Solid Mechanics Formulas Cheat Sheet, Cheat Sheet of Applied Solid Mechanics

Download Solid Mechanics Formulas Cheat Sheet and more Cheat Sheet Applied Solid Mechanics in PDF only on Docsity! Formulas in Solid Mechanics Tore Dahlberg Solid Mechanics/IKP, Linköping University Linköping, Sweden This collection of formulas is intended for use by foreign students in the course TMHL61, Damage Mechanics and Life Analysis, as a complement to the textbook Dahlberg and Ekberg: Failure, Fracture, Fatigue - An Introduction, Studentlitteratur, Lund, Sweden, 2002. It may be use at examinations in this course. Contents Page 1. Definitions and notations 1 2. Stress, Strain, and Material Relations 2 3. Geometric Properties of Cross-Sectional Area 3 4. One-Dimensional Bodies (bars, axles, beams) 5 5. Bending of Beam Elementary Cases 11 6. Material Fatigue 14 7. Multi-Axial Stress States 17 8. Energy Methods the Castigliano Theorem 20 9. Stress Concentration 21 10. Material data 25 Version 03-09-18 1. Definitions and notations Definition of coordinate system and loadings on beam Loaded beam, length L, cross section A, and load q(x), with coordinate system (origin at the geometric centre of cross section) and positive section forces and moments: normal force N, shear forces Ty and Tz, torque Mx, and bending moments My, Mz Notations Quantity Symbol SI Unit Coordinate directions, with origin at geometric centre of x, y, z m cross-sectional area A Normal stress in direction i (= x, y, z) σi N/m2 Shear stress in direction j on surface with normal direction i τij N/m2 Normal strain in direction i εi Shear strain (corresponding to shear stress τij) γij rad Moment with respect to axis i M, Mi Nm Normal force N, P N (= kg m/s2) Shear force in direction i (= y, z) T, Ti N Load q(x) N/m Cross-sectional area A m2 Length L, L0 m Change of length δ m Displacement in direction x u, u(x), u(x,y) m Displacement in direction y v, v(x), v(x,y) m Beam deflection w(x) m Second moment of area (i = y, z) I, Ii m 4 Modulus of elasticity (Young’s modulus) E N/m2 Poisson’s ratio ν Shear modulus G N/m2 Bulk modulus K N/m2 Temperature coefficient α x y z T T M M N y z y z xM Tz Mz My NxM Ty L A q x( ) K− 1 1 Rotation of axes Coordinate system Ωηζ has been rotated the angle α with respect to the coordinate system Oyz Principal moments of area I1 + I2 = Iy + Iz Principal axes A line of symmetry is always a principal axis Second moment of area with respect to axes through geometric centre for some symmetric areas (beam cross sections) Rectangular area, base B, height H Solid circular area, diameter D Thick-walled circular tube, diameters D and d y z z y d A Iη = ⌠⌡A ζ2 dA = Iy cos 2α + Iz sin 2α − 2Iyz sinα cosα Iζ = ⌠⌡A η2 dA = Iy sin 2α + Iz cos 2α + 2Iyz sinα cosα Iηζ = ⌠⌡A ζη dA = (Iy − Iz) sinα cosα + Iyz (cos 2α − sin2α) = Iy − Iz 2 sin 2α + Iyz cos 2α I1,2 = Iy + Iz 2 ± R where R =√ Iy − Iz2  2 + Iyz2 sin 2α = − Iyz R or cos 2α = Iy − Iz 2R y z H B Iy = BH 3 12 and Iz = HB3 12 y z D Iy = Iz = πD 