Parametric study of peak circumferential stress at the saddle support

Int. J. Pres. Ves. & Piping 48 (1991) 183-207

Parametric Study of Peak Circumferential Stress at the Saddle Support Ong Lin Seng Nanyang Technological University, School of Mechanical and Production Engineering, Nanyang Avenue, Singapore 2263 (Received 19 March 1991; accepted 1 April 1991)

ABSTRACT A parametric study is presented to determine the peak circumferential stress at the saddle support of an unstiffened horizontal cylindrical vessel. The proposed parameteric formula consists of various multiplying factors. These factors are introduced to account for the influences of saddle flexibility, location of saddle support, support spacing, saddle support angle, saddle width, and vessel dimensions. Parameteric data are generated from a well proven analysis which is based on a contact stress formulation using a complete shell bending theory. The saddle top plate and wear plate, which extend beyond the saddle horn, are important features in lowering the magnitude of peak stress at the saddle support. Therefore, the effect of the saddle top plate and wear plate is studied in great detail. The validity of the proposed parametric formula is corroborated by two sets of examples. The first set relates to rigid saddle supports and the second set relates to flexible supports. Comparsion of results with theory and experiment show that the established parameteric formula is able to predict the peak stress at the saddle support with high degree of accuracy and confidence.

NOTATION a b c

Distance of saddle support from one end Width of saddle support Distance between two saddle supports

183 Int. J. Pres. Ves. & Piping 0308-0161/91/$03-50 (~) 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland

184 E I ka kc kij kr

ks k~

k~,j L

O F Rij t /'ij tr tw Ol

P O~o

Ong Lin Seng

Young's modulus of elasticity Second m o m e n t of area, sectional modulus A factor to account for the location of support from one end A factor to account for the support spacing The k, value at location (i) with respect to location (j) A stress reduction factor associated with extended plate above the saddle horn A factor to account for the effect of saddle configuration A factor to determine the peak circumferential stress at the saddle support The k~ at location (j) Length of vessel Total support reaction at each support Radius of vessel The rigidity factor at location i Thickness of vessel Thickness of plate segment (ij) Thickness of saddle top plate Thickness of wear plate Angle of saddle plate extension Half saddle embracing angle Specific weight of fluid content in vessel Peak circumferential stress on the outside surface, at the saddle support

1 INTRODUCTION Saddle supports are the most frequently used type of support for horizontal vessels and pipelines. The saddle support usually embraces 120° or 150° of the vessel circumference. As the support is relatively rigid compared to the vessel shell, the support reaction on the vessel is in a way similar to that of a rigid die acting on a semi-infinite elastic solid. A great concentration of forces arises from this loading condition, particularly along the outer edges of the saddle support. The highest stress is the circumferential stress located at the top-most position of the saddle support, commonly referred to as the saddle horn. The peak stress at the saddle horn can be reduced quite remarkably by employing a flexible saddle support. The flexible saddle support refers to one which has a certain degree of flexibility in its saddle horn region. The use of a flexible saddle support will greatly reduce the pinching effect of

Peak circumferential stress at the saddle support

185

the saddle on the vessel wall and it results in a moderation of the stress localisation along the saddle-vessel junctions. Although it is advantageous to use a flexible saddle, there is a limit to the design of flexible saddle support because the saddle support must be strong enough to carry the fully loaded vessel, particularly when the vessel is subjected to hydraulic testing. The design of saddle support therefore lies between strength and flexibility. The British Standard document 2 provides details of saddle support dimensions. It is recommended in this document that the saddle support be fabricated from an 'I' beam of varying depth and stiffened at regular intervals at right angles to the web. The saddle support constructed in this manner inevitably incorporates some degree of saddle flexibility across the width of the saddle support. The effect is the moderation and reduction of discontinuity stresses along the circumferential edges of the support. At the saddle horn, the sudden transition of the saddle support to the shell is moderated by a saddle top plate which extends slightly above the saddle horn (see Fig. 1). The extended saddle plate will greatly reduce the discontinuity stresses at the saddle horn. In BS5500, the thickness of the saddle plate is recommended to be the same as the thickness of the vessel. The angular extension of the saddle plate above the saddle horn is between 9-1 and 11-5 ° for smaller diameter vessels (600-1150mm diameter), and between 6.6 ° and 8-12 ° for larger diameter vessels (1200-3000 mm diameter). The code states that if the saddle plate subtends an angle not less than 6 ° above the saddle horn, the reduced stresses in the shell at the edge of the saddle can be obtained by replacing the thickness of shell in the relevant equation by the combined thickness of shell and saddle plate.

/3 .' half saddle angle

I

. . . .

saaale

~ : extended plate angle

"~'--~:T~I'J~-'~--'\\\\~,~\\\\\\

*Q

*Q

Fig. 1. Basic dimensions of a saddle supported vessel.

saddle

186

Ong Lin Seng

Several analytical attempts have been made over the years to provide solutions for determining the peak stress at the saddle horn. A few well known researchers in this area are Krupka, 3 Tooth, 4'-~ Lakis, 6 etc. Despite the available theoretical analyses, the semi-empirical design method suggested by Zick 7 in 1954 is still presently used by codes and designers. The reasons are first, that no comprehensive parametric study has been performed based on accurate theoretical analysis and second, that stresses in the vessel calculated according to Zick's m e t h o d though giving a stress several times lower than the real one, never cause vessel to fail in the saddle support region. For the latter, Tooth ~ had show that stresses at the saddle support at collapse are much higher than the induced stresses due to the use of a rigid saddle support. Although Krupka 3 had proposed a set of parametric curves to determine the peak stress at the saddle horn, his parametric curves were obtained from a semi-bending shell theory, which ignores the effect of axial bending. Besides, his theoretical model was for a rigid lug pressing on to a cylinder. In other words, he did not model the original problem of a cylindrical vessel supported by two saddle supports. The more accurate analyses were attempted independently by Tooth, 4'5 and Lakis 6 and their students and collaborators. They modelled the original problem mathematically and adopted a complete shell bending theory. In recent years, the rapid advance and progress of the finite element method and the ready availability of finite element packages have prompted some researchers to solve the saddle support problem by the finite element methods. One of these examples is a finite element study to determine the optimum saddle support location, based on a minimum stress criterion. However, to the author's knowledge, in all these finite element studies the saddle supports were modelled as solid (elastic) blocks. The support modelled in this way is very rigid compared to the vessel. Such a model will be representative of a concrete saddle support but not of fabricated saddle supports made from steel sections. A fabricated support designed in accordance with the BS5500 would not be absolutely rigid across the saddle width. The results of these finite element studies are therefore not applicable to saddle built according to BS5500. The most suitable and accurate theoretical approach for the fabricated saddle support is that of Tooth. 4 The theoretical approach of Tooth 4 had been verified by a large amount of experimental test data, 5 obtained from both model vessels and field vessels. To date, no parametric study has been carried out based on the approach of Tooth. 4 The main reason is that there are too many interacting factors to be considered. These factors are: saddle plate

