—- : { fi i cesnsninss aA ; WP ei ? 1 8 H Ce AS A; ; nn \ \Aisads Vids Wee j ceeeeemnassnnssioenitsscimatete - sateen UNITED STATES See EXPERIMEN TAL MODEL BASIN NAVY YARD, WASHINGTON, D.C. THE CRITICAL EXTERNAL PRESSURE OF CYLINDRICAL TUBES BY R. von MISES ZEITSCHRIFT DES VEREINES DEUTSCHER INGENIEURS, (VOL. 58, No. 19, MAY, 1914 P 750) TRANSLATED AND ANNOTATED BY D.F WINDENBURG TAL,MO MENTAL MODE) By ee ERECTED 1898 Siw > CONSTRUCTION AND REPAIR \ EG. NAVY DEPARTMENT i | 7 ) we NE 307 on fh | _ AUGUST, I93I REPORT N& 309 “4 rn a Po ae WANE LET Te ba Sat 4 " iS ‘ an? ed ‘ oe ein: SATUOAN oa RORY a 42 6 atl a id as NON iil (MONI ed 5 0 0301 0043 THE CRITICAL EXTERNAL PRESSURE OF CYLINDRICAL TUBES. By Re von Mises. zeitschrift des Vereines Deutscher Ingenieurs, ( vo1.58, No. 19 May, 1914,p.750). Translated and annotated by D. F. Windenburg. U.S. Experimental Model Basin, Navy Yard, Washington, D.C. Report No. 309 August, 1931. yu 7 + Pan tee ‘tac 2 THE CRITICAL EXTERNAL PRESSURE OF CYLINDRICAL TUBES. By Re. von Mises. Zeitschrift des Vereines Deutscher Ingenieure, (Vo1l.58, No.19, May ,1914, pe750) Translated and annotated bygDie ike Windenburg. (2) A circular, cylindrical boiler flue in the pressure chamber of a steam boiler is loaded similarly to a vertically, centrally loaded column: as long as the pressure remains under a certain limit we have stable equilibrium and a uni- form contraction of the material on all sides. For greater pressures, we have a buckling or bulging of the tube. (See C. Bach: "Elastizitat und Festigkeit", 1905 pe273). Observations of the critical pressure, corresponding to the great prac- tical significance of the question, have already been abundantly made. The corre- sponding theory, quite similar to the Euler buckling theory, was developed first for the tube of infinite length only. Here, the Pp CE SCM aie i instability pressure is given by Be-RERSeTAL(M MRS ea sera a) where "a" designates the inner radius, 2h the shell thickness of the tube (Fig. 1), E the modulus of elasticity and 9 Poisson's ratio. This equation was first developed by Bresse in Fig. 1. Cylinari PaBEy ie igere cy 1829. (See Love-Timpe "Lehrbuch der Elastizi- pressure p. tat", Leipzig 1907, p. 637). The complete ex- pression for tubes of finite length was treated by R. Lorenz ("Physikal. Zeit- schrift", 1911, p. 257) who in order to avoid the difficulties of computation in- troduced a series of neglected terms. R.V. Southwell (Phil.Mag., Vol.25, 1913, p.687) has also recently given an approximate solution. The following development is from the rigorous theory of thin elastic shells andderives the accurate expression for the critical pressure. For practi- cal computations, the results can be easily simplified according to the circum- (1), ssistant Physicist, U.S. Experimental Model Basin, Washington, D.C. ~2< stances. In most cases, the annexed table will be sufficient. The values given here differ as much as 20% or more from the Lorenz values and in the direction of a better agreement with observation. The essential characteristic of the buckling phenomena, the unlimited increase of the number of lobes as the length of the tube decreases, has been previously emphasized in the investigations mentioned above. cv The extension of the investigation to the oF Gin. 57) region above the elastic limit, and the possibility \ AN nN of a theory for the corrugated tube, will be dis- YD, LaKA cussed in a section at the close of this article. 1. The Elastic Equations. Let a point of the cylindrical surface (See Fig. 2) have the coordinates x, measured in the di- Fig. 2. Stresses and displacements in a volume element. rection of the axis, and @ , measured on the cir- cumference of the cross section. Let the elastic displacement of this point bé u in the direction of the axis, v in the direction of the tangent to the circle, and w in the direction of the radius, measured in- ward. We then express the strains €,, and €. for the u- and v- directions and the angular change 7 in the u-v plane as follows: e= Ze, Soe ea a ceoa((%) (See Love, The Mathematical Theory of Elasticity, p.543) where as above, a de- notes the radius of the cylinder. Let 2h designate the shell thickness, E the elastic modulus, ando= 0.3, Poisson's ratio. With