Principal Stresses and Strain and Theories of Failure Strength of Materials Prof. A. S. PATIL Department of Mechanical Engineering Sinhgad Academy of Engineering, Pune Strength of Materials

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. Strength of Materials

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 Strength of Materials

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) Strength of Materials

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. Strength of Materials

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. Strength of Materials

ANALYTICAL METHOD Normal stress on plane AE = B σy τ σx θ σθ τθ Shear stress on plane AE = Strength of Materials

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 Strength of Materials

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 Strength of Materials

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: Strength of Materials

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° Strength of Materials

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 Strength of Materials

Case 4 – Member subjected to simple shear stress ( 𝜎 𝑥 , 𝜎 𝑦 =0) τ θ σθ τθ For Principal stress, 𝜃 𝑝 =45,135 Strength of Materials

Orientation of Maximum Shear Planes 90 Strength of Materials

Principal Planes & Maximum Shear Planes 45 x p = s ± 45 Strength of Materials