Shafts

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Shafts

  1. 1. A. SHAFTS1.0 INTRODUCTIONShafts are used to transmit motion, torque and/or power in any combination.They are also subjectto lateral loads. These can be constant or fluctuating. The sizing of shafts, therefore, is usually determined as a function oftorsionally Induced stresses (shear stresses), bending stresses (tensile or compressive stresses) and the nature of the load(constant or fluctuating).2.0 DETERMINATION OF STRESSES FOR SOLID CYLINDRICAL SHAFTS(a) NomenclatureLet d = shaft diameter, in.H.P. = horsepowerKm = shock factor for bending loadsKt = shock factor for torsional loadsM = bending moment, In-lbs.N = revolutions per minute (RPM)S = shear stress, lbs/in²Smax = maximum allowable shear stress, lbs/in²T = torque, in-lbs.Tmax = maximum allowable torque, In-lbs.(b) Relation Between Torque and HorsepowerT = 63,025 (H.P.) (1)N(c) Torsional LoadingFor shafts under steady torsional loads (and no bending loads),d = (5.09 T)1/3 (2)SIn the particular case of a shaft in which Smax = 12,000 psi (for example in the case of #303 stainless steel and a graduallyapplied load), equations (1) and (2) can be combined to yield:Tmax = 2353 d3 (3)and(H.P.)max = 0.037 d3N (4)Table 1 shows maximum safe torsional loads based on #303 stainless steel with Smax = 12,000psi and a gradually applied load:T12D200 Catalog
  2. 2. TABLE 1MAXIMUM SAFE TORSIONAL LOADS BASED ON #303 STAINLESS STEELWITH Smax = 12,000 PSI AND GRADUALLY APPLIED LOADShaftDiameter(inches)1/16 3/32 1/8 5/32 3/16 7/32 1/4 5/16 3/8 1/2Tmax 0.56 1.89 4.60 8.93 15.64 24.38 36.77 71.46 124.1 295The maximum torques in this table can be converted to horsepower for a given value of N (RPM) by the power Nomogramgiven in the Designer Data Section of this book.(d) Combined Torsional and Bending LoadsCombined bending and torsion arises as a result of component weight, belt tension, gear-tooth forces, etc. In that case shaftsize can be determined from the equation:d = [ 5.1 ((KmM)² + (KtT)² )1/2]1/3 (5)Smaxwhere the values of the shock factors (Km, Kt) are given in the following table:TABLE 2SHOCK FACTORS VS. TYPE OF LOADType of Load Km KtGradually appliedload1.0 1.0Minor shocks,sudden load1.4—2.0 1.0—1.5Heavy shocks,suddenheavy loads2.0—3.0 1.5—3.0If we arbitrarily consider only minor shock loads and assume that Km = Kt = (2)½ ≅ 1.41 and (as before) Smax 12,000 psi, thecombined allowable bending and torsional moments can be related to shaft size as shown in the following table:T13
  3. 3. TABLE 3TABLE OF COMBINED LOADING OF SHAFTS(Based on #303 stainless Steel With Smax = 12,000 psi. end Minor Shocks)ShaftDia.Moment Inch-Pounds1/16TortionBending00.3960.0400.3940.0790.3880.1190.3780.1590.3620.1980.3420.2380.3170.2770.2830.3170.2360.3560.1730.39603/32TortionBending01.340.1341.330.2681.310.4021.280.5361.230.6701.160.8041.070.9380.9531.070.8031.210.5721.3401/8TortionBending03.250.3253.240.6503.200.9753.091.302.981.632.861.952.602.282.312.601.952.921.433.2505/32TortionBending06.320.6326.281.266.181.896.022.525.793.165.473.795.064.424.515.053.785.682.766.3203/16TortionBending011.11.1110.92.2210.83.3310.54.4410.35.559.566.668.827.777.868.886.589.994.7311.107/32TortionBending017.21.7217.13.4416.85.1616.56.8815.88.6014.810.313.812.012.713.810.715.57.5317.201/4TortionBending026.02.6025.85.2025.47.8024.810.423.813.022.515.620.818.218.520.815.623.411.326.005/16TortionBending050.55.0650.310.149.615.248.220.246.325.243.730.340.435.436.140.430.445.52.1950.503/8TortionBending087.78.7787.417.585.926.383.735.180.443.875.952.670.261.462.670.252.678.938.487.701/2TortionBending020820.6207.441.620562.420083.2192104181124.8168145.6151166.4127.5187.293.5208.005/8TortionBending0406.240.6404.281.2398121.8387.5162.4372.3203.0351.9243.6325.1284.2290.2324.8244.0365.4177.5406.203/4TortionBending070270.2698.4140.4687.8210.6669.6280.8643.3351.0607.9421.2561.5491.4501.3561.6421.1631.8305.97020If