the simplifying substitution ee ee ew ew ee em Oem we ew ee ew ee wow oes we can easily find the values for the various longitudinal and transverse forces, namely, the normal forces Tj and Tz and the shear force S, Fig.2, as follows: (Translator's note: These forces are really stresses multiplied by the thick- ness). T, = c(€,+0Cg), T = c(Ee oc, ), S= (1-0) 7 ------(3) Since the shear modulus is str TS)° the solution of the first two e- quations of (3) for q and G2 yields the well know simple relation © i T) -9T, 3 (Tee ae py oe th (Translator's note: In the text the shear modulus is given Bl +O) ). iL = -3- The shear forces N, and Ng that act in the w- direction cannot be ex- pressed by means of displacements, but can be determined only through bending equilibrium. From the previously considered resultant stresses, three different stress moments, Gj, Go, and H,arise. The moment G, is due to the normal stresses in the cross-section of which Tj was the resultant. It causes in the first line a curvature Kj of the cylinder-generator. Similarly, the normal stresses in the longitudinal cross section give a bending moment Go, that changes the curvature 2 of the line of intersection by an amount Kj. We have then (See Love, p.543) _ ow —1 (aw 1 dy Noe ear 9 Ko ee <3 pe ae SSStzS (4) (Translator's note: In the text £ is given in the place of =.) and considering the cross contraction (Poisson's ratio) similar to eq.(43) (See Love,p.530,eq-37) Gi = -o B(x +0K>), Go = -o B(x, 40K) -w22-- (5) The factor win eq. (5) together with the 2h contained in ce, eq. (2), gives the moment of inertia of the rectangular cross-section of height 2h and breadth unity. The shearing stresses whose resultant is Nj, produce a moment H about the x-axis that tends to make the rectangular element dx a dgof the tangential plane into an oblique quadrilateral. The deformation in this sense is (See Love, pe543) and the moment H becomes (See Love, p.530, eqe37) a= eB (1 -o)w ~at-eukueduanesst (7) (Translator's note; In the text h® is given in place of n*). An equally large turning moment about the v-direction acts in the long- itudinal section. 2e The Conditions of Equilibrium. The equilibrium of the forces in the u- and v-directions requires (See Love, p-535, eq-(45) ) (Translator's note: See also Prescott's "Applied Elasticity", p.549, eqe17-98, and pe550, eq. 17.102). -4- The last term in the second equation is based on the fact that the two longitudinal sections bordering the element are not parallel. In order to be able to set up the equations for the w - direction, we must know the value of the radius of curvature after the deformation. This is, = Aas De however, Q@=@ Sea and since (conditions of equilibrium in the w = di- rection), cites of eke note: See Prescott's "Applied Elasticity", p.549, equa- tion 17.101), p—it — ON2 Tz —Q, must hold, it follows, by the omission of Soe “ey quantities of anes order: a BPE EM IS nn (9) a Ox aop Q The two unknown shearing forces Ny and No are determined with the help of the equations showing that the summation of the moments about the u- and v- directions are equal to zero. (See Love, p.536, eq.46) at SG, es JH —~A6G,_ WV = = a arial Ne 1 Oe Se SE if 9) Equations (8), (9), and (10), in combination with the elastic relations of the previous section, give the complete statement for every problem dealing with cylindrical shells. A particular solution corresponding to the symmetrical compression through the external pressure p is obtained with u = v = 0. It fol- lows then that S = Ny = Np = Gy = Gp = H= O, and from eq. (9): Tp = -ap. Fur- ther eq. (3) gives T] =o Tp and eq. (1) Cy = - oe = =, w= oop . (Translator's note: The results are easily obtained since w is independent of x and 9 ). If we subtract this value of w from the actual displacement u, Vv, WwW, then the remaining part satisfies the same conditions, provided that the first term in the parenthesis in eq. (9) is dropped. (Translator's note: This means that the uniform radial compressive displacement is neglected in comparison with displacements due to buckling. That is, the displacements due to buckling are measured from the original position of the neutral axis of the shell as is shown in Figs. 4 to 6. Subtracting acp from w in (1) and substituting the resulting value of To obtained from (3) in (9) gives equation (9") directly). In place of equation (8) we have, therefore, HE lt © = iP pO eensasee (9: 2 x ade 7 ae a We can now, through the elimination of Ny and No» derive the following three equilibrium equations from equations (8), (9'), and (10): alii OS ea ao aan eames See iy * Opax g ig 28 28 ERI El Wh eey yey opr | OPOX ane? (0 an a (IT) 06 __ 9? 