Analysis of Cylindrical Tanks with Flat Bases by Moment Distribution Methods

Transcription

1 May, Anaysis of Cyindrica Tanks with Fat Bases by Moment Distribution Methods T. SyopSis by AminGhai, HE moment distribution method is used to find the moments and the ring tension in the was and the bases of the foowing two types of cyindrica tank : (a) Cyindrica tank on a rigid fat foundation. (b) Cyindrica tank with fat base supported on a cyindrica shaft of smaer diameter. For the first type a tria and error method is used to determine the width of the ringshaped part of the base which wi ift up from the foundation surface. n the second type the variation of the thickness in the overhanging part of the base is taken into account. Graphs are presented in the appendix to faciitate the soution. The design of each type is iustrated by a numerica exampe. M.Sc., Ph.D. n order to appy the method we need to compute the fixedend moments, the stiffness and the carryover factors for each eement. This method of moment distribution was used by Mtirkus,Gy.* in Hungary and by Lavery, J. H.2 in Austraia for the anaysis of certain types of cyindrica tanks. Some of the data given by MBrkus wi be used in this study. 3. Stiffness, CarryoverFactorsandFixedend Moments in Cyindrica Was (a) Stiffness The moment which causes unit rotation at a hinged end of a cyindrica wa varies according to the different conditions of support at the far end. n the foowing the stiffness factors are given for the three cases shown in Fig.. t is convenient to express the stiffness S 2. ntroduction The usua procedure of the moment distribution method of Hardy Cross coud be used to take account of the continuity of the was of cyindrica tanks with their roofs or bases. A vertica eement of the wa is considered together with a radia eement of the roof or the base. The method invoves the cacuation of moments at the ends of the eements under artificia conditions of restraint, then a distribution of unbaanced moments by arithmetica proportion when the artificia restraints are removed. The fixedend moments per unit ength deveoped at the edge of the cyindrica wa due to the iquid pressure, and those deveoped at the edge of the circuar pate are determined, the unbaanced moment is distributed between the connecting eements in proportion to their stiffness. The term stiffness here means the moment needed at the end of the cyindrica wa or the pate produce to unit rotation of this end. Aso, if a moment is distributed to one end of the cyinder (or the pate) whie the other end is hed fixed, a fraction of the distributed moment is carried over to the fixed end of the cyinder (or the pate). * The index numbers refer to the items in the ist of references at the end of the artice Fig. terms in of E where E the is 2 ( p) moduus of easticity and p Poisson s ratio. n other words the moment which causes a unit rotation at an edge of stiffness S is equa to S. 2 ( p) For the three cases of Fig. we have : case a (Fig. a) sinh PH cosh PH sin PH cos PH S = 2pt3 ( cos2 PH + cosh2 PH case b (Fig. b) S = 2pt3 case c (Fig. C) S = 2pt3 E sin (3H cos PH sinh ph sin2 P H sinh2 PH sin2 PH + sinh2 ph sinh PH cosh PH cos PH

