ANALYSIS OF TORSIONAL RIGIDITY OF CIRCULAR BEAMS WITH DIFFERENT ENGINEERING MATERIALS SUBJECTED TO ST. VENANT TORSION

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1 MPACT: nternational Jornal of Research in Engineering & Technology (MPACT: JRET) SSN(E): -884; SSN(P): Vol., sse, Feb 05, -46 mact Jornals ANALYSS OF TORSONAL RGDTY OF CRCULAR BEAMS WTH DFFERENT ENGNEERNG MATERALS SUBJECTED TO ST. VENANT TORSON M. C. AGARANA & O. O. AGBOOLA Deartment of Mathematics, College of Science & Technology, Covenant University, Ota, Nigeria ABSTRACT Many engineering strctres, sch as airlane wings, beams and shafts are sbjected to higher torsional forces today de to advancement in Strctral Engineering, in terms of size and technology. n this aer, we analyzed the resistance of circlar beams, of different engineering materials, to their corresonding twisting moments. We obtained the torsional rigidity for the different beams as the ratio of twisting moment to the angle of twist er nit length. t is observed that torsional rigidity of the beams is a fnction of their areas and the engineering material they are made of. Secifically it is observed that the circlar beam made of brass engineering material has the greatest torsional rigidity among the twelve engineering materials considered. KEYWORDS: Beams, Torsional Rigidity, Twisting Moment, St. Venant Torsion, Brass NTRODUCTON When a beam is transversely loaded in sch a manner that the resltant force asses throgh the longitdinal shear central ais, the beam only bends and no torsion will occr. When the resltant force acts away from the shear central ais, then the beam will not only bend bt also twist. [,, 4] Torsion is twisting abot an ais rodced by the action of two oosing coles acting in arallel lanes [5]. Another name for coles is torqe or twisting moment. Torsional rigidity of a beam is a ratio of moment to the angle of twist er nit length [6]. When torsion is alied to a strctral member, its cross-section may war in addition to twisting. f the member is allowed to war freely, then the alied torqe is resisted entirely by torsional shear stresses (called St. Venant s torsional shear stress). f the member is not allowed to war freely, the alied torqe is resisted by St. Venant s torsional shear stress and waring tension. This behavior is called non-niform torsion [,,, 4]. Beams of non-circlar section tends to behave non-symmetrically when nder torqe and lane sections do not lane. Also the distribtion of stress in a section is not necessarily linear[]. St. Venant s theory is sally alied when the cross-section is non-deformable ot of its lain or those deformations are very small [0]. lane z Consider a circlar beam with length l, with one of its bases fied in the y lane, while the other base (in the l ) is acted on by a cole whose moment lies along ais. The beam twists throgh an angle determined by the magnitde of alied cole and the modls of rigidity of the beam. The amont of twist rodced can ths be sed to determine the alied force [5]. mact Factor(JCC): This article can be downloaded from www. imactjornals.s

2 4 M. C. Agarana & O. O. Agboola Saint-Venant (885) was the first to rodce the correct soltion to the roblem of torsion of bars sbjected to moment coles at the ends n material science, shear modls or modls of rigidity, denoted by µ or G, is defined as the ratio of shear stress to the shear strain [6]. FORMULATON OF THE PROBLEM For a beam of constant circlar cross-section sbjected to torsion, the St. Venant s torsion is given by [,,, and 4] T SV df r m () dz Where, φ is the angle of twist (twist angle), µ isthemodls of rigidity, TSV is St. Venanttorsion, ρ is the olar moment of inertia, z isthedirectionalong ais of themember. df dz Twist rate By symmetry, anysection of the beam erendiclar to z ais remains erendiclar to this ais dring deformation and the action of the cole will merelyrota teeach section throgh some angleφ, called the angle of twist [5]. Theamont of rotation wil lclearlydeendonthedistance of the section from the base z 0, and since the deformations are small, the amont of rotationφ is roortional to the distance of the section from the fied base. Theangle of twist can then be written as f az, () Where a isthe twist er nitlength. t is therelative anglar dislacement of a air of crosssectionsthat are nitdistanceaart. Letw be the dislacementalong z - ais. w 0, if the crosssection of the beam remainlaneafter deformation. Sincew is indeendent of z, we write w at µ (, y) () Where µ T isthetorsión fnction nde Coernics Vale:.0 - Articles can be sent to editor@imactjornals.s

