CHAPTER 8 TORSION Summary For a solid or hollow shaft of uniform circular cross-section throughout its length,the theory of pure torsion states that Tt G0 万=R=D where T is the applied external torque,constant over length L; J is the polar second moment of area of shaft cross-section .πD4 32 for a solid shaft and (dfor a hollow shat; 32 D is the outside diameter;R is the outside radius; d is the inside diameter; t is the shear stress at radius R and is the maximum value for both solid and hollow shafts; G is the modulus of rigidity(shear modulus);and 6 is the angle of twist in radians on a length L. For very thin-walled hollow shafts J=2nr3t,where r is the mean radius of the shaft wall and t is the thickness. Shear stress and shear strain are related to the angle of twist thus: t=- -R=Gy Strain energy in torsion is given by T2L GJ82 U= 2G= 2L 4Gx volume for solid shafis For a circular shaft subjected to combined bending and torsion the equivalent bending moment is M.=[M+√/(M2+T)] and the equivalent torque is T。=√/(M2+T2) where M and T are the applied bending moment and torque respectively. The power transmitted by a shaft carrying torque T at o rad/s Tw. 176
CHAPTER 8 TORSION Sommary For a solid or hollow shft of uniform circular cross-section throughout its length, the theory of pure torsion states that T T GO J R=L -=- where Tis the applied external torque, constant over length L; J is the polar second moment of area of shaft cross-section x(D4 - d 4, for a hollow shaft; xD4 32 32 = - for a solid shaft and D is the outside diameter; R is the outside radius; d is the inside diameter; T is the shear stress at radius R and is the maximum value for both solid and hollow shafts; G is the modulus of rigidity (shear modulus); and 8 is the angle of twist in radians on a length L. For very thin-walled hollow shafts J = 2nr3t, where T is the mean radius of the shaft wall and t is the thickness. Shear stress and shear strain are related to the angle of twist thus: GB L T=-R=G~ Strain energy in torsion is given by U=- x volume for solid shafis 2GJ 2L For a circular shaft subjected to combined bending and torsion the equivalent bending moment is Me = i[M + J(Mz +T ')I and the equivalent torque is where M and T are the applied bending moment and torque respectively. The pa~er transmitted by a shaft carrying torque Tat o rad/s = To. T, = +J( M +T 2, 176
$8.1 Torsion 177 8.1.Simple torsion theory When a uniform circular shaft is subjected to a torque it can be shown that every section of the shaft is subjected to a state of pure shear(Fig.8.1),the moment of resistance developed by the shear stresses being everywhere equal to the magnitude,and opposite in sense,to the applied torque.For the purposes of deriving a simple theory to describe the behaviour of shafts subjected to torque it is necessary to make the following basic assumptions: (1)The material is homogeneous,i.e.of uniform elastic properties throughout. (2)The material is elastic,following Hooke's law with shear stress proportional to shear strain. (3)The stress does not exceed the elastic limit or limit of proportionality. (4)Circular sections remain circular. (5)Cross-sections remain plane.(This is certainly not the case with the torsion of non- circular sections.) (6)Cross-sections rotate as if rigid,i.e.every diameter rotates through the same angle. Resisting Applied shear Complementory shear Applied torque T Fig.8.1.Shear system set up on an element in the surface of a shaft subjected to torsion. Practical tests carried out on circular shafts have shown that the theory developed below on the basis of these assumptions shows excellent correlation with experimental results. (a)Angle of twist Consider now the solid circular shaft of radius R subjected to a torque T at one end,the other end being fixed(Fig.8.2).Under the action of this torque a radial line at the free end of the shaft twists through an angle 0,point A moves to B,and AB subtends an angle y at the fixed end.This is then the angle of distortion of the shaft,i.e.the shear strain. Since angle in radians arc radius arc AB=R0=Ly Y=R0/L (8.1) From the definition of rigidity modulus shear stress t G= shear strain y
§8.1 Torsion 177 8.1. Simple torsion theory When a uniform circular shaft is subjected to a torque it can be ShOWn that every sectiOn of the shaft is subjected to a state of pure shear (Fig. 8.1 ), the moment of resistance developed by the shear stresses being everywhere equal to the magnitude, and opposite in sense, to the applied torque. For the purposes of deriving a simple theory to describe the behaviour of shafts subjected to torque it is necessary to make the following basic assumptionS: (1) The material is homogeneous, i.e. of uniform elastic properties throughout. (2) The material is elastic, following Hooke's law with shear stress proportional to shear strain. (3) The stress does nOt exceed the elastic limit or limit of proportionality. (4) Circular SectiOnS remain circular. (5) Cross-sectioDS remain plane. (This is certainly nOt the case with the torsion of DODcircular SectiOnS.) (6) Cross-sectioDS rotate as if rigid, i.e. every diameter rotates through the same angle. Fig. 8.1. Shear system set up on an elem-ent in thesufface of a shaft subjected to torsion. Practical tests carried out on circular shafts have shown that the theory developed below on the basis of these assumptions shows excellent correlation with experimental results. (a) Angle of twist Consider now the solid circular shaft of radius R subjected to a torque T at one end, the other end being fixed (Fig. 8.2). Under the action of this torque a radial line at the free end of the shaft twists through an angle 9, point A moves to B, and AB subtends an angle y at the fixed end. This is then the angle of distortion of the shaft, i.e. the shear strain. SinCe angle in radians = arc + radius arc AB = R8 = Ly y = R8/ L (8.1) From the definition of rigidity modulus shear stress T shear strain y G=
