o - * - & éi_::: .i-flJ LASSIFIED e DE _. A noen To: QasHFCATION CHA CENTRAL ORNL=-2442 THERMAL STRESS ANALYSIS OF THE ART HEAT EXCHANGER CHANMNELS AND HEADER PIPES D. L. Platus CENTRAL RESEARCH LiBRARY DOCUMENT COLLECTION LIBRARY LOAN COPY DO NOT TRANSFER TO ANOTHER PERSON If you wish someone else to see this document, send in name with document and the library will arrange o loan. TSTARCH LIBRARY CULLECTION 2 cy. 774 C-84 - Reactors - Special AEC RESEARCH AND DEVELOPMENT REPORT of Aircraft Reactors M-3679 (20th ed., Rev.) eatures 3 OAK RIDGE NATIONAL LABORATORY operated by UNION CARBIDE CORPORATION for the U.S5. ATOMIC ENERGY COMMISSION :i A =& ‘&"-';;é?_ D ST ORNL -2442 C-84 - Reactors =~ Special Features of Aircraft Reactors M-3679 (20th ed., Rev.) This document consists of 38 pages. Copy 77 of 227 copies. Series A. Contract No. W~7405-eng-20 REACTOR PROJECTS DIVISION THERMAL. STRESS ANALYSIS OF THE ART HEAT EXCHANGER CHANNELS AND HEADER PIPES D. L. Platus DATE ISSUED fiflflp'4£1QEg - T g TENE M g "TGAK R.lDGE NATIQNAL LABORATORY i Qak Rldge Tennessee ____ m:,,-“ 1N £ 4 - 4 R s 05 fl i ”“rvfi%Wflwwmwwrw%fl,Q; 3 yy5kL 03kL22? ? Are TR AR ' b ogE e T e T, e . e . K AL AR oy e WA K g s SRR A T 5 m’c‘)r e i th T TS ST, ~ [, Tyt A Rt AN i oENX ) a"{?a«,‘é'«t- -"‘?"o,“"- S AT T e _ NOMENCLATURE Subscripts on Deflections: Szmbols: X3¥s2 Xp? Zp - H { D, I stényZ ép, én deflections due to in-plane bending of channel deflection due to out-of-plane bending of channel deflection due to rigid-body rotation of plane of channel about y-axis deflections due to relative thermal expansion of channel deflections due to deformations of header pipe rectangular coordinates coordinates of point b distance from origin to point b angle between negative x-axis and ob radius of curve of channel maximum radius of header pipe length of header pipe directions in xz-plane, parallel and normal to plane of curve, respectively deflections parallel to x~-, y-, and z- axes, respectively deflections in plane and normal to plane of curve, respectively §0, 88, &7 angular deflections with rotation vectors parallel to x-, y-, S Yp 28y and z- axes, respectively angular deflections with rotation vectors parallel to p and n, respectively forces parallel to x- and z- axes, respectively . . 4 0 — R e i) iy g g S T e B T R SN TR et R & T T NN : IR i M I BN AR e [ R VRS W Y et .- At e e et T o LT g s BRI R L R A T R T e F, N forces parallel to p and n, respectively X y M MffimrjMn mements tangent, radial, and normal to plane of curve at sz moments parailel to X-, y-, and z- axes,; respectively fa MN moments paralilliel to p and n, respectively equator; respactively MH resuttant bending moment acting on header pipe MHX9MHy x= and y= components of MH 69615@2 normal stresses GNpqn stresses due to ipn-piane bending of channel OpsC, gtresses due to out-of-plane bending of channel T shear stress or twisting stress C a functicn of the cress section used in calculating shear stress: K a function of the cross section used in calculating torsional rigidity Ip moment of inertia of channel cross section about an axis radia. to the curve IN moment of inertia of channel cross section about an axis normal. to plane of curve ZP section modulus of channel about an axis radial to the curve ZN section modulus of the channel about an axis normal to the curve IH moment of inertia of cross section of header pipe JH polar moment of inertia of cross section of header pipe v peisson's ratic E modusus of siasticity G shear modulus, G = §?f§$?7) By B ov==8sp coaefficients in set of linear algebraic equations a linear coefficient of expansion; angle describing direction of MH T temperature B R AR A ek S g " g . B et g - g W e o SErtD) L BN e e WD o Erssy o SRR > SRR SR Ll T e s g e a N g M; ey - o ,“'mtm ‘fi PN e ol -'.