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i.e. 1 cm on the stress diagram represents 60 MN/m2.

The two principal stresses in the plane of the surface are then:

o,(= 5.25 cm) = 315 MN/m2 ct2(= 2.0 cm) = 120 MN/m2

The third principal stress, normal to the free (unloaded) surface, is zero, i.e. <73 = 0

The directions of the principal stresses are also obtained from the stress circle. With reference to the 0° gauge direction, o\ lies at 9\ — 15° clockwise cr2 lies at (15° + 90°) = 105° clockwise with 03 normal to the surface and hence to the plane of o\ and cr2.

N.B. - These angles are the directions of the principal stresses (and strains) and they do not refer to the directions of the plane on which the stresses act, these being normal to the above directions.

It is now possible to determine the value of the third principal strain, i.e. that normal to the surface. This is given by eqn. (14.2) as

210 x 109

The complete Mohr's three-dimensional stress and strain representations can now be drawn as shown in Figs. 8.45 and 8.46.

Fig. 8.45. Mohr stress circles.
Fig. 8.46. Mohr strain circles.

Problems

8.1 (B). Given that the following strains exist at a point in a three-dimensional system determine the equivalent stresses which act at the point.

exx = 0.0010 Yxy = 0.0002 £Vy = 0.0005 Yzx = 0.0008 £„=0.0007 Kv, =0.0010

8.2 (B). The following cartesian stresses act at a point in a body subjected to a complex loading system. If E = 206 GN/m2 and u = 0.3, determine the equivalent strains present.

[764.6, 182, 291, 1388, 631, 883.5, all x 10"6.]

8.3 (B). Does a uniaxial stress field produce a uniaxial strain condition? Repeat Problem 8.2 for the following stress field:

[No; 1092, -327.7, -327.7, 0, 0, 0, all x 10"6.]

8.4 (C). The state of stress at a point in a body is given by the following equations:

If equilibrium is to be achieved what equations must the body-force stresses X, Y and Z satisfy?

[—(a + 2«); -(p + 2ey)\ -(« + 2qx + 3kz2).] 8.5 (C). At a point the state of stress may be represented in standard form by the following:

Show that, if body forces are neglected, equilibrium exists.

8.6 (C). The plane stress distribution in a flat plate of unit thickness is given by:

Show that, in the absence of body forces, equilibrium exists. The load on the plate is specified by the following boundary conditions:

where w is the width of the plate.

If the length of the plate is L, determine the values of the constants a J? and c and determine the total load on the edge of the plate, x = w/2.

0 0

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