AISI 1213 steel is used, Smax = 8000 psi. and the diameters given in the tables have to be increased by 15%.ExampleA 1/50th H.P. motor drives a shaft rotating at 315 RPM and is subject to a bending moment of 6 in-lbs, the load application beingsudden. Determine shaft size for a #303 stainless steel shaft.From equation (1),T = 63,025 (H.P.) = ( 63,025) (0.02 ) = 4 in-lbs.N (315)Referring to Table 3 (Combined Loading), we find that the combination of an allowable torsional moment in excess of 4 in-lbs.(actual 4.44) together with an allowable bending moment in excess of 6 in-lbs. (actual 10.3) first occurs for a 3/16" shaftdiameter.If there were no bending moment and the load were to be applied gradually (Km = Kt = 1) Table 1 shows that a 1/8” shaftdiameter would be sufficient.T14f
  4. 4. Precision ground stainless steel shafts in undersize, nominal and oversize dimensions, as well as cold-drawn precisionlow-carbon steel shafts are available from SDP, as featured in our catalog. A new line of case hardened steel (C1060), casehardened stainless steel (440C) and thru hardened stainless steel (416) is being offered for use with bronze and needle bearings.(e) Combined Torsional and Bending Loads for Fluctuating Loads With Stress ReversalIn such cases fatigue failure governs the design. Various failure theories have been proposed (e.g. Soderberg, Mises—Henckyetc.). All of these result in formulas involving both the average loading and the reversing component of the loading. Agreementwith experiment varies depending on the nature of the material and failure mode. The equations resulting from these severaltheories also involve yield-point stresses and endurance limits and the various considerations governing the applicability of aparticular set of equations extends beyond the scope of this discussion. For a fuller treatment the reader is referred to thefollowing literature:(i) R.M. Phelan “Fundamentals of Mechanical Design”, Third Edition, McGraw- Hill, New York,N.Y., 1970. Chapter 6.(ii) J.E. Shigley: “Mechanical Engineering Design”, Third Edition, McGraw-I-fill, New York, N.Y.,1977, Chapter 13.(iii) M.F. Spotts: “Design of Machine Elements”, Third Edition, Prentice-Hall, Englewood Cliffs,New Jersey, 1961, Chapter 3.3.0 HOLLOW CYLINDRICAL SHAFTSIn the case of hollow cylindrical shafts under torsion and/or bending, we can proceed as follows. Suppose a hollow shaft has anoutside diameter, d0, and an inside diameter, dj, To size such a shaft for torsional and/or bending loads, we can convert thedesign calculation to that for an equivalent solid shaft of the same material by noting that the stress is inversely proportional tothe ratio of moment of inertia (of the shaft cross-section) to shaft radius.Equating this ratio for both the hollow shaft and the equivalent solid shaft of diameter deq we have:π (d04 — d14) = πd3eg.32d0 32Solving for deq, we have:deq = ( d04 — d14 )1/3 (6)d0T15

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