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Torsion of a Multiwebbed Rectangular Tube

Torsion of a Multiwebbed Rectangular Tube 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

Torsion of a Multiwebbed Rectangular Tube

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Publisher
Emerald Publishing
Copyright
Copyright © Emerald Group Publishing Limited
ISSN
0002-2667
DOI
10.1108/eb032366
Publisher site
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Abstract

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

Journal

Aircraft Engineering and Aerospace TechnologyEmerald Publishing

Published: Dec 1, 1953

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