20°F 1 0°% es 4 4 9xz OX0D Ope ara a Aer Aaa (ITT } If the forces and moments are expressed through the deformations, and these through the displacements u, v, and w, we have in equations I to III the ne- cessary equations for the determination of u, v, and We 3. Expression for Displacements. Since the equations I to'III become linear and homogeneous after the in- troduction of u, v, and w, and do not contain the coordinates x, pexplicitly, we can substitue sine and cosine terms for the dependent variables, u, v, win a well known manner. Let us write = A sin ngsin = ax = B cos ngcos £S wo2e--(11) ee ax W= sin ngcos waa The change of sine and cosine is governed by the equations themselves as we will see below. The term n can be used to signify a real integral number, and the terma@a quantity such that — makesan uneven integral number, where Z is the free length of the tube. For only if this expression is an uneven integer are the radial displacements w for x = + & equal to zero. It is sufficient for us to consider the case in which the uneven integral number has the value 1 (one), since all other cases (multiple buckling) are then easily explained. We have then ee =-(12) If we substitute equation (11) in equations (1), (4), and (6), we get ~ aA - _ 7B+! -@B ) & ¢, Ak C2 Tea tee) cha bal 02 cos np sin panama a) lap = — 2(n+8) aw = ~2+8) cos n poi Let us introduce these values in equations (3), (5), and (7). For sim- plification we will write (1-@)(nB+1) = C, nB + 1—a@aA = D. Then Pee, Bake beOu, als == [(i-ayen p-a2)—C(n?-a2)]w = - - -----(1) =o= 62 = 4% [oat+(n2- eine S, )) ew --~~-(13) ashe = £5 [ac a)(n?—/)+ a?C] ur The substitution of the expressions (13) in Eqs. (I) to (III) with the simplifications 2 Ao sx, pg srg, Lyte si eupe. Sees (14) gives the following three linear equations in C and D: DN ssa Glee |e VIN (ei) D[on?-a?}+ Cla*tn?+a,x] SMAQQ AIM GIG Wits Bherwe (rr') D[on?-a?-a] + C[n’+a*-|+a52] (lane) aan (rrr) Here the coefficients of the four x = terms which appear are desig- nated by the abbreviations aj, 4, zs aye These values are as follows: a, = we + a? Ay =—(N?-1)(n2+2?)-oa? lead Mie’ Aor el” Oba eee ee (15) a = —_@? QA, = ar ne+ar-s 3 /-o 4 ( ) In order that these three equations (I*) to (III') shall be ccnsistent the determinant of their coefficients must vanish. (Translator’s note: See "Theory of Equations", Dickson, p.l21). This gives the required relation be- tween x and y. 4. Equation for the Buckling Pressure. The equation for the determinaticn of the critical value of y, namely, (l-) 72+2a@2 = —(n2+a?) a(1-C) on?— a2 N2+a?+a,x A2x we (16) on?-a*-o@ = °+@*-1+a3x agxt+y(I-n?) has, as one can sasily determine, the form VUA+ Bxjie= (C\ opm nee eae (17) The coefficients A to E can easily be determined separately. We obtain A equal to (1 = n”) times the two-rowed determinant to the left in equation (16) above, with 4; set equal to zero. A= (1 -n®) (n” +a@*) [(1-0)n? + 2@* + n¥*o - a] = (1 - n?) (n? +a2)* In the same manner, except for the factor 1 - nN”, B is equal to ay times the first member of equation (16). B= (1 - n*) [n? + oe [pe + (2 - a4] The absolute term C of eq. (17), except for the sign, has the value of the determinant eq. (16) if x and y are set equal to zero in it. ples 2 Nr+a@z2 =C=a)) G2 || C&S ee Gnd) on?-ato n*+ar-) | — BiG ia le) The coefficient E of x” is also found ina simple manner. ~E =|(I-o) ne+ 2a*] 42 |= a4(n?+a?) [c-2) nt+ 2a] a 3 The determination of D is somewhat more cumbersome, since it requires the combination of two determinants. (1a) n2 +2ae =(n2+ a2) AG) ce -D= (e) a, Gp onr-a-o n? +a2-/ Q Cl-o) n2+2a2 —(n*+a*) (l-0) a? + Onna n?+ar nO) ag a4 ave, computation leads to the val -D=(n*s a4)" ant[n2+(240) a2] [n?