3 Analysis of Torsional Rigidity of Circlar Beams with Different Engineering Materials Sbjected to St. Venant Torsion 5 Consider a article originalty at (, y, z ), since f az, for the dislacement of this article - a zy, v az, w 0. The angle of twist f can also be written as [5] f T l (4) m r WhereT is the alied torqe and l is the length of the beam. Figre : Twisting of Circlar Section Assmtions The bar is straight and of niform cross section. The material of the bar has niform roerties. Figre : Showing the Dislacement Along and The only loading is the alied torqe which is alied normal to the ais of the bar. The bar is stressed within its elastic limit. mact Factor(JCC): This article can be downloaded from

4 6 M. C. Agarana & O. O. Agboola The Elerian strain is given in terms of dislacement, v, w by [5] e ij ( i, j + j, i ) (5) Where, i j i,,, ; j i j,,,, v and y and w z Þ - azy - a - f v - az a f The reresentative strains, in matri form, are as follows [5] e ij æ f µ ö 0 0 (- + T,) f 0 0 ( µ + T,) f ( µ f µ T,) ( T,) 0 ç çè ø (6) Where µ µ T T,, µ µ T T,, µ µ T, T,, µ µ T, T, The stress-strain relation is given as [5] E dij ( eij + ekk dij ) + - (7) Where d ij isthe stress tensor, E istheyong smodls and isthepoisson s ratio. Sb stitting the resectivestrains in the stress-strainrelation, weget d ij æ 0 0 f ( T µ, ) ö ( µ f T, ) + ça( T µ µ, - ) f ( T, + ) 0 çè ø (8) Where is a lameconstant, given as nde Coernics Vale:.0 - Articles can be sent to editor@imactjornals.s

5 Analysis of Torsional Rigidity of Circlar Beams with Different Engineering Materials Sbjected to St. Venant Torsion 7 E ( + ) (9) Recall that T µ is a fnction of and. \ fori, T µ, 0 fori, T µ, 0 Þ T µ + T µ (0),, 0 This is the Lalace eqation. Hence, µ T is a harmonicfnction. Since µ T is a harmonicfnction in thesimlyconnected región R, reresentingthecross-section of thebeam, there eist sananalyticfnction µ T + iy of the comle variable + iy where y (, y) is a harmonic conjúgate of T µ. ThefnctionssatisfytheCachy-Riemann eqations, namely µ T y (i.e., T µ, y, ) () And µ T y - T (i.e., µ,, - y ) () The harmonic conjgate y of µ T isgiven as üï y ( + ) on C ï ïïý and () Ñ y 0 in R ï ïïþ This is a Dirichlet roblem. ANALYSS The torsional rigidity of a beam is defined as a ratio of moment to the angle of twist er nit length [5]. The torsional rigidity of a beam with circlar cross-section is given as [, 5] D M (4) f mact Factor(JCC): This article can be downloaded from

6 8 M. C. Agarana & O. O. Agboola WhereM is the reslting momento on the srface l and is given as [8]: òò ( d - d) (5) R M da where d and d are stress tensorswith ç y d f æ ö y - ç è ø and d f æ + ö ç ç è ø. On the srface l, we have j (,, 0), hence 0. Therefore, M 0 and M 0 òò ( d d) (6) R Þ M M - da Letsconsidertheharmonicfnction [8] ( ) y c - + k (7) wherec, k are constants. Þ c ( - ) + k ( + ) onthebondary, or ( - c ) + ( + c ) k (8) The crve defined by this eqation is an ellise a + (9) b f we choose c < and a k - c, b k + c, then c a a - b + b, k a a b + b - ( ) Þ a b - + a b a + b a + b So, d - f a becomes a + b nde Coernics Vale:.0 - Articles can be sent to editor@imactjornals.s

7 Analysis of Torsional Rigidity of Circlar Beams with Different Engineering Materials Sbjected to St. Venant Torsion 9 Similarly, d f b becomes a + b From eqation (6); the torsional momen to becomes f æ ö M b da a da + a + b ç è ø òò òò (0) R R f ( a ) + b a + b () where and are the moments of inertia of the ellitical section abot the - and - aes. Bt ab and 4 a b, so we have 4 M f a b a + b () For a beam with circlar cross-section; a b radis ( r) 4 4 f a a f a f r \ M a + a. () The torsional rigidity of the circlar beam can be written as D 4 M r (4) f bt A r A Þ r (5) A 4 r (6) Sbstirting eqation (6) into (4), wehave D A (7) mact Factor(JCC): This article can be downloaded from

8 40 M. C. Agarana & O. O. Agboola D A (Torsional rigidity) (8) D Þ A E ( + ) (9) \ D A E ( - v) 4 (0) For a circlar beam; the moment of inertia abot the - ais is given as r () 4 Also, the moment of inertia abot the y - ais isgiven as r () 4 Ths, the olar moment of inertia abot the - ais is given as r r + () 4 4 r (4) Hence, St. Venanttorsion becomes T SV r m dq (5) d With the bondary conditions f ( ) 0 at 0, f ( ) f at l; To find the twist anglef, we integratea long the length of thebeam as shownbelow df T d d d ò ò (6) SV m f T SV, m and constants along the beam. nde Coernics Vale:.0 - Articles can be sent to editor@imactjornals.s

