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A rational shape design of externally pressurized torispherical dome ends under buckling constraints
Cumpurers & Srrucrures Vol. 43. No. 5, pp. 839-851. 1992 Printed in Great Britain.
0
0045-7949/92 SS.00 + 0.00 1992 Pergamon PressLtd
A RATIONAL SHAPE DESIGN OF EXTERNALLY PRESSURIZED TORISPHERICAL DOME ENDS UNDER BUCKLING CONSTRAINTS M. F. YANG,? C. C. t
Department
$ Department
LIANG~
and C. H.
CHEN~
of Civil Engineering, Chung Cheng Institute of Technology, Ta-Shi, Taiwan 33509, Republic of China of Naval Architecture and Marine Engineering, Chung Cheng Institute of Technology, Ta-Shi, Taiwan 33509, Republic of China (Received
30 April 1991)
Abstract-In this work, the search for the optimal form for torispherical dome ends under external pressure load is conducted. Based on the fabrication and strength requirements, a group of ‘compromised’ contours are considered. With the adoption of both the BS5500 and ASME Section VIII PressureVessel codes, a reasonable buckling pressure range is proposed. The geometry of the dome end is described by the four-centered ellipse method which is commonly used in engineering drawing. A minimum weight optimization problem is studied by the discrete backtrack programming method; the optimal forms are obtained under buckling constraints. The elastic buckling analysis of the dome end is carried out by the finite element method using doubly-curved truncated shell elements. Two different sized dome end examples are studied. The developed optimal search procedure is found to be very efficient and easy-to-use for the applications, such as torispherical dome and subjected to externally pressurized loading.
NOTATION
D E F K pb PL P” PI+’
R
the length of major axis of a torispherical dome end the length of minor axis of a torispherical dome end the diameter of cylinder the modulus of elasticity objective function the ratio of the length of major axis to the length of minor axis for a torispherical dome end the critical buckling presure of torispherical dome end the lower bound of critical buckling pressure the upper bound of critical buckling pressure the design pressure of torispherical dome end the radius of spherical cap the radius of toroidal knuckle the thickness of shell the total weight of a torispherical dome end the total weight of a hemispherical dome end design variable the lower bound value of design variable x, the upper bound value of design variable x, Poisson’s ratio specific gravity the angle of spherical cape the angle of toroidal knuckle 1.
INTRODUCHON
Pressure vessels are very important in shell structures, and a majority of them are axisymmetric. It is common practice to construct the pressure vessels from various combinations of cylinders, cones and domes. Applications may be found in boilers, nuclear vessels, storage tanks, gas holders, underwater vessels, offshore drilling rigs, missiles, spacecraft and so on. In this paper we are only concerned with the structural integrity of the dome ends of the pressure hull.
Many experimental and theoretical buckling analyses of domes and structures have been carried
out [l-15]. Early experimental results about the collapse of a steel torispherical head under external pressure were investigated in [4,5]. Jones’ work [4] in 1962, primarily studied the effects of external pressure on the thin-walled torispherical, toriconical, and ellipsoidal pressure vessel heads including the determination of the collapse pressure and the shellto-head junction stresses. In 1964, Bart’s (51 reported the test results for a series of 24 dished heads subjected to hydrostatic pressure on the convex force, in which 10 dished heads were torispheres; the shape, thickness, and yield strength measurements are tabulated together with the actual collapse pressures. A review of the recent experimental investigations can be found in [l, 7,9]. In [l], 50 hemi-ellipsoidal domes were cast in solid urethane plastic with profiles which varied from oblate to prolate shapes. The domes were subjected to external hydraulic pressure in a test tank and tested to failure. All the domes appeared to fail through elastic instability, the oblate shapes failing axisymmetrically, and the prolate shape failing asymmetrically in a labor manner. In [I the technical basis of the external pressure section of the British Pressure Vessel Code BS5500 was presented, in which dome collapse was considered. Some comparisons are given with an equivalent American Code ASME Section VIII. Galletly et al. [9] studied the test results which were obtained from 24 externally pressurized torispherical steel shells. The knuckle radius-to-diameter ratio of the domes
839
840
M. F. YANGet al.
varied from 0.06 to 0.18 and the spherical cap radius-to-thickness ratios were between 75 and 335. Initial shape and thickness measurements were carried out on all the torispheres, and a summary of this info~ation was given in [9]. The BOSOR 5 [ 161shell buckling program was employed to predict the buckling/collapse pressure of all the domes; both perfect domes and those with axisymmetric imperfections were considered. Some other important theoretical buckling analysis of torispherical domes under external pressure load can be found in [3,8,9, 1I, 17,183. Among the above references the work of Galletly and Aylward(171, investigated the effect of elastic buckling pressure of unstiffened an cylinder with torispherical/ hemispherical end closures. Hishida and Ozawa [8] analyzed three-dimensional the axisymmetric boundary value problem of pressure vessels with torispherical drumheads by the use of the ‘indirect fictitious-boundary integral method’. Also in 1987, Ross [3] suggested a method of improving the structural efficiency of domes by inverting the dome ends of submarine pressure hulls so that they were concave inwardly to the effects of external pressure. During the period 1986-1988 Galletly et al. conducted research works [9, 11, 181 into perfect clamped torispherical shells subjected to external pressure. The torispherical shells were analyzed using the BOSOR 5 shell buckling program. The buckling with collapse pressure, modes of failure, and the development of plastic zones in the shell wall were also determined. For further supplementary details on the externally pressurized torispherical dome ends one should refer to [IS, 191. Many industrial applications, such as submersible vehicle design, are weight-c~tical, especially as opertional diving depth increases, it follows that designers will seek maximum strength of the structure with minimum weight. Minimum weight design of cylindrical and conical shells of the main pressure hull have been studied in detail by considering the primary failure modes in the literature [2632]. But studies concerned with the optimal shape design of torispherical dome ends which withstand the external pressure are still limited. The optimal geometric shape of externally pressurized torispherical dome ends under buckling constraints have not attracted as much attention as spherical caps or hemispheres over the last few decades, although doubly curved ends have been widely used across a range of industries. Previous works relating to the search for optimal configurations of torispherical dome ends under internal pressure can be found in [20-241. It is probably worth noting that the optimal design of pressure vessel dome ends under buckling constraints is, in general, not very frequently undertaken in the literature 1251.It seems that, for external pressurized cases, the optimization of torispheres under buckling constraints was attempted only in [12, 15, 191. [I91 used a discrete version of dynamic programming to
optimize elastic-plastic externally pressurized torispheres and the influence of stepwise mass distribution along the median on the failure load. A significant increase up to 50% in the external pressure was obtained, the optimally-shard torispheres were quite sensitive to axisymmetric deviations in geometry. In another paper Blachut [ 151used a direct search technique, which is a complex method of Box, and he combined this method with the BOSOR 5 code as a solver to study the optimal geometries of an elastic or elastic-plastic externally pressurized torispherical dome end of constant thickness and prescribed weight under static stability criteria. The Blachut end closures, which will withstand maximum or minimum buckling pressures and their response to initial axisymmetric imperfection, are examined and compared to those of hemispherical closures of the same weight and material properties. From Ross’s work [I] relating to dome failure test we see that the domes can fail in three major modes under uniform external pressure, namely, first if the dome ends are oblate hemi-ellipsoids, they can buckle axisymmetrically with the nose denting inward. Second, if the domes are prolate hemi-ellipsoids, they can buckle asymmetrically in a labor form. A third possible failure mode is through axisymmetric yield. Experimental tests have shown that for thin-walled domes under uniform external pressure, the mode of failure is most likely to be buckling, where very often the buckling pressure is a small fraction of the pressure to cause axisymmetric yield. This feature is considerably worsened by the effects of initial outof-circularity, which may have a very serious detrimental effect on the buckling resistance of the dome [l-3]. An attempt, therefore, is made in this paper to find an optimal, externally pressurized dome end within a class of torispherical shells under static buckling constraints based on elatic theory for perfect vessels. For imperfectly shaped vessels, which buckle inelastically, the experimental buckling pressure in this case may only be a small fraction of the predicted value of perfect vessel and that will not be covered in this paper. Often, as studied in [23], the cylindrical pressure vessels are closed by axisymmetric dome ends, and the shapes of the dome ends are chosen based on the requirements of the fabrication and structural strength. In addition, the dome ends should be as shallow as possible from the fabrication standpoint, while stress analysis demands a higher contour for the dome end so as to achieve a smooth membrane stress transition from the end to the cylinder while, at the same time, also minimizing bending effects at their junctions. Probably, under this situation the simplest dome end for a cylindrical pressure vessel is a flat plate. But fortunately, stresses in such a plate element as well as the bending stresses at its junction to the cylinder are unacceptably high. Consequently, the spherical or ellipsoidal dome ends are normally used. However, even the hemispherical dome end shows
841
Optimal form of torispherical dome ends favorable stress distribution and minimal bending effects at the junctions, but its fabrication may have some problems. For structures between the flat plate and the hemispherical dome end, representing the two ‘extremes’ in shape, there are the so-called ‘compromised’ contours, including ellipsoidal and torispherical dome ends. Based on this sort of ‘compromised’ idea, we can search the optimal configuration of externally pressurized torispherical dome ends from the two ‘extremes’ in this paper. The British Pressure Vessel Code BS5500, which was first issued in 1976, was designed to withstand external pressure [7]. This adopted BS5500 method was, and still is, quite different from that of ASME Section VIII Pressure Vessel Code [26]. The BS5500 method gives the design pressure Pw of externally pressurized dome ends which is not allowed to exceed l/8.33 of the elastic buckling pressure P,, while the corresponding ratio for ASME Section VIII is l/19.4. Therefore, from a practical design standpoint, a reasonable buckling pressure range is proposed between 8.33P,,. and 19.4P, in this paper. Three commonly used dome geometries, namely hemispherical dome, ellipsoidal dome, and torispherical dome (Fig. 1) can be well described by the four-centered ellipse method [27]. This ellipse method is commonly used in engineering drawing and the mathematical description is given in the Appendix. The four-centered ellipse method is very useful in the design and construction of the dome structure. In this paper, the torispherical dome end is simulated by doubly curved truncated elements. The finite element mesh configuration can be re-constructed by the four-centered ellipse method in the optimal process (Fig. 2). The optimal shape of a torisphe~cal dome under external pressure is studied in this paper by using the discrete backtrack programming method. The proposed numerical method is similar to the well-known branch-and-bound method, and the accuracy and efficiency of this scheme have been studied by Yuan et al. [28].