‘4&;‘,'\1" 3 THERMAL STRESS ANALYSIS OF THE ART HEAT EXCHANGER CHANNELS AND HEADER PIPES This report summarizes the study which was made to determine the stresses; deflections, and the forces and moments acting on the ART heat exchanger channels and header pipes due to relative thermal expan- sion between the channels and the pressure shell at full power operation. Introduction Figure 1 shows a sketch of a channel and header pipes, and a portion of the pressure shell to which they are connected. During full power operation the temperature of the channel will be higher than that of the pressure shell, and thereby produce relative thermal expansion. The resulting forces and moments will cause deformation of the channel, the header pipes, and the thermal sleeves which connect the pipes to the pres- sure shell, ‘The stresses due to these loads will be transmitted to the incoming NaK piping. Figure 2 shows the idealized system used for the analysis. The channel was treated as a semi-circular-arc curved-beam connected directly to the header pipes, which were treated as cantilever beams. This analysis assumes that there is no deformation in the incoming NaK piping, or in the - thermal sleeves. It is expected that this assumption will yield an ade~ quate initial estimate for the analysis of the channel. Because of symmetry it was sufficient to consider only one-half of the channel and one header pipe. The channel was assumed fixed at the mid-point and the deflections due to the relative thermal expansion were applied to the system. Elastic theory was assumed for all calculations. g e——————— —, e e T T o o TRl ek Gt S TR L AR e L o . St m e rneeierdy At TR ; 3 SRR G i IR N e e AT SRR AR ‘ . o crob AR R R R ) 1 FIG.{- SKETCH OF CHANNEL, HEAPER PIPES AND ORIENTATION I[N REACTOR INLET NaoK PIPE | \‘ I P . . LT ) O 2 O O . 4 ORNL-LR-DWG. 27494 CHANNEL c e = / PRESSURE < / SHELL 1] " / ) A % A SECTION A-A / ] / LINER (XI) SOUTH // | HEADER ; — 1 .. THERMAL SLEEVES OUTLET ‘ NaK PIPE T ORNL-LR-DWG. 27495 FIG.2— SKETCH BEAM STRUCTURE USED TO APPROXIMATE CHANNEL AND HEADERS g -5 “ YN s Thermal Expansions The total vertical expansion of the channel relative to the pressure shell was reported in ART Design Memo 8«D-5 as 82 mils. This was assumed to be distributed equally between the sections above and below the mid-point, so that 41 mils was the vertical deflection used in the calculations. For the radial expansion, the channel was assumed to be at an average temperature of 1&250F’and the pressure shell at 1240°F, Taking the radial position of the header pipes to be 19.59 inches, this gives a radial deflection of 32 mils. Method of Analysis Deflection equations were written to determine the forces and moments acting on the channel at point b, from the thermal expansions applied between points a and c. The coordinate system is shown in Fig. 3. Since the channel is free to grow radially, forces were not applied in the y-direction. The modes of deformation included in- plane and out-of-plane bending of the channel from flexure and torsion and deformation of the header pipe by flexure and torsion. The deflections for point b due to deformation and rotation of the channel may be expressed by the following equations, in which the sub- scripts refer to the modes of deflection. fx = SXP + SxN + éxr Jz = 8zP + SzN +>cSzr do = dep, + é‘eN §g = &gy + 4@, dr = §rp + Sy at wBm o w % (1) (2) (3) (1) (5) ORNL -LR-DWG. 27496 FIG.3- COORDINATE SYSTEM SHOWING FORCES AND MOMENTS ACTING ON CHANNEL Afi@,__il:]\EADER PIPE = Fo The relative thermal expansions applied between points a and c must be equal to the differences in the deflections of point b caused by deformations of the channel and those caused by deformations of the header pipe. Hence, the following relations may be written, in which the subscfipts T and H refer to the appiied relative therm@l expansions and the deflections due to deformation of the header pipe, réspectivelyl, Sxp + Sxy + 8%, - 8xy = Iy (6) Szp + Sz +dx, = - Iz, (7) Jep + S@N - 86, = 0 (8) Sgy, + 59 - gy = 0 (9) §rp +&7y =Sry = O (10) By expressing the deflections in Eqs (6) through (10) in terms of the loads acting on the channel at point b, a set of equations results from which these loads may be determined. Since the rigid body rotation of the plane of the channel about the y-axis is an unknown in addition to the five loads ij Fz’ Mk, My and Mz’ an additional equation is re- quired, and may be written by summing moments about the y-axis. EMy = M& +Fz -Fx =0 (11) Deflections From In-Plane Bending of Channel It is seen from Fig. 3; that the force and moment producing in- plane bending of the channel are P and MN° These may be resolved into forces and moments parallel to the coordinate axes. P = F_ sing - F, cos ¢ (12) My = M_sin ¢ + M, cos @ (13) 1. The applied relative thermal expansions are taken as positive for both the x and z directions, Note also that the deformation of the header pipe in the z direction has been neglected. ‘I 1 5 - Lag The deflections duvue to these loads with respect to the p~ and . 2 n- axes are given by 2 5 Pr MNI § r - == (14) P I EL; ~ EI D My - Pr L Y, = - BT, © 2 BT, (15) Resplving these deflections into components along the coordinate axes, §xp = ~&p cos ¢ (16) §z, = &p sin ¢ (17_ d’eP = dY_ sin ¢ (18) §rp = &Y, cos ¢ (19) Substituting Eqs (12) through (15) into Egs (16) through (18), 5 CSDXP =£f %E(Fx cos § - F, sin @) cos ¢ (20) + %TE (MX sin ¢ + M, cos Q) cos ¢ 2 c?zP = -iz-fif— (F, cos ¢ - F, sin @) sin ¢ N (21) o , +-E——I-§ (MX sin ¢ + M, cos @) sin ¢ 23 See Refo l’ Part l’ Ppa 7"’80 2 cSGP = EI (F, cos g - F_ sin @) sin ¢ (22) + g-—;’—N (M, sin @ + M cos @) sin ¢ 2 §7p = EIN (F, cos ¢ - F, sin @) cos ¢ (23) + g:_%fi (M, sin ¢ + M, cos @) cos ¢ Deflections From Out-of-Plane Bending of Channel The force and moments producing out-of-plane bending of the channel are N, MP’ and My. Resolving N and MP along the coordinate axes, N = F_sin g + F, cos @ (24) M, = M sin ¢ - M, cos ) (25) The deflections due to these loads with respect to the n, p, and y 5 axes are given by e Mre 11 Sn = N [EEI +6+_7T )GK] T’Cfi;*fif} ~5- [g ET—-G_K-(Q—— b 3, See Ref. 1, Part 2, pp. 13-16, . 5 . X . & -10- 1 1y M 1 SV ="1'\I'§'(E"“ 1; %MPI'(ET_ GK) - (EI "“é'fi) (27) P 2 r N 1 N1l M 1 v 1.1 dty = = [g BT, ~ (2 - é') fi{'] =z @IP "d'K) ty My GIP * E}"K’) (28) Resolving (Jn and § \}&) into components parallel to the coordinate axes §xy =dn sin ¢ (29) ‘ §2y =dn cos ¢ (30) §7y =3y, sin ¢ (31) foy = - Y, cos ¢ € ‘ Substituting Eqs (24) through (27) into Egs (28) through (32), Je; ="l:2'2"(Fxsin¢+F cos¢)[ E_];"" (2,.17) ] _ (33) i o L ‘ ’g(Mx cos¢=MZ sin @) “'fi%;'”“@fi)"”gMyr (f?