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Kon-41.2010 Machine design basics B (4 cr) Machine elements Strength calculation Symbols and units Stresses Failure theories Static load Fatigue loads Stress concentration factors Reversed stress (mean stress zero) Smith diagrams (non-alloy structural steels) Engineering materials Steels Cast irons 10 Aluminium 11 Copper alloys 11 Physical properties of steels and cast irons 12 Physical properties of materials 13 Bolted joint 14 Stresses of a bolt during tightening 14 Torque required to tighten the bolt 15 Welded connections 17 Stresses in fillet weld 17 Simple calculation method 17 Parallel keys 18 Interference fits 19 Spring design 20 Helical extension and compression springs 20 Belleville springs 21 Rubber springs 22 Gears 23 Helical gears (external gears) 24 Forces on gear teeth 25 Mechanical power transmission 26 Narrow V-belt drives (SFS 3527) 27 Datum lengths of narrow V-belts and datum diameters of pulleys 28 Rolling bearings 30 Equivalent dynamic bearing load (constant) 32 Lubrication and lubricant classification 33 Lubrication mechanisms 33 Oil classification 34 Design of pressure vessels 36 Pressure equipment directive 36 Nominal design stress 36 Cylindrical and spherical shells 36 Dished ends 38 Machine Elements/SK Strength calculation Symbols and units Quantity Acceleration Force Gravity Moment of inertia Torque Mass Rotation speed Power Work Symbol a E F G J Mv, T m n P W SI-unit m/s2 N/mm2, MPa N N kgm2 Nm kg r/min, r/s W Nm, J Radius Diameter Length r d l m, mm m, mm m, mm Modulus of elasticity Quantity Area Pressure Density Stress (tensile, com- Symbol A p ρ σ SI-unit m2 Pa, N/m2, bar kg/m3 N/mm2, MPa τ ∆l (δ) ε t v ω α η µ N/mm2, MPa m, mm s m/s rad/s rad/s2 - pression, bending) Shear stress Extension Strain Time Velocity Angular velocity Angular acceleration Efficiency Friction coefficient Stresses F A Tensile stress σ= ♦ Hooke’s law σ = Eε = E∆l / l Shear stress τ= F A Surface pressure p= F A F projected area D Bending stress σ= M W Torsion stress τ= Mv Wv B Machine Elements/SK W Wz = Wy = ≈ 0, 1d Wv Cross-section area A πd ≈ 0, 2d 16 πd A= πd 32 π( D − d ) Wz = Wy = 32 D π( D4 − d ) 16 D ( D4 − d ) ≈ 0, D A= π( D2 − d ) σ Rm ReH ReL σ = F/A tensile stress cross-section area δ length change (extension) = δ/L strain A Modulus of elasticity E = tan β ε β ε ReH ReL Rm upper yield strength lower yield strength tensile strength Fig Stress-strain –diagram (low carbon steel) Failure theories Distortion energy theory, effective stress σ vert = σ + τ (1) Maximum shear stress theory, effective stress σ vert = σ + 4τ (2) Machine Elements/SK Static load A Ductile (tough) material Effective stress σ vert ≤ σ sall = ReL n (3) where ReL is a yield strength and n safety factor Normally n = 1,2 B Brittle material Effective stress σ vert ≤ Rm n (4) where Rm is a tensile strength and safety factor n = Fatigue loads a) Fully reversed c) Fluctuating Fig Fatigue loads b) Repeated Machine Elements/SK Stress concentration factors Bending Torsion Fig Stress concentration factor for a shaft shoulder The maximum stress (bending) σmax = Kft σnim (5) σnim is a nominal stress, Kft is a stress concentration factor Kft = + q(Ktt - 1) (6) where q is a notch sensitivity of the material (steel S355: q ≈ 0,9) and Ktt geometric stress concentration factor (fig 3) Machine Elements/SK k1 Surface roughness Ra = 0,3 0,9 0,6 0,8 1,6 3,2 0,8 0,7 6,3 0,6 0,5 Rolled, forged or casted 25 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 Tensile strength Rm (N/mm2) Fig Surface quality factor k1 k2 0,9 0,8 0,7 0,6 10 20 30 40 50 60 70 80 90 100 110 120 d (mm) Fig Size factor k2 Reversed stress (mean stress zero) Bending or tensile-compression load (mean stress σm = 0) n= k1k2σ w K ftσ nim (7) Torsion load (mean stress τm = 0) n= k1k2τ w K fvτ nim (8) In other cases the safety factor is calculated using Smith diagram Table Physical properties of structural steels Steel Tensile (N/mm2) Bending (N/mm2) Torsion (N/mm2) Re σw Rte σtw τvs τvw S235 (Fe 37) 235 175 335 195 170 135 E295 (Fe 50) 295 230 410 250 205 175 S355 (Fe 52) 355 245 490 265 240 215 Machine Elements/SK Notched specimen Shape Stress concentration factor Kf Bending Kft Torsion Kfv Groove 1,5 1,3 1,8 Retaining ring groove 2,5 3,5 2,5 3,5 Shoulder fillet ≈ 1,5 r/d = 0,1 and d/D = 0,7 ≈ 1,25 r/d = 0,1 and d/D = 0,7 Transverse hole 1,4 1,8 d/D = 0,14 1,4 1,8 d/D = 0,14 End-milled keyway * 2,6 ≈ 2,3 Sled-runner keyway * 2,5 2,5 Shaft-hub connection: interference fit 1,7 1,9 1,3 1,4 Shaft-hub connection: key 2,4 1,5 1,6 * Stress concentration factor depends on corner radius and material Fig Preliminary design values for stress concentration factors Machine Elements/SK Smith diagrams (non-alloy structural steels) Raaka-ainekäsikirja Muokatut teräkset uudistettu painos Metalliteollisuuden Kustannus Oy 2001 361 s ISBN 951-817-751-1 N/mm2 400 ReH = 355 S355 E295 300 245 230 295 S235 235 200 175 σw 100 100 200 300 400 σm (N/mm2) -σw -100 -175 -200 -245 -230 Tensile - compression -300 a) N/mm2 Rte= 490 S355 500 E295 400 410 S235 335 300 N/mm2 300 S355 265 200 σtw 215 250 200 175 135 195 τvw 100 200 300 400 500 σm (N/mm2) -σtw -100 -τvw 170 100 100 E295 S235 τvs = 240 205 100 200 τm 300 (N/mm2) -100 -135 -195 -175 -200 -200 -250 -265 -215 Torsion Bending -300 b) Fig c) Machine Elements/SK Engineering materials Steels According to SFS-EN 10027-1 Steels designated according to their application and mechanical or physical properties Principal symbols: • S structural steel • P steels for pressure purposes • L steels for pipelines • E engineering steel ̇ followed by a number being the specified minimum yield strength (N/mm2), e.g S235, E295 ̇ for steel casting the name shall be preceded by the letter G ̇ additional symbols for impact strength etc, e.g S355J2 Table Structural steels SFS-EN 10025 v 2004 S235JR S235J0 S235J2 S275JR S275J0 S275J2 S355JR S355J0 S355J2 S355K2 S185 E295 3) E335 3) E360 3) Yield strength 1) ReH (N/mm2) 235 235 235 275 275 275 355 355 355 355 185 295 335 360 1) Nominal thickness ≤ 16 mm Tensile strength 2) Impact strength SFS-EN 10025 SFS 200 Rm (N/mm2) 360 510 360 510 360 510 430 580 430 580 430 580 510 680 510 680 510 680 510 680 310 540 490 660 590 770 690 900 KV (J) / t (°C) 27 / 20 27 / 27 / -20 27 / 20 27 / 27 / -20 27 / 20 27 / 27 / -20 40 / -20 -/-/- v 1991 Fe 360 B FN Fe 360 C Fe 360 D2 Fe 430 B Fe 430 C Fe 430 D2 Fe 510 B Fe 