SHAFTS 1, FUNDAMENTALS 1.1: Definitions a) Shafts:Is a machine element intended to transmit torque along their axes and for supporting rotating elements. Since the transmission of torque is associated with applied forces, eg gear teeth forces, belt forces etc, shafis are usually loaded in shear bending in addition to torque. b) Axle: A fixed or rotation m/c element intended to support stationary or rotating elements but transmit no torque. Axle are mainly subjected to bending c) Spindle: A short shaft 4) Pins: Short Axles e) Journal: —_ Parts of shaft or axle accommodating bearing stepped end of the shaft 1) Line shafts: | Main shafis. Are the ones that are driven by prime moves and power from there shafis transmitted to other element by belt or chain 2) Countershafts: are there shafts between a line shaft and a driven machine 1.2 Types of shafts Shaft may be classified as: a) Plain shafts (constant diameter) or Stepped shafts b) Solid shaft or Hollow shafts ©) Round shafts or Profiled shafts Special types of shafts: Crank shafts: used to convert linear into circular motion and vice versa ‘Camshaft: used to convert circular motion into non harmonic linear motion e.g to actuate the valve in IC engines. Flexible shafts: used to transmit motion between two points where the rotational axes are at angle with each other [e.g. speedometer shaft in car] Shaft may be hollow when there is a strict requirement in weight saving. e.g. a hollow shaft with 2 Ratio of 0.75 is lighter by about 50% of a solid shaft of equal strength and rigidity.Fig 3.5 Hollow Shaft (iN. Fig. 3.2 Cranked Shaft — 7 aX Fig. 3.4 Full Shaft - ane U Fig 3.6 Profiled Shaft2, SHAFT MATERIALS. ‘Any material used must satisfy strength, rigidity, wear resistance and other function plus environmental conditions. For normal applications, mild steel (st 37, st 50 and st 60) are commonly used. For higher strength/surface treatments requirement, material used are plain and alloy steels. Other possible shaft materials are, High strength cast Iron [Nodular cast Iron] for complex Geometry eg crankshafts, shaft with large flanges and heavy shafts. 3. DESIGN CONSIDERATION OF SHAFT Complete design of a shaft or axle demand attention on a) Design for strength [static and fatigue loading] b) Design for rigidity [Stiffness with both bending and torsion] ©) Vibration especially for critical speeds 4) The other production, environmental and economic factor In most application strength is the main determining factor. 3.1 Determination of shaft sizes in the basis of strength: 3.1.1 General loading cases: a) Torque, T causes Torsional stress t ») Transverses forces: Causes Bending stress and direct shear stresses ©) Axial forces: causes tensile stresses or compressive stresses Usually the weight of shaft and element components fixed on it (e.g, bearings) are neglected because their magnitude is small compared to external loads. 3.1.2 Cases of nature of loading a) Static or steady loading: may be encountered under: i) Pure torque ii) Simple bending iii) Axial (Tensile/Compressive)stresses iv) Direct Shear stress y) Combined torsion, bending and Axial loading, ) Fluctuating loads: Fluctuating bending, fluctuating torsion or combined fluctuating bending and torsion or steady loading: i) Pure torque (T) For a shaft transmitting Power at a rotational angular velocity «, torque is: T = £LeT => T= torque (Nm), P= Power (watts), N= Speed in r.p.m Torsional shear stress [Tmax] ei Tmax = 7 . f a olar section modulus for circular cross-section Z) = “= (solid shaft) = 16T ae Sane For hollow shaft. tmax = ri where k = = For safe operation Tmax S Tallow factor of safety. Note that: taitow= 0.570 a, = yield stress of material Examples 1. A line shaft rotating at 200 r.p.m is to transmit 25 h.p. The shaft may be assumed to be made of, mild steel with allowable shear stress of 42“; Determine the diameter of the shaft. 