[:;-"aj;f) Gy = 2w, sin g+, con 9) [P b+ (3 - ) | oin 6 I'2 1 1 - % (M_cos ¢ - M, sin ?) (fi'fl;' + ‘fi) sin ¢ (34) e T 1 Tyl . +_2—M.V[Q ;E"i; u(?m-z—) fi}Slnfd P wl]l= P am 3T 1‘2 1 1 _—é-(chosgj-MZs:.nQfl) ET1;+§'I€>C°S¢ 2 r T 1 my 1 _2_1My‘|:2 E'f;'(e'z)GK]°°S¢ + I‘2 1 1 Z’_(Fx 51n¢+FZ cos @) -E—I—I:-i--(-fi(—)s:_ngfi dry ] . 1 1 . _gr(MXcosgfi-Mz sin @) fi;+°§f(-> sin ¢ T 1 L . +§My(fifli-;-a{')51n¢ 2 r . 1 1 CSQN = - = (Fx sin @ + F, cos @) (fi; + ————GK> cos ¢ +{TTI'(MX cos § - M sin ) EJI‘-I-:--i--é]-i-(-) cos ¢ r 1 1 - 'fé’My (E——IP - Efi) cos ¢ Jzy, =r5(FX sin § + F_ cos ¢)\:%E%I-;+ (T - 2) & (35) (36) (37) Deflections From Rigid-Body Rotation of Channel About y-axis ‘ ; : - The x- and z- deflections from rigid-body rotation of the channel about the y-axis are given by ‘ Sz = b 8F_ cos @ (38) SXI_ = ob 5’¢r sin ¢ -(39) Deflections From Deformation of Header Pipe Since the loads acting on the channel are transmitted to the header pipe in the opposite directions, the deflections of the header pipe may be written in terms of these loads. 3 2 FJ M_{ ‘PXH - - E]I-H ( 3 ye) (k0) Jog = i (41) " ET, 1 FXFQ : Jf; = ;EI—H(—Q——-M},E) (42) " vy = 2—-; (43) H \ Solution of Deflection Equatipn§ Substituting Egs (20)#(£3), (33)-(37), (38), (39), and (40)-(L3) into Eqs (6)~(11) gives six equations in six unknowns. These can be written in terms of coefficients 84717 through 8,47 a8 follows: “ 4, Note that the distance from the origin to b has been denoted by ob instead of r. ©Since the channel is not c¢ircular, the actual distances have been used in Eqs (38) and (39) instead of an average radius. The discrepancy involved here is small since ©b = 24.1 inches and r has been taken as 21.9. ' A .;an -1%- a F o +ta, F o+ 813 M+ aq) My tagg M +a ¢r =£XT ayy Fo+ a5, Fz+a25 M, + a5 My+a25 M, + asg ¢r=..QS’ZT F + = a5l F‘X+a32 Z-&--:a.35 Mx+8'51+ M.Y a55 MZ 0 23,51Fx+a.52 Fz+sa.55Mx+a5lL My+a55 MZ = 0 a6l FX-+EJ.62 FZ +a61+ My = 0 7 where, 3 3 - 5[Tr 1 (3 )g_ .2 o a1 T ¥ FT *’(T“QGK Sln¢+gEI cos P + iy P N H a ~r51T—4]-'—+3T"-2~]-=—sin¢c§°¢-’gisin¢cos¢ 12 - 5 EI; "\I GK ~ EI, 2 2 r 1 1 r . a1335‘f°"‘2=’(§"j; +§)sin¢cos¢+fi-&-51n¢cos¢ 20y 1 A1 > r %1, T 7|2 "ET“‘(E’E)GR']Sln¢'2EI P H 2 2 r 1 1 2 r 2 8.15m-2- ifi*;+61-351n¢+fi§008 5 l o = 1 7t [E fi“*@g’.g) afi] sin ¢ cos ¢ - | BT sin ¢ cos ¢ 37 1 > p 8y, =T [E-E—I—+(-h—-2 'G"'I'{'] c052¢+g%f1;_-sin¢ 25 8ol 25 %06 31 52 23 5.5,4. 35 a1 B ahB i f i ° (_l__ ! 2 EIP d _ . r (...1_-_._ 2 EIP' kg mmy 2 1 2 r . 2 _GK> cos“f - E——IN51H ¢ o\l - (2 - 2) GK]COS 2 . r : -fi)sn.ngflcosgfl-fifl;sn.ngfcos{é 2 +'G]'I-T) sin¢cos¢+%—§ sin @ cos ¢ 2 1 2 2 + EK) cos ¢ - %‘T"" sin ¢ 1 . +(-}f) s:.n¢cos¢+gfi-§-—1; sin @ cos ¢ | 2 Ty 1 . fl - ( “'é")'é‘fi]sm@l“zEzH -(2-%1) (—:}Lfi] cos ¢ a—é—) tos ¢ ALY 8,5!4_ 55 861 6o 8.6,4. ] ] i i ] amw LY 1 1 4 i (EI ") ot Moy H H T, ( 1 ET, 1 I rd 1 d ( r (..___l 2 EIP gr (E__ + l’“) sin?p + T - cosld + - 1 Ip P - ék) sin ¢ 2 2 r 1 . 2 r 2 5 (-—-—-——E + C?K) sin“g + ETI\; cos“¢ 2 2 1 . 5 (ET" + Ef) sin @ cos @ - %T§ sin @ cos ¢ 1 1 . 7t 5%) sin @ cos ¢ +’gf E§E sin @ cos ¢ P 7%%) sin ¢ GK =16~ Results Equations (44) were solved with the aid of an IBM-650., The numerical data and values of the coefficients are given in Appendix A. The following results were obtained: ¥ F, = 505.9 Ibs. FZ = = L|-2607 le s Moo= - 5462 in-1bs. (45) My = 391 in-lbs, Moo= - 6620 in-lbs. ¢r = - %,L68x107° radian Calculation of Deflections The y~deflection at b relative to point a can be calculated from > the loads producing in-plane bending”. 