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# درس عن تصميم الcylinders

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#1 م/عبدالظاهر استاذ الكترونيات تاريخ التسجيل: Oct 2007 الدولة: مصر المشاركات: 295 معدل تقييم المستوى: 12  درس عن تصميم الcylinders

CYLINDERS

Cylinders find many applications, two of the most common categories being :
- fluid containers such as pipes and pressure vessels - interference-fitted bearing bushes, sleeves and the like.
Cylinders can act as beams or shafts eg. ( load building blocks which have already been examined ) but in the present context cylinders are loaded primarily by internal and external ( gauge ) pressures pi and po due to adjacent fluids or to contacting cylindrical surfaces.
The following notes examine cylinder safety and derive appropriate design equations. Since the analysis presupposes an elastic material, superposition may be applied if other loading mechanisms such as bending or torsion occur. The treatment ignores end effects and the axial variation of pressures and stresses - so all cross- sections are loaded and stressed identically, and cylinder length is of no consequence. Secondary stresses arising from incompatibility between cylinder and end closure are examined in the context of pressure vessels. The stress state in the wall is essentially triaxial and initial analysis gives the principal components without need for resolution :
• axial stress, σa, sometimes called the longitudinal stress
• tangential stress, σt, also referred to as the hoop or circumferential stress
• radial stress, σr . . . . all of which are taken to be positive tensile.
Stresses are identical in cylinders which have geometrically similar cross-sections and which are loaded by the same pressures - absolute size is irrelevant insofar as stresses are concerned. Geometric similarity is expressed by a single ratio between two of the three dimensions - inside diameter (or 'bore') Di, outside diameter Do and wall thickness t. Cylinders are usually considered to be either thick - in which stress concentration due to relative curvature is significant - or thin in which stress concentration is negligible. Other differences are :

Thin Thick Limiting proportions (approx) Di/t > 20 Di/t < 25 Analytic treatment simple approximation accurate Statically determinate yes no Stress state membrane - ie. biaxial triaxial Stress - radial, σr zero varies with radius - tangential, σt uniform varies with radius - axial, σa uniform uniform

The limiting proportions - the dividing line between thin and thick - is not hard and fast. Whether we choose to treat a particular cylinder as thin or as thick is a matter only of the accuracy we seek from the mathematical model together with the effort we're prepared to expend in solving the model. For example, if in a particular design the thin model results in a cylinder with Di/t = 7 then we'd either
• repeat the calculations using the thick model if an accurate solution was sought, or
• accept the result of the thin model knowing that it was in significant error due to the neglect of curvature- induced stress concentration.
We shall address the error associated with the thin cylinders below. The Pressure Vessel Code adopts the latter approach as the Code incorporates large design factors to cater for other imponderables.

Physically, thin cylinders are not suitable when the external fluid pressure is much greater than the internal, unless the cylinder is supported or stiffened against local buckling. We shall not consider thin cylinders with net external pressures sufficient to cause buckling, unless the cylinders are obviously adequately supported as in figure (a) below. Axial stress

Cylinders are classed as being either :
• open - in which there is no axial component of wall stress, or
• closed - in which an axial stress must exist to equilibrate the fluid pressure.
Open cylinders are typified by interference-fitted bushes figures (a) and (b) below, in which there are no longitudinal pressures and so no axial stresses. A fluid container sealed by a piston is open (c) - in this case an external axial force Fa is mandatory for equilibrium of the overall piston-and-cylinder assembly (a bicycle pump is a common example). If Ai is the internal circular area then a free body of either piston or cylinder end (c) must have Fa = pi Ai and there is no need for axial wall stresses to equilibrate the fluid pressure.
If an axial stress does exist then it is uniform across the cylinder wall, no matter whether the cylinder is thin or thick. The stress may be found easily from equilibrium in the axial direction, aided by a free body viewed in side sectional elevation. Considering the free body of one end of an internally pressurised closed cylinder (d) the fluid pressure resultant pi Ai is equilibrated by the wall stress resultant σa Aa - where the annular wall area is Aa = Ao - Ai in which Ao is the outside circular area.
In the more general case where an external fluid pressure also exists :

( i) σa = ( pi Ai - po Ao ) / Aa
A free body (e) of part of a pipe which connects two vessels (not shown) might be thought open, since no ends are evident and the fluid pressure is self-equilibrating across the free body. However each connected vessel acts as a pipe closure, so the pipe is in fact closed and axial stresses must occur. Thin cylinders The tangential stress is uniform across the wall; so, from equilibrium of the free body : σt = ( pi Di - po Do ) / 2 t
Since Di / t > > 1, this equation with ( i), yields :

( 1) σt = Δp Di / 2 t where Δp = pi - po
σa = Δp Di / 4 t (CLOSED) or σa = 0 (OPEN) The radial stress is zero, the tangential stress is always the principal of greatest magnitude, and the axial stress is either zero in the case of open thin cylinders or half the tangential stress in closed thin cylinders. The Mohr's circles thus appear as shown. Derivation of the design equations corresponding to the various failure theories is left as an exercise for the reader :-

( ii) n Δp Di / t = 2 Sy [ maximum shear stress; distortion energy (open) ] = ( 4/√3 ) Sy [ distortion energy (closed) ] = 2 Sut [ modified Mohr ]
So much for thin cylinder theory - let us now look at the more realistic but somewhat more elaborate
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م/عبدالظاهرسعيد #2 م/عبدالظاهر استاذ الكترونيات تاريخ التسجيل: Oct 2007 الدولة: مصر المشاركات: 295 معدل تقييم المستوى: 12  رد: درس عن تصميم الcylinders

