docFundamental Mechanics of Materials Equations
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Fundamental Mechanics of Materials Equations
Fundamental Mechanics of Materials Equations Basic definitions π π ππππ Average normal stress in an axial member π πππ ππππ F π π‘ππ’ avg β«½ A πΎ πππππ Average direct shear stress π ππ’ πΏ π₯ ππππ‘π V avg β«½ πΌ πππβπ AV π πβπ Average bearing stress π πππππ F b β«½ π π‘βππ‘π Ab Average normal strain in an axial member π₯π π₯π€ π₯π‘ β§avg β«½ ππ‘ππππ π£πππ π = ππ ππ L π π€ π‘ πΎ = πβππππ ππ πππππ ππππ 90° Six rules for constructing shear-force and bending-moment diagrams Rule 1: β¬V β«½ P0 Rule 2: β¬V β«½ V2 β«Ί V1 β«½ x2 β«x w( x ) dx 1 Rule 3: Rule 4: dV β«½ w( x ) dx β¬ M β«½ M 2 β«Ί M1 β«½ x2 β«x V dx 1 Rule 5: Rule 6: dM β«½V dx β¬M β«½ β«ΊM 0 Flexure Average normal strain caused by temperature change β§T β«½ β¬ T Hookeβs Law (one-dimensional) β«½ E β§ and β«½ G Poissonβs ratio β§lat β― β«½ β«Ίβ§ long Flexure formula My Mc M x β«½ β«Ί or max β«½ where β«½ I I S Unsymmetric bending of arbitrary cross sections Relationship between E, G, and Ξ½ E Gβ«½ 2(1 β«Ή ) Definition of allowable stress allow β«½ failure or allow β«½ failure FS FS Factor of safety Unsymmetric bending of symmetric cross sections M y z Mz y My I z tan β«½ x β«½ β«Ί Iy Iz Mz I y FS β«½ failure actual or FS β«½ failure actual Axial deformation Deformation in axial members FL FL β«½ or β«½ β i i AE i Ai Ei Force-temperature-deformation relationship β«½ FL β«Ή β¬TL AE Torsion Maximum torsion shear stress in a circular shaft Tc max β«½ J where the polar moment of inertia J is defined as J β«½ [ R4 β«Ί r 4 ] β«½ [ D 4 β«Ί d 4 ] 2 32 πππππ Angle of twist in a circular shaft π2 π1 = π1 π2 TL TL π1 π1 = π2 π2 β«½ or β«½ β i i JG i Ji Gi Power transmission in a shaft π€ππ‘π‘π = ππ/π P β«½ T βπ = 6600 ππ β ππ/π β‘ I z z β«Ί I yz y β€ β‘β«Ί I y β«Ή I yz z β€ β₯ My β«Ή β’ y β₯M x β«½ β’ 2 β’β£ I y I z β«Ί I yz β₯β¦ β’β£ I y I z β«Ί I y2z β₯β¦ z Horizontal shear stress associated with bending Sβ«½ I c ππππππ ππ‘π πππππ πΈπ΅ π= πΈπ΄ βππ¦ ππ΄ = π πΌ βπππ¦ ππ΅ = πΌπ VQ π€βπππ π = βπ¦οΏ½π π΄π It Shear flow formula VQ qβ«½ I Shear flow, fastener spacing, and fastener shear relationship πππππ π πππππ π‘ππππ qs β± n f Vf β«½ n f f A f π= = or π πΌ For circular cross sections, 1 3 Qβ«½ d (solid sections) 12 2 1 Q β«½ [ R3 β«Ί r 3 ] β«½ [ D 3 β«Ί d 3 ] (hollow sections) 3 12 H β«½ Beam deflections Elastic curve relations between w, V, M, ΞΈ, and v for constant EI Deflection β«½ v dv Slope β«½ β«½ dx d 2v Moment M β«½ EI 2 dx dM d 3v Shear V β«½ β«½ EI 3 dx dx dV d4v Load w β«½ β«½ EI 4 dx dx Fundamental Mechanics of Materials Equations Plane stress transformations