510 C Fe 510 D2 Fe 510 DD2 Fe 310-0 Fe 490-2 Fe 590-2 Fe 690-2 v 1986 Fe 37 B 2) Nominal thickness < mm Classification by impact strength (SFS-EN 10027-1) Test temperature Impact strength (J) °C 20 -20 -30 -40 -50 -60 27 J JR JO J2 J3 J4 J5 J6 40 J KR KO K2 K3 K4 K5 K6 60 J LR LO L2 L3 L4 L5 L6 3) Engineering steels Fe 44 B Fe 52 C Fe 33 Fe 50 Fe 60 Fe 70 Machine Elements/SK Steels designated according to chemical composition (Examples in tables 2…4) Non-alloy steels • letter C and the carbon content % multiplied by 100 Non-alloy steels (with Mn ≥ %), non-alloy free-cutting steels and alloy steels (except high speed steels) where the content, by weight, of every alloying element is < % • carbon content % multiplied by 100 • chemical symbols indicating the alloy elements (in decreasing order) • numbers indicating the values of contents of alloy elements Alloy steels (except high speed steels) • letter X • carbon content % multiplied by 100 • chemical symbols indicating the alloy elements (in decreasing order) • numbers indicating the values of contents of alloy elements Table Quenched and tempered steels (SFS-EN 10083) Material Re (N/mm2) Rm (N/mm2) 370 450 650 800 630 780 700 850 900 1100 1000 1200 C 45 25 CrMo 42 CrMo 34 CrNiMo (40 mm < d < 100 mm) ̇ ̇ heat treatment including hardening and annealing in relative high temperature (500…700 °C) shafts, couplings, gears, bolts and nuts Table Case hardening steels ̇ ̇ ̇ SFS-EN 10084 Re (N/mm2) Rm (N/mm2) Hardness HB 20NiCrMo2-2 16MnCr5 20NiCrMo5 18CrNiMo7-6 490 590 690 780 740 1030 790 1080 1030 1370 1080 1330 265 285 345 370 higher carbon content in thin surface layer high wear resistance and fatigue strength and bending strength gears and shafts Table Stainless steels ̇ ̇ ̇ SFS-EN 10088-2 Yield strength Rp0,2 (N/mm2) Tensile strength Rm (N/mm2) Modulus of elasticity E (N/mm2) X2CrNi19-11 X2CrNi18-9 X5CrNi18-10 X2CrNiMo17-12-2 X3CrNiMo17-13-3 200 200 210 220 220 500 650 500 650 520 720 520 670 530 730 200 000 200 000 200 000 200 000 200 000 corrosion resistant ductile at low temperatures pipes, vessels, valves, machinery in process industry, containers and tanks 25 Machine Elements/SK Forces on gear teeth Transmitted load (tangential load) Ft = M v1 M v P P = = = πd1n1 πd n r1 r2 (19) Mv1,2 is a torque on a gear, n1,2 rotational speed, P power and d1,2 pitch diameter (1 pinion, gear) Radial force Fr = Ft tan α t = Ft tan α n / cos β (20) Axial force Fa = Ft tan β (21) On spur gears the teeth are straight and aligned with the axis of the gear, the helix angle β = Fn α Fr β FN b Ft β Fa Fn Fig Forces on gear teeth: Ft tangential force, Fr radial force and Fa axial force Gear ratio i= n1 ω1 d z2 = = = n2 ω2 d1 z1 (22) where index is for the driving gear (pinion) and index for the driven gear Driving Driven r1 r2 pitch point n1 n2 a Fig Two gears in mesh 26 Machine Elements/SK Mechanical power transmission P1 n1 P2 n2 Gear Motor Coupling Coupling Driven machine Gear ratio i = n1/n2 Fig Mechanical power transmission Fig Gear coupling a) b) flexible part Fig Flexible couplings (KUMERA) 27 Machine Elements/SK Narrow V-belt drives (SFS 3527) If the diameter of the small pulley dp is known, calculate the diameter of the large