2 dy = pr’ (]21 ) 1) el (46) EEIN EIN Substituting P and M from Egs (12) and (13) into Eq (46), N r5 TE &y = - EETE (Fx cos §§ =~ F, sin @) - (g - %) §T§ (M.X sin @ + MZ cos @) (L47) 5. See Ref. 1, pp. 7-8. S SN ~17- Using the values (L45) for the forces and moments in Eq (L47), 8y = - 0.0373 in. This result indicates that the channel will expand 37 mils towards Shell V, at the equator; in addition to the free relative thermal expansion reported in ART Design Memo 8-D-5. | Since there are no forces on the channel in the y-direction, the y- deflection at b, relative to the actual coordinate system, is zero. The x-deflection at b can be calculated by summing the in-plane and cut-of-plane x-deflections for the channel, or from the deflection of the header pipe. Taking the latter, and using Eq (40), 1 [F 2 u g x = - o= X .= - 0.00299 in. BT 3 5 The angular deflections at b can be calculated from the deformation cf the header pipe, using Eqs (41), (42), and (43). M x -3 Se - &1 1.785x10"° rad. F f° , 1 _ 8¢) = *E-"f-}-I' —— - M‘YO = 5.657x10 rad. M { &y = - % = 2.813x107° rad. GdH Stresses in Channel at Header The stresses in the channel at point b (Fig. 2) can be calculated from the moments, M. My, and My. From Eqs (12), (13), (24), and (25), with the values from (L495), -18- P = F sing¢ - F_ cos g - 708.5 1bs. ] My = M_sin g+ M, cos @ - 8563 1in-lbs, N = F_sin¢g + F, cos @ = 16.22 1bs. i = 572 in-le - il MP . sin ¢ - Mx cos‘¢ The maximum bending stresses due to MN and MP are calculated as follows: - +- --M-E- e + 3 op - + 83 psi P Oy = - E§ = 8234 psi The maximum shear stress due to My may be estimated by an 6 approximate method , This stress occurs at or near the inside corner of the channel. (See Fig. 4) MyC : Trax = ) wWhere K = a function of the cross section of the chanmnel used in calculating the torsional rigidity7 i 0.09295 inF ___._._E:;ZD {1 + [0,118 1n (1 - %) - 0.238 -21-); tan h %-g} (49) 1+ 5 Q i See Ref. 2, pp. 170-181, 7. The evaluation of K is given in Appendix B. w where D = diameter of the largest circle inscribed in the cross . section = 0.45 in. r = radius of curvature of the boundary at the point . (negative when Eq 49 is used) = - 0.110 in, A = area of the section = 2.582 in? ¢ = angle through which a tangent to the boundary rotates in turning or traveling around the reentrant portion, measured in radians = TI/2 For these values, ¢ = 0.8612 in. Using Eq (48) with the above values of C and K, T = 3623 psi max Stresses in Channel at Equator The moments tangent, radial, and normal to the curve at the equator may be expressed from equilibrium conditions, (see Figs. % and 5). M, = MP + Nr = 217.2 in-lbs. - = It - M = - 391 in-1bs. = i o= - MN +lPr'= - 6954 in-lbs. The maximum bending stresses due to Mr and Mh given by o =t = * 57 psi - 6687 psi Q i | li ORNL -LR-DWG. 27497 FIG.4-CROSS-SECTION OF CHANNEL ORNL -LR-DWG. 27498 M FIG.5-SKETCH SHOWING PRINCIPAL MOMENTS ) ON CHANNEL AT EQUATOR g TS Nl TR a. L "'3 .lfi‘i: i The meximum shear stress due to M, can be calculated by Eq (48). t Using the same C/K ratio as before, tfiax = 2013 psi . Stresses in Header Pipe Bending stresses are produced in the header pipe from Fx’ MX and Mya Referring to Fig. 3 the moment in the y-direction at the fixed end of the pipe due to Fx and My is given by MHy = M -F f = - 385% in-1bs. The moment in the x-direction at the fixed end 1is MHx = Mx = - 5)4-62 in~1bs, Taking the vector sum of these moments gives a maximum bending ¥ -'r'-.z‘;\z' moment, MH = - 6684 in-1bs. The maximum bending stress due to this moment is given by — H . o, =1 — = % 6282 psi . H The maximum shear stress