K=
-g_
I-=
@= tan-1 OD = tan-‘K OA (pz g-0
Fig. 2. Geometry of torispher~~i dome end.
2. GEOMETRIC DESCRIPTION OF TORISPHERICAL DOME ENDS
Consider first an elastic torispherical dome end of constant thickness t (Fig. 2) under static external pressure. It can be assumed that there is no flange and the torispherical dome end is fully clamped at the edge. By using the four-centered ellipse method, the geometry of torispherical dome end can be constructed (Fig. 2). If we know the ratio K (=b/a), then the parameters 8, rp, r, and R can be determined from the following equations which can be derived as (Appendix) 8 = tan-~(~/ff) = tan-’ K cp =n/2-a r=-
a 2[
/
d
1
1 + sin 0 - cos f3 cos* t?
R=~[l+~]
Torisphere ,-Sphere
where
Fig. 1. Typical dome end profiles.
(1) (2)
then the total weight of the torispherical with constant t is [15, 191 I
f(K,Q)
R= f(K,O)
and y is the specific gravity.
(3)
(41 dome end
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M. F. YANG et al.
Nodql
circle
Harvey’s study [29] the torispherical dome end should have a large knuckle radius in order to minimize the hoop stresses in this region. Many presssure vessel contraction codes recognize this fact and therefore specify a minimum permissible knuckle radius. For instance, the British Pressure Vessel Code BS5500 and the ASME Version VIII specify the minimum value of the knuckle radius as 6% of the crown radius, r/D > 0.06 (that is K > 0.21). Therefore, the lowest limit on the knuckle shallowness (r/D = 0.06) is chosen in this paper. Blachut [l 1, 151shows that an unexpected drop in the buckling strength with the range 0.45 < r/D < 0.5 (that is 0.95 < K < 1.00). Thus, the range of r/D is selected to be 0.06 < r/D < 0.45 (that is 0.21 < K < 0.95) here.
1
3. FINITE ELEMENT rf
Fig. 3. The doubly curved truncated shell element.
In [15, 191 the ratios r/D, R/D, and thickness t were selected for the design variables of the optimum shape design of torispherical dome end. However, in this work, when the four-centered ellipse method is used to construct the torispherical dome end, the radius of toroidal knuckle r and the radius of spherical cap R have certain relations which an be expressed as functions of the ratio K and the angle of spherical cap 8. Therefore, the K and thickness t are selected for design variables. The choice of boundary values for K and t depends heavily on practical application. However, the following factors must be taken into account. From
ANALYSIS
The elastic buckling analysis of externally pressurized torispherical dome ends is carried out using the finite element analysis method with doubly curved truncated elements. This truncated element is a tapered constant meridional curvature one, with two nodal circles at the ends. Each nodal circle has four degrees of freedom (ai, vi, wi and e,), this leads to a total of eight degrees of freedom in each element, as shown in Fig. 3. For solving eigenvalues and eigenvectors, the power method with the precision of the eigenvalues being set to 0.1%, is employed based on [2]. For the buckling analysis, 20 doubly curved truncated elements are considered. If the ratio of the minor radius to the major radius K is renewed for each of the finite elements meshes, these can then be re-constructed by r, R, 6 and cp as used in the
x
I’ 11 _1
1
1 b
n : nodal @I : element
number number
n=1,2,...,21 n=1,2,...,20 01
.i Fig. 4. The finite element idealization of torispherical dome end.
843
Optimal form of torispherical dome ends four-centered ellipse method. Figure 4 shows the typical generation for finite element mesh. Ten elements have been used to describe the spherical cap part, and ten other elements for toroidal knuckle. It is worthy of note that a finer mesh is used near the toroidal knuckle since the stress distribution in this region is very high. 4. MATHEMATICAL
FORMULATION
An externally pressurized torispherical dome end is designed for the minimum weight objective satisfying specified design requirements. The preassigned parameters for this design problem are the mechanical properties of the shell material and the manufa~turing limitations on the geometric parameters. The following optimization problem can then be formulated. 1. Objective function
g, = (PL - P6)IPb G 0, where, PL is the lower bound of critical buckling (Pr = 8.33P,). (c) The upper bound on design variables
where upper lower (d)
xi are design variables (i = 1,2), xUf is the bound value of design variable xi and xL8is the bound value of design variables x,. The lower bound on design variables g, = (XL1- Xi)/(XQ -XL<) < 0.
Accompanying engineering demands for high pressure vessels with and without large sizes are often the economic ones of weight reduction to save materials, reduce fabrication costs, and to enhance shipping and erection procedures. Thus, objective function F(Z) for weight is considered to be minimized in this study minimize
where Pb is the critical buckling pressure and P, is the upper bound of critical buckling pressure, (Pv = 19.4PW, where P, is the maximum work pressure). (b) The lower bound on buckling pressure
F(Z) = W,/ W,
where W, is the weight of torispherical dome end, W, is the weight of hemispherical dome end, and WI = fnyD2t. 2. Design variables The actual dimensions of the torispherical dome end are considered as the independent design variables (Fig. 4). These variables are: (a) the thickness of the dome shell, xI = t, (b) the ratio of the minor axis to the major axis, x, = K = b/a = tan 8, (c) the major axis of the dome end, x, = a. From engineering design considerations, the length of major radius a is specified which must coincide with cylindrical shell at the junction. Thus, there are two independent design variables taken for this problem, and the vector 2 will be given as
5. THE OPTIMIZATION METHODTHE BACKTRACK PR~RAMMING METHOD For actual structural applications, in many cases, the design variables, such as section sizes, sheet thickness, and the depth of a member section, are of the discrete type [30,31]. The structural optimization involves the choice of size, shape, and physical properties of structures. The optimal torispherical dome end design problem reveals that the discrete optimization problem is a nonlinear constrained one. The adopted discrete backtrack programming method is a scheme designed to solve nonlinear constrained function minimization problems by a systematic search procedure. The main idea of the backtrack programming method is that a partial search is carried out for each variable and, if all possibilities are calculated, a backtrack cycle is made and a new partial search will go on. For a detailed theoretical background of the backtrack programming method the reader can refer to [28,32]. the backtrack programming method adopts the intervalhalving procedure (IHP) for efficient search. A flow chart for the torispherical dome end is given in Fig. 5. 4. MODEL
3. Constraints The upper and lower bounds are imposed on all the design variables. Based on the stand~int of practical design, a reasonable buckling pressure range is proposed. The BS5500 design pressure is not allowed to exceed P&33, and the corresponding ratio for ASME is PJ19.4. The constraints g, are represented by the following: (a) The upper bound on buckling pressure g, = (P* - PlJ)IPU G 0,
DESCRIPTION
As an example, a submarine dome end design case is presented. Most submarine design is weight critical, especially as operational diving depth increases as the weight effect may influence the dive and float operation considerably. Submarine pressure hulls are generally formed by using cylindrical shells and conical shells for central sections, and to~sphe~~l shells for head domes (Fig. 6). Two reference models of the torispherical dome end shell of a submarine were considered. Both models have the same elastic modulus of
M.
F. YANG et al.
‘-i
START
Chooae the discrete values of the design variables: X+
i = l....,n k = l...u..,m
t &=
Xim, ,i = l,..., n
t i=l
t Xiu=Kk
‘es
I
Xik=Xik+A
1
Xiu = Xik Fig.
5(a).
Optimal form of torispherical dome ends
Optimum reeutte
compute Xnu from FMlu ,..., Xnu) = F,,Wlm, . ...Xnmj
Xnu > Xnm,
F,= F(Xlu,...JCtw)
Yes
-
Xnu = Xn,u+l
t
0 C
Celculete the parameter of four centered ellipsed method: e,rlr,r,R.ew(lb-(4)
Note: i : the number of deeiin vefidea k
t
I
I
Calculete the critiil Preeeure, Pb
: the number
of diirete
due
mi:theleetnumberofithdeeignveriebIe bXi=Xik-Xi,k-1
1
buckling
I
A Return
Fig. 5. Algorithm for the backtrap programming method.
