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Μνημοσύνη Κορωναίος
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8 Tale A. Properties of Plane Figures. Rectangle 6. ircle A. Rigt Triangle 7. Hollow ircle A. Triangle 8. Paraola a 4. Trapezoid 9. Paraolic Spandrel a A ( a + ) ( a + ) A 6 a + a a a 6 ( + ) 6( a ) ( a 4 a ) r d d D r R Zero slope Zero slope π d A πr 4 πr πd π A ( R r ) D d 4 ( π ) π R r 4 ( 4 4 ) A 8 A π D d 64 ( 4 4) Semicircle 0. General Spandrel r, πr A 4r π 8 r π 8 9π 4 πr 8 4 Zero slope n n A n + n n + + n + 4n + 79

9 Fundamental Mecanics of Materials quations Appendi ommon Greek letters α Alpa µ β γ Beta Gamma Mu Nu Pi, δ Delta ρ Ro ε θ κ λ psilon Σ, σ Sigma Teta Kappa amda ν π τ φ ω Tau Pi Omega Basic definitions Aerage normal stress in an aial memer F σ ag A Aerage direct sear stress V τ ag A V Aerage earing stress F σ A Aerage normal strain in an aial memer δ εlong d t εlat or or d t Aerage normal strain caused temperature cange εt α T Aerage sear strain π γ cange in angle from rad Hooke s law (one-dimensional) σ ε and τ Gγ Poisson s ratio εlat ν ε long Relationsip etween, G, and ν G ( + ν) Definition of allowale stress σ failure τ σ allow or τ allow FS FS Factor of safet σ failure τ FS or FS σ τ actual Aial deformation Deformation in aial memers failure actual failure F F i i δ or δ A i A i i Force-temperature-deformation relationsip F δ + α T A Torsion Maimum torsion sear stress in a circular saft Tc τ ma J were te polar moment of inertia J is defined as: π π J [ R 4 r 4 ] [ D 4 d 4 ] Angle of twist in a circular saft T T i i φ or φ JG JG Power transmission in a saft P Tω Power units and conersion factors W N m 550 l ft 6,600 l in. p s s s re Hz re π rad s rpm π rad 60 s i i i 88

10 Gear relationsips etween gears A and B TA TB R φ R φ R ω R ω RA RB RA DA N A Gear ratio R D N B A A B B A A B B B Si rules for constructing sear-force and ending-moment diagrams Rule : V P 0 Rule : V V V w( ) d Rule : dv d w ( ) Rule 4: M M M V d Rule 5: dm d V Rule 6: M M 0 Fleure Fleural strain and stress ε σ ρ ρ Fleure Formula M Mc M σ or σ ma were S S z Transformed-section metod for eams of two materials [were material () is transformed into an equialent amount of material ()] M M n σ σ n transformed transformed Bending due to eccentric aial load F M σ A z Unsmmetric ending of aritrar cross sections z z z zz σ M M + + z z z z z or ( M z + M z ) ( M z+ M z z ) z σ + z z z z M z+ M z z tan β M + M z z Unsmmetric ending of smmetric cross sections Mz M z σ z B tan β M z M z c Bending of cured ars σ M r ra r ( n r) ( r ) c n were r n A A da r Horizontal sear stress associated wit ending VQ τ H were Q Σ iai t Sear flow formula VQ q Sear flow, fastener spacing, and fastener sear relationsip qs nv n τ A f f f f f For circular cross sections, Q r d (solid sections) Q [ R r ] [ D d ] (ollow sections) Beam deflections lastic cure relations etween w, V, M, θ, and for constant Deflection d Slope θ (forsmall deflections) d M d Moment d dm V d Sear d d dv oad w d 4 4 d d Plane stress transformations t σt σn τtn θ τnt τnt τtn σn θ Stresses on an aritrar plane σn σ cos θ + σ sin θ + τ sinθcosθ σt σ sin θ + σ cos θ τ sinθcosθ τ ( σ σ )sinθcos θ + τ (cos θ sin θ) nt σt n 89