due to Mz is given by Mer T = — = - 3111 psi . max JH -22- - R, - ey B i Bk oo % b . : Thermal stresses are produced in the header pipe from radial and axial temperature gradients, in addition to those caused by relative thermal expansion of the channel. Figure 6 shows the north and south headers and pipes with the approximate temperatures of the surrounding fluids. The axial temperature gradients should be small except in the regions close to the headers which are subjected to cross-flow from the fuel. Because of the azimuthal variation in heat transfer coefficients and the complicated geometry of these regions, it is possible that thermal cycling of these small sections of pipe could occur. Although these temperature fluctuations would be difficult to calculate, small thermal shields welded to the headers might be warranted., Because the layer of fuel surrounding the header pipes over most of their lengths is very thin, an estimate of the radial temperature profiles may be calculated using simple conduction with semi-infinite slab geometry. With the temperatures indicated in Fig. 6, these cal- culations give temperature differences across the walls of 230F and 52?F for the north and south header pipes, respectively. The stresses in the outside walls due to these gradients can be calculated by QEAT =TT 43) % X L-v ( giving - 4574 psi for the north header pipe, and 6176 psi for the south. The bending and twisting stresses may now be combined with the stresses due to the radial temperature gradient in the header pipe in order to get the maximum normal and shear stresses, Figure 7 shows the fixed end of the header pipe, the direction of the resultant bending moment, and the bending stresses due to MH' Since the twisting stresses 23 + 1I500°F NaK ORNL -LR-DWG. 27499 FIG.6 ~SKETCH SHOWING NORTH AND SOUTH HEADER PIPES APPROXIMATE FLUID TEMPERATURES TENSION MAXIMUM STRESSES FOR SOUTH HEADER | PIPE X N —" 2 COMPRESSON > ORNL -LR-DWG. 27500 MAXIMUM STRESSES FOR NORTH HEADER PIPE FIG.7- FIXED END OF HEADER PIPE SHOWING BENDING STRESSES AND ELEMENTS: AT2AWHICH MAXIMUM STRESSES OCCUR -2l and temperatuniafggfifnt stresses are uniformly distributed around the header pipe, the direction of MH determines the location of the maximum stresses in the pipe. For the north header pipe, the temperature gradient stresses are compressive at the outside wall, so the maximum stresses occur where the bending stress is a maximum in compression, This element is at the outside wall, 900 clockwise from the direction of - MH and an angle @ counterclockwise frbm the positive y-axis, where For the south header, the temperature gradient stresses are tensile at the outer wall, so the maximum stresses occur where the bending stress is a maximum in tension. This point is 180° from the point of maximum stress in the north header pipe, or an angle @ counter clock- wise from the negative y-axis, when the south header pipe is oriented as in Fig. 7 (eg the z-axis is directed away from the equator).. The maximum stresses can be calculated for the north and south header pipes from the combined stresses acting on the elements in the outer walls of the header pipes, as shown in Fig. 7. The stresses acting on these elements are shown in Fig. 8. The maximum normal and shear stresses are given by + F (4 o, - 0O Qjmax: (12 2) *'”L2 (45) ORNL -LR-DWG. 27501 T,= -11,258 __l_, T ==3 G, = —~4574 ‘_T_‘ 5, = 12,860 7=~ 31 G, =6176 FIG.8 ~ELEMENTS UNDER MAXIMUM STRESS IN OUTSIDE WALL OF (a) NORTH HEADER PIPE; (b) SOUTH HEADER PIPE (STRESSES IN PSi) . v -26-~ For the north header pipe, o max T, max i For the south header pipe, Q i max S § max - 12,480 psi 4565 psi. 14,080 psi 4565 psi. References D. L. Platus and B. L. Greenstreet, "Deflection Equations for Various Loadings of Circular-Arc Curved Beams", CF 57-4-G6, R. J. Roark, Formulas for Stresses and Strain, Third Edition, 1954, pp 170-181. S. Timoshenko and J. N. Goodier, Theory of Elasticity, Second Edition, 1951, p. 287. S. Timoshenko, Strength of Materials, Part II, 1941, pp 270-271. 