4 64 y z Dd Iy = Iz = π 64 ( D 4 − d 4 ) 4 Thin-walled circular tube, radius R and wall thickness t (t << R) Triangular area, base B and height H Hexagonal area, side length a Elliptical area, major axis 2a and minor axis 2b Half circle, radius a (geometric centre at e) 4. One-Dimensional Bodies (bars, axles, beams) Tension/compression of bar Change of length N, E, and A are constant along bar L = length of bar N(x), E(x), and A(x) may vary along bar Torsion of axle Maximum shear stress Mv = torque = Mx Wv = section modulus in torsion (given below) Torsion (deformation) angle Mv = torque = Mx Kv = section factor of torsional stiffness (given below) y z R t Iy = Iz = πR 3t y z H 2B/ 2B/ Iy = BH 3 36 and Iz = HB3 48 y z a a a a a Iy = Iz = 5√3 16 a 4 y z 2a b2 Iy = πab 3 4 and Iz = πba 3 4 y z a e Iy =    π 8 − 8 9π    a 4 ≅ 0,110 a 4 and e = 4a 3π δ = NL EA or δ = ⌠⌡0 L ε(x)dx = ⌠⌡0 L N(x) E(x)A(x) dx τmax = Mv Wv Θ = Mv L GKv 5 Section modulus Wv and section factor Kv for some cross sections (at torsion) Torsion of thin-walled circular tube, radius R, thickness t, where t << R, Thin-walled tube of arbitrary cross section A = area enclosed by the tube t(s) = wall thickness s = coordinate around the tube Thick-walled circular tube, diameters D and d, Solid axle with circular cross section, diameter D, Solid axle with triangular cross section, side length a Solid axle with elliptical cross section, major axle 2a and minor axle 2b Solid axle with rectangular cross section b by a, where b ≥ a for kWv and kKv, see table below y z R t Wv = 2πR 2t Kv = 2πR 3t t(s) (s) AArea s Wv = 2Atmin Kv = 4A2 ⌠ ⌡s [t(s)]− 1 ds y z Dd Wv = π 16 D 4 − d 4 D Kv = π 32 (D 4 − d 4) y z D Wv = πD 3 16 Kv = πD 4 32 y z a / 2a / 2a Wv = a 3 20 Kv = a 4 √3 80 y z 2a b2 Wv = π 2 a b 2 Kv = πa 3b 3 a 2 + b 2 y z a b Wv = kWv a 2b Kv = kKv a 3b 6 Non-homogeneous boundary conditions (a) Displacement δ prescribed (b) Slope Θ prescribed (c) Moment M0 prescribed (d) Force P prescribed Beam on elastic bed Differential equation EI = constant bending stiffness k = bed modulus (N/m2) Solution Boundary conditions as given above Beam vibration Differential equation EI = constant bending stiffness m = beam mass per metre (kg/m) t = time Assume solution w(x,t) = X(x)⋅T(t). Then the standing wave solution is where µ4 = ω2m /EI Boundary conditions (as given above) give an eigenvalue problem that provides the eigenfrequencies and eigenmodes (eigenforms) of the vibrating beam w(*) = δ x x=L x x=L O z O 0 0MM x x=LP P (a) (b) (c) (d) d dx w(*) = Θ − EI d2 dx 2 w(*) = M0 − EI d3 dx 3 w(*) = P EI d4 dx 4 w(x) + kw(x) = q (x) w(x) = wpart(x) + whom(x) where whom(x) = {C1 cos (λx) + C2 sin (λx)} e λx + {C3 cos (λx) + C2 sin (λx)} e − λx ; λ4 = k 4EI EI ∂4 ∂x 4 w(x , t) + m ∂2 ∂t2 w(x , t) = q (x , t) T(t) = e iωt and X(x) = C1 cosh (µx) + C2 cos (µx) + C3 sinh (µx) + C4 sin (µx) 9 Axially loaded beam, stability, the Euler cases Beam axially loaded in tension Differential equation N = normal force in tension (N > 0) Solution New boundary condition on shear force (other boundary conditions as given above) Beam axially loaded in compression Differential equation P = normal force in compression (P > 0) Solution New boundary condition on shear force (other boundary conditions as given above) Elementary cases: the Euler cases (Pc is critical load) Case 1 Case 2a Case 2b Case 3 Case 4 EI d4 dx 4 w(x) − N d2 dx 2 w(x) = q (x) w(x) = wpart(x) + whom(x) where whom(x) = C1 + C2√ NEI x + C3 sinh √ NEI x + C4 cosh √ NEI x T(*) = − EI d3 dx 3 w(*) + N d dx w(*) EI d4 dx 4 w(x) + P d2 dx 2 w(x) = q (x) w(x) = wpart(x) + whom(x) where whom(x) = C1 + C2√ PEI x + C3 sin √ PEI x + C4 cos √ PEI x T(*) = − EI d3 dx 3 w(*) − P d dx w(*) P L, EI L, EI P L, EI P L, EI P L, EI P Pc = π2EI 4L2 Pc = π2EI L2 Pc = π2EI L2 Pc = 2.05 π2EI L2 Pc = 4π2EI L2 10 5. Bending of Beam Elementary Cases Cantilever beam x z w(x) L, EI P w(x) = PL3 6EI    3 x 2 L2 − x 3 L3    w(L) = PL3 3EI d dx w(L) = PL2 2EI x z w(x) L, EI M w(x) = ML2 2EI    x 2 L2    w(L) = ML2 2EI d dx w(L) = ML EI x z w(x) L, EI q = Q/L w(x) = qL4 24EI    x 4 L4 − 4 x 3 L3 + 6 x 2 L2    w(L) = qL4 8EI d dx w(L) = qL3 6EI x z w(x) L, EI q 0 w(x) = q0 L 4 120EI    x 5 L5 − 10 x 3 L3 + 20 x 2 L2    w(L) = 11 q0 L 4 120EI d dx w(L) = q0 L 3 8EI x z w(x) L, EI q 0 w(x) = q0 L 4 120EI    − x 5 L5 + 5 x 4 L4 − 10 x 3 L3 + 10 x 2 L2    w(L) = q0 L 4 30EI d dx w(L) = q0 L 3 24EI 11 6. Material Fatigue Fatigue limits (notations) Load Alternating Pulsating Tension/compression Bending Torsion The Haigh diagram σa = stress amplitude σm = mean stress σY = yield limit σU = ultimate strenght σu, σup = fatigue limits λ, δ, κ = factors reducing fatigue limits (similar diagrams for σub, σubp and τuv, τuvp) Factors reducing fatigue limits Surface finish κ Factor κ reducing the fatigue limit due to surface irregularities (a) polished surface ( κ = 1) (b) ground (c) machined (d) standard notch (e) rolling skin (f) corrosion in sweet water (g) corrosion in salt water ± σu σup ± σup ± σub σubp ± σubp ± τuv τuvp ± τuvp u a m up up up u Y U Y 300 600 900 1200 MPa (a) (b) (c) (d) (e) (f) (g) U 1.0 0.8 0.6 0.4 0.2 14 Volume factor λ (due to process) Factor λ reducing the fatigue limit due to size of raw material (a) diameter at circular cross section (b) thickness at rectangular cross section Volume factor δ (due to geometry) Factor δ reducing the fatigue limits σub and τuv due to loaded volume. Steel with ultimate strength σU = (a) 1500 MPa (b) 1000 MPa (c) 600 MPa (d) 400 MPa (e) aluminium Factor δ = 1 when fatigue notch factor Kf > 1 is used. Fatigue notch factor Kf (at stress concentration) Kt = stress concentration factor (see Section 12.8) q = fatigue notch sensitivity factor Fatigue notch sensitivity factor q Fatigue notch sensitivity factor q for steel with ultimate strength σU = (a) 1600 MPa (b) 1300 MPa (c) 1000 MPa (d) 700 MPa (e) 400 MPa 20 40 60 80 10 20 30 40 (a) (b) 100mm 50mm 1.0 0.8 40 80 1200 (a) (b) (c) (d) (e) Diameter or thickness in mm 1.0 0.9 0.8 Kf = 1 + q (Kt − 1) 1 102 5 (a) (b) (c) (d) (e) q r 1.0 0.8 0.6 0.4 0.2 0.1 0.5 Fillet radius in mm 15 Wöhler diagram σai = stress amplitude Ni = fatigue life (in cycles) at stress amplitude σai Damage accumulation D ni = number of loading cycles at stress amplitude σai Ni = fatigue life at stress amplitude σai Palmgren-Miner’s rule Failure when ni = number of loading cycles at stress amplitude σai Ni = fatigue life at stress amplitude σai I = number of loading stress levels Fatigue data (cyclic, constant-amplitude loading) The following fatigue limits may be used only when solving exercises. For a real design, data should be taken from latest official standard and not from this table.1 Material Tension Bending Torsion alternating pulsating alternating pulsating alternating pulsating MPa MPa MPa MPa MPa MPa Carbon steel 141312-00 110 110 110 170 150 150 100 100 100 141450-1 140 130 130 190 170 170 120 120 120 141510-00 230 141550-01 180 160 160 240 210 210 140 140 140 141650-01 200 180 180 270 240 240 150 150 150 141650 460 Stainless steel 2337-02, Aluminium SS 4120-02, ; SS 4425-06, 1 Data in this table has been collected from B Sundström (editor): Handbok och Formelsamling i Hållfasthetslära, Institutionen för hållfasthetslära, KTH, Stockholm, 1998. 1 2 3 4 5 6 70 a log N ai D = ni Ni ∑ i = 1 I ni Ni = 1 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± σu = ± 270 MPa σub = ± 110 MPa σu = ± 120 MPa 16 Principal strains and principal directions at three-dimensional stress state Use shear strain The determinant gives three roots (the principal strains) I = unit matrix Direction of principal strain εi (i = 1, 2, 3) is given by nix, niy and niz are the elements of the unit and vector ni in the direction of εi ( T means transpose) Hooke’s law, including temperature term (three-dimensional stress state) α = temperature coefficient ∆T = change of temperature (relative to temperature giving no stress) Effective stress The Huber-von Mises effective stress (the deviatoric stress hypothesis) The Tresca effective stress (the shear stress hypothesis) εij = γij / 2 for i ≠ j Strain matrix E =    εx εxy εxz εyx εy εyz εzx εzy εz   | E − ε I | = 0 (E − εi I) ⋅ ni = 0 ni T ⋅ ni = 1 εx = 1 E [σx − ν(σy + σz)] + α ∆T εy = 1 E [σy − ν(σz + σx)] + α ∆T εz = 1 E [σz − ν(σx + σy)] + α ∆T γxy = τxy G γyz = τyz G γzx = τzx G σe vM = √σx2 + σy2 + σz2 − σxσy − σyσz − σzσx + 3τxy2 + 3τyz2 + 3τzx2 =√12{(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2} σe T = max [ | σ1 − σ2 | , | σ2 − σ3 | , | σ3 − σ1 | ] = σmax pr − σmin pr (pr = principal stress) 19 8. Energy Methods the Castigliano Theorem Strain energy u per unit of volume Linear elastic material and uni-axial stress Total strain energy U in beam loaded in tension/compression, torsion, bending, and shear Mt = torque = Mx Kv = section factor of torsional stiffness Mbend = bending moment = My β = shear factor, see below Cross section Shear factor β β is given for some cross sections in the table (µ is the