Equatorial Bending Of An Elliptic Toroidal Shell
Thin-Walled Structures 96 (2015) 286–294Contents lists available at ScienceDirectThin-Walled Structuresjournal homepage: www.elsevier.com/locate/twsEquatorial bending of an elliptic toroidal shellAlphose Zingoni n, Nosakhare Enoma, Nishalin GovenderDepartment of Civil Engineering, University of Cape Town, Rondebosch 7701, Cape Town, South Africaart ic l e i nf oa b s t r a c tArticle history:Received 5 August 2015Received in revised form18 August 2015Accepted 18 August 2015The exact differential equations for the axisymmetric bending of elliptic toroidal shells are difficult tosolve. In this paper, and by considering a semi-elliptic toroid, we present an approximate bending solution that is valid in regions adjacent to the horizontal equatorial plane. The formulation accuratelysimulates edge effects which may arise from loading and geometric discontinuities located in theequatorial plane of elliptic toroids. In particular, the developed closed-form results provide a very effective means for evaluating the state of stress in the relatively narrow zones experiencing mid-side localeffects in complete elliptic toroidal vessels subjected to hydrostatic loading, and for calculating the deformed shape of the shell midsurface.& 2015 Elsevier Ltd. All rights reserved.Keywords:Toroidal shellElliptic toroidShell of revolutionBending theory of shellsShell analysisDiscontinuity stressesContainment vessel1. IntroductionThe membrane theory of shells is a greatly simplified yet veryeffective basis for estimating stresses and deformations in thoseregions of the shell over which the loading and geometry do notchange too rapidly. However, and as is well-known, the theorybecomes inadequate at or in the vicinity of supports, concentratedloadings, shell junctions or discontinuities in shell geometry(thickness, slope, radii of curvature), loading and material properties. Novozhilov [1] has called these locations “lines of distortion” in reference to the existence of a bending effect locally disturbing the membrane state of stress in these regions. Discontinuity problems in shells of revolution have been the subjectof many investigations, and a good body of closed-form resultsexists for the more common types of shells and loading conditions[2–5].The performance of containment shells is usually assessed withregard to their stress and deformation response in the linearelastic range [2,3], their vibration characteristics and dynamic response, as well as their nonlinear buckling and postbuckling behaviour within the elastic and plastic ranges of material behaviour.Metal shells are particularly susceptible to buckling on account oftheir thin-ness (radius-to-thickness ratios typically in excess of500). Numerical studies have been carried out on the bucklingcapacity of vertical cylindrical steel tanks [6–11], horizontalnCorresponding author. Fax: þ27 21 650 3293.E-mail address: alphose.zingoni@uct.ac.za (A. 170263-8231/& 2015 Elsevier Ltd. All rights reserved.cylindrical and near-cylindrical vessels [12–14] and conical tanks[15–17]. The buckling capacity of multi-segmented shells underexternal water pressurisation has also been investigated [18], ashas the elastic buckling of certain unusual mathematical forms forshells [19,20].Junction stresses in various shell assemblies and multi-segmented vessels have been the subject of intensive studies over thepast 15 years [21–25]. Mechanics phenomena around shell intersections and at shell-branching locations have also been of interest[26,27]. The presence of ring beams at shell junctions has a considerable influence on the behaviour of the shell; some effortshave also been directed towards understanding ring–shell interactions [28,29]. A more comprehensive review of recent studies onthe statics, dynamics and stability of various types of liquid-containment shells under a variety of loading conditions