178 Mechanics of Materials §8.1 Fig.8.2. (8.2) where t is the shear stress set up at radius R. Therefore equating eqns.(8.1)and(8.2), RO T L-G a-2(-) (8.3) where t'is the shear stress at any other radius r. (b)Stresses Let the cross-section of the shaft be considered as divided into elements of radius r and thickness dr as shown in Fig.8.3 each subjected to a shear stress t'. Fig.8.3.Shaft cross-section. The force set up on each element stress x area =t'x 2nr dr (approximately)
178 Mechanics of Materials $8.1 T .. Fig. 8.2. T y=- G where T is the shear stress set up at radius R. Therefore equating eqns. (8.1) and (8.2), where T' is the shear stress at any other radius r. (b) Stresses Let the cross-section of the shaft be considered as divided into elements of radius r and thickness dr as shown in Fig. 8.3 each subjected to a shear stress z'. Fig. 8.3. Shaft cross-section. The force set up on each element = stress x area = 2' x 2nr dr (approximately)
$8.2 Torsion 179 This force will produce a moment about the centre axis of the shaft,providing a contribution to the torque =(t'X2πrdr)Xr =2πx'r2dr The total torque on the section T will then be the sum of all such contributions across the section, i.e. T= 2nt'r2 dr 0 Now the shear stress t'will vary with the radius r and must therefore be replaced in terms of r before the integral is evaluated. From eqn.(8.3) GO t'= L T= 2 L R 2nr3dr L 0 The integral 2dr is called the polar second moment ofarea J,and may be evaluated asa standard form for solid and hollow shafts as shown in $8.2 below. T GO or 万= (8.4) Combining egns.(8.3)and(8.4)produces the so-called simple theory of torsion: Tt GO 了=R=L (8.5) 8.2.Polar second moment of area As stated above the polar second moment of area J is defined as 2nr3dr
$8.2 Torsion 179 This force will produce a moment about the centre axis of the shaft, providing a contribution to the torque = (7' x 2nrdr) x r = 2n7'r2 dr The total torque on the section T will then be the sum of all such contributions across the section, i.e. T = 2nz'r2dr J i 0 Now the shear stress z' will vary with the radius rand must therefore be replaced in terms of r before the integral is evaluated. From eqn. (8.3) R .. 0 = E L jkr3 dr 0 The integral 5 0" 2nr3 dr is called the polar second moment of area J, and may be evaluated as a standard form for solid and hollow shafts as shown in $8.2 below. GO L .. T=-J or T GO JL - =- Combining eqns. (8.3) and (8.4) produces the so-called simple theory of torsion: T z G8 -- - J-R-L 8.2. Polar second moment of area As stated above the polar second moment of area J is defined as J = 2nr3dr 0 i
180 Mechanics of Materials $8.3 For a solid shaft, -[ 2πR4 元D4 or 4 32 (8.6) For a hollow shaft of internal radius r, =R-ror五0-的 (8.7) For thin-walled hollow shafts the values of D and d may be nearly equal,and in such cases there can be considerable errors in using the above equation involving the difference of two large quantities of similar value.It is therefore convenient to obtain an alternative form of expression for the polar moment of area. Now 2πr3dr=Σ(2πrdr)r2 =2Ar2 where A(=2xr dr)is the area of each small element of Fig.8.3,i.e.J is the sum of the Ar2 terms for all elements. If a thin hollow cylinder is therefore considered as just one of these small elements with its wall thickness t=dr,then J=Ar2 =(2nrt)r2 =2r3t(approximately) (8.8) 8.3.Shear stress and shear strain in shafts The shear stresses which are developed in a shaft subjected to pure torsion are indicated in Fig.8.1,their values being given by the simple torsion theory as Ge -TR Now from the definition of the shear or rigidity modulus G, t=Gy It therefore follows that the two equations may be combined to relate the shear stress and strain in the shaft to the angle of twist per unit length,thus G0 t =Gy (8.9)
180 For a solid shafi, Mechanics of Materials nD* or - 4 32 2n~4 =- For a hollow shaft of internal radius r, J=2n r3dr=2n - i [:I: x x = -(R4-r4) or -(D4-d*) 2 32 $8.3 For thin-walled hollow shafis the values of D and d may be nearly equal, and in such cases there can be considerable errors in using the above equation involving the difference of two large quantities of similar value. It is therefore convenient to obtain an alternative form of expression for the polar moment of area. Now J = 2nr3dr = C(2nrdr)r’ = AY’ 0 i where A ( = 2nr dr) is the area of each small element of Fig. 8.3, i.e. J is the sum of the Ar2 terms for all elements. If a thin hollow cylinder is therefore considered as just one of these small elements with its wall thickness t = dr, then J = Ar’ = (2nrt)r’ = 2xr3t (approximately) (8.8) 8.3. Shear stress and shear strain in shafts The shear stresses which are developed in a shaft subjected to pure torsion are indicated in Fig. 8.1, their values being given by the simple torsion theory as GO L 7=-R Now from the definition of the shear or rigidity modulus G, r = Gy It therefore follows that the two equations may be combined to relate the shear stress and strain in the shaft to the angle of twist per unit length, thus