Peak circumferential stress at the saddle support

187

extension, flexibility of saddle, saddle support angle, saddle width, locations of saddle supports, and vessel dimensions. All these factors are important as they significantly affect the level of peak stress in the vessel. Exclusion of any of these factors will render parametric study restrictive and impractical. More in-depth study of the problem must therefore be carried out before a meaningful parametric study can be done. As the extended saddle top plate is an inherent design feature of a saddle support, it has to be studied in a greater detail. Stanley and Mableson ~° carried out a series of photoelastic tests to determine the optimum dimensions of the extended plate. In their investigation, all model vessels were supported by unwelded saddle supports. They noticed that without the extended plate, the maximum circumferential stress would occur at the saddle horn. When the extended plate was incorporated in the saddle support, the stress at the saddle horn reduced greatly. The peak stress was eventually located at the edge of the extended plate as the thickness of the extended plate increased. They concluded from their tests that the o p t i m u m dimensions of the extended plate was 1-4 times the vessel thickness and 10° angular extension above the saddle horn. A related theoretical work in this subject area was carried out by Krupka. 3 Krupka presented the results for a saddle support which had a saddle angle of 100° and a saddle plate overhanging 5 ° . In his case, the saddle plate was welded to the vessel wall. From his graphs presented in Ref. 3, it can be seen that the optimum thickness of the saddle plate is 1.33 times the vessel thickness. This results seems to agree well with that of Stanley and Mableson, 1° in spite of the fact that the saddle support configurations are different in these two cases. Devoid of comprehensive theoretical and experimental studies, the conclusion drawn regarding the o p t i m u m dimensions of the extended plate is subject to further verification. In summary, there is no great difficulty in determining the peak saddle horn stress----either by theoretical or finite element approaches. However, it is a pity that the results of this research work are not made available to the pressure vessel designers. With this consideration, this paper strives to provide a parametric formula to determine the peak circumferential stress at the saddle support. All geometric dimensions of the saddle support and the vessel will be taken into consideration. 2 ANALYSIS F O R F I N D I N G T H E P E A K C I R C U M F E R E N T I A L STRESS A T S A D D L E The m e t h o d of analysis is similar to that of the author's previous paper H which solves the saddle-vessel problem. In this m e t h o d , the

Ong Lin Seng

188

interfacial pressure distribution between the vessel and the saddle is found through a contact stress formulation. Once the contact pressure distribution between the vessel and the saddle is found, the contact pressures are then treated as external loads acting on the cylindrical vessel. The stress and displacement quantities anywhere on the vessel are eventually found by assuming that the cylindrical vessel is simply supported at both ends. For the problem of a cylinder simply supported at both ends, the solution of Ong 12 can be followed. It is to be noted that there is no loss in accuracy in this solution m e t h o d by assuming that the vessel is simply supported at both ends. Because the resultant reaction forces at the simply supported ends are statically zero, the contact forces of the support on the vessel balance the resultant external loads on the vessel. To include the effect of extended plate in the analysis,~the flexibility matrix associated with the extended plate has to be formulated. Here, the solution procedure of Ong tl will not be detailed, but the required flexibility matrix associated with the extended plate will be presented. The assumptions made in the analysis are as follows: (a)

(b)

The extended saddle plate behaves like a curve cantilever beam with its built-in end at the saddle horn (see Fig. 2). The extended plate will be subjected to varying interracial pressures and shears. The saddle support is radially rigid at the saddle horn. This assumption would be invalid for a saddle support which is not centre

--~-

of cylinder i'@\.," , "\ \ \ I \',~

I

L I

\/

',OI

~, • ',

Saddle horn .'1

~. .

"~O, X/ \\

.x \',

/ shellthickness

".

/

\ \

/ / \ \

/

/,~

extended plate I

~

~r

trl

b

section x x

Fill. 2. E x t e n d e d saddle top plate as a built-in cantilever.

-t

Peak circumferential stress at the saddle support

(c)

189

absolutely rigid at its saddle horn. A stress reduction factor will be introduced in the parametric formula to account for the flexibility of the saddle support. The contact pressure distribution between the saddle and the vessel varies along the saddle arc but is uniform across the saddle width. This assumption will be justifiable for a fabricated saddle support which is not rigidly stiffened across the saddle width. Moreover, in the extended plate region, the contact pressure between the extended plate and the vessel is likely to be uniform across the width of the saddle support.

The extended plate can be analysed as a curve cantilever built-in at the saddle horn (see Fig. 2). The included angle of the curve cantilever is a~, which is the amount of angular extension of the extended plate above the saddle horn. The thickness of the extended plate is tr. According to the analysis, 11 it is required to formulate the flexibility matrix of the extended plate. For this purpose, the extended plate is sub-divided into N number of equal divisions. The flexibility matrix (Fs) associated with the extended plate can be formulated and expressed in a matrix form as shown below. Fs

= FFRR I_FFR

FRT] FTrJ(:N×2N)

(1)

Matrix F R R relates to the influence of normal contact pressure on radial displacements; matrix F I T relates the influence of tangential shear on tangential displacements. Matrices FRT and F F R relate the interaction of force and displacement between radial and tangential actions. The elements of the flexibility matrix are presented in the Appendix. The stress reduction associated with the extended plate can be defined by a stress reduction factor kr as follows: kr ~-

The peak stress (with the extended plate) The stress at saddle horn (without the extended plate)

(2)

The kr values are generated for four different saddle support angles (90 °, 120 °, 150 ° and 180°) with varying thickness ratio (tr/t) and four different values of ko (defilaed by equation 4a). These values are tabulated in Table 4. Although the reduction factor kr is derived on the basis that the extended plate has a uniform thickness, it can be modified for use in an extended plate with a step change in thickness. It can also be used for estimating the reduction factor associated with a flexible saddle

190

Ong Lin Seng

support. The procedure in obtaining a modified kr value for these cases is described in the next section.

