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STRESSING Failure of Cylindrical Tubes Under External Pressure This applies only for steel tubes and has been determined by experiment. Thus to find the failing pressure of a thin tube, The Various Types of Failure Considered and determine its critical length. Suitable Formulae for Each Case Developed If this is less than the length of the tube use formula in (I) If greater use formula in (II). By R. C. C. Ringrose Formulae for the failure of thin tubes and cylinders REFERENCES TO LITERATURE NDER external pressure cylinders fail in (i) Very long tube of length 'l' Timoshenko. Theory of Elastic Stability. McGraw Hill. 1936. the following ways: R. V. Southwell. Phil. Trans. Roy. Soc. 1913. Pure tension, compression (due to hoop H. E. Saunders and D. F. Windenburg. Trans. Am. Soc. Mech. Engr., Vol. 53, No. 15. 1931. and radial stresses). Roark. Formulas for Stress and Strain. McGraw Hill. Flexural instability (Euler buckling). (ii) Short tube of length 'l' Elastic instability (buckling of the shell of the This formula also applies to a long tube con cylinder). strained at intervals of length 'l'. The ends are Notation assumed to be held circular. The failing pressure r=mean radius of the cylinder. 1a. Thick cylinders p is given by t=wall thickness. If the walls of the cylinder are relatively thick l=length of tube. compared with its radius (wall thickness greater E=Young's Modulus for the material. than 1/10/R), under external pressures hoop and v=Poisson's ratio. radial stresses arc induced in the walls. (iii) Thin tubes with closed ends These can be determined by Lami's method The external pressure here is both lateral and and give: longitudinal, and the ends are assumed to remain TABL E l–SUMMAR Y OF STRESSING CASES circular. Failure occurs when the skin breaks up Type of Stress into lobes and the failing pressure is given by Case or failure Formula 1. Thick cylinder Radial and hoop stresses r =internal radius p =internal pressure 1 1 r =external radius p =external pressure 2 2 s =loop stress s = radial compressive stress 2 1 1b. Semi-thick cylinders 2. Semi-thick Compressive hoop A moderately thick cylinder may collapse under where n is the no. of lobes formed in the tube by stress cylinder external pressure at stresses above the propor buckling. tional limit, but below the yield point. This is similar to the failure of a short strut. To find min. p: plot p against for integral 3. Thin cylinder Then the collapsing pressure is given by (a) Compressive hoop stress values of n >2. Then min value of p for the given (b) Buckling in this group of curves is the required value. (c) Long tube Skin instability (d) Short tube (iv) Curved panel under uniform radial pressure Sy=compressive yield point Skin instability length e For a curved panel af radius of curvature 'r' This formula holds only when propor subtending angle 2a at the centre the failing pressure is given by (e) Wit h closed Skin instability tional limit. ends (a) Straight edges hinged (curved edges free) c. Thin cylinders In thin cylinders there is no radial stress, and they can fail under a hoop compressive stress of (b) Straight edges clamped (curved edges free) Failure however is much more likely to occur due to instability (flexural and elastic). 2. Flexural instability where K is defined by the equation K tan α cot Ka=1. Failure due to flexural instability is analogous (f) Curved Skin instability panels sub to the buckling of a long strut (Euler effect) under tending angle 2a at Note compressive loading, i.e. the walls of the cylinder centre 1. When stress checking a thin cylinder, the flex. (1) Straight edges hinged cylinder should be capable of satisfying all the The formula for this (due to Föppl) is thin cylinder cases. (2) Straight Skin instability 2. For long cylinders, reinforced at intervals of edges clamped length 'l' by frames, it has been found by experiment that the strength of the frames I=Momcnt of inertia of cross section of the should be at least 10 per cent stronger than cylinder wall. The failing pressure of a short, thin tube the shell lying between the frames assuming p=failing load per inch run across the wall depends upon (among other things) the length. both cylinder and frame to fail under buck As the magnitude of the length increases the ling i.e. using Föppl's equation: 3. Elastic instability failing pressure decreases proportionately until a Failure here is due to local buckling of the skin value is reached above which the failing pressure when the walls arc very thin and it is from flat is independent of the length. where suffices c, s denote cylinder and frames plate theory that the following formulae are This value 'l' is known as the critical length obtained. and is given by the formula respectively. 138 Aircraft Engineering
Aircraft Engineering and Aerospace Technology – Emerald Publishing
Published: May 1, 1951
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