The influence of slenderness ratio and stress concentration in taps on load calculations to thermal expansion in U-shaped compensators of thermal network Текст научной статьи по специальности «Физика»

Journal of Siberian Federal University. Engineering & Technologies 1 (2015 8) 11-32

УДК 697.443.3

The Influence of Slenderness Ratio and Stress Concentration in Taps on Load Calculations to Thermal Expansion in U-shaped Compensators of Thermal Network

Yuri L. Lipovka, Vitaliy I. Belilovets and Alex Y. Lipovka*

Siberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia

Received 23.07.2014, received in revised form 21.10.2014, accepted 04.02.2015

This article is about calculations of U-shaped regions of water heat network. The universal calculation algorithm is used for various geometric schemes and simple self-compensating pipeline sections. The influence of slenderness ratio and stress concentration factor in the smooth curved bends on voltages and maximum permissible flight compensated shoulders of U-shaped regions for different geometric configurations was taken into account in this work.

Keywords: radial U-shaped compensator, radial compensators, self-compensation for thermal expansion of pipeline, slenderness ratio of pipe, heating network.

Влияние коэффициентов гибкости и концентрации напряжения в отводах на расчет нагрузки от температурных расширений в П-образных компенсаторах тепловой сети

Ю.Л. Липовка, В.И. Белиловец, А.Ю. Липовка

Сибирский федеральный университет Россия, 660041, Красноярск, Свободный, 79

В настоящей статье рассматривается вопрос расчета П-образных участков водяной тепловой сети на самокомпенсацию температурных расширений. Приведен универсальный алгоритм расчета, применимый для различных геометрических схем простых, плоских, неразрезных самокомпенсирующихся участков трубопровода (Г-, 2-, П-, Л-образные конфигурации). Изложено влияние коэффициента гибкости и коэффициента концентрации

© Siberian Federal University. All rights reserved Corresponding author E-mail address: alex.lipovka@gmail.com

напряжений в гнутых гладких отводах на значения напряжений и на предельно допустимый вылет компенсируемых плеч П-образных участков различных геометрических конфигураций.

Ключевые слова: радиальный П-образный компенсатор, радиальные компенсаторы, самокомпенсация температурных расширений трубопровода, коэффициент гибкости трубы, тепловая сеть.

Abstract

The most vulnerable elements in the radial compensators of pipeline are taps. The cross-sectional pipe wall ovalisation and increasing ductility in bending as compared with straight pipes appear in taps. Taps are pipeline's elements with sharp change in shape. This change leads to a concentration of additional stresses in these elements arising under the influence of forces, their flattened cross-section.

Calculation model radial compensator, based on the thermal expansion, can be represented as a rod frame with rigid corners. In this case the taps geometry, their flexibility and concentration of bending stresses in them are not taken into account. Differences in results with calculated model, which reflects features of taps, are relevant.

The article has three computational models U-shaped pipe sections with aboveground way laying in the horizontal plane. There are following assumptions: pillars are absolutely rigid, resistance to friction forces movable bearings in longitudinal thermal expansion of pipeline is not considered.

Materials and Methods

When calculating pipeline to compensate radial thermal expansion compensators are determined their dimensions, in which longitudinal bending stresses arising from the elastic deformation of pipe will not exceed the permissible values. Rod model (pipeline's length exceeds the outer diameter of more than an order of magnitude) is used as a design scheme of pipeline. n-shaped radial compensator is a simple (line called simple if it throughout its length from one to the other fixpoint has no branches) and a flat (line called flat if the center line is located in the same plane) plot calculated self-compensating pipe fixed between two fixed pillars. Computational model of this pipeline's section under the influence of stress is a statically indeterminate system. Statically indeterminate system is called such a system in which the action of arbitrary load not all longitudinal and transverse forces and moments can be found from the equations of equilibrium of a rigid body or a solids system. Additional equations, which should express conditions of strain compatibility system, are introduced for calculations. Statically indeterminate system is characterized with number of extra links that is the largest number of links that can be removed at the same time without disturbing the geometric immutability and immobility system. In order to obtain additional equations, it's necessary to select a base system. To achieve this goal n-defined statically indeterminate system is transformed into a statically determinate with removing unnecessary links from it. The resulting system is called statically determinate basic. Elimination of any links does not change internal forces and deformation of the system, if it makes additional forces and moments, which are the reaction dropped connections. Thus, if you apply a given load and response remote connections to basic system, this and considered systems will be equivalent. In considered system the directions of available hard links, including those relationships discarded in the transition to basic system, there can be no movement, and therefore in basic system moving in the

directions of dropped connections must be zero. And for this reactio n must have dropped connections strictly drfioed values. .Zero movement condition in the direction of any i-th connection of n-dropped on basisof the superposition principle has the form

