Zicks Formula

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Stresses in Large Horizontal CylindricalPressure Vessels on Two Saddle Supports

Original paper published in September 1951 "THE WELDING JOURNAL RESEARCH SUPPLEMENT." This papercontains revisions and additions to the original paper based upon questions raised as to intent and coverage.

Approximate stresses that exist in cylindrical vessels supported on two saddles at various conditions and design of stiffening for vessels which require it

by L. P. Zick

INTRODUCTION

The design of horizontal cylindrical vessels with dishedheads to resist internal pressure is covered by existing

codes. However, the method of support is left prettymuch up to the designer. In general the cylindrical shell ismade a uniform thickness which is determined by themaximum circumferential stress due to the internalpressure. Since the longitudinal stress is only one-half of this circumferential stress, these vessels have available abeam strength which makes the two-saddle supportsystem ideal for a wide range of proportions. However,certain limitations are necessary to make designsconsistent with the intent of the code.

The purpose of this paper is to indicate the approximatestresses that exist in cylindrical vessels supported on twosaddles at various locations. Knowing these stresses, it is

possible to determine which vessels may be designed forinternal pressure alone, and to design structurally adequateand economical stiffening for the vessels which require it.Formulas are developed to cover various conditions, and achart is given which covers support designs for pressurevessels made of mild steel for storage of liquid weighing42 lb. per cu. ft.

HISTORY

In a paper 1 published in 1933 Herman Schorer pointedout that a length of cylindrical shell supported bytangential end shears varying, proportionately to the sineof the central angle measured from the top of the vessel

can support its own metal weight and the full containedliquid weight without circumferential bending moments inthe shell. To complete this analysis, rings around the entirecircumference are re uired at the su ortin oints to

L. P. Zick is Chief Engineer with the Chicago Bridge & Iron Co., Oak Brook, Ill.

Reprinted from Welding Journal Research Supplement, 1971

transfer these shears to the foundation without distortingthe cylindrical shell. Discussions of Schorer's paper by H.C. Boardman and others gave approximate solutions for thehalf full condition. When a ring of uniform cross section issupported on two vertical posts, the full condition governsthe design of the ring if the central angle between the postintersections with the ring is less than 126, and thehalf-full condition governs if this angle is more than 126.

However, the full condition governs the design of ringssupported directly in or adjacent to saddles.

Mr. Boardman's discussion also pointed out that theheads may substitute for the rings provided the supportsare near the heads. His unpublished paper has been usedsuccessfully since 1941 for vessels supported on saddlesnear the heads. His method of analysis covering supportsnear the heads is included in this paper in a slightlymodified form.

Discussions of Mr. Schorer's paper also gave successfuland semi-successful examples of unstiffened cylindricalshells supported on saddles, but an analysis is lacking. The

semi-successful examples indicated that the shells hadactually slumped down over the horns of the saddles whilebeing filled with liquid, but had rounded up again wheninternal pressure was applied.

Testing done by others, 2,3 gave very useful results in theranges of their respective tests, but the investigatorsconcluded that analysis was highly indeterminate. In recentyears the author has participated in strain gage surveys of several large vessels. 4 A typical test setup is shown inFig. 1.

In this paper an attempt has been made to produce anapproximate analysis involving certain empiricalassumptions which make the theoretical analysis closelyapproximate the test results.

SELECTION OF SUPPORTS

When a cylindrical vessel acts as its own carrying beamacross two symmetrically placed saddle supports, one-half of the total load will be carried by each support. Thiswould be true even if one support should

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The most economical location and type of supportgenerally depend upon the strength of the vessel to besupported and the cost of the supports, or of the supportsand additional stiffening if required. In a few cases theadvantage of placing fittings and piping in the bottom of the vessel beyond the saddle will govern the location of the

The pressure-vessel codes limit the contact angle of eachsaddle to a minimum of 120 except for very small vessels.In certain cases a larger contact angle should be used.Generally the saddle width is not a controlling factor; so anominal width of 12 in. for steel or 15 in. for concrete maybe used. This width should be increased for extremelyheavy vessels, and in certain cases it may be desirable toreduce this width for small vessels.

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at or near the saddles. Vessels must also be rigid enough tosupport normal external loads such as wind.

Where liquids of different weights are to be stored orwhere different materials are to be used, a rough designmay be obtained from the chart and this design should be

checked by the applicable formulas outlined in thefollowing sections. Table I outlines the coefficients to beused with the applicable formulas for various support typesand locations. The notation used is listed at the end of thepaper under the heading Nomenclature.

