 # Advanced Mechanics of Materials and Applied Elasticity: Analysis of Stress

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## 1.5 Definition and Components of Stress

Stress and strain are most important concepts for a comprehension of the mechanics of solids. They permit the mechanical behavior of load-carrying components to be described in terms fundamental to the engineer. Both the analysis and design of a given machine or structural element involve the determination of stress and material stress–strain relationships. The latter is taken up in Chapter 2.

Consider a body in equilibrium subject to a system of external forces, as shown in Fig. 1.1a. Under the action of these forces, internal forces are developed within the body. To examine the latter at some interior point Q, we use an imaginary plane to cut the body at a section a–a through Q, dividing the body into two parts. As the forces acting on the entire body are in equilibrium, the forces acting on one part alone must be in equilibrium: this requires the presence of forces on plane a–a. These internal forces, applied to both parts, are distributed continuously over the cut surface. This process, referred to as the method of sections (Fig. 1.1), is relied on as a first step in solving all problems involving the investigation of internal forces. Figure 1.1 Method of sections: (a) Sectioning of a loaded body; (b) free body with external and internal forces; (c) enlarged area D A with components of the force D F.

A free-body diagram is simply a sketch of a body with all the appropriate forces, both known and unknown, acting on it. Figure 1.1b shows such a plot of the isolated left part of the body. An element of area DA, located at point Q on the cut surface, is acted on by force D F . Let the origin of coordinates be placed at point Q, with x normal and y, z tangent to DA. In general, D F does not lie along x, y, or z.

Decomposing D F into components parallel to x, y, and z (Fig. 1.1c), we define the normal stress s x and the shearing stresses t xy and t xz :

Equation 1.5 These definitions provide the stress components at a point Q to which the area D A is reduced in the limit. Clearly, the expression D A 0 depends on the idealization discussed in Section 1.1. Our consideration is with the average stress on areas, which, while small as compared with the size of the body, is large compared with interatomic distances in the solid. Stress is thus defined adequately for engineering purposes. As shown in Eq. (1.5), the intensity of force perpendicular, or normal, to the surface is termed the normal stress at a point, while the intensity of force parallel to the surface is the shearing stress at a point.

The values obtained in the limiting process of Eq. (1.5) differ from point to point on the surface as D F varies. The stress components depend not only on D F , however, but also on the orientation of the plane on which it acts at point Q. Even at a given point, therefore, the stresses will differ as different planes are considered. The complete description of stress at a point thus requires the specification of the stress on all planes passing through the point.

Because the stress (s or t) is obtained by dividing the force by area, it has units of force per unit area. In SI units, stress is measured in newtons per square meter (N/m2), or pascals (Pa). As the pascal is a very small quantity, the megapascal (MPa) is commonly used. When U.S. Customary System units are used, stress is expressed in pounds per square inch (psi) or kips per square inch (ksi).

It is verified in Section 1.12 that in order to enable the determination of the stresses on an infinite number of planes passing through a point Q, thus defining the stresses at that point, we need only specify the stress components on three mutually perpendicular planes passing through the point. These three planes, perpendicular to the coordinate axes, contain three hidden sides of an infinitesimal cube (Fig. 1.2). We emphasize that when we move from point Q to point Q' the values of stress will, in general, change. Also, body forces can exist. However, these cases are not discussed here (see Sec. 1.8), as we are now merely interested in establishing the terminology necessary to specify a stress component. Figure 1.2 Element subjected to three-dimensional stress. All stresses have positive sense.

The general case of a three-dimensional state of stress is shown in Fig. 1.2. Consider the stresses to be identical at points Q and Q' and uniformly distributed on each face, represented by a single vector acting at the center of each face. In accordance with the foregoing, a total of nine scalar stress components defines the state of stress at a point. The stress components can be assembled in the following matrix form, wherein each row represents the group of stresses acting on a plane passing through Q(x, y, z):

Equation 1.6 We note that in indicial notation (refer to Sec. 1.17), a stress component is written as t ij , where the subscripts i and j each assume the values of x, y, and z as required by the foregoing equation. The double subscript notation is interpreted as follows: The first subscript indicates the direction of a normal to the plane or face on which the stress component acts; the second subscript relates to the direction of the stress itself. Repetitive subscripts are avoided in this text, so the normal stresses t xx , t yy , and t zz are designated s x , s y , and s z , as indicated in Eq. (1.6). A face or plane is usually identified by the axis normal to it; for example, the x faces are perpendicular to the x axis.

