1. Principal Stresses and Strain and Theories of Failure
Strength of Materials
Prof. A. S. PATIL
Department of Mechanical Engineering
Sinhgad Academy of Engineering, Pune
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2. Agenda
Normal & shear stresses on any oblique plane. Concept of principal planes,
derivation of expression for principal stresses & maximum shear stress,
Position of principal planes & planes of maximum shear.
Graphical solution using Mohr’s circle of stresses.
Principal stresses in shaft subjected to torsion, bending moment & axial thrust (solid as well as hollow), Concept of equivalent torsional and bending moments.
Theories of elastic failure: Maximum principal stress theory, maximum shear stress theory, maximum distortion energy theory, maximum strain theory -their applications & limitations.
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3. 6.1 STRESS ON AN OBLIQUE PLANE
Case 1 – Member subjected to axial load
Normal and Shear force on the plane at an angle Ɵ :-
Normal and Shear stress on the plane at an angle Ɵ
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4. Case 2 :- A body subjected to general two dimensional stress system
Stress element showing two-dimensional state of stress
METHODS FOR DETERMINATION OF THE STRESSES ON AN OBLIQUE SECTION OF A BODY
1. Analytical method
2. Graphical method (Mohr’s circle)
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5. All the parameters are shown in their +ve sense in the Fig.
NOTATIONS A E D C B σy τ σx θ σθ τθ
σx  Normal Stress in x- direction
σy  Normal Stress in y- direction
τ Shear Stresses in x & y – directions
θ  Angle made by inclined plane wrt vertical
σθ Normal Stress on inclined plane AE
τθ Shear Stress on inclined plane AE
θP  Inclination of Principal planes
σP  Principal stresses
θS  Inclination of Max. shear stress planes [θS = θP + 450].
All the parameters are shown in their +ve sense in the Fig.
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6. All the parameters are shown in their +ve sense in the Fig.
SIGN CONVENTIONS A E D C B σy τ σx θ σθ τθ
Normal stresses, σ  Tensile stresses +ve.
Shear Stresses, τ, in x – direction & Inclined Plane  Clockwise +ve.
Shear Stresses, τ, in y – direction  Anti-Clockwise +ve.
Angle, θ  measured w r t vertical, Anti-Clockwise +ve.
All the parameters are shown in their +ve sense in the Fig.
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7. ANALYTICAL METHOD Normal stress on plane AE =B σy τ σx θ σθ τθ
Shear stress on plane AE =
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8. PRINCIPAL PLANES There are no shear stresses on principal planes
the planes where the normal stress () is the maximum or minimum
the orientations of the principal planes (p) are given by equating τ = 0
At p .
. .
Which gives two values of Ɵ differing by 90°.
Thus two principal planes are mutually perpendicular
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9. PRINCIPAL STRESSES
Principal stresses are the normal stresses () acting on the principal planes (planes which are at an angle of Ɵp and Ɵp+90, where the shear stress is zero).
where
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10. MAXIMUM SHEAR STRESS (max)
To find maximum value for shear stress and its plane (s), differentiate the equation of shear stress and equate to zero
orientations of the two planes (s) are given by:
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11. MAXIMUM SHEAR STRESS (max)
gives two values (Ɵs1 and Ɵs2) differs by 90°
Thus maximum shear stress occurs on two mutually perpendicular planes
In terms of principal stresses
Also, 𝜃 𝑠 = 𝜃 𝑝 +45°
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12. Case 3 – Member subjected to bi-axial load (τ = 0)
Principal stresses are at Ɵp=0 and Ɵp=90
σ1 , σ2 = 𝜎 𝑥 , 𝜎 𝑦
Max. shear stress A E D C B σy σx θ σθ τθ
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13. Case 4 – Member subjected to simple shear stress ( 𝜎 𝑥 , 𝜎 𝑦 =0)τ θ σθ τθ
For Principal stress,
𝜃 𝑝 =45,135
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14. Orientation of Maximum Shear Planes
90
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15. Principal Planes & Maximum Shear Planes
45 x
p = s ± 45
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