Types of stresses and theories of failure (machine design & industrial drafting )
- 1. Subject :- Machine Design And Industrial Drafting Topic :- Types Of Stresses And Theories Of Failure • Made By :- - Bhagyesh Patel_150990119001 - Sanjay Chaudhary_150990119002 - Rajkumar Desai_150990119003 - Divyangsinh Raj_150990119004 - Naved Fruitwala _150990119006
- 2. Types Of Stresses 1. Tensile stress 2. Compressive stress 3. Bending stress 4. Direct shear stress 5. Torsional shear stress 6. Bearing pressure 7. Crushing stress 8. Contact stress
- 3. Tensile Stress • When a mechanical component is subjected to two equal and opposite axial pulls (called axial tensile forces), then the stress induced in it is called tensile stress • Example : Connecting rods
- 4. Compressive Stress • When a mechanical component is subjected to two equal and opposite axial pushes then the stress induced in it is called force compressive stress • Example:- connecting rod, links of structure, columns.
- 5. Bending Stress • When a beam is subjected to a bending moment M as shown in Fig. , the stress induced in it is known as bending stress. • The bending stress is nothing but a tensile and a compressive stress. The fibers on one side of neutral surface are subjected to tensile stresses while on other side are subjected to compressive stresses.
- 6. • The bending stress is any fiber at a distance y from the neutral axis is given by, • • The maximum bending stress is induced at a extreme fibre i.e. at a greatest distance from the neutral axis and is given by, • = bending stress at a distance y from the neutral axis, • • M = bending moment acting on section AA, N-mm • I = moment of inertia of cross-section about the neutral axis XX, • • y = distance of the fibre from the neutral axis, mm • • = distance of the extreme fibre from the neutral axis, mm • • Z = section modulus, = • • . b M y I max 2 d y b maxy maxy max I y 4 mm 3 mm 2 N mm
- 7. • For rectangular cross-section : • Let, • d = distance perpendicular to neutral axis, mm • b = distance parallel to neutral axis, mm • From Fig., • And 4 64 d I 0 max 2 d y 3 3 max 12 6 2 bd I bd Z dy
- 8. • For circular cross-section : • Let, d = diameter of the cross-section, mm • From Fig., • • And • • For hollow circular cross-section : • • Let, = outside diameter of the cross section, mm • • = inside diameter of the cross section, mm • • From Fig. , • • • And • Examples of components subjected to bending stress : Shafts, axles, levers, etc. • • 4 64 d I max 2 d y 4 3 max 64 32 2 d I d Z dy id od 4 4 0 64 id d I 0 max 2 d y 4 4 0 4 4 0 0max 0 64 64 32 2 i i d d d dI Z dy d
- 9. • Direct Shear Stress : • When a mechanical component is subjected to two equal and opposite force acting tangentially across the resisting section, it tends to shear off across the section. The stress induced in such section is known as direct shear stress. • The direct shear stress acts along the plane. • • Fig. shows two plates held together by means of a rivet. The direct shear stress induced in the rivet is given by, • • Where, = direct shear stress, s d P A 2 s d P A d 2 N mm
- 10. • = direct shear force acting across the section, N • A = cross-sectional area of the rivet, • a rivet subjected to a double shear. The direct shear stress induced in this case is, • • Examples of components subjected to direct shear stress : Rivets, bolts, knuckle pin, cotter, etc. sP 2 mm 2 s d P A
- 11. Torsional shear stress • When a mechanical component is subjected to the action of two equal and opposite couples acting in parallel planes , then the component is said to be in torsion and shear stress induced in it is known as torsional shear stress.
- 12. Torsional shear stress: For circular cross-sections: • From fig. For hollow circular cross-sections: • From fig. 0 4 4 0 4 4 0 4 4 0 0 max Outside diameter of cross sections Inside diameter of cross sections ( ) 64 ( ) 64 ( ) 32 2 i i xx i yy i xx yy d d d d I and d d I d d J I I d r 4 4 4 max 64 64 32 2 xx yy xx yy d I and d I d J I I d r
- 13. Bearing pressure • A localized compressive stress at the area of contact between two components having relative motion between them is known as bearing pressure .
- 14. • Consider a pin and eye loaded as shown in fig. the actual distribution of the bearing pressure between the eye and pin will not uniform. 2 2 N Where, Pb =average bearing pressure , ; mm P= Force acting on the pin, N; A= projected area of contact,mm ; d= diameter of pin,mm; l= length of pin in eye,mm; Bearing pressure Projects area of contact b force P P P A dl Bearing pressure
- 15. Example of components subjected to a bearing pressure: Bush and Pin Threads of power screws
- 16. Crushing Stress (Bearing Stress) • A localized compressive stress at the area of contact between two components having no relative motion between them is known as crushing stress or bearing stress. • It is a special case of bearing pressure with no relative motion between the components in contact. • Crushing Stress = Force Projected area of contact • Examples of components subjected to crushing stress are : Rivets, threads of nut and bolt, key and shaft, etc.
- 17. Contact Stress • When two bodies having curved surface (like : two spheres or two cylinder or cylinder and flat surface) are pressed against each other, the point or line contact between the two bodies changes to area contact. The stress developed in the two bodies in the contact zone is known as contact stress or Hertz contact stress. • Examples of contact stress : contact zone of wheel and rail, cam and follower, mating gear teeth, rolling contact bearing, etc.
- 18. Theories Of Failure • When a machine component is subjected to a uniaxial stress, it is easy to predict the failure because the stress and the strength can be compared directly. There is only one value of stress and one value of strength • But the problem of predicting the failure of a component subjected to biaxial or triaxial stresses is more complicated. This is because there are multiple stresses, but still only one significant strength. • The different theories of failure have been proposed to predict the failure of the components subjected to biaxial or triaxial stresses and a shear stress.