pulley Dp using the speed ratio i n1 n2 (1) Dp = idp (2) i= If the required centre distance of a V-belt system E and diameters Dp and dp are known, the length of the V-belt is ( Dp − dp )2 L ≈ 2E + π ( Dp + dp ) + (3) 4E If the length L differs from the standard datum length Lp (SFS-ISO 4184), the new centre distance of a V-belt system is Ep = E + Lp − L (4) The recommended centre distance of a V-belt system is E = 0,75 1,0(dp + Dp) Initial tension The initial tension of the belt is critical because it ensures that the belt will not slip under the design load The too high tension can damage the belts and bearings The proper belttensioning can be calculated according to the standard Adjustment for the centre distance The adjustable length for the mounting is y = 20 30 mm depending on the belt profile The adjustable length for the tension is x = 0,03Lp (SFS-ISO 155) Lj n1 Dp β v dp E x y Fig Adjustable lengths of the centre distance between the pulley shafts 28 Machine Elements/SK Datum lengths of narrow V-belts and datum diameters of pulleys Standard datum lengths Ld of narrow V-belts are in the table Datum diameters dd of pulleys are in the table Grooves of pulleys are in the figure The datum width wd is characterizing the groove profile The groove angle α of the pulley is 34 or 38° (SFS-ISO 4183) Table Standard datum lengths of narrow V-belts and distribution according to the sections, dimensions in millimeters (SFS-ISO 4184) Nominal datum length Ld (= Lp) Section 630 710 800 900 1000 1120 1250 1400 1600 1800 2000 2240 2500 2800 3150 3550 4000 4500 5000 5600 6300 7100 8000 9000 10000 11200 12500 + + + + + + + + + + + + + + + + SPZ Tolerances SPA + + + + + + + + + + + + + + + + SPB + + + + + + + + + + + + + + + + + SPC + + + + + + + + + + + + + + + + + Fig Grooves of the pulleys (mm) (SFS-ISO 4183) Max difference between the lengths of the belts of the same set ±6 ±8 ±8 ± 10 ± 10 ± 13 ± 13 ± 16 ± 16 ± 20 ± 20 ± 25 ± 25 ± 32 ± 32 ± 40 ± 40 ± 50 ± 50 ± 63 ± 63 ± 80 ± 80 ± 100 ± 100 ± 125 ± 125 10 16 Section wd b (min.) h (min.) e f (min.) α = 34°, dd: SPZ 8,5 12 ≤80 SPA SPB SPC 11 14 19 2,75 3,5 4,8 11 14 19 15 19 25,5 11,5 16 ≤118 ≤190 ≤315 α = 38°, dd: >80 >118 >190 >315 29 Machine Elements/SK Table Datum diameter dd (SFS-ISO 4183) Datum diameter dd (= dp) Recommendation 1) Nominal diam mm 50 53 56 60 63 67 71 75 80 85 90 95 100 106 112 118 125 132 140 150 160 170 180 190 200 212 224 236 250 265 280 300 315 335 355 375 400 425 450 475 500 530 560 600 630 1) + ∗ Z SPZ + Tol ±0,8 % A SPA Radial and axial runout B SPB C SPC D E mm 0,2 + ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ + + + ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ + + ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ + + ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ + + + + + + + ∗ ∗ ∗ ∗ ∗ ∗ ∗ + + + ∗ ∗ ±0,8 % ±0,8 % ±0,8 % ∗ ±0,8 % ∗ ∗ ∗ only classical V-belts (Z, A E) narrow and classical V-belts 0,3 0,4 0,5 0,6 + + + + + 30 Machine Elements/SK Rolling bearings Fig a) deep groove ball bearing, b) self-aligning ball bearing, c) angular contact ball bearing, d) cylindrical roller bearing, e) needle roller bearing, f) spherical roller bearing, g) taper roller bearing, h) thrust