2 66 The Stmctura Engineer t is the wa thickness and H its height. is a dimensioness factor. The term QH equation in the x, y, z system of coordinates PH = where R is the wa radius... (4) For cyindrica was having big vaues of PH ( QH> x), the terms between brackets in equations (), (2) and (3) tend to unity. The stiffness at one end wi be the same whatever the conditions at the other end, and wi be equa to S = 2pt3.... (5) Most eevated tanks as we as some grounded tanks have dimensions which give vaues of ph> x, and the stiffness of the was can be easiy cacuated by equation (5). (b) Carryover Factor f a moment is appied at the edge of a cyinder whie the other edge is fixed (Fig. b), a fraction of this moment wi be carriedover to the fixed edge. The ratio between the appied moment and the moment deveoped at the far fixed edge is the " carryover factor." This factor depends upon the vaue PH, it may be a fraction with positive or negative sign. n Tabe beow the carryover factors are given for vaues of PH between and 6. Tabe.Carryover Factors for Cyindrica Was where W is the defection ; q the intensity of oading ; D is the fexura rigidity Ed3 D= 2 ( $)... (8) in which d is the pate thickness. This differentia equation (7) can be expressed in a poar system of coordinates in which the centre of the pate coincides with the origin of the system. n the case of circuar symmetrica oading the differentia equation wi be, d4w 2 d3w d2w dw ~ + ~ ~ ~ d, z + $ d, = D ' w in this equation indicates the defection of a points which ie on a circe of radius r. The soution of the differentia equation (9) in its genera form is A, Az, A3 and A4 are the integration constants which are to be determined from the edge conditions of the pate. The principa moments wi be acting in a radia direction Mr, and in a tangentia direction Mt. Their vaues per unit ength are : ' (9) andmt=d(+ dw Y dr p) d2w For ong cyinders (ph > x), the carryover factor is very sma, which means that a moment appied at one edge dies before it reaches the other end. (c) Fixedend Moments The fixing moment at the bottom of a cyindrica wa having its top edge free, and fied with iquid of specific gravity y, coud be expressed by the reation : There exist tabes and curves, 3, 4, 5 and 6 which give the moment as we as the ring tension in cyindrica was fixed at their bases and subjected to trianguar oading, with the top edge under various conditions. 4. Stiffness, CarryoverFactorsandFixedend Moments in Circuar Pates The bending of a circuar pate oaded symmetricay with respect to its centre has been exhaustivey treated by many authors (see for exampe references 2, 7, 8 and 9. With the usua assumptions cmsidered in the eastic theory of pates, it coud be shown that the defection of the pate is governed by the differentia 4 ** The term.368 = 43( pa), in which is taken equa to + * The term to /3 ( p:) in which p. is taken equa When a circuar pate is buit continuous with a cyindrica wa or with another ringshaped sab, the radia moments in the wa and the sab per unit ength of their common edge must be equa. Aso, the rotation in the radia direction (g) of the edge of the pate must be equa to the rotation at the edge of the cyinder. Simiar to what was considered when deaing with cyindrica was, the stiffness at the edge of a circuar pate is equa to the radia moment appied at this edge to et it rotate a rotation cquas unity. f a radia moment (Mr) is appied at edge of a ring shaped sab of radii a and b (Fig. 2), whie the P Fig. 2 other edge 2 being fixed, a fraction of this moment (Mr2) wi be carried over to the fixed edge. The

3 May, ratio between the appied radia moment per unit ength and the corresponding vaue of the moment deveoped at the far edge is the carryover factor f2. The carryover factor depends upon the ratio between the two radii, and may be a vaue ess or more than unity. The radia moments at a fixed edge due to a oading on the sab are caed the fixedend moments. Tabes and graphs are avaiabe, 2 and 8, which give the fixedend moments, stiffness and carryover factors in circuar and ringshaped sabs for various oadings and edge conditions. These were obtained from the basic equations (o), () by choosing the integration constants which suit the edge conditions. The vaues given in tabe 2 beow were cacuated by Mhrkus (S), they are given here since they wi be used ater in this discussion. and the base may cause a ringshaped part of the base near its edge to bend as shown in Fig. 3, whereas the inside circuar part of the base may remain fat. This is a noninear probem as regarding the bending of the circuar pate, the conditions at the edges of the deformed part of the base are changing with the defection, and the deformation of the pate wi not be proportiona to the oad appied on it. The stiffness of the defected ringshaped part of the base depends upon the dimension b (Fig. 3), and this depends upon the unknown moments which are deveoped at the edge of the base. A tria and error method wi be used here ; first the radius b wibe assumed and then corrected to satisfy the conditions of the probem. These conditions are that at a circe of radius b, the defection, the moment and the sope of the defected surface are zero. The defected Tabe 2.Stiffness, Carryover Factors and Fixedend Moments in Ringshaped Sabs of constant thickness (Fig. 2) T Outer radius Stiffness at edge Fixedend moments due nner radius E d3 E to a distributed oad Q Carryover _ a factor M fi2 a in terms of b b M2 in terms of T b q b2 a 4 a2.o o Sign Convention for Moments Any system of signs coud be foowed, simiar to those used in the usua moment distribution anaysis of beams or frames. n this discussion, because of symmetry, the anaysis is carried out on one haf of the structure, namey the eft haf. An externa moment appied at the end of an eement is positive when it tends to rotate this end in the cockwise direction. 6. Variation of the Momentsawayfromthe Edges The moment distribution serves to cacuate the continuity moments at the intersection of circuar sabs with cyindrica was or between eements of ringshaped sabs supported aong annuar rings. Starting from these moments the variation of stresses throughouthe cyinder or the sab can be easiy cacuated. Data is avaiabe, and 4, which give the moments and ring tension at different heights in cyindrica was subjected to trianguar oading as we as for radia moments appied at the edge. Aso, there exist tabes and curves, 2 and S, for the radia and the tangentia moments at different radii of a ringshaped sab due to a distributed oad on the sab or due to radia moments appied at the edges. The fina ring tension and moments in the wa or the sab can be obtained by superposition. 7. CyindricaTank on RigidFatFoundation When a cyindrica tank is constructed on an absoutey rigid foundationsuch as soid rockor if the tank is constructed on a thick stiff pain concrete footing, and the wa is buit monoithic with the base, the continuity moment between the wa +&4 4 Fig.3.Cyindricatank on a rigidfoundation part of the base can be considered as if totay fixed at the circe of radius b, but oaded in a manner that the radia moment at the fixation is zero. This may be seen by summing the two moment diagrams for the two oadings shown in Figs. 4a and b. A right assumption of the distance b shoud satisfy the condition that the moment at radius b is zero. This can be quicky checked and the assumption modified unti the right vaue of b is reached. Not more than three trias shoud be necessary. The procedure of cacuation is fuy expained during the soution of the foowing numerica exampe. ExampLe : (a) Given Data Tank diameter 2R = 4' ; tank height. H..= 6' ; foor thickness d = " ; wa thickness t = " ;