Thick cylinders

In the absence of bending and direct shear, the radial and tangential stresses are circumferentially symmetric (independent of θ) but vary with radius r. It is convenient to normalise the radius by the constant internal radius, setting :

( iii) γ ≡ ( r / ri )2 ; 1 ≤ γ ≤ γo = ( ro / ri )2 = ( Do / Di )2
where γo is a constant, representing the proportions of the cylinder. Consider the element, defined by r, δr and δθ, and of unit width perpendicular to the sketch plane. For radial equilibrium :
σr r δθ - ( σr + δσr )( r + δr ) δθ + 2 σt δr sin(δθ/2) = 0
which, on expanding with neglect of second order terms, gives :

( iv) ( σt - σr ) δr = r δσr
As σt and σr cannot be found individually from this equilibrium equation, the problem is statically indeterminate - so compatibility and the constitutive law must be invoked. The compatibility requirement, first postulated by Lamé, that initially plane transverse cross-sections remain plane when loaded, implies that the axial strain, εa, is constant across the cross-section - ie. is independent of radius. With the elastic constitutive law, this may be expressed as :
E εa = σa - ν ( σt + σr ) = constant, ie. independent of r,
and since σa is itself constant and independent of r, it follows that :

( v) σt + σr = constant = 2 σm where σm is some constant, as yet undefined.
This latter compatibility/constitutive equation enables resolution of the indeterminacy. Thus, eliminating σt between ( iv) and ( v) :
δσr / ( σr - σm ) = - 2 δr / r and integrating with γ given by ( iii )
( vi) σr = σm - constant / r2 = σm - σv / γ where σv is an integration constant.

The form of the stress distribution is thus established but the two unknown constants σm and σv must be evaluated before the distribution is defined completely. From ( vi) and from equilibrium of elements at the inner and outer surfaces :
inside σr = σm - σv = - pi
outside σr = σm - σv /γo = - po
which may be solved for the unknown constants in terms of the known pressures :

( 2) σm = ( pi - γo po ) / ( γo - 1 ) ; σv = ( pi - po ) γo / ( γo - 1 ) and = σm + pi
Combining ( v) and ( vi), and recalling ( i), yields the final stress distribution equations :
( 3) σt = σm + σv / γ
σr = σm - σv / γ ≤ 0
σa = σm (closed) or σa = 0 (open)
by which all stress components are expressible in terms of location, r ( ie. γ ), once the constants σm and σv have been evaluated from ( 2) in terms of proportions and loading. The stress variations given by ( 2) and ( 3) for a particular closed cylinder are sketched here, and the Mohr's circles corresponding to the bore ( γ = 1 ) and to some other location in the wall ( γ > 1 ) appear below - the similarity between the Mohr's circles for thick and thin cylinders is noticeable. The tangential /radial stress distribution is seen to take the form of a horn disposed symmetrically about σm and of opening radius σv at the bore, confirming the general conclusion that the bore is the critical location - as expected since curvature and stress concentration are most severe there. If this cylinder were ductile and the maximum shear stress failure criterion is applied, then the equivalent stress is 180 MPa at the bore and only 20 MPa at the outside - evidently the material in a thick cylinder is not used effectively throughout the wall.
The advantages of the normalised treatment used here may be appreciated by comparing ( 2) and ( 3) with the corresponding equations in the literature.
The pressures or overall proportions ( γo ) may not be known initially, in which case equations ( 3) are usually applied in the context of the problem, with σm and σv retained as unknown constants whose solution is sought first.

( The nomenclature should be noted here: σa is used in these Notes to represent principal stress amplitude and hence the radius of Mohr's circles - see the nomenclature explanation - however σa symbolises axial stress in cylinder theory and necessitates σv being used here to represent Mohr's circle radius. )
For the particular case of pressure po external to a solid shaft, ri = 0, both γo and γ tend to infinity, in which case ( 2) and ( 3) degenerate to :
( 3a) σt = σr = - po ie. stresses are everywhere uniformly compressive for solid shafts. Design equations

The following equations have been derived from ( 2) and ( 3) with the appropriate failure theory. The detailed proofs, left to the reader, exemplify the difficulties associated with the application logic of piecewise- defined theories such as the maximum shear stress and modified Mohr.
For DUCTILE materials the concept of equivalent stress, σe, is relevant. If the distortion energy failure theory is applied, then using ( 3) :
σe = [ { ( σt - σr )2 + ( σr - σa )2 + ( σa - σt )2 } / 2 ]1/2
= [ ( σm - σa )2 + 3 ( σv / γ )2 ]1/2 where the appropriate σa is given by ( 3)

The maximum equivalent stress, defined as σ* = σemax, occurs when γ is a minimum ie. at the bore where γ = 1. Resulting design equations are therefore :
( vii) Distortion energy
OPEN σ* = [ ( pi - γo po )2 + 3 γo2 ( pi - po )2 ]1/2 / ( γo - 1 )
CLOSED σ* = √3 | pi - po | γo / ( γo - 1 )

