Failure Theories Des Mach Elem Mech. Eng. Department Chulalongkorn University

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1 Failure Theories Review stress trasformatio Failure theories for ductile materials Maimum-Shear-Stress Theor Distortio-Eerg Theor Coulomb-Mohr Theor Failure theories for brittle materials Maimum-Normal-Stress Theor Modificatios of the Mohr Theor 00 Des Mach Elem Mech. Eg. Departmet Chulalogkor Uiversit

2 Stress trasformatio Bedig momet θ Torque P At a poit, there is ol o stress state,, Usig differet coordiate, idetical stress state ca be writte b differet,, e,, At a proper coordiate, 0 ad ol, eist. This coordiate is called Pricipal coordiate Idetical stress state, but is displaed b differet coordiate sstem cos θ si θ cos θ si θ si θ cos θ

3 Pricipal stresses Dimesio Dimesio Pricipal stress, Maimum shear stress ma ± / Pricipal stress,, Pricipal stresses are the solutio of the followig equatio I I I 0 I z I z z z z z z CW, I z z z z z CCW, Maimum shear stress ma

4 Failure Theories For simple load, failure ca be kow b simple test tesio test, compressio test. For the combiatio of loadig modes, failure theor is required to predict the failure. There is o uiversal theor of failure for the geeral case of material properties ad stress state. Cosideratio are separated depeded o metal behavior ductile or brittle. Data used i the failure theories are based o the simple test tesio test, Compressio test. Ductile Materials ε 0.05 Elogatio 5% f Maimum shear stress theor MSS Distortio eerg theor DE Ductile Coulomb-Mohr DCM Brittle Materials ε < 0.05 Elogatio < 5% f Maimum ormal stress theor MNS Brittle Coulomb-Mohr BCM Modifier Mohr MM

5 Maimum shear stress theor The maimum shear stress MSS theor predicts that ieldig begis wheever the maimum shear stress i a elemet equals or eceeds the maimum shear stress i a tesio-test specime of the same material whe that specime begis to ield. MSS theor is also referred to as the Tresca or Guest theor. Stress state at a poit Tesio-test P A ma S ma ma S S Yieldig begis Icorporate a factor of safet ma ma S S or S ad Ss 0. 5S or S

6 Maimum shear stress theor Dimesio - Plae stress Cosider at pricipal directio Assumig that pricipal stress A B No stress i the ormal plae, hece the other pricipal stress 0 A B Yield coditio A 0 - A B B A S A B S B S Yield if a stress state is outside the oield regio

Distortio-Eerg theor The distortio eerg theor predicts that ieldig occurs whe the distortio strai eerg per uit volume reaches or eceeds the distortio strai eerg per uit volume for ield i simple tesio

7 Distortio-Eerg theor The distortio eerg theor predicts that ieldig occurs whe the distortio strai eerg per uit volume reaches or eceeds the distortio strai eerg per uit volume for ield i simple tesio or compressio of the same material. The distortio eerg theor is also called the vo Mises or vo Mises-Heck theor or the octahedral-shear-stress theor Agular distortio elemet At pricipal directio av Pure volume chage Pure agular distortio Strai eerg per uit volume Strai eerg for producig ol volume chage Distortio eerg

8 Distortio-Eerg theor Strai eerg per uit volume F F s L Work F s ε work F ds Area uder F -S graph work F ds dε Area uder -ε graph V A L simple tesio: strai eerg per uit volume u dε ε u u E [ ε ε ε ] [ ν ] Hooke s law [ ν ] ε z E ε ν z E ε z z ν E [ ] [ ]

9 Distortio-Eerg theor [ ] ν E u ν E u av v ν E u v strai eerg per uit volume strai eerg per uit volume Distortio eerg ν E u u u v d

10 Distortio-Eerg theor 4 The distortio eerg theor predicts that ieldig occurs whe the distortio strai eerg per uit volume reaches or eceeds the distortio strai eerg per uit volume for ield i simple tesio or compressio of the same material. Distortio strai eerg at a poit Tesio-test ν E u d At ield S 0 d S E u ν Yieldig begis whe S / S E E ν ν Vo Mises stress

11 Distortio-Eerg theor 5 / Vo Mises stress I z coordiate, the vo Mises stress ca be calculated from [ ] / 6 z z z z Vo Mises stress Yield stregth S S Yieldig begis whe Icorporate a factor of safet

Distortio-Eerg theor 6 D - Plae stress Cosider at pricipal directio Assumig that pricipal stress A B No stress i the ormal plae, hece the other pricipal stress 0 Vo Mises stress / A A B B The oield

12 Distortio-Eerg theor 6 D - Plae stress Cosider at pricipal directio Assumig that pricipal stress A B No stress i the ormal plae, hece the other pricipal stress 0 Vo Mises stress / A A B B The oield regio of the distortio eerg theor is wider tha the regio of the Maimum shear stress theor. The predictio from the distortio eerg agrees well with all data for ductile behavior. Hece, it is the most widel used theor for ductile materials ad is recommeded for desig problems. Yield if a stress state is outside the oield regio

13 Distortio-Eerg theor 7 Dimesio - Plae stress pure shear 0 vo Mises stress equatio [ ] 6 / z z z z / Yieldig begis whe / S S S Shear ield stregth S S s

14 Eample A material has the ield stregth S c S t 00 MPa, ad ε f Determie the factor of safet of the followig cases. Mpa a b c d e 0 0 0

15 Coulomb-Mohr Theor Ductile Materials Ca be used for materials whose stregths i tesio ad compressio are ot equal. Use data from tesio test ad compressio test to draw Mohr s circles Draw failure eveloped taget to the circles Yield if a stress state is outside the evelope Triagles OB i C i are similar, therefore BC BC C C BC BC C C S t S c B C S t B C B C S c origi - C St origi - C origi - C Sc Yieldig begis Icorporate a factor of safet S t S c S t S c

16 Coulomb-Mohr Theor Ductile Materials D - Plae stress Cosider at pricipal directio Assumig that pricipal stress A B No stress i the ormal plae, hece the other pricipal stress 0 A B Yield coditio A 0 - A B B S A S t A t B S c B S c Yield if a stress state is outside the oield regio

Maimum-Normal-Stress Theor Brittle The maimum ormal stress MNS theor states that failure occurs wheever oe of the three pricipal stresses equals or eceeds the stregth.

17 Maimum-Normal-Stress Theor Brittle The maimum ormal stress MNS theor states that failure occurs wheever oe of the three pricipal stresses equals or eceeds the stregth. Pricipal stress Yieldig begis S ut or Suc Icorporate a factor of safet S ut or S uc Note Yield stregth of the brittle materials ca ot be observed, hece the ultimate tesile stregth or ultimate compressive stregth are used istead

18 Modificatios of the Mohr Theor Brittle Brittle-Coulomb-Mohr Plae stress factor of safet Modified Mohr Plae stress factor of safet A B Yield coditio A B Yield coditio A 0 A S ut A 0 A S ut - A B B S ut B S A uc B S uc - A B A B A S ut - A B A B > S S S S uc ut B S uc ut A uc B B S uc

19 Modificatios of the Mohr Theor Brittle

20 Selectio of Failure Criteria

FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES

LECTURE Third Editio FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES A. J. Clark School of Egieerig Departmet of Civil ad Evirometal Egieerig Chapter 7.4 b Dr. Ibrahim A. Assakkaf SPRING 3 ENES