4 68 The Structwa Engineer (d) Carryover Factor This factor wi be used to find the radia moment carried over to the inner edge of the ring sab (at radius b) when a moment is appied at the outer edge a from Tabe 2 for =.25 f2 =.584 b (e) Fixedend Moments wa : By substituting in equation (6) we get Fig. 4.Radia bending moment diagram on a circuar ring sab. weight of iquid = 62.5 b./ft.3 ; weight of concrete = 5 b./ft.3 ; Poisson s ratio p = 6 The tank is assumed to be paced on an absoutey rigid fat foundation. (b) Loads The bent part of the base pate wi be oaded downwards by the weight of the iquid pus its own weight Q 62.5 X X 2 = 25 b./ft. (c) Stiffness Factors.368H.368 x 6.4 wa = /2 x.83 = 5.25 > x 4Ft 5 25.*. sz = 2Pt3 = 2 X L.. X.833 = base : n order to estimate the stiffness of the ring (part 2 3, Fig. 5), the dimension b must be assumed x 2 x.83 x 6 M2 = (&) base : By interpoation a b =.25 and =.8 b a = 393 b.ft./ft. from Tabe 2 we get M23 = 25 X 62 X.53 = 525 b.ft./ft. M32 = + 25 X 22 X.35 = b.ft./ft. The distribution procedure is shown in Fig. 6a. The moment obtained at radius b = b.ft./ft., which shoud be zero if the assumption of the radius b was correct. For the second tria we take b = 8. A simiar cacuation wi give the moments shown in Fig. 6b, with the moment at radius b = b.ft./ft. Hence the right vaue of b, which gives zero moment at radius b, must ie between 6 and 8. A reasonabe vaue to be assumed for the next tria may be taken by making inear interpoation between the two previous trias, this gives ; 733 b = 6 + (86) = The moment distribution for the third tria with b = 6.8 is done in Fig. 6c, the corresponding vaue of the radia moment at b = b.ft./ft., which is very sma, and no more trias need to be considered. The variation of ring tension and moments are obtained by the hep of graphs from references and 8, Y Fig. 5 As a first tria we take b =.8a = 6. a 2 b 6.25 By interpoation from Tabe 2 we get the vaues are tabuated beow. Diagrams for the ring tension and the bending moments are shown in Fig. 7. t may be seen from this exampe that the base is subjected to radia moments ony near its outer edge, and they diminish very quicky away from the edge. Hence it is possibe to construct the midde part of the base with a reduced thickness. We can aso see