may be seenin a recent survey [30].Toroidal shells have mostly been studied with pressure-vesselapplications in mind, though liquid-containment applications havealso been of interest. The classical solutions for pressurised circularand elliptical toroids may be seen in texts on linear shell analysis[2–5]. Even where toroidal shells with uniform geometry aresubjected to internal pressure only, the membrane solution becomes inadequate in the vicinity of the horizontal circles furthestfrom the equatorial plane, owing to the vanishing of the curvaturein one of the principal planes [31].Sutcliffe [32] tackled the stress analysis of both circular andelliptical toroidal shells subjected to internal pressure. While accurate for the purpose, the formulation is somewhat cumbersome
A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294for practical implementation. Galletly [33] considered the elasticbuckling of an elliptic toroidal shell subjected to uniform internalpressure, and confirmed that internally pressurised elliptical toroids, unlike circular toroids, may possibly buckle, depending onthe axes ratio of the elliptical cross-section. The study was alsoextended to plastic buckling [34], for which the post-bucklingbehaviour of the shell was noted to be stable.Redekop [35] studied the buckling behaviour of an orthotropictoroidal shell of elliptical cross-section, while Yamada et al. [36]considered the free vibration response of a toroidal shell of ellipticsection. Xu and Redekop [37] also considered the free vibration ofelliptic toroidal shells, but with orthotropic properties. Zhan andRedekop [38] studied toroidal tanks with cross-sections made upof combinations of circular arcs of different radii (ovaloid shape),and observed the vibration, buckling and collapse behaviour ofthis type of toroidal vessel.In this paper, we will focus attention on the thin elliptic toroidal shell. Noting the lack of a convenient analytical solution forthe axisymmetric bending of an elliptic toroidal shell, we aim atdeveloping a practical means for estimating bending-disturbanceeffects that may arise in the mid-side locations (herein referred toas “equatorial” locations) of vertically elongated thin elliptic toroids, where the vertical semi-axis b of the ellipse is greater thanthe horizontal semi-axis a . Specifically, we aim to develop andpresent a set of closed-form expressions for interior shell stressesdue to axisymmetric bending moments and shearing forces applied in the equatorial plane of the elliptic section as uniformlydistributed edge actions.The formulation is intended for use in quantifying (i) thejunction effects in the vicinity of the equatorial plane of subseaelliptic–toroidal shell structures (where a horizontal plate deckmay be attached to the inner walls of the toroid to provide aninterior working platform extending right round the torus), or (ii)the edge effects in the vicinity of supports where the elliptic toroidal vessel is used as an elevated circular tank supported onclosely-spaced vertical columns attached at both the intrados andextrados of the torus. The relatively weak edge effects associatedwith partial filling of the tank may also be quantified on the basisof this solution. We will begin by defining the geometry of theelliptic toroid.2. Geometrical preliminariesFig. 1 shows the relevant geometrical parameters of an elliptictoroidal shell. To generate the torus, an ellipse of semi-axes a(horizontal) and b (vertical) is rotated about a vertical axis Y Ythat lies at a distance A ( a ) from the local vertical axis y y ofthe ellipse. The equatorial plane (horizontal plane of symmetry) is287denoted by E E . In what follows, we will take the generatorcurve (or meridian) of the toroidal shell as the ellipse to the left ofthe axis Y Y . Let P be any point on the generator meridian. Theradius of curvature of the ellipse at point P is denoted by r1 and thecorresponding