If, on the other hand, the maximum shear stress theory is applied, it may again be shown that the maximum equivalent stress occurs at the bore, where the principals from ( 2) and ( 3) are :
σt = [ ( γo + 1 ) pi - 2 γo po ] / ( γo - 1 ) ; σr = - pi ; σa = [ pi - γo po ] / ( γo - 1 ) or 0

Application of the theory requires selection of the minimum and maximum principal stresses, with neglect of the intermediate, and it is apparent that σt may or may not exceed either σr or σa. It is necessary therefore to consider all feasible combinations of the principals, and this results in :
( viii) Maximum shear stress OPEN ordered application limits, po/pi value of principals minimum maximum ( γo - 1 ) σ* σr ≤ σa ≤ σt 0 ( 1 + 1/γo ) / 2 2 γo ( pi - po ) σr ≤ σt ≤ σa ( 1 + 1/γo ) / 2 1 ( γo - 1 ) pi σt ≤ σr ≤ σa 1 infinity - ( γo + 1 ) pi + 2 γo po CLOSED - 0 infinity 2 γo | pi - po |
BRITTLE cylinders subscribe to the modified Mohr failure theory for which the safety factor is given by the maximum of three terms, each of which is linear in the principal stresses :
1/n = max ( σmax/St , - σmin/Sc , σmax/St - (σmax+ σmin)/Sc ) where σmax, σmin = max, min ( σt, σr, σa)

The implications of this piecewise defined theory may be appreciated with reference to a closed cylinder of geometry γo = 3 in a material whose strengths are St = 200, Sc = 700 MPa. The cylinder is subjected to a constant internal pressure of 10 MPa while the external pressure increases from 0 to 30 MPa. The sketch traces the three principal stresses at the critical bore - the two extremes are the tangential and the radial (which is constant). Also shown are the three terms of the modified Mohr equation - the maximum of these comprises four linear segments a-b-c-d-e. The tangential extreme, which is tensile initially at a, becomes progressively more compressive as the external pressure rises until at d when the pressures are identical and the stresses are all compressively equal to the pressures. In a-d the varying tangential principal is maximum while the constant radial is minimum. These roles are reversed in d-e. The three modified Mohr terms change accordingly - in a-b the first term is the maximum of the three, in b-c the third term dominates, in c-d-e the second term. As a result of the interaction between the individually simple terms the safety factor changes in a complex fashion, increasing from 10 to 70 then remaining constant before falling monotonically with further rise in external pressure. Repeating this analysis algebraically (problem #8) leads to a four-piece design equation.

An identical approach for brittle OPEN cylinders results in a similar design equation, viz :
( ix) Modified Mohr value of OPEN ordered application limits, po/pi ( γo - 1 ) Suc / n principals minimum maximum where m = Suc / St σr ≤ σa ≤ σt 0 1/γo [ ( γo + 1 ) pi - 2 γo po ] m " 1/γo ( 1 + 1/γo ) / 2 [ ( γo + 1 ) m - 2 ] pi - 2 ( m - 1 ) γo po σr ≤ σt ≤ σa ( 1 + 1/γo ) / 2 1 ( γo - 1 ) pi σt ≤ σr ≤ σa 1 infinity - ( γo + 1 ) pi + 2 γo po The failure loci corresponding to ( vii), ( viii), and ( ix), are illustrated for the case of γo = 4; loci for other proportions are of similar shape. The absence of upper limits for ductile closed cylinders is due to triaxiality - approximately equal principals with small shear, analogous to hydrostatic loading. Error of thin cylinder approximations The error which arises from the use of the thin cylinder equations will now be evaluated for ductile cylinders subjected to internal pressure and subscribing to the distortion energy failure theory. Setting w = t/Di, and employing ( ii) & ( iii), the error in the maximum equivalent stress is :
Error = 1 - σethin / σ*thick
= 1 - 2 ( 1 + w ) / [ 1 + 3 ( 1 + 2w )4 ]1/2 Open
= 1 - ( 1 + w ) / ( 1 + 2w )2 Closed

The plot of these errors indicates that a thin/thick transition at say w = 0.04 corresponds to an error of about 10 %, and that the thin cylinder equations always underestimate the equivalent stress.
The design of thick cylinders is illustrated by this example.
While there may be little difference between theories in analysis situations - when equivalent stresses or safety factors are being calculated - the equations' forms often lead to substantial differences in the dimensions predicted by the theories in synthesis situations. Thus note the large difference in the diameters predicted by the thin and thick theories in part (a) of this example, and by the maximum shear stress and distortion energy ductile failure theories in parts (b) and (c).
Stress concentration at the bore of thick cylinders can lead to very wasteful use of material - see the above example with internal and external pressures of 120 and 40 MPa. It is not uncommon to be faced with a design which at first glance seems impossible - thus in the selection of a fixed bore cylinder, increasing the OD in an effort to "beef it up" merely increases γo and stress concentration, so preventing a solution from being obtained. It may be possible to choose a stronger material, however a more satisfactory solution - especially when weight is an issue - is to use a compound cylinder, which we now examine.
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م/عبدالظاهرسعيد #3 م/عبدالظاهر استاذ الكترونيات تاريخ التسجيل: Oct 2007 الدولة: مصر المشاركات: 295 معدل تقييم المستوى: 12  رد: درس عن تصميم الcylinders

Cylinders - some applications  Evening view of LNG ( Liquified Natural Gas ) Process Train, Burrup Peninsula, Karratha in the North West of Western Australia ( Woodside Petroleum Ltd ). The Train is marked with a cross in the site view on the right.
Click the photos to enlarge them.
Cylinders in the form of tubes, pipes, tanks and pressure vessels clearly play a crucial role in this plant - as they do in most processing plant.