5 May, Tabe3.RingTension and BendingMoments in the Was distance from top edge '.2H.4H.6H due to iquid pressure on a cyindrica.: B wa hinged at the bottom 2 a due to restraining moment29 b. at the bottom tota (b./ft.),,, due to iquid pressure on a cyindrica wa hinged at the bottom H r" W. a due to restraining moment29 the bottom tota (b.ft./ft.) b. at that the tangentia moments in the base are sma and need not be considered. 8. Cyindrica Tank with Fat Base Supported on Cyindrica Shaft of SmaerDiameter Water towers of medium capacities are often made of the type shown in Fig. 8. The fat base may be fi %F "h fb) second +,h/ b (8' (bzeio) Fig. 6.Yament distribution : Soution of Exampe. preferred to other types because of the simpicity of its shuttering and construction. A considerabe reduction of the stresses in the base is achieved by taking the diameter of the supporting shaft ess than that of the tank wa. A suitabe choice of the overhanging ength is necessary to obtain vaues of the positive and negative radia moments which require the minimum base thickness. n the foowing this type of tank wi be anaysed by a moment distribution method. The joint between the tank wa and the base (joint 2, Fig. 8) can rotate and aso can move downwards. A direct moment distribution cacuation coud be appied if the stiffness, the carryover factor and the fixedend moments of the overhanging part are evauated taking into account that the outer edge can move downwards but is not free to rotate. For the sake of economy as we as good appearance the cantiever part of the base sab is usuay made tapered with the greater thickness at the inner edge. The stiffness of this part is greaty affected by the variation of the thickness and consequenty the bending moments and the ring tension in the tank wa wi aso be affected. A method for the estimation of the stiffness, the carryover factors, and the fixedend moments in ringshaped sabs of varying thickness is presented in the appendix. These vaues are cacuated by the Author and potted in curves (Figs. 3, 4 and 5) for sabs of various ratios of inner to outer diameters, aso for various ratios of the thickness at the inner and outer radii. The fixed end moments are given due to a concentrated oad P per unit ength on the outer edge (which represents the oad from the wa and the roof), and for uniformy distributed oad Q per unit area, (which represents the sefweight of the sab base and the weight of the iquid above it). The anaysis of this type of tank by moment distribution wi be expained whie soving a numerica exampe. Exampe 2 : The concrete water tower shown in Fig. 8 of 2, gaons capacity is supported on a cyindrica shaft 28' diameter, cast monoithic with the tank base. The thickness of the shaft wa near the top is 8". The tank is supposed to be covered by a roof weighing 4 b./sq.ft., simpy supported on the tank was. t is required to find the bending moments in the sab base, and the moments and ring tension on the was of the tank and the shaft. Weight of the contained

6 7 The Structura Engimir Fig. 7.Ring tension and bending moment diagram for a tank on a rigid foundation (Exampe ) iquid 62.5 b./cu.ft. ; weight of concrete materia 5 b./cu.ft. Poisson's ratio p = 6 (a) Loads (see Fig. 9a) Weight of roof, acting verticay on the top edge of the wa 7 P = 4 X = 34 b./ft. = 34 b./ft. 2 8 Own weight of wa = 5 x ; x 22 = 22 b./ft. t Tota vertica oad at the outer edge of the base sab = 254 b./ft. Distributed oad on the overhanging part of the base (part 2 3, Fig. 9a) incuding its own weight = 4 = 62.5 x 22 + E) (b) Stiffness Factors 2 x 2 The stiffness of a eements E terms of 2( p"). X 5 = 59 b./ft.2 Distributed oad on part 3 3 incuding own weight 26 q2 = 62.5 X 22 + X 5 = 7 b./ft.2 2 wi be estimated in 795 t 26 68P.5Sf Q52 \ S (h) 2kbu/;on and carryover fucfors MP 4 /Y2 3 M3 3 M3 3 ' rcom f f FE ffs 525 t 6'6' U95 f 78 h uo c t //9 f f P (5 f9 5 c f 7 t /oo + U a c 2 c fo 4. i JZUO f 2669 Fig. 8

