Torsionally loaded shafts are among the most commonly used structures in engineering. For instance, the drive shaft of a standard rear-wheel drive automobile, depicted in Fig., serves primarily to transmit torsion. These shafts are almost always hollow and circular in cross section, transmitting power from the transmission to the deferential joint at which the rotation is diverted to the drive wheels. As in the case of pressure vessels, it is important to be aware of design methods for such structures purely for their inherent usefulness.
Shearing stresses and strains
Not all deformation is elongation or compressive, and we need to extend our concept of strain to include shearing," or distortional, effects. To illustrate the nature of shearing distortions, first consider a square grid inscribed on a tensile specimen as depicted in Fig. Upon uniaxial loading, the grid would be deformed so as to increase the length of the lines in the tensile loading direction and contract the lines perpendicular to the loading direction. However, the lines remain perpendicular to one another. These are termed normal strains, since planes normal to the loading direction are moving apart.
Now consider the case illustrated in Fig.(b), in which the load P is applied transversely to the specimen. Here the horizontal lines tend to slide relative to one another, with line lengths of the originally square grid remaining unchanged. The vertical lines tilt to accommodate this motion, so the originally right angles between the lines are distorted. Such a loading is termed direct shear. Analogously to our definition of normal stress as force per unit area1, or σ = P/A,
We write the shear stress τ as
Normal stresses act to pull parallel planes within the material apart or push them closer together, while shear stresses act to slide planes along one another. Normal stresses promote crack formation and growth, while shear stresses underlie yield and plastic slip. The shear stress can be depicted on the stress square as shown in Fig.(a); it is traditional to use a half-arrowhead to distinguish shear stress from normal stress. The yx subscript indicates the stress is on the y plane in the x direction.
The τyx arrow on the +y plane must be accompanied by one in the opposite direction on the −y plane, in order to maintain horizontal equilibrium. But these two arrows by themselves would tend to cause a clockwise rotation, and to maintain moment equilibrium we must also add two vertical arrows as shown in Fig.(b); these are labelled τxy, since they are on x planes in the y direction. For rotational equilibrium, the magnitudes of the horizontal and vertical stresses must be equal:
Hence any shearing that tends to cause tangential sliding of horizontal planes is accompanied by an equal tendency to slide vertical planes as well. Note that all of these are positive by our earlier convention of + arrows on + faces being positive. A positive state of shear stress, then, has arrows meeting at the upper right and lower left of the stress square. Conversely, arrows in a negative state of shear meet at the lower right and upper left.
The strain accompanying the shear stress τxy is a shear strain denoted . This quantity is a deformation per unit length just as was the normal strain ɛ , but now the displacement is transverse to the length over which it is distributed (see Fig). This is also the distortion or change in the right angle:
This angular distortion is found experimentally to be linearly proportional to the shear stress at sufficiently small loads, and the shearing counterpart of Hooke's Law can be written as
where G is a material property, called the shear modulus.
For isotropic materials (properties same in all directions), there is no Poisson-type effect to consider in shear, so that the shear strain is not influenced by the presence of normal stresses. Similarly, application of a shearing stress has no influence on the normal strains. For plane stress situations (no normal or shearing stress components in the z direction), the constitutive equations as developed so far can be written:
It will be shown later that for isotropic materials, only two of the material constants here are independent, and that
Hence if any two of the three properties E, G, or ν, are known, the other is determined.
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