Page 234 - Mechanical Behavior of Materials

Section 6.2 Plane Stress 235

most severe stresses at a given location can be found. The stresses involved are called principal
stresses, and the particular directions in which they act are the principal axes. Both principal normal
stresses and principal shear stresses are of interest. The main purpose of this chapter is to present
the procedures for determining these principal stresses and their directions.
Treatment of the topic begins with the relatively simple case of plane stress, where the stresses
acting on one orthogonal plane are zero. We next extend the topic to the general case of three-
dimensional states of stress, and then conclude the chapter by considering states of strain. The
degree of detail is limited to what is needed as background for later chapters. Full detail can be found
in any of a number of relatively advanced books on mechanics of materials and similar subjects, such
as Boresi (2003), Timoshenko (1970), and Ugural (2012).
The material presented in this chapter is especially needed for Chapter 7, which employs
principal stresses to analyze the effect of complex states of stress on yielding of ductile materials
and fracture of brittle materials. It is also needed as background for a number of other chapters later
in the book, as we consider such topics as brittle fracture of cracked members, failure due to cyclic
loading, plastic deformation of materials and components, and time-dependent behavior.

6.2 PLANE STRESS

Plane stress is of practical interest, as it occurs at any free (unloaded) surface, and surface locations
often have the most severe stresses, as in bending of beams and torsion of circular shafts.
Consider any given point in a solid body, and assume that an x-y-z coordinate system has been
chosen for this point. The material at this point is, in general, subjected to six components of stress,
σ x , σ y , σ z , τ xy , τ yz , and τ zx , as illustrated on a small element of material in Fig. 6.1. If the three
components of stress acting on one of the three pairs of parallel faces of the element are all zero,
then a state of plane stress exists. Taking the unstressed plane to be parallel to the x-y plane gives

σ z = τ yz = τ zx = 0 (6.1)

Equilibrium of forces on the element of Fig. 6.1 requires that the moments must sum to zero about
both the x- and y-axes, requiring in turn that the components of τ yz and τ zx acting on the other two
planes must also be zero.
z
σ z

τ yz
τ zx
σ y
τ
xy y
σ x

x

Figure 6.1 The six components needed to completely describe the state of stress at a point.