7 ,May, (i) Tank wa (eement 2 ) :.368 H.368 x ph= ~ _ 8 dr.tj7 X s2 = 2 p 3 = (;) x 2 x 7 = (ii) Base sab (eement 2 2) : thickness at outer radius d = 9 Thickness at inner radius =d+d.d =9+7=26 7 d 2 = Ratio of inner to outer radii = c = = From the curves of Fig. 3 or from Tabe 5 (see the appendix) we get s23 = (&)3 X 3% X 4.7 =.49 3 s32 = ( X X 5. =.65 (iii) Base sab (eement 3 3 ) : The radia moment which causes unit rotation at the edge of a circuar pate of radius c. R E d3 ( + P)* d3 where s = ( + p). c. K 3 :. s3 = (S> x i3.5 x.667 =.88 (iv) Shaft wa (eement 3 4) : s3 = 2 pt3 = 2 x x (A) = $3.5 x.258 The distribution factors are cacuated in the usua way and given in Fig. 9b. (ii) Base sab (eement 2 3) : From the tabe or the graphs given in the appendix we get due to concentrated oad P ; M =.27 X 254 X 3.5 = 2,4 M23 due to distributed oadq ; M =.64 X 59 X 3.52 =,25 k32 due to concentrated oad P ; M =.92 X 254 X 3.5 = 8,8 due to distributed oad q ; M =.52 X 59 X 3.52 = 9,8 (iii) Base sab (eement 3 3 ) : 3,65 b.ft./ft. 7,98 b.ft./ft. The radia fixing moment at the edge of a circuar sab of radius = c R, uniformy oaded by a oad q2 is equa to ; M = q2 (?) (3) M33 = 7 X 3.5* 38,7 b.ft./ft. 8 The distribution process is carried out in the usua way and is shown in Fig. 9c. Because of the reativey arge vaues of the carryover factors, the convergence is comparativey sow, but this does not add much difficuty to the probem. After the moments,at the joints are obtained, the variation of the fing tension and the moments in the tank wa is obtained by superimposing the effect of the iquid pressure on a wa fixed at the base pus the effect of the reaxed moment (45324 = 8 b.ft./ft.) during thc distribution process. The moment and ring tension aong the shaft wa are obtained by considering the effect of a moment (= 27 b.ft./ft.) acting at the top. The variation of, the ring tension and the moment in a ong cyindrica wa (ph > x) due to a radia moment MO appied at the edge (Fig. ) is given by the equations (c) CarryoverFactors The carryover factors in the tank and shaft was are zero. The carryover factors in the base from 2 to 3 and from 3 to 2 are f23 =.26 andfs2 =.795 (taken from Tabe 5 in the appendix). (d) FixedendMoments (i) Tank wa (eement 2 ) = + 45 b.ft./ft.... (6) * This can be easiy proved by the theory of bending of circuar pates, see Timoshenko, Strength of Materias, Part, p.32. * See Timoshenko, Strength of Materias, Part ; p.4. Fig.

8 72 The Structura Engineer Fig..Moments and ring tension diagrams for a water tower (Exampe 2) T= epxsirt px,.... t (4) andm = MO epxcos px..... (5) The variation of the radia and tangentia moments in the base sab 3 3' is obtained by adding the moments due to a distributed oad on a simpy supported circuar sab pus the effect of the restraining moment aong the support (2939 b.ft./ft.). A radia moment M O appied on the edge of a circuar sab wi cause constant radia and tangentia moments equa to MO. The radia and tangentia moments at different radii of a simpy supported sab oaded by a uniform oad Q are given in Tabe 4. n the overhanging part of the sab the tangentia moments are usuay sma and need not be considered. The reinforcement bars used as distributors to the main stee in the radia direction are normay sufficient to resist the tangentia moments. (p=;) Tabe4.Radiaand Tangentia Moments in auniformy oaded, simpy supported Circuar Sab yi R ~~ Radia moments in terms of q.h* Tangentia moments in terms of q.rs Of the bending moments and the ring moments for circuar rings tapered towards the edge tension in the tank Of the above are shown are needed when cacuating the stresses in a cyindrica in Fig.. tank with a fat base supported on an inner circuar support, as the tank of Fig Appendix : Stiffness, Carryover Factors and Because the inner and outer radii of the ring (Fig. FixedendMomentsinCircuarRingshaped 2a) are usuay neary equa, i.e. c is some factor round SabsofVariabeThickness unity, the tangentia moments in the ring are sma and the ony important moments are in the radia Circuar pates of variabe thickness require even direction. Hence, such a ringshaped sab is more or for the simpe cases very tedious cacuations. The ess acting as overhanging radia beams'. The stiffness and carryover factors and the fixedend variation of the thickness of the ring has an effect on its statica behaviour which is assumed for simpicity * The accuracy of this assumption is checked at the end of the as the same effect on a radia beam with the same appendix. variation of the thickness.