+ 35 a] +[n2+(-0)a?] [N24 2(/+0) a] (Translator's note: This value can be readily checked if we make the substit- ution n® +a@* = Z and collect in powers of Z.) Thus eq. (17) is completely solved. In order to write this expression in a simpler form, we can make the substitution of the quotient az apart Ra OOS Club astudiaides ster ha" as (he) in place of @. After this simplification we get the following equation (A) Yili ce ee = ar IEEE @2 x — Int 40° oe tia [qeeye — 2n* {i++} ae Q} +(U-aQ) (i +2o)eff+at 2 8 ie aye G o+(i+0)@/ (Translator's note; In the text the last term in the bracket on the left hand oO side is given Ta If we now replace O by its value 0.3, equation (A) takes the form: 0.9] ax nt y[i+z(1-03@) (I+ 1.86@)] = Az @2 + cay [ @agpe 2024 13Q) (I+0.35@) + (J- 0.39) (I+/.6@)] +o =a P_(OPSOQEM Baas (A) (Translator's note: In the text, the last term in the bracket on the left hand side is given 0.430) Since y is equal to P» except for a known factor, we have in equation (A) the required expression. 5. Simplification of the Equation. The equation (A) represents an hyperbola in an x-y coordinate system. =O= In the particular case Q= O (tube of infinite length), equation (A) reduces to y(l+x)=x(n®-1) 0 wenn nn nn------------ (19) From the definition in eq. (14) it follows that y represents the mag- nitude of the elastic compressive strain of the tube wall under the pressure p. Even though we go beyond the proportional limit of the material !see further be- low), y for iron or other metal will smount to not more than a few thousandths at the most. Now the borizontal asymptote of the hyperbola, eq. (19), has the ordinate n* - 1, which is at least 3. We see, therefore, that the partic- ular part of the hyperbola in question has only a very slight curvature and can therefore be represented by a straight line with sufficient accuracy. We omit, accordingly, the last member to the right in eq. (A) and bring the value of y, considerably shortened, on the other side. (Translator's note; The omission of 2 the x~ term is completely justified since x is always very small and the coeffi- cients of the x and xe putation for the E.M.B. model S III 125D50T1 gives x = 2.632 x 1076, x? = 6.927 x 10712, 9 = 0.3815, and using n = 16, eq. (A') reduces to y(1l + 3.98 x 107°) = 519.2 x 1079 4 668.5x + 116.9x* = (519.2 + 1759.2 + 0.00081) x 1076 terms are of the same order of magnitude. An actual com- Since the xe) term contributes only 0.00081 as compared with 1759.2 for the x term it is surely negligible.). If we remove, then, the factor {n® - 1)? from the co= efficient of x we have: Veo 7 + x[n2— -l+- h2- a nz—/ woere A a Age 9 [3+0+(i-0%) 9] =eE(i+o)-o *fa(I+2a) +(1-0 o*)(1-o9)(1+ 42 14.07 = 9)] (franslator's note: Tn the text, the denominator of A,is given 1 - OF ie In general, it is sufficient to regard @ as a small quantity and to neglect higher powers of @ in computing XA. (Translator's note: The values of @ for E.M.B. models S IJI 2000D 50T1 to S III 125D 50T1, which represent lengths of from 2D to 0.125D, vary from 0.024 to 0.33 and their squares vary from 0.0006 to 0.11. The neglect of e* terms causes an error of less than 1% for models whose frame spacing is equal to one half the diameter or greater. However, for the E.M.B. model SX 154D 50T1 with a frame spacing of 0.137D, the error is 17%. This frame spacing corresponds to that commonly used in submarine design. The slight error noted in (A) where 7 is given in place of gis is unimportant since it is multiplied by Q* in the last term of (B) and disappears.) It becomes. evident then that the coefficient of @ Will become divisible by n® - 1 and we find in place of equation (B) y RESO? +x fn? - 1+ p2aplen® - 1-0) ] oSeososs {c) If, in eq. (C), we substitute the value of Q from eq.