The layout of a medium sized coal-fired boiler is sketched - the arrowed man gives an idea of the boiler size.

Pulverised coal is transported in a continuous jet of air through each burner and burns in the combustion chamber ( the large chamber between burners and the man in the sketch ). The hot gaseous combustion products then follow a tortuous path produced by baffles, giving up their heat generally in counterflow to the steam, water and combustion air before exhausting through the boiler stack ( not shown ).
The boiler is constructed mainly of interconnected cylinders such as the superheater tubes, the generating tubes which completely cover the walls of the combustion chamber ( the 'water walls' ), various larger diameter headers ( not identified ), and the large drums. Water and steam circulate by natural convection through the tube network, the steam finally collecting in the upper ( steam ) drum, from whence it is drawn off for use.
click the sketch to enlarge it This is a view inside a boiler similar to the above during manufacture.

This shows a gas pass downstream from the combustion chamber. Again cylinders are featured - water/steam tubes completely cover the walls of the passage so that the gases cannot give up heat unless it's transferred to the water/steam.
The tradesperson is applying insulation to selected areas of the wall. This photograph shows the combustion chamber of an operating boiler similar to the above.

The pulverised coal-air mixture is pouring continuously out of the burners and commences to burn immediately it enters the combustion chamber.
The water wall in which the burners are mounted is clearly shown - the arduous conditions under which the high pressure cylindrical tubes operate can be appreciated. Some ash build-up on the tubes is apparent.  This sketch is of a pair of angular contact ball bearings mounted back-to-back.
The fits of each bearing's inner race on the shaft and outer race in the housing are critical to the smooth operation of the bearing.
If fits are too loose then the bearing is sloppy and cannot offer the guidance it's designed to provide.
If there is too much interference then the balls are tightly constricted - ball loads, friction and noise all increase, and wear life decreases.

Each bearing race is a cylinder, as are the shaft ( albeit probably a solid cylinder ) and the housing ( a cylinder of large OD ). The compound cylinder theory presented later in this chapter is necessary to understand how these statically indeterminate mating cylinder pairs behave on assembly.
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م/عبدالظاهرسعيد #4 م/عبدالظاهر استاذ الكترونيات تاريخ التسجيل: Oct 2007 الدولة: مصر المشاركات: 295 معدل تقييم المستوى: 12  رد: درس عن تصميم الcylinders PRESSURE VESSELS

Air receivers and LNG tanks are two common examples of pressure vessels. They are usually cylindrical in form and hence employ cylinder theory in their design, but there are numerous practical aspects which transcend the basic theory - end closures have to be fixed to the cylinder, holes have to be cut and inlet/outlet pipes ( branches or nozzles ) attached, weld flaw probabilities have to be acknowledged, and so on. Some of these aspects are now examined, as a lead- up to the practical design of pressure vessels.
We consider only internally pressurised, welded steel unfired vessels operating at room temperature and above, thus avoiding the complexities associated with
• buckling due to external pressure and manufacturing inaccuracies
• the dangerous partnership of low temperatures and welding which always requires special precautions in design and construction.
Unfired implies the absence of any energy source in the vessel. AS 1210 Code for Unfired Pressure Vessels (the Code) forms the legal background to most contracts between manufacturer and purchaser; these notes aim also to provide a brief introduction to this Code - sufficient for the outline design of a simple vessel. One must be careful not to regard AS 1210 as a design cookbook - engineering judgement remains a sine qua non, and pitfalls abound for the indiscriminate user. Some of the equations developed in the following notes are amended slightly in the Code, and the nomenclature also differs.
AS 1210 adopts a simple approach to vessel design based on thin cylnder theory and the neglect of secondary complexities such as the essential incompatibility between intersecting cylinders. This approach requires substantial safety factors, despite the secondary effects being minor within the Code's scope. Appendix A of AS1210 stipulates that the design tensile stress of the plate material, S, shall be :-

( i) S = minimum ( Sy /1.6 , Su /4 , SyT /1.5 , Scr /1.6 )
in which Sy and Su are the tensile yield and ultimate at room temperature, and the last two terms are applicable only at design temperatures above 50oC, SyT being the yield strength and Scr the 100 000 hr creep rupture strength at the design temperature.
Minimum mechanical properties are laid down in AS 1548 Steel Plates for Boilers and Unfired Pressure Vessels, a steel being referenced typically as 'AS1548-2-430-H' in which the '2' refers to Section 2 of AS1548 which deals with a particular type of steel, '430' denotes the minimum ultimate (MPa), and 'H', if it appears, indicates that properties have been verified by high temperature testing rather than being inferred from room temperature properties. Note that properties are slightly thickness- dependent.