A^OTAifc+AtfiAO, (1)

p(fc)

where Ajt - movement in the direction of (-th cemmunicatoon system caused by the reaction of the k-th connection;

AjF - movement in the directidn of i-th communication syetrm caused by the si multaneou s action of all extoenul load, tn tBe mechf d cf ccaktiof dorces k-th cocnecCion usunlly dencded Xk. With this notation, and in the powor of Hooke't law movemene d;. can be written as

Hk

= Sik*tc (2)

where 5jfc - siingle (cr speaific) moving in ^he; direceion of i-rh communication system cause d by the reaction of Xk a lt i( e. reaction, which aoincidf s wiih tire! direceion of Ah Intuit uniiyv. f^ulbs^ttiLtuttieeiut (2) into (1), we obtain

<5*3 + biF= 0. (3)

p(fc)

The physical meaning for equation (3): moving in 1he direction of basic systrm i-th dropped conneectiona is zero. Writing expreisions, similar to (3), for she cntire se1 of drojrji^d connectro ns, we obrain the system oC canonicnl equations force meihod, which can f e represented as a single equationc

I

Ci^^ic^'^/c + A;F) = 0- (4)

fCt)

The tnral numOdr of te^ttnes it determined ^i_"tiiL do; gf ne.^ of the sostrm'e nedundancy and does nnot drpend on its ipetifif charactfristice. Single movement s^y-fleunc is determined accordinng to next ioomula

= Yf

vM M,

<ec e = l d—j^^1 ' S5)

where f - single movement or the i-th direction, caused by a oingle exposure, npplted at the po int kt

Mt - btadrng mr men- from a single exposure, applied at the point i; Mk - bendingmoment due; to impactof the unit applied to the point k; IK - slenderness ratio elemeni; v - basic elemeni stiffness to elemenni's stiffne s s; l - integrable elemenf lengtle. The main problem of calculation for pipeline as a statically indeterminate system is formulated as follows: for a ginen geomutric sclieme, the temperature difference betweenn hot and cold pipeline cnd size of pipes constituting portion ia required to determine the effort and strain the sy stem. In

calculating the temperature effects on simple pipelines as the main unknown is usually accepted in forces and moments of extra links, and therefore, this method is called the method of calculating forces. According to the method forces, one of the fixed pillars calculated area is considered breakout, and it's applied to elastic deformation forces and bending moment replacing a dropped pillar.

To determine elastin fonces arising in pipelmt with thermal expansion, the authors use the method of the elastic center. This mnshod is well considered in [1,2] and presents one of modifications of the force methoe, whéch consists in the fact that all side coefficients of canonical equations (i.e., coefficients 5ik, in which i ^ k) become zero. TTliis is achieved by moving basic unknowns of fixpeint dropped into the elsstic center of gravity calculated of pipeline. The applidatioe poins eor basic unknowns is connected to the point of placing a dcopped pillar infinioely rigid hypothetical console. The pipeline's axis will be endowed with a certain distribution of the elastic mass proportional to iSs stiffness (Fig. 1).

Beedien momeet from forces of elastic detormation in any section of pipeline is determined with the fosmula

M = ty- yfPf - {x - x2)Py , (6)

whe ro x, y - cooadinates oC conside red secCion in the oaiginal coardinate sestem; x0c y0 - coofdcnr2es oC gravity center calculated elaftic pipeline section; Pf, Py - elastic forces calculated center line of pipeline.

The basis for caleulating 2he fotces of elastic deformalien Cas been put the Castigliano's theorem. Loadis static end srrain energy equal to wook of external forces

Ax =

au

dE'

au

Ay = dP'

ury

(7)

(8)

y

Fig. 1. Computational scheme U-shaped compensator to the method of elastic center

Note: Xl, X2, Mx - longitudinal and transverse forces and bending moment respectively; A - the application point force and moment, replacing the dropped fixed pillar; ynT - the point elastic center of gravity of the system with respect to coordinate; Px, Py - vectors elastic forces relative to the coordinate axes X and Y, respectively; x0, y0 - coordinates of the elastic center of gravity relative to the coordinate axes XX and Y, respectively.

here U - strain energy ;

P*, Py - the same, in the formula (6);

Ax, Ay - displacement for the application point of force in its direction along respective axes. Displacement Ax, Ay calculated thermal extensions

Ax = aAtLx , (9)

= aAtL* ' (10) where a - coefficient of linear thermal expansion; At - design temperature drop;

Lx, Ly - lengths of axial line section of prorections on axes. Values of thr basic unk nown Px = XC P- = JX-?