ball bearing, i) cylindrical roller thrust bearing, j) spherical roller thrust bearing (SKF) Fig Bearing housing (SKF), rolling bearing and adapter sleeve with nut and locking device Locating bearing Non-locating Fig Bearing arrangement Locating bearing Non-locating 31 Machine Elements/SK Basic rating life equation C p L10 = ⎛⎜ ⎞⎟ ⎝ P⎠ (1) where L10 is the basic rating life (millions of revolutions) C the basic dynamic load (N) P the equivalent dynamic bearing load (N) p = for ball bearings and 10/3 for roller bearings For bearings operating at constant speed the basic rating life (operating hours) is 1000000 ⎛ C ⎞ p L10 h = (2) ⎜ ⎟ 60n ⎝ P ⎠ where n is the rotational speed (r/min) Adjusted rating life (million of revolutions) C p Lna = a1a23 ⎛⎜ ⎞⎟ ⎝ P⎠ (3) where a1 is the life adjustment factor for reliability a23 the combined factor for material and lubrication The index n represents the difference between the requisite reliability and 100 % Table Values for life adjustment factor a1 Reliability % a1 90 95 0,62 96 0,53 97 0,44 98 0,33 99 0,21 Fig The viscosity ν1 required at the operating temperature to ensure adequate lubrication 32 Machine Elements/SK Fig Factor a23 as a function of the viscosity ratio κ = ν/ν1 ν is the actual viscosity of the lubricant If the lubricant contains EP-additives, higher values may be obtained (shaded area) Equivalent dynamic bearing load (constant) P = XFr + YFa where Fr Fa X Y (4) is the radial bearing load (N) the axial bearing load (N) the radial load factor for the bearing the axial load factor for the bearing Table Load factors for deep groove ball bearings (normal clearance) Fa/Fr ≤ e Fa/C0 e X Y 0,025 0,04 0,07 0,13 0,25 0,5 0,22 0,24 0,27 0,31 0,37 0,44 1 1 1 0 0 0 Fa/Fr > e X Y 0,56 0,56 0,56 0,56 0,56 0,56 1,8 1,6 1,4 1,2 C0 is a basic static load rating (a total permanent deformation of rolling element and raceway is approximately 0,0001 of the rolling element diameter) 33 Machine Elements/SK Lubrication and lubricant classification Lubrication mechanisms In fluid film lubrication the rubbing surfaces are completely separated by a thick film of lubricant Fluid films are formed in three ways: hydrodynamic, elastohydrodynamic or hydrostatic film (fig 1) In elastohydrodynamic lubrication (EHD) the viscosity of lubricant increases, as the pressure on an oil increases and elastic deformation of two surfaces occurs due to the pressure of lubricant Film and pressure is formed by motion of lubricated surfaces Hydrodynamic lubrication EHD-lubrication p pEHD u1 hc u2 u Film is formed by pumping fluid under pressure Hydrostatic lubrication F pressure pT u h p F h pP Fig Fluid film lubrication Friction coefficient µ Boundary lubrication Mixed lubrication Fluid film lubrication Hydrodynamic bearing Hydrostatic bearing Speed Fig Effect of speed on bearing friction velocity film thickness pressure load hmin 34 Machine Elements/SK The relationship between the roughnesses of the surfaces and the film thickness is important The film thickness increases as the speed is increased, the lubricant viscosity is increased, the load is decreased, or the geometric conformity of the mating surfaces is improved Boundary lubrication occurs when speeds