What Is Axial Loading? What Is Stress?
The deformation caused in a body by external forces or other actions generally varies from one point to another, i.e., it is not homogeneous. In fact, a homogeneous deformation is rare. It occurs, for example, in a body with isostatic supports under a uniform temperature variation or in a slender member under constant axial force.
Rod BC of the example considered in the preceding section is a two-force member and, therefore, the forces FBC and F'BC acting on its ends B and C (Fig. 1.5) are directed along the axis of the rod. We say that the rod is under axial loading.
An actual example of structural members under axial loading is provided by the members of the bridge truss shown in Photo 1.1.
Returning to rod BC of Fig. 1.5, we recall that the section we passed through the rod to determine the internal force in the rod and the corresponding stress was perpendicular to the axis of the rod; the internal force was therefore normal to the plane of the section (Fig. 1.7) and the corresponding stress is described as a normal stress.
Thus, formula (1.5) gives us the normal stress in a member under axial loading:
σ
=P/A
We should also note that, in formula (1.5), s is obtained by dividing the magnitude P of the resultant of the internal forces distributed over the cross section by the area A of the cross section; it represents, therefore, the average value of the stress over the cross section, rather than the stress at a specific point of the cross section.
To define the stress at a given point Q of the cross section, we should consider a small area DA. Dividing the magnitude of DF by DA, we obtain the average value of the stress over DA. Letting DA approach zero, we obtain the stress at point Q:
σ = lim dF/dA as dA approaches infinity (1.6)
In general, the value obtained for the stress s at a given point Q of the section is different from the value of the average stress given by formula (1.5), and s is found to vary across the section. In a slender rod subjected to equal and opposite concentrated loads P and P' , this variation is small in a section away from the points of application of the concentrated loads, but it is quite noticeable in the neighborhood of these
It follows from Eq. (1.6) that the magnitude of the resultant of the distributed internal forces is
∫dF = ∫σ dA lower limit = A
But the conditions of equilibrium of each of the portions of rod require that this magnitude be equal to the magnitude P of the concentrated loads. We have, therefore,
P = ∫dF = ∫σ dA lower limit = A
which means that the volume under each of the stress surfaces must be equal to the magnitude P of the loads. This, however, is the only information that we can derive from our knowledge of statics, regarding the distribution of normal stresses in the various sections of the rod.
The actual distribution of stresses in any given section is statically indeterminate. To learn more about this distribution, it is necessary to consider the deformations resulting from the particular mode of application of the loads at the ends of the rod.
In practice, it will be assumed that the distribution of normal stresses in an axially loaded member is uniform, except in the immediate vicinity of the points of application of the loads. The value s of the stress is then equal to save and can be obtained from formula (1.5).
However, we should realize that, when we assume a uniform distribution of stresses in the section, i.e., when we assume that the internal forces are uniformly distributed across the section, it follows from elementary statics† that the resultant P of the internal forces must be applied at the centroid C of the section.
This means that a uniform distribution of stress is possible only if the line of action of the concentrated loads P and P' passes through the centroid of the section considered. This type of loading is called centric loading and will be assumed to take place in all straight two-force members found in trusses and pin-connected structures, such as the one considered in Fig. 1.1.
However, if a two-force member is loaded axially, but eccentrically we find from the conditions of equilibrium of the portion of member that the internal forces in a given section must be equivalent to a force P applied at the centroid of the section and a couple M of moment M = Pd. The distribution of forces—and, thus, the corresponding distribution of stresses—cannot be uniform. Nor can the distribution of stresses be symmetric.