(18) and the val- ues of x and y from eq. (14) and place O = 0.3, we finally obtain the equation for the critical pressure, p: EE 3 P= yaa A 0.73E | bee sera ja Wie ia Ita (Translator's note: Equation (C) is given incorrectly in the text where the de- nominator 1 - 2Qis omitted. This mistake follows from the use of 1 -Q 2 for (1 -Q)* in the denominator of Aj. This same error also affects the final formule ND. The denominator of the last term, given as 1 + (26 2 in the text, actually becomes al eng when the corrected equation (C) is used.). If we omit the fraction that stands in the parenthesis beside n® = abe we have the approximate solution of Southwell, which has been given above. Tor the general range of application, eq. (D) gives directly, as can be seen,a use-~ ful approximation. (YTranslator’s note: Southwell's equation and eq. (D) become identical when in (D) we neglect 1 in comparison with (24)? and omit the frac- tion notea above, and in Southweli's equation we place Z equal Toes 4 This 4 16 value of Z is derived by Southwell (loc. cit.) for the ideal type of end con- straints which merely keep the ends circular without imposing any other restric- tions upon the types of distortion, and was also used by Cook, = Phi).Mag.,Oct., 1925, pp.844-8. The value ot (4a) ? varies from 40 for the E.M.B. model series III 20G0D50T1, to 2.8 for the E.M.B. model series III 125D50T1.) The quantities that stand on the right hand side of eq.(D) are all given directly with the exception of n: that is, the length of the tube, 1; the radius, a; the wall thickness, 2h; and the elastic modulus, Kk. Concerning nan, which must be a whole number, more will be said later. 6. Discussion of Results. If we set Q= © in eq. (C), we obtain for the tube of infinite length, ye (a? 3°) se In the x-y-coordinate system the lines converge at the origin and thosewith the greater slopes have the greater values of n. The smallest value other than zero is for n = 2, that is, y = 3x, or, if we substitute in eq. (D), f =COand n= 2 we get Dis 2.19 E(B)". -10- This is identical with the above-mentioned equation (a) with o = 0.3. If a/Z and, therefore, @ have values other than zero, equation (C) represents a particular straight line which will cut the axis of ordinates in some point for each value of n. The ordinates of the point of intersection - equal to the term which is free from x in eq. (C) - decrease with increasing n and constant a/1, aco2 while simultaneously the slopes of the lines - represented by the co- efficient of x - increase. It fol- lows, therefore, that the straight lines represent a portion of a pol- = ygon that bends downward. The point x = 0, y = 0, is at- z jE ACEEEAS 2 1012 14 16 18 20 tained only for the polygon,n =© ,. Fige 3. The least critical pres- Now n, however, signifies the num- sure for various wall thicknesses ey 2h and tube lengths 1. ber of lobes that appear in the cir- cumference at the time of collapse. the figures 4 to 6 show the shape of the de- formation of the circle through w = C sin n 9 when n = 2, 3, 4e (Translator’s note: wis the displacement in the radial direction.). We have, therefore, the er the number of incipient waves. (Translator's note: This can be seen directly from Fig. 3). Of course, in the resulting deformation, not all the waves will ap- pear completely formed, but each wave will have a length equal to the circumfer- ence divided by n. Oe Coes Figs. 4 to 6, Deformations of the four times the scale for the In Fig. 3, the polygons are shown corresponding to the values of a/l = 0.5, 0.4 0.3, .2, -0.1, and the straight line for Z=0o é Fig. 7 shows the lines to circular cross section for n equals 2 to 4. particular region in question to about h/a = 0.014. In both figures, the lengths along the x-axis (on which x Zz so is given simultaneously) are given in values of 1002 ° 3 la a exiees Table 1 contains those values of the whole number n for the above given values of a/f and for intervals equal to 0.002 of the ratio h/a, which give the least critical pressure. The table is made from direct readings from Figs.3 and 7. Further, Table 2 also contains the magnitudes of the least critical pressures, for the same values of a// and h/a. Equation (D) is the basis for calculations, but improvements are taken into consideration with regard to the more exact equations (4) and (B). The values of p given in the table are valid for an elastic modulus of E = 2,000,000 kg /om® (28 x 10° lbs. per sq. in.) and can be changed for metals with another modulus E' in the ratio &'/E. In both tables those values are cut off at the lower right hand side which correspond to stress of more than 1800 kg/cm® (25,600 lbs.per sqein.) YA Re = 1800 -------(21) In the same way, a dotted line is ar drawn in Fig. 3, which corresponds to eq. (21), namely. 