Creator:R. Epain and B. Vodar Date Created: Place Created: Keywords:elastic loading,thick-walled high pressure cylinders Context:report by R. Epain and B. Vodar ************************************************** ELASTIC LOADING OF THICK-WALLKD HIGH PRESSURE CYLINDERS by R. iuPAIN and B.VODAR We consider a thick-walled cylinder submitted to uniformly uistributed internal and external pressures and a uniformly distributed longitudinal load and we establish the relations between these loads such that the cylinder does not undergo plastic deformation. For this purpose we use the criteria of Von Kises and of Treses as well as a linearised form of the criterium of the intrinsic curve of lOhavCaquot. We describe a graphic method which allows the resolution of these problems in a more varied manner than that of the calculations. Wr finish with several remarks on the conditions and limits in the use of this method. A hollow cylinder of circular cross-section ( figure 1 ) is submitted to internal , external ^ and longitudinal uniformly distributed pressures, and the limiting relations, i e , the limit in which the vessel undergoes no plastic deformation, between these quantities are established. These relations will be referred to as elastic loaning conditions. ^ (X) Throughout this paper we will use "conditions of elastic loading" for the French "condition de portance 6lastique". A bettor expression ai^it Oe "elastic limit load". Wo will establish these relations for three criteria of plasticity : the criterium of Von Rises that of Hohr-Caquot and finally, that of Tresca . The woll known formulas of Lame give the radiklj3^circuaferontial OJ y and longitudinal Creatresses as Inunctions of p , and J^ in the following form t where k • ^»the ratio of eztemal radius to the internal radius of the tL cylinder. Substituting these expressions into tne corresponding criteria we obtain the desired conditions of elastic loading. References 1 Mechanlk der festen Korpor im plastisch defomablen Zustand-Von Miseo, R, GCttinger Kachrichten 1915 • 2 Definition du dctoainc 4lastique dans les corps isotropes-Caquot, A, Proc. of the 4 th Congr. Intern. Appl. Hech. Ca.bridge iyj>5, p. 24 . 3 H&aoire sur l'dcoulement des corps solides. Tresca - HAnoires presents par divers savants 186a, volume 18, pages 773 & 799- 4 MAcoires sur l'equilibre int^rieur des corps solides homogenes.Lanr4 &Clapeyron H&toires presents par divers savants 1833, 1) Relationsof elastic loading for the criterium of Von Hioea 5 On expressing the principal stresses as functions of the pressures Rvp^yfe in the relation of Von Mises, given by (o% — $ we can make the following remarks t 1) plastic flow is only possible if P lies on the elliptic cylinder; 2) since the hydrostatic load is represented by a vector parallel to Z, only the component of Oi^ in the ..lane TT is necessary for determining whether the material does or does not remain elastic. 3) The ellipse corresponding to k »oo having a finite dimension confirms the known result that a finite state of loau is sufficient to create a plastic deformation in a cylinder od infinite thickness. The second remark leads to the establishment of a graphic method permitting the resolution of two types of proW-ems relative to elastic loading. On the first graph A three equidistant axes p^ } p^ p^ are traced as well as the axes V for different values of K. For these same values one traces on transparent paper a series of graphs B representingthe corresponding ellipses. The number of tiiese graphs is limited both by the allowed interpolations and the fact that k = 4 constitutes a limiting value m practice. Finally one obtains a now simplification by scaling the designs to ^3/2 and on letting Oa =1. The first problem is the following J " given k and (J^ determine whether this cylinder remains elastic under the loads fa ; P6 ' " * 0116 superimposes the graphs A en B making the axes V coincide and one traces the projection of OP on the plane TT whose comj/onents onto p^ } p^ , p' are respectively fy^ , J^j-tfcylinder does or does not remain elastic according to vdlAber p falls inside ( see figure 3 ) or on the ellipse. In particular one finds the following well-known result t the maximum interna) pressure p. that a cylinder can withstand elastically in the case where O is zero is equal to t fe for a closed cylinder, for an open cylinder, for a plane otn^tn condition. The second typo of problem can be stated as " given a cylinder characters zed by k and tfo and submitted to an external pressure J^ determine pg such that h. will be maximum and find this value". One begins as before, then one /A J traces OC = jh,^ and A paralled to ^ and tangent to the ellipse. One then deduces that ^ = CJo CD and ^ « qz DP. ( see figure 4 ). The detercin .tion of the oxtrema ca be done by the preceding method and this allows one to obtain fcue results shown in the table below* MAXIMUM VALUE GIVEN VALUE fc Pe fc ' Pi Pi Pe CORRESIOMDIMG VALUE fe Pe Pi Pe P« Pi : P-2L bxcr. re n VilFrr - 7 - 2 - iluLATIOflS OF uLASTIC LOADING COHflriSHOKmMG TO TiU, CKIT^ilUM Of Tiiu IiiTRIHSXC CURVa OF . -OHR-CAQUOT - For simplification we utilize a linearized intrinsic curve obtained by drawing the tangents D to the circles of diameters 0& and Oc. where Oo and OJ, are the absolute values of the elastic limits for pure tension