, (11)

Jxojyo Jxy 0

p (12)

Py _~T~J ~j 2dtitJ ' (12)

JxOjyO Jxy 0

where Ax, Ae - calculated thermal expansion of area under consideration conduit in the direction x and y axes, respectively;

Et - modulus of elasticity the pipe material at the calculation tempe rature; J - inertia moment o0 she cross sectio n of pipe wall

7 = 6e[Ai4-(Ai-2S)4], (13)

where DH - outer diameter of prpeline; 5 - pipe wall thickness;

Jx0, ./y0 - central inertia moments of the reduced length for the centerline of pipeline section

f yds y f ds

Jxo=Jyy Y~y°y JT ' (14)

— f yds y J ds }y0 = Jx Y~Xo fY1

(15)

here; x, y, x0, y0 - the same, in the formula (6); K - the same, in the formula (5);

Jxy00 - central centrifugal inertia reduced length for the centerline of pipeline section xyds f ds

ixyds i ds

JxyO = I -~a x0y0 I -¡^,

(16)

Stiffness reduction factor is introduced in integrating alo ng the curved pipeline sections, so here straights K = 1, and for curved K <1.

Elastic center coordinates can be determined from equations r ds

0 XT sv 0 K

n ds

0 y~K sx

^li-^ ■ (18) J K

where Sx , Sy - stark; i nn^rrtiiiEi mo ments oo reduced lebgth for the cente rli ne of calculated pipeline section relative to x and y nxes, respectively; Lnp - ^ecCuc;e;d lengt- oft ^xiai^]_i.ne seciion. Karman researches have shown thnt the curcnSure bends causes ovalisation tlieir original cross-seclional area itnc^ stiffnesc reduction. Flattening on initial sound secSiion causes essential changes in distribution ef bjeatdinv stress compased with bending soiid becms. To deteemine the coefficient of flexibelity Karmatni useal ties rneshy method, foUowed by c eotution art" Iht Rltz mrShod. The decision was ssceived in form of trigoaometric eerier. It is allrgr- lhai eesults obfained by T. Karman significantly diverged from the ^x.sp^i.^it^en^;^ nS^tn. l^olrt; tCr^ts ins the derivation of the coefficient of flexibility in order to msthematical ralculateonn Kato-an made a oumber of assumptions and, in particular, he

disregarded pipe radum relations to bend radum, considering this reletionslup vs a very small amount. He al so ded nyl ronside r displacement of the neutral laye r. Reported assumptions may be valid only for a relatively small pipe curves of" curvature, i.e. a aargm radius oh curved (4 - 5 outer diameter) [3]. At the momeni thy calculating problem of pipe elbows hae produee d many works. For example, in [4, 5] these ase analytical eolutions for cusve s pipe s. MaCr eials [6, 7] are devoted to solution curves of pipes using t-e finite rlement method.

Karman coefficiemt for cusved smooih retractem we calculate according to [8]

K = S/CV), (19)

where Kp - slenderness ratio, excluding c onstraint defo rmatio n e nds of the curved portion pipeline _ 1,65

kp _ r x,ino,s ' (20)

liere X - geometric characteristic flexibility re moval

4RS

A_<R-R)-' (21)

where R - radius of curvvture for tap;

Dh15 - tire same, in the formula (13); ce - dimensionless parameter

PR2

M _ 3-6^F7R-^x ' (22)

EpRn - S)S

whe re P - excess internal pressure in pipeline ; R - this same, in the formula (21); DH,5 - tine same, in the formula (13); Et - the same, in the formula (U);

C - coefficienr reflecting the unrasiness of deformation at tire rndr of curved element (tap), at X < 1,65 calculated with the formula

f = ■

^ „ r- .- ^r-

(23)

1 + ip1'5

here y - angulau parameter

p = V^R/l - s) , (24)

where rt - the same, in the formula (21)

Z»H,5 - tine in the formula (13)); if - central angle of tap in radians. Whan X > 1,6a value for <r is set equal to i.O. Slendemess ratio bent pipe with straight sections at the enda wtth X > 2i2 ls 1.0, while X < 22,f2 is celculated with formula (19).

When bending tap amies the influence of forces, flattened their cross section, there tire significant local stresses. If longitudinal stresses, calculated in the usual theora oibending, are denoted by c, then the marimum longitudinal stresres san ie riete rmined with tie formula

(7sn|t,: = igCi , (25)

where i0 - concentiatihh ratio of longitudinal ttresses in tap;

0,9

S ~ 2/. ' (26)

A /3

where X - the same, in the formula (20).