3 P A R A M E T R I C STUDY The parametric study presented in this paper is for a thin-walled cylindrical vessel of thickness, t, radius, r, length, L, supported by two welded-on saddle supports with a width, b, a total embracing angle, 2fl, and a distance, a, from the nearest end (refer to Fig. 1). The extended plate dimensions are expressed by the thickness ratio (tr/t), and the angular extension (re) above the saddle horn. The total saddle support reaction per saddle support Q. Prior to the parametric study, it is of paramount importance to identify a set of dimensional groups which governs the varying parameters. This is by no means easy when the theoretical formulation of the problem is a complex one. In such a case, intuitive and logical deduction or reference to some simple but similar problem may help to identify these dimensional groups. For the present problem, Krupka 3 has identified some geometrical parameters from this theoretical study of the saddle support problem. The author had tested those dimensional groups based on the present theoretical approach and found them to be suitable. Some of these dimensional groups will be used here, whereas others are expressed in slightly different manner, for generating the parametric data. The peak circumferential stress can be expressed in a similar manner as that of Krupka 3 by the following expression.

O~o=ka.kc.k,.ke~.O~

(3)

where O~o =the peak circumferential stress on the outside surface Q =total reaction force at saddle support = (L~rrZp)/2 k~ =a factor associated with saddle support angle ka ---a factor to account for the location of support from one end kc =a factor to account for the support spacing ks =a factor to account for the effect of wear plate and different saddle support design The factor k~ is obtained by varying the saddle support angle, and keeping all the other parameters constant. Similarly, the factor ka is obtained by varying the extent of the vessel overhang (a) and the factor kc is obtained by varying the support spacing (c), with the other

Peak circumferential stress at the saddle support

191

dimensions remaining unchanged. These three factors (k,, ka, kc) are generated and tabulated in Tables 1-3 respectively. The dimensional parameters for these factors are listed below. T o find k , - - T a b l e (1), calculate

-r4

0- b

(4a)

T o find ka--Table (2), calculate kl

a ~ r

(4b)

T o find k~----Table (3), calculate k2

c ~

(4c)

r

In BSI document BS5276: Part 2, 2 the saddle support proportion for

b/r (saddle width over shell radius ratio) is between 0.3 and 0-5. For a range of vessel dimensions with r/t = 10-300, it means that the value ko will vary between 6 and 60 (i.e. k0 = 6-60). The parametric data generated in Table 4 cover this range of dimensions adequately. The stress reduction factor ks for the combined effect of wear plate, saddle top plate and saddle support can be evaluated by a procedure described below. Define:

ks = kss. ksr

(5)

TABLE 1 The Effect of Saddle Support Angle k,

Saddle angle

k - r ~]~ o- b

2~ (degrees)

5

60 70 80 90 100 110 120 130 140 150 160 170 180

1.610 1-456 1-307 1.155 1.023 0.902 0.789 0-689 0.598 0.516 0-447 0-377 0.317

10

20

2.382 3.465 2 - 1 5 2 3.131 1 . 9 3 3 2.815 1 - 7 1 0 2.491 1 . 5 1 7 2-212 1.341 1.958 1-177 1"721 1-031 1.512 0.899 1-321 0.779 1-148 0-673 1.003 0.576 0.856 0.487 0.727

40

4.834 4.372 3"933 3"483 3-095 2.741 2"413 2.123 1-859 1.619 1-417 1.213 1-033

60

5.805 5-253 4-728 4.190 3.725 3.302 2-910 2.559 2-245 1.956 1.717 1.471 1.256

80

6.483 5-869 5-285 4.685 4.168 3-696 3-259 2.871 2.518 2"198 1-929 1.655 1.414

100

120

6.888 6-238 5.618 4.982 4.433 3-933 3-468 3.056 2.682 2.342 2-057 1.765 1-509

7-137 6-464 5-823 5.164 4.596 4-078 3-600 3.170 2-783 2-430 2.135 1-833 1.568

Ong Lin Seng

192

TABLE 2 The Effect of the Location of Saddle Support (k a) /,,

Saddle angle

k ~ = ~ X/~ r ~ r

2l~ (degrees)

0.1

0.2

0.3

0.4

0.5

0.6

6(} 7(} 80 90 100 110 120 130 14(1 150 160 170 180

0-358 0.335 0.319 0-312 0.307 0-304 0.306 0.312 0-322 0-338 0-359 0.387 (I.424

0.522 0.499 0.485 0.484 0.485 I}.491 0.505 I).524 0.548 (/.579 0.615 0.655 0.704

0.654 0.638 0-629 0.634 0.643 0.656 0.676 0.702 0.731 0.764 0.797 0-831 0.870

0-769 0.760 0-756 0-766 (}.776 0-790 0-809 0-832 0-856 0-882 0-905 0-827 0-953

0.858 0.855 0.853 0.864 0.872 0.881 0"895 0.911 0.927 0.943 0.956 0.967 0.984

0.925 0.924 (t.923 0.933 (I.938 0.942 (}'951 (}.960 0.968 (}.977 0"983 (}.988 1-000

0.7

0.970 0-970 0-969 0-977 0-98(} 0.981 0-984 0-989 0.993 0.997 1.000 I-(X)O . .

0.8

0-990 0.991 0.989 (}-997 0.997 (}.995 0.997 1-000 1.000 1.000 . . -.

0.9

1.O

0-996 0.997 (I.995 1-000 1.000 1-000 1-000 ---. .

1.000 1.000 1.000 ------

.

in which k s s = t h e stress reduction factor associated with saddle support design, ksr = the stress reduction factor associated with the saddle top plate and wear plate which extend above the saddle horn. The factors kss and ksr can be determined as follows: TABLE 3 The Effect of Support Spacing (kc) /7,

Saddle

k = c ~r 2 r

Angle

eo (degrees)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

60 70 80 90 100 110 120 130 140 150 160 170 180

1.128 1.095 1.072 1.058 1.049 1.047 1.053 1.060 1.073 1.091 1.109 1.126 1.139

1.160 1.162 1.152 1.143 1.136 1.134 1.138 1-140 1-143 1-140 1.143 1-137 1.120

1.174 1.188 1.195 1.197 1.195 1-190 1.182 1.172 1.160 1.146 1.130 1.112 1-092

1.157 1-175 1.189 1-199 1.200 1-195 1-182 1.168 1.149 1.127 1.105 1.083 1-065

1.131 1.147 1-162 1-172 1.175 1.172 1.165 1-150 1-131 1-110 1-089 1-062 1-046

1.100 1.114 1.126 1.134 1-137 1-136 1-132 1.120 1.107 1.088 1.068 1.046 1.030

1.064 1.074 1.082 1.088 1.090 1.090 1-086 1-081 1.072 1-062 1.046 1-030 1.020

1.030 1.037 1.042 1.046 1.046 1.046 1.040 1.040 1.038 1.032 1-026 1-021 1-010

1.010 1.013 1-014 1.016 1.016 1.016 1.014 1.014 1.013 1.010 1.010 1.008 1.005

1-000 1.000 1-000 1-000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Peak circumferential stress at the saddle support