9 May, dx. (8) ::: ( + x) ( + d 43 SA= DA which is the stiffness per unit ength of the outer edge of the sab. The stiffness at edge B is the moment required for a unit rotation at edge B, whie edge A is hed against rotation (but free to move in the vertica direction). The vaue of the stiffness per unit ength of edge B is The vaues of the integra in equation (8) were cacuated for different vaues of d and c (= c), and then the stiffnesses S, and S, were cacuated, they are given in Tabe 5 and potted in graphs (Fig. 3). The vaues given are for ring sabs having the ratio of the inner to the outer radii c =.6,.7,.8,.9 and., and vaues of d between zero and 2. Fig. 2b (a) Stiffness Consider an eement of the ring sab of Fig. 2b between two vertica radia sections with a width unity at the outer edge. The thickness of this eement varies ineary between d at the outer edge and (d + dd ) at the inner edge. The stiffening effect of the adjacent eements of the ringshaped sab on the bending deformation of the eementa radia beams can be taken into account by increasing the moment of inertia of each beam in the ratio The fex ~ P2> ura rigidity of the eementa beam at edge A which is aso equa to the fexura rigidity of the sab at the outer edge. The ength of the eementa beam is equa to the difference between the two ring radii = R ( C) = c R The thickness of the eementa beam at any point (X = X., see Fig. 2b) between A and B = d + dd x, and the breadth at the same point = ( c). x = G %. The fexura rigidity at x is E Ex = ( xc ) (d + dd ~)3. (7) 2 ( p ) The stiffness of the eementa beam at edge A is the moment which causes a unit rotation at A. t coud be proved () that this moment is equa to Fig. 3.Stiffness SA and S, at edges A and B of ringshaped sabs of variabe thickness

10 74 The Engineer (b) Carryover Factors.....A moment appied at edge A of the eementa beam Fig. 2b wi be carried over to edge B with its fu vaue. f a moment S, per unit width is appied at A, th,e moment per unit width wi be deveoped at B is equa to S,. With the sign convention used, C the carryover factor from A to H fa* =.... (2) C By a simiar way, the carryover factor from R to A fb+ = c.... (2 ) [f M,= P M, = (PZM,) C (23) (ii) Due to a distributed oad q per unit area (Fig. 5) MA = (c) Fixedend Moments The F.E.Ms. are cacuated for the two oadings : (i) A concentrated oad P per unit ength on the sab. (ii) A distributed oad q per unit area of the sab. For both oadings the edge A is supposed to be restrained in direction but free to sette downwards. By considering a radia eement of the sab it coud be proved () that the F.E.Ms. per unit ength of the edges A and B are as foows : (i) Due to a concentrated oad P per unit ength on edge A (Fig. 4).. (24) The vaues of the fixedend moments M, and M, are cacuated by equations (22), (23), (24) and (25) and given in Tabe 5 and potted in graphs Figs. 4 and 5. They are expressed in terms of P for a concentrated oad P per unit ength on the edge A, and in terms of qzz for a distributed oad g per unit area. The sabs considered have the ratio c between.6 and. and d between zero and 2.. Fig. 4.Fixedend moments MA and MB at edges A Fig. 5.Fixedend moments MA and MB at edges and B of a ringshaped sab of variabe thickness due A and B of a ringshaped sab of variabe thickness due to a concentrated oad P per unit ength of edge A. to a distributedoad of q unit area.