9-element model pipel ine sect ion

Cnlculntian algorithms to compensate for thermal expansion of pipeline sections certain geomerric configurarions are presented in references for design of heat networks in section strength calculations. If you submit a calculation algorithm based on using a certain number of standard elements that are building design model, tha aesulting algorithm is applied to self-compensating schemes pipelines of various geometric configurations. Further, consider the scheme of 9 elements. This circuit includes straight 5 pipes and 4 of the same smooth curved drainages random rotation angle. We give as an illustrative example n-shaped pipeline section between two fixed pillars. Mentally divide it into nine elements: five straight pipes and four identical taps. The following is an algorithm based on self-compensation for thermal expansion of sector and its design scheme. This algorithm may be applied to other design schemes by removing unnecessary components, changes in steering angle taps (Fig. 2).

Elements ll, l3, l5, l9 are parallel to the coordinate axis X.

Distance from ende of curved element (tap) from its? center of gravity in thd direction of the reference cootdinate system, according to design scheme of curved rlement (Fig. 3), deeermined by the formula

a = R

= re

r = R

' . V sm — s

if -f (resin

V

-cos

p\cos/?

' . P Sin — 2

V

2 57,296 2f v V

-—oss- ,

,2t9i6 m (p

57,296

( p p p\ cosP

sinB — ( 2sin----—os -)-57,—29n

H 2 2 57,292 " '

. P n (P b P P PVsinS nin — rcss — \nsin

n 2 57,296 rccs 2

57,296

(27)

(28) (29)

X

Fig. 2. Calculation scheme 9-element n -shaped pipeline section

Note: numbers on the diagram below to identify numbers of sections; y^r - elastic center of gravity; l1, l2, l3,..., t9 elements of calculated area; Pe Py - elastic resistance forces (basic unknowns); x0, y0 - distance —rom the center o— gravity and elastic axes.

Fig. 3. Calculation scheme curved element (tap)

Note: a,b,c,d - distance from ends of the curved element (tap) to its center of gravity; R - radius of tap's curvature; 9 - central angle of tap; p - chord angle curved element (tap) to its axis of ordinates.

d=R

<P e . ( ■ < < V\sinfS

i — cosp + I 2sin— — ^——cos—)-

t^K 2 57,296 2) p

S7,296

(30)

where engles (j>, /0 are measured in degrees.

Projection lengths taps op the coordinateaxei are defined as Per axle x

Lix = c + d, (31)

where c, d- tine same, in the formuln (27) and (28); / - tap) number (index takes values h, 2, 3, 4). Per axle y

Liy = a + b, (32)

where a, b - the same , in the formula (29) and (350); / - the same, in tne formule (27). The tength of the axial line for calculated total pipe section is determined by the formula

¿dp = l! + l3 + h + l7 + l9 ri 4Z0TB , (33)

whete S, it jfrrr, lln l9 - lsngth of straight pipes, corresponding calculation scheme (Fig. 2); l0TB - length axial line for tap

_ pR

lmB = 57,2!?)62 ' (34)

wRere <p, R tha oame, iin tulie; formulas (27) - (30); Of - the same, in the formula (19). Center coordinafes oe gravity of individunl according to calculation scneme area (Fig. 2)

wtih resp ec4 to 0he x-axis dnfined by the formulae Relative to the axis x

h

^ "2" ' (35)

xc2 = l1 + d, (36)

1

xx = ( + Llx + 2 liCo1(p , (37 )

xc4 = ( = L,+ + =co+(p + c, (38)

1

x+ = = + ^lx + (ico+p p + 2L + 2 ( 5 . (39)

x:c6 = (,= L+ + (ico+p + L2x + (5 -1 d , (40)

1

x+ = (,+ Lio + (ico+p + L2: + (5 + Lio +2 (7cosp , (41)

xc8 = p + Llx L l3coscp + L2x + /,5 -I- L3x + l7coscp + c, (42)

1

=c9 = li + cs L l3sosp + =2l + ls+ c3s 3 scosp f LAx + 22 lg, (43) RelaCive to the axis y

1 t

ycl =-llSinO° = 0 , (4 4)

yc2 = b (47) 1

bc3 = ¿íy + I2 hs mP, (46 )

3=c4 = ¿íy "t" I3SÍ01L + a, (47)

yc5 = ¿ly + liSÍS(p + ¿sy , (48)

yc6 c= Lly S- l3pinc + a, (49) 1

yc7 ■ Lly + l333nc - - l73in(cI (50)

ycs = Lly + Z3 pinc — l 7 p in. — a, (51)

yc9 = /3sin< - / 7s m< , (52)

whe re xc1 - xc9; ycl -yc9 - section elements;

