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Aircraft Engineering and Aerospace Technology
, Volume 25 (12): 1 – Dec 1, 1953

/lp/emerald-publishing/torsion-of-a-multiwebbed-rectangular-tube-z6pxzhvwes

- Publisher
- Emerald Publishing
- Copyright
- Copyright © Emerald Group Publishing Limited
- ISSN
- 0002-2667
- DOI
- 10.1108/eb032366
- Publisher site
- See Article on Publisher Site

STRESSING Torsion of a Multi-webbed In order to determine the constants B and C we consider the twist of the two end cells, thus: Rectangular Tube An Exact Solution of the Problem Treated Approximately by Mr Mansfield By W. H. Wittrick, M.A., Ph.D., A.F.R.Ae.S. A comparison of Eq. (8) with the recurrence Eq. (1) shows that the conditions for determining REFERENCE TO LITERATURE B and C are Introduction (1) Mansfield, E. H. Torsional Stresses in Multi-webbcd S =(\-p)S PREVIOUS paper by Mansfield1 con Rectangular Cylinders. AIRCRAFT ENGINEERING, Vol. XXV, 0 1 January 1953, p. 20. sidered the problem of" the torsion of a doubly symmetrical rectangular box con where S S , S and S are the values obtained 0; t n n+l taining a large number of equidistant identical by substituting /=0 , 1, n and («+l ) respectively webs. Mansfield overcame the practical difficulty into Eq. (7). On solving these two equations we Notation involved in solving the large number of simul obtain taneous equations by an ingenious approximate 2a=widt h of section 5=Ca-'"+ 1 >=-/3[(a-l)(a"-l)+ o(a"+a)]- 1 method consisting of the replacement of the webs 26=dept h of section by an equivalent continuous medium. This device Substitution into Eq. (7) now gives the expression /i=are a enclosed by Section=4a6 resulted in the derivation of a simple differential equation instead of the system of simultaneous «=numbe r of cells equations. t =thickness of front and rear spar webs The purpose of the present paper is to show r =thickncss of top and bottom skins that an exact solution of the problem is readily 2 obtainable, and that, using the theory of recur i=thickncss of each internal web Eq. (10) gives the shear flow Si in terms of the rence equations, simple results for the shear flows twist per unit length r . In order to determine r G=modulu s of rigidity and torsional rigidity may be determined. These in terms of the applied torque T we have r=twis t per unit length of tube results are derived for an arbitrary number of internal webs. !T=applied torque St=shear flow in top and bottom skins of /th Method of Solution On substituting Eq. (10) and summing the two cell geometrical progressions involved we obtain the Consider a tube (FIG. 1) of rectangular section r=at/nbt equation 2ax2b which is subdivided into n cells by a a=l+r+(r2+2r)1/2 series of (« — 1) equidistant identical internal webs. Let the thickness of the front and rear spar webs cj)=at /bt„ be t the thickness of the top and bottom skins t lt 2 p=t/t and the thickness of each internal web t. Further, if a torque T is applied, let the shear flow in the top and bottom walls of the /th cell be S . The Eqs. (10) and (11) provide a complete solution upward shear flow in the web separating the /th Eq. (1) then becomes to the problem. They can be simplified slightly and (/— l)th cells is then S — S_ { t v as follows: Using the well-known formula for the twist of -Vi+2(1+/-)A,-A = 0 (4) (+1 the /th cell we then obtain the equation Writ e ^.=0^/6/0 The solution of this recurrence equation is Then Eqs. (2) and (9) give A =5a i +Ca < (5) i 1 2 r=p(f>/n where a and <z are the roots of the quadratic x 2 where A (=4ab) is the total area enclosed by the equation Also, from Eq. (6), tube, T is the twist per unit length of the tube, and G is the modulus of rigidity, which is assumed uniform throughout. This equation may be written in the form It is obvious that a a =l . Hence, if we write -5,_!+2( l +r)S -S =tAGr/nb.... (1) i l+1 1 2 i.e. (a-\)2=2a (f>/n (12) where r=atl>ibt» (2) On using Eq. (12), we can write Eq. (11) in the Eq. (1) is a recurrence equation, which holds if form 2</<( « — 1). To obtain its solution we substitute the two roots are a =a and a =l/a . Eqs. (3) x 2 and (5) then give the following expression for the shear flow S . Also Eq. (10) can be written in the form The upward shear flow in the internal web separating the /th and (/+l)th cells is equal to (S -Si). On using Eqs. (12), (15) and (16) this may be written in the form 372 Aircraft Engineering

Aircraft Engineering and Aerospace Technology – Emerald Publishing

**Published: ** Dec 1, 1953

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