7 to 64% higher than the corresponding theoretical values obtained from eqn (18). This shows that there is no evidence of imperfection sensitivity for ring-stiffened cylinders under external pressure. However, it was not possible to ascertain the type (stable/unstable) of post-buckling behaviour of the specimens, since no measurements of the radial displacements were reported in [3]. The theoretical results in Fig. 4a-d indicate a slightly stable post-buckling behaviour with no apparent influence of the initial imperfections for the four specimens. In general, this is in agreement with the experimental results. However, it should be recalled that these curves are valid in the elastic range only. When strains exceed or even approach the proportional limit of the material, the buckling behaviour might be different. In such cases, a refinement
of the present analysis would be necessary to include plastic or elasto-plastic behaviour. It is of interest to note here that in the literature one finds two different views about the post-buckling behaviour of stiffened cylinders under external pressure. The first view is in [7j, where it is stated that “only a small elastic post-buckling reduction in resistance is found and this only for long cylinders”, while in[8] it is shown that the post-buckling’behaviour is stable for very short cylinders (small value of the Batdorf parameter Z), then unstable for medium range and finally slightly unstable or neutral for very long cylinders. Similar results are found in [4]. The analyses in [4], [7] and [S] apply the results of an equivalent unstiffened cylinder to the stiffened one. In general, the theoretical results of this paper as well as the experimental results of [3] indicate that imperfections have no significant effect on the buckling of ring-stiffened cylinders under external pressure. The different findings reported in [4] and [8] are perhaps due to extending the results of unstiffened cylinders to those of stiffened ones. Such an extension might be adequate to estimate the buckling pressure; however, for the non-linear range beyond buckling
Imperfection sensitivity of stiffened cylinders the behaviours of the unstiffened and the stiffened cylinders are quite different [9], and are therefore not valid. The results of this paper may have a significant practical importance. The current specifications [5,6] limit the initial imperfections (overall out-of-roundness) by D,, - Dti. d 0.01 D,,,,,, . It is sometimes very expensive to satisfy such limits, especially for offshore structures where imperfections may arise during manufacturing and installation. The results from this work indicate that the above stringent requirements should be reviewed with a view to relax them. 5. CONCLUSIONS
In this paper the influence of initial imperfections on the elastic-buckling behaviour of ring-stiffened cylinders under external pressure is examined; the external loading included both axial and radial pressures. It is concluded that the presence of initial imperfections has no apparent influence on the buckling behaviour; this is confirmed and verified by experimental results on ring-stiffened cylinders having appreciable amounts of initial imperfections. The results of this work may result in reducing construction costs by relaxing the current limits on initial imperfections.
69 REFERENCES
1. S. Kendrick, The buckling under external pressure of ring-stiffened circular cylinders. Trans. R. Inst. Naval Architects 107,139-156 (1965). 2. S. S. Seleim and J. Roorda, Theoretical and experimental results on the post-buckling of ring-stiffened cylinders. Me& Struct. Mech. J. 15, 69-87 (1987). 3. C. D. Miller and R. K. Kinra, External pressure tests of ring-stiffened fabricated steel cylinders. 13th Annual Offshore Technology Conference, Houston, Texas, May &7, pp. 371-386 (1981). 4. J. W. Hutchinson and J. C. Amaxigo, Imperfection sensitivity of eccentrically stiffened cylindrical shells. AIAA Jnl5, 392-401 (1967). 5. American Society of Mechanical Engineers Boiler and Pressure Vessel Code, Section VIII, Divisions 1 and 2 and Section III, Division 1 (1980). 6. American Petroleum Institute, Recommended Practice for Planning, Designing and Constructing Fixed Ofshore Platforms. API RP 2A. 1lth Edn. Jan. (1980). 7. L. iI. Donnell, Effect ‘of imperfections on buckling of thin cylinders under external pressure. J. appl. Mech. 23, 569-575 (1956). 8. B. Budiansky and J. C. Amaxigo, Initial post-buckling behaviour of cylindrical shells under external pressure. J. Math. Phys. 47, 223-235 (1968). 9. M. Esslinger and B. Geier, Buckling and post-buckling behaviour of discretely stiffened thin-walled circular cylinders. Proc. of the European Colloquium of Mechanics, EURO MECH 15, Universite d’Orsay, France, Sept. 15-18, pp. 246-253 (1969).
