Formul me-3074683 Erdi Karaçal Mechanical Engineer University of Gaziantep

Formul me-3074683 Erdi Karaçal Mechanical Engineer University of Gaziantep

• 1.
  1. Stress Analysis Moment of Inertias 1. Atalet moment of inertia; 2. Polar moment of inertia; Ix = ò y 2 dA ò 2 Iy = x dA ò 2 2 Jz = (x + y )dA Shape Ix Iy J Rectangle bh3/12 hb3/12 1b2h (b2+h2 ) Triangle bh3/36 hb3/36 æ ö ç ¸ çè ø¸ h2+b2 bh 18 Circle πd4/64 πd4/64 πd4/32 Stresses Normal Stresses Shear Stresses Axial Tensile F σ = A Torsional t Tr = J · t = 16T/πd3 for solid circular beam Compression F σ = A Bending b Mc σ = I 32M σ = · b 3 πd for solid circular beam Transverse (Flexural) t = VQ Ib , - Q = A y · max t = 4V/3A for solid circular beam · t max = 2V/A for hollow circular section · t max = 3V/2A for rectangular beam σ + σ - σ - σ Principle stresses æ ö σ = + + ç ¸ è ø t 2 x y x y 2 1,2 xy 2 2 2 t xy x y tan2φ = σ - σ Max. and min. shear stresses 2 - σ - σ = + + æ ö ç ¸ è ø x y 2 t t 1,2 xy 2
• 2.
  σ' = σ1- σ1σ2 + σ2 or t 2 2 Von-Mises stresses 2 2 σ' = σx + 3 xy (for biaxial) Stresses States Triaxial stress state 1 2 3 1 σ σ + σ ε = - μ E E 2 1 3 2 σ σ + σ ε = - μ E E 3 1 2 2 σ σ + σ ε = - μ E E Stress in Cylinders Thick-Walled (t/r>1/20) Wessels (internally and externally pressurized cyclinders): 2 2 2 2 2 i o o i p a - p b - a b (p - p )/r σ = t 2 2 b - a 2 2 2 2 2 i o o i p a - p b + a b (p - p )/r σ = r 2 2 b - a 2 i p a σ = l 2 2 b - a · If the external pressure is zero (po=0); 2 2 æ ö ç ¸ è ø a p b i σ = 1+ t 2 2 2 b - a r 2 2 æ ö ç ¸ è ø a p b i σ = 1- r 2 2 2 b - a r At the inner surface; r=aÞ σr = -pi 2 2 b + a σ = p t i 2 2 b - a At the outer surface; r=bÞ σr = 0 2 2a σ = p t i 2 2 b - a · If the internal pressure is zero (pi=0); 2 2 æ ö ç ¸ è ø b a σ = -p 1+ t o 2 2 2 b - a r 2 2 æ ö ç ¸ è ø b a σ = -p 1- r o 2 2 2 b - a r
• 3.
  At the innersurface; r=aÞ σr = 0 2 2b σ = -p t o 2 2 b - a At the outer surface; r=bÞ σr = -po 2 2 b + a σ = -p t o 2 2 b - a a=inside radius of the cylinder b=outside radius of the cylinder pi=internal pressure po=external pressure Thin-Walled Wessels(t/r<1/20): i t pd σ = 2t i l pd σ = 4t Curved Members In Flexure: · ò A r = dA ρ · My σ = Ae(r - y) Þ o o o Mc σ = Aer , i i i Mc σ = Aer Press and Shrink Fit: · 2 2 b + a σ = -p it 2 2 b - a · 2 2 c + b σ = -p ot 2 2 c - b · æ ö ç ¸ è ø d 2 2 bp c + b = +μ E c - b o 2 2 o o · bp æ b + a ö = - -μ ç ¸ è ø d 2 2 i 2 2 i E b - a i · æ ö æ ö ç ¸ ç ¸ è ø è ø = - +μ - μ d d d 2 2 2 2 bp c + b bp b + a = + o i 2 2 o 2 2 i E c - b E b - a o i · ( ) ( ) é ù ê ú êë úû d 2 2 2 2 E c - b b - a ( ) if E = E = E; interface pressure = p = o i 2 2 2 b 2b c - a
• 4.