centre of curvature by O1. For the three-dimensional toroidal surface, there would be two principal radii of curvature (being the maximum and minimum values of curvature) atany given point P , and these occur in planes perpendicular to eachother. The first principal radius of curvature of the toroidal surfaceat point P is the radius of curvature r1 ( PO1) of the generatorellipse at that point, while the second principal radius of curvatureat point P , denoted by r2, is equal to the distance PO2, where O2 isthe point at which the surface normal at P intersects the axis ofrevolution Y Y of the torus.Point P itself may be defined by an angular coordinate ϕ , whichis the angle measured from the upward direction of the axis ofrevolution of the torus to the surface normal at point P . The range0 ϕ 2π covers all points on the toroidal surface, with 0 ϕ πdescribing points in the outer region of the torus, and π ϕ 2πdescribing points in the inner region of the torus; the coordinatesϕ π /2 and ϕ 3π /2 define points on the equatorial plane, whichof course correspond to the extrados and intrados of the toruswith respect to the axis Y Y .For the outer region of the torus (0 ϕ π ), the principal radiiof curvature are given by [3]r1 r2 a2b2(a22sin ϕ b2 cos2 ϕ3/2)(positive)(1a)Aa2 sin ϕa2 sin2 ϕ b2 cos2 ϕ(1/2)(positive)(1b)while for the inner region (π ϕ 2π ), these becomer1 r2 a2b2(a22sin ϕ b2 cos2 ϕ3/2)(negative)Aa2 22sin ϕa sin ϕ b2 cos2 ϕ((2a)1/2)(positive)(2b)The values of r1 and r2 at the extrados of the torus ( ϕ π /2) andthe intrados ( ϕ 3π /2), which correspond to the outer and innersides of the elliptical section, will be required in due course.Evaluating these, we obtainr1 b2a(3a)r2 A a(3b)at the extrados, andyPAr1,r2YbO1 φO2Eb2a(4a)r2 A a(4b)r1 AEat the intrados.3. Governing equationyYaFig. 1. Geometrical parameters of an elliptic toroidal shell.Fig. 2 shows a bending element of an axisymmetrically-loadedshell of revolution in the {ϕ, θ} coordinate system. Here, themeridional angle ϕ identifies the position of a point along a given
288A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–2942r22 d Q ϕ r22r drd r22 dQ ϕ cot ϕ 2 2 2tr1 dϕdϕ tr1 dϕtr1 dϕ tr1θQφNφ r d2r d r2 dr22 2 2 dϕ tr1 dϕ tr1 dϕdθMφ cot ϕ r νr1 dr2 2 ( r1 νr2 ) cot ϕ t r1 dϕNθMθRMθ d r2 r2 1 Q ϕ r1EV ν (cot ϕ) dϕ t t sin2 ϕ r2NθMφ dMφdφQφ dQφφr1Nφ dNφFig. 2. Element of a shell of revolution under axisymmetric bending.meridian of the shell (as already defined), while the coordinate θ isthe circumferential angle measured from an arbitrary verticalplane to the vertical plane containing the meridian in question, thetwo vertical planes intersecting at the axis of revolution of theshell. The element is in equilibrium under the loads acting over itssurface (external loading on the shell) and the forces and momentsacting along its four edges (internal actions in the shell), but in thefigure, the surface loading is not shown, since we will only beconcerned with edge effects.The actions {Nϕ, Nθ } are direct forces per unit length of theedge, acting in the direction of the tangent to the shell meridian atany given point (henceforth called the meridional direction), andthe direction of the tangent to the horizontal circumferential circlepassing through that point (henceforth called the hoop direction),respectively. The actions {Mϕ, Mθ } are bending moments per unitlength of the edge, as seen in the vertical meridional section andthe horizontal hoop section respectively. The action Q ϕ is a shearforce per unit length of the edge, acting in the meridional section;there is no shear force in the horizontal section (Q θ 0), owing toaxisymmetry. All meridional actions are shown incremented in thedirection of increasing ϕ , but the hoop actions do not change withrespect to θ owing to axisymmetry.Considering equilibrium of the shell element, and making useof strain–displacement relations as well as Hooke’s law, we mayexpress