MAXlMUM LONGlTUDlNAL STRESS

The cylindrical shell acts as a beam over the twosupports to resist by bending the uniform load of the vesseland its contents. The equivalent length of the vessel (seeFigs. 2 and 3) equals (L + (4H/3)), closely, and the totalweight of the vessel and its contents equals 2Q. However,

it can be shown that the liquid weight in a hemisphericalhead adds only a shear load at its junction with thecylinder. This can be approximated for heads where H Rby representing the pressure on the head and thelongitudinal stress as a clockwise couple on the headshown at the left of Fig. 3. Therefore the vessel may betaken as a beam loaded as shown in Fig. 3; the momentdiagram determined by statics is also shown. Maximummoments occur at the mid-span and over the supports.

settle more than the other. This would also be true if adifferential in temperature or if the axial restraint of thesupports should cause the vessel acting as a beam to bow upor down at the center. This fact alone gives the two-supportsystem preference over a multiple-support system.

Thin-wall vessels of large diameter are best supported nearthe heads provided they can support their own weight andcontents between supports and provided the heads are stiff enough to transfer the load to the saddles. Thick-wall vesselstoo long to act as simple beams are best supported where themaximum longitudinal bending stress in the shell at thesaddles is nearly equal to the maximum longitudinal bendingstress in the shell at the saddles is nearly equal to themaximum longitudinal bending stress at mid-span, providedthe shell is stiff enough to resist this bending and to transferthe load to the saddles. Where the stiffness required is notavailable in the shell alone, ring stiffeners must be added

Figure 2 indicates the most economical locations andtypes of supports for large steel horizontal pressure vesselson two supports. A liquid weight of 42 lb. per cu. ft. wasused because it is representative of the volatile liquidsusually associated with pressure vessels.


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Tests have shown that except near the saddles a cylindri-cal shell just full of liquid has practically no circumferen-tial bending moments and therefore behaves as a beamwith a section modulus I/c = r2t.

However, in the region above each saddle circumferential

bending moments are introduced allowing the unstiffenedupper portion of the shell to deflect, thus making itineffective as a beam. This reduces the effective crosssection acting as a beam just as though the shell were splitalong a horizontal line at a level above the

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saddle. (See Fig. 4 (a).) If this effective arc is representedby 2 ( in radians) it can be shown that the sectionmodulus becomes:

Strain gage studies indicate that this effective arc isapproximately equal to the contact angle plus one-sixth of the unstiffened shell as indicated in Section A-A of Fig. 4.Of course, if the shell is stiffened by a head or completering stiffener near the saddle the effective arc, 2 , equals

the entire cross section, and I/c = r2

t.

K1 is a constant for a given set of conditions, butactually varies with the ratios A/L and H/L R/L fordifferent saddle angles. For convenience, K 1 is plotted inFig. 5 against A/L for various types of saddle supports,assuming conservative values of H = 0 when the mid-spangoverns and H = R when the shell section at the saddlegoverns. A maximum value of R/L = 0.09 was assumedbecause other factors govern the design for larger values of this ratio. As in a beam the mid-span governs for thesmaller values of A/L and the shell section at the saddlegoverns for the larger values of A/L; however, the point

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where the ending stress in the shell is equal at mid-spanand at the saddle varies with the saddle angle because of the reduced effective cross section. Fig. 5A in App. Bgives acceptable values at K 1.

This maximum bending stress, S 1, may be either tensionor compression. The tension stress when combined withthe axial stress due to internal pressure should not exceedthe allowable tension stress of the material times the effi-ciency of the girth joints. The compression stress shouldnot exceed one half of the compression yield point of thematerial or the value given by

which is based upon the accepted formula for buckling of short steel cylindrical columns*. The compression

*See also per UG-23(b) ASME Code Section VlIl Div. 1.

Since most vessels are of uniform shell thickness, thedesign formula involves only the maximum value of thelongitudinal bending stress. Dividing the maximummoment by the section modulus gives the maximum axialstress in lb. per sq. in. in the shell due to bending as abeam, or

(b) MOMENT DIAGRAM IN FT.-LBS


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stress is not a factor in a steel vessel where t/r 0.005 andthe vessel is designed to be fully stressed under internalpressure.

TANGENTIAL SHEAR STRESS

Figure 4 (d) shows the total shear diagram for vesselssupported in saddles away from the heads.

