ITTERNAL PRESSURE.] 



APPLIED MECHANICS. 



799 



might be increased until it attained sufficient magnitude 

 to burst or rend asunder the material. Vessels destined 

 to sustain considerable internal pressure, are generally 

 made circular in section ; and a few simple considerations 

 will enable us to ascertain the amount of strain, which 

 the pressure in such vessels throws upon the material of 

 the casing, and hence to compute the strength of material 

 that should be employed in their construction in order 

 that they may sustain a given pressure. 



Let us suppose that a steam-boiler is of cylindrical 

 form as in section 1 (Fig. 59), having pressure exerted 

 on it equally, on every part of its surface, as indicated 

 by the arrows. One effect of this pressure, if it exceeded 

 the strength of the material containing it, would be to 

 rend it asunder at some point A, section 2 ; or it might 

 rend it at two opposite points, B and C, section 3, so as 

 to force one-half of the casing away fiom the other. 

 Now, if the material were of perfectly uniform consis- 

 tency and strength throughout, there is no reason why 

 one point A, or any two points B and C, should be 

 selected for breakage more than any other points ; but 

 as, in practice, materials are never perfectly uniform, so, 

 in theory, we may suppose the casing to be somewhat 

 weaker at A, or at B and C, than elsewhere, and trace 

 the influence of the pressure to effect a breakage at such 



Fig. 99. 



points. Were we to suppose the whole of the vessel to 

 be filled up with solid matter, except a thin film of fluid 

 in the middle, as shown in section 4, exerting a pressure 

 on every point of the flat side of the solid portion ex- 

 posed to it, we should involve no change in the con- 

 ditions, for the two points of the circumference, H and 

 K, terminating the film of fluid, would have to resist 

 exactly the same bursting-strain as if the rest of the 

 casing were filled with Huid instead of solid matter. 

 Farther, as this section would apply at any part of the 

 length of the cylinder, we may take a belt or portion of 

 its length, one inch in breadth, and trace the effect on it 

 an effect which would be repeated equally on every such 

 belt one inch wide throughout the whole length of the 

 cylinder, if the material were homogeneous throughout. 

 Knowing the pressure which the internal fluid exerts on 

 every square inch of the casing, we see, by section 4, that 

 the bursting pressure on a belt of casing one inch wide, 

 is to be measured by the pressure on a surface repre- 

 sented by the line H K that is, on a surface having for 

 length the diameter, and one inch in width. If this 

 pressure burst the casing at one point E, section 6, so as 

 to cause one-half to turn round the opposite point L as a 

 fulcrum or centre ; then, by the well-known principle of 

 the lever, that a uniform pressure, acting at every point 

 of the arm of a lever, has the same effect to turn it 

 round its fulcrum, as if it were all collected into one 

 force acting at the middle point of the arm, we see that 

 the bursting pressure is equivalent to a force acting at 

 D (as marked by the arrow), while the cohesive force of 

 tbe casing acts at E, double the distance from the ful- 



crum L. Hence we conclude that the bursting force, as 

 : resisted at one point of the circumference, is half the 

 pressure on the diameter, or the pressure on half the 

 diameter, or on the radius. Again, if the vessel open at 

 two points F and G, section 6, we see that the whole 

 bursting force, marked by the central arrow, being re- 

 sisted by two equal forces at F and G, each of those 

 forces need be only half the bursting force ; so that, in 

 this case, we find that the effect to open the circum- 

 ference is equal to half the pressure on the diameter as 

 before. 



To apply these considerations in practice, let us sup- 

 pose that a boiler of English wrought-iron, 6 feet in 

 diameter, has to sustain an internal pressure of 100 Ibs. 

 per square inch, and ascertain the necessary thickness 

 of the plate to resist this force. Taking a belt of it 1 

 inch wide, 



Half the diameter, 3 feet or .... 36 ins. 



With a width of 1 in. 



Has a surface of 36 sq. ins. 



Multiplying by pressure per sq. in. . 100 Ibs. 



The pressure on half the diameter is . 3600 Ibs. 

 which is the bursting strain on any belt of the circum- 

 ference 1 inch wide. 



By Table IV., English wrought-iron 

 having 1 sq. in. area of section, sus- 

 tains 18,600 Ibs. ; and by a simple 

 proportion, 18,600 Ibs. : 3,600 Ibs. : : 

 1 q. in : 0'2 sq. in. nearly. We find 

 therefore, that the sectional area of 

 the belt of circumference must be 0'2 

 oq. in. ; or that, as it is 1 inch wide, 

 its thickness must be 0-2 in., or -^ 

 or ^ of an inch. In practice, boilers 

 are made of numerous plates riveted 

 together; and as the material is cut 

 away by the rivet-holes, it is weaker 

 at these places than elsewhere. It 

 would, therefore, be advisable to make 

 the strength of the plate at least double 

 that calculated, which would give a 

 thickness of about J of an inch. 



In the case of hydraulic presses 

 which are made of cast-iron, and have 

 to sustain enormous pressures, it is 

 necessary to make the material of great 

 thickness. There is, however, a limit to the pressure 

 which such cylinders can sustain ; for it is found that after 

 a certain thickness of iron has been attained, the actual 

 substance of the iron becomes compressed, so that the 

 inner surface is considerably extended, while the outer 

 surface sustains scarcely any bursting force. Additional 

 thickness of metal does not therefore contribute propor- 

 tional increase of strength. Indeed, the practical difficulty 

 of casting thick masses of iron sound and solid in texture, 

 renders it almost impossible to construct hydraulic presses 

 of any very great force. It is doubtful whether, in any 

 case, it is advisable to exceed a thickness of 6 or 7 inches 

 when cast-iron is the material employed. The same 

 remarks apply to a certain extent in the case of ordnance : 

 but in making large guns or mortars, a considerable ad- 

 vantage results from the process of manufacture. The 

 gun is cas.t solid, and afterwards the heart is bored out. 

 Most of the unsound and spongy portions of the metal 

 are thus removed, and the shell left is much sounder 

 than it would be, were it cast hollow, as hydraulic presses 

 generally are. 



Vessels for containing fluids exerting pressure, are like- 

 wise subject to a bursting strain in the direction of their 

 length, which is resisted by the cohesion of the casing. 

 Taking the case of the boiler 6 feet in diam., and 0'2 in. 

 thick, we may easily calculate what pressure it would bear 

 before bursting in the direction of its length. The whole 

 boiler may be taken as a hollow or tubular bar of wrought- 

 iron, capable of sustaining a certain weight hung to it, 

 equivalent to the pressure on one of its circular ends, tend- 

 ing to force or blow it to a distance from the other end. 