Nϕ , Nθ , Mϕ and Mθ in terms of displacements, and thenreduce the ensuing relationships to the well-known Reissner–Meissner differential equations for the axisymmetric bending ofgeneral shells of revolution [2,3] d2V r2 r d r2 dV D D sin ϕ 2 D 2 cos ϕ sin ϕdϕ r1 dϕ dϕ r1 r1 r cos2 ϕ dD V r1r2 (sin ϕ) Q ϕ ν cos ϕ D sin ϕ D 1 r2 sin ϕ dϕ (5a)(5b)In these equations, the variable V is the angular rotation of theshell meridian as seen in the meridional section, while Q ϕ is theshear force as already defined. The material properties E (Young'smodulus of elasticity) and ν (Poisson's ratio) are assumed to beconstant. The parameter t denotes shell thickness, while D is theflexural rigidity of the shell, given byD Et 312 (1 v2)(6)For thin shells of revolution which are not shallow (non-shallow shells shall be taken as those in which bending phenomenaoccur in locations for which ϕ is at least π /6 from the locationsϕ 0 and ϕ π ), second-derivative terms of V and Q ϕ are muchbigger than first-derivative and zero-derivative terms. As an approximation, we may therefore drop all first-derivative and zeroderivative terms on the left-hand sides of Eqs. (5), so that theequations becomer2d2V 1 Qϕ2Ddϕd2Q ϕdϕ2 r12r22(7a)EtV(7b)In the present problem of the prolate (vertically-elongated)elliptical cross-section (b a), this approximation is particularlyjustified, since we intend to investigate bending phenomena inregions that are adjacent to the equatorial plane ( ϕ π /2),where the sides of the elliptical section remain fairly steep for aconsiderable distance on either side of the equatorial plane.Combining Eqs. (7a) and (7b), we obtain the fourth-order differential equationd 4Q ϕdϕ 4 4λ 4 Q ϕ 0(8)where λ is a slenderness parameter defined as follows:λ4 r14 Et4Dr22(412 22) rr t 3 1 ν2(9)For the elliptical toroidal shell, the principal radii of curvature r1and r2 are functions of ϕ (as given by Eqs. (1) and (2)), so theslenderness parameter λ is also a function of ϕ , implying that thesecond term of the above fourth-order differential equation has avariable coefficient. Given the complexity of this coefficient, theexact solution of Eq. (8) is rather difficult to obtain.4. Approximate general solutionNow for a prolate elliptical profile, the radii of curvature r1 andr2 vary rather slowly in the neighbourhoods of the equatorial
A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294plane, which are the regions of present interest. Based on thisobservation, we will introduce the further approximation that r1and r2 are practically constant in the narrow zones adjacent to theequatorial plane and within which bending disturbances areconfined. These constant values of r1 and r2 will be taken as thevalues at the location ϕ π /2 (for the outer part of the toroidalshell) and the location ϕ 3π /2 (for the inner part of the toroidalshell), as already given by Eqs. (3) and (4).With this simplification, the slenderness parameter λ in Eq. (9)is now a constant, and the fourth-order differential equation (Eq.(8)) now has constant coefficients. This allows us to take thegeneral solution of Eq. (8) in the well-known form [3]Q ϕ e λϕ ( C1 cos λϕ C2 sin λϕ) e λϕ ( C3 cos λϕ C4 sin λϕ)(10)where C1, C2 , C3 and C4 are constants of integration to be evaluatedfrom the boundary conditions of the shell. Based on the wellknown behaviour of spherical and cylindrical shells, let us tentatively assume that for the semi-elliptical toroidal shell, the bending disturbance emanating from either the outer edge or the inneredge is of a decaying character. Furthermore, let us assume thatany adjacent edges of the shell are sufficiently far apart to allowdecoupling of edge effects. This permits us to retain only two ofthe constants appearing in Eq. (10), and to rewrite the solution asQ ϕ Ce λψ sin (λψ β )(11)where ψ is now the meridional angle measured from the normalat the edge of the shell to the normal at the point