846
M. F. YANG et al. Table 1. The principle design data of reference dome
+gT-J-~;~-~I
Torispherical dome end Length of major axis, a
Model 1
Model 2
63.98 in (1.625m)
91.73 in (2.330 m)
0.61
0.53
(II) : Cone
Ratio of minor axis to major axis, K = b/a
(III) : Cylinder
Thickness, t
1.26 in (0.032 m)
1.77 in (0.045 m)
Radius of toroidal knuckle, r
29.28 in (0.744 m)
34.35 in (0.872 m)
(I) : Torisphere
Fig. 6. Typical submersible pressure hull structure. 3.0 x 10’ lbf/in2
(207.0 GN/m2) and the same Poisson’s ratio of 0.3. The fixed major axis a for model 1 is 63.98in (162SOcm) and 91.73in (233.0cm) for mode1 2. The principal dimensions for the two reference models are listed in Table 1. Based on previous discussions, the range of the ratio K is selected as 0.21
Angle of toroidal knuckle, cp Radius of spherical cap, R
58.62”
61.99”
95.91 in (2.436 m)
156.83 in (3.985 m)
32.38”
28.01”
Angle of spherical cap, 0
7. NUMERICAL
RESULTS
AND DISCUSSION
The optimal comparison between the results of the torispherical dome end and the two reference dome end models are shown in Table 3. The comparisons
Table 2. The upper and lower bounds of design variables for optimum study Torispherical dome end
Side constraint
Major axis, a
Fixed
Ratio of minor axis to major axis,
Upper bound Lower bound Discrete value, AK
K =b/a
Thickness, t
Upper bound Lower bound Discrete value, At
Mode1 1
Mode1 2
63.98 in (1.625 m)
91.73 in (2.330 m)
0.95 0.21 0.01
0.95 0.21 0.01
2.56 in (0.065 m) 0.85 in (0.022 m) 0.01 in
3.66 in (0.093 m) I .22 in (0.03 1m) 0.01 in
Table 3. The numerical comparison table of the torispherical dome ends Model 1
Model 2
Ref. dome
Opt. dome
Ref. dome
Opt. dome
63.98 in (1.625 m)
63.98 in
(I ,625 m)
91.73 in (2.330 m)
91.73 in (2.330 m)
0.61
0.30
0.53
0.36
Thickness, t
I .26 in (0.032 m)
1.51 in (0.038 m)
1.77 in (0.045 m)
1.89 in (0.048 m)
Radius of toroidal knuckle, r
29.28 in (0.743 m)
11.49 in (0.292 m)
34.35 in (0.872 m)
20.61 in (0.523 m)
Angle of toroidal knuckle. cp
58.617”
73.301”
61.987”
70.201”
95.91 in (2.436 m)
194.15 in (4.931 m)
156.8 in (3.985 m)
230.57 in (5.856 m)
38.383”
16.699”
28.013”
19.799
873.46 Ibf (3.885 kN)
2486.90 lbf (11.062 kN)
2345.59 lbf (10.433 kN)
8216.604 psi (56.65 MN/m2)
3800.292 psi (26.20 MN/m’)
6273.851 psi (43.25 MN/m’)
2
2
2
Torispherical dome end Major axis, a Ratio of minor axis top major axis, K =b/a
Radius of spherical cap, R Angle of spherical cap, 0 Total weight of dome end
913,06 Ibf (4.061 kN)
Reduction ratio of weight
4.34%
Critical buckling pressure, Pb
The number of lobes of critical buckling
5.68% 3935.04 psi (27.13 MN/m2) 2
Optimal form of torisphericaldome ends
c-
a=
63.98” model
Reference r= R= t= 0 = (P =
1
Dome:
Optimum r= R= t0= (P =
29.29 in. = 0.743 m 95.91 in. = 2.436 m 1.26 in. = 0.032 m 31.3630 56.6170
Dome:
11.49 in. = 0.292 m 194.15 in. = 4.931 m 1.51 in. = 0.036 m 16.69s” 73.3OP
Fig. 7. The dome configuration comparision of model 1 for minimum weight design.
between referenced dome ends and optimal dome ends are displayed in Figs 7 and 8. The buckling modes for reference dome ends and optimum dome neds are shown in Figs 9 and 10. For a hemispherical dome end, the thickness effect of the dome end to the buckling pressure are studied and shown in Fig. 11 and Table 4. If thickness is kept constant, the influence of ratio K to the buckling pressure is also studied. The results are displayed in Fig. 10 and Table 5.
a=
From Table 3 and Figs 7 and 8, the ratios of K are found to be 0.30 and 0.36 for optimum model 1 and model 2, while the ratios of K are chosen as 0.61 and 0.53 for the reference models. The critical buckling pressures are 3800.292 psi (26.2 MN/m’) and 3935.04 psi (27.13 MN/m*) for optimum models, and for reference being models 8216.60 psi (56.65 MN/m*) and 6273.85 psi (43.25 MN/m*), respectively. Due to the fact that the ratios of k of the optimum models are smaller than that of reference
91.73” model
Reference r= R= t= 0= 60=
Dome:
34.35 in. = 0.672 m 156.66 in. = 3.965 m 1.77 in. = 0.045 m 26.013’ 61.967’
2 Optimum r= R= t= 0= P=
Dome:
20.61 in. = 0.523 m 230.57 in. = 5.656 m 1.69 in. = 0.046 m 19.799’ 72.201’
Fig. 8. The dome configuration comparison of model 2 for minimum weight design.
848
M. F. YANGet
Refsrence
Dome
Optimum
Dome
K=0.30 t=t.u in.=O.OSB m P,-3600.29 pd (=25.20 lai/m9
K=0.61
t=1.20 in.=o.o32 m P,=6216.06 psi
(=66.66 m/m9
No. of
al.
Lobes=2
No. of
Lob-=2
Fig. 9. Buckling mode for model 1. Optimum Dome
K=0.69
K=O.: t=1.-.it h.=O.O46 P,=Fmio.04 pei mk’l _, (=z27.ta . ..- - __,
t=1.77 in.=O.O46 m P,=6273.66 psf (=43.26 MN/m’) No. of Lobes=2
Nn
m
I I
nt lnh..=S
Fig. 10. Buckling mode for model 2
models, the shape of the optimum dome ends of two models tend to be flatter than the reference models, while buckling pressures are lower than the reference models. The weight is reduced by 4.34% for model 1 and 5.68% for model 2. The reason for that is we set the preassigned buckling pressure bound: 8.33P, 6 Pb < 10.4P,, and select the minimum weight type of objective function. If, on the other hand, the total weight of the torispherical dome ends are kept constant without the pressure bound requirement and then if we search for the optimum shape with the maximum critical buckling pressure, the critical buckling pressure could be raised. This problem has already been studied by Blachut [15]. From the practical viewpoint, each pressure vessel has a certain work environment, for example, the *o ‘Z P *-0
.
1
, .-~, - ,
~I
: model 1 in a = 63.98
S-
.~~
1
K = 1.00
’ - - : model 6*
2
a = 91.73
in
K = 1.00
I
E
6
(
-
; 2
_
4-
: ‘3 5 zB
2
0L 0.0
0.4
0.8
1.2 1.6 2.0 THICKNESS(in.)
2.4
2.8
3.2
Fig. 11. The buckling pressure via thickness of a hemispherical dome end (I psi = 6894.76 N/m*, 1 in = 0.0254 m).
maximum operation depth of a submarine is lOOft (304.8 m), then the design pressure P, is 444.44 psi (3.06 MN/m*). The BS5500 method will not allow the buckling pressure to be lower than 8.33 times the design pressure P,, and this leads to the lower bound of buckling pressure being 3702.18 psi (25.52 MN/m*). A dome end, at which buckling pressure value is much higher than 3702.18 psi in this
Table 4. The buckling pressure table at some corresponding thickness of hemispherical dome end (A)
a =
63.98 in, K = 1.00 Model 1 1 (in) PA(n) (psi) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.28 3.0
361.32(14) 1442.39 (15) 3213.64(15) 5651.91 (15) 8827.85 (13) 12874.68 (11) 17554.03 (10) 23172.47 (9) 28911.32 (9) 35737.3 1 (9) 43619.84 (8) 51764.25 (7) 63217.10(7) 72567.76 (7) 84779.08 (7j
(B) a = 91.73 in, K = 1.00 Model 2 t (in) PA(n) (psi) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Pb, critical buckling pressure. (n) the number of lobes. 1 in = 0.0254 m Conversion factors: 6894.76 N/m2.
184.19(15) 704.19(15) 1575.12 (15) 2786.56(15) 4331.54(15) 6182.74(13) 8442.41 (13) 11024.95 (12) 13953.88 (12) 17431.68 (10) 20924.07 (10) 24960.59 (10) 30068.39 (8) 34020.55 (9) 40114.12(10)
and
1 psi =
Optimal form of torispherical dome ends
Table 5. The buckling pressure table at some corresponding ratio K = (b/a) of torispherical dome ends
(A) I = 1.268 in, a = 63.98 in Model I K=bla
P,(n)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
(B) t = 1.77 in, a = 91.73 in Model 2 K = b/a Ph(n)
477.81 (1) 1245.36 (1) 2575.91 (1) 4214.41 (1) 5974.42 (2) 8006.73 (2) 10207.94 (2) 12306.44 (3) 14305.97 (4) 14060.97 (12) 11077.44(11) 8895.36 (I 1) 7373.55 (IO) 6232.25 (9) 5388.57 (9) 4714.42 (8) 4165.56 (8) 3775.33 (8) 3372.46 (7) 3054.72 (7)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
450.14 (1) 1190.43 (1) 2440.04 (1) 4014.13 (1) 5708.25 (2) 7660.88 (2) 9784.04 (2) 11795.17(2) 13748.55 (4) 13716.78 (12) 10409.31 (12) 8538.17 (10) 7057.35 (IO) 5974.79 (9) 5154.76 (9) 4512.89 (8) 3987.24 (8) 3578.80 (8) 323 1.96 (7) 2921.53 (7)
Ph. critical buckling pressure, psi. (n) the number of lobes. Conversion factors: I in = 0.0254 m 6894.76 N/m*.
