U. S. NAVAL INSTITUTE, ANNAPOLIS, MD. SHRINKAGES (APPLYING CLAVARINO'S EQUATIONS). By LIEUT. J. H. GLENNON, U. S. Navy. Let R represent the radius of the bore, R' the radius of the surface of contact of a hoop or jacket with the cylindrical inner body (which may be either a simple tube or a compound tube put together in any way whatever, provided always that the modulus of elasticity is and remains constant throughout, and that the elastic strength of the metal is not at any place exceeded at present or at any future stage), and R" the external radius of the hoop (which likewise may be simple or compound with the same proviso). Let P, P', and P" represent any three simultaneous pressures that exist at these respective radii in the assembled gun. By (13), Meigs-Ingersoll's Elastic Strength of Guns (1891) we have for the elongation of any radius r in the inner body. (modulus of elasticity, E,), caused by the pressures P and P', The compression of the diameter is evidently the double of this with a negative sign, or 2R'5PR-4P'R2-P'R'2 ̄ ̄ (b) U. S. NAVAL INSTITUTE, ANNAPOLIS, MD. SHRINKAGES (APPLYING CLAVARINO'S EQUATIONS). By LIEUT. J. H. GLENNON, U. S. Navy. Let R represent the radius of the bore, R' the radius of the surface of contact of a hoop or jacket with the cylindrical inner body (which may be either a simple tube or a compound tube put together in any way whatever, provided always that the modulus of elasticity is and remains constant throughout, and that the elastic strength of the metal is not at any place exceeded at present or at any future stage), and R" the external radius of the hoop (which likewise may be simple or compound with the same proviso). Let P, P', and P" represent any three simultaneous pressures that exist at these respective radii in the assembled gun. By (13), Meigs-Ingersoll's Elastic Strength of Guns (1891) we have for the elongation of any radius r in the inner body (modulus of elasticity, E,), caused by the pressures P and P', r PR-P'R' 4R''R' (P-P') sr = 3ER-R 34 + (R11 — R2) r2 The extension or elongation of the exterior radius of this inner body (substituting R' for r) then is The compression of the diameter is evidently the double of this with a negative sign, or 2R5PR-4P'R-P'R'27 —24R' = — (b) Proceeding now to the hoop (modulus of elasticity, E); the elongation or extension of any radius r in its thickness is, by (13), Meigs-Ingersoll's Elastic Strength of Guns, r P'R'-P" R''2 4RR (P-P") Ar= 3E (R''2 — R'3) ra The extension or elongation of the interior radius of the hoop (substituting R' for r) then is, The extension of the interior diameter of the hoop caused by the pressures P' and P" then is, The various simultaneous pressures that may occur in the bore, at the surface of contact, and outside the hoop simply move the surface of contact in or out. An increase in the compression of the exterior diameter of the inner body is accompanied therefore by an equal decrease in the extension of the inner diameter of the hoop (neglecting the very small, secondary, quantities caused by varying pressures upon the thin ring of metal whose thickness is half the shrinkage). The shrinkage is the sum of the compression of the exterior diameter of the inner body and the extension of the interior diameter of the hoop (or jacket) and is (adding (b) and (d) and denoting the relative shrinkage or shrinkage per inch of diameter by ), This formula is true for any simultaneous pressures that may occur at the points indicated. The shrinkage is most readily calculated by supposing the |