Thomas Industrial Library

12.9 PLANE STRESS AND PLANE STRAIN

The general state of stress and strain is three-dimensional but there exist particular geometric configurations that can be treated differently.

Plane Stress

The two-dimensional, or biaxial, stress state is also called plane stress. Plane stress requires that one principal stress be zero. This condition is common in some applications. For example, a thin plate or shell may also have a state of plane stress away from its boundaries or points of attachment. These cases can be treated with the simpler approach of equations 12.7.

Plane Strain

There are principal strains associated with the principal stresses. If one of the principal strains (say 3) is zero, and if the remaining strains are independent of the dimension along its principal axis, n 3, it is called plane strain . This condition occurs in particular geometries. For example, if a long, solid, prismatic bar is loaded only in the transverse direction, regions within the bar that are distant from any end constraints will see essentially zero strain in the direction along the axis of the bar and be in plane strain. (However, the stress is not zero in the zero-strain direction.) A long, hydraulic dam can be considered to have a plane strain condition in regions well removed from its ends or base at which it is attached to surrounding structures.

12.10 APPLIED VERSUS PRINCIPAL STRESSES

We now want to summarize the differences between the stresses applied to an element and the principal stresses that may occur on other planes as a result of the applied stresses. The applied stresses are the nine components of the stress tensor (equation 12.5a, p. 348) that result from whatever loads are applied to the particular geometry of the object as defined in a coordinate system chosen for convenience. The principal stresses are the three principal normal stresses and the three principal shear stresses defined in Section 12-8. Of course, many of the applied-stress terms may be zero in a given case. For example, in a tensile-test specimen the only nonzero applied stress is the x term in equation 12.5a (p. 348), which is unidirectional and normal. There are no applied shear stresses on the surfaces normal to the force axis in pure tensile loading. However, the principal stresses are both normal and shear.

In a tensile-test specimen, the applied stress is pure tensile and the maximum principal normal stress is equal to it in magnitude and direction. But a principal shear stress of half the magnitude of the applied tensile stress acts on a plane 45 ° from the plane of the principal normal stress. Thus, the principal shear stresses will typically be nonzero even in the absence of any applied shear stress. This fact is important to an understanding of why parts fail. The most difficult task for the machine designer is to correctly determine the locations, types, and magnitudes of all the applied stresses acting on the part. The calculation of the principal stresses is then pro forma using equations 12.5 to 12.7 (pp. 348-350).