in question; Cand β are new constants of integration. For the shell edge locatedin the equatorial plane, the relationship between ϕ and ψ is simplyψ π3π ϕ; ψ ϕ 22(12a, b)where the first equation refers to the outer edge of the semi-elliptic toroidal shell, and the second refers to the inner edge (closerto the axis of revolution).Equilibrium considerations of the shell element yieldNϕ Q ϕ cot ϕ 0(13)δ 1 dV 1 dVνMϕ D V cot ϕ D r1 dϕr2 r1 dϕ Mϕ 1 ν dVν dV D Mθ D V cot ϕ ν Mϕ r1 dϕr1 dϕ r2(19)(20)(neglecting the first term in relation to the second).At this stage, we may now make use of the general solution forQ ϕ (Eq. (11)) to eliminate Q ϕ from the above relationships. Theresults are as follows:Nϕ 0(21) r Nθ Cλ 2 e λψ {cos (λψ β ) sin (λψ β )} r1 (22)Mϕ 2λ 3D r22 λψ Ce {cos (λψ β ) sin (λψ β )}Etr1 r12 Mθ ν MϕCλ r2 ( r2 sin ϕ) e λψ {cos (λψ β ) sin (λψ β )}Et r1 V 21 r22 d Q ϕ 2 Et r1 dϕ23D r22 d Q ϕ 2 Etr1 r1 dϕ 3Similarly, r dQ ϕ 1 dr2r dQ ϕNθ 2Q ϕ 2 r1 dϕr1 dϕ r1 dϕ V (18)neglecting the second term (in V ) in relation to the first term (inthe first derivative of V ), particularly as the second term alsocontains the factor cot ϕ ( 0 in the neighbourhood of the equatorial plane). Making use of Eq. (15) to eliminate V , and using theapproximation that, for the prolate elliptical profile, r2 and r1 arepractically constant in the narrow zone experiencing edge effects,we obtainδ ignoring the second term in Q ϕ in relation to the first term in thefirst derivative of Q ϕ , particularly as the rate of variation of r2 withrespect to ϕ is also very small in the neighbourhood of theequatorial plane.In evaluating the edge effects, compatibility conditions involving the horizontal displacement δ of the shell edge (which is amovement perpendicular to the axis of revolution of the torus),and the meridional rotation V of the shell edge, will be required.From Eq. (7b), we may write V as follows:(17)Using curvature–rotation relations and Hooke's law, the bending moment Mϕ may be expressed in terms of the rotation V asfollows:since cot ϕ 0 in the vicinity of the toroidal shell edges( ϕ π /2; 3π /2). For the hoop stress resultant, we obtain,(14) r dQ1( r2 sin ϕ) r2 dϕϕEt12892Cλ2 r22 λψ ecos (λψ β )Et r12 (23)(24)(25)(26)5. Boundary conditions and generalised edge effectsFig. 3 shows the edges of the semi-elliptical toroidal shellsubjected to uniformly distributed bending moments and horizontal shearing forces, where {Me1, He1} are the axisymmetricactions applied at the outer edge, and {Me2, He2 } are the axisymmetric actions applied at the inner edge. The followingtreatment applies equally to the two edges. In the formulation, we(15)From strain–displacement and stress–strain considerations, thehorizontal displacement δ may be expressed in terms of the stressresultants Nϕ and Nθ as follows:δ 1( r2 sin ϕ) ( Nθ νNϕ ) Et1 ( r2 sin ϕ) NθEtsince Nϕ 0. Using result (14) to eliminate Nθ , we obtain(16)Fig. 3. Bending and shearing edge actions on a semi-elliptic toroidal shell.
290A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294will therefore denote the applied edge bending moments and edgeshearing forces simply by Me and He , with ψ denoting the meridional angle from the edge in question (outer or inner).When Me only is applied at a given edge (in the absence of He ),we want to choose the constants C and β such that the followingboundary conditions are satisfied: r Nθ 2λ 2 e λψ (cos λψ ) He r1 ( Mϕ )ψ 0 Me(27a)Mθ ν Mϕ( Q ϕ )ψ 0 0(27b)δ Mϕ 4λ 3D r22 λψ e (sin λψ ) HeEtr1 r12 (40)(41)(42)2λ r2 ( r2 sin ϕ) e λψ (cos λψ ) HeEt r1 (43)Applying these conditions to Eqs. (11) and (23), we obtainβ 0(28a)Etr1 r12 MeC 2λ 3D r22 2λ2 r22 λψ e (cos λψ sin λψ ) HeEt r12 (44)At the edge (ψ 0), we have(28b)Substituting these values of β and C into Eqs. (21)–(26), weobtain the interior stress resultants, bending moments and deformations due to the edge bending moment Me as follows:Nϕ 0Nθ V δ e Ve (29)Etr1 r1 λψ e (cos λψ sin λψ ) Me2λ2D r2 2λ r2 r2 HeEt r1 (45)2λ2 r22 HeEt r12 (46)(30)6. Shell stresses due to edge actionsMϕ e λψ (cos λψ sin λψ ) Me(31)Mθ ν Mϕ(32) r2 δ 12 (sin ϕ) e λψ (cos λψ sin λψ ) Me 2λ D (33)The bending-related shell stresses σϕ (in the meridional direction) and σθ (in the hoop direction) are obtained by combining thedirect stresses due to the stress resultants Nϕ and Nθ (which areconstant across the shell thickness) with the flexural stresses dueto the bending moments Mϕ and Mθ (which vary linearly across theshell thickness from a positive value on one side of the shellmidsurface to a negative value of the same magnitude on theopposite side of the shell midsurface). Thus,V r1 λψ e(cos λψ ) Me λD (34)σϕ Nϕ6Mϕ6Mϕ 2 2ttt(47a)σθ 6Mϕ Nθ6MN 2θ θ ν 2 tt t t(47b)At the edge (ψ 0), we have r2 δe 12 Me 2λ D Ve (35) r1 M λD e(36)When He only is applied at a given edge (in the absence of Me ), wewant to choose the constants C and β such that the followingboundary conditions are satisfied:( Mϕ )ψ 0 0(37a)( Q ϕ )ψ 0 He(37b)Applying these conditions to Eqs. (11) and (23), we obtainβ C π42 He(38a)(38b)Substitution of these values of β and C into Eqs. (21)–(26) givesthe interior stress resultants, bending moments and deformationsdue to the edge shear He as follows:Nϕ 0(39)where the first equation makes use of the fact that Nϕ 0 (Eqs.(29) and (39)) and the second equation makes use of the relationship Mθ νMϕ (Eqs. (32) and (42)). In the notation , the uppersign refers to the inner surface of the shell (with respect to theglobal axis of revolution of the torus), while the lower sign refersto the outer surface.For the stresses due to the edge bending moment Me , wesubstitute the results for Mϕ and Nθ (as given by Eqs. (30) and (31))into the above expressions, and then eliminate the parameters Dand λ using Eqs. (6) and (9) respectively, leading to the results:σϕ 6Me λψe (cos λψ sin λψ )t2(48a)Er1 r1 Me e λψ (cos λψ sin λψ )2λ2D r2 6M ν 2 e e λψ (cos λψ sin λψ )t1/22Me λψ 2 e 3 1 ν2(cos λψ sin λψ ) tσθ {()} 3ν (cos λψ sin λψ ) (48b)For the stresses due to the edge shearing force He , we substitute
A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294the results for Mϕ and Nθ (as given by Eqs. (40) and (41)) intorelations (47), and then eliminate the parameters D and λ usingEqs. (6) and (9) respectively, leading to the resultsσϕ 224λ3D r2 λψ e (sin λψ )Et 3r1 r12 He 6He1/4{ 3( 1 ν )}2{()}{()}(49b)6He27.2. Effect of HeThe magnitudes of the stresses due to He (both meridional andhoop) not only depend on the shell thickness t , but more significantly, are directly proportional to r2 . Now r2 A a for theouter edge of the semi-elliptic toroidal shell, and r2 A a for theinner edge (see Eqs. (3b) and (4b)). It therefore follows that, for thesame horizontal shear force He applied at the inner and outeredges of the semi-elliptic toroidal shell, the peak values of theinduced shell stresses on the outer side of the torus will differfrom those on the inner side, since the r2 values differ; the outerside will experience larger stresses than the inner side.We also observe that only the parameters A and a of the elliptictoroid (but not b) have an influence on the peak values of thestresses due to He . The rate of decay of stresses with distance fromthe shell edge, as well as the wavelength of the oscillations, willalso be different between the outer and inner sides, owing to thedifferent values of λ .For He , we may write the stresses for the outer and inner edgesmore explicitly as follows:Outer edge6He1/4{ 3( 1 ν )}2 1 (A a)1/2e λψ sin λψ t 3/2 (50a)(50b) 1 (A a)1/2e λψ sin λψ t 3/2 3ν1/4{ 3( 1 ν )}2The magnitudes of the stresses due to Me (both meridional andhoop) are not dependant on the principal radii r1 and r2 (and hencethe parameters a and b) of the elliptic torus. For a given appliedMe , the magnitudes of the induced shell stresses only depend onthe thickness t of