LECTURE 18 Members Subjected to Torsional Loads Torsion of circular shafts Definition of Torsion: Consider a shaft rigidly clamped at one end and twisted at the other end by a torque T = F.d applied in a plane perpendicular to the axis of the bar such a shaft is said to be in torsion. Effects of Torsion: The effects of a torsional load applied to a bar are (i) To impart an angular displacement of one end cross – section with respect to the other end. (ii) To setup shear stresses on any cross section of the bar perpendicular to its axis. GENERATION OF SHEAR STRESSES The physical understanding of the phenomena of setting up of shear stresses in a shaft subjected to a torsion may be understood from the figure 1-3. Fig 1: Here the cylindrical member or a shaft is in static equilibrium where T is the resultant external torque acting on the member. Let the member be imagined to be cut by some imaginary plane ‘mn'. Fig 2: When the plane ‘mn' cuts remove the portion on R.H.S. and we get a fig 2. Now since the entire member is in equilibrium, therefore, each portion must be in equilibrium. Thus, the member is in equilibrium under the action of resultant external torque T and developed resisting Torque Tr . Fig 3: The Figure shows that how the resisting torque Tr is developed. The resisting torque Tr is produced by virtue of an infinites mal shear forces acting on the plane perpendicular to the axis of the shaft. Obviously such shear forces would be developed by virtue of sheer stresses. Therefore we can say that when a particular member (say shaft in this case) is subjected to a torque, the result would be that on any element there will be shear stresses acting. While on other faces the complementary sheer forces come into picture. Thus, we can say that when a member is subjected to torque, an element of this member will be subjected to a state of pure shear. Shaft: The shafts are the machine elements which are used to transmit power in machines. Twisting Moment: The twisting moment for any section along the bar / shaft is defined to be the algebraic sum of the moments of the applied couples that lie to one side of the section under consideration. The choice of the side in any case is of course arbitrary. Shearing Strain: If a generator a – b is marked on the surface of the unloaded bar, then after the twisting moment 'T' has been applied this line moves to ab'. The angle ‘g' measured in radians, between the final and original positions of the generators is defined as the shearing strain at the surface of the bar or shaft. The same definition will hold at any interior point of the bar. Modulus of Elasticity in shear: The ratio of the shear stress to the shear strain is called the modulus of elasticity in shear OR Modulus of Rigidity and in represented by the symbol Angle of Twist: If a shaft of length L is subjected to a constant twisting moment T along its length, than the angle q through which one end of the bar will twist relative to the other is known is the angle of twist. Despite the differences in the forms of loading, we see that there are number of similarities between bending and torsion, including for example, a linear variation of stresses and strain with position. In torsion the members are subjected to moments (couples) in planes normal to their axes. For the purpose of desiging a circular shaft to withstand a given torque, we must develop an equation giving the relation between twisting moment, maximum shear stress produced, and a quantity representing the size and shape of the cross-sectional area of the shaft. Not all torsion problems, involve rotating machinery, however, for example some types of vehicle suspension system employ torsional springs. Indeed, even coil springs are really curved members in torsion as shown in figure. Many torque carrying engineering members are cylindrical in shape. Examples are drive shafts, bolts and screw drivers. Simple Torsion Theory or Development of Torsion Formula : Here we are basically interested to derive an equation between the relevant parameters Relationship in Torsion: 1 st Term: It refers to applied loading ad a property of section, which in the instance is the polar second moment of area. 2 nd Term: This refers to stress, and the stress increases as the distance from the axis increases. 3 rd Term: it refers to the deformation and contains the terms modulus of rigidity & combined term ( q / l) which is equivalent to strain for the purpose of designing a circular shaft to with stand a given torque we must develop an equation giving the relation between Twisting moments max m shear stain produced and a quantity representing the size and shape of the cross – sectional area of the shaft. Refer to the figure shown above where 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, the moment of resistance developed by the shear stresses being every where equal to the magnitude, and opposite in sense, to the applied torque. For the purpose of deriving a simple theory to describe the behavior of shafts subjected to torque it is necessary make the following base assumptions. Assumption: (i) The materiel is homogenous i.e of uniform elastic properties exists throughout the material. (ii) The material is elastic, follows Hook's law, with shear stress proportional to shear strain. (iii) The stress does not exceed the elastic limit. (iv) The circular section remains circular (v) Cross section remain plane. (vi) Cross section rotate as if rigid i.e. every diameter rotates through the same angle. Consider now the solid circular shaft of radius R subjected to a torque T at one end, the other end being fixed Under the action of this torque a radial line at the free end of the shaft twists through an angle q , point A moves to B, and AB subtends an angle ‘ g ' at the fixed end. This is then the angle of distortion of the shaft i.e the shear strain. Since angle in radius = arc / Radius arc AB = Rq                  = L g [since L and g also constitute the arc AB] Thus, g = Rq / L      (1) From the definition of Modulus of rigidity or Modulus of elasticity in shear Stresses: Let us consider a small strip of radius r and thickness dr which is subjected to shear stress t'. The force set up on each element = stress x area = t' x 2p r dr (approximately) This force will produce a moment or torque about the center axis of the shaft. = t' . 2 p r dr . r = 2 p t' . r2. dr The total torque T on the section, will be the sum of all the contributions. Since t' is a function of r, because it varies with radius so writing down t' in terms of r from the equation (1). Where T = applied external Torque, which is constant over Length L; J = Polar moment of Inertia [ D = Outside diameter ; d = inside diameter ] G = Modules of rigidity (or Modulus of elasticity in shear) q = It is the angle of twist in radians on a length L. Tensional Stiffness: The tensional stiffness k is defined as the torque per radius twist i.e, k = T / q = GJ / L Power Transmitted by a shaft : If T is the applied Torque and w is the angular velocity of the shaft, then the power transmitted by the shaft is