### Sign Convention

Referring again to Fig. 1.2, we observe that both stresses labeled t yx tend to twist the element in a clockwise direction. It would be convenient, therefore, if a sign convention were adopted under which these stresses carried the same sign. Applying a convention relying solely on the coordinate direction of the stresses would clearly not produce the desired result, inasmuch as the t yx stress acting on the upper surface is directed in the positive x direction, while t yx acting on the lower surface is directed in the negative x direction. The following sign convention, which applies to both normal and shear stresses, is related to the deformational influence of a stress and is based on the relationship between the direction of an outward normal drawn to a particular surface and the directions of the stress components on the same surface.

When both the outer normal and the stress component face in a positive direction relative to the coordinate axes, the stress is positive. When both the outer normal and the stress component face in a negative direction relative to the coordinate axes, the stress is positive. When the normal points in a positive direction while the stress points in a negative direction (or vice versa), the stress is negative. In accordance with this sign convention, tensile stresses are always positive and compressive stresses always negative. Figure 1.2 depicts a system of positive normal and shear stresses.

### Equality of Shearing Stresses

We now examine properties of shearing stress by studying the equilibrium of forces (see Sec. 1.4) acting on the cubic element shown in Fig. 1.2. As the stresses acting on opposite faces (which are of equal area) are equal in magnitude but opposite in direction, translational equilibrium in all directions is assured; that is, SFx = 0, SFy = 0, and SFz = 0. Rotational equilibrium is established by taking moments of the x-, y-, and z-directed forces about point Q, for example. From SMz = 0,

(– t xy dy dz)dx + (t yx dx dz)dy = 0

Simplifying,

Equation 1.7a Likewise, from SMy = 0 and SMx = 0, we have

Equation 1.7b Hence, the subscripts for the shearing stresses are commutative, and the stress tensor is symmetric. This means that shearing stresses on mutually perpendicular planes of the element are equal. Therefore, no distinction will hereafter be made between the stress components t xy and t yx , t xz and t zx , or t yz and t zy . In Section 1.8, it is shown rigorously that the foregoing is valid even when stress components vary from one point to another.

### Some Special Cases of Stress

Under particular circumstances, the general state of stress (Fig. 1.2) reduces to simpler stress states, as briefly described here. These stresses, which are commonly encountered in practice, are given detailed consideration throughout the text.

1. Triaxial Stress. We shall observe in Section 1.13 that an element subjected to only stresses s 1, s 2, and s 3 acting in mutually perpendicular directions is said to be in a state of triaxial stress. Such a state of stress can be written as

Equation a The absence of shearing stresses indicates that the preceding stresses are the principal stresses for the element. A special case of triaxial stress, known as spherical or dilatational stress, occurs if all principal stresses are equal (see Sec. 1.14). Equal triaxial tension is sometimes called hydrostatic tension. An example of equal triaxial compression is found in a small element of liquid under static pressure.
2. Two-Dimensional or Plane Stress. In this case, only the x and y faces of the element are subjected to stress, and all the stresses act parallel to the x and y axes, as shown in Fig. 1.3a. The plane stress matrix is written

Equation 1.8  Figure 1.3 (a) Element in plane stress; (b) two-dimensional presentation of plane stress; (c) element in pure shear.

Although the three-dimensional nature of the element under stress should not be forgotten, for the sake of convenience we usually draw only a two-dimensional view of the plane stress element (Fig. 1.3b). When only two normal stresses are present, the state of stress is called biaxial. These stresses occur in thin plates stressed in two mutually perpendicular directions.
3. Pure Shear. In this case, the element is subjected to plane shearing stresses only, for example, t xy and t yx (Fig. 1.3c). Typical pure shear occurs over the cross sections and on longitudinal planes of a circular shaft subjected to torsion.
4. Uniaxial Stress. When normal stresses act along one direction only, the one-dimensional state of stress is referred to as a uniaxial tension or compression.