193

3.1 Rigidity of the saddle support design, k~ Rigid saddle support: kss = 1.0. Flexible saddle support: ks~ = khr. i_k,h/

(6)

in which referring to Fig. 3, khr = the the kcr = the k,h = the

stress reduction factor at the saddle horn with reference to rigid line k, value at section r, k, value at section h (i.e. at the saddle horn).

To determine khr select a section (normal to the saddle arc) below the saddle horn at which it can be regarded as radially rigid. The line cutting through this section is named here as the rigid line (see Fig. 3). Above the rigid line, the saddle section is assumed to be flexible and below the rigid line the saddle section is regarded as rigid. The factor khr c a n be determined by treating the flexible saddle section as an extended plate, with its thickness equal to the base thickness of the flexible section. Thus, khr can be obtained in the same way as that for k~ from Table 4. Since only part of the saddle flexibility is being taken into consideration, the factor kh~ obtained in this manner will be larger than the actual value. This procedure will ensure a conservative prediction of the peak stress.

J

I

cylinder (t)

"\"..

I I

tI

rigid line

x\ \ \

k,

.. sn ""

" ' - I J//'7>I

-t

ws

k .nr / ///~"//~~ ' / 6 "" -. tsh

®

/\ I

\

tws= t + tw tsh=t+tr+tw

flexible section

"®

.... plato J [ saddle top plate

l

rigid section

F i g . 3. F l e x i b l e s a d d l e s u p p o r t

with top plate and wear plate.

194

Ong Lin Seng

In eqn (6), the factor khr is multiplied by a ratio ( k , r / k ~ , h ) so as to change its reference from the rigid line to the saddle horn. The stress reduction factor k~,~ can also be obtained by a slightly different concept of approach. In this approach, a factor called the 'rigidity factor', R.j, is defined as follows: k~j Rij - kij(/ii ~ oo)

(7)

----kii k,j

where kij = the kr factor at location i, with reference to location j which is assumed to be absolutely rigid; Rij = rigidity factor at location i, due to segment (ij) which is not absolutely rigid; tii = the thickness of segment (ij). The saddle horn rigidity factor can therefore be expressed by: R,r = k,r. k,_____/~ k,h

(8)

which is equal to kss in eqn (6) for the flexible saddle. For rigid saddle support, it can be easily verified that Rii-- 1.0. The rigidity factor is therefore an alternative way for quantifying the rigidity (or flexibility) of a joint or section. This concept of rigidity factor can be used to assess the rigidity of a joint attributed to varying or step change in thickness.

3.2 Saddle top plate and wear plate (k~,) For the purpose of analysis, the wear plate and the saddle top plate are considered wholly as one entity. Thus we have an extended plate with a step change in thickness at the edge of the top saddle plate. Using the concept of rigidity factor and referring to Fig. 3, the stress reduction factor ksr can be easily evaluated as follows: ksr = (kws. g~h). (ksh. g hr)

(\kws

k0h (kh

• k~h • k , ~ /

= khs- ks2 • khr • k~-----z~

"

kor)

" k~h/

(9)

Peak circumferential stress at the saddle support

195

In general, by combining eqn (5) and (9),

= (kws. k h). (R h. R r)

(10)

It has to be mentioned that the above procedure may not be applicable in the case where the peak stress in the vessel lies within the saddle support section or within the region of extended plate. From Table 4, it can be observed that when the peak stress lies within the extended plate region, further extension of the extended plate only alters the peak stress slightly. In other words, there is no great change in stress reduction by further extending the extended plate when the peak stress occurs within the extended plate region. The way to determine the location of the peak stress is to compare the obtained kr value with the minimum value in the same column of Tables 4. If the thickness ratio (tr/t) corresponding to k~ is less than that for the minimum kr, then the peak stress occurs within the extended plate region. Otherwise, the peak stress occurs at the edge of the extended plate. Parametric curves are plotted for representative cases of k~, ka and kc shown in Figs 4, 5 and 6 respectively. These curves can be used for graphical interpolations of parametric data. For the factor k~, a typical graph is plotted in Fig. (7), for the saddle angle of 150 ° and ko = 20. This graph is plotted to show the general behaviour of the kr curve. Generally, tabulated data are preferable as interpolated results can be determined accurately.

4 EXAMPLES Two sets of examples are presented in this section to illustrate the use of the proposed parameteric formula. The first set relates to saddle supports which are rigidly stiffened at the saddle horn and the second set relates to saddle supports which are of flexible constructions. The theoretical and experimental data are quoted from Tooth. 5 For all cases, the vessels are fully filled with water. Comparisons of results between experiment, Zick's BS5500 and present parametric study are shown in Table 5 for the rigid saddles and in Table 6 for the flexible saddles.

4.1 Rigid saddle supports Table 5 tabulates the results for three different vessels. These vessels have rigid saddle supports, meaning that the saddle supports are rigidly stiffened at the saddle horns. The saddle support configurations are

TABLE 4 T h e Stress R e d u c t i o n F a c t o r (kr) ot

tr/t

2fl =

oo

215 = 120° 0.25 0.50 0.75 1.00 1-25 1-50 1.75 2.00 2.50 3.00 3.50 4.00 5.00 2fl = 150° 0-25 0-50 0.75 1.00 1.25 1.50 1.75 2-00 2.50 3.00 3.50 4-00 5.00 oc

15°

ko

ko

ko

10

20

40

60

10

20

40

60

10

20

40

60

0-937 0-854 0.698 0-725 0-749 0.786 0-815 0-834 0.857 0-869 0.876 0-880 0.884 0.887

0.906 0.828 0.692 0.659 0.699 0.741 0.780 0.808 0.841 0.858 0-867 0.873 0-879 0-888

0.889 0.819 0.708 0.578 0.638 0.688 0.734 0.774 0.823 0.849 0.864 0-873 0-882 0.889