11 May; Tabe 5.Stiffness, carryover factors and fixed end radia moments in ringshaped sab of variabe thickness*.... c'= c d' ~.2, p,,.3,,,. 8, $ 9,.4.,. $ T F.E.Ms. due to Stiffness factor Carryover factor P/unit ength ~ SA in terms of DA.oooo g SB in terms of DA.oooo ~. 24 B ~ ~ fab from A to B.oooo,>,,.,? $ 9,,.25,,,,,*.4286 B t.6667 t t t fba from B to A.oooo.9,,.8, S, p.6,) ' on outer edge! MA in terms of P AfB in terms of P ~ ~ Os distributed oad qunit area n!a M, in terms in terms of of : * Thc outer edge is free to move downwards but restrained ir direction.

12 76 The Structura Engineer (d) Check on Origina Assumption The accuracy of the anaysis of ringshaped sabs by the consideration of eements taken as separate radia beams wi be checked here by the comparison of the vaues obtained by this anaysis with those obtained by an exact sab anaysis for the case of a sab of constant thickness. Consider the sab shown in Fig. 6, it is required to find the stiffness at end B, The moment at the outer edge wi be ==. D 2c D 2c (Mr)r= R R ( 4 2 ) ( + c) (3) The carryover factor from B to A Consider the case of a ring sab having c =.6 and p = 6. Substituting these vaues in equations (3) and (32) we get or the radia moment per unit ength of the circumference at B which makes a unit rotation at this end whie the outer edge is restraint against rotation. n this case there is no oading on the sab. The equation which defines the defected surface of the ring is : W = A + Azogr + A4r2gr (see equation ). The constants A, A2, A3 and A4 are to be determined from the foowing edge conditions atedgeb,r=cr;o=andw=o,] at edge A, r = R ; = andq =, where 8 is the rotation and Q is the shear. * The constants A to A4 which satisfy these conditions are As= 2 R ( C ) The radia moment The stiffness S, is equa to the which causes unit rotation, Substituting for R by? we get c S, = and A4=O J moment at edge B D ( + c2) ( c2).. (3) c C!+C) and 2 x fba = ( +.6).355 The corresponding vaues obtained by consideration of an eementa radia beam are (see Tabe 5 for c =.6 and d = ) : and fb* =.6 Comparison between the vaues obtained by the two methods shows the degree of accuracy in the origina assumption adopted for the determination of the vaues given in Tabe 5 and the graphs of Figs. 3, 4 and 5. Acknowedgment This investigation was carried out at the Department of Civi Engineering, Leeds University, party in the course of an anaysis concerning the Structura Behaviour of Concrete Tanks. The writer woud ike to express his sincere thanks to Professor R. H. Evans, D.Sc., Ph.D., M..Struct.E., M..C.E., M..Mech.E., and to Dr. E. Lightfoot,M.Sc.,Ph.D.,A.M..Struct.E., A.M..C.E. whose guidance and carefu supervision enabed this paper to be written. References. MBrkus, Gy., Anaysis of Circuar Storage Tanks with Pane Covers and Foor Pates by the Moment Distribution Method, Viziigyi Kozkmenyek (Hydrauic Proceedings), 953,, Budapest. (n Hungarian with German and Engish abstracts). 2. Lavery, J. H., Continuity in Eevated Cyindrica Tank Structures, The Journa of the nstitution of Engineers, Austraia, Vo. 2, 948, October and November. 3. Sater, G., Design of Circuar Concrete Tanks, Transactions of the American Society of Civi Engineers, Vo. 5, Reinforced and Prestressed Concrete Tanks, pubished by the Concrete Association of ndia, Bombay, 953. (A reprint of the PortaEd Cement Association Chicago). 5. Gray, W. S., Reinforced Concrete Reservoirs and Tanks, Concrete Pubications Ltd., London. 6 North, J. C. Cyindrica Reinforced Concrete Surface Tanks, New Zeaand Engineering, December, Timoshenko, S., Theory of Pates and Shes, McGraw Hi Book Co., London, Markus Gy., Anaysis of Circuar Pates by the Moment Distribution Method, Viziigyi Kozkmenyek (Hydrauic Proceedings), 952,, Budapest. (n Hungarian with German and Engish,+bstracts). 9. Oravas, G., Anaysis of Coar Sabs for Shes of Revoutions, Proceedings of the American Society of Civi Engineers, Vo. 82, March, Ghai, A., Ph.D. Thesis, The Structura Anaysis of Circuar and Rectanpar Concrete Tanks, University Leeds, 957.