/1, /3, /5, l i9 - length =or icnoi-ritispconcJ.^ng calculation scheme (Fig. 2)) elements; Lix - Ln - the same, in the formula (31); Aj,, L2c - the same, in the formula (f 2); a, be, c, d - the same, in the formulas (27) - (30); cp - the same, in the formulaf (27) - (30). Static inertia momento of length pipeHne c ts mponents in original coordinate system, according to calculation ichtme atea (Fig. 2!)), ase detorminnd by formulas Relative to cha axis x

II (53)

SX2 ~ tb3C2 , (54)

$x3 = ¿3yc3, (55)

Sx4 P0TByc4 , (56)

^x5 = 's^C (57)

Sx6 ^7TByc6 , (58)

Sx7 = hyC7. (59)

Sxs = ¿OTB>c8, (60)

Sx 9 = MTc9< (61) Relative to the axis y

Syl = llxcl , (62)

= IoHBxc2 , (63)

Sy3 = ItxC3, (64)

Syt — IOTB^C4 I (65)

SyS = I^BBc5^ (66)

Sy6 — ^oti, (67)

Sy7 — I?Xc 7- (68)

Sy8 — loTB-fcS, (69)

Syg = Igecg, (70)

where xc 1 - xc9; —1 -yc9 - section elements;

l1, l2, l3, l4, l5, l6, l7, l8, l9 - lenggth for correspo nding calculation scheme (Fig. 2) elements; l0TB - tlie same, in the formula (34). Coordinates elastic center of gravity relative to origin of the coordinate system according to calculation scheme area (Fig. 2) have the following form The axis x

Syl + Sy2 + Sy2 + Sy4 + Sy§ + Syfr + SyJ + SyQ + Syg

x0 = -j-, (71)

The axis>>

Sxl + Sx2 + Sx3 + Sx4 + Sx5 + Sx6 + Sx7 + Sx8 + Sxg ____

3o =-j---(72)

where - Sy9 - the sanTe, in the formulas (62) - (70); Sx1 - Sx9 - the saite, in tlie formulas (53) - (61); Lnp - tlie same, in the formula (33). Coefficients for calculating inertia moments of its own alements f and 7 of the calculation scheme area (Fig. 2) relative to initial coordinate system defined by tlie formulas The axis x

sin2m

Cxi =-^2-, (73) - 21 -

The axis y

cos2<p

Cyi = -J2-. (74)

where cp - tlie same;, in tlie formulas (27) - (30).

Factor to calculate the inertia moment of its own centrii—gal element 3 the calculation scheme area (Fig. 2) relative to the anitial coordinate system iscalculated by the formula

sinicosi

Qryl(3) = -12-, (75)

where p - the same, in the formulas (27) -(30).

Factor to salculate the inertia moment of its own centrifugal element 7 the calculation scheme area (Fig. 2) atiative ts the initial coordinate snstem is salculated ley the fotmula

_ smcpcosg) (76)

0*3/1(7) - dfe -

where cp - the same , in the formulns (27) - (30).

Coefficie nts for calculvting inertia moments of its own curvilinear elements (taps) relative to the initial coordinate sysCem defined by thue farmulas The axis x

( 4 r < (91 4 r < v

Cxr = I 5n,29S-sib9 - n siby I sib9ff + -siby n sn^S-sii9 — U , (77)

\ V 2 1 2 <2 2(5n,29S)

The axiay

( 4r<v t r l 4r< sV

Cyr = r5n,29SvsSib9<2-nSib<vJccs9m U -smsv n 57^,29S<iiiir^ u 2(5n"296) , )78)

whie;re p, is - the same , (ii fao Jfcs]rmiii^i3i!5> (27i - (niO)».

Coefficients for circultting ^r^ejnrtia moment of its; own centiifugai curvifinear elements 2 and 4 of the calculation scheme pection relative to the initial coordinate system is calculateg by the formula

I 4 r <v \ Cey9(r;4) = ti5зn,2ií)S)s^siiгr ^ — síb<vJ iriii/Jccs/?, (79)

where cp, f - tge sante, in the 4osmulas (27) - (30).

Coefficients for calculating inertia moment of its own centrifugal curvili ne ar elements 6 and 8 of the calcuiatian scheme recaton relative (o tho initial coordinate system is calculated by fhe formula

/ 4 r <0 g

Cgyi(e;r) = -a ^Er■/t,2<r><5 — o^iii'^ — sib((J ^ciic/dcitiamt?, (80)

whece cp^, if — the same, in ihe formulas (27) - (30).

Inertia moments of element lengths for pipeline in t—e original coocdinate syste m relative to the x-axis are deiem-ned by formulas

Tlie axis X

2 ¿I/I (0 ) , 2

, 2 sin2 (00) 21

Jxi = =1 [S —12)— + ,ci2 )- (81)

= 2 2 2 2 Ф \

Jx2=K\R С*2+Ус2 57296)'

(82)

Ухз = /з(/з2Сх1+Усз2)' (83)

R

- ( 2 R Ф \ Jx4 --¿{R CxR+y^4 gJ^J.