0.887 0.814 0-716 0.589 0.587 0-650 0.702 0.740 0.797 0.831 0.850 0.862 0.874 0.889

0.804 0.870 0.789 0-662 0-524 0.529 0-591 0.638 0.698 0.730 0.749 0.760 0.771 0.784

0.951 0.851 0.773 0-667 0.545 0.433 0.524 0.570 0.652 0.699 0.727 0.744 0.763 0.786

0-845 0.928 0.777 0.690 0.585 0.480 0.432 0.474 0.582 0.650 0-693 0.719 0.748 0.787

0.868 0.831 0.796 0.723 0.623 0.523 0.432 0.412 0.53l 0.613 0-666 0-701 0-734 0.788

0.896 0.861 0.746 0.638 0.530 0.425 0.372 0.440 0.534 0.590 0.623 0.643 0.666 0.688

0.955 0.825 0.723 0.616 0.527 0.438 0.357 0.340 0.459 0.536 0.584 0.614 0.648 0.691

0.890 0.795 0.712 0.611 0.528 0.456 0.387 0.324 (}.357 0.455 0-523 0.569 0.622 0.693

0-840 0-785 0.709 0.620 0.529 0.486 0.407 0.350 0.288 0.394 0.472 0.529 0.597 0.694

0.929 0.851 0-695 0.707 0-734 0-766 0-796 0-818 0.843 0-856 0.863 0.867 0.871 0.876

0.886 0-833 0-691 0.642 0.686 0.725 0.766 0.795 0.831 0-849 0-860 0.866 0.872 0.878

0.851 0.822 0.702 0.559 0.619 0.670 0.714 0.755 0.806 0.834 0.849 0.859 0.869 0.880

0.863 0.817 0.712 0.587 0.569 0.633 0.684 0.723 0.782 0.817 0.838 0.851 0.869 0.879

0.834 0.852 0.788 0.664 0.527 0.497 0.562 0.611 0.674 0.707 0.727 0.738 0.750 0.764

0.804 0.816 0.778 0.674 0.551 0.437 0.484 0.546 0.632 0.681 0.710 0.727 0.746 0.767

0.773 0.778 0-772 0.693 0.587 0-484 0.388 0-454 0.565 0.635 0-675 0.705 0.734 0.770

0.758 0.755 0.776 0.714 0.620 0.521 0.432 0.390 0.511 0.593 0.648 0-683 0.722 0-771

0.931 0.854 0.723 0.665 0.556 0.450 0.358 0.391 0.492 0.552 0.588 0.611 0.634 0.662

(}.903 0.821 0-710 0.644 0.555 0.465 0.381 0.309 0.419 0.500 0.552 0.585 0.622 0.667

0.881 0.799 0.703 0.625 0.559 0.484 0.411 0.346 0-317 0.418 0.490 0.539 0.596 0.671

0.885 0.790 0.706 0.614 0.564 0.500 0.435 0.374 0.273 0.356 0.438 0.498 0.571 0.672

0-925 0-841 0.689 0-682 0-712 0-750 0.783 0.805 0-831 0.844 0.852 0.856 0.859 0-864

0.900 0.795 0.628 0.616 0.671 0.716 0-756 0-784 0.818 0-834 0-843 0-848 0-867 0-874

0.872 0.798 0-692 0.554 0.599 0.652 0.703 0.744 0.797 0-825 0-841 0-850 0-860 0.875

0-882 0.807 0.704 0.583 0.553 0.619 0.671 0.714 0.779 0-815 0-835 0-848 0-861 0.878

0.826 0.851 0.786 0.663 0.527 0.452 0.520 0.572 0-639 0.677 0-699 0.711 0.725 0.739

(}.786 0.808 0.774 0.669 0.548 0.436 0.442 0.507 0.597 0.649 0.680 0.699 0.719 0.746

0.752 0.769 0.769 0.686 0.581 0.457 0.389 0.415 0.528 0-600 0.644 0.675 0.707 0.749

0.735 0.745 0.777 (I.701 0.614 (}.517 0.429 0.354 0.477 0.562 0-619 0.656 0.698 0.752

0.892 0.852 0.742 0.625 0.522 0.422 0.335 0.336 0.442 0.506 0.544 0.568 0.594 0.625

0.847 0.824 0-724 0.603 0.518 0.433 0.355 0.289 0.370 (}-456 0.510 0.545 0.584 0.633

0.826 0.801 0-710 0.605 0.512 0.444 0.378 0.319 0.270 0-373 0.447 0.497 0.556 0.638

0.854 0.801 0.713 0.618 0.524 0.457 0.398 0.344 0.252 0-315 0.399 0.460 0.535 0.642

0.861 0.810 0.667 0-644 0-677 0.712 0.746 0.770 0.799 0.813 0.821 0-826 0.831 0.836

0-808 0-782 0-660 0-583 0-631 0-676 0.720 0.752 0.790 0-810 0.821 0.828 0.834 0.842

0.802 0.760 0.665 0-534 0-565 0.619 0.671 0.714 0.768 0.797 0-813 0-823 0-835 0.845

0.816 0.781 0-671 0-560 0-517 0.582 0.635 0.681 0.747 0.784 0-805 0-818 0-831 0.848

0.811 0.821 0-764 0-648 0-517 0.400 0.461 0.515 0.586 0.625 0.648 0.661 0.675 0-692

0.781 0.785 0.752 0.655 0.538 0.428 0-390 0-457 0-549 0-604 0-636 0-655 0-676 0-701

0.744 0.744 0.742 0.667 0.567 0.468 0.381 (}.370 0.485 0.559 0.605 0.635 0.667 0.707

0.724 0.718 0.742 0.684 0.594 0.502 0.417 0-344 0-434 0.519 0.576 0-614 0.656 0.711

0.924 0.788 0.735 0.652 0.544 0.439 0.349 0.278 0.367 0.434 0.476 0-501 0.529 0.560

0.900 0.741 0.692 0.627 0.538 0-448 0.367 0.299 0.298 0.385 0-441 0-478 0-519 0.570

0.887 0.693 0.674 0.602 0.537 0.462 0.391 0.329 0.231 0.309 0.383 0.434 0.496 0.576

0.885 0.683 0.664 0.615 0.541 0.475 0.411 0.352 0.257 0.250 0.332 0.393 0.469 0-579

18o °

0.25 0.50 0.75 bOO 1.25 1.50 1-75 2.00 2.50 3.00 3-50 4.00 5.00 oo

10°

90 °

0-25 0'50 0-75 1-00 1"25 1-50 1-75 2-00 2"50 3.00 3"50 4.00 5.00

2fl =

5°

Peak circumferential stress at the saddle support

197

8

_._.. 213=60 °

I

.,. ,,..--" f

J

~

i.,-" ~

/

~ 2

f

I

~

.~..