Where the shell is held round, the tangential shearingstresses vary directly with the sine of the central angle ,as shown in Section B-B of Fig. 4, and the maximumoccurs at the e uator.

However, if the shell is free to deform above the saddle,the tangential shearing stresses act on a reduced effectivecross section and the maximum occurs at the horn of thesaddle. This is approximated by assuming the shearscontinue to vary as the sin but only act on twice the arcgiven by ( /2 + /20) or ( - ) as shown in Section A-A of Fig. 4. The summation of the vertical components of theseassumed shears must equal the maximum total shear.

The maximum tangential shear stress will occur on the

center side of the saddle provided the saddle is beyond theinfluence of the head but not past the quarter point of thevessel. Then with saddles away from the heads the maximumshear stress in lb. per sq. in. is given by

Values of K 2 listed in Table I for various types of supportsare obtained from the expressions given for the maximumshears in Fig. 4, and the appendix.

Figure 4 (f) indicates the total shear diagram for vesselssupported on saddles near the heads. In this case the headstiffens the shell in the region of the saddle. This causes

most of the tangential shearing stress to be carried acrossthe saddle to the head, and then the load is transferred back to the head side of the saddle by tangential shearingstresses applied to an arc slightly larger than the contactangle of the saddle. Section C-C of Fig. 4 indicates thisshear distribution; that is, the shears vary as the sin andact downward above angle and act upward below angle

. The summation of the downward vertical componentsmust balance the summation of the upward verticalcomponents. Then with saddles at the heads the maximumshear stress in lb. per sq. in. is given by

in the shell, or

in the head.

Values of K 2 given in Table I for different size saddles at theheads are obtained from the expression given for the maximumshear stress in Section C-C of Fi .4 & a endix.

The tangential shear stress should not exceed 0.8 of theallowable tension stress.

CIRCUMFERENTIAL STRESS AT HORN OFSADDLE

In the plane of the saddle the load must be transferredfrom the cylindrical shell to the saddle: As was pointedout in the previous section the tangential shears adjust theirdistribution in order to make this transfer with a minimumamount of circumferential bending and distortion. The

evaluation of these shears was quite empirical except forthe case of the ring stiffener in the plane of the saddle.Evaluation of the circumferential bending stresses is evenmore difficult.

Starting with a ring in the plane of the saddle, the sheardistribution is known. The bending moment at any pointabove the saddle may be computed by any of the methodsof indeterminate structures. If the ring is assumed uniformin cross section and fixed at the horns of the saddles, themoment, M , in in.-lb. at any point A is given by:

This is shown schematically in Fig. 6. Note that must bein radians in the formula.

The maximum moment occurs when = . Substitutingfor and K 6 for the expression in the brackets divided by

, the maximum circumferential bending moment in in.-lb.is

When the shell is supported on a saddle and there is noring stiffener the shears tend to bunch up near the horn of the saddle, so that the actual maximum circumferentialbending moment in the shell is considerably less than Mas calculated above for a ring stiffener in the plane of thesaddle. The exact analysis is not known; however, stressescalculated on the assum tion

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that a wide width of shell is effective in resisting thehypothetical moment, M , agree conservatively with theresults of strain gage surveys. It was found that thiseffective width of shell should be equal to 4 times the shellradius or equal to one-half the length of the vessel,whichever is smaller. It should be kept in mind that use of this seemingly excessive width of shell is an artificewhereby the hypothetical moment M is made to rendercalculated stresses in reasonable accord with actual

When the saddles are near the heads, the shears carry tothe head and are then transferred back to the saddle. Againthe shears tend to concentrate near the horn of the saddle.Because of the relatively short stiff members this transferreduces the circumferential bending moment still more.

To introduce the effect of the head the maximummoment is taken as

Where K 3 equals K 6 when A/R is greater than 1. Values of K3 are plotted in Fig. 7 using the assumption that thismoment is divided by four when A/R is less than 0.5.

The change in shear distribution also reduces the directload at the horns of the saddle; this is assumed to be Q/4for shells without added stiffeners. However, since thisload exists, the effective width of the shell which resiststhis direct load is limited to that portion which is stiffenedby the contact of the saddle. It is assumed that 5t each sideof the saddle acts with the portion directly over the saddle.See A endix B.