  2. Deflection Analysis F k = y , k=spring constant T GJ k = = θ l , k=Torsional spring rate k = AE for tension or compression loading l Castigliano’s Theorem: Strain Energy Axial Load F2L U = 2AE Direct Shear Force F2L U = 2AG Torsional Load T2L U = 2GJ Bending Moment ò M2 U = dx 2EI Flexural Shear ò CF2 U = dx 2GA , C is constant C=1.2 for rectangular shape C=1.11 Circular C=2.0 for thin walled tubular, Buckling Consideration: Slenderness ratio=æ ö l k çè ø¸ , I k = A æ ö çè ø¸1 2π2EC Sy l = k Euler column U Total energy F Force on the deflection point θ Angular deflection ¶ ¶ y = U F Tl θ = GJ
• 5.
  · ³ Þ( ) æ ö æ ö l l P Cπ E çè ø¸ èç ø¸ 2 cr 2 1 Critical Unit Load = = k k A l/k or 2 2 P = Cπ EI cr l Johnson's Column æ l ö æ l ö P æ S ö æ 1 öæ l ö çè < Critital Unit Load = S - k ø¸ èç k ø¸ A ç 2π ¸ è ø èç CE ø¸èç k ø¸ · Þ 2 2 cr y y 1 = 1. Both ends are rounded-simply supported ÞC=1 2. Both ends are fixed ÞC=4 3. One end fixed, one end rounded and guided ÞC=2 4. One end fixed, one end free ÞC=1/4
• 6.
  3.Design For StaticStrength Ductile Materials 1. Max. Normal Stress Theory (MNST): · If, σ1 > σ2 > σ3 · y 1 S n = σ 3. Distortion Energy Theory · If, σ1 > σ2 > σ3 · 2 2 2 (σ1 - σ2 ) + (σ2 - σ3 ) + (σ3 - σ1) σ' = 2 · For biaxial stress state; 2 2 σ' = σx + 3τxy · y S n = σ' 2. Max. Shear Stress Theory (MSST): · Yield strength in shear (Ssy)=Sy/2 · ( ) t 1 3 max σ - σ = 2 for biaxial stress state; 1 2 2 =σ + 4τ 2 t max x xy S sy max n = τ Brittle Materials 1. Max. Normal Stress Theory (MNST): 3. The Modified Mohr Theory (MMT) · If, σ1 > σ2 > σ3 · ut 1 S n = σ · If, σ1 > σ2 > σ3
• 7.
  or uc 3 n = S σ · uc 3 S uc ut 1 ut 3 S = S - S σ -1 Sσ · 3 3 S n = σ 2. The Column Mohr Theory (CMT) or Internal Friction Theory (IFT): · uc 3 S uc 1 ut 3 S = S σ -1 Sσ · 3 3 S n = σ
• 8.
  5. Design forFatigue Strength Endurance limit for test specimen (Se’); · For ductile materials: Se’=0.5 Sut if Sut<1400 MPa Se’=700 MPa if Sut ³ 1400 MPa · For irons: Se’=0.4 Sut if Sut<400 MPa Se’=160 MPa if Sut³ 400 MPa · For Aliminiums: Se’=0.4 Sut if Sut<330 MPa Se’=130 MPa if Sut³ 330 MPa · For copper alloys: Se’ » 0.4 Sut if Sut<280 MPa Se’ » 100 MPa if Sut³ 280 MPa Se = ka kb kc kd ke Se’ Sf=10c Nb é ù ê ú ë û 1 0.8S u e b = - log 3 S é( 0.8S ) 2 ù ê u ú êë e úû c = log S
• 9.
