5 failure theories 2015 bach khoa

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CHAPTER 5: FAILURE THEORIES Failure – A part is permanently distorted (bị bóp méo) and will not function properly A part has been separated into two or more pieces Material Strength Sy = Yield strength in tension, Syt = Syc Sys = Yield strength in shear Su = Ultimate strength in tension, Sut Suc = Ultimate strength in compression Sus = Ultimate strength in shear = 0.67 Su CHAPTER 5: FAILURE THEORIES A ductile material deforms significantly before fracturing Ductility is measured by % elongation at the fracture point Materials with 5% or more elongation are considered ductile Brittle material yields very little before fracturing, the yield strength is approximately the same as the ultimate strength in tension The ultimate strength in compression is much larger than the ultimate strength in tension FAILURE THEORIES – DUCTILE MATERIALS Yield strength of a material is used to design components made of ductile material • Maximum shear stress theory (Tresca 1886) (Thuyết bền: ứng suất tiếp lớn – TB3) (max )component > ( )obtained from a tension test at the yield point  = Sy   = Sy Failure To avoid failure (max )component Sy <  = Sy max = Sy 2n Design equation  =Sy n = Safety factor FAILURE THEORIES – DUCTILE MATERIALS SPECIAL CASES A special planar state of stress τxy= τ σx= σ  TB    4 2    Sy   Purely shear state of stress τxy= τ    Sy   2n 2n FAILURE THEORIES – DUCTILE MATERIALS • Distortion energy theory (von Mises-Hencky) (Thuyết bền: Thế biến đổi hình dáng lớn – TB4) Simple tension test → (Sy)t t (Sy)h Hydrostatic state of stress → (Sy)h h >> (Sy)t Distortion contributes to failure much more than change in volume h h t (total strain energy) – (strain energy due to hydrostatic stress) = strain energy due to angular distortion > strain energy obtained from a tension test at the yield point → failure FAILURE THEORIES – DUCTILE MATERIALS The area under the curve in the elastic region is called the Elastic Strain Energy U = ½ ε 3D case UT = ½ 1ε1 + ½ 2ε2 + ½ 3ε3 Stress-strain relationship ε1 = ε2 = 1 E 2 E  ε3 = E UT = 2E v v 2 E 1 E  v E v v 3 E 3 E  v E (12 + 22 + 32) - 2v (12 + 13 + 23) FAILURE THEORIES – DUCTILE MATERIALS Distortion strain energy = total strain energy – hydrostatic strain energy Ud = UT – Uh UT = 2E (12 + 22 + 32) - 2v (12 + 13 + 23) (1) Substitute 1 = 2 = 3 = h Uh = 2E (h2 + h2 + h2) - 2v (hh + hh+ hh) Simplify and substitute 1 + 2 + 3 = 3h into the above equation Uh = 3h2 2E (1 – 2v) = (1 + 2 + 3)2 (1 – 2v) 6E Subtract the hydrostatic strain energy from the total energy to obtain the distortion energy + v ( –  )2 + ( –  )2 + ( –  )2 3 U d = UT – U h = 6E (2) FAILURE THEORIES – DUCTILE MATERIALS Strain energy from a tension test at the yield point 1= Sy and 2 = 3 = U d = UT – Uh = Utest = (Sy) 1+v 6E Substitute in equation (2) (1 – 2)2 + (1 – 3)2 + (2 – 3)2 1+v 3E To avoid failure, Ud < Utest (1 – 2)2 + (1 – 3)2 + (2 – 3)2 ½ < Sy FAILURE THEORIES – DUCTILE MATERIALS (1 – 2)2 + (1 – 3)2 + (2 – 3)2 ½ < Sy 2D case, 3 = ½ (12 – 12 + 22) < Sy =  Where  is von Mises stress ′ = Sy n Design equation FAILURE THEORIES – DUCTILE MATERIALS τxy= τ Pure torsion,  = 1 = – 2 (12 – 2 1 + 22) = Sy2 3 = Sy2 Sys = Sy / √ → Sys = 577 Sy or Relationship between yield strength in tension and shear If y = 0, then 1, 2 = x/2 ±     Sy 3n [(x)/2]2 + (xy)2 the design equation can be written in terms of the dominant component stresses (due to bending and torsion)  TB4    3 2     τxy= τ Sy 3n σ x= σ FAILURE THEORIES – DUCTILE MATERIALS Distortion energy theory Sy ′ = n Maximum shear stress theory Sy max = 2n • Select material: consider environment, density, availability → Sy , Su • Choose a safety factor n Size Weight Cost The selection of an appropriate safety factor should be based on the following:  Degree of uncertainty about loading (type, magnitude and direction)  Degree of uncertainty about material strength  Uncertainties related to stress analysis  Consequence of failure; human safety and economics  Type of manufacturing process  Codes and standards FAILURE THEORIES – DUCTILE MATERIALS Design Process  Use n = 1.2 to 1.5 for reliable materials subjected to loads that can be determined with certainty  Use n = 1.5 to 2.5 for average materials subjected to loads that can be determined Also, human safety and economics are not an issue  Use n = 3.0 to 4.0 for well known materials subjected to uncertain loads FAILURE THEORIES – DUCTILE MATERIALS Design Process • Select material, consider environment, density, availability → Sy , Su • Choose a safety factor • Formulate the von Mises or maximum shear stress in terms of size • Use appropriate failure theory to calculate the size ′ = Sy n max = • Optimize for weight, size, or cost Sy 2n FAILURE THEORIES – BRITTLE MATERIALS One of the characteristics of a brittle material is that the ultimate strength in compression is much larger than ultimate strength in tension Suc >> Sut Mohr’s circles for compression and tension tests  Suc 3 Stress state 1 Sut  Tension test Compression test Failure envelope The component is safe if the state of stress falls inside the failure envelope 1 > 3 and 2 = FAILURE THEORIES – BRITTLE MATERIALS Modified Coulomb-Mohr theory (Thuyết Bền Morh – TB5) 3 or 2 3 or 2 Sut Safe Suc Sut Safe I Sut Safe Sut II -Sut Safe III Suc Cast iron data Suc Three design zones 1 FAILURE THEORIES – BRITTLE MATERIALS Modified Coulomb-Mohr theory Zone I Sut 1 > , 2 > and 1 > 2 Sut 1 = n 2 I Design equation Sut II -Sut Zone II III 1 > , 2 < and 2 < Sut Sut 1 = n Suc Design equation Zone III 1 > , 2 < and 2 > Sut 1 ( 1 2 – )– = Sut Suc Suc n Design equation 1 ...CHAPTER 5: FAILURE THEORIES A ductile material deforms significantly before fracturing Ductility is measured by % elongation at the fracture point Materials with 5% or more elongation... test at the yield point  = Sy   = Sy Failure To avoid failure (max )component Sy <  = Sy max = Sy 2n Design equation  =Sy n = Safety factor FAILURE THEORIES – DUCTILE MATERIALS SPECIAL CASES... Consequence of failure; human safety and economics  Type of manufacturing process  Codes and standards FAILURE THEORIES – DUCTILE MATERIALS Design Process  Use n = 1.2 to 1 .5 for reliable