MECHANICS OF SOLIDS TORSION TUTORIAL 2 TORSION OF THIN WALLED SECTIONS AND THIN STRIPS

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1 MECHANICS OF SOLIDS ORSION UORIAL ORSION OF HIN WALLED SECIONS AND HIN SRIPS Yu shuld judge yur prgress by cmpleting the self assessment exercises. On cmpletin f this tutrial yu shuld be able t d the fllwing. Derive the maximum shear stress in varius sectins with thin walls. Calculate the maximum shear stress in varius sectins with thin walls. Derive the maximum shear stress in slid thin strips. Calculate the maximum shear stress in slid thin strips. It is assumed that students ding this tutrial already understand the basic principles f shear stress due t trsin. Students must als be able t perfrm basic differentiatin and calculus frm their maths studies.

2 ORSION IN HIN WALLED VESSELS and HIN SRIPS 1. CIRCULAR SECIONS When a circular sectin shaft is subjected t a trque, the shear r stress at any radius r is given by τ J J is the plar secnd mment f area. his applies t slid r hllw shafts. πd d Fr a hllw shaft J. D is the uter diameter and d the inner diameter. D d D d he mean diameter is D m. he wall thickness is t Figure 1 π π J Dm t Dm t D mt t Dm Fr a thin wall defined as t D/0 the last term is π negligible and we simplify t J D mt Dm Dm Dm he maximum shear stress ccurs at radius r t and is given by. π πd t D t m m he crss sectinal area at the mean diameter is A = πd m /. Substituting we get τ max At. NON CIRCULAR SECIONS he equatin just derived fr thin walled circular sectins may be applied t nn-circular sectins such as shwn belw. A is the crss sectinal area taken at the middle f the wall. Figure

3 WORKED EXAMPLE N.1 A rectangular tube has utside dimensins 60 mm x 0 mm and has a wall mm thick. Calculate the maximum shear stress when a trque f 000 Nm is applied. SOLUION he dimensins at the centre f the wall are 58 mm x 8 mm s A = 58 x 8 = 0 mm 000 τ 0. MPa 6 A t x 0 x 10 x x 10 WORKED EXAMPLE N. A hllw hexagnal tube has a crner radius f 5 mm and a wall thickness f 1 mm. Calculate the maximum shear stress when a trque f 00 Nm is applied. SOLUION Figure Gemetry tells us that the radius t the inner crner is he area f a hexagn is r R R A he mean radius is. mm t = 6.8 mm A x 1550 mm A 00 t x 1550 x 10 τ 6 x 1x MPa SELF ASSESSMEN EXERCISE N.1 1. A hllw circular tube is 60 mm uter diameter and has a wall mm thick. Calculate the maximum shear stress when a trque f 50 Nm is applied. (Answer 17.1 MPa) A rectangular tube has utside dimensins 0 mm x 0 mm and has a wall mm thick. Calculate the maximum shear stress when a trque f 00 Nm is applied. (Answer 70.5 MPa) A hllw hexagnal tube has a crner radius f 0 mm and a wall thickness f mm. Calculate the maximum shear stress when a trque f 50 Nm is applied. (Answer 0 MPa)

4 . HIN RECANGULAR SECION his sectin deals with the trsin f slid thin rectangular strips. In tutrial 1 it was stated that L and θ and when the strip is very thin α = β = 1/ αbd βbd G Fr thin sectins it is usual t use t fr the depth s the equatins becme: L and θ he first equatin may be derived anther way as fllws. Bt Bt G Let the crss sectin be B by t. Within this sectin cnsider a thin rectangular hllw sectin with wall thickness dy and distance y frm the lng centreline (clured red in the diagram). We may apply the frging thery t this layer. he area A (y) is apprximately t Ay yb y yb yt y yb Figure Acrss the slid sectin the shear stress varies linearly frm τ max t +τ max as shwn. he shear stress at distance y frm the centreline is fund by ratis as τ y = y τ max /t Figure 5 he shear stress in the vertical sides is assumed negligible. Frm the previus sectin we have: τ max A t s fr ur thin rectangle this becmes d τ y A dy Rearrange t get the prtin f rque carried in this thin rectangle. d = τ y A y dy. Substitute A y = yb and d = τ y y B dy. Substitute τ y = y τ max /t and d = 8 B y τ max dy/t. he ttal trque carried by the thin rectangular strip is fund by integrating. t/ 8Bτ max 8Bτ max y 8Bτ max t Bt d y dy t t t Rearrange and If we let B Bt J this becmes B t J r Cmpare with frmula fr a circular sectin τ max J t/ y

5 WORKED EXAMPLE N. Cmpare the maximum shear stress fr the tw sectins shwn with the same uter dimensins when a trque f 5 Nm is applied. SOLUION Figure 6 SOLID SECION t = 10 mm x MPa L t 0.1x 0.01 HOLLOW SECION t = wall thickness = 1 mm A = 99 x 9 = 891 mm 5 A t x 891x 10 τ 6 x 1x 10.8 MPa. OPEN HIN SECIONS he shear flw runs thrugh the thin sectin parallel t the lng edge s the thery fr a thin strip can be applied t each cmpnent that is a thin strip. We take the effective plar secnd mment f area as the sum f Bt / fr each cmpnent. If the sectin is made up frm thin rectangular sectins as shwn B1t1 Bt B1t J Figure 7 If the sectin is circular, B is the circular length. Fr the cmplete circle shwn B= πd. D is the mean diameter. he trsin equatins becme τ max Bt Figure 8 and θ L G Bt

6 WORKED EXAMPLE N. Cmpare the maximum shear stress fr a thin pen tube and clsed tube 60 mm diameter and wall thickness mm. SOLUION Clsed tube 8.8 x 10-9 πdmt π x 60 x x 10 6 Open tube.98 x 10-9 Lt π x 60 x x 10 he clsed sectin prduces the smaller stress s it must be the strnger f the tw. WORKED EXAMPLE N. A sectin with the dimensin shwn is made frm steel plate 1.5 mm thick. Calculate the maximum shear stress and the angle f twist fr the sectin shwn when a trque f 11. Nm is applied. G = 8 GPa. SOLUION Figure 9 Since the thickness is the same we may take B as the mean perimeter. B = (0-0.65) + (8-1.5) = 95.5 mm x MPa Bt x 10 x 1.5 x 10 L x 11. x 1 θ.19 rad/m GBt 9 8 x 10 x 95.5 x 10 x 1.5 x 10

7 SELF ASSESSMEN EXERCISE N. 1. Cmpare the maximum shear stress prduced in a thin strip and a hllw rectangular sectin as shwn fr a given trque. (Answer τ =.75 x 10 6 fr the strip and τ = x 10 6 fr the hllw sectin). w tubes are made frm mm thick metal sheet by rlling them int a cylinder. One has the seam welded and the ther des nt. he tubes are t transmit a trque f 0 Nm and the maximum shear stress must nt exceed MPa. What must be the rati f the mean diameters f each if they are t have equal trsinal strength? (Answer 16/1). An L shaped sectin 0.6 m lng is made frm thin steel plate mm thick. he sectin is 60 mm lng in bth sides. Calculate the maximum shear stress and angle f twist when a trque f Nm is applied. G = 80 GPa. (Answer 5. MPa and rad)