FlightGlobal.com
Home
Premium
Archive
Video
Images
Forum
Atlas
Blogs
Jobs
Shop
RSS
Email Newsletters
You are in:
Home
Aviation History
1931
1931 - 1039.PDF
SEPTEMBER 25, 1931 71 THE AIRCRAFT ENGINEER SUPPLEMENT TO FLIGHT By equation (1) this integral may be written I 2G~ therefore IT 4GA,rt2J t If t is constant and S is the mean perimeter, (2) becomes ITS6 = (2A) We now pass to the more general case in which such sec- tions as C and D in Fig. 1 have to be considered. The discussion of the general case will be divided into three parts: (i) The general equations of the torsion problem ; (ii) Cylinders with multiply connected cross sections ; (iii) The stresses in multiply connected cross sections. (i) General Equations. The equations for any cylindrical prism under torsion are well known, but are given here for the sake of completeness. If coordinate axes are taken so that the Z axis is parallel to the generators of the cylinder, then the displacement («, v, w) of any point in the prism from its unstrained position is given by u = — Qyz, v = %zx, w = %<f>(xy) Where 0 is the angle of twist per unit of length of the cylinder. by the shortest distance between the two contours gives the resultant shear strain per unit angle of twist at that point. It follows therefore that the resultant shear stress fs at any point X in Fig. 3 is given by, XL" _ \T" /s = G6^T-B (3> Where TA is the value of T on contour A and TB is the value of Y on contour B, Sn being the length of the normal to the curve A (or B) at the point X intercepted between the curves A and B.J Another important property of *F, to be inferred from" (3), is that it is constant round any boundary of the section. From this it follows that if TA and TB are the values of ¥ on the inside and outside boundaries of a thin-walled hollow section, then the shear stress at any point is given by equation (3) above. It should further be noted also that T satisfies the differen- tial equation and that the components of shear stress parallel to the x and y axes are G6 -—, and — G0 — respectively. This cy ox is important in the calculation of the torque. (ii) Cylinders with multiply-connected Cross Sections. The cross-sections of cylinders may be classified, for the The torsion problem is to find the function tj> which satisfies all necessary conditions. It is known that <f> must satisfy the equation + = 0 This being so, it follows that there is a conjugate function V su?h that d<f> _ dty , d<f> _ d<l> d* dy dy dx whi'''j satisfies the same equation. Frrtkermore. with the function ^ there is associated a UIK ion Y such that F ' our present purpose the most important property of * i>' hat if lines of constant *F be plotted on the cross-section °i tl>; prism (as contours are on a map) as, for instance, for the i\ otangle in Fig. 3, then the difference between the values °' -i for two consecutive contours near a given point, divided present purpose, according to a scheme borrowed from hydromechanical theory. Consider Fig. 4. A section such as (a) is termed singly ' connected ; one such as (b) doubly connected ; one such as (c) triply connected ; one such as (d) quadruply connected and so on. It is observed that the number of connection, is equal to the number of boundaries, and therefore to the number of values that W may assume at the boundaries of the section. The explanation of this mode of classification by reference to " degrees of connectedness," is as follows :— In (a) any closed circuit c can be contracted to a point without passing outside the material of the cross section; in (b) one circuit can be drawn which cannot be contracted to a point without passing out of the material of the cross- section ; in (c) two such circuits, which are irreconcilable, can be drawn ; and in general, in an »-triply connected cross- section (n — 1), such circuits can be drawn. Circuits are irreconcilable when they cannot be made to coincide with one another without being displaced out of the material of the cross section. For instance, in (c), Fig. 4, the two
Sign up to
Flight Digital Magazine
Flight Print Magazine
Airline Business Magazine
E-newsletters
RSS
Events
