By Prof L S Srinath
This publication is designed to supply a great origin in Mechanics of Deformable Solids after an introductory direction on energy of Materials. This variation has been revised and enlarged to make it a finished resource at the topic. Exhaustive therapy of crucial themes like theories of failure, strength tools, thermal stresses, tension focus, touch stresses, fracture mechanics make this a whole supplying at the topic.
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Using this in the third ( ) ( ) and fourth equations and solving, nx = ± 1/ 2 , ny = ± 1/ 2 . ( ) Hence, s1 = –1 is in the direction +1/ 2, − 1/ 2, 0 . It should be noted that the plus and minus signs associated with nx, ny and nz represent the same line. e. n3 = n1 ¥ n2. 6 directions. 1⎛ 1 ⎞ 1− ⎜ ⎟ 2⎝ 3⎠ 1/ 2 , nz = 1 2 ⎛ 1 ⎞ ⎜⎝ 1 + ⎟ 3⎠ 1/ 2 For the given state of stress, determine the principal stresses and their ⎡0 1 1 ⎤ [tij] = ⎢1 0 1 ⎥ ⎢ ⎥ ⎢⎣1 1 0 ⎥⎦ Solution \ l 1 = 0, l2 = –3, l3 = 2 f (s) = –s 3 + 3s + 2 = 0 = (–s 3 – 1) + (3s + 3) = –(s + 1) (s 2 – s + 1) + 3(s + 1) = (s + 1) (s – 2) (s + 1) = 0 s 1 = s2 = –1 and s3 = 2 Since two of the three principal stresses are equal, and s3 is different, the axis of s3 is unique and every direction perpendicular to s3 is a principal direction associated with s1 = s2.
Case (iii) s1 = s2 = s3 This is a trivial case since this is the isotropic or the hydrostatic state of stress. Mohr’s circles collapse to a single point on the s axis. See Appendix 1 for the graphical determination of the normal and shear stresses on an arbitrary plane, using Mohr’s circles. 19 PLANES OF MAXIMUM SHEAR From Sec. 17 and also from Fig. 16 for the case s1 > s2 > s3, the maximum 1 1 shear stress is (s1 – s3) = t2 and the associated normal stress is (s1 + s3). 39) in Sec. 18, one gets nx = ± 1/2 , n y = 0 and n z = ±1 2.
The coordinates (x, y, z) of the point Q are then n x = Tx, n n y = T y, z =Tz Since nx2 + n 2y + nz2 = 1, we get from the above two equations. 2 x2 + y + z 2 = 1 2 2 2 σ1 σ2 σ3 y n Q x n z Q P Fig. 55) This is the equation of an ellipsoid referred to the principal axes. This ellipsoid is called the stress ellipsoid or Lame's ellipsoid. 22). These are the extermum values. If two of the principal stresses are equal, for instance Analysis of Stress 37 s1 = s2, Lame’s ellipsoid is an ellipsoid of revolution and the state of stress at a given point is symmetrical with respect to the third principal axis Pz.