11 or σ + σ σ σ σ n + cosθ + τ sinθ σ + σ σ σ σ t cosθ τ sinθ σ σ τ nt sinθ + τ cosθ Principal stress magnitudes σ p, p σ + σ σ σ ± τ + Orientation of principal planes τ tanθ p σ σ Maimum in-plane sear stress magnitude τ ma σ σ σ ± τ or τma + σ + σ σ ag σ σ tanθs τ note: θ θ ± 45 Asolute maimum sear stress magnitude σma σmin τ as ma Normal stress inariance σ + σ σ + σ σ + σ n t p p Plane strain transformations Strain in aritrar directions ε ε cos θ + ε sin θ + γ sinθcosθ or n ε ε sin θ + ε cos θ γ sinθcosθ t γ ( ε ε )sinθcos θ + γ (cos θ sin θ) nt ε + ε ε ε γ εn + cosθ + sin θ ε + ε ε ε γ εt cosθ sin θ γ ( ε ε )sinθ + γ cosθ nt Principal strain magnitudes ε p, p ε + ε ε ε γ ± + Orientation of principal strains γ tanθ p ε ε s p σ p p Maimum in-plane sear strain γ ma ± ε + ε ε ag Normal strain inariance ε ε γ or γ εp ε p + ε + ε ε + ε ε + ε n t p p Generalized Hooke s law Normal stress/normal strain relationsips ma ε [ σ νσ ( + σ z )] ε [ σ νσ ( + σ z )] εz [ σz νσ ( + σ )] σ [( νε ) + νε ( + ε z)] ( + ν)( ν) σ [( νε ) + νε ( + ε z)] ( + ν)( ν) σ z [( νε ) z + νε ( + ε )] ( + ν)( ν) Sear stress/sear strain relationsips γ τ; γ z τz; γ z τz G G G were G ( + ν) Volumetric strain or Dilatation V ν e ε + ε + ε z ( σ + σ + σ z ) V Bulk modulus K ( ν) Normal stress/normal strain relationsips for plane stress ε ( σ νσ ) ε ( σ νσ ) σ ( ε + νε ) or ν ν ε z ( σ + σ ) σ ( ε + νε ) ν ν ε z ( ) ν ε + ε Sear stress/sear strain relationsips for plane stress γ τ or τ Gγ G 80

12 Tin-walled pressure essels Tangential stress and strain in sperical pressure essel pr pd pr σt εt ( ν) t 4t t ongitudinal and circumferential stresses in clindrical pressure essels pr pd pr σlong εlong ( ν) t 4t t pr pd pr σoop εoop ( ν) t t t Tick-walled pressure essels Radial stress in tick-walled clinder a pi po a( pi po) σ r a ( a ) r or a pi po a σ r a r a r ircumferential stress in tick-walled clinder a pi po a( pi po) σ θ + a ( a ) r or a pi po a σ + a r θ + a r Maimum sear stress a p p ( ) ( i o) τma σθ σ r ( a ) r ongitudinal normal stress in closed clinder a pi po σ long a Radial displacement for internal pressure onl a pi δr ( ν) r + ( + ν) ( a ) r [ ] Radial displacement for eternal pressure onl po δr ( ν) r + ( + ν) a ( a ) r [ ] Radial displacement for eternal pressure on solid clinder ( ν) pr o δ r ontact pressure for interference fit connection of tick clinder onto a tick clinder p c δ ( )( c a ) ( c a ) ontact pressure for interference fit connection of tick clinder onto a solid clinder p c δ ( c ) c Failure teories Mises equialent stress for plane stress σm σ p σ pσ p + σ p σ σσ + σ + τ olumn uckling uler uckling load P cr π ( K) uler uckling stress π σ cr ( K/ r) Radius of gration r A Secant formula P ec K σ ma + sec A r r / / P A 8

13 Simpl Supported Beams Beam Slope Deflection lastic ure θ θ ma P P θ θ 6 ma P 48 P ( 4 ) 48 for 0 θ a P θ θ θ + P ( ) 6 Pa ( a ) 6 Pa a at P ( ) 6 a for 0 M θ θ θ θ + M M 6 ma at M 9 M ( + ) 6 w 4 θ θ ma w θ θ 4 ma 5 84 w 4 w ( + ) 4 5 w wa θ ( a) 4 θ θ a 6 θ θ wa θ + ( a ) 4 w 7 w 0 0 θ 60 θ + w 0 45 wa (4 7a + a ) 4 ma a at at 0.59 w 0 4 w ( 4a + a a 4 a + a ) a for 0 wa ( 6 + a a) a for w 0 ( ) 60 8

14 antileer Beams Beam Slope Deflection lastic ure 7 P ma P θ ma ma P P ( ) 6 θma P 8 ma θma P θ ma 8 ma 5 48 P P ( ) for 0 P (6 ) for 48 9 M ma M θ ma ma M M θma 0 w ma θ ma w θ ma 6 ma 4 w 8 w (6 4 + ) 4 w 0 ma θ ma w 0 θ ma 4 ma w w 0 ( ) 0 8

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θ p = deg ε n = με ε t = με γ nt = μrad

θ p = deg ε n = με ε t = με γ nt = μrad IDE 110 S08 Test 7 Name: 1. The strain components ε x = 946 με, ε y = -294 με and γ xy = -362 με are given for a point in a body subjected to plane strain. Determine the strain components ε n, ε t, and