are low or applied loads are very high For this type of lubrication EP-additives are required to prevent welding of the contact and adhesive wear Role of lubricant ̌ ̌ ̌ ̌ ̌ reduce friction and power loss reduce wear cooling prevent corrosion eliminate harmful particles • wear particles • deposits Fig Internal combustion engine (Neste Oil) Additives of lubricants ̌ ̌ ̌ ̌ ̌ ̌ ̌ pour point depressants • lower the temperature at which a mineral oil is immobilized by wax viscosity index improvers • reduce the effect of temperature on viscosity foam inhibitors oxidation inhibitors rust inhibitors detergents and dispersants • reduce deposits of sludge in internal combustion engines antiwear and extreme pressure (EP) agents Oil classification SAE viscosity classification for engine and automotive gear oils is given in tables and ISO viscosity classification for industrial oils is given in table Performance classification for engine and automotive gear oils is given in tables and 35 Machine Elements/SK Table SAE viscosity grades for engine oils SAE grade Visc cP max Pumping temp max 0W 5W 10W 15W 20W 25W 20 30 40 50 60 6200/-35 °C 6600/-30 °C 7000/-25 °C 7000/-20 °C 9500/-15 °C 13000/-10 °C - -40 °C -35 °C -30 °C -25 °C -20 °C -15 °C - Viscosity mm2/s (100 °C) max 3,8 3,8 4,1 5,6 5,6 9,3 5,6 9,3 12,5 16,3 21,9 < 9,3 < 12,5 < 16,3 < 21,9 < 26,1 Table API engine oil classification Gasoline engine oil categories Diesel engine oil categories (SA …SH), SJ, SL, SM -better performance → (CA…CE), CF, CG, CH, CI -better performance → The performance requirements for each classification are defined in terms of performance in engine tests (protection against wear, oxidation, deposits and corrosion) Table SAE viscosity grades for axle and manual transmission oils SAE grade 75W 80W 85W 90 140 250 Max temperature for a viscosity 150000 cP Viscosity mm2/s (100 °C) max -40 °C -26 °C -12 °C 4,1 7,0 11,0 13,5 24,0 41,0 24,0 41,0 Table API gear oil classification API-type Service conditions GL-1 Gear oils without EP additives GL-2 Mildly fortified gear oils for worm wheels GL-3 Lubricant with light EP for non-hypoid gears and bevel wheels GL-4 Medium EP effect lubricant for moderate load hypoid gears GL-5 High EP effect lubricant for hypoid gear drives GL-1, GL-4 and GL-5 are in common use Table ISO viscosity classes ISO VG (ISO 3448) 22 220 32 320 46 460 68 680 10 100 1000 15 150 1500 ISO viscosity class (ISO VG) is a kinematic viscosity (mm2/s) at temperature +40 °C, allowed variation ±10 % 36 Machine Elements/SK Design of pressure vessels Pressure equipment directive Directive 97/23/EC applies to the design, manufacture and conformity assessment of pressure equipment and assemblies with a maximum allowable pressure PS greater than 0,5 bar Pressure equipment means vessels, piping, safety accessories and pressure accessories Where applicable, pressure equipment includes elements attached to pressurized parts, such as flanges, nozzles, couplings, supports, lifting lugs, etc Vessel means a housing designed and built to contain fluids under pressure including its direct attachments up to the coupling point connecting it to other equipment Piping means piping components intended for the transport