(84)

. 2 sinW) , , J=s = ls[( 52—УТ- -"—Ч (85)

ч 12

R

_ R2 2 2 < ) == — Í21 2(5-2 22 3/26 50C76J'

(86)

y'^yRt—C*1—yC7()- (87)

R

— R(( 6 < R

Лв Я (92+У20 (-—б6

(88)

Z 2 sín2(00) Л ilOo.; 1= l U2 s- — У29Ч (89)

where Jxl -1 - sectüen ellemontu;

,/1) - the s=m-s m fOe formulas (27) - (3O);

К- - the same, in the formula (19) ;

yc1 -J-- - t-e eame, in the formulai (4— - (52);

J, l( -( 0, S9 - length for corresponding calculation tcheme (Fig. 2) elemen=s; (1 - rhe same, in the formulas (27) - (3O); C= -1 -ie same, iie the farmala (73); — - tlie ^au^e, in -tlie eorrmüa (77). The axisi

( 2cos2(00) Л /yiR-Mi--+ x2i2)i (9O)

Jyi^^Cy--22 (9l)

/уз = h-h2Cyi + Xc(2)) (92)

R

- R1 ( 2 Ф \

(93)

(94)

(95)

(96)

(97)

(98)

whe re - Jy9 - section elemente;

R - the same, in the formulas (27) - (30); K - the samo, in ahe formula ((9t; xt - xc9 - this seme, in the foomulas (35) - (43); /i, h 1), I7i )9 - Jfji1 rOTresponding c^^^ntH-DHtiLoir schemt (Fig. 2) alements; cp - the same e in tho formuias (27) -(30); C, - the sense in the fonmula (7)« Cf2 - the same, in the formula (78) . Ct^nteerg^i^it^^:!1 inertia moments tor length of pipnline nomponents en the origcnal cooedinete system ((I^eiLtt. Is) ere deiermined n^ty formats

(99)

(100)

(101)

(102)

(103)

(104)

(10))

( 2 sin (0°)cos(0°) \ Jxy9 = *9 1*9 --+ *C9yC9 I , (107)

where Jxy1 - Jxn - section elements ;

R - the same, in the formulas (27) - (30); K - nhe same, in the formula (1 9); -a i - *f - the same, in the formulas (35) - (43); yc1 -y)9 - the same, in the formulas (44) - (52); T, S3, X, l7, l9 - for corresponding calculation sclieme (Fig. 2) elements; (f) - the same, in the formulas (27) - (30); C^p), Cgf - the same, in the formulas (75) and (76); Cnj,2(2.4t, CxJ,2(6.8) - the same, i nthe foimutas (79) and (80) . Central ineetia momenfs calxulated for total length of pipeline relative to the axis passing through the elastic center of gravity, according to calculation scheme area (Fig. 2) calculated by the formulas The axis x

9

Ao = ^Ai — ^npyo2' (108)

¿=1

The axis y

9

Zyo = h^- J--J;p:rC;2 , (109)

t=i

weere Ja - the same, in the formulas <Ci=»1) - (892)); Js - ihe ssme, in ahhe formulas (90) - (98); Lai - the same; in t-e fe (mula ((33); x0, ys - tltiis same, in tfe formulas (72) h (72). Centaal centriLlfiiih^ inertia moment total calculated pipaline section relative to the axis passing

through the elas-c cen-er of gravity, according to calculation scheme section (Fig. 2)is defined as

9

t-rr n 7xet — eer xoyr> (110)

1=1

where Jxyi - the same, in the; formulas (99)) - (117);

Lnp, x„, y0 - thte is ame, in Ihe formulas (g3 ), (7 1), (7 2). Catculaled Semperatura elor^g;((t;ion for total c^lc;u].aL^e;d. pipeline section are determined by fomudar

The axis x

Ax n £ii(9i — toXp -e e1c -e ifcosp + e2x —15 + e3x + i7cos<p + egx + I9S, (111)

The axis y

Ay = eA{t1 — t2){Lly + l3sin< + L2y — L3y — l7sin<p — LAy), (112)

where e - primary pipeline stretching coefficient;

A - linear thermal expansion of piping material at an estimated coolant temperature (pipe wall);

ti - calculated coolant temperature (pipe wall); t2 - installation temperature;

lb /3, l5, l7, l9 - fer corresponding calculation scheme (Fig. 2) elements; LRx - L4n -9 the same, in the formula (31); Lly, Thy - the same, in the formula (32); cp - the same, in the eormuhas (27) - (30). Distance from the gravity's center of the arc for the curved element (tap) to the center of curvature along the bisector (Fig. 3) is determined by the formula

