60. At a point within a solid in a state of strain, the stresses upon a pair of rectangular planes through it are given on BB' the intensity of the normal component stress is 40, on AA' the intensity of the normal component stress is 30, and the tangential component stresses are each of intensity 10. Find the obliquity to BB' of the planes of principal stress, and find the principal stresses. 61. In 60 find the stress on a plane inclined at 46° 43′ to BB'. Deducting 31° 43′, we find the plane to be inclined at 15° to the plane of greatest principal stress. Using results of 60, and t being then zero, RANKINE'S METHOD OF ELLIPSE OF STRESS. The preceding method of finding the stress upon any particular plane through a point at which the state of strain is known, is too tedious to be readily remembered or applied, and becomes intricate when the stresses are some thrusts and others tensions. We now proceed to a general method. Having already proved that there is a pair of principal stresses at a point, we proceed, upon the supposition that these are given, to find the stress on a third plane through the point. Equal-like principal stresses. If the pair of principal stresses at a point be like (both thrusts or both tensions), and be of equal intensity, the stress on any third plane through the point is of that same intensity, and is normal to the plane. Let AA' and BB' be the planes of principal stress at the point o, and let the intensities of the principal stresses, p and q, be equal and alike (both thrusts). cc' is any third plane through o, inclined at 0 to AA'. OAB is a small triangular prism at o, having its faces in those planes. This prism is in equilibrium under the three forces the total thrusts under OA, OB, and AB. Then RO represents the total stress on AB in direction COR. Every plane through o is a plane of principal stress. Each point in a fluid is in this state of strain. I D Equal-unlike principal stresses.-If the pair of principal stresses at a point be unlike (one a thrust and the other a tension), and be of equal intensity, the stress on any third plane through the point is of that same intensity, and is inclined at an angle to the normal to the plane of principal a stress, equal to that 9 which the normal q to this third plane makes therewith, but upon the opposite side. PP PP ↑↑ PPPP Fig. 27. -20.. Let AA′ and BB′ be the planes of principal stress at the point o; p and q the intensities of the principal stresses of equal value, p being a thrust and q a tension; and cc' any third plane through o inclined at o`to aa'. OAB, a small triangular prism at o bounded by these three planes, is in equilibrium ̄under the three forces, viz., the amounts of stress on its faces OA, OB, and AB. Lay off OD = total stress parallel to ox, = p. OA, and OE = total stress parallel to Oy in the direction of q, 9. OB. Complete the parallelogram ODRE. Then RO represents the total stress on AB in direction and amount. That is, RO is inclined at the same angle to the axis ox as ON is, but on the opposite side. Hence the inclination of RO to the normal oN is 2 0. Consider the triangle of forces OER, we have OE drawn from o in the direction of q, then ER drawn from E in the direction of p; hence RO, taken in the same order, is the direction of ". If be greater than 45°, r is like q. If equals 45°, r is entirely tangential to AB. Hence, if the principal stresses at a point be equal and unlike the stress on a third plane, is of that same intensity, is like the stress on the plane it is least inclined to, and its direction is inclined to the axis at the same angle as the normal is, but upon the opposite side. If the new plane be inclined at 45°, the stress is entirely tangential. The principal stresses at a point within a solid in a state of strain being given, to find the intensity and obliquity of the stress at that point on a third plane through it. AA and BB' are the planes of principal stress at o; p and q are the principal stresses. Let Ρ be the greater, and let them be both positive, say both tensions. It is required to find, the intensity of the stress upon cc', and Y, the angle it makes with ON, the normal to cc'. O is the inclination of cɗ to AA, the plane of greatest principal stress. Of two unequal quantities the greater is equal to the sum of their half sum and their half difference, while the lesser equals their difference. A PPPP V V V V Fig. 28. p-q an identity, We may look upon the plane AA' as bearing two separate tension of intensity p; and on the plane BB' as bearing a |