1psi =
and
case, is not necessary and will be uneconomic. On the other hand, the design of the dome end must consider the practical applications. The flat dome and case, as in a submarine, is suitable for fabrication, but it is not suitable for military use. Since the flat dome ends structure is not suitable for the layout of torpedo tubes as the dome end space will be smaller in this type of configuration. The size of the dome end space can be influenced by the ratio K. From Table 4 and Figs 11 and 12 for a constant ratio K hemispherical dome end, the critical buckling pressure is to be raised with the increase of the thickness. Similar results can also be obtained from the various shapes of dome end. Also, the thickness is the other important influencing factor of the total weight of the dome end. However, the larger the thickness, the greater the weight. This situation must be taken into account.
a = 63.98
-:
in.
model
2
a = 91.73
in.
t = 1.77 in.
_,0.0
0.2
0.4
0.6
0.8
1.0 K=a/b
1.2
1.4
1.6
1.8
2.0
Fig. 12. The buckling pressure via the slope K ( = b/a) of dome end (I psi = 6894.76 N/m*, 1 in = 0.0254 m).
849
From Table 5 and Fig. 10, the maximum critical buckling pressure was found at K = 0.88 (that is R/D = 0.55). This value agrees with the results of Blachut [15]. When the ratio of K deviates from the value of 0.88, the buckling pressure drops. From Table 5, when K is below 0.3, the dome ends are rather flat and buckle axisymmetrically with the nose denting inward and, while K is higher than 0.3, the
dome ends buckle asymmetrically in a labor form. These phenomena also agree well with Ross and Mackney’s experimental results [l]. 13.CONCLUSIONS The four-centered ellipse method was adopted to describe the shape of the torispherical dome end and to make the shape optimization problem easier. This approach uses only two design variables: the ratio of minor axis to major axis K and thickness t. The discrete backtrack programming method combined with the doubly curved truncated shell elements were used. The above optimal serch procedure is found to be very efficient and easy-to-use for applications such as torispherical dome end subjected to externally pressurized loading. For further study, the optimum design of torispherical dome end may consider such factors as strength, space use, cost, fabrication, safety, and so on. Under such kind of considerations, the multiobjective function may need to be adopted for the optimal design problem. For instance, the objective function of the minimum weight and maximum critical buckling pressure can be taken into account at the same time. REFERENCES
1. C. T. F. Ross and M. D. A. Mackney, Deformation and
stability studies of thin-walled domes under uniform external pressure. J. Strain Anal. 18, 167-172 (1983). 2. C. T. F. Ross, Finite Element Programs for Axisymmet ric Problems in Engineering. Ellis-Horwood, Chichester
(1984). 3. C. T. F. Ross, Design of dome ends to withstand uniform external pressure. .I. Ship Res. 31, 139-143 (1987). 4. E. 0. Jones, The effects of external pressure on thin shell pressure vessel heads. J. Engng Industry 205-2 19 (1962). 5. R. Bart, An experimental study of the strength of standard flanged and dished ellipsoidal convex to pressure. J. Engng Industry 188-192 (1964). 6. P. Stanley and T. D. Campbell, Very thin torispherical pressure vessel ends under internal pressure: test procedure and typical results. J. Strain Anal. 16, 171-186 (1981). S. Kendrick, The technical basis of the external pressure section of BS5500. Trans ASME J. Press. Vess. Tech. 106, 143-149 (1984). M. Hishida and H. Ozawa, Three-dimensional axisymmetric elastic stresses in pressure vessels with torispherical drumheads (comparison of elasticity, photoelasticity, and shell theory solution). J. Strain Anal. 20, 183-191 (1985). G. D. Galletly, J. Kruzelecki, D. G. Moffat and B. Warrington, Buckling of shallow torispherical domes subjected to external pressure-a comparison of
850
M. F. YANGet al.
experiment, theory, and design codes. J. Stain Anal. 22, 163-175 (1987). 10. M. W. Uddin, Buckling of general spherical shells under external pressure. 1n1. J. Mech. Sci. 29,469-481 (1987). 1I. J. Blachut and G. D. Galletly, Clamped torispherical shells under external pressure-some new results. 1. Strain Anal. 23, 9-24 (1988). 12. G. D. Galletly and A. Muc, Buckling of fibre reinforced plastic/steel torispherical shells under external pressure. Proc. Inst. Mech. Engrs 202, 409-420 (1988). 13. D. Boote and D. Mascia, On the nonlinear
analysis methodologies for thin spherical shells under external pressure with different finite-element codes. J. Ship Res. 33, 318-325 (1989). 14. C. D. Miller, Research related to buckling design of nuclear containment. Nucl. Engng Design 79, 217-227 (1984). 15. J. Blachut, Search for optimal torispherical end closures under buckling constraints. Int. J. Mech. Sci. 31, 623-633 (1989). 16. D. Bushnell, BOSOR 5-program for buckling of elastic-plastic complex shells of revolution including large deflections and creep. Compur. Struct 6, 221-239
APPENDIX
The torispherical dome end geometry can be described by using a four-centered ellipse method [2i’]. Joint A and D lay off m eaual to m--m. This is done eranhicallv as indicated in Fig. Al by swinging from A aro&d to A’-with 0 as the -center, where now m from m’ is the required distance DA’, With D as center, an arc from A’ to the diagonal - m will locate F. Beside m by a perpendicular crossing A0 at G and intersecting i%? produce (if necessary) at H. Make m’ equal to m and m’ equal to m. Then G, G’, H and H’ will be centers for four tangent circle arcs forming a curve approximating the shape of an ellipse. The geometric relations from above description are X0 = major axis = a t5a = minor axis = 6
--
rF=AD-DO=a-b m’=i3;i=OD+m
(1976).
17. G. D. Galletly and R. W. Aylward, Buckling under external pressure of cylinders with either torispherical or hemispherical end closures. Press. Vess. Piping 1, 139-154 (1973). 18. G. D. Galletly, J. Blachut and J. Kruzelecki, Plastic buckling of externally pressurised dome ends. In Advances in Marine Structures (Edited by C. S. Smith and J. D. Clarke), pp. 238-261. Elsevier, London (1986). 19. J. Blachut, Optimally shaped torispheres with respect to buckling and their sensitivity to axisymmetric imperfections. Comput. Struct. 29, 975-981 (1988). 20. E. H. Mansfield, An optimum surface of revolution for pressurized shell. Int. J. Mech. Sci. 23, 57-62 (1981). 21, W. Szyszkowski and P. G. Glockner, A rational design of thin-walled pressure vessel ends. Tram ASME
-
DA’=m=a-b
aLi’==H
and the lengths of x
xI_rT_iiij --=_
24. J. Kowalski, Minimum mass design of boiler drums by nonlinear programming. Tram ASME J. Press. Vess. Tech. l&7,-83-87 (1985). 25. A. Gaiewski and M. Zvczkowski. Ootimal structural design* under stability fonstraints. &t Mechanics of Elclstie St~iliry (Edited by H. H. E. Leipholz and G. E. Oraras). Kluwer (1988). 26. ASME Boiler and Pressure Vessel Code, Section VIII. Pressure Vessels Division (1977). 27. T. E. French, C. J. Vierckand and R. J. Foster, Engineering Drawing and Graphic Technology, 3rd Edn. McGraw-Hill (I 987). 28. K. Y. Yuan, C. C. Liang and Y. C. Ma, The estimation of the accuracy and efficiency of the backtrack programming method for discrete-variable structural optimization problems. Comput. Struct. 36, 21 l-222 (1990). 29. J. F. Harvey, Pressure Component Construction: Design and Materials Application. Van Nostrand Reinhold, New York (198Oj.’ 30. A. M. Branat, S. Jendo and S. Owczarek, Criteria and method of Structural Optimization. Martinus Nijhoff (1986). 31. R. H. Gallagher and 0. C. Zienkiewicz, Optimum Structural Design. Rainbow Bridge, San Francisco, CA (1974). 32. J. Farkas, Optimum Design of Metal Structures. John Wiley, New York (1984).
-DF
2 =-
a
I +sin6-cost3
2[
J. Press. Vess. Tech 109, 386-373 (1987). 22. J. Middleton and J. Petmska, Optimal pressure vessel shape design to maximize limit load. Engng. Comput. 3, 287-294 ( 19868). 23. W. Szyszkowski and P, G. Glockner, Design for bucklefree shapes in pressure vessels. Trans ASME J. Press. Vess. Tech. 107, 387-393 (1985).
is
case
1
1
Let r be the radius of toroidal knuckle part and R the radius of spherical part, where both r and R can be derived from the above relationships. They can be found as the following r=XiT =-
=-
=iTJ
-AT cos 0 a
2[
I+sinB-cose co.52I3
1
R=mC-J
A
Fig. Al. A four-centered approximate ellipse.
Optimal form of torispherical dome ends
=gj+r
R =f(a,
851 0) =f(a,
K) =f(a,
b).
646)
For the hemi-spherical dome
a-r ==+fr
8 = 450
rsinfI+a-r _ sin 9
then
=&[1+S].
and
K=l
(A21
r=R=a=b.
From eqns (Al) and (A2), both r and R are functions of a and 6. Let ratio K be the minor axis to major axis K=tan0
=b
(A3)
a
For the 2 : 1 ellipsoid, b =a/2
then K = l/2.
then it gives cp +I r =f(a, 6) =f(a,
(A4) K) =f(a,
b)
(AS)
If the ratio K is known, then 0 can be calculated from eqn (A3), the angle cpcan be calculated from eqn (A4), and r and R can be calculated from eqs (A5) and (A6), respectively. The geometry of torispherical dome end can be sketched by the parameters r, R, 0 and cp.