2 sin%

p = 57,296-(R, (113)

V

Bending moments in cross sections at ends of linehr elements and in the middle taps according to calculation scheme area (Fig. 2) are determined as

M= = ~y0Px+ *oPy , (114)

M2 = -y0Px - = - *o)Py , (115)

M3 = — - (R - p)sin| - y°] Px - S?2 + d - (R - p(cos j- x-] Py , ( 1 16)

M4 = (^2- - yo)3x - (i + -ix - *o) Py - (1 17)

M5 = (L3y + l3Sincp - yo)Px - (/3 -I- Px* + (3Cos- - x0)Py , ( 11 8)

M 6 = (l31 + l4Pincp + a + (R — p^sm*-2- — 3/4] Px-

P 0 (119)

M 2 + Li* + l3cosp + - — (R — p)c-s p — x-X Iy ,

— = (L^ + l3sinp -Is L^,-; - R°(px -M(l= + L3i + liCnS(p JS L2x l0(P( >

—8 = (L + l3Sin<( + L2y y y0(Px -m (l= + + l3CnC— 1 L2* + _ x0Xpi <

M9 = [L3x + l-iP>i-7.(p + a + (R (( p)sin— y y-] cx -

( 1 2 2)

P

(120)

(121)

y [/3 + + liCos(p + L2x + / +d + (R — p)cnp — y x-- ] Pp i M30 = (L^ + l^m— — y0)P( —

— II3 + L3( 1 l3Cop(p + L2i l5 + l3i — x0)p: i

- 26-

-ll = ^y -I- hsincp — l7sin<p — y0)Px — — (il + Lix + lpC0S(p + L2x + ° + L3x + l7cos<p — x0)p

(124)

Ail2 = |ily = l-smp — l7sin<p — a — (R — p) — y0] Px —

( P -

— = Plx = l-c(osp = P2x = ls + P-x + l7cosp = c — (fl — p)cosp — x0- Ps

—l- = (l-sin(p - l7SiS(p - yo)Px -

-(ll + ¿lx < l-coS<p + p2x + l5 + L-x + l7coSty + p4x <

(125)

(126)

0)ry ,

—l4 = Pl-Sisp - ^sisp - y0)Px -

-(P — Zlx — l-COSp — Za^xx — l5 "I" ¿3x — l7c0SP — Z4x — l9 - x°)p

(127)

where Mi - M14 - numbers of sections according to calculation scheme (Fig. 2); P-, Py - the same, in the formulas (11) and (12); A, l3, /5, /7, l9 - for corresponding calculation scheme (Fig. 2) elements; £is - LAx - the same, in the formula (31); Liy, L2y - tlie same, in the formula (32); 9 - the same, in the formulas (27) - (30); p - the same, in the formula (113); x0, y0 - the same, in the formulas (71) and (72). Section modulus of the pipe wall defined as

where J - the same, in the formula (13); DH - the same, in the formula (13). Bending compensation voltage in the i-th section, according to calculation scheme area (Fig. 2), determined by the formula

Consider three computational model U-shaped compensators located in the horizontal plane and are not clamped by the ground. For each computational model variale is the radius of curvature tap. Please find enclosed characteristics of computational models below. In calculations of fixed points were considered to be absolutely rigid and do not take into account the resistance of friction forces movable pillars.

(128)

(129)

where Mt - the same, in the formulas (114) - (127); W - the same, in the formula (128).

Results of research for computational models of U-shaped compensators

Table 1. Characteristics of computational models U-shaped compensators

Calculated value Dimension Value

External pipe diameter / nominal wall thickness compensator number 1 m 159/4,5

Fly / back width compensator number 1 (excluding taps size) m 3/1,5

Allowable compensation voltage compensator number 1 MPa 146

External pipe diameter / nominal wall thickness compensator number 2 mm 219/6

Fly / back width compensator number 2 (excluding taps size) m 4/2

Allowable compensation voltage compensator number 2 MPa 154

External pipe diameter / nominal wall thickness compensator number 3 mm 426/7

Fly / back width compensator number 3 (excluding taps size) m 6/3

Alloweble compensation voltage compensator number 3 MPa 150

Pre-stretch coefficient nondimensional quantity 1

Calculated coolant temperatuee (the pipe wall) 0C 130

Installation temperature 0C -20

The elastic modulus ofpiping material at a working temperature MPa 196000

Excesdive internal pressure MPa 1,6

Linear thermal expansion of piping material at the estimated temperature of coolant (the pipe wall) mm/m0C 0,0125

Stoength reduction factor of weld joint action at any load other than the bending moment nondimensional quantity 1

Reduction factor of weld joint strength at bending moment nondimensional quantity 0,9

Rated allowable stress ofpiping material at the operating temperature MPa 140

W IK

is

I«

Voltage, «

]№

90 3t 10 60 50 41 30 20 10 0

^ rated, voltage

. allowable stress for lie section in the horizontal plane __ allowable stress for the sectionin the vertical plane

— rated voltage (ex eluding the impact of taps geometry: flexibility and coefficients of stress concentration!