213=90 o

.....--. -

~

~

2B = 120 °

~

213= 150 °

/-. 213= 180 °

0 0

20

40

60

80

100

120

m

k

r

Fig. 4. T h e effect o f s a d d l e s u p p o r t a n g l e .

ka 1.1

1 0°

,~

~

....

0.9

/

0.8

-

"

0.7 0.6

/ Z/ . . ~../7-"

/,, ~';

0.5 0.4 0.3

~2~

~ , 2 B :: 6C

:9o

K" 1

fl = 1 ~-0"

0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 5. T h e effect o f s u p p o r t l o c a t i o n f r o m e n d .

0.9

1

Ong Lin Seng

198 kc

1.2 1,18

,,"" 1.16

.213 =

.....

12~)°

ss S°

1.14 1.12

::

1.1

", N,%,

=~0

-.

",

1.08 1.06 1.04

213 = 18C

" " ' " "'.,..~...

1,02 1

0.98 0.2

0

0.4

0.6

0.8

I

1.2

1.4

1.8

1.6

~-

2

k=

r

Fig. 6. The effect of support spacing (kc).

k (stress r e d u c t i o n f a c t o r ) r

1.1

l

0.9

213 = 150 o

~j ~-..

T h e p e a k stress o c c u r s at the h o r n

1% = 2 0 0(=5

°

0.8

............ 0.7

........-

~,=10

............ ~ = I 5

0.6 0.5

':~--'"

...'" ..........

T h e p e a k s t r e s s o c c u r s at the e d g e

0.4 0.3 0.2

T h e p e a k stress o c c u r s at 2 a b o v e h o r n

0.1 0

,

I 1

,

I 2

,

I 3

o

.....

,

I 4

,

I

,

5

Fig. 7. The effect of extended plate.

I

6

,

I

i

tr/t

o

TABLE 5 C o m p a r i s o n of Results b e t w e e n T h e o r y , E x p e r i m e n t a n d P a r a m e t r i c S t u d y - - R i g i d l y Stiffened Saddle No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Items

Vessel I

Radius (r) (mm) Thickness (t) (mm) Length (L) (mm) Location of (a) Saddle angle (2fl °) Saddle width (b) (mm) Extended angle (o¢) degrees) Thickness ratio (tr/t) Factor k0 Factor kl Factor k 2 Factor k~ Factor ka Factor k c Factor k s Total force at support (Q) (kN) Stiffened saddle (experiment 5) (MPa) Stiffened saddle (theory 5) (MPa) BS55005 (MPa) Stiffened saddle (parametric) (MPa)

1 830 26.6 54 860 6 805 154 762 4 1.0

19-92 0.448 2-718 1.090 0.920 1.000 0.693 2 831 -310.3 -355.0 -194.7 -335-23

Vessel 2

455 3.3 7 320 1 410 150 102 --

Vessel 3

455 4-67 7 320 1 410 150 102 --

--

--

52.38 0.264 0.842 1.828 0.697 1.123 1.0 27 -290.5 -301.8 -120.7 -302.29

44.03 0.314 1-002 1-687 0.780 1-110 1-0 27 -174-7 -180-5 -63-1 -183.19

TABLE 6 C o m p a r i s o n of Results b e t w e e n T h e o r y , E x p e r i m e n t a n d P a r a m e t r i c S t u d y - - F l e x i b l e Saddle No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Items

Radius (r) (mm) Thickness (t) (mm) Length (L) (mm) Location of support (a) Saddle angle (2fl°) Saddle width (b) (mm) Factor k0 Factor k~ Factor k 2 Factor k~ Factor k a Factor k c Factor ks Total force at support (Q) (kN) Flexible saddle (experiment 5) (MPa) Flexible saddle (theory 5) (MPa) BS55005 (MPa) Flexible saddle (parametric) (MPa)

Vessel 1

1 830 26-6 54 860 6 805 150 762 19.92 0.448 2-718 1-148 0.912 1.000 0.494 2 831 -186.2 -210-0 -194.7 -249.49

a These are estimated values based on assumed saddle dimensions.

Vessel 2

Vessel 3

455 3.3 7 320 1 410 150 102 52-38 0.264 0.842 1-828 0.697 1-123 0-566" 27 -120-0 -155.4 -120.7 -171-0

455 4.67 7 320 1 410 150 102 44-03 0-314 1.002 1.687 0-780 1.110 0-502 a 27 -70-4 -61.4 -63.1 -91-96

200

Ong Lin Seng [Ref5]

Vessel 1 ~

extended plate (26 6 mm)

70~

Rigid saddles

Vessel 2 and Vessel 3

~

\ web stiffeners

bl

L3° I wcafplatc ~ .~ . ~ (26.6mm) 6 / ///

Flexible saddles

9,ore

**12.7mm

al r- ~

stiffened

©-_

l l ~

///

-

saddle

topplate

I I

<,9,mml

~/" ~/"

\

+d,,n° C ]

~12.7

mm

dl

Fig. 8. Saddle support configurations.

shown in Fig. 8(a) for reference. In the following, only the detailed calculations for Vessel 1 are explained and shown in steps. 2 r ~ _ 1830,18/1830 = 19.2 k°=v 762 ¥26.6 From Table 1, with saddle angle = 154 °, k, = 1.090. (a)

(b)

kl =

a ~/~_ 6805, 2~ = 0-448 r 1830 ~/1830 From Table 2, k a = 0.920. (c)

kz = Cr ~/~ = 411830250, =/-2-6.6 1~8 3t 0 2 - 7 2

From Table" 3, since k2 > 2, therefore kc = 1. (d) With 4 ° extended plate, i.e. c~ = 4 ° tr/t = 1, and k0 = 19-92, from Table 4, ksr = kr = 0.693 (obtained by interpolation). Since the saddle is rigid at the saddle horn, ks = 0.693. (e) The support reaction, Q Q = ½~rrZLp = 2831 kN

(f)

The peak circumferential stress at the saddle support can be evaulated using eqn (3) as follows: o,0 = (0.920)(1-0)(0-693)(1-090) 2 831 0 0 0 , / 2 6 . 6 = 335.23 MPa 26.62 ~t 1830

The above result agrees very well with the theoretical result? The results of the other two vessels also agree very well with the theory (see Table 5).