28~ L e b r T “;; 3 - _i‘ -“ - Appendix A Numerical Data . 6 . X, = - 19.59 in. G = 5.8x10° psi Z, = 14.08 in. IP = 20.72 inh r = 21.9 in. Iy = 2.46 J‘.n)+ ob = 24,125 in. K = 0.09295uinu / = 7.50 én. Ip = 1,53 inu 2-1/2 inch E = 15x10° psi Iy = 3,06 in Sch 40 Pipe Y, = 0.3 SXT = 0.032 in. QS‘ZT = 0.0ll-l ine Numerical Values of Coefficients Used in Egs (b&4) a;; = 2558°7x10-§ 8y, = 3212.6x10'6 a = 3012.9x10" " a,. = u693.5x10‘6 12 6 PP 6 a = - 206,19x10 a., = = 299,76x10" 13 6 23 6 a = - 112,53x10 a, = - 154,86x10 14 6 ol _6 815 = 161.18x10 8,5 = 206,19x10 ag = 14.079 s = 19.59 8 = - 206.19x10"6 31 _6 a = - 299,88x10 52 -6 a = 21.173%x10 33 6 a = 16.55%x10 3l 6 g5 = - 15.787x10 — Lt -29- 81 Blyo auB nhn 8..)4_5 %46 261 B¢o g6l - 112.55x10'6 a _6 o1 - 154,88x10 8o 6 2 16-55X10 8.55 52.u6hx10': 8c), - 11.896x10 8 1.0000 14.08 19.59 1.0000 il Il 161.18x10" 6 206.17x10'6 - lh.787x10-6 - ll.896x10-6 ll.986)n:.10_6 vt i s Appendix B Evaluation of Torsional Rigidity Factor, K For a narrow rectangular beam of length b and width c, K can be approximated using the membrane analogy to give K = %bc2 (B1) 9 Similarly, for a narrow trapezoidal section”, 1 2 2 K = i§-bl(cl + C2)(Cl + c2) (B2) . For a cross=-section built up of narrow elements, K can be approximated by summing the K's for the individual elementslo. Thus, for the channel (Fig. 4), by summing two trapezoidal sections and one rectangle, Zvc” + b (c) + )+ c5) (B3) 8. See Ref. 4, p. 270. 9. See Ref. 4, p. 271. r lO. See Refo 5, po 2870 M\ -%1- Using Eq (B3) with - b = 5.00 ino Cl = O-lO ino - c = 001_25 in. C2 = 0050 in. bl = 3,45 in, 0.09295 inh -~ i - W00~ vin B~ po = - - o i O c..u::«:m_*uzzb:bmwnmzww;‘g—.?pr‘cr_.:npd?dpcpummn:>mhjuu 81, 82-84. 85-86. 87. 88. 89. 90-92. C-84 INTERNAL DISTRIBUTION Amos Billington Blankenship Blizard Boch Borkowski Boyd Breeding Briggs Cardwell Center (K-25) Charpie Cotton Culler DeVan Douglas Emlet (K-25) Ferguson Fraas Frye, Jr. Furgerson Gray Greenstreet . Grimes Guth S. Harrill L. Heestand W. Hoffman Hollaender Householder Jordan Kasten Keilholtz Kelley Kerrebrock Lane - = DA DY RO KPP G O = m o 37. 38. 39. 40, 4]1. 42. 43. 44, 45, 46. 47. 48. 49. 50. 51. 52. 53. 54, 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65-67. 68-75. ., 76, 77-79, 80. EXTERNAL DISTRIBUTION Division of Research and Development, AEC, ORO Air Force Ballistic Missile Division AFPR, Boeing, Seattle AFPR, Boeing, Wichita AFPR, Curtiss-Wright, Clifton AFPR, Douglas, Long Beach AFPR, Douglas, Santa Monica ORNL-2442 T - Reactors - Special Features e of Aircraft Reactors M-3679 (20th ed., Rev.) Livingston MacPherson MacPherson Manly McNally Meghreblian Morgan Murphy Murray (Y-12) Nelson atriarca Perry Savage Savolainen Schultheiss Senn Shipley Skinner Snell Stanek . Swartout Taboada . Tallackson Taylor Trauger Weinberg Winters Yarosh OR - Y-12 Technical Library, Dccument ‘Referencde Section - Laboratory Records Department Laboratory Records ORNL R.C. 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