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° Average normal strain caused by temperature change β§T β«½ β¬ T Hooke’s Law (one-dimensional) β«½ E β§ and β«½ G Poisson’s ratio β§lat β― β«½ β«Ίβ§ long Relationship between E, G, and ν 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 FS β«½ failure actual or FS β«½ failure actual Axial deformation Deformation in axial members FL FL β«½ or β«½ ∑ 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 or β«½ β«½∑ i i JG i Ji Gi Power transmission in a shaft π€ππ‘π‘π = ππ/π Pβ«½T βπ = 6600 ππ β ππ/π 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 Rule 3: dV β«½ w( x ) dx Rule 4: β¬ M β«½ M 2 β«Ί M1 β«½ Rule 5: dM β«½V dx β¬M β«½ β«ΊM 0 Rule 6: x2 ∫x 1 w( x ) dx x2 ∫x V dx 1 Flexure 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 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 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 Horizontal shear stress associated with bending Sβ«½ I c ππππππ ππ‘π πππππ πΈπ΅ π= πΈπ΄ −ππ¦ ππ΄ = π πΌ −πππ¦ ππ΅ = πΌπ VQ π€βπππ π = ∑π¦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, θ, 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 Hooke’s 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 or 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) ππ₯ + ππ¦ ππ₯ − ππ¦ + cos 2π + ππ₯π¦ sin 2π 2 2 ππ₯ + ππ¦ ππ₯ − ππ¦ ππ‘ = − cos 2π − ππ₯π¦ sin 2π 2 2 ππ₯ − ππ¦ sin 2π + ππ₯π¦ cos 2π πππ‘ = − 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§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₯ − ππ¦ sin 2π + cos 2π + 2 2 2 πΎπ₯π¦ ππ₯ + ππ¦ ππ₯ − ππ¦ sin 2π ππ‘ = − cos 2π − 2 2 2 ππ₯ − ππ¦ πΎπ₯π¦ πΎππ‘ =− sin 2π + cos 2π 2 2 2 Principal strain magnitudes ππ = −π (π + ππ¦ ) 1−π π₯ 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§p1, p 2 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 πππππππ−ππ’π‘π πππ = 0 πππππππ−πππ πππ = −π 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 or 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 SIMPLY SUPPORTED BEAMS Beam Slope Deflection 3 2 1 θ1 = −θ 2 = − 2 PL 16 EI Pb( L2 − b 2 ) θ1 = − 6 LEI 4 vmax = − 5 PL 48 EI vmax = − ML2 9 3 EI β 3β at x = L ββ1 − β 3 ββ β 10 11 wL3 θ1 = −θ 2 = − 24 EI 13 wa 2 θ1 = − (2 L − a ) 2 24 LEI θ2 = + 16 wa 2 24 LEI Px (3L2 − 4 x 2 ) 48 EI for 0 ≤ x ≤ L 2 6 v=− Pbx 2 2 (L − b − x2 ) 6 LEI for 0 ≤ x ≤ a 9 v=− Mx (2 L2 − 3Lx + x 2 ) 6 LEI 12 vmax 5wL4 =− 384 EI 14 v=− v=− wx ( L3 − 2 Lx 2 + x3 ) 24 EI wx ( Lx3 − 4aLx 2 + 2a 2 x 2 + 4a 2 L2 24 LEI wa 3 v=− (4 L2 − 7 aL + 3a 2 ) −4a 3 L + a 4 ) for 0 ≤ x ≤ a 24 LEI wa 2 2 2 (2 x3 − 6 Lx 2 + a 2 x + 4 L2 x − a 2 L) v = − at x = a (2 L − a ) 24 LEI 3 7 w0 L 360 EI w0 L3 θ2 = + 45 EI θ1 = − v=− at x = a 8 ML θ1 = − 3EI ML θ2 = + 6 EI 3 Pa 2b 2 v=− 3LEI Pa ( L2 − a 2 ) θ2 = + 6 LEI 7 Elastic Curve 17 w0 L4 vmax = −0.00652 EI at x = 0.5193L 15 18 v=− for a ≤ x ≤ L w0 x (7 L4 − 10 L2 x 2 + 3 x 4 ) 360 LEI CANTILEVER BEAMS Beam Slope Deflection Elastic Curve 20 19 θ max PL2 =− 2 EI 22 21 vmax PL3 =− 3EI 23 θ max = − 2 PL 8 EI 24 vmax = − 3 5 PL 48 EI 26 25 θ max = − ML EI 28 ML2 =− 2 EI 29 θ max 31 θ max for 0 ≤ x ≤ L 2 for L ≤ x ≤ L 2 Mx 2 v=− 2 EI 30 vmax wL4 =− 8 EI 32 w0 L3 =− 24 EI Px 2 v=− (3L − 2 x ) 12 EI PL2 (6 x − L ) v=− 48 EI 27 vmax wL3 =− 6 EI Px 2 v=− (3L − x ) 6 EI vmax wx 2 (6 L2 − 4 Lx + x 2 ) v=− 24 EI 33 w0 L4 =− 30 EI w0 x 2 v=− (10 L3 − 10 L2 x + 5 Lx 2 − x 3 ) 120 LEI