Peak circumferential stress at the saddle support

201

4.2 Flexible (or non-rigid) saddle supports Table 6 tabulates the results for the same three vessels supported on flexible saddle supports• The saddle support designs of these three vessels are shown in Fig. 8 ( c ) - ( d ) .

4.2.1 Vessel i (see Fig. 8(a)) Vessel 1 has a wear plate welded to the vessel in additional to the saddle top plate. The saddle top plate is 3 ° above the saddle horn and the wear plate is 3 ° above the saddle top plate. Since the saddle support is not absolutely rigid at the saddle horn, the rigidity factor Rh~ has to be determined. Referring to the drawing of this saddle support, the rigid line is chosen at 5 ° below the saddle horn. The equivalent thickness at the rigid line section is about two times the basic shell thickness. The equivalent thickness of the fabricated section can be d e t e r m i n e d by equating the second area of m o m e n t of the rigid line section to that of a rectangle with width equal to the saddle width (b). For a~ = 5 °, tr/t = 2, k0 = 19"92, from Table 4, khr = 0-784. Using the notations adopted in Fig. 3 for the present problem, eqn (8) gives, Rhr =

kh~ • k,~ k~h

(1321

= 0.784. \ 1 . 1 4 8 / = 0.902 where ksh = kws = k,h = k,~ =

0"854 0.770 1-148 1.061

(for tr/t = (for tr/t = (for 213 = (for 2/3 =

1"72 and a" = 3 °) 1-00 and a~ = 3 °) 150 °) 156 °)

Rsh = ksh k , h _ (1-148~ • k,s - 0"854. \ 1 " - - ~ / = 0"924 Using eqn (9), ksr = kws. ksh. Rsn • Rhr

= (0"854)(0-770)(0-924)(0-902) = 0-548 The stress reduction factor associated with the saddle support is therefore: ks = kss. ksr = (0.902). (0.548) = 0.494

Ong Lin Seng

202

The peak circumferential stress evaulated from eqn (3) as follows:

at the saddle s u p p o r t can be f-----

%o = (0.912)(1-0)(0.494)(1.148) 2 831 000 ~ / 2 6 . 6 = 249-49 M P a 26.62 ~/1830 The result is slightly higher than the theoretical values as in this example we have not considered the total flexibility of the saddle support.

4.2.2 Vessels 2 and 3 The flexible saddle supports used for Vessels 2 and 3 were similar in design, as shown in Fig. 8(d). It is u n f o r t u n a t e that the detailed dimensions of the saddle supports are not available in Ref. 5. T o assess the stress reduction factor due to these saddle supports, estimations of the u n k n o w n dimensions are made. T h e purpose is to show the numerical p r o c e d u r e used for assessing the flexibility of the flexible saddle. For this support configuration, the rigid line is chosen at 15 ° below the saddle horn. It is n o t e d that for this saddle configuration, the section above the rigid line has an almost uniform thickness. For Vessel 2, the base thickness at the rigid line section is estimated to be about three times the basic shell thickness. With tr = 15 °, tr/t = 3, k0 = 52.38, Table 4 gives, khr = 0.380 (interpolated value); therefore, k~r = (0.380) (2.721) R,~ = khr. k , , .\ ~ j = 0-566 Vessel 3 has a thicker shell thickness than Vessel 2, since both use the support of similar design, the thickness ratio for this vessel at the rigid line section is therefore, tilt = (3)(3.3/4-67) = 2-12. With a~ = 15 °, tr/t = 2.12, k 0 = 4 4 " 0 3 , Table 4 gives khr = 0-341 (interpolated value); therefore, Rhr =

k~r = (0.341) (2"483] = 0.502 k,~. k,h " \1.687/

It is seen that the peak circumferential stress in the vessel, at the saddle support location, can be reduced by about 50% through the use of flexible saddle supports. The results of Vessels 2 and 3 are tabulated in Table 6. It can be seen that the parametric results, based on the estimated saddle dimensions, agree quite well with the theoretical results. A m o r e elaborate m e t h o d of predicting the saddle flexibility is by sectioning the saddle into few m o r e parts and making use of the concept of rigidity factor to find the

Peak circumferential stress at the saddle support

203

stress reduction factor. This can be done if the detailed dimensions of the saddle supports are known.

5 DISCUSSION

5.1 The effect of the extended plate (a)

It can be concluded from Table 4 that generally, the peak stress at the saddle horn can be reduced by 25 to 40% with the use of saddle top plate which extends not less than 5 ° above the saddle horn and has the same thickness as that of vessel. (b) From the analysis of the extended plate, it is discovered that for a thin extended plate (tr/t < 0.25), the peak stress remains at the saddle horn. Subsequent thickening of the plate relocates the peak stress to a new location, about 2 ° above the saddle horn but still within the plate region. Continuous thickening of the plate eventually brings the peak stress to the edge of the plate. It is interesting to note that the peak stress does not progress in a continuous manner from the saddle horn to the edge of plate, but it occurs at the three positions mentioned in the above. These changes of the peak stress positions can also be realised from Fig. 4, which shows the behaviour of kr with respect to the thickness ratio (tJt). It can be observed from Fig. 4 that there exist an inflexion point in the lower range of (tr/t). This inflexion point signifies the change of the peak stress location from the saddle horn to the position 2° above it. The minimum (lowest value) of the curve signifies the change of the peak stress from within the extended plate to its edge. The minimum point of the curve also corresponds to the optimum dimensions of the extended plate, basing on a minimum stress criterion and no restriction on the plate dimensions. (c) The optimum dimensions of the extended plate is mainly governed by the angular extension of the extended plate and the dimensional parameter ko. Generally, with 5 ° plate extension, the optimum thickness ratio lies within 0.75 and 1.5; with 10° plate extension, the optimum thickness ratio lies within 1.25 and 2.5; and with 15° plate extension, the optimum thickness ratio lies within 1.75 and 3-0. The optimum thickness ratio tends to be lower with a lower value of ko, which corresponds to a saddle support with a wider extended plate. The optimum dimensions

Ong Lin Seng

204

(d)

(e)

of the extended plate proposed by Stanley & Mableson '° cannot be generalised as they would be only valid for a limited range of dimensions. When the peak stress occurs within the extended plate region, extending the arc length of the extended plate would only change the stress reduction factor slightly. For such a case, the best way to reduce the magnitude of the peak stress further is to thicken the extended plate. From Table 4, it can be observed that the kr value does not vary greatly with respect to the change of saddle support angle. Thus for intermediate saddle support angles, the respective kr may be obtained from the relevant table which is closer to the intermediate saddle support angle, without the need for interpolation between tables of results.