2. A solid shaft is transmitting IMW at 240 r-p.m Determine the diameter of shaft if the maximum torque exceed the mean torque by 20%. Take the maximum allowable shear stress as oo, 3. Find the diameter of a solid steel shaft to transmit 20kW at 200 r.p.m. The allowable shear stress may be taken as 45 Ifa hollow shaft is to be used in place of the solid shaft ,find inside and outside diameters when the ratio of inside to outside diameter is 0.5(i) Bending ie Simple Bending For bending stress to be calculated bending moment caused by transverse forces [distributed or concentrated] must be known. Bending moment My is tead from bending moment diagram at the point of interest [usually were My is Max) For circular shaft/Axle, bending stress pmax i8 given a8: Oymax = Fs Z,= section modulus mls For circular cross-section: Zp = 32M, omax = as Somax is maximum on the surface of the shaft ie stress distribution is as shown below: 32My ay where k = Sit For hollow shaft pmax = iq] z Examples: 1. A shaft made of mild steel carries two pulleys each weighing 1500N supported at a distance of Imetre from the ends respectively. The supported length of shaft is 3 metres. Determine the diameter of the shaft if the safe stress of the material for shaft is 60%; 2. A pair of wheels of'a railway wagon carries a load of SOKN on each axle box, acting at a distance of 100mm outside the wheel base. The gauge of the rail is 1.4m.Find the diameter of the axle between the wheel, if stress is not to exceed 100(ii) tensile/compressive normal stress The stress is caused by axial load [Tension or compression] Fa Stress, 03 = Fa =axial force, A= cross sectional area ae ee 4 40g = 2 for solid shaft for hollow shaft (iv) Shear stress [due to transverse force]. Practical shafts are usually slender (L >> ) because of the slenderness, the shear stress produced by a force perpendicular to the axis of the shaft, F,is much sinaller than the bending stress caused by the same force hence shear stress is neglected in practical calculations. (v) Combined torsion, bending and Axial loading. When shaft is subjected to the combined twisting moment, bending moment and Axial forces then the shaft must be designed on the basis of two moments and axial load. Various theory have been suggested to account for the elastic failure of the materials when they are subjected to the various types of combined stresses to calculate equivalent stresses. Equivalent Stress: Since the Shaft can suffer axial, bending and torsional stresses simultaneously: the Equivalent stress [Static] maximum stress is given by: eq =V (Ca +05)? + Trax) Where for Distortion Energy Theory (Von Misses Theory) n=4 for Maximum Shear Stress Theory (Tresea yield Criterion) Dimensioning for maximum static stress (Dimensioning for an even shaft) Ifaxial load is neglected Gis (ay +3 sy For Distortion Energy Theory 3 3 oq (22) +4(24)" For Maximum Shear Stress Theoryfed IM? + in| For Distortion Energy Theory ad> [= M,? +T?| For Maximum Shear Stress Theory 3.2. Shaft sizing based on stiffness [Rigidity] Deflection of shaft transmitting power must be within a certain limit in order shafts to transmit power with uniform steady motion. The stiffness or rigidity of shafting must be considered with respect to both torsion and lateral deflection. 3.2.1 Torsional rigidity: The relation between the angle of twist, the length and diameter of shaft is given by TE ase nd* J = 55 for solid shaft and (ds ~ di) aa for hollow shaft Where T= Torque of shaft L=length of shaft G = modulus of rigidity in shear J= polar moment of inertia of cross-sectional area about axis of rotation. 0 = torsion deflection (angle of twist in radians) Torsional Rigidit SS3.2.2 Lateral deflection. In some shafts lateral rigidity is the criteria of design and diameter of shafts are then determined by the permissible lateral deflection. Lateral deflection can hinder proper bearing operation, accurate machine tool performance, satisfactory gear tooth action, shaft alignment and other similar requirements The lateral deflection may be found by the use of the appropriate equation from mechanics of materials Some of the important cases are as given below a) Cantilever with a concentrated load at the free and deflection, y = = b) b) Simply supported beam with @ concentrated load at the middle, y = FE ©) if the shaft is of variable cross-section, the deflection may be determined from fundamental equation for elastic curve of a beam namely = a M = bending moment at section x E= modulus of Elasticity, 1= Moment of Inertia. : { P Lateral deflection: