Mechanical

LAME’S ELLIPSOID: Let Pxyz be a coordinate frame of reference at point P, parallel to the principal axes at P. On a plane passing through P with normal n, the resultant stress vector is  and its components  = , ,= ; Let PQ be along the resultant stress vector and its length be equal to its magnitude, i.e. PQ=   The coordinates (x, y, z) of the point Q are then x= ;Y=, Z= ; Since ++=1; we get from the above two equations ++=1; This is the equation of an ellipsoid referred to the principal axes. This ellipsoid is called the stress ellipsoid or Lame's ellipsoid. One of its three semiaxes is the longest, the other the shortest, and the third inbetween (Fig.1.22). These are the extermum values. If two of the principal stresses are equal, for instance =; 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. If all the principal stresses are equal, ==; Lame's ellipsoid becomes a sphere. Each radius vector PQ of the stress ellipsoid represents to a certain scale, the resultant stress on one of the planes through the centre of the ellipsoid. It can be shown (Example 1.11) that the stress represented by a radius vector of the stress ellipsoid acts on the plane parallel to tangent plane to the surface called the stress-director surface, defined by ++=1; The tangent plane to the stress-director surface is drawn at the point of intersection of the surface with the radius vector. Consequently, Lame’s ellipsoid and the stress-director surface together completely define the state of stress at point P.[pic 1][pic 2][pic 3][pic 4][pic 5][pic 6][pic 7][pic 8][pic 9][pic 10][pic 11][pic 12][pic 13][pic 14][pic 15][pic 16][pic 17][pic 18][pic 19][pic 20][pic 21][pic 22][pic 23][pic 24]