and pure compression ( see figure 5 )• There is plastic flow at a point in the wall of the cylinder if the local values of the congtcaints are such that the circle of I'iohr constructed by the major Q^j and the minor GJ^ stresses is tangent to or cuts the lines ID • In tiie limiting case where this circle io tangent to the lines D the figure 5 shows that one has t radius of the Kohr circle - Ofci-Cm b Coft — Om+OW, 3 Z 2L where QhtZ? represents the abscissa of the conter of this circle t while ^ 3=5 (Xtj ib are the equations representing the tangents D« On noting that one can urite •— --~ and Ob 4-at I? CO&o< , in the preceding equation, the necessary condition '.hat the cylinder roiauins^eTastio is expressed by the inequality crM - 2L 0i, (5>(r, f for <5>^>(7i -P'+Wb =or fT «>0i>(5 The equations represent six planes in the ap-.ee fa > fif > pe tfe now seek to determine the contours foraad by their traces on tne planes perpendicular to the line p^. xs fy ^Og * In order to do this wc will utilize a newcoordinate system T, V, Z where the axis Z coincides with the line pt~ = pe while W is at the intersection of the plane formed by the coordinates p^ and p e and the plane 77" perpendicular to Z and passing through the origin. Under these conditions the .new orthonormal vectors are written as functions of the old coordinate system in the following way I K * %s fe 7vi p! -Jfel Inversely the old orthonormal vectors J^ ^ and ^ are given in the new eysteu by I if- y^+y^ It follows that the aquations of the six planes described above are written in the new ooordinate system as indicated below ( for (Jo Z^Z dc )l M v=:-V3— 3&*-m y/+ f-^/ ^2 = ± 07, P.-p = ±07 ft'/ , ri' re 2 -Ps + P< - - 07 • - 11 - In the space V, V, Z defined in the preceding section the equations of these planes ore reduced to i V__L .iii+L W =±|/X~(jr. Since the coordinate Z no longer appears in these expressions, those planes are normal to the plane 7r • They form an irregular hexagonal prim inscribed in the elliptic cylinder of Hiaos. The magnitude of the elastic domain is once again, as in the case of Hisea, independent of the hydrostatic component of the load vector. The intersection of this prism with the plane 7T gives the contour of the elastic domain. The figure 8 shows that this contour can be traced in a remarkably simple way, the other half of the hexagon being symmetric with respect to the origins. J For k » 3, side 1 of the hexagon almost coincides with the perpendicular to pe (the position of 1 in the figure corresponding to k = OO ). This shows that increasing k above the value 3 adds only a very small gain to the elastic loading. For k «= l,the hexagon is reduced to the line ^ ( aero surface ). The study of the maxima allows one to rediscover the following well-known result t for a given p there exisb> contrary to the criterium of Miaea an Infinite number of values of p^ for which p^ is maximum and equal to * This results from the fact that the tangent of the contour drawn paraJlel to p coincides with side 2 of the hexagon. The extremities of the load vectors corresponding to the cases of cylinders opon, closed, and in plane strain condition end respectively at the points a, b, and c of the figure 8« 12 - 4 - C0BDITI0K3 Ox ftPPLICA^OE AftU Vfrll^ OP VfripiTi OF TLu^ HJTiOD? In the use of these methods we have implicitly assumed that the loads increase proportional to the same parameter. This condition is automatically satisfied for the cases of open and closed cylinders ss well as for a cylinder in plane strain condition. However, it Is no longer true for the case of shrink fits, where one applies first p followed than by p^- • The method presented here is nevertheless always valid on the condition that it is utilized in two stops. In that vihich concerns the limits of validity of these methods we can make the following remarks. For a hydrostatic load p^ sz p — p^ — p€ j of large magnitude the relation stress-strain should no longer be linear, in other words, Hookas law should cease to be valid. The work of Bridgaan 6 on the compressibility of pure iron shows that already at 12,000 Atmospheres there exists small divergences from linearity. Further if the deformations become large the relations between the oomponents of the deformation tensor and the spatial derivatives of the components of the displacements become quadratic. At this point, the Lead equations which are formed from the linear forms at these relations are no longer valid, and the relations i of elastic loading, which are derived from them, must be entirely reconsidered* Thus, oven, if the criterium of plasticity used, as in the case for the criteria of Wisea, Kohr-Caquot and Tresca, implies the condition that a hydrostatic constraint does not cause plastic deformation, it doos not automatically result from tBi. that a fctartrtl. W p^ = R. = = pe proteot* the cyl!^ from all plastic flow* On the other hand, if the loads though vez-y large are not isotropic ( p. T^. pe ) one can think that a plastic law governs the deformation beyond the elastic regime of Hookee law. On other words, there is no phase of a nan linear lawconsequently, the relations of elastic loading should remain valid* ^E^T^ (6) The Physics of High Pressure Bridgnan P.W* 0. Bsll & Sons London 2nd 3d. 1949 p. 154. Fiq 1 Fiq. Z Fig 3 Fiq 4 Fig.5 Fig.6 Fiq 7 f;3.8