C~rnpulers& Swxtures Vol. 34, No. I, PP. 6369, 1990 Printed in Great Britain.
IMPERFECTION SENSITIVITY OF STrFFENED CYLINDERS SUBJECTED TO EXTERNAL PRESSURE SALAM S. SELEIM and JOHN B. KENNEDY Department of Civil Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4 (Receiwd 19 llecember 1988) Abstract-The effect of initial imperfections on the elastic-buckling behaviour of ring-stiffened cylinders under external pressure is examined using the potential energy approach. Experimental results in the open
literature of four ring-stiffened cylinders tested under external pressure were used to compare and verify the th~reti~l analysis. Both the theoretical and experimental results indicate no apparent influence of initial imperfections on the buckling behaviour within the elastic range. This was so in spite of the presence of an appreciable amount of initial imperfections. The results from this work indicate that less stringent requirements in the manufacturing of ring-stiffened cylinders may be tolerated, leading to possible economy.
NOTATION
initial out-of-roundness. For a ring-stiffened cylinder, shown in Fig. 1, the amount and shape of the initial imperfections depend mainly on the manufacturing procedure as well as the relative dimensions of the shell and the stiffening rings. In this paper attention is focussed on the effect of initial imperfections on the buckling hehaviour within the elastic range of ring-stiffened cylinders subjected to external pressure. Two modes of buckling instahility are considered, nameIy shell and general instahility; each mode is dealt with exclusively assuming no interaction between the two modes. The analysis is carried out using the Rayleigh-Ritz approach in conjunction with the potential energy method.
mean radius of the shell cross-sectional area of a stiffening ring (A, = b/z) f width of a stiffening ring (see Fig. 1) b’ Ef3,‘12(1- v2) D Young’s modulus E eccentricity of centroid of a stiffening ring ; height of a stiffening ring (see Fig. 1) second moment of inertia of a stiffening ring centre-line spacing between stiffening rings : L overall length of a ring-stiffened shell m,, number of longitudinal half waves at buckling ncr number of circumferential waves at buckling number of stiffening rings N PC-r theoretical buckling pressure experimental buckling pressure P f wall thickness of shell Fmm/L ct v Poisson’s ratio
2.POTENTLU ENERGY EXPRESSIONS
Consider a circular cylindrical shell of radius a, overall length L and wall thickness t stiffened with N rings of cross-sectional area A, and spaced a distance I apart as shown in Fig. 1. Under the action of a radial pressure P, and an axial pressure P2, the total displacements U, V and W’,in the directions shown in Fig. 2, of an arbitrary point are composed of the prebuckling displacements UO, V,, and W, and the additional displacements due to buckling II, u and w. The prebuckling defo~ations under the action of the
1. INTRODUCI’ION
In the study of stiffened cylinders the discrepancy between theoretical and experimental results is usually attributed to initial imperfections. For circular cylinders the term “initial im~rf~tions” refers to
Fig. 1. A typical ring-stiffened cylinder. CM3w-E
Fig. 2. Displacements of a shell element.
63
SALAM S. SELEIM and JOHNB. KENNEDY
64
uniform external pressure can be assumed to have the following forms: U,,X= const.,
D
iJ,,0 = 0
In
w2 &2
(1)
b=z
Kw; - 2~0,~) dx d0
+
0 SH
0
+
(w.08
+
w)2
a2
J
+ 2vw,,(w,@ + w) + 2(1 - v)
I
(4)
L
U
v, = 0 W, = const.