1- -

4 - U—

t=* / \ -/-/— H r==l

VI 13 4 S i I t S a 11 U 15 U 15

Fig. 4. Graph of stress distribution compensator number 1 at radius of the outer diameter of tap pipe to one

1 h ' h H î

t b t y

Fig. 5. A graph ofvoltage compensator number 1 and number 2 at radius ofthreetap pipe outer diameters

Voltage, MPa

Section on length calculated area

Fig. 6. Graph of stress distribution compensator number 3 in radius of three tap pipe outer diameters

Allowable compensation voltage was determined in all cases [8].

Below are graphs of the stress distribution on calculated cross sections of these models U-shaped compensators.

If you pay attention, you can see taat the maximum voltage with and witUout consideration of coefficients are comparable in the first two computational models within approximately tap equal to three outer diameters of pipe. For the third calculation model similar graph is as follows

1

2

Fig. 7. A plot of the maximum total departure adjacent compensated shoulders the radius of tapis curvature for calculation model compensator number 1

Fig. 8. A plot of the maximum total departure adjacent compensated shoulders the radius of tops curvature for calculation model compensator number 2

It is evident that a similar comparison with the first model calculations on the graph above is not observed. Now pay attention to plots oa the maximum total departure from adtacent compensated shoulder radius of curvaturetaps.

These are graphs the first two computational models for compensators. It is seen that lines on graphs in both cases intersect at about 3.5 diameters (note markers on charts). Here is a chart of the third calculation model

Intersection happens here at 4 diameters. Therefore there is a lack of comparability graphs of stress distribution on cross sections of area.

Total length 3PQ rompe ns a ting shoulders, m ' ■

including coefficients flexibility taps and stress concentran on

excluding taps and flexibility coefficients of stress concentration

0 400 800 1200 1KB 2000 2400 2800 3200 Tapiadfrs, mm

Fig. 9. A plot of tlie maximum total departure adjacent compensated shoulders the radius of taps curvature for calculation model compensator number 3

Fig. 10. Plots of the maximum total departure adjacent compensated shoulders the radius of taps curvature for calculation model compensator number 3 with a thickness of 12 mm (left) and 7 mm (right)

If the third calculation model to increase the wall thickness from 7 to 12 mm, the point of intersection graphs of maximum total departure from adjacent compensated shoulder radius of curvature taps shifts to the origin

Findings

Chart analysis suggests the following conclusions:

1. Most unfavorable from the viewpoint of safety factors is the use of taps in which the radius of curvature is one outside diameter of pipe

2. Increasing the radius of curvature taps from one to two outer diameters increases the maximum departure compensated shoulderr twice.

3. In calculations coefficients of flexibility taps and stress concentrations should take into account because ignoring the latter leads to incorrect results.

4. In case of tap radius reduction and increasing the wall thickness of pipe in calculation model there is comparing the coefficients of flexibility and taps them stress concentration to unity.

References

[1] Николаев А.А. Справочник проектировщика. М.: Издательство литературы по строительству, 1965. 359 с.

[2] Лямин А.А., Скворцов А.А. Проектирование и расчет конструкций тепловых сетей. М.: Издательство литературы по строительству, 1965. 293 с.

[3] Камерштейн А.Г., Рождественский В.В., Ручимский М.Н. Расчет трубопроводов на прочность. М.: Государственное научно-техническое издательство нефтяной и горно-топливной литературы, 1963. 424 с.

[4] Orynyak S.A. Radchenko // International Journal of Solids and Structures 44 (2007) 14881510.

[5] Kolesnikov A.M. // Arch. Mech. Warszawa, 2011. 63, 5-6, pp. 507-516.

[6] Fonseca E.M.M., FJM. Q. de Melo and M.L.R. Madureira // International Journal of Manufacturing Science and Engineering / International Science Press July-December 2011. Vol. 2. No. 2. P. 109-114.

[7] Giordano A. and Guarracino F. // University of Naples "Federico II" - ITALY / 2002 ABAQUS Users' Conference.

[8] Нормы расчета на прочность трубопроводов тепловых сетей. РД 10-400-01. М.: Госгортехнадзор России, 2001. 45 с.