Corrosion

In design, corrosion which occurs over the life of a vessel is catered for by a corrosion allowance, c, whose design value depends upon the vessel duty and contents' corrosiveness - for example 1 mm is typical for air receivers in which condensation of air moisture is normally inevitable. It is important to realise that when dimensions in any formula refer to a corrodable surface, then the dimensions inserted into the formula are those at the end of the vessel's life, when all the corrosion allowance has been eaten away. So, if a plate is of nominal thickness T now, and is subject to corrosion on one side, then ( T - c ) must be substituted whenever nominal thickness appears in an equation. Similarly a tube of current bore Di which corrodes will have a bore of ( Di + 2c ) at the end of its life.
Welded Joint Efficiency

Welded joints are not as strong as the parent plate unless welds are thoroughly inspected and, if flawed, repaired during manufacture - all of which is expensive. This strength reduction is characterised by the weld or joint efficiency η = joint strength / parent strength - which varies from 100% for a perfect weld (ie. virtually seamless) through 75-85% for a tolerably good weld.
Consider the three bars illustrated below which are all of the same width b and material strength S and which are loaded by the tensile load P. The first bar is seamless of thickness ts and the second welded with joint efficiency η and thickness tw locally at the joint. The safety factor in this second bar is Sbts / P away from the joint and ( η S) btw /P at the joint. If in design these safety factors are to be identical (ie. the bar is not to be weakened by the joint) then tw must equal ts /η. However it is most unusual for the economics to justify increase of bar thickness locally in way of the joint - the whole bar must therefore be of thickness tw to cater for an isolated joint inhomogeneity, as shown in the third sketch. The implications of joint efficiency upon cost of raw plate materials are thus readily apparent, however it should be appreciated that away from the joint region the material really needs only to be as thick as a seamless bar. Pressure vessel cylinders are usually made from flat plates which are rolled then welded along longitudinal joints.
On the other hand, circumferential joints are used to attach end closures ( dished ends or heads ) to the cylinder, and to weld together rolled plates for a long vessel if plate size availability or rolling machine capacity are restricted.

Weld types and efficiencies usually differ for longitudinal and circumferential joints, and therefore the joint stresses in a vessel must satisfy both the requirements :-
longitudinal joint, efficiency η l , circumferential stress σ c = p D / 2t ≤ η l S circumferential joint, efficiency η c , longitudinal stress σ l = p D / 4t ≤ η c S

In the design context, having selected the material and hence the design stress from ( i), these inequalities may be solved for the plate necessary minimum or calculation thickness, t :-

( 1) t = maximum ( pD/2η lS , pD/4η cS )
In this formula, AS 1210 takes the diameter to be the mean at the wall mid-surface ( D = Di + t ) though it quotes the equation conveniently in terms of Di . As the circumferential stress is twice the longitudinal it follows that the first of these inequalities is usually controlling - provided that η c is greater than half η l, which is commonly the case. A plate thickness T, such that ( T - c) ≥ t, would be chosen from a commercially available range such as the BHP plate size schedule, referred to in Problem 3 below.
The Code defines four classes of vessel manufacture - 1, 2 ( A & B ) and 3 - and provides details of the weld types and efficiencies which are permissible in each class. Some of the major differences between the classes are summarised in the table below; other requirements such as certification of welders' skills need not be gone into here. The sketch indicates that the most expensive Class 1 vessels are mandatory if failure is potentially lethal, no matter what the thickness. Class 2 vessels having a wall thickness greater than 32 mm are inadmissible, and so on. All steel vessels are hydrostatically tested after manufacture to 150% of the design pressure.

Class of vessel 1 2A 2B 3 Plate thickness, T mm no limit T ≤ 32 T ≤ 12 η l (double butt mandatory) 1.0 0.85 0.75 0.65 η c - - - depends on weld type, usually less than η l - - - Heat treatment required - - - not required - - - Radiographic examination required spot - - - not required - - - A pressure vessel is shown entering a heat treatment furnace - clearly the cost associated with heat treatment is not negligible.

It is apparent that in the design process, the selection of class (if scope for choice exists), and of weld type, involves an economical balance between welding costs ( including inspection and repairing flaws during manufacture, if carried out ) and the cost of plate - a categorical decision cannot be made until the relative or total cost of all potential candidates can be compared.
Welds should preferably not be situated adjacent to anomalies such as stress concentrations or inhomogeneities due to other welds, especially major structural welds. Hence the staggering of the two longitudinal welds in the sketch above.
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م/عبدالظاهرسعيد #5 م/عبدالظاهر استاذ الكترونيات تاريخ التسجيل: Oct 2007 الدولة: مصر المشاركات: 295 معدل تقييم المستوى: 12  رد: درس عن تصميم الcylinders Thin Shells of Revolution - Heads

The stresses in a pressurised thin axisymmetric shell of revolution will now be considered so that the behaviour of dished ends may be appreciated.
The shell is formed conceptually by rotating the meridian, a curved line of selected shape lying in the r-z meridional plane, about the z-axis. The resulting surface of revolution is clothed by a small, symmetrically disposed thickness t, and the resulting shell loaded by internal pressure p. If the meridian were a straight line for example, parallel to the z-axis and distant D/2 from it, then the shell would be a cylinder of diameter D.