Internal pressure stresses do not add directly to the localbending stresses, because the shell rounds up underpressure. Therefore the maximum circumferentialcombined stress in the shell is compressive, occurs at thehorn of the saddle, and is due to local bending and directstress. This maximum combined stress in lb. per sq. in. is

iven b

or

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be equal to 1.50 times the tension allowable provided thecompressive strength of the material equals the tensilestrength. In the first place when the region at the horn of the saddle yields, it acts as a hinge, and the upper portionof the shell continues to resist the loads as a two-hingedarch. There would be little distortion until a second pointnear the equator started to yield. Secondly, if rings areadded to reduce this local stress, a local longitudinalbending stress occurs at the edge of the ring underpressure. 5 This local stress would be 1.8 times the designring stress if the rings were infinitely rigid. Weld seams inthe shell should not be located near the horn of the saddlewhere the maximum moment occurs.

EXTERNAL LOADS

Long vessels with very small t/r values are susceptible todistortion from unsymmetrical external loads such as wind.It is assumed that vacuum relief valves will be providedwhere required; so it is not necessary to design against afull vacuum. However, experience indicates that vesselsdesigned to withstand 1 lb. per sq. in. external pressure cansuccessfully resist external loads encountered in normal

Assume the external pressure is 1 lb. per sq. in. in theformulas used to determine the sloping portion of theexternal pressure chart in the 1950 A.S.M.E. UnfiredPressure Vessel Code. Then when the vessel is unstiffenedbetween the heads, the maximum length in feet betweenstiffeners (the heads) is given approximately by

When ring stiffeners are added to the vessel at thesupports, the maximum length in feet between stiffeners isgiven by

ADDITIONAL STRESS IN HEAD USED ASSTIFFENER

It seems reasonable to allow this combined stress to* Note: For multiple supports:

L=Twice the length of portion of shell carried bysaddle

If L 8R use 1st formula.

When the head stiffness is utilized by placing the saddleclose to the heads, the tangential shear stresses cause anadditional stress in the head which is additive to thepressure stress. Referring to Section C-C of Fig. 4, it canbe seen that the tangential shearing stresses have horizontalcomponents which would cause varying horizontal tensionstresses across the entire height of the head if the headwere a flat disk. The real action in a dished head would bea combination of ring action and direct stress; however, forsimplicity the action on a flat disk is considered reasonablefor design purposes.

Assume that the summation of the horizontalcomponents of the tangential shears is resisted by thevertical cross section of the flat head at the center line,


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and assume that the maximum stress is 1.5 times the aver-age stress. Then the maximum additional stress in the headin lb. per sq. in. is given by

or

This stress should be combined with the stress in thehead due to internal pressure. However, it is recommendedthat this combined stress be allowed to be 25% greaterthan the allowable tension stress because of the nature of the stress and because of the method of analysis.

RING COMPRESSION IN SHELL OVER SADDLE

Figure 8 indicates the saddle reactions, assuming thesurfaces of the shell and saddle are in frictionless contactwithout attachment. The sum of the assumed tangentialshears on both edges of the saddle at any point A is alsoshown in Fig. 8. These forces acting on the shell band

directly over the saddle cause ring compression in the shellband. Since the saddle reactions are radial, they passthrough the center 0. Taking moments about point Oindicates that the ring compression at any point A is givenby the summation of the tangential shears between and

This ring compression is maximum at the bottom, where= . Again a width of shell equal to 5t each side of the

saddle plus the width of the saddle is assumed to resist thisforce. See Appendix B. Then the stress in lb. per sq. in. dueto ring compression is given by

or

The ring compression stress should not exceed one-half of the compression yield point of the material.

WEAR PLATES

The stress may be reduced by attaching a wear platesomewhat larger than the surface of the saddle to the shelldirectly over the saddle. The thickness t used in the

formulas for the assumed cylindrical shell thickness maybe taken as (t 1 + t2) for S 5 (where t 1 = shell thickness and t 2= wear plate thickness), provided the width of the addedplate equals at least (b + 10t 1) (See Appendix B).

The thickness t may be taken as (t 1 + t2) in the formulafor S 2, provided the plate extends r/10 inches above thehorn of the saddle near the head, and provided the plateextends between the saddle and an adjacent stiffener ring.(Also check for S 2 stress in the shell at the equator.)

