Principal Stresses and Strain and Theories of Failure

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 1

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 2

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 3

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 4

NOTATIONS σy 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 C σx E B τ θ σθ θ D σx A τ τθ Shear Stress on inclined plane AE τ σy 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 5

SIGN CONVENTIONS σy C σx E B τ θ σθ τ θ D σx A τ σy 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 6

ANALYTICAL METHOD Normal stress on plane AE = σy C σx E B τ θ σθ θ D A τ τ σx Shear stress on plane AE = σy Strength of Materials 7

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 8

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 9

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 10

MAXIMUM SHEAR STRESS ( max) gives two values (Ɵs 1 and Ɵs 2) differs by 90° • Thus maximum shear stress occurs on two mutually perpendicular planes In terms of principal stresses Strength of Materials 11

• σy σx C E B τ θ σθ θ D σx A σy Strength of Materials 12

• C E B τ θ σθ τ θ D τ A Strength of Materials 13

Orientation of Maximum Shear Planes Maximum Shear s 2 s 1 x 90 Strength of Materials 14

Principal Planes & Maximum Shear Planes 45 Principal plane x Maximum shear plane p = s ± 45 Strength of Materials 15