These shells are similar to thin cylinders in that radial stresses are negligible and the membrane stresses :
• the circumferential or hoop stress σθ (that is σt of cylinder theory) and
• the meridional stress σφ (analogous to σa of cylinder theory)
can both be found from equilibrium since they, and the loading, are axisymmetric. Consider the element located at the point A in the r-z plane as shown, and defined by φ, δφ and δθ. The local surface normal cuts the z-axis at the point B, AB being defined as the radius rθ. The centre of curvature lies at C on the normal, AC being the instantaneous radius of curvature of the meridian, rφ.
The components of the pressure and stress resultants along the outward normal are :

pressure : p δA = p ( r δθ ) rφ.δφ
meridional stress : - 2 σφ ( t r δθ ) sin δφ/2
circumferential stress : - 2 σθ ( t rφ δφ ) sin δθ/2 . sin φ

Taking limits and noting that r = rθ sin φ, then equilibrium of the element requires that :-
( ii) σ&theta / r&theta + σφ / rφ = p / t the so-called membrane equation Furthermore, for vertical equilibrium of the dish area above the hoop :-
( iii) π r2 p = 2 π r t σφ sin φ
Solving equations ( ii) and ( iii) gives the stress components in terms of rθ and rφ , which are in turn functions solely of the meridional geometry - its shape and location with respect to the rotation axis :
( 2) σφ = ( p /2t ) rθ ; σθ = σφ ( 2 - rθ /rφ )
Some typical specialisations of ( 2) are as follows :-
Cylinder
rθ = D/2 and rφ tends to infinity - therefore from ( 2)
( iv) σθ = 2σφ = pD/2t ie. the thin cylinder equations once again.

Sphere
rθ = rφ = D/2 and so from ( 2)
( v) σθ = σφ = pD/4t
The sphere is an ideal end closure since the stresses are less than those in other shapes, however the degree of forming necessary renders it impractical except for very high pressures when the manufacturing cost may be justified.

Ellipsoid
An elliptical meridian of semi -major and -minor axes a, b and eccentricity ε = √[ 1 - (b/a)2 ], is rotated about the minor axis to form the head of the cylinder whose diameter is D = 2a as sketched below.
The location of an element on the ellipse is defined most directly by the radius r from the rotation axis, however it will be found more convenient to define the alternative independent variable u = √[ 1 - ε2 ( r/a )2 ] where u = f (r) and b/a ≤ u ≤ 1.
The geometry of the ellipse may be invoked to derive the radii of interest ( 2) in terms of u (ie. in terms of r) :-
rφ = (a2/b) u3 ; rθ = (a2/b) u
The stresses at the r-element follow immediately from ( 2) as :-

( vi) σφ = ( pa2/2bt) u ; σθ = σφ ( 2 - 1/u2 ) where u = f ( r ) as defined above

These stresses are graphed for a = 2b, the most common proportions for practical ellipsoidal ends. The prominent feature of this stress pattern is the tensile- to- compressive transition of the hoop stress at about 80%D. Consequences of compressive behaviour include :
• a propensity for local buckling.
• an increase in the equivalent stress - assuming the maximum shear stress theory for example with the third principal stress (the radial) zero,
• a significant incompatibility between cylinder and head, due to different senses of the hoop (ie. diametral) strain - under pressure the cylinder tends to expand, and the ellipsoidal end to contract diametrically at the junction. Bending moments in the walls of the cylinder and head are therefore set up at the junction, with corresponding bending stresses. These relatively minor secondary stresses cannot be explained by membrane theory, but other work shows that they become insignificant at a distance of about five times the wall thickness from the junction.
Tensile strains in the cylinder at the junction tend to relieve the compression in the dished end and for this reason the stress concentration factors K cited in AS 1210 are less than might be inferred from the above graph - for example :-
( vii) S ≥ K pD/2tη where K = [ 2 + (a/b)2 ] / 6
ie. for a/b = 2, K = 1 and not the 1.5 graphed above.

Torisphere Another common shape for a dished end is torispherical. This consists of a spherical central portion of radius R and a toroidal knuckle of radius r, where R/r is often 12 or thereabouts, and R is about 95 % of the cylinder diameter. Junctions of the torus with both sphere and cylinder give rise to geometric singularities and hence to secondary bending stresses as discussed above. The AS 1210 stress concentration factor, M, reflects this behaviour :-
( viii) S ≥ M pR/2tη ; M = [ 3 + ( R/r )1/2 ] / 4
- the greater the deviation from a sphere ( R/r = 1), the larger the factor. Noticeably, the highly stressed region again extends outwards from about 80% D. Torispherical ends are often preferred to ellipsoidal since the depth of drawing is less and hence they are slightly cheaper - about 10% on average - but this is often outweighed by their higher stress concentration and consequent lesser allowable pressure for a given size.

Dimensions are taken to the wall mid-surface in ( vii), ( viii).
The great majority of heads are seamless, being manufactured from discs
either pressed, or spun in the larger sizes. Theoretically for a seamless head η = 1 in ( vii), ( viii) - however to allow for thinning which results from drawing, the pressure ratings in one commercial brochure are based on η = 0.875 for thicknesses ≥ 25 mm and 0.85 for thicknesses < 25 mm.
Heads are always provided with a short ' flat ' ie. the heads are manufactured with a short integral cylinder, to avoid the junction weld coinciding with stress concentration due to the head- to- cylinder incompatibility and the small head radii of curvature.