000026 = 8.19 x 10~¢ We will return to the significance of this demarcation immediately. az 04 06 08 10 12 44 ook Fig. 7- Fig. 3 to a larger scale. If the values of h, a, and 2 are such that the tables are not sufficient, we must determine the pressure p for various values of n from eq. (D). The determinative val- ue is then the least of these values. Teble 1. In Fig. 8 are represented on the co- The number of lobes n around the puroumberence, SOrEvautens 11 thiekn 2h t iS le ee Ina ordinates a/l and 100 h/a the lim- iting lines that separate, for ex- a ample, the region in which n = 3 gives the smallest value of p from the region in which n = 2 or q is de- terminative. The use of Fig.8 saves Table 2. Critical pressure p in kg. per sqe cm. for garious wall thick- many trials. nesses 2h and tube lengths 1 for E= 2 x 10~ kg. per Sqe CMe -l2- 7- Comparison with Experiment. The comparison of our results with the accumulated observations of the buckling pressure of boiler flues becomes very difficult for several reasons. First,the exact value of the elastic modulus E is almost never given for the experi- ments. This is of particu- lar importance if the aver- age pressure lies near the elastic limit, so that E sinks far below the custom- o az ag 6 OB 0 ALE LB «RO gah ary value. Second, the ex- Eee Tone eee ee perimental arrangements pro- vide no certainty that the ends of the tubes can be considered as fixed, in the sense that w= 0 is set for e=+ 2/2. Analogous to the case of ordinary buck- ling, we will, by not having sufficiently rigid ends, introduce e greater value of & as the effective length ("Massgebende Lange"). Finally, the given wall thicknesses are often to be regarded as the average for somewhat variable magni- tudes while for the proposed problem it appears rather that the minimum value should be used. There are to be considered: Experiments by Fairbairn (1858), by Rich- ards (1881, discussed by Wehage, referred to by Bach), the experiments of the Danzig shipyards,(1887 to 1892) and those of A.P.Carman (1905). All those cases were eliminated for which the average stress on the surface of the material of the tube was above the proportional limit, that is, about 1800 atmospheres. (Translator's note: An atmosphere is slightly greater. than a kg./om*).This core responds in eq.(22) to the imposed upper limit on y of 3.19 x 10°4, The number of experimental results to be compared, therefore is not very great. For the Carman experiments (Physical Review 21, 1905, p.381) conducted with very small, thick-walled brass tubes, this limitation permits only the tubes of very great length to be considered: They give, as was noted by Carman himself very good agreement with the formule (a) for (=a, For the experiments of the Danzig shipyards (Z.1894, pe689), only two lie within the valid region of our derivation, namely: 2a = 100 em, ¢ = 106.2 cm, 2h= 0.81 cm, p= 24 atm. =i5— 2a = 100cm, Geis 198 cm 2h= 1.14 cm, p= 32 atm. If we set up the quotient h/a and a//, a glance at Fig. 8 or at Table 1 shows that in the first case we must use for computation n = 5 and in the second n=4. Our formula (D) gives then, using E = 2 x 10°ke./em®, p = 30 for the fird tube and p = 39 for the second, which is about 23 per cent more than the observed values. In order to give an estimate of this variation, it might be remarked tha for the first tube the observed value p= 24 would result if the wall thickness were about 0.4 mm less and the effective ("massgebende") tube length about 10% greater. For the second tube, eq. (D) would give DiS 32 atm. if instead of 11.4 _mm as wall thickness, 10.8 mm be used. These are differences which may very pro- bably be assumed in the whole range of the experiment since it was sought to at- tain the most probable conditions in practice, not exact experimental conditions. Concerning the five experiments of Richards (Engineering, 1881,I p.429. Compare Wehage, Dingler's Polyt. Jour. 1881, Vol.242, p.236), two experiments, ace cording to the descriptions of the authors, were made on old, second-hand tubes that were lapped and riveted and therefore must have shown marked departure from the circular form. For a third experiment, the tube already had an observed bulge. For these three cases, our formula gives approximately twice as high a collapsing pressure as that actually observed. For the two other experiments (welded and strap jointed tubes) the results were exactly as for the tubes of the Danzig Navy Yard. The experimental values are: 2a = 96.5 em, l = 218.5 cm, 2h = 1.27 cn, P 2a =137-1 cm, 2 = 91.4 em, 2h = 0.635 cm, op 31.6 atm; 9.0 atm. The computation by formula (D) gives in the first case p = 45 atm. for n = 3, and in the second p = 11.9 atm. for n = 7- In order to obtain the observed values from the formula, it is sufficient to increase the length about 10%, from consideration of the defective support, and decrease the wall thickness about 0.5 mm from consideration of the inequalities, the welds, etc. The experiments of Fairbairn (Trans. Royal Society, London, 1858,p.389), which were not undertaken with actual boiler flues, but with small, carefully pre- pared models, furnish for the most part results which completely agree with our computations. We choose three examples at random, or, to be exact, from those which, according to Fairbairn’s data, show the most uniform lobe formation. 2a = 6 inches, a/Z = 0.1 h/a = 0.00717 pexpt. = 4.6 at p calc. = 4.8 etn. 2a = 8 inches a/Z = 0.133 h/a = 0.00537 pexpt. = 2.75 at p calc. = 2.85 atm. 2a = 8 inches afi = 0.1 h/a = 0.00537 pexpt. = 2.18 at p calc. = 2.15 atm. Also Fairbairn observed the increase of the number of lobes with the de- crease in the length of the tube in agreement with our theory. In general we might say that the available experimental data do not per- mit decisive conclusions, but point to the usefulness of the formula in design. Concluding Remarks. It is possible to object to the practical application of the formula here developed, since very frequently the collapse of boiler flues follows only from pressures that do not satisfy eq. (21). In these cases, the proportional limit is exceeded and our derivation is no longer valid. The behavior here is quite similar to the ordinary buckling phenomena. For rods which are not very slender, the Euler formula is no longer applicable but must be replaced with em- pirical formulas. We might, however, refer here to the expedient that is applied with suc- cess in the theory of buckling. The formulas for the critical pressure remain the same if in place of E another suitable value is substituted. In the first ap- proximation we might substitute E, from the slope - in general variable --of the stress-strain curve. It is more accurate, as v. Karman (Mitteilungen fiber Forsch- ungsarbeiten, Vol. 81) has shown, to choose an intermediate value between the slope value and the elastic constant E. (Translator'’s note: The resulting mod- ulus E’ that should be used in eq. (D) is in general an intermediate value between the two moduli E - within the proportional limit - and Ey - at failing stress - and can be expressed as EB’ eh ee - See v. Karman, loc. cit. p20). The ( JE +./Ey)* place at which the slope is to be chosen must of course be found through a pre- vious estimate. Another range of application, for which the above derivation is again not valid can also be pointed out. For corrugated tubes the principal results can be applied. It is only necessary to introduce in the fundamental equations an increased bending rigidity in the circular section; In the expression %q.(5) a value must be substituted for Kp that represents the increase of the moment of -15- inertia calculation. Summary. From the elastic theory of thin, elastic shells, an accurate formula has been derived for the buckling pressure of smooth boiler flues. For the ma-~ jority of cases, formula (D) is considered sufficiently accurate: ni 2n2-1.3 A = > Ep, + 0. 73E/[n rea CICA Geyer das (o) where, 2a is the diameter, 2h the wall thickness, / the tube length, E the elas- tic modulus, and where n (the number of lobes) is that whole number which makes the pressure p a minimum value. The Tables 1 and 2, and the Figures 4, 7, and 8 facilitate the application. Comparison with the present partly imperfect experiments leads us to expect that the new formula will prove true within the proportional limit, as does the Euler formula for buckling load. Concerning the extension of the theo- ry to non-elastic cases and to corrugated tubes, the concluding remarks above contain a suggestion as to possible procedure in regard to this problem. (Trans- lator's note: It should be borne in mind that formula (D) differs from the form- ula given by von Mises in the German text and quoted by v. Sanden and Gtinther, (Werft und Reederei, 1920, Heft 10, p. 217) and Johow-Foerster, (“Hilfsbuch ftr den Schiffbau", Berlin, 1928, p. 929), where the denominator of the last term is given as l +(pe) instead of (24) - 1. The difference is negligible for models with moderately long frame spacing, (Z = a or greater), since (Ta) is large com- pared with unity and the entire fraction is small compared with n®, However,this difference becomes very great when attempts are made to apply the formula to the region of short frame spacings used in submarine design. The following table shows experimental and computed values for several E.M.B. models: E.M.B. COLLAPSING PRESSURE, lbs.per sq.in. Theoretical MODEL Experi- mental NUMBER SIII1LOOOD50T1 SIII 500D50T1 SIII 250D50T1 SHI54D50T9OU2 D E = 430,000,000 lbs. per sq. Von Mises has pointed out t 3 formula (D) is not appldeanie for stress- =O es beyond the proportional limit unless the corrected value of E be used. It should also be noted that in this region, the values of ¢2/a must necessarily be small. Hence ge becomes fairly large and its square cannot be neglected. Therefore the more exact formula (B) must be used instead of the approximate formula (D) which was obtained from (B) by neglectinge*. The two formulas may give values differing by 20% or more as shown in the above table. it is evident, therefore, that formula (D) does not apply to short frame spaces. The attempts of the Germans(Johow-Foerster,"Hilfsbuch fiir den Schiffbau” Berlin, 1928, p.929) to apply this formula to short frame spacings by using coef- ficients of 0.4 to 0.6 depending upon the shell thickness were based on the as- sumption that the discrepancies were due to variations from circular form. It should be noted also that these coefficients were applied to the incorrect form- ula (D) which gives values at least 20% below those given by the exact formula (B) in this region. If it is desired to apply these instability formulas to submarine de- sign, the exact formula (B) should be used and also the corrected value of the elastic modulus. This latter value is, of course, very difficult to obtain. It requires an accurate knowledge of the elastic curve and the stresses at the time of collapse, since Ey varies rapidly as the yield point of the material is ap- proached. On the other hand, the application of a constant multiplying factor de- pendent upon the shell thickness only is but a makeshift device for extracting reasonable answers from a formula in a region where it was never intended to be used and where it can not possibly give reliable results. If any multiplying face tor is to be used, it should at least be a function of the stresses at the time of collapse and not of thickness only. Tests conducted at the U.S. Experimental Model Basin show that formula (D) gives very good results for models whose Jength is equal to or greater than the radius, erring slightly on the side of safety. For shorter models, formula (D) gives values which are considerably too high.) References. (1) Bach, C., "Elastizitdt und Festigkeit”, 1905, p. 273. (2) Love-Timpe, "Lehrbuch oer Elastizitat”, Leipzig, 1907, Pe O37. (3) Lorenz, R., Physikal. Zeitschrift, 1911, p. 257. (4) Southwell, R.V., Phil. Mag., Vol.25, 1913, p. 687. (5) Love, A.E.H., "The Mathematical Theory of Elasticity", Fourth Edition. (6) Prescott, J., "Applied Elasticity", 1924. (7) Dickson, "Theory of Equations". (8) Cook, G., Phil. Mag., Oct. 1925, pp. 844-8. (9) Carman, A.P., Phys. Review 21, 1905, p. 381. (10) Zeitschrift 1894, p. 689. (Danziger Werft). (11) Richards, Engineering, 1881, I., p. 429. (12) Wehage, Dingler's Polyt. Jour., 1881, Vol.242, p. 236. (13) Fairbairn, W., Trans. Royal Society, London 1858, p.389. (14) Karman, T. v., Mitteilungen uber Forschungsarbeiten No.81, 1910. (15) Johow-Foerster, "Hilfsbuch fur den Schiffbau", Berlin. (16) v.Senden umd Gunther, Werft und Reederei, No. 10, 1920. ofS ot, ER as paet raleqrel ls wen x Kae hits o et Teo HR spate, ssh et previo ttwalr Then : hanes wet, ees OF he ee “ e : opt ROE tS abees : ie acon “eostzaat) Bed : atPret rou ir ee tye, we al mn iain 4 ry marin an: dui Ahi