of fluids, when connected together for integration into a pressure system Piping includes in particular a pipe or system of pipes, tubing, fittings, expansion joints, hoses, or other pressure-bearing components as appropriate The pressure equipment must satisfy the essential requirements Pressure equipment must be designed, manufactured and checked, and if applicable equipped and installed, in such a way as to ensure its safety when put into service in accordance with the manufacturer's instructions, or in reasonably foreseeable conditions Nominal design stress The maximum allowed value of the nominal design stress is (other than austenitic steels, A < 30 %) ⎛ Rp0,2 / t Rm / 20 ⎞ ⎟ (1) f d = min⎜⎜ ; 2,4 ⎟⎠ ⎝ 1,5 where Rp0,2/t is the 0,2 % proof strength at temperature t (yield strength ReH may be used in lieu of Rp0,2) and Rm/20 is the tensile strength at temperature 20 °C For testing category the nominal stress shall be multiplied by 0,9 Numbers 1,5 and 2,4 are safety factors Equations for austenitic steels are in standard SFS-EN 13445-3 Mechanical properties of steels for pressure purposes at elevated temperatures is given in table Cylindrical and spherical shells The required thickness of cylindrical shells shall be calculated from the equation (SFS-EN 13445-3 /3/) e= pDi fz − p (2) where p is the calculation pressure, Di the inside diameter of the pressure vessel, the design stress f ≤ fd and z the weld joint coefficient The weld joint coefficient is related to the testing group (z = 1; 0,85 or 0,7) 37 Machine Elements/SK Table Mechanical properties of steels for pressure purposes ReH (N/mm2) Rm (N/mm2) > 16 > 40 t ≤ 100 mm1) t ≤ 16 mm ≤ 40 mm ≤ 60 mm 360 480 215 SFS-EN 10028-2 225 235 P235GH 410 530 245 SFS-EN 10028-2 255 P265GH 265 460 580 285 SFS-EN 10028-2 290 P295GH 295 510 650 2) SFS-EN 10028-2 335 345 P355GH 355 Fine grain (N/mm2) > 16 > 35 steels Standard t ≤ 70 mm1) t ≤ 16 mm ≤ 35 mm ≤ 50 mm P275N SFS-EN 10028-3 390 510 275 275 265 P355N SFS-EN 10028-3 490 630 355 355 345 P460N SFS-EN 10028-3 570 720 460 450 440 1) In standards mechanical properties for product thickness up to t = 150 mm 2) Product thickness ≤ 60 mm Steel Standard > 60 ≤ 100 mm1) 200 215 260 315 > 50 ≤ 70 mm1) 255 325 420 Table 0,2 % proof stress at elevated temperatures Steel 20 t / mm ≤ 60 ≤ 60 ≤ 60 ≤ 60 ≤ 35 ≤ 35 ≤ 35 P235GH P265GH P295GH P355GH P275NH P355NH P460NH Temperature °C 150 200 250 300 0,2 % proof stress / N/mm2 190 180 170 150 130 215 205 195 175 155 250 235 225 205 185 290 270 255 235 215 245 226 196 177 147 304 284 245 226 216 402 373 333 314 294 50 100 206 234 272 318 264 336 - 350 400 Standard 120 140 170 200 127 196 265 110 130 155 180 108 167 235 SFS-EN 10028-2 SFS-EN 10028-2 SFS-EN 10028-2 SFS-EN 10028-2 SFS-EN 10028-3 SFS-EN 10028-3 SFS-EN 10028-3 If the outside diameter De is known, the required thickness shall be calculated from the equation e= pDe fz + p (3) The equations are valid for e/De not greater than 0,16 Tolerances and fabrication allowances shall be additional (fig 1) c2 c2 c1 c0 c1 c0 e c0 c1 c2 er e er eord ε Fig Wall thickness en ea ε eord en ea the minimum required thickness without allowances the corrosion allowance the absolute value of possible