9 Analysis of Torsional Rigidity of Circlar Beams with Different Engineering Materials Sbjected to St. Venant Torsion 4 f TSV d m ò (7) TSV \ f (8) m f length of the beamalongz - ais isl, then f T SV (9) m l From eqation (), the dislacement is given as at µ (, y) at µ (, ) (40) Dislacement along and aes are - a and a (4) resectively, since dislacement along - iszero. f the length of the beam along - ais isl - f and f (sincef az T z T z -, m m ) (4) f the length of the beam along - ais isl, then T l T l -, m m (4) From eqation (), the harmonic conjúgate y, of the torsión fnction which is a fnction of and, is considered for the different vales of and and lotted on a grah as shown in Figre. Table : The Following Chart Gives Tyical Vales for the Modls of Rigidity, Yong S Modls and Poisson Ratios for Different Engineering Materials [6, 9] Engineering Materials Modls of Rigidity (Psi X 0 6 ) Yong s Modls (Psi X 0 6 ) Poisson Ratio ( ) Beryllim coer Brass mact Factor(JCC): This article can be downloaded from

10 4 M. C. Agarana & O. O. Agboola Table : Contd., Bronze Coer ron (Malleable) Magnesim Molybdenm Monel Nickel silver Nickel steel Titanim Zinc For this aer, we consider a circlar beam, for twelve different engineering materials of diameter.m, angle of twist of 0 o, length of 0m and the Torqe (T) as the St. Venant Torsion (T SV ). Table shows the calclated vales of olar moment ( ), St. Venant Torsion ( T SV ) and torsional rigidity (D ). i.er 0.6 o m, f 0, l 0m, T TSV Table : Calclated Vales of Polar Moment, St. Venant Torsion TSV Engineering Materials m T SV D Beryllim coer Brass Bronze Coer ron Magnesim Molybdenm Monel Nickel silver Nickel steels Titanim Zinc Dislacement along and and Torsional Rigidity D for circlar beam of different engineering materials, withl 0m, 0.4 Table : Dislacement Along and for Circlar Beam with Different Engineering Materials nde Coernics Vale:.0 - Articles can be sent to editor@imactjornals.s

11 Analysis of Torsional Rigidity of Circlar Beams with Different Engineering Materials Sbjected to St. Venant Torsion 4 The Relationshi between µ, T SV and D When two variables and y are related, they are said to be correlated []. n order to define the linear relationshi and the amont of linear relationshi between µ, TSV coefficient. We dedced the following: and D, we adot the concet of regression and correlation µ T SV (44) and R (45) Similarly, we have 9 D µ And + (46) R (47) Where r is the correlation coefficient, ( r ). Recall that D Torsional rigidity, Polar moment of inertial abot - ais, T SV St. Venant torsion, m Modls of rigidity The Relationshi between the Cross-Sectional Area (A) and Torsional Rigidity (D) n this section we are interested in finding ot if there is any relationshi between the cross sectional area of the beams and torsional rigidity of the beams of different engineering materials. As shown in tables (4)-(8), where r is the radis, we considered five cases where the diameter of the cross sectional area is given vales.,.,., 4. and 5. resectively. Table 4: Torsional Rigidity of Beams with Different Materials When Cross-Sectionalareais.097m r A E v D BeryllimCoer *0^6 Brass *0^6 Bronze *0^6 Coer *0^6 ron *0^6 Magnessim *0^5 Molybdenm *0^6 Monel *0^6 Nickel Silver *0^6 Nickel Steels *0^6 mact Factor(JCC): This article can be downloaded from

12 44 M. C. Agarana & O. O. Agboola Table 4: Contd., Titanim *0^6 Zinc *0^5 Table 5: Torsional Rigidity of Beams with Different Materials When Cross-Sectionalareais.807m Engineering Material R A E v D BeryllimCoer *0^7 Brass *0^7 Bronze *0^7 Coer *0^7 ron *0^7 Magnessim *0^6 Molybdenm *0^7 Monel *0^7 Nickel Silver *0^7 Nickel Steels *0^7 Titanim *0^7 Zinc *0^7 Table 6: Torsional Rigidity of Beams with Different Materials When Cross-Sectionalareais m Engineering Material R A E v D BeryllimCoer *0^7 Brass *0^8 Bronze *0^8 Coer *0^7 ron *0^8 Magnessim *0^7 Molybdenm *0^8 Monel *0^8 Nickel Silver *0^7 Nickel Steels *0^8 Titanim *0^8 Zinc *0^7 Table 7: Torsional Rigidity of Beams with Different Materials When Cross-Sectional Area is.8544m Engineering Material R A E V D BeryllimCoer *0^8 Brass *0^9 Bronze *0^9 Coer *0^8 ron *0^8 Magnessim *0^7 Molybdenm *0^8 Monel *0^8 Nickel Silver *0^8 Nickel Steels *0^8 Titanim *0^8 Zinc *0^8 nde Coernics Vale:.0 - Articles can be sent to editor@imactjornals.s