5.2 The behaviour of various factors

5.2.1 The factor k, (Fig. 4) The factor k, is higher for smaller support embracing angle and higher value of k0. It is because a smaller support embracing angle and a higher k0 which corresponds to a saddle support with narrower width, will intensify the effect of load localisation at the saddle support.

5.2.2 The factor ka (Fig. 5) The effect of the distance of support from one end (or overhang) is an important factor in controlling the level of peak stress at the saddle support. The study shows that the location of support relative to the end, changes the level of peak stress quite significantly. As the vessel is subjected to fluid loading, the overhang of vessel from the support would behave like a loaded cantilever. In this case, the end stiffening effect is dominant only when the support is placed near to the end, otherwise the cantilever action prevails. Figure 5 seems to imply that by placing the support nearer to the end of vessel would benefit from a lower peak stress. However, placing the support too near to the end could create high axial stresses at the centre of the vessel. The vessel may fail at its mid-section by buckling due to these high axial stresses. Therefore, one must also check the axial stress at the centre of vessel when deciding the location of the saddle supports. The bending stress at the vessel mid-section can be calculated quite quickly and accurately by the simple beam theory, as presented by G.3.3:BS5500.

Peak circumferential stress at the saddle support

205

5.2.3 The factor kc (Fig. 6) It can be seen from Fig. 6 that the support spacing does not change the peak stress greatly compared to the effect of vessel overhang from the support. From Table 3, it shows that ignoring this effect will only result mostly, in a 20% difference in the peak stress value. The effect of support spacing can be disregarded when the dimensional parameter k2 is greater than 2. 5.3 Flexible saddle and concrete saddle

The present parametric study is not applicable to concrete saddle supports which are considered rigid across their saddle widths. For the concrete or solid saddle supports, very high localised stresses are induced along the edges of the saddle arcs. For such a problem, the analysis must take into account the compatibility between the saddle and the vessel in the axial direction. That is, the contact pressure between the saddle support and the vessel can no more be assumed uniform across the saddle width. The best location of the concrete saddle support would be at one which would result in minimum restraint between the saddle support and the vessel. If the concrete supports are placed too near to the end, then the peak stress would occur at the inner edge (towards the vessel centre) of the support. If the concrete supports are too close to each other, then the peak stress would occur at the outer edge of the support. Most of the field elected vessels are rarely supported by concrete saddle supports because they are not recommended by design codes. 6

CONCLUSION

A parametric formula accompanied by tabulated data and graphs are presented in this paper to determine the peak circumferential stress in the cylindrical vessel at the saddle support location. Various factors which govern the level of peak stress at the saddle support are considered and included in the parametric formula by means of factors. Comparison of results with well established theory and published experimental results show that the present parametric study is accurate and can be used with confidence. REFERENCES 1. British Standard Institution, Unfired fusion welded pressure vessels, BS5500: 1986.

206

Ong Lin Seng

2. British Standard Institution, BS 5276: Part 2, Specification for saddle supports for horizontal cylindrical pressure vessels, 1983. 3. Krupka, V., Analysis for lug or saddle-supported cylindrical pressure vessels. First International Conference on Pressure Vessel Technology, Delft, 1969, Part 1,491-500. 4. Duthie, G. C., White, G. C. & Tooth, A. S., An analysis for cylindrical vessels under local loading--application to saddle supported vessel problem. Journal of Strain Analysis, 17 (3) (1982) 157-68. 5. Tooth, A. S., Duthie, G. C., White, G. C. & Carmichael, J., Stresses in horizontal storage vessels--a comparison of theory and experiment. Journal of Strain Analysis, 17, (3) (1982) 169-76. 6. Lakis, A. A. & Dore, R., General method for analysing contact stresses on cylindrical vessels. International Journal of Solid Structures, 14 (1978) 499-516. 7. Zick, L. P., Stresses in large horizontal cylindrical vessels on two saddle supports. The Welding Research Supplement, IX (1951) 435-44. 8. Tooth, A. S. & Jones, N., Plastic collapse loads of cylindrical pressure vessels supported by rigid saddles. Journal of Strain Analysis, 17 (3) (1982) 187-98. 9. Widera, G. E. O., Sang, Z. F. & Natarajan, R., On the design of horizontal pressure vessels. Journal of Pressure Vessel Technology, 110 (1988). 10. Stanley, P. & Mableson, A. R., An exploratory two-dimensional study of the stress in saddle-supported cylindrical vessels. Proceedings Fourth International Conference on Pressure Vessel Technology, London, 1980, Part 2, 135-48. 11. Ong, L. S., Analysis of twin-saddle-supported vessel subjected to nonsymmetric loadings. Int. J. Pres. Ves. & Piping, 35 (1988) 423-37. 12. Ong, L. S., A cqlnputer program for cylindrical shell analysis. Int. J. Pres. Ves. & Piping, 30 (1987) 131-49.

APPENDIX Referring to Fig. 2, A1 = 0 i - 0j

C1 = cos (0i - 0j)

S 1 = sin ( 0 i - 0j)

A2 = a~ - 0i

C2

$2 = s i n (c~ - 0,)

A 3 = t:t" -

C 3 = c o s (~" -

0i

rs=r + ~ 2

= COS ( ~ ' - - 0 i ) 0j)

$3 = s i n (0: - 0j)

I = b t3 12

E.--7/FRRij = ½(C1A3 - C2S3)

(for i < j)

rs

= ½(CIA2 - C3S2)

(for i > j)

(AO

Peak circumferential stress at the saddle support EI ~, r

t~s3

ij = A 3 - S, - $2 - S3 + ½ ( C , A 3 + C2S3)

(for i < j)

C3S2)

(for i > j)

= A 2 + $1 - Sz - S~ + ~ ( C ~ A 2 +

-~f FTRij = 1 - C3 - ½(S~A3 + $2S3) Is

= C1 - C3 - ½($1A2 +

$2S3)

207

(A2)

(for i < j) (for i > j)

(A3)

FRT = F r R T In the above matrices FRRij, FrFij and FI'Rij relate the displacement at node i due to a unit load at node j. F R R and F r r are both symmetric matrices.