www.EngineeringEBooksPdf.com Fundamental Equations of Mechanics of Materials Axial Load Shear Normal Stress Average direct shear stress s = P A tavg = Displacement Transverse shear stress L P(x)dx A L0 (x)E d = d = ⌺ V A VQ It t = PL AE Shear flow q = tt = dT = a ⌬TL Torsion VQ I Stress in Thin-Walled Pressure Vessel Shear stress in circular shaft t = Cylinder Tr J s1 = where p c solid cross section p J= (c - ci 4) tubular cross section o Stress Transformation Equations sx + sy P = Tv = 2pf T txЈyЈ = L f = T(x)dx L0 J(x)G f = ⌺ sx - sy s1,2 = T = 2tA m T 2A m txy (sx - sy)>2 sx + sy tmax = savg = Normal stress My s = I Unsymmetric bending tan a = cos 2u + txy sin 2u sin 2u + txy cos 2u tan 2us = - Bending M yz Mz y + , Iz Iy sx - sy { Maximum in-plane shear stress Shear Flow s = - tan 2up = TL JG q = tavg t = + Principal Stress Average shear stress in a thin-walled tube tavg pr 2t s1 = s2 = sxЈ = Angle of twist pr 2t s2 = Sphere J= Power pr t Iz tan u Iy A a sx - sy 2 b + t2xy (sx - sy)>2 txy a sx - sy A sx + sy b + t2xy Absolute maximum shear stress smax tabs = for smax, smin same sign max smax - smin for smax, smin opposite signs tabs = max www.EngineeringEBooksPdf.com Geometric Properties of Area Elements Material Property Relations Poisson’s ratio Generalized Hooke’s Law s - n(sy ex = E x s - n(sx ey = E y sz - n(sx ez = E gxy = txy, gyz = G where h b + sz ) r d 4v EI dx d3v EI dx d2v EI dx = + sz ) + sy) C h 1 tyz, gzx = tzx G G scr = Secant formula smax a h(a A= C h + b) x 2a + b a+b b h Trapezoidal area y r = w(x) = V (x) A = p2r 4r 3p C x Ix = pr Iy = pr Semicircular area y = M(x) A = pr Ix = r p2EI (KL)2 x C p2E , r = 2I>A (KL >r)2 5a P ec L P = c + sec a bd A 2r A EA r A= b Conservation of energy Ue = Ui C 8b N2L constant axial load 2AE L M dx Ui = bending moment L0 2EI L fsV dx transverse shear Ui = L0 2GA L T dx Ui = torsional moment L0 2GJ Iy = Circular area Energy Methods Strain energy 3h Triangular area Buckling Critical stress x b M EI Pcr = 36 bh bh A= dM = V dx Critical axial load Ix = Rectangular area Relations Between w, V, M Elastic Curve Iy = 12 bh 12 hb Ix = x C E G = 2(1 + n) dV = w(x), dx A = bh y Plat n = Plong ab zero slope a Semiparabolic area Ui = A= b C zero slope www.EngineeringEBooksPdf.com 4a a Exparabolic area ab 3 10 b 4 pr 4 pr Average Mechanical Properties of Typical Engineering Materialsa (SI Units) Density R (Mg/m3) Moduls of Elasticity E (GPa) Modulus of Rigidity G (GPa) 2014-T6 6061-T6 2.79 73.1 27 414 414 172 469 469 2.71 68.9 26 255 255 131 290 290 Gray ASTM 20 Malleable ASTM A-197 7.19 67.0 27 – – – 179 7.28 172 68 – – – 8.74 101 37 70.0 70.0 8.83 103 38 345 1.83 44.7 18 Structural A-36 7.85 200 Structural A992 7.85 200 Stainless 304 7.86 Tool L2 Yield Strength (MPa) SY Tens Comp.b Shear Ultimate Strength (MPa) Su Tens Comp.b Shear %Elongation in 50 mm specimen Poisson’s Ratio N Coef of Therm Expansion A (10–6)/°C 290 10 0.35 23 186 12 0.35 24 669 – 0.6 0.28 12 276 572 – 0.28 12 – 241 241 – 35 0.35 18 345 – 655 655 – 20 0.34 17 152 152 – 276 276 152 0.30 26 75 250 250 – 400 400 – 30 0.32 12 75 345 345 – 450 450 – 30 0.32 12 193 75 207 207 – 517 517 – 40 0.27 17 8.16 200 75 703 703 – 800 800 – 22 0.32 12 4.43 120 44 924 924 – 1,000 1,000 – 16 0.36 9.4 Low Strength 2.38 22.1 – – – 12 – – – – 0.15 11 High Strength 2.37 29.0 – – – 38 – – – – 0.15 11 Plastic Kevlar 49 1.45 131 – – – – 717 483 20.3 2.8 0.34 – Reinforced 30% Glass 1.45 