The thickness t may he taken as (t 1 + t2) in the first termof the formula for S 3 provided the plate extends r/10 inchesabove the horn of the saddle near the head. However, (t1 2 +t22) should be substituted for t 2 in the second term. Thecombined circumferential stress (S 3) at the top edge of thewear plate should also be checked using the shell platethickness t 1 and the width of the wear plate. When checkingat this point, the value of K 3 should be reduced byextrapolation in Figure 7 assuming equal to the centralangle of the wear plate but not more than the saddle angle

lus 12 .DESIGN OF RING STIFFENERS

When the saddles must be located away from the headsand when the shell alone cannot resist the circumferentialbending, ring stiffeners should be added at or near thesupports. Because the size of rings involved does notwarrant further refinement, the formulas developed in thispaper assume that the added rings are continuous with auniform cross section. The ring stiffener must be attachedto the shell, and the portion of the shell reinforced by thestiffener plus a width of shell equal to 5t each side may beassumed to act with each stiffener. The ring radius isassumed equal to r.

When n stiffeners are added directly over the saddle asshown in Fig. 4 (c), the tangential shear distribution isknown. The equation for the resulting bending moment atany point was developed previously, and the resultingmoment diagram is shown in Fig. 6. The maximummoment occurs at the horn of the saddle and is given inin.-lb. for each stiffener by

Knowing the maximum moment M and the moment atthe top of the vessel, M t, the direct load at the point of maximum moment may be found by statics. Then thedirect load at the horn of the saddle is given in pounds by

or

If n stiffeners are added adjacent to the saddle as shownin Fig. 4 (b), the rings will act together and each will beloaded with shears distributed as in Section B-B on oneside but will be supported on the saddle side

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by a shear distribution similar to that shown in SectionA-A. Conservatively, the support may be assumed to betangential and concentrated at the horn of the saddle. Thisis shown schematically in Fig. 9; the resulting bendingmoment diagram is also indicated. This bending momentin in.-lb. at any point A above the horn of the saddle isgiven by

For the range of saddle angles considered M ismaximum near the equator where = . This moment andthe direct stress may be found using a procedure similar tothat used for the stiffener in the plane of the saddle.Substituting for and K 6 for the expression in thebrackets divided by 2 , the maximum moment in each ringadjacent to the saddle is given in in.-lb. by

Knowing the moments M and M t, the direct load atmay be found by statics and is given by

or

Then the maximum combined stress due to liquid loadin each ring used to stiffen the shell at or near the saddleis given in lb. per sq. in. by

where = the area and I/c = the section modulus of thecross section of the composite ring stiffener. When a ringis attached to the inside surface of the shell directly overthe saddle or to the outside surface of the shell adjacent tothe saddle, the maximum combined stress is compressionat the shell. However, if the ring is attached to theopposite surface, the maximum combined stress may beeither compression in the outer flange due to liquid ortension at the shell due to liquid and internal pressure.

The maximum combined compression stress due toliquid should not exceed one-half of the compression yieldpoint of the material. The maximum combined tensionstress due to liquid and pressure should not exceed theallowable tension stress of the material.

DESIGN OF SADDLES

Each saddle should be rigid enough to prevent theseparation of the horns of the saddle; therefore the saddleshould be designed for a full water load. The horn of thesaddle should be taken at the intersection of the outer edgeof the web with the top flange of a steel saddle. Theminimum section at the low point of either a steel orconcrete saddle must resist a total force, F, in pounds,equal to the summation of the horizontal components of

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the reactions on one-half of the saddle. Then

The effective section resisting this load should be limitedto the metal cross section within a distance equal to r/3below the shell. This cross section should be limited to thereinforcing steel within the distance r/3 in concrete

saddles. The average stress should not exceed two-thirdsof the tension allowable of the material. A low allowablestress is recommended because the effect of thecircumferential bending in the shell at the horn of thesaddle has been neglected.

The upper and lower flanges of a steel saddle should bedesigned to resist bending over the web(s), and the web(s)should be stiffened according to the A.I.S.C. specificationsagainst buckling. The contact area between the shell andconcrete saddle or between the metal saddle and theconcrete foundation should be adequate to support thebearing loads.

Where extreme movements are anticipated or where the

saddles are welded to the shell, bearings or rockers shouldbe provided at one saddle. Under normal conditions a sheetof elastic waterproof material at least 1/4 in. thick betweenthe shell and a concrete saddle will suffice.

Nomenclature


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Appendix

The formulas developed by outline in the text aredeveloped mathematically here under headings cor-responding to those of the text.

The pertinent assumptions and statements appearing in thetext have not been repeated.