negative tolerance on nominal thickness (from material standards) the allowance for possible thinning during manufacturing process the required thickness with allowances the additional thickness resulting from the selection of the ordered thickness the ordered thickness the nominal thickness (on drawings) the analysis thickness, used for the check of the strength 38 Machine Elements/SK The required thickness of spherical shells shall be calculated from one of he following two equations pDi e= (4) fz − p e= pDe fz + p (5) Dished ends The following requirements are limited in application to ends for which all following conditions are met (see fig 2): • • • r ≤ 0,2Di r ≥ 0,06 Di r ≥ 2e • • • h ≥ 3,5e a) e ≤ 0,08De ea ≥ 0,001De R ≤ De b) H h ≥ 3,5e H r De r De R = De e R = 0,8De e r = 0,1De r = 0,154De H = 0,193De-0,445e H = 0,225De-0,635e V ≈ 0,1(De-2e)3 V ≈ 0,1298(De-2e)3 Fig Dished ends: a) Klöpper-end, b) korbbogen-end The required thickness e shall be greatest of es, ey and eb e = max(es , ey , eb ) (6) pR fz − 0,5 p βp(0,75R + 0,2 Di ) ey = f es = ⎡ p ⎛D ⎞ eb = (0,75R + 0,2 Di ) ⎢ ⎜ i⎟ ⎢⎣111 f b ⎝ r ⎠ (7) (8) ⎛ ⎞ 0,825 ⎤ ⎜⎝ 1,5 ⎟⎠ ⎥ ⎥⎦ (9) where fb = fb = Rp0,2 / t 1,5 1,6 Rp0,2 / t 1,5 (10) (for cold spun seamless austenitic stainless steel) (11) 39 Machine Elements/SK Formulae for calculation of factor β ⎛1⎞ Z = log10 ⎜ ⎟ ⎝Y ⎠ ⎞ ⎛e Y = min⎜ ;0,04 ⎟ ⎠ ⎝R N = 1,006 − X = r Di [6,2 + (90Y )4 ] ♦ X = 0,06: β 0,06 = N ( −0,3635Z + 2,2124 Z − 3,2937 Z + 1,8873) ♦ 0,06 < X < 0,1: β = 25[(0,1 − X ) β 0,06 + ( X − 0,06) β 0,1 ] ♦ X = 0,1: β 0,1 = N ( −0,1833Z + 1,0383Z − 1,2943Z + 0,837) ♦ 0,1 < X < 0,2: β = 10[(0,2 − X ) β 0,1 + ( X − 0,1) β 0,2 ] ♦ X = 0,2: β 0,2 = max 0,95(0,56 − 1,94Y − 82,5Y );0,5 [ ] The calculation method for β is iterative Computer procedure is recommended References Painelaitteet Turvatekniikan keskus (TUKES) http://www.tukes.fi/painelaitteet/esitteet_ ja_oppaat/ painelaiteopas.pdf 2.12.2004 16 s Heikkilä E & Huhdankoski E Rautaruukin paineastiakäsikirja 1999, painos Raahe: Rautaruukki Oy 1999 176 s ISBN 952-5010-27-9 SFS-EN 13445-3 Lämmittämättömät painesäiliöt Osa 3: Suunnittelu Unfired pressure vessels Part Design Suomen Standardisoimisliitto 2002 708 s Teollisuusputkistot ja painelaitesäädäntö Kunnossapitokoulu n:o 71 Kunnossapito 10 2001 s Hovi K Paineastiat, putkistot ja niiden koneenosat Julkaisussa: Airila M et al (toim.) Koneenosien suunnittelu 4, WSOY 1985 S 13 165 ISBN 951-0-13223-3 SFS-EN 13445-2 Lämmittämättömät painesäiliöt Osa 2: Materiaalit Unfired pressure vessels Part Materials Suomen Standardisoimisliitto 2002 101 s Other standards: SFS-EN 13480 Parts 1…5 Metalliset teollisuusputkistot Metallic industrial piping SFS-EN 12952 Osat 1…8 Vesiputkikattilat Water-tube boilers SFS-EN 12953 Osat 1…8 Tulitorvikattilat Shell boilers [...]... spring Fig 7 Cylindrical rubber spring (torsion loading) 23 Machine Elements/SK Gears Gears are used to transmit torque and angular velocity in many applications There is a wide variety of gear types to choose from Spur gears Helical gears Spur gears, internal set Bevel gears Worm and worm gear Rack and pinion Crossed helical gears 24 Machine Elements/SK Helical gears (external gears) Normal module mn,