Generalized Hookeβs Law Normal and shear stresses on an arbitrary plane Normal stress/normal strain relationships 1 β§x β«½ [ β΄x β«Ί β― (β΄y β«Ή β΄z )] E 1 β§y β«½ [ β΄y β«Ί β― (β΄x β«Ή β΄z )] E 1 β§z β«½ [ β΄z β«Ί β― (β΄x β«Ή β΄y )] E Shear stress/shear strain relationships 1 1 1 β₯xy β«½ βΆ xy β₯ yz β«½ βΆ yz β₯zx β«½ βΆ zx G G G where β΄n β«½ β΄x cos 2 βͺ β«Ή β΄y sin 2 βͺ β«Ή 2 βΆ xy sin βͺ cos βͺ βΆ nt β«½ β«Ί(β΄ x β«Ί β΄y )sin βͺ cos βͺ β«Ή βΆ xy ( cos 2 βͺ β«Ί sin 2 βͺ) or ππ₯ + ππ¦ ππ₯ β ππ¦ + cos 2π + ππ₯π¦ sin 2π 2 2 ππ₯ + ππ¦ ππ₯ β ππ¦ ππ‘ = β cos 2π β ππ₯π¦ sin 2π 2 2 ππ₯ β ππ¦ sin 2π + ππ₯π¦ cos 2π πππ‘ = β 2 Principal stress magnitudes ππ = 2 β΄x β«Ή β΄y β β΄x β«Ί β΄y ββ 2 ββ β«Ή βΆ xy β«Ύ βββ β 2 2 β Orientation of principal planes βΆ xy tan 2βͺp β«½ (β΄x β«Ί β΄y ) 2 Maximum in-plane shear stress magnitude β΄ p1, p 2 β«½ 2 β β΄x β«Ί β΄y ββ 2 βΆ max β«½ β«Ύ ββ β«Ή βΆ xy β ββ 2 ββ β΄avg β«½ or Gβ«½ βΆ max β«½ β΄p1 β«Ί β΄p 2 2 β΄x β«Ή β΄y Pressure vessels Axial stress in spherical pressure vessel pr pd β΄a β«½ β«½ 2t 4t Longitudinal and hoop stresses in cylindrical pressure vessels pr pd pr pd β΄long β«½ β΄hoop β«½ β«½ β«½ 2t 4t t 2t Plane strain transformations Normal and shear strain in arbitrary directions β§n β«½ β§x cos 2 βͺ β«Ή β§y sin 2 βͺ β«Ή β₯xy sin βͺ cos βͺ β₯nt β«½ β«Ί2( β§x β«Ί β§y )sin βͺ cos βͺ β«Ή β₯xy (cos 2 βͺ β«Ί sin 2 βͺ) β§x β«Ή β§y β β§x β«Ί β§y ββ2 β β₯xy ββ2 β β«Ή βββ ββ β«½ β«Ύ βββ β 2 β β 2 ββ 2 ππ§ = πΎπ₯π¦ ππ₯ + ππ¦ ππ₯ β ππ¦ sin 2π + cos 2π + 2 2 2 πΎπ₯π¦ ππ₯ + ππ¦ ππ₯ β ππ¦ sin 2π ππ‘ = β cos 2π β 2 2 2 ππ₯ β ππ¦ πΎπ₯π¦ πΎππ‘ =β sin 2π + cos 2π 2 2 2 Principal strain magnitudes ππ = βπ (π + ππ¦ ) 1βπ π₯ or Orientation of principal strains β₯xy tan 2βͺp β«½ β§x β«Ί β§y β β§x β«Ί β§y ββ2 β β₯xy ββ2 β ββ ββ 2 βββ β«Ή βββ 2 βββ β«Ή β§y 2 Normal strain invariance β§x β«Ή β§y β«½ β§n β«Ή β§t β«½ β§p1 β«Ή β§p 2 Mises equivalent stress for plane stress 1/ 2 β΄M β«½ [ β΄2p1 β«Ί β΄ p1 β΄p 2 β«Ή β΄ 2p 2 ] Column buckling Euler buckling load Pcr β«½ β΄cr β«½ or β₯max β«½ β§p1 β«Ί β§p 2 πππππππβππ’π‘π πππ = 0 πππππππβπππ πππ = βπ Failure theories β² 2 EI ( KL )2 Euler buckling stress Maximum in-plane shear strain β₯max β«½β«Ύ 2 β§x β§avg β«½ Normal stress/normal strain relationships for plane stress 1 β§x β«½ ( β΄x β«Ί β―β΄y ) E E β΄x β«½ (β§x β«Ή β―β§y ) 1 β«Ί β―2 1 β§y β«½ ( β΄y β«Ί β―β΄x ) or E E β΄y β«½ (β§y β«Ή β―β§x ) 1 β«Ί β―2 β― β§z β«½ β«Ί (β΄x β«Ή β΄y ) E Shear stress/shear strain relationships for plane stress 1 or β₯xy β«½ βΆ xy βΆ xy β«½ Gβ₯ xy G 2 Absolute maximum shear stress magnitude β΄ β«Ί β΄min βΆabs max β«½ max 2 Normal, stress invariance β΄x β«Ή β΄y β«½ β΄n β«Ή β΄t β«½ β΄ p1 β«Ή β΄p 2 β§p1, p 2 