(u$
(3
>I
Considering the radial direction with initial imperfection w, which has the same shape as the buckling deformation w, the strains of the imperfect shell can be written as:
dx d9
v,~ -
w -
5 (w,@@ +
w)
*
de
>
s
Nor 2n
(ut,+Kw:,-2wv,B)de
+2a x
1
VJ -
9
0
2n
(2)
Kwt,-2wo,,)2d0
(u$+
w + 2a (u; + Kw$ - 2~0.8)
(6)
s0
s 2%
+v,
y/f
vb=g
(w,m
3
where K = 1 + 2w,/w is the imperfection parameter. The change in curvature of the stiffening ring in the 8 direction is assumed to be the same as that of the shell, i.e., wsX; this is a reasonable assumption since in the vicinity of the ring the shell bends with it as if it were an additional flange. The extensional strain at the centre of gravity of the ring, l,, is taken as:
+
w)~
where No,., No, and NoS are the membrane force resultants prior to buckling and are defined as:
P2a
Nox =- 2
(w,eo + w)
(3)
E,,, 2X L u’=2(1 __g) s o S[( o
uJ+
1 *
(A,+tl) --
Nos = where e is the eccentricity of the centre of gravity of the ring measured from the neutral axis of the shell. Using eqns (2) and (3), and ignoring those terms that will vanish during the minimization of the potential energy, the extensional and bending strain energies of the shell U, and iJ, and those of the r th stiffening ring V, and V,, respectively, can be written as follows:
PI at1
-_ NOe=
r,= to- ;
(7)
de
0
(8)
t P, aA,l (A, + tl) J
The buckling displacements u, u and w are assumed to have the following forms:
v=BsinnB.sin~
(9)
w=CcosnB.siny I
+
where n is the number of waves in the circumferential direction and m is the number of half waves in the longitudinal direction. The expressions in eqns (9) correspond to simply-supported boundary conditions in as much as they allow unrestricted end rotations. The initial imperfection w. is assumed to have the same shape as the radial buckling displacement w as follows:
vfx + Kw; 2
mxx w, = Co cos nfl * sin -. L +NOx;
(v;
+
Kw;) dx d0
(10)
The total work done by the external pressure Wr is equal to the sum of the work done in the axial
65
Imperfection sensitivity of stiffened cylinders direction W, plus that done in the radial direction W,. For the above assumed buckling shapes, Kendrick [l] showed that W, reduces to zero. The work done in the radial direction of the imperfect shell is 8iven in [Z] as follows:
+ 2nZA,(1 - v2) tL
a23 =
--n
l-f-
!.?(I
3-g~
-V)
1
[
- 2wu,, - ; GV@- O%,V,~ dx df?’ (11) >
x A,h(l+~(l--n’))+n(a-l)~I:
where 3 =w*+w. Now, the total potential energy Ur of the imperfect stiffened shell is Ur= v,+
u,+
v,+ vb+ wr.
ff,=
- &(4n)+
a33= 1 f &[&++(*2-
(12)
Substituting eqns (8X (9) and (IO) into eqn (12) yields a non-linear expression in A, B and C. For equilibrium, the total potential energy U, must be stationary, i.e.,
au,
au,
z=O,
x-0
and
au, ==O.
12vix2n) -$$$‘A,&;
+21f/* (l IL
1)2f2CL2(pl2-v)]
- G)(l - nr; I a2tL
+u -v2) -iiJ-A,(l+i(I
-n2))1
(13)
Equations (13) give a set of three non-linear equations in A, B and C, which can be solved numerically to obtain the buckling pressure and the amplitudes of the buckling deformations. The equations can also be solved ~alyti~lly if terms including quadratic and higher orders of the relatively small displacements U, , W, , I( and v are ignored in favour of similar terms of the radial displacement w. BY doing so, eqns (13) reduce to the fohowing:
a3.$=&(9u’+9nd+2n%~v}
+ 3n4A,(1 - ~3 J/45; 4a2tL aj5 = 3a2, ; as6= 3aw; b, = cd,;
I
b 2- -
0
a2, 0 aI2 412 6 a22 a23 a,3 63
a23
a33
~34
a35
0 0
-4,;
bz=(l
-*~a)&-;&;
a3t
and where:
[I
b, = Co b,
(14)
63
where:
a,,=.‘+&jyl
+g.y-nm,; a
af2 = -- “2” (1 +v) -&-v) [ aI3
=@$---
t&d 12n2
(1
--v)+&;
1
=
1
641 Et
v2)‘ ,
and
1,. =
P,a(l -v*) Et J
-?A,; The expression A is equal to unity except for m = N + 1 when it reduces to tl(A, + tl). However, the case when m = N + 1 corresponds to the shell instability mode for which the area of stiffener A,
SALAM S. SELEIM and JOHNB. KENNEDY
66
does not usually affect the buckling behaviour and can be ignored. Thus it is practically reasonable to set a equal to unity in all cases. Eliminating A and B from eqns (14), and considering only first order terms of the pressure parameters I, and 1, and up to and including third order terms of the modal amplitude C, results in the following equation: [6,0 + &,A, +
bw
+ hl+ ~*1WoC2
+ [a,,,+ 6,,1, + 42L21C3 +
[b,, 1, + &2~2lG = 0 (16)
where: 6 10 =
-&'12P,3P23 -
611=
+d3p22
+d2P33
3.