72.4 – – – – 90 131 – – 0.34 – 26d 6.2d – 0.29e – 36d 6.7d – 0.31e – Materials Metallic Aluminum Wrought Alloys Cast Iron Alloys Copper Alloys Red Brass C83400 Bronze C86100 Magnesium Alloy Steel Alloys Titanium Alloy [Am 1004-T61] [Ti-6Al-4V] Nonmetallic Concrete Wood Select Structural Grade Douglas Fir 0.47 13.1 – – – – 2.1c White Spruce 3.60 9.65 – – – – 2.5c a Specific values may vary for a particular material due to alloy or mineral composition,mechanical working of the specimen,or heat treatment For a more exact value reference books for the material should be consulted b The yield and ultimate strengths for ductile materials can be assumed equal for both tension and compression c Measured perpendicular to the grain d Measured parallel to the grain e Deformation measured perpendicular to the grain when the load is applied along the grain www.EngineeringEBooksPdf.com Average Mechanical Properties of Typical Engineering Materialsa (U.S Customary Units) Specific Weight (lb/in3) Moduls of Elasticity E (103) ksi Modulus of Rigidity G (103) ksi Yield Strength (ksi) SY Tens Comp.b Shear Ultimate Strength (ksi) Su Tens Comp.b Shear 2014-T6 0.101 10.6 3.9 60 60 25 68 68 6061-T6 0.098 10.0 3.7 37 37 19 42 42 Gray ASTM 20 0.260 10.0 3.9 – – – 26 Malleable ASTM A-197 0.263 25.0 9.8 – – – Red Brass C83400 0.316 14.6 5.4 11.4 11.4 Bronze C86100 0.319 15.0 5.6 50 0.066 6.48 2.5 Structural A-36 0.284 29.0 Structural A992 0.284 29.0 Stainless 304 0.284 Tool L2 %Elongation in in specimen Poisson’s Ratio N Coef of Therm Expansion A (10–6)/°F 42 10 0.35 12.8 27 12 0.35 13.1 96 – 0.6 0.28 6.70 40 83 – 0.28 6.60 – 35 35 – 35 0.35 9.80 50 – 35 35 – 20 0.34 9.60 22 22 – 40 40 22 0.30 14.3 11.0 36 36 – 58 58 – 30 0.32 6.60 11.0 50 50 – 65 65 – 30 0.32 6.60 28.0 11.0 30 30 – 75 75 – 40 0.27 9.60 0.295 29.0 11.0 102 102 – 116 116 – 22 0.32 6.50 0.160 17.4 6.4 134 134 – 145 145 – 16 0.36 5.20 Low Strength 0.086 3.20 – – – 1.8 – – – – 0.15 6.0 High Strength 0.086 4.20 – – – 5.5 – – – – 0.15 6.0 Plastic Kevlar 49 0.0524 19.0 – – – – 104 70 10.2 2.8 0.34 – Reinforced 30% Glass 0.0524 10.5 – – – – 13 19 – – 0.34 – 3.78d 0.90d – 0.29e – 5.18d 0.97d – 0.31e – Materials Metallic Aluminum Wrought Alloys Cast Iron Alloys Copper Alloys Magnesium Alloy Steel Alloys Titanium Alloy [Am 1004-T61] [Ti-6Al-4V] Nonmetallic Concrete Wood Select Structural Grade Douglas Fir 0.017 1.90 – – – – 0.30c White Spruce 0.130 1.40 – – – – 0.36c a Specific values may vary for a particular material due to alloy or mineral composition,mechanical working of the specimen,or heat treatment For a more exact value reference books for the material should be consulted b The yield and ultimate strengths for ductile materials can be assumed equal for both tension and compression c Measured perpendicular to the grain d Measured parallel to the grain e Deformation measured perpendicular to the grain when the load is applied along the grain www.EngineeringEBooksPdf.com MECHANICS OF MATERIALS www.EngineeringEBooksPdf.com www.EngineeringEBooksPdf.com MECHANICS OF MATERIALS NINTH EDITION R C HIBBELER Prentice Hall Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.EngineeringEBooksPdf.com Vice President and Editorial Director, ECS: Marcia Horton Senior Acquisitions Editor: Norrin Dias Editorial Assistant: Sandra L Rodriguez Marketing Assistant: Jon Bryant Senior Managing Editor: Scott 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