Flat plate end closures are not suitable in larger sizes, though often used for doors - being flat, there are no membrane stresses and pressure is resisted solely by plate bending. We have noted already that, for a given load, bending stresses are generally much larger than direct (eg. membrane) stresses. Flat plates therefore have to be much thicker than dished ends for similar duty.
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م/عبدالظاهرسعيد #6 م/عبدالظاهر استاذ الكترونيات تاريخ التسجيل: Oct 2007 الدولة: مصر المشاركات: 295 معدل تقييم المستوى: 12  رد: درس عن تصميم الcylinders

Compensation

Compensation, or reinforcement,is the provision of extra stress-transmitting area in the wall of a cylinder or shell when some area is removed by boring a hole for branch attachment. The principle is sketched. The left sketch shows part of a cylinder's longitudinal section; the major circumferential stress acts across the critical longitudinal plane. The nominal thickness is T, and a hole of diameter Db is bored - dimensions being reckoned in the fully corroded condition. The stress-transmitting area removed is A = Dbt where the calculation thickness t is given by ( 1).
The figure on the right demonstrates compensation for area removal by providing equal area for alternate force paths in otherwise unused material of cylinder and branch. Not all the branch wall can be devoted to compensation since the internally pressurised branch is a cylinder in its own right, with calculation and nominal thicknesses, tb and Tb, determined in a manner identical to the main shell.

Provided that the longitudinal welds in both shell and branch do not lie in the critical longitudinal planethen - from a compensation point of view - both t and tb would be reckoned from ( 1) with η = 1. The thickness differences ( T - t ) and ( Tb - tb ) contribute to compensation - though reinforcement is ineffective beyond the limits Ln normal to the vessel wall, and Lp from the branch centreline parallel to the wall, as shown below for a set-in branch :- AS 1210 gives the limits as :-
( ix) Ln = maximum [ 0.8 ( DbTb )1/2 + Tr , minimum ( 2.5T, 2.5Tb + Tr ) ]
or ( DbT )1/2 for a flanged-in head
( x) Lp = maximum [ Db , Db/2 + T + Tb + 2c ]

Usually the first of the Lp limits, Db, controls. However a compensating area cannot contribute to more than one branch, so if the spacing of two branches Db1 and Db2 is less than ( Db1+ Db2 ), then by proportion Lp1 = Db1 . spacing/( Db1+ Db2 ).
Furthermore, if the branch is attached to a dished end, then no compensation area is effective if it lies outside the aforementioned 80% limits. If the head is torispherical, the hole should lie in the spherical portion and t will be given by ( v). If the head is ellipsoidal, then AS 1210 defines an equivalent sphere for the application of ( v), since the hole will not lie close to the rim region of sharp curvature which dictates the thickness via the stress concentration factor in ( vii).

Within the Ln, Lp limits, compensation requires that :-
( xi) A1 + A2 + A3 + A4 + A5 ≥ A = Dbt
The inward protrusion '3' is subjected to corrosion on three surfaces but there is no pressure differential across it; it will not exist for a set-on branch. The sketch indicates that:-
( xii) A1 = ( 2Lp - Db - 2tb ) ( T - t ) ; A2 = 2Ln ( Tb - tb ) etc.
These equations should be compared carefully with the corresponding equations in AS 1210 to clarify the restrictions implicit in the Code.
Compensation should be disposed symmetrically about the hole and as close to the hole as possible. It is usually more economical to increase the branch thickness than to provide a separate reinforcing ring, however such an increase should not be excessive.

It will be appreciated that the principle of compensation is very simple and ignores inevitable stress concentrations - hence the substantial safety factors mentioned above. AS 1210 lays down branch size upper limits beyond which this simplistic approach is no longer permissible, namely :-
Maximum branch bore = minimum ( Di/2, 500 mm ) if Di ≤ 1500 mm, or
= minimum ( Di/3, 1000 mm) if Di > 1500 mm

Strictly, reinforcement ought to be checked in all planes which contain the branch centreline; AS 1210 defines A = FDbt where F is a factor which varies between unity for the longitudinal plane considered above, and 1/2 ( ie. σa/σt ) for the transverse plane. Generally such a check is unnecessary. A single opening, 75 mm diameter or less in plate 6 mm thick or greater, does not require compensation. Short pipes of this size are sometimes welded straight into the shell as in the photograph, however not only is the pipe prone to damage, but if the pipe thickness differs appreciably from the shell thickness then welding could be awkward.
Each pipe is equipped with a flange welded to the outboard end - once the vessel has been installed permanently, the mating pipe can be attached by bolting the flanges together with a gasket between them to prevent leakage (refer eg. to the chapter on Threaded Fasteners).Alternatively, and perhaps more cheaply, a flange may be welded directly into the shell to obviate the need for a short pipe.
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م/عبدالظاهرسعيد #7 م/عبدالظاهر استاذ الكترونيات تاريخ التسجيل: Oct 2007 الدولة: مصر المشاركات: 295 معدل تقييم المستوى: 12  رد: درس عن تصميم الcylinders

Pipes and flanges

Long pipe runs which convey fluids between items of plant such as pressure vessels are usually made up from short lengths of pipe welded together, however the pipes are not welded permanently into the vessels but rather attached in a manner which permits easy separation. One of the simplest demountable attachments is the screwed connection ( a) though these are suitable only for relatively low pressures, due to the difficulty of making good a leaking thread. The most common demountable connection involves welding a flange to each of the pipes, then connecting the flanges by nuts and bolts ( b). A gasket of some relatively soft material is interposed between the flanges to fill any imperfections in their faces, thus preventing fluid leakage (further details are given in the chapter on Threaded Fasteners).
The higher the fluid pressure and temperature, the more robust must be the flanges and bolting to contain them. AS 2129 Flanges and Bolting for Pipes, Valves and Fittings contains tables of dimensions for flanges and other pipe fittings. Each Table ( D, E, F . . . T ) lists for the various nominal pipe sizes the minimum dimensions which are suitable for a certain limiting combination of fluid pressure and temperature. Thus for steel in the range -50 to 232 oC, the maximum pressure pFT permitted by each flange Table is : Table D E F H J K R S T pFT (MPa) 0.69 1.38 2.07 3.45 4.83 6.20 8.27 12.4 19.3