- 19. Max Principal Stress Theory (Rankine Theory) • This theory states that the failure of the mechanical component subjected to biaxial or triaxial stresses occur when the max principal stress reaches the ultimate or yield strength of the material. • According to this theory, the failure occurs when, σ1 ≥ Su or Sy • For safety against failure, σ1 ≤ Su or Sy • Considering factor of safety, σ1 ≥ Su or Sy Nf or σ1 ≤ σall • Where, σ1 = max principal stress • σall = allowable stress • Su = ultimate strength of the material • Sy = yield strength of the material • Sut = ultimate tensile strength of the material • Suc = ultimate compressive strength of the material • Nf = factor of safety
- 20. Cont... • For the biaxial stress system, the two principal stresses σ1 and σ2 are plotted on X and Y axes, for all combinations of σ1 and σ2 the shaded area represents the safe zone.
- 21. Application Of Max Principal Stress Theory • This theory ignores the possibility of failure due to shear stress. • The ductile materials are relatively weaker in shear. • Hence, this theory is not useful for ductile materials. • However, for the brittle materials which are relatively strong in shear, this theory is used.
- 22. Max Shear Stress Theory (Tresca And Guest Theory) • This theory states that the failure of the mechanical component subjected to biaxial or triaxial stresses occur when the max shear stress at any point reaches the ultimate or yield strength in shear of the material. • According to this theory, the failure occurs when, τmax ≥ Ssy or τmax ≥ 0.5Syt • For safety against failure, τmax < Ssy
- 23. Cont... • Considering Factor of safety, τmax < Ssy NF or τmax < 0.5 Syt NF or τmax < τall • Where, τmax= max shear stress • 𝜏all = allowable shear stress • Ssy = yield strength of the material in shear = 0.5Syt • Syt = yield strength of the material in tension • Nf = factor of safety
- 24. Cont... • For the biaxial stress system, the two principal stresses σ1 and σ2 are plotted on X and Y axes, for all combinations of σ1 and σ2 the shaded area represents the safe zone.
- 25. Application Of Max Shear Stress Theory • The ductile materials are weaker in shear. • Hence, this theory which accounts for shear failure, is used for ductile materials. • If this theory is used the results are on safer side. • Hence, when there exists some uncertainty in loading or assumptions are made in design for simplification, this theory is preferred.
- 26. Distortion Energy Theorem (Von Mises And Hencky Theory) • When any mechanical component is subjected to stresses, it undergoes a change in volume as well as shape. The total energy required to produce this change is stored in a material and is known as total strain energy (U). • The total strain energy is made of two parts : 1. Strain energy producing the change in volume (Uv) 2. Strain energy producing the distortion (Ud), known as distortion energy.
- 27. Cont… • This theorem states that the failure of the mechanical component subjected to biaxial or triaxial stresses occurs when the distortion energy per unit volume at any point in a component reaches the limiting distortion energy per unit volume at the yield point in simple tension. This theory is also known as Octahedral Stress Theory. • The distortion energy per unit volume in a component is given by, Ud = 1+v 3E (σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2 2
- 28. Cont… • The distortion energy per unit volume at yield strength is given by, Udy = 1+v 3E Syt 2 • According to this theory, the failure occurs when, Ud ≥ Udy • Now equating we get, 1 + v 3E (σ1 − σ2)2+(σ2 − σ3)2+(σ3 − σ1)2 2 = 1 + v 3E Syt 2
- 29. Cont… • For a biaxial stresses putting σ3 = 0 we get, (σ1−σ2)2+(σ2)2+(σ1)2 2 ≥ Syt 2 σ1 2 + σ2 2 − σ1. σ2 ≥ Syt 2 σ1 2 + σ2 2 − σ1. σ2 1 2 ≥ Syt • Hence for safety against failure, σ1 2 + σ2 2 − σ1. σ2 1 2 < Syt • Considering factor of safety, • σ1 2 + σ2 2 − σ1. σ2 1 2 ≤ Syt Nf
- 30. Cont... • For the biaxial stress system, the two principal stresses σ1 and σ2 are plotted on X and Y axes, for all combinations of σ1 and σ2 the shaded area represents the safe zone.
- 31. Application Of Distortion Energy Theory • For a ductile material, the distortion energy theory is very accurate and more close to the actual failure than any other theory. • Hence, this theory is widely used for a ductile materials when accurate results are required. • This theory is also known as Von Mises And Hencky Theory or Shear Energy Theory
- 32. Coulomb-Mohr Theory • It is used for predicting the failure of the mechanical components made of material whose tensile strength and compressive strength are not equal.
- 33. Cont… • Figure shows two Mohr circles : one for uniaxial tension test and one for uniaxial compression test. • The tangents 𝐴1 𝐵1 and 𝐴2 𝐵2 to two Mohr circles are the failure lines. • The shaded area represents the safe zone.
- 34. Cont… • According to Coulomb-Mohr Theory, for safety against failure, σ1 St − σ2 Sc ≤ 1 • Considering factor of safety, σ1 St − σ2 Sc ≤ 1 Nf • Where, σ1, σ2= max and min principal stresses in mechanical component. • St = tensile strength of the material • Sc = compressive strength of the material • Nf = factor of safety
- 35. Cont... • For the biaxial stress system, the two principal stresses σ1 and σ2 are plotted on X and Y axes, for all combinations of σ1 and σ2 the shaded area represents the safe zone.