inertia calculation. 



■15- 



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: 



yO = 



ZE 



0.73 E In 



'-I + 



2ng-/.3.1 /t^ 



J a-2 



cnM[i+(^rj'^ " L mr-i-i^' (d) 



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 £ a minimum value. The Tables 1 and 2, and the Figures Jt 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 GQnther, 

 (Werft und Reederel, 192O, Heft 10, p. 217) and ,Tohow-Foerster, ("Hilf sbuch ftlr 

 den Schiffbau", Berlin, 1928, p. 929 ) > where the denominator of the last term is 

 given as 1 +('— ^ instead of f—\ - 1. The difference is negligible for models 

 with moderately long frame spacing, (/ = a or greater), since ( 2£.) is large com- 

 pared with unity and the entire fraction is small compared with n^. Hov/ever,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. 



MODEL 



L 



d 



Experi- 



Theoretical 1 



Eq.(B) 



Eq.(D) 



Incorrect 





mental 







Eq.(D) 



NUIffiER 













SIIIIOOOD5OTI 



1.0 



% 



51.7 



51.8 



51.5 



SIII 500D50TI 



0.5 



96 



100.3 



101.2 



97.7 



siii 250D50TI 



0.25 



140 



232.9 



243.8 



214.7 

 33b.b 



SX154D50T99U1 



0.137 



170 



425.5 



497.1 



SIII 125D^OTl 



0.125 



195 



40b. 



479.1 



312.1 



E = 30.000,000 lbs. per sq. in. 

 Von Mises has pointed out that formula (D) is not applicable for stress- 