the shell. This total lack of dependence on theelliptic parameters a and b is a surprising result.It means that for the same moment Me applied at the inner andouter edges of the semi-elliptic toroidal shell, the peak values ofthe induced shell stresses will be the same. However, the rate ofdecay of the stresses with distance from the respective shell edge,as well as the wavelength of the oscillations of the variation, willnot be the same, because the value of the shell slendernessparameter λ (which governs both the rate of decay and the wavelength of the oscillations) at the outer edge differs from that atthe inner edge – see Eq. (9).2 sin λψ 1 σθ 2He 3/2 (A a )1/2e λψ t cos λψ σϕ 1/4{ 3( 1 ν )}7. Analytical observations7.1. Effect of Me1/4{ 3( 1 ν )}1/4{ 3( 1 ν )}Inner edgeσϕ σθ sin λψ 1/4 3ν2(49a) 2λ r 2 λψ24λ3D r 22 λψ(cos λψ ) He ν(sin λψ ) He e et r1 Et 3r1 r12 1/4 1 3ν1/2 2He cos λψ ( r 2 ) e λψ 3 1 ν 2 t 3/2 3 1 ν2 1 σθ 2He 3/2 (A a )1/2e λψ t cos λψ 1 ( r2 )1/2e λψ sin λψ t 3/2 291(51a)1/4{ 3( 1 ν )}2 sin λψ (51b)7.3. Comparisons with cylindrical and spherical shell solutionsNot surprisingly, the derived theoretical results for the elliptictoroidal shell are quite similar to those for the bending of a circularcylindrical shell, and for the bending of a non-shallow sphericalshell when use is made of the Geckeler approximation [2,3]. Bothof these problems also result in a fourth-order governing differential equation of the same form as Eq. (8), with a general solutionof the same form as Eq. (10) or Eq. (11).Looking at the expressions for Nθ , V , δ and Mϕ (Eqs. (14), (15),(17) and (18) respectively), and replacing r1 dϕ (for the curvedmeridian) by dx (for the cylindrical shell), one can transform theresults for the elliptic–toroidal shell to those for a cylindrical shell.However, the cylindrical-shell model will not exactly replicate thebehaviour of the toroid, since dx (parallel to the axis of revolution)is only an approximation for the real r1 dϕ , an approximation thatis very good in the neighbourhood of ϕ π /2 (errors are of theorder of ψ 2/6, where ψ is the angle from the edge), and that becomes better as r1 becomes larger, and exact as r1 approaches infinity. When r1 reaches infinity, the r2 for the toroidal shell becomesthe radius of the cylinder. If one wanted to use the cylinder solution to approximate the behaviour of the elliptic toroidal shell inthe equatorial zones, the equivalent cylindrical shells for the outerand inner edges of the semi-elliptic toroid are the two cylinderswhich are tangential to the elliptic toroid at the extrados( r2 A a ) and the intrados ( r2 A a ).Alternatively, one may replace the elliptic toroidal shell withthe equivalent spherical shell in the vicinity of the equatorialplane, and then use the Geckeler approximation. The equivalentspherical shells for the outer and inner edges of the semi-elliptictoroid are the two spheres which are tangential to the elliptictorus at the extrados and the intrados; these spheres are of radiiA a and A a respectively.For the outer edge of the elliptic toroid, the spherical-shellsolution is a better approximation of the stress distribution in theelliptic toroid than the cylindrical-shell solution (the sphericalsurface and the elliptic–toroidal surface have the same value of r1at their tangent circle), but for the inner edge of the elliptic toroid,the cylindrical-shell approximation is better than the sphericalshell solution, owing to the inability of the sphere to model themeridional curvature of the elliptic torus on its inner side (the