  · ka :surface factor, ka=aSut b Surface Finish Factor a Factor b Ground 1.58 -0.065 Machined or Cold Drawn 4.51 -0.265 Hot Rolled 57.7 -0.718 As Forged 272 -0.995 · kb : size factor; kb=1 if d£ 8 mm and kb= 1.189d-0.097 if 8 mm<d£ 250 mm for bending & torsional loading. For non-rotating element, -0.097 kb = 1.189deq deq=0.37d For pure axial loading, kb=1 and Se’=0.45Sut For combined loading, a =1.11 if Sut £ 1520 MPa and a =1 if Sut ³ 1520 MPa for ductile materials. · kc : reliability factor · kd : temperature effects, kd=1 if T£ 3500 and kd=0.5 if 3500<T£ 5000 · ke : stress concentration factor, ke=1/Kf Kf=1+q(Kt-1) Kt : geometric stress concentration factor, q=notch sensitivity. Modified Goodman Soderberg Infinite Life Finite Life Infinite Life Finite Life
• 10.
  1 a m e u n = σ σ + S S n = 1 σ + σ a m f u S S 1 a m e y n = σ σ + S S 1 a m f y n = σ σ + S S · Fa=(Fmax-Fmin)/2 · Fm=(Fmax+Fmin)/2 6. Tolerances and Fits Clearance fit: TF=Cmax-Cmin Cmax=DU-dL Cmin=DL-dU Transition fit: TF=Imax+Cmax Cmax=DU-dL Imax=dU-DL Interference fit: TF=Imax-Imin Imax=dU-DL Imin=dL-Du Tolerances on the shaft, on the hole and on the fit
• 11.
  TS=dU-dL TH=DU-DL TF=TH+TS 7. Design of Power Screws æ ö ç ¸ è ø Fd L +πd μ m m R m T = 2πd - μL æ ö ç ¸ è ø Fdπd μ - L m m L m T = 2πd +μL Considering μ = tanρ ; m ( ) R Fd T = tanλ + ρ 2 m ( ) L Fd T = tanρ - λ 2 If μ ³ tanλ or m > L πd m or r > l or TL>0, then screw is self locking. If the friction between the stationary member and the collar of the screw is taken into consideration; m ( ) c c R Fd μ d F T = tanλ + ρ + 2 2 m ( ) c c L Fd μ d F T = tanρ - λ + 2 2 Condition for self locking is : TL>0 T FL Efficiency of screws: o ε = = T 2πT R R when collar friction is negligible, we tanλ obtain e as, ε = tanλ ( + ρ ) Thread Stresses: · Bearing Stresses b ( 2 2 ) r 4pF σ = πh d - d or b Fp m σ = πd th where p t = 2 · Shear Stresses For Screw Thread For Nut Thread
• 12.
  ts 2F = πd h r tn 2F = πdh · Bending Stresses The maximum bending stress, 6F mh σ = πd Stresses on the body of screw: · Tensile or Compressive stresses x F t σ = A 2 t t πd A = 4 r m t d + d d = 2 · Shear Stresses 16T = πd t R xy 3 t · Combined stresses: Based on distortion energy theory; Sy σ' = σx 2 + 3 t 2 xy n = σ' Based on maximum shear stress theory; t 2 t 2 x xy 1 S max =σ + 4 2 t sy max n = 8. Design of Bolted Joints Feb and Fep are forces shared by the bolt and by the members respectively. C = k Fe=Feb+Fep Feb=CFeFep=(1-C)Fe stiffness ratio: b k + k b m Fb=Fi+CFe Fm=Fi-(1-C)Fe
• 13.
