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1.8 Variation of Stress Within a Body

As pointed out in Section 1.5, the components of stress generally vary from point to point in a stressed body. These variations are governed by the conditions of equilibrium of statics. Fulfillment of these conditions establishes certain relationships, known as the differential equations of equilibrium, which involve the derivatives of the stress components.

Consider a thin element of sides dx and dy (Fig. 1.9), and assume that s x , s y , s xy , and t yx are functions of x, y but do not vary throughout the thickness (are independent of z) and that the other stress components are zero. Also assume that the x and y components of the body forces per unit volume, Fx and Fy , are independent of z and that the z component of the body force Fz = 0. This combination of stresses, satisfying the conditions described, is the plane stress. Note that because the element is very small, for the sake of simplicity, the stress components may be considered to be distributed uniformly over each face. In the figure they are shown by a single vector representing the mean values applied at the center of each face.

Figure 1.9

Figure 1.9 Element with stresses and body forces.

As we move from one point to another, for example, from the lower-left corner to the upper-right corner of the element, one stress component, say s x , acting on the negative x face, changes in value on the positive x face. The stresses s y , t xy , and t yx similarly change. The variation of stress with position may be expressed by a truncated Taylor's expansion:

Equation a

020equ01.jpg


The partial derivative is used because s x is a function of x and y. Treating all the components similarly, the state of stress shown in Fig. 1.9 is obtained.

We consider now the equilibrium of an element of unit thickness, taking moments of force about the lower-left corner. Thus, SMz = 0 yields

Neglecting the triple products involving dx and dy, this reduces to t xy = t yx . In a like manner, it may be shown that t yz = t zy and t xz = t zx , as already obtained in Section 1.5. From the equilibrium of x forces, SFx = 0, we have

Equation b

020equ03.jpg


Upon simplification, Eq. (b) becomes

Equation c

020equ04.jpg


Inasmuch as dx dy is nonzero, the quantity in the parentheses must vanish. A similar expression is written to describe the equilibrium of y forces. The x and y equations yield the following differential equations of equilibrium for two-dimensional stress:

Equation 1.13

01equ13.jpg


The differential equations of equilibrium for the case of three-dimensional stress may be generalized from the preceding expressions as follows:

Equation 1.14

01equ14.jpg


A succinct representation of these expressions, on the basis of the range and summation conventions (Sec. 1.17), may be written as

Equation 1.15a

01equ15a.jpg


where xx = x, xy = y, and xz = z. The repeated subscript is j, indicating summation. The unrepeated subscript is i. Here i is termed the free index, and j, the dummy index.

If in the foregoing expression the symbol u2202.gif/u2202.gifx is replaced by a comma, we have

Equation 1.15b

01equ15b.jpg


where the subscript after the comma denotes the coordinate with respect to which differentiation is performed. If no body forces exist, Eq. (1.15b) reduces to t ij,j = 0, indicating that the sum of the three stress derivatives is zero. As the two equilibrium relations of Eqs. (1.13) contain three unknowns (s x , s y , t xy ) and the three expressions of Eqs. (1.14) involve the six unknown stress components, problems in stress analysis are internally statically indeterminate.

In a number of practical applications, the weight of the member is the only body force. If we take the y axis as upward and designate by r the mass density per unit volume of the member and by g, the gravitational acceleration, then Fx = Fz = 0 and Fy = –r g in Eqs. (1.13) and (1.14). The resultant of this force over the volume of the member is usually so small compared with the surface forces that it can be ignored, as stated in Section 1.1. However, in dynamic systems, the stresses caused by body forces may far exceed those associated with surface forces so as to be the principal influence on the stress field.*

Application of Eqs. (1.13) and (1.14) to a variety of loaded members is presented in sections employing the approach of the theory of elasticity, beginning with Chapter 3. The following sample problem shows the pattern of the body force distribution for an arbitrary state of stress in equilibrium.

Example 1.2. The Body Forces in a Structure

The stress field within an elastic structural member is expressed as follows:

Equation d

022equ01.jpg


Determine the body force distribution required for equilibrium.

Solution

Substitution of the given stresses into Eq. (1.14) yields

022equ01a.jpg

The body force distribution, as obtained from these expressions, is therefore

Equation e

022equ02.jpg


The state of stress and body force at any specific point within the member may be obtained by substituting the specific values of x, y, and z into Eqs. (d) and (e), respectively.

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