0
0045-7949/92 SS.00 + 0.00 1992 Pergamon PressLtd
A RATIONAL SHAPE DESIGN OF EXTERNALLY PRESSURIZED TORISPHERICAL DOME ENDS UNDER BUCKLING CONSTRAINTS M. F. YANG,? C. C. t
Department
$ Department
LIANG~
and C. H.
CHEN~
of Civil Engineering, Chung Cheng Institute of Technology, Ta-Shi, Taiwan 33509, Republic of China of Naval Architecture and Marine Engineering, Chung Cheng Institute of Technology, Ta-Shi, Taiwan 33509, Republic of China (Received
30 April 1991)
Abstract-In this work, the search for the optimal form for torispherical dome ends under external pressure load is conducted. Based on the fabrication and strength requirements, a group of ‘compromised’ contours are considered. With the adoption of both the BS5500 and ASME Section VIII PressureVessel codes, a reasonable buckling pressure range is proposed. The geometry of the dome end is described by the four-centered ellipse method which is commonly used in engineering drawing. A minimum weight optimization problem is studied by the discrete backtrack programming method; the optimal forms are obtained under buckling constraints. The elastic buckling analysis of the dome end is carried out by the finite element method using doubly-curved truncated shell elements. Two different sized dome end examples are studied. The developed optimal search procedure is found to be very efficient and easy-to-use for the applications, such as torispherical dome and subjected to externally pressurized loading.
NOTATION
D E F K pb PL P” PI+’
R
the length of major axis of a torispherical dome end the length of minor axis of a torispherical dome end the diameter of cylinder the modulus of elasticity objective function the ratio of the length of major axis to the length of minor axis for a torispherical dome end the critical buckling presure of torispherical dome end the lower bound of critical buckling pressure the upper bound of critical buckling pressure the design pressure of torispherical dome end the radius of spherical cap the radius of toroidal knuckle the thickness of shell the total weight of a torispherical dome end the total weight of a hemispherical dome end design variable the lower bound value of design variable x, the upper bound value of design variable x, Poisson’s ratio specific gravity the angle of spherical cape the angle of toroidal knuckle 1.
INTRODUCHON
Pressure vessels are very important in shell structures, and a majority of them are axisymmetric. It is common practice to construct the pressure vessels from various combinations of cylinders, cones and domes. Applications may be found in boilers, nuclear vessels, storage tanks, gas holders, underwater vessels, offshore drilling rigs, missiles, spacecraft and so on. In this paper we are only concerned with the structural integrity of the dome ends of the pressure hull.
Many experimental and theoretical buckling analyses of domes and structures have been carried
out [l-15]. Early experimental results about the collapse of a steel torispherical head under external pressure were investigated in [4,5]. Jones’ work [4] in 1962, primarily studied the effects of external pressure on the thin-walled torispherical, toriconical, and ellipsoidal pressure vessel heads including the determination of the collapse pressure and the shellto-head junction stresses. In 1964, Bart’s (51 reported the test results for a series of 24 dished heads subjected to hydrostatic pressure on the convex force, in which 10 dished heads were torispheres; the shape, thickness, and yield strength measurements are tabulated together with the actual collapse pressures. A review of the recent experimental investigations can be found in [l, 7,9]. In [l], 50 hemi-ellipsoidal domes were cast in solid urethane plastic with profiles which varied from oblate to prolate shapes. The domes were subjected to external hydraulic pressure in a test tank and tested to failure. All the domes appeared to fail through elastic instability, the oblate shapes failing axisymmetrically, and the prolate shape failing asymmetrically in a labor manner. In [I the technical basis of the external pressure section of the British Pressure Vessel Code BS5500 was presented, in which dome collapse was considered. Some comparisons are given with an equivalent American Code ASME Section VIII. Galletly et al. [9] studied the test results which were obtained from 24 externally pressurized torispherical steel shells. The knuckle radius-to-diameter ratio of the domes
839
840
M. F. YANGet al.
varied from 0.06 to 0.18 and the spherical cap radius-to-thickness ratios were between 75 and 335. Initial shape and thickness measurements were carried out on all the torispheres, and a summary of this info~ation was given in [9]. The BOSOR 5 [ 161shell buckling program was employed to predict the buckling/collapse pressure of all the domes; both perfect domes and those with axisymmetric imperfections were considered. Some other important theoretical buckling analysis of torispherical domes under external pressure load can be found in [3,8,9, 1I, 17,183. Among the above references the work of Galletly and Aylward(171, investigated the effect of elastic buckling pressure of unstiffened an cylinder with torispherical/ hemispherical end closures. Hishida and Ozawa [8] analyzed three-dimensional the axisymmetric boundary value problem of pressure vessels with torispherical drumheads by the use of the ‘indirect fictitious-boundary integral method’. Also in 1987, Ross [3] suggested a method of improving the structural efficiency of domes by inverting the dome ends of submarine pressure hulls so that they were concave inwardly to the effects of external pressure. During the period 1986-1988 Galletly et al. conducted research works [9, 11, 181 into perfect clamped torispherical shells subjected to external pressure. The torispherical shells were analyzed using the BOSOR 5 shell buckling program. The buckling with collapse pressure, modes of failure, and the development of plastic zones in the shell wall were also determined. For further supplementary details on the externally pressurized torispherical dome ends one should refer to [IS, 191. Many industrial applications, such as submersible vehicle design, are weight-c~tical, especially as opertional diving depth increases, it follows that designers will seek maximum strength of the structure with minimum weight. Minimum weight design of cylindrical and conical shells of the main pressure hull have been studied in detail by considering the primary failure modes in the literature [2632]. But studies concerned with the optimal shape design of torispherical dome ends which withstand the external pressure are still limited. The optimal geometric shape of externally pressurized torispherical dome ends under buckling constraints have not attracted as much attention as spherical caps or hemispheres over the last few decades, although doubly curved ends have been widely used across a range of industries. Previous works relating to the search for optimal configurations of torispherical dome ends under internal pressure can be found in [20-241. It is probably worth noting that the optimal design of pressure vessel dome ends under buckling constraints is, in general, not very frequently undertaken in the literature 1251.It seems that, for external pressurized cases, the optimization of torispheres under buckling constraints was attempted only in [12, 15, 191. [I91 used a discrete version of dynamic programming to
optimize elastic-plastic externally pressurized torispheres and the influence of stepwise mass distribution along the median on the failure load. A significant increase up to 50% in the external pressure was obtained, the optimally-shard torispheres were quite sensitive to axisymmetric deviations in geometry. In another paper Blachut [ 151used a direct search technique, which is a complex method of Box, and he combined this method with the BOSOR 5 code as a solver to study the optimal geometries of an elastic or elastic-plastic externally pressurized torispherical dome end of constant thickness and prescribed weight under static stability criteria. The Blachut end closures, which will withstand maximum or minimum buckling pressures and their response to initial axisymmetric imperfection, are examined and compared to those of hemispherical closures of the same weight and material properties. From Ross’s work [I] relating to dome failure test we see that the domes can fail in three major modes under uniform external pressure, namely, first if the dome ends are oblate hemi-ellipsoids, they can buckle axisymmetrically with the nose denting inward. Second, if the domes are prolate hemi-ellipsoids, they can buckle asymmetrically in a labor form. A third possible failure mode is through axisymmetric yield. Experimental tests have shown that for thin-walled domes under uniform external pressure, the mode of failure is most likely to be buckling, where very often the buckling pressure is a small fraction of the pressure to cause axisymmetric yield. This feature is considerably worsened by the effects of initial outof-circularity, which may have a very serious detrimental effect on the buckling resistance of the dome [l-3]. An attempt, therefore, is made in this paper to find an optimal, externally pressurized dome end within a class of torispherical shells under static buckling constraints based on elatic theory for perfect vessels. For imperfectly shaped vessels, which buckle inelastically, the experimental buckling pressure in this case may only be a small fraction of the predicted value of perfect vessel and that will not be covered in this paper. Often, as studied in [23], the cylindrical pressure vessels are closed by axisymmetric dome ends, and the shapes of the dome ends are chosen based on the requirements of the fabrication and structural strength. In addition, the dome ends should be as shallow as possible from the fabrication standpoint, while stress analysis demands a higher contour for the dome end so as to achieve a smooth membrane stress transition from the end to the cylinder while, at the same time, also minimizing bending effects at their junctions. Probably, under this situation the simplest dome end for a cylindrical pressure vessel is a flat plate. But fortunately, stresses in such a plate element as well as the bending stresses at its junction to the cylinder are unacceptably high. Consequently, the spherical or ellipsoidal dome ends are normally used. However, even the hemispherical dome end shows
841
Optimal form of torispherical dome ends favorable stress distribution and minimal bending effects at the junctions, but its fabrication may have some problems. For structures between the flat plate and the hemispherical dome end, representing the two ‘extremes’ in shape, there are the so-called ‘compromised’ contours, including ellipsoidal and torispherical dome ends. Based on this sort of ‘compromised’ idea, we can search the optimal configuration of externally pressurized torispherical dome ends from the two ‘extremes’ in this paper. The British Pressure Vessel Code BS5500, which was first issued in 1976, was designed to withstand external pressure [7]. This adopted BS5500 method was, and still is, quite different from that of ASME Section VIII Pressure Vessel Code [26]. The BS5500 method gives the design pressure Pw of externally pressurized dome ends which is not allowed to exceed l/8.33 of the elastic buckling pressure P,, while the corresponding ratio for ASME Section VIII is l/19.4. Therefore, from a practical design standpoint, a reasonable buckling pressure range is proposed between 8.33P,,. and 19.4P, in this paper. Three commonly used dome geometries, namely hemispherical dome, ellipsoidal dome, and torispherical dome (Fig. 1) can be well described by the four-centered ellipse method [27]. This ellipse method is commonly used in engineering drawing and the mathematical description is given in the Appendix. The four-centered ellipse method is very useful in the design and construction of the dome structure. In this paper, the torispherical dome end is simulated by doubly curved truncated elements. The finite element mesh configuration can be re-constructed by the four-centered ellipse method in the optimal process (Fig. 2). The optimal shape of a torisphe~cal dome under external pressure is studied in this paper by using the discrete backtrack programming method. The proposed numerical method is similar to the well-known branch-and-bound method, and the accuracy and efficiency of this scheme have been studied by Yuan et al. [28].