  Stiffness of bolt:b b b A E k = L Stiffness of members: 1 1 1 1 = + +..........+ k k k k m 1 2 n · Shigley and Mishke approach; For cone angle of a = 300 , 1.813E d i æ ö ç ¸ è ø i i i k = 1.15L + 0.5d ln 5 1 1 1 1 = + +..........+ k k k k 1.15L + 2.5d m 1 2 n If L1=L2=L/2 and materials are same, æ ö çè ø¸ m 1.813Ed k = 2.885L + 2.5d 2ln 0.577L + 2.5d For cone angle of a = 450 , i i i ( ) ( ) æ ö ç ¸ è ø i πE d k = 5 2L + 0.5d ln 2L + 2.5d If L1=L2=L/2 and materials are same, æ ö çè ø¸ m πEd k = L + 0.5d 2ln 5 L + 2.5d · Wileman approach; (Bid/L) km = EdAie where Ai and Bi are constants related to the material. For Steel Ai=0.78715 and Bi=0.62873, for Aliminium Ai=0.79670 and Bi=0.63816, for Gray cast iron Ai=0.77871 and Bi=0.61616. · Filiz approach; π-B d 5 1 L æ öæ ö çè ø¸èç ø¸ m eq 2 π 1 k = E d e 2 1-B E E where 1 2 eq 1 2 E = E +E 2 æ ö çè ø¸ 1 0.1d B = L 8 æ L ö ç 1 ¸ è ø 1 2 B = 1- L
• 14.
  Static loading; Fb£ SyAt or Fb < SpAt ( Sp = 0.85Sy ) Fm ³ 0 ( 1-C ) nFe £ F i £ SpAt - CnFe n=load factor of safety N ns N Critical load per bolt = i ce F F = 1-C Dynamic Loading: CnF e a t σ = 2A i m a t F σ = + σ A s 1 a m e u n = σ σ + S S æ ö ç ¸ è ø A S CnF S t u e u F = - +1 imax n 2N S s e where: ns is strength factor of safety and N is the number of bolts Constraints: · 0.6Fp £ Fi £ 0.9Fp where Fp = AtSp · æ ö ç ¸ è ø e ut imax t ut e cF n S F = A S - +1 2N S F cF · e £ £ e (1- c) F A S - i t p N N c = p D · 3.5d £ cb £ 10d and b b N 9 . Design of Riveted Joints
• 15.
  Shearing of Rivets: t F = A , F=Force on each rivet πd2 A = 4 Secondary Shear Force: Mr å i i N 2 i 1 F'' = r Bearing (compression) Failure: s F = - A , A=td, t=thickness of the plate Plate Tension Failure: s F = A , A = ( w -Nd) t w=width of plate N=number of rivets on the selected cross section Primary Shear Force: å N i 1 F F' = A 10 . Design of Welded Joints · Primary Shear Stress t F ' = A · J = 0.707hJu · Secondary Shear Stress t Mr '' = J · I = 0.707hIu · Bending Stress Mc σ = I Table 9-3 Minimum weld-metal properties
• 16.
  AWS electrode Number Tensile Strength Yield Strength MPa Percent Elongation E60xx 420 340 17-25 E70xx 480 390 22 E80xx 530 460 19 E90xx 620 530 14-17 E100xx 690 600 13-16 E120xx 830 740 14 Table 9-5 Fatigue-strength reduction factors Type of Weld Kf Reinforced butt weld 1.2 Toe of transverse fillet weld 1.5 End of parallel fillet weld 2.7 T-butt joint with sharp corners 2.0
• 17.
  Table 9-1 TorsionalProperties of Fillet Welds* Weld Throat Area Location of G Unit Polar Moment of Inertia *G is centroid of weld group; h is weld size; plane of torque couple is in the plane of the paper; all welds are of the same size.
• 18.
  Table 9-2 BendingProperties of Fillet Welds* Weld Throat Area Location of G Unit Moment of Inertia *Iu, unit moment of inertia, is taken about a horizontal axis through G, the centroid of the weld group; h is weld size; the plane of the bending couple is normal to the paper; all welds are of the same size