Stress equilibrium equation

Stresses is a continuous function of the location in a body. Therefore, the stress at any point is interrelated with the stress at other points. When force is acting along the x-axis of a solid, normal stress vis developed at the contacting surface.

The solid is a continuum and internal force is exerted by the contacting particles on the others. As a result, stress develops in the entire solid body rather than just the contacting point. Considering the rate of stress development per unit length along x-axis as the increment in stress across the length dx is:

If the point on solid is at rest and in equilibrium, the stress developed along the x- axis due to the aforementioned external force and internal force will be balanced by a stress of the same magnitude but in a different direction. Therefore, a’_x can be expressed in term of a_x.

Similarly, the following normal stress components can be defined based on Eq. (2.4): Six shear stress components can be expressed in a similar fashion:

A total of 18 stress components can be expressed in the above-derived forms. All these unknown stress components can now be determined by knowing only nine of them, as shown in Table 2.2.

Other than the stresses developed on six planes, body forces may exist too. Let fx,f_y and/_; be the body forces (force per volume) acting along the x, у and z axes, as shown in Fig. 2.5.

Under static equilibrium, the summation of forces acting on an infinitesimal part of the solid, as shown in Fig. 2.5, along the x-axis is zero. The involved forces are shown in Fig. 2.6.

TABLE 2.2

Unknown stress components for solid in equilibrium

		Plane лгу-plane (1 & 3)	yz-plane (2 & 4)	лгг-plane (5 & 6)
Direction	positive x
	negative л
	positive у
	negative у
	positive z
	negative z

FIGURE 2.5 Components of stress in 3-D.

FIGURE 2.6 Stresses that act along .v-axis.

FIGURE 2.7 Stresses with the ability to cause rotation about x-axis.

Similarly, equating the summation of forces along у and z axes to zero yields the following:

Eqs. (2.5), (2.6) and (2.7) are equilibrium equation for stresses considering body forces. Equating the summation of moment about the л-axis induced by forces (Fig. 2.5) to zero results in the following (Fig. 2.7):

FIGURE 2.8 Complementary pairs of shear stresses.

Neglecting the 4^th order products, because they are insignificant, gives the following:

Similarly, the following relationships are produced by taking the summation of moment about the v and z axes as zero:

Eqs. (2.8), (2.9) and (2.10) show that for each pair of shear stress (say r*_v) developed. a complementary shear stress (say r_vv) of the same magnitude will also be developed on their adjacent face. These complementary shear stresses act in the direction that can stabilise the rotation that would be caused by the developed shear stress pair alone, as shown in Fig. 2.8.

Therefore, the stress components in Eq. (2.3) can be expressed as a symmetrical matrix:

Now, the stress at a point is defined completely by six independent components (three normal and three shear) instead of nine independent components (three normal and six shear).