Jouravski factor, see Section 12.3 One-Dimensional Bodies) Elementary case: pure bending Only bending momentet Mbend is present. The moment varies linearly along the beam with moments M1 and M2 at the beam ends. One has Mbend(x) = M1 + (M2 M1)x /L, which gives The second term is negative if M1 and M2 have different signs The Castigliano theorem δ = displacement in the direction of force P of the point where force P is applied Θ = rotation (change of angle) at moment M u = σ ε 2 Utot = ⌠⌡0 L   N(x)2 2EA(x) + Mt(x)2 2GKv(x) + Mbend(x)2 2EI(x) + β T(x)2 2GA(x)    dx β µ 6/5 3/2 10/9 4/3 2 2 A/A A/Awebweb β = A I2 ⌠ ⌡A    SA’ b    2 dA L, EI MM1 2 M1 M M2 x − Utot = L 6EI {M1 2 + M1 M2 + M2 2} δ = ∂U ∂P and Θ = ∂U ∂M 20 9. Stress Concentration Tension/compression Maximum normal stress at a stress concentration is σmax = Kt σnom, where Kt and σnom are given in the diagrams Tension of flat bar with shoulder fillet Tension of flat bar with notch Tension of circular bar with shoulder Tension of circular bar with U-shaped fillet groove K t 0 B b P nom = bh P P r r/b B/b thickness h 3.0 2.5 2.0 1.5 1.0 2.0 1.5 1.2 1.1 1.05 1.01 0.1 0.2 K t 0 nom = bh P r r/b B/b PP B b thickness h 3.0 2.5 2.0 1.5 1.0 2.0 1.2 1.1 1.05 1.01 0.1 0.2 0.3 0.4 0.5 0.6 K t 0 nom = r PP D d r/d d2 D/d 4 P 3.0 2.5 2.0 1.5 1.0 2.0 1.5 1.2 1.1 1.05 1.01 0.1 0.2 0.3 K t 0 nom = r PP D d r/d d2 D/d 4 P 3.0 2.5 2.0 1.5 1.0 1.2 1.1 1.05 1.01 0.1 0.2 0.3 0.4 0.5 0.6 21 Torsion Maximium shear stress at stress concentration is τmax = Kt τnom, where Kt and τnom are given in the diagrams Torsion of circular bar with shoulder Torsion of circular bar with notch fillet Torsion of bar with longitudinal keyway Torsion of circular bar with hole K t 0 nom = r D d r/d d D/d MM 3 vv Mv16 3.0 2.5 2.0 1.5 1.0 2.0 1.3 1.2 1.1 0.1 0.2 0.3 K t 0 nom = r D d r/d d M M 3 D/d v v vM16 3.0 2.5 2.0 1.5 1.0 1.2 1.05 1.01 0.1 0.2 0.3 0.4 0.5 0.6 K t 0 r/d r d 8 d 3nom = v /4 7d d 16M 4.0 3.5 3.0 2.5 2.0 0.05 0.10 K t 0 nom = d MM 3 D d d/D M v v v D 16 D 6 2 2.0 1.5 1.0 0.1 0.2 0.3 24 10. Material data The following material properties may be used only when solving exercises. For a real design, data should be taken from latest official standard and not from this table (two values for the same material means different qualities).1 Material Young’s ν α106 Ultimate Yield limit modulus strength tension/ bending torsion E compression GPa K-1 MPa MPa MPa MPa Carbon steel 141312-00 206 0.3 12 360 >240 260 140 460 141450-1 205 0.3 430 >250 290 160 510 141510-00 205 0.3 510 >320 640 141550-01 205 0.3 490 >270 360 190 590 141650-01 206 0.3 11 590 >310 390 220 690 141650 206 0.3 860 >550 610 Offset yield strength Rp0.2 (σ0,2) Stainless steel 2337-02 196 0.29 16.8 >490 >200 Aluminium SS 4120-02 70 23 170 >65 215 SS 4120-24 70 23 220 >170 270 SS 4425-06 70 23 >340 >270 1 Data in this table has been collected from B Sundström (editor): Handbok och Formelsamling i Hållfasthetslära, Institutionen för hållfasthetslära, KTH, Stockholm, 1998. 25