PllPzzP33;
-2P,3h*q23 -
+Pd3
Equation (16) can be used to study the effect of initial imperfection on the buckling behaviour of ring-stiffened cylinders under any combination of axial and lateral pressure effects. The case of external lateral pressure (1, = 0) was studied by Seleim and Roorda [2], who found that there is no significant effect of initial imperfections on the buckling behaviour. The case of external axial pressure (1, = 0) is not a practical one, since stringer-stiffened (i.e. using axial stiffeners) rather than ring-stiffened cylinders are commonly used for such a case of loading. The case of external uniform pressure (a, = 1,) is a common case for marine structures and some nuclear reactor components and will now be examined further.
+p23q12
2Pl,P23%3
+p:2433
To study the elastic buckling behaviour of ringstiffened cylinders under a uniform pressure, one may rewrite eqn (16) as follows:
-P22413)
-P:3qll
+ 2P,,P*,cl23
+ 2P,,P33ql*
BUCICLING BEHAVIOUR(I, = 1, = A)
-PllP**q33 1 =
_
-P22P23%G 62 = 422(P:3
-PIIP33)
630 = PII (P23P24 - P2lP34 + P23P35)
631
=P24h1(1*3
-PL3P24
+P23c?lI
+P34(2PL*q12 +P3dPIIQ23 632 =
-P12413
-P134,2)
-P**%*) -P12%3
-P,3412);
-PllP34!722;
6O, = b,k+3P22 + (1 -
602 =
-PLSPSd;
+P23qll
-Pl2P23) ~‘HP:,
-;(P:*
+ b2(P,,P23
-P,2P,3)
-P,,P**h
-PllP22);
and where pij, qii and & are defined using the coefficient matrix [a,] of eqns (14), as follows:
,
,
,
(a)
(17)
,
,
I
a,,=min
6
( > -2
6I1
,
m = 1 or m = N + 1 and n = 2,3,. . . (18)
au =pij + q$, + C&I.,; i = l-3, j = l-6.
L,,
+ SOCO
where 6, =6,, +a,, and 60=60, +ao2. The terms . . contaming the coefficients &,,, 6*,, 6,, and 6,, were omitted since they were found to have insignificant effect (approximately f 1%) on the buckling pressure parameter a. For a perfect ring-stiffened cylinder, the buckling pressure can be obtained from eqn (17) by setting Co = 0 and minimizing the linear part with respect to m and n. To simplify the process of minimization, Kendrick [1] suggested that one may consider the case when m = 1 which corresponds to the general instability mode (Fig. 3a) and m = N + 1 which corresponds to the shell instability mode (Fig. 3b). This is acceptable since it was assumed that these two modes do not interact. The critical buckling pressure parameter 1, and the corresponding critical values mcr and ner can be derived from the following expression obtained by Kendrick [ 11:
2PL*P%q12;
+PL2(P12P34
+ d30C3
6,C --P,#22);
+ q33(P:2
6 20 =d2P36; 621 =
6lOC
Lcl
(b)
Fig. 3. Basic buckling modes of ring-stiffened cylinders under external pressure. (a) General instability mode (G.I.M.). (b) Shell instability mode (WM.).