K=
-g_
I-=
@= tan-1 OD = tan-‘K OA (pz g-0
Fig. 2. Geometry of torispher~~i dome end.
2. GEOMETRIC DESCRIPTION OF TORISPHERICAL DOME ENDS
Consider first an elastic torispherical dome end of constant thickness t (Fig. 2) under static external pressure. It can be assumed that there is no flange and the torispherical dome end is fully clamped at the edge. By using the four-centered ellipse method, the geometry of torispherical dome end can be constructed (Fig. 2). If we know the ratio K (=b/a), then the parameters 8, rp, r, and R can be determined from the following equations which can be derived as (Appendix) 8 = tan-~(~/ff) = tan-’ K cp =n/2-a r=-
a 2[
/
d
1
1 + sin 0 - cos f3 cos* t?
R=~[l+~]
Torisphere ,-Sphere
where
Fig. 1. Typical dome end profiles.
(1) (2)
then the total weight of the torispherical with constant t is [15, 191 I
f(K,Q)
R= f(K,O)
and y is the specific gravity.
(3)
(41 dome end
842
M. F. YANG et al.
Nodql
circle
Harvey’s study [29] the torispherical dome end should have a large knuckle radius in order to minimize the hoop stresses in this region. Many presssure vessel contraction codes recognize this fact and therefore specify a minimum permissible knuckle radius. For instance, the British Pressure Vessel Code BS5500 and the ASME Version VIII specify the minimum value of the knuckle radius as 6% of the crown radius, r/D > 0.06 (that is K > 0.21). Therefore, the lowest limit on the knuckle shallowness (r/D = 0.06) is chosen in this paper. Blachut [l 1, 151shows that an unexpected drop in the buckling strength with the range 0.45 < r/D < 0.5 (that is 0.95 < K < 1.00). Thus, the range of r/D is selected to be 0.06 < r/D < 0.45 (that is 0.21 < K < 0.95) here.
1
3. FINITE ELEMENT rf
Fig. 3. The doubly curved truncated shell element.
In [15, 191 the ratios r/D, R/D, and thickness t were selected for the design variables of the optimum shape design of torispherical dome end. However, in this work, when the four-centered ellipse method is used to construct the torispherical dome end, the radius of toroidal knuckle r and the radius of spherical cap R have certain relations which an be expressed as functions of the ratio K and the angle of spherical cap 8. Therefore, the K and thickness t are selected for design variables. The choice of boundary values for K and t depends heavily on practical application. However, the following factors must be taken into account. From
ANALYSIS
The elastic buckling analysis of externally pressurized torispherical dome ends is carried out using the finite element analysis method with doubly curved truncated elements. This truncated element is a tapered constant meridional curvature one, with two nodal circles at the ends. Each nodal circle has four degrees of freedom (ai, vi, wi and e,), this leads to a total of eight degrees of freedom in each element, as shown in Fig. 3. For solving eigenvalues and eigenvectors, the power method with the precision of the eigenvalues being set to 0.1%, is employed based on [2]. For the buckling analysis, 20 doubly curved truncated elements are considered. If the ratio of the minor radius to the major radius K is renewed for each of the finite elements meshes, these can then be re-constructed by r, R, 6 and cp as used in the
x
I’ 11 _1
1
1 b
n : nodal @I : element
number number
n=1,2,...,21 n=1,2,...,20 01
.i Fig. 4. The finite element idealization of torispherical dome end.
843
Optimal form of torispherical dome ends four-centered ellipse method. Figure 4 shows the typical generation for finite element mesh. Ten elements have been used to describe the spherical cap part, and ten other elements for toroidal knuckle. It is worthy of note that a finer mesh is used near the toroidal knuckle since the stress distribution in this region is very high. 4. MATHEMATICAL
FORMULATION
An externally pressurized torispherical dome end is designed for the minimum weight objective satisfying specified design requirements. The preassigned parameters for this design problem are the mechanical properties of the shell material and the manufa~turing limitations on the geometric parameters. The following optimization problem can then be formulated. 1. Objective function
g, = (PL - P6)IPb G 0, where, PL is the lower bound of critical buckling (Pr = 8.33P,). (c) The upper bound on design variables
where upper lower (d)
xi are design variables (i = 1,2), xUf is the bound value of design variable xi and xL8is the bound value of design variables x,. The lower bound on design variables g, = (XL1- Xi)/(XQ -XL<) < 0.
Accompanying engineering demands for high pressure vessels with and without large sizes are often the economic ones of weight reduction to save materials, reduce fabrication costs, and to enhance shipping and erection procedures. Thus, objective function F(Z) for weight is considered to be minimized in this study minimize
where Pb is the critical buckling pressure and P, is the upper bound of critical buckling pressure, (Pv = 19.4PW, where P, is the maximum work pressure). (b) The lower bound on buckling pressure
F(Z) = W,/ W,
where W, is the weight of torispherical dome end, W, is the weight of hemispherical dome end, and WI = fnyD2t. 2. Design variables The actual dimensions of the torispherical dome end are considered as the independent design variables (Fig. 4). These variables are: (a) the thickness of the dome shell, xI = t, (b) the ratio of the minor axis to the major axis, x, = K = b/a = tan 8, (c) the major axis of the dome end, x, = a. From engineering design considerations, the length of major radius a is specified which must coincide with cylindrical shell at the junction. Thus, there are two independent design variables taken for this problem, and the vector 2 will be given as
5. THE OPTIMIZATION METHODTHE BACKTRACK PR~RAMMING METHOD For actual structural applications, in many cases, the design variables, such as section sizes, sheet thickness, and the depth of a member section, are of the discrete type [30,31]. The structural optimization involves the choice of size, shape, and physical properties of structures. The optimal torispherical dome end design problem reveals that the discrete optimization problem is a nonlinear constrained one. The adopted discrete backtrack programming method is a scheme designed to solve nonlinear constrained function minimization problems by a systematic search procedure. The main idea of the backtrack programming method is that a partial search is carried out for each variable and, if all possibilities are calculated, a backtrack cycle is made and a new partial search will go on. For a detailed theoretical background of the backtrack programming method the reader can refer to [28,32]. the backtrack programming method adopts the intervalhalving procedure (IHP) for efficient search. A flow chart for the torispherical dome end is given in Fig. 5. 4. MODEL
3. Constraints The upper and lower bounds are imposed on all the design variables. Based on the stand~int of practical design, a reasonable buckling pressure range is proposed. The BS5500 design pressure is not allowed to exceed P&33, and the corresponding ratio for ASME is PJ19.4. The constraints g, are represented by the following: (a) The upper bound on buckling pressure g, = (P* - PlJ)IPU G 0,
DESCRIPTION
As an example, a submarine dome end design case is presented. Most submarine design is weight critical, especially as operational diving depth increases as the weight effect may influence the dive and float operation considerably. Submarine pressure hulls are generally formed by using cylindrical shells and conical shells for central sections, and to~sphe~~l shells for head domes (Fig. 6). Two reference models of the torispherical dome end shell of a submarine were considered. Both models have the same elastic modulus of
M.
F. YANG et al.
‘-i
START
Chooae the discrete values of the design variables: X+
i = l....,n k = l...u..,m
t &=
Xim, ,i = l,..., n
t i=l
t Xiu=Kk
‘es
I
Xik=Xik+A
1
Xiu = Xik Fig.
5(a).
Optimal form of torispherical dome ends
Optimum reeutte
compute Xnu from FMlu ,..., Xnu) = F,,Wlm, . ...Xnmj
Xnu > Xnm,
F,= F(Xlu,...JCtw)
Yes
-
Xnu = Xn,u+l
t
0 C
Celculete the parameter of four centered ellipsed method: e,rlr,r,R.ew(lb-(4)
Note: i : the number of deeiin vefidea k
t
I
I
Calculete the critiil Preeeure, Pb
: the number
of diirete
due
mi:theleetnumberofithdeeignveriebIe bXi=Xik-Xi,k-1
1
buckling
I
A Return
Fig. 5. Algorithm for the backtrap programming method.