Maximum Longitudinal Stress

Referring to Fig. 3, the bending moment in ft.-lb. at thesaddle is

effective . If equals any central angle

measured from the bottom, the moment of inertia is

The section modulus for the tension side of the equivalentbeam is

The bending moment in ft.-lb. at the mid span is

The section modulus = r2t, and

Tangential Shear Stress

Section B-B of Fig. 4 indicates the plot of the shearsadjacent to a stiffener. The summation of the verticalcomponents of the shears on each side of the stiffener mustequal the load on the saddle Q. Referring to Fig. 4 (d) thesum of the shears on both sides of the stiffener at any pointis Q sin / r. Then the summation of the verticalcomponents is given by

The maximum shear stress occurs at the equator when sin= 1 and K 2 = 1/ = 0.319.

Section A-A of Fig. 4 indicates the plot of the shears

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arc rsin

Referring to Section A-A of Fig. 4 the centroid of the where

or

where

Then the stress in the shell at the saddle in lb. per sq. in. isgiven by


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in an unstiffened shell. Again this summation of thevertical components of the shears on each side of thesaddle must equal the load on the saddle. Then the totalshear at any point is

and the summation of the vertical components is given

by

The maximum shear occurs where = and

Section C-C of Fig. 4 indicates the shear transfer acrossthe saddle to the head and back to the head side of thesaddle. Here the summation of the vertical components of the shears on arc acting downward must equal the

summation of the vertical component of the shears on thelower arc ( ) acting upward. Then

or

Finally

The maximum shear occurs where 2 = and

Circumferential Stress at Horn of Saddle

See under the heading Design of Ring Stiffeners.

Additional Stress in Head Used as Stiffener

Referring to Section C-C of Fig. 4, the tangential shearshave horizontal components which cause tension acrossthe head. The summation of these components on thevertical axis is

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Then assuming this load is resisted by 2rt h. and that themaximum stress is 1.5 times the average

where

Wear Plates

The ring compression at any point in the shell over thesaddle is given by the summation of the tangential shearsover the arc = ( ) shown in Section A-A or C-C of Fig.4 or in Fig. 8. Then

The ring compression becomes a maximum in the shellat the bottom of the saddle. Or if = this expressionbecomes

Then

Design of Ring Stiffeners

Stiffener in Plane of Saddle

Referring to Fig. 6, the arch above the horns of thesaddle resists the tangential shear load. Assuming this archfixed at the top of the saddles, the bending moment may befound usin column analo .

If the arch is cut at the top, the static moment at anypoint A is


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Then the M s /EI diagram is the load on the analogouscolumn

The area of this analogous column is

The centroid is sin / r , and the moment of inertiaabout the horizontal axis is

The load on the analogous column is

The moment about the horizontal axis is

Then the indeterminate moment is

The distance from the neutral axis to point A is given

by

Finally, the combined moment is given by

This is the maximum when = ; then

FinallyBecause of symmetry the shear stress is zero at the top

of the vessel; therefore, the direct load in the ring at thetop of the vessel, Pt, may be found by taking moments onthe arc about the horn of the saddle. Then

The direct load, P , at = , the point of maximummoment may be found by taking moments about the

center. Then

Substituting the value above for P t, and solving for P ,gives

or

where

If the rings are adjacent to the saddle, K 6 and K 7 may befound in a similar manner, except that the static structurewould become the entire ring split at the top and loaded asindicated in Fig. 9.

Design of Saddles

The summation of the horizontal components of theradial reactions on one-half of the Saddle shown in Fig. 8must be resisted by the saddle at = . Then thishorizontal force is given by

Then

The bending at the horn would change the saddlereaction distribution, and increase this horizontal force.

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After the article had been published, certain refinementsseemed desirable; therefore, the following has been addedto take greater advantage of the inherent stiffness of thesevessels. The methods outlined in the paper will giveconservative results.

The effective width of shell has been limited to 10t inorder to prepare the chart of Fig. 2. It has been shown 5

that this effective width may be taken as 1.56 r t . Thatis, where 5t each side of the saddle or stiffener has beenused, the more liberal value of 0.78 rt each side could beused.

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Appendix B

The values plotted in Fig. 5 for K 1 cover conservativelyall types of heads between H = 0 and H = R. More liberalvalues are given in Fig. 5A for hemispherical and 2 to 1ellipsoidal heads for values of H/L between 0 and 0.1. Theminimum values of K 1 given in Table I have not beenlisted for specific values of R/L and H/L; so they areconservative. Specific minimum values of K 1 may be readfrom Fig. 5A.

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