Imperfection sensitivity of stiffened cylinders
67
Table 1. Mechanical properties of test material Static yield Specimen
ASTM
Heat
Thickness
Dynamic yield stress
stress
Modulus of elasticity
specification number t (in) (ksi) x lo-’ F, $ (ksi) F,§ (ksi) 35.2 0.517 A36 85886 39.3 28.2 1 35.2 0.517 A36 85886 39.3 28.2 2 40.3 0.260 A36 71554 3 45.3 28.2 39.1 0.261 A36 71554 4 44.6 29.0 t These specimens corresponds to specimen nos 3, 4, 17 and 18, respectively, in the original text [4]. $ Crosshead speed zero. 0 Crosshead speed 0.5 in/min with 8 in gauge length. Note: 1 in = 25.4 mm, 1ksi = 6.895 MPa. ll0.t
Table 2. Geometry of test specimen Specimen no. 1 2 3 4
Mean diameter
Thickness
Overall length
D (in)
Bay length
1 (in)
L (in)
I (in)
15.50 15.51 15.73 15.71
0.517 0.517 0.260 0.261
192.0 192.0 192.0 192.0
44.82 60.76 31.69 48.67
Ring size b * h (in) 0.513 x 0.513 x 0.385 x 0.272 x
3.04 3.06 1.30 1.45
Note: 1 in = 25.4 mm.
Using m = m,, and n = nor in eqn (17) yields
(19)
A computer solution program was developed based on the above theoretical analysis. Equation (19) will now be used to study the effect of initial imperfections on the buckling behaviour of four ring-stiffened cylinders tested under a uniform external pressure. 4. EFFECT
OF IMPERFECTIONS
In order to compare and examine the analysis presented herein, the results of four ring-stiffened cylinders tested under external pressure will be used. The results are part of a test program carried out by Miller and Kinra [3] on 20 ring-stiffened fabricated steel cylinders. The four selected specimens were
made from typical offshore platform steel A36 following routine platform fabrication procedure. The mechanical properties and the geometries of the four specimens are shown in Tables 1 and 2, respectively. Using eqn (18), the buckling pressure and the corresponding critical values ma and n,, were computed for the four selected specimens. A comparison between the experimental and theoretical results is shown in Table 3. Specimens 1 and 2 failed by the shell instability mode (Fig. 3b), while specimens 3 and 4 failed by the general instability mode (Fig. 3a). All the four specimens failed at pressures higher than those predicted by eqn (18) as shown in Table 3. The post-buckling behaviour of the four specimens is shown in Fig. 4a-d as derived from eqn (19). The ratios between the stresses at failure to the dynamic yield stress were 0.94, 0.80, 0.20 and 0.16 for specimens 14, respectively [4]. The behaviour of specimens 3 and 4 was clearly elastic; however, specimens 1 and 2 exhibited inelastic behaviour.
Table 3. Experimental and theoretical buckling results
Specimen no.
Experimental buckling pressure -P, (psi)
Measured initial imperfection 2, t (in)
1 2
2120 1790
0.084 0.052
0.163 0.101
PJPC, 1.07 1.31
&I 4 3
3 4
545 402
0.102 0.094
0.392 0.360
1.62 1.64
2 2
tc,= Lx
- 4, 4
.
Note: 1 psi = 0.006895 MPa, 1 in = 25.4 mm.
W
zs 478.3 878.38 > 468.5 1102.2
Failure mode Shell instability General instability
SALAM
S. SELEIM and JOHNB. KENNEDY
2.o1(a)
c,h
2’o(b)
0.00
1.5-
0.05
1.5 -
0.00
0.10
0.05
0.15
0.10
0.20
J 0.5
0.15 0.20
0.0
0.1
0.2
0.3
0.4
(x5
1
2’o------a00
0.05
L”L 1.0
0.10 0.15 0.20
z
0.0
0.2
0.4
0.6
aa
Fig. 4. (a) Buckling behaviour of specimen no. 1 @A.M.). (b) Buckling behaviour of specimen no. 2 (S.I.M.). (c) Buckling behaviour of specimen no. 3 (G.I.M.). (d) Buckling behaviour of specimen no. 4 (G.I.M.).