846
M. F. YANG et al. Table 1. The principle design data of reference dome
+gT-J-~;~-~I
Torispherical dome end Length of major axis, a
Model 1
Model 2
63.98 in (1.625m)
91.73 in (2.330 m)
0.61
0.53
(II) : Cone
Ratio of minor axis to major axis, K = b/a
(III) : Cylinder
Thickness, t
1.26 in (0.032 m)
1.77 in (0.045 m)
Radius of toroidal knuckle, r
29.28 in (0.744 m)
34.35 in (0.872 m)
(I) : Torisphere
Fig. 6. Typical submersible pressure hull structure. 3.0 x 10’ lbf/in2
(207.0 GN/m2) and the same Poisson’s ratio of 0.3. The fixed major axis a for model 1 is 63.98in (162SOcm) and 91.73in (233.0cm) for mode1 2. The principal dimensions for the two reference models are listed in Table 1. Based on previous discussions, the range of the ratio K is selected as 0.21
Angle of toroidal knuckle, cp Radius of spherical cap, R
58.62”
61.99”
95.91 in (2.436 m)
156.83 in (3.985 m)
32.38”
28.01”
Angle of spherical cap, 0
7. NUMERICAL
RESULTS
AND DISCUSSION
The optimal comparison between the results of the torispherical dome end and the two reference dome end models are shown in Table 3. The comparisons
Table 2. The upper and lower bounds of design variables for optimum study Torispherical dome end
Side constraint
Major axis, a
Fixed
Ratio of minor axis to major axis,
Upper bound Lower bound Discrete value, AK
K =b/a
Thickness, t
Upper bound Lower bound Discrete value, At
Mode1 1
Mode1 2
63.98 in (1.625 m)
91.73 in (2.330 m)
0.95 0.21 0.01
0.95 0.21 0.01
2.56 in (0.065 m) 0.85 in (0.022 m) 0.01 in
3.66 in (0.093 m) I .22 in (0.03 1m) 0.01 in
Table 3. The numerical comparison table of the torispherical dome ends Model 1
Model 2
Ref. dome
Opt. dome
Ref. dome
Opt. dome
63.98 in (1.625 m)
63.98 in
(I ,625 m)
91.73 in (2.330 m)
91.73 in (2.330 m)
0.61
0.30
0.53
0.36
Thickness, t
I .26 in (0.032 m)
1.51 in (0.038 m)
1.77 in (0.045 m)
1.89 in (0.048 m)
Radius of toroidal knuckle, r
29.28 in (0.743 m)
11.49 in (0.292 m)
34.35 in (0.872 m)
20.61 in (0.523 m)
Angle of toroidal knuckle. cp
58.617”
73.301”
61.987”
70.201”
95.91 in (2.436 m)
194.15 in (4.931 m)
156.8 in (3.985 m)
230.57 in (5.856 m)
38.383”
16.699”
28.013”
19.799
873.46 Ibf (3.885 kN)
2486.90 lbf (11.062 kN)
2345.59 lbf (10.433 kN)
8216.604 psi (56.65 MN/m2)
3800.292 psi (26.20 MN/m’)
6273.851 psi (43.25 MN/m’)
2
2
2
Torispherical dome end Major axis, a Ratio of minor axis top major axis, K =b/a
Radius of spherical cap, R Angle of spherical cap, 0 Total weight of dome end
913,06 Ibf (4.061 kN)
Reduction ratio of weight
4.34%
Critical buckling pressure, Pb
The number of lobes of critical buckling
5.68% 3935.04 psi (27.13 MN/m2) 2
Optimal form of torisphericaldome ends
c-
a=
63.98” model
Reference r= R= t= 0 = (P =
1
Dome:
Optimum r= R= t0= (P =
29.29 in. = 0.743 m 95.91 in. = 2.436 m 1.26 in. = 0.032 m 31.3630 56.6170
Dome:
11.49 in. = 0.292 m 194.15 in. = 4.931 m 1.51 in. = 0.036 m 16.69s” 73.3OP
Fig. 7. The dome configuration comparision of model 1 for minimum weight design.
between referenced dome ends and optimal dome ends are displayed in Figs 7 and 8. The buckling modes for reference dome ends and optimum dome neds are shown in Figs 9 and 10. For a hemispherical dome end, the thickness effect of the dome end to the buckling pressure are studied and shown in Fig. 11 and Table 4. If thickness is kept constant, the influence of ratio K to the buckling pressure is also studied. The results are displayed in Fig. 10 and Table 5.
a=
From Table 3 and Figs 7 and 8, the ratios of K are found to be 0.30 and 0.36 for optimum model 1 and model 2, while the ratios of K are chosen as 0.61 and 0.53 for the reference models. The critical buckling pressures are 3800.292 psi (26.2 MN/m’) and 3935.04 psi (27.13 MN/m*) for optimum models, and for reference being models 8216.60 psi (56.65 MN/m*) and 6273.85 psi (43.25 MN/m*), respectively. Due to the fact that the ratios of k of the optimum models are smaller than that of reference
91.73” model
Reference r= R= t= 0= 60=
Dome:
34.35 in. = 0.672 m 156.66 in. = 3.965 m 1.77 in. = 0.045 m 26.013’ 61.967’
2 Optimum r= R= t= 0= P=
Dome:
20.61 in. = 0.523 m 230.57 in. = 5.656 m 1.69 in. = 0.046 m 19.799’ 72.201’
Fig. 8. The dome configuration comparison of model 2 for minimum weight design.
848
M. F. YANGet
Refsrence
Dome
Optimum
Dome
K=0.30 t=t.u in.=O.OSB m P,-3600.29 pd (=25.20 lai/m9
K=0.61
t=1.20 in.=o.o32 m P,=6216.06 psi
(=66.66 m/m9
No. of
al.
Lobes=2
No. of
Lob-=2
Fig. 9. Buckling mode for model 1. Optimum Dome
K=0.69
K=O.: t=1.-.it h.=O.O46 P,=Fmio.04 pei mk’l _, (=z27.ta . ..- - __,
t=1.77 in.=O.O46 m P,=6273.66 psf (=43.26 MN/m’) No. of Lobes=2
Nn
m
I I
nt lnh..=S
Fig. 10. Buckling mode for model 2
models, the shape of the optimum dome ends of two models tend to be flatter than the reference models, while buckling pressures are lower than the reference models. The weight is reduced by 4.34% for model 1 and 5.68% for model 2. The reason for that is we set the preassigned buckling pressure bound: 8.33P, 6 Pb < 10.4P,, and select the minimum weight type of objective function. If, on the other hand, the total weight of the torispherical dome ends are kept constant without the pressure bound requirement and then if we search for the optimum shape with the maximum critical buckling pressure, the critical buckling pressure could be raised. This problem has already been studied by Blachut [15]. From the practical viewpoint, each pressure vessel has a certain work environment, for example, the *o ‘Z P *-0
.
1
, .-~, - ,
~I
: model 1 in a = 63.98
S-
.~~
1
K = 1.00
’ - - : model 6*
2
a = 91.73
in
K = 1.00
I
E
6
(
-
; 2
_
4-
: ‘3 5 zB
2
0L 0.0
0.4
0.8
1.2 1.6 2.0 THICKNESS(in.)
2.4
2.8
3.2
Fig. 11. The buckling pressure via thickness of a hemispherical dome end (I psi = 6894.76 N/m*, 1 in = 0.0254 m).
maximum operation depth of a submarine is lOOft (304.8 m), then the design pressure P, is 444.44 psi (3.06 MN/m*). The BS5500 method will not allow the buckling pressure to be lower than 8.33 times the design pressure P,, and this leads to the lower bound of buckling pressure being 3702.18 psi (25.52 MN/m*). A dome end, at which buckling pressure value is much higher than 3702.18 psi in this
Table 4. The buckling pressure table at some corresponding thickness of hemispherical dome end (A)
a =
63.98 in, K = 1.00 Model 1 1 (in) PA(n) (psi) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.28 3.0
361.32(14) 1442.39 (15) 3213.64(15) 5651.91 (15) 8827.85 (13) 12874.68 (11) 17554.03 (10) 23172.47 (9) 28911.32 (9) 35737.3 1 (9) 43619.84 (8) 51764.25 (7) 63217.10(7) 72567.76 (7) 84779.08 (7j
(B) a = 91.73 in, K = 1.00 Model 2 t (in) PA(n) (psi) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Pb, critical buckling pressure. (n) the number of lobes. 1 in = 0.0254 m Conversion factors: 6894.76 N/m2.
184.19(15) 704.19(15) 1575.12 (15) 2786.56(15) 4331.54(15) 6182.74(13) 8442.41 (13) 11024.95 (12) 13953.88 (12) 17431.68 (10) 20924.07 (10) 24960.59 (10) 30068.39 (8) 34020.55 (9) 40114.12(10)
and
1 psi =
Optimal form of torispherical dome ends
Table 5. The buckling pressure table at some corresponding ratio K = (b/a) of torispherical dome ends
(A) I = 1.268 in, a = 63.98 in Model I K=bla
P,(n)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
(B) t = 1.77 in, a = 91.73 in Model 2 K = b/a Ph(n)
477.81 (1) 1245.36 (1) 2575.91 (1) 4214.41 (1) 5974.42 (2) 8006.73 (2) 10207.94 (2) 12306.44 (3) 14305.97 (4) 14060.97 (12) 11077.44(11) 8895.36 (I 1) 7373.55 (IO) 6232.25 (9) 5388.57 (9) 4714.42 (8) 4165.56 (8) 3775.33 (8) 3372.46 (7) 3054.72 (7)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
450.14 (1) 1190.43 (1) 2440.04 (1) 4014.13 (1) 5708.25 (2) 7660.88 (2) 9784.04 (2) 11795.17(2) 13748.55 (4) 13716.78 (12) 10409.31 (12) 8538.17 (10) 7057.35 (IO) 5974.79 (9) 5154.76 (9) 4512.89 (8) 3987.24 (8) 3578.80 (8) 323 1.96 (7) 2921.53 (7)
Ph. critical buckling pressure, psi. (n) the number of lobes. Conversion factors: I in = 0.0254 m 6894.76 N/m*.