292A. Zingoni et al. / Thin-Walled Structures 96 (2015) 286–294spherical surface and the elliptic–toroidal surface have differentmagnitudes of r1 at their tangent circle, and the signs of the curvature are also different).8. Numerical resultsAs an example, consider a semi-elliptical toroidal shell with theparametersa 10 m ; b 20 m ; A 30 m ; t 0.05 m ; E 200 109 N/m2;ν 0.3These parameters are possible proportions for a subsea marineobservatory of circular plan shape, where the overall structuraldiameter is 80 m and the height is 40 m. The principal radii ofcurvature and shell slenderness parameters at the two edges follow from Eqs. (3), (4) and (9):Outer edge: r1 40 m ; r2 40 m ; λ 36.3568Inner edge: r1 40 m ; r2 20 m ; λ 51.4163A bending moment Me 100 kNm/m is first applied at both theouter shell edge and the inner shell edge. The variations of meridional and hoop stresses with the angular coordinate ψ from theshell edge are given by Eqs. (48), which apply for both the outeredge and the inner edge, but using the λ value applicable for theedge in question. The arc length s (in metres) from the shell edgeis simply given by s r1 ψ 40ψ , where angle ψ is in radians.Based on the theoretical formulation that has been developed inthis paper, Fig. 4(a) shows plots of meridional and hoop stressesversus distance s from the outer edge, while Fig. 5(a) shows plotsof meridional and hoop stresses versus distance s from the inneredge.As a means for validating the theoretical results, a finite-element modelling of the semi-elliptic toroidal shell using the FEMprogramme ABAQUS [39] was performed, for an edge loading ofFig. 4. Application of Me1 at the outer edge: (a) analytical results; (b) FEM results.Fig. 5. Application of Me2 at the inner edge: (a) analytical results; (b) FEM results.100 kNm/m . Two-node axisymmetric shell elements were
sponse, as well as their nonlinear buckling and postbuckling be-haviour within the elastic and plastic ranges of material behaviour. Metal shells are particularly susceptible to buckling on account of their thin-ness (radius-to-thickness ratios typically in excess of 500). Numerical studies have been carried out on the buckling
CCS Discrete Math I Professor: Padraic Bartlett Lecture 9: Elliptic Curves Week 9 UCSB 2014 It is possible to write endlessly on elliptic curves. (This is not a threat.) Serge Lang, Elliptic curves: Diophantine analysis. 1 Elliptic
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Pipe Bending Systems Useful information How do I find the right bending tool Pipe bending pliers no. 242 Manual pipe bender - Draw bending operation Manual bending tool sets no. 243 Manual bending tool sets no. 244 Manual bending tool sets no. 278 Universal manual pipe bender no. 276 Manual pipe bender - Pushing bending operation no. 2456
There are several types of tube bending method adapted to manufacturers' purpose. In fact, press bending or rotary draw bending method is commonly used to form bend of pipe or tube. B. 3 Roll bending method Press bending is simplest and cheapest method of bending cold tube and pipe. This method is almost the first bending
applications. Smooth degree-3 curves, known as elliptic curves, were used in Andrew Wiles’s proof of Fermat’s Last Theorem [11]. The points on elliptic curves form a group with a nice geometric description. Hendrick Lenstra [5] exploited this group structure to show that elliptic curves can be used to factor large numbers with a relatively .
Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA. Keywords: Quantum cryptanalysis, elliptic curve cryptography, elliptic curve discrete log-arithm problem. 1 .
Pipe Bending Revision 2.0 September 2014 1.0 Bending Processes for Pipe Fitting 1.1 Pipe and Tube Bending Pipe bending machines are typically human powered, pneumatic powered, hydraulic assisted, or electric servo motor. In the pipe bending operation the tube may be supported internally or externally to preserve the cross section of the pipe.
Freeform-bending processes, like three-roll-push bending, shape the work piece kinematically, thus the bending contour is not dependent on the tool geometry. The main goal of this project is composition of deformation in tube or pipe bending theory and the effects of Rotary Tube bending machine parts