These limits are halved linearly from 232 to 427 oC and drop to zero at 532 oC as sketched. So, when steel flanges and fittings have to be selected for a design pressure p and temperature T, their size is dictated by a Table for which :
( xiii) pFT ≥ p . maximum [ 1 , 390/( 622 - T ) , 210/( 532 - T ) ] ; T ( oC) < 532
A small bore pipe is often demountably attached to a vessel by means of a flange butt-welded to the pipe and mating with a pad which is welded into or formed on the vessel wall, ( c) above. Clearly the pad dimensions must match the flange. Studs are screwed into holes tapped in the pad, since through- bolts like those of ( b) would allow leakage.
Acceptable pad forms are defined in AS 1210. Pipe wall thicknesses, if required, may be found from ( 1) using design stresses for pipe materials cited in AS 1210, in conjunction with manufacturers' lists or AS 1835.

Inspection Openings

Openings are required to monitor the condition of the vessel's interior if subject to corrosion, and may be necessary also for manufacture. The size and disposition of the opening(s) depend upon the duty and size of the vessel - in a small vessel a single handhole or a flanged- in inspection opening may be adequate whereas large vessels require elliptical manholes, often with reinforcement / seating rings, though heads may be flanged inwards (reverse knuckle) to provide a seating surface. The minor axis of an elliptical opening in a cylindrical shell should lie parallel to the longitudinal axis of the shell. The opening is sealed usually by an internal door, a gasket and one or two bridges and studs. The door is ellliptical to permit its removal, if necessary for remachining a damaged gasket seating surface.
The studs provide the initial sealing force, ie. the initial seating pressure on the gasket face before the fluid is pressurised. When the fluid pressure later rises, the door tends to be self- sealing as the pressure load on the door increases the gasket contact pressure. The load on the studs therefore decreases, however the Code specifies that the door must withstand simultaneously bending by both fluid pressure and maximum possible stud (or bolt) tightening. The flat door calculation thickness t is thus given by :-
( xiv) ( C1. fluid pressure . door area + C2. bolt stress . bolt area ) / t2 ≤ S ; C1, C2 constant
The door is equipped with a locating spigot to aid its engagement when closing. If the door is heavy then provision must be made for supporting it during opening or closing - any such support must not interfere with even take-up of the gasket, nor must it hinder easy access to the vessel. The designer of the door support must visualise the door's detailed operation.
The choice of gasket material depends upon the vessel duty - fluid, temperature and pressure - and the flanges' surface finish and rigidity. The stiffer the gasket, the greater must be the initial seating force and hence door thickness.
Allowable stresses for studding materials are quoted in AS 1210, which stipulates that the core areas and not the effective stress areas should be used in stress calculations :

Screw size (mm) M8 M10 M12 M16 M20 M24 M30 M36 Core area (mm2) 32.8 52.3 76.2 144 225 324 519 759

Bridges, which must be weaker than the studs, are designed as simple beams with cross- sectional proportions of depth/width ≥ 3 if rectangular.
Supports

A horizontal pressure vessel (length L, diameter D mm) is commonly mounted on two saddle supports - more would result in static indeterminacy and difficulty in predicting the load distribution in the event of foundation settlement. Each support should extend at least 120o around and approximately √( 30D ) along the vessel [ BS 5500 ] in order to transmit the reaction gradually into the shell wall. One support is attached to the vessel to prevent axial movement, the other is not attached but merely supports the vessel's weight, thus permitting free longitudinal expansion of the vessel when thermal strains occur.
The safety of any artefact must be verified under all possible circumstances - not just in normal duty but also during manufacture, erection, test, aberrant service and so on. Under hydrostatic test for example, a pressure vessel is subjected to a superposition of loads - internal pressure plus bending due to the distributed weight w of shell and water charge. A simply supported beam model of the vessel indicates that the supports are optimally located at 0.207L from the ends, corresponding to bending moment magnitudes of 0.0214wL2 at both the centre and at the supports - but the model neglects possible distortion which may occur when concentrated loads are applied to the relatively thin shell of a pressure vessel. To lessen this possibility, supports should be situated within D/4 of the ends to take advantage of the stiffening afforded by the heads, although this location will lead to bending stresses larger than those arising from the optimum location.
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م/عبدالظاهرسعيد #8
 Seagmarsela بانتظار التفعيل بالبريد تاريخ التسجيل: Aug 2013 المشاركات: 1 معدل تقييم المستوى: 0  самаритянка 11.11.2011

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#9
 matilrora عضو جديد تاريخ التسجيل: Aug 2013 المشاركات: 1 معدل تقييم المستوى: 0  hello

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#10
 علي فؤاد ابراهيم عضو جديد تاريخ التسجيل: Dec 2014 المشاركات: 6 معدل تقييم المستوى: 0  رد: درس عن تصميم الcylinders

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