1psi =
and
case, is not necessary and will be uneconomic. On the other hand, the design of the dome end must consider the practical applications. The flat dome and case, as in a submarine, is suitable for fabrication, but it is not suitable for military use. Since the flat dome ends structure is not suitable for the layout of torpedo tubes as the dome end space will be smaller in this type of configuration. The size of the dome end space can be influenced by the ratio K. From Table 4 and Figs 11 and 12 for a constant ratio K hemispherical dome end, the critical buckling pressure is to be raised with the increase of the thickness. Similar results can also be obtained from the various shapes of dome end. Also, the thickness is the other important influencing factor of the total weight of the dome end. However, the larger the thickness, the greater the weight. This situation must be taken into account.
a = 63.98
-:
in.
model
2
a = 91.73
in.
t = 1.77 in.
_,0.0
0.2
0.4
0.6
0.8
1.0 K=a/b
1.2
1.4
1.6
1.8
2.0
Fig. 12. The buckling pressure via the slope K ( = b/a) of dome end (I psi = 6894.76 N/m*, 1 in = 0.0254 m).
849
From Table 5 and Fig. 10, the maximum critical buckling pressure was found at K = 0.88 (that is R/D = 0.55). This value agrees with the results of Blachut [15]. When the ratio of K deviates from the value of 0.88, the buckling pressure drops. From Table 5, when K is below 0.3, the dome ends are rather flat and buckle axisymmetrically with the nose denting inward and, while K is higher than 0.3, the
dome ends buckle asymmetrically in a labor form. These phenomena also agree well with Ross and Mackney’s experimental results [l]. 13.CONCLUSIONS The four-centered ellipse method was adopted to describe the shape of the torispherical dome end and to make the shape optimization problem easier. This approach uses only two design variables: the ratio of minor axis to major axis K and thickness t. The discrete backtrack programming method combined with the doubly curved truncated shell elements were used. The above optimal serch procedure is found to be very efficient and easy-to-use for applications such as torispherical dome end subjected to externally pressurized loading. For further study, the optimum design of torispherical dome end may consider such factors as strength, space use, cost, fabrication, safety, and so on. Under such kind of considerations, the multiobjective function may need to be adopted for the optimal design problem. For instance, the objective function of the minimum weight and maximum critical buckling pressure can be taken into account at the same time. REFERENCES
1. C. T. F. Ross and M. D. A. Mackney, Deformation and
stability studies of thin-walled domes under uniform external pressure. J. Strain Anal. 18, 167-172 (1983). 2. C. T. F. Ross, Finite Element Programs for Axisymmet ric Problems in Engineering. Ellis-Horwood, Chichester
(1984). 3. C. T. F. Ross, Design of dome ends to withstand uniform external pressure. .I. Ship Res. 31, 139-143 (1987). 4. E. 0. Jones, The effects of external pressure on thin shell pressure vessel heads. J. Engng Industry 205-2 19 (1962). 5. R. Bart, An experimental study of the strength of standard flanged and dished ellipsoidal convex to pressure. J. Engng Industry 188-192 (1964). 6. P. Stanley and T. D. Campbell, Very thin torispherical pressure vessel ends under internal pressure: test procedure and typical results. J. Strain Anal. 16, 171-186 (1981). S. Kendrick, The technical basis of the external pressure section of BS5500. Trans ASME J. Press. Vess. Tech. 106, 143-149 (1984). M. Hishida and H. Ozawa, Three-dimensional axisymmetric elastic stresses in pressure vessels with torispherical drumheads (comparison of elasticity, photoelasticity, and shell theory solution). J. Strain Anal. 20, 183-191 (1985). G. D. Galletly, J. Kruzelecki, D. G. Moffat and B. Warrington, Buckling of shallow torispherical domes subjected to external pressure-a comparison of
850
M. F. YANGet al.
experiment, theory, and design codes. J. Stain Anal. 22, 163-175 (1987). 10. M. W. Uddin, Buckling of general spherical shells under external pressure. 1n1. J. Mech. Sci. 29,469-481 (1987). 1I. J. Blachut and G. D. Galletly, Clamped torispherical shells under external pressure-some new results. 1. Strain Anal. 23, 9-24 (1988). 12. G. D. Galletly and A. Muc, Buckling of fibre reinforced plastic/steel torispherical shells under external pressure. Proc. Inst. Mech. Engrs 202, 409-420 (1988). 13. D. Boote and D. Mascia, On the nonlinear
analysis methodologies for thin spherical shells under external pressure with different finite-element codes. J. Ship Res. 33, 318-325 (1989). 14. C. D. Miller, Research related to buckling design of nuclear containment. Nucl. Engng Design 79, 217-227 (1984). 15. J. Blachut, Search for optimal torispherical end closures under buckling constraints. Int. J. Mech. Sci. 31, 623-633 (1989). 16. D. Bushnell, BOSOR 5-program for buckling of elastic-plastic complex shells of revolution including large deflections and creep. Compur. Struct 6, 221-239
APPENDIX
The torispherical dome end geometry can be described by using a four-centered ellipse method [2i’]. Joint A and D lay off m eaual to m--m. This is done eranhicallv as indicated in Fig. Al by swinging from A aro&d to A’-with 0 as the -center, where now m from m’ is the required distance DA’, With D as center, an arc from A’ to the diagonal - m will locate F. Beside m by a perpendicular crossing A0 at G and intersecting i%? produce (if necessary) at H. Make m’ equal to m and m’ equal to m. Then G, G’, H and H’ will be centers for four tangent circle arcs forming a curve approximating the shape of an ellipse. The geometric relations from above description are X0 = major axis = a t5a = minor axis = 6
--
rF=AD-DO=a-b m’=i3;i=OD+m
(1976).
17. G. D. Galletly and R. W. Aylward, Buckling under external pressure of cylinders with either torispherical or hemispherical end closures. Press. Vess. Piping 1, 139-154 (1973). 18. G. D. Galletly, J. Blachut and J. Kruzelecki, Plastic buckling of externally pressurised dome ends. In Advances in Marine Structures (Edited by C. S. Smith and J. D. Clarke), pp. 238-261. Elsevier, London (1986). 19. J. Blachut, Optimally shaped torispheres with respect to buckling and their sensitivity to axisymmetric imperfections. Comput. Struct. 29, 975-981 (1988). 20. E. H. Mansfield, An optimum surface of revolution for pressurized shell. Int. J. Mech. Sci. 23, 57-62 (1981). 21, W. Szyszkowski and P. G. Glockner, A rational design of thin-walled pressure vessel ends. Tram ASME
-
DA’=m=a-b
aLi’==H
and the lengths of x
xI_rT_iiij --=_
24. J. Kowalski, Minimum mass design of boiler drums by nonlinear programming. Tram ASME J. Press. Vess. Tech. l&7,-83-87 (1985). 25. A. Gaiewski and M. Zvczkowski. Ootimal structural design* under stability fonstraints. &t Mechanics of Elclstie St~iliry (Edited by H. H. E. Leipholz and G. E. Oraras). Kluwer (1988). 26. ASME Boiler and Pressure Vessel Code, Section VIII. Pressure Vessels Division (1977). 27. T. E. French, C. J. Vierckand and R. J. Foster, Engineering Drawing and Graphic Technology, 3rd Edn. McGraw-Hill (I 987). 28. K. Y. Yuan, C. C. Liang and Y. C. Ma, The estimation of the accuracy and efficiency of the backtrack programming method for discrete-variable structural optimization problems. Comput. Struct. 36, 21 l-222 (1990). 29. J. F. Harvey, Pressure Component Construction: Design and Materials Application. Van Nostrand Reinhold, New York (198Oj.’ 30. A. M. Branat, S. Jendo and S. Owczarek, Criteria and method of Structural Optimization. Martinus Nijhoff (1986). 31. R. H. Gallagher and 0. C. Zienkiewicz, Optimum Structural Design. Rainbow Bridge, San Francisco, CA (1974). 32. J. Farkas, Optimum Design of Metal Structures. John Wiley, New York (1984).
-DF
2 =-
a
I +sin6-cost3
2[
J. Press. Vess. Tech 109, 386-373 (1987). 22. J. Middleton and J. Petmska, Optimal pressure vessel shape design to maximize limit load. Engng. Comput. 3, 287-294 ( 19868). 23. W. Szyszkowski and P, G. Glockner, Design for bucklefree shapes in pressure vessels. Trans ASME J. Press. Vess. Tech. 107, 387-393 (1985).
is
case
1
1
Let r be the radius of toroidal knuckle part and R the radius of spherical part, where both r and R can be derived from the above relationships. They can be found as the following r=XiT =-
=-
=iTJ
-AT cos 0 a
2[
I+sinB-cose co.52I3
1
R=mC-J
A
Fig. Al. A four-centered approximate ellipse.
Optimal form of torispherical dome ends
=gj+r
R =f(a,
851 0) =f(a,
K) =f(a,
b).
646)
For the hemi-spherical dome
a-r ==+fr
8 = 450
rsinfI+a-r _ sin 9
then
=&[1+S].
and
K=l
(A21
r=R=a=b.
From eqns (Al) and (A2), both r and R are functions of a and 6. Let ratio K be the minor axis to major axis K=tan0
=b
(A3)
a
For the 2 : 1 ellipsoid, b =a/2
then K = l/2.
then it gives cp +I r =f(a, 6) =f(a,
(A4) K) =f(a,
b)
(AS)
If the ratio K is known, then 0 can be calculated from eqn (A3), the angle cpcan be calculated from eqn (A4), and r and R can be calculated from eqs (A5) and (A6), respectively. The geometry of torispherical dome end can be sketched by the parameters r, R, 0 and cp.