All specimens had an appreciable amount of initial imperfections ranging from 10 to 39% of the shell wall thickness. The initial out-of-roundness factor y f&,)/0.01&,,) was 2.11, 1.29, 2.54 (r =(LXand 2.36 for specimens l-4, respectively [3]; the values of y far exceeded the limit value “y < 1.0” set by the current specifications [5,6]. In spite of such excessive initial imperfections, all specimens failed at pressures ranging from 7 to 64% higher than the corresponding theoretical values obtained from eqn (18). This shows that there is no evidence of imperfection sensitivity for ring-stiffened cylinders under external pressure. However, it was not possible to ascertain the type (stable/unstable) of post-buckling behaviour of the specimens, since no measurements of the radial displacements were reported in [3]. The theoretical results in Fig. 4a-d indicate a slightly stable post-buckling behaviour with no apparent influence of the initial imperfections for the four specimens. In general, this is in agreement with the experimental results. However, it should be recalled that these curves are valid in the elastic range only. When strains exceed or even approach the proportional limit of the material, the buckling behaviour might be different. In such cases, a refinement
of the present analysis would be necessary to include plastic or elasto-plastic behaviour. It is of interest to note here that in the literature one finds two different views about the post-buckling behaviour of stiffened cylinders under external pressure. The first view is in [7j, where it is stated that “only a small elastic post-buckling reduction in resistance is found and this only for long cylinders”, while in[8] it is shown that the post-buckling’behaviour is stable for very short cylinders (small value of the Batdorf parameter Z), then unstable for medium range and finally slightly unstable or neutral for very long cylinders. Similar results are found in [4]. The analyses in [4], [7] and [S] apply the results of an equivalent unstiffened cylinder to the stiffened one. In general, the theoretical results of this paper as well as the experimental results of [3] indicate that imperfections have no significant effect on the buckling of ring-stiffened cylinders under external pressure. The different findings reported in [4] and [8] are perhaps due to extending the results of unstiffened cylinders to those of stiffened ones. Such an extension might be adequate to estimate the buckling pressure; however, for the non-linear range beyond buckling
Imperfection sensitivity of stiffened cylinders the behaviours of the unstiffened and the stiffened cylinders are quite different [9], and are therefore not valid. The results of this paper may have a significant practical importance. The current specifications [5,6] limit the initial imperfections (overall out-of-roundness) by D,, - Dti. d 0.01 D,,,,,, . It is sometimes very expensive to satisfy such limits, especially for offshore structures where imperfections may arise during manufacturing and installation. The results from this work indicate that the above stringent requirements should be reviewed with a view to relax them. 5. CONCLUSIONS
In this paper the influence of initial imperfections on the elastic-buckling behaviour of ring-stiffened cylinders under external pressure is examined; the external loading included both axial and radial pressures. It is concluded that the presence of initial imperfections has no apparent influence on the buckling behaviour; this is confirmed and verified by experimental results on ring-stiffened cylinders having appreciable amounts of initial imperfections. The results of this work may result in reducing construction costs by relaxing the current limits on initial imperfections.
69 REFERENCES
1. S. Kendrick, The buckling under external pressure of ring-stiffened circular cylinders. Trans. R. Inst. Naval Architects 107,139-156 (1965). 2. S. S. Seleim and J. Roorda, Theoretical and experimental results on the post-buckling of ring-stiffened cylinders. Me& Struct. Mech. J. 15, 69-87 (1987). 3. C. D. Miller and R. K. Kinra, External pressure tests of ring-stiffened fabricated steel cylinders. 13th Annual Offshore Technology Conference, Houston, Texas, May &7, pp. 371-386 (1981). 4. J. W. Hutchinson and J. C. Amaxigo, Imperfection sensitivity of eccentrically stiffened cylindrical shells. AIAA Jnl5, 392-401 (1967). 5. American Society of Mechanical Engineers Boiler and Pressure Vessel Code, Section VIII, Divisions 1 and 2 and Section III, Division 1 (1980). 6. American Petroleum Institute, Recommended Practice for Planning, Designing and Constructing Fixed Ofshore Platforms. API RP 2A. 1lth Edn. Jan. (1980). 7. L. iI. Donnell, Effect ‘of imperfections on buckling of thin cylinders under external pressure. J. appl. Mech. 23, 569-575 (1956). 8. B. Budiansky and J. C. Amaxigo, Initial post-buckling behaviour of cylindrical shells under external pressure. J. Math. Phys. 47, 223-235 (1968). 9. M. Esslinger and B. Geier, Buckling and post-buckling behaviour of discretely stiffened thin-walled circular cylinders. Proc. of the European Colloquium of Mechanics, EURO MECH 15, Universite d’Orsay, France, Sept. 15-18, pp. 246-253 (1969).