E 2(1 β«Ή β― ) β²2 E ( KL r )2 Radius of gyration r2 β«½ I A 2 ] β«½ [ β΄x2 β«Ί β΄x β΄y β«Ή β΄y2 β«Ή 3 βΆ xy 1/2 2 x = Ξ£xi Ai Ξ£Ai Ξ£yi Ai Ξ£Ai y = I = Ξ£ ( I c + d 2 A) Table A.1 Properties of Plane Figures 1. Rectangle yβ² 6. Circle y y β x A = bh bh3 12 hb3 Iy = 12 hb3 I yβ² = 3 h 2 b x = 2 bh3 Ixβ² = 3 y = x h C β y xβ² b Ix = Οd 2 4 Οr 4 Οd 4 Ix = Iy = = 4 64 r A = Οr 2 = x C d 2. Right Triangle 7. Hollow Circle y yβ² y bh 2 h y = 3 b x = 3 bh3 Ixβ² = 12 A= β x h β y C x xβ² b bh3 36 hb3 Iy = 36 hb3 I yβ² = 12 Ix = Ο 2 (D β d 2 ) 4 Ο I x = I y = ( R4 β r 4 ) 4 Ο (D4 β d 4 ) = 64 A = Ο ( R2 β r 2 ) = R r x C d D 3. Triangle 8. Parabola yβ² bh 2 h y = 3 (a + b) x = 3 bh3 Ixβ² = 12 yβ² A= a y β x h β y C x xβ² y β x bh3 36 bh 2 (a β ab + b 2 ) Iy = 36 h 2 xβ² b2 2bh A= 3 3b x = 8 yβ² = Ix = x h C β y xβ² b y = 3h 5 y = 3h 10 Zero slope b 4. Trapezoid 9. Parabolic Spandrel a h x β y C (a + b)h A= 2 1 2a + b y = h 3 a+b h3 (a 2 + 4 ab + b 2 ) Ix = 36 (a + b) ( yβ² ) y β x h 2 xβ² b2 bh A= 3 3b x = 4 yβ² = Zero slope h β y C b x xβ² b 5. Semicircle 10. General Spandrel A= y, yβ² r C β y x xβ² 696 yβ² Οr 2 2 4r y = 3Ο I x β² = I yβ² = Ix = Οr 4 8 ( Ο8 β 98Ο )r 4 y β x h n xβ² bn bh h A= x n +1 β y xβ² n +1 x = b n+2 yβ² = Zero slope C b y = n +1 h 4n + 2 SIMPLY SUPPORTED BEAMS Beam Slope Deflection 3 2 1 ΞΈ1 = βΞΈ 2 = β 2 PL 16 EI Pb( L2 β b 2 ) ΞΈ1 = β 6 LEI 4 vmax = β 5 ML ΞΈ1 = β 3EI ML ΞΈ2 = + 6 EI 3 PL 48 EI Pa 2b 2 v=β 3LEI Pa ( L2 β a 2 ) ΞΈ2 = + 6 LEI 7 vmax = β ML2 9 3 EI β 3β at x = L ββ1 β β 3 ββ β 11 wL3 ΞΈ1 = βΞΈ 2 = β 24 EI 13 wa 2 ΞΈ1 = β (2 L β a ) 2 24 LEI ΞΈ2 = + 16 wa 2 24 LEI Px (3L2 β 4 x 2 ) 48 EI for 0 β€ x β€ L 6 v=β 2 Pbx 2 2 (L β b β x2 ) 6 LEI for 0 β€ x β€ a 9 v=β Mx (2 L2 β 3Lx + x 2 ) 6 LEI 12 vmax 5wL4 =β 384 EI 14 v=β v=β wx ( L3 β 2 Lx 2 + x3 ) 24 EI wx ( Lx3 β 4aLx 2 + 2a 2 x 2 + 4a 2 L2 24 LEI wa 3 v=β (4 L2 β 7 aL + 3a 2 ) β4a 3 L + a 4 ) for 0 β€ x β€ a 24 LEI wa 2 2 2 (2 x3 β 6 Lx 2 + a 2 x + 4 L2 x β a 2 L) v = β at x = a (2 L β a ) 24 LEI 3 7 w0 L 360 EI w0 L3 ΞΈ2 = + 45 EI ΞΈ1 = β v=β at x = a 8 10 Elastic Curve 17 w0 L4 vmax = β0.00652 EI at x = 0.5193L 15 18 v=β for a β€ x β€ L w0 x (7 L4 β 10 L2 x 2 + 3 x 4 ) 360 LEI CANTILEVER BEAMS Beam Slope Deflection Elastic Curve 20 19 ΞΈ max PL2 =β 2 EI 22 21 vmax PL3 =β 3EI 23 ΞΈ max = β 2 PL 8 EI 24 vmax = β 3 5 PL 48 EI 26 25 ΞΈ max = β ML EI 28 ML2 =β 2 EI 29 ΞΈ max 31 ΞΈ max for 0 β€ x β€ L 2 for L β€ x β€ L 2 Mx 2 v=β 2 EI 30 vmax wL4 =β 8 EI 32 w0 L3 =β 24 EI Px 2 v=β (3L β 2 x ) 12 EI PL2 (6 x β L ) v=β 48 EI 27 vmax wL3 =β 6 EI Px 2 v=β (3L β x ) 6 EI vmax wx 2 (6 L2 β 4 Lx + x 2 ) v=β 24 EI 33 w0 L4 =β 30 EI w0 x 2 v=β (10 L3 β 10 L2 x + 5 Lx 2 β x 3 ) 120 LEI