13 Analysis of Torsional Rigidity of Circlar Beams with Different Engineering Materials Sbjected to St. Venant Torsion 45 Table 8: Torsional Rigidity of Beams with Different Materials When Cross-Sectional Area is.77m RESULT DSCUSSONS Engineering Material R A E V D BeryllimCoer *0^8 Brass *0^9 Bronze *0^9 Coer *0^8 ron *0^8 Magnessim *0^8 Molybdenm *0^9 Monel *0^8 Nickel Silver *0^8 Nickel Steels *0^8 Titanim *0^8 Zinc *0^8 The nmerical calclations were carried ot for a circlar beam of length l, with one of its bases fied in the ylane, while the other base (in the lane z l) is acted on by a cole whose moment lies along the z-ais ( -ais). As an illstration, the length l of the beam is taken to be 0m, the diameter,.m, the angle of twist, 0 o and the olar, 0.4. (i.e l0m, r0.6m, Æ0 0, 0.4. Twelve different engineering materials with different vales of E, mand ʋ were considered. The reslts are shown on the varios tables and figres. t is observed from table, that circlar beams of brass engineering material has the highest torsional rigidity nder St. Venanttorsion, while circlar beam made of magnesim engineering material has the lowest torsional rigidity. Clearly from the vale of R in eqation (45), which is aroimately 0.9, it shows there is a strong ositive correlation between the modls of rigidity and St. Venant torsion, and eqation (44) shows there is a linear relationshi between modls of rigidity and St. Venant torsion. Also, that the higher the modls of rigidity of the engineering material the higher the St. Venant torsion. However, the vale of R in eqation (46), which is aroimately - 0.0, shows that aart from the fact that torsional rigidity and modls of rigidity have a negative correlation, the correlation is very weak. This imlies that the vale of the modls of rigidity has a little negative effect on the resistance of twist of a circlar beam. The effect of the cross sectional area of the circlar beam is shown in tables (4),(5),(6),(7) and (8).t can easily be seen that the cross-sectional area of the beams affect the torsional rigidity of the beams. The wider the cross-sectional area the higher the torsional rigidity. CONCLUSONS This work deals with the analysis of torsional rigidity of circlar beams with different engineering materials sbjected to St. Venant torsion. The torsional rigidity of these beams were calclated as a ratio of twisting moment to the angle of twist er nit length. t is shown that the circlar beam made of brass engineering material has high torsional rigidity, relatively, when sbjected to St. Venant s torsion. Concerning the cross-sectional area of circlar beams made of different materials; it was dedced that the wider the cross sectional area the higher the torsional rigidity and vice versa. This henomenon is of great imortance, esecially in the field of civil and mechanical engineering. REFERENCES. Trahair, N.S. (977). The Behavior and Design of Steel Strctres, Chaman and Hall, London. mact Factor(JCC): This article can be downloaded from

14 46 M. C. Agarana & O. O. Agboola. Mc Gire, W. (968). Steel Strctres, Prentice Hall.. Nethercot, D.A., Salter, P.R. and Malik, A.S. (989). Design of members sbjected to combined bending and torsion, The Steel Constrction nstitte Material/Chater 7.df 5. Agarana, M. C., Torsional Rigidity of Beams of given area with different Cross sections, abacs, The Jornal of the Mathematical Association of Nigeria volme 7, nmber, (00), Machine Design Reference Handbook, MdGraw Hill Marks Standard Hanbook 7. Lectre 07, Torsion of Circlar Sections, 8. The Engineering Toolbo, 9. Pia Hannewald (006), Strctral Design and Bridges, TRTA-BKN. Master Thesis, SSN 0-497, SRN KTH/BKN/EX--6 SE 0. Agarana, M.C. (006), Beginning Statistics, Nile Ventres Pblisher, Lagos, Nigeria (SBN Beams sbjected to torsion and bending, htt//: nde Coernics Vale:.0 - Articles can be sent to editor@imactjornals.s