III. Method of Sections.14 Stress
As stated above, external forces acting on a body give to internal resisting forces. The external forces deform the body;15 the internal forces tend to retain its original shape and volume.
To solve problems of strength of materials it is necessary to know how to determine internal forces and deformations in a body. The internal forces at any section of a body are determined by the method of sections. The idea of this method is as follows.
Consider a body which is in a state of equilibrium under the action of forces. If, for instance, we are interested in the internal forces acting at a section, we imagine the body cut through this section and one of the two parts removed, say, the right one. The remaining left-hand part will then be acted on by the external forces. In order for this part of the body to remain in equilibrium, it is necessary to apply internal forces over the entire section.
These forces represent the action of the removed right-hand part of the body on the remaining left-hand part. Being internal forces for the entire body, they play the role of external forces for the isolated part. The magnitude of the resultant of the internal forces can be determined from the condition of equilibrium16 of the isolated part. The law of distribution of internal forces over the section is not in general known. To solve this problem, it is necessary to know in each particular case how the body deforms under the action of external forces. Thus, the method of sections only allows us to determine the sum of the internal forces acting at the section in question. The sum of these forces may reduce to a single force, to a couple or, in the general case, to a force and a couple.
If an infinitesimal17 area ΔA is isolated at the section, it may be said, assuming the internal forces to be acting at all points in the section, that this area is acted on by an infinitesimal force ΔP. The ratio of the internal force ΔP to the magnitude of the isolated area ΔA gives the average stress on this area
.
Thus, the stress (which characterizes the intensity of internal forces) is defined as the force per unit area. The stress is expressed in newtons per square metre (N/m2). Reducing the area to zero, i. e., passing to the limit, we obtain the true stress at a given point, say, the centre of the area ΔA. Consequently, the true stress at a given point is
.
If the internal forces (elastic forces) are known to be uniformly distributed over the section, in this simplest case the stress is calculated by dividing the total elastic force acting at the section by the entire cross-sectional area, i. e.,
.
Since
the force has a direction, the stress will also have a direction. In
the general case, the stress (
)
on a given area
will make an angle
with this area. Resolving this stress into two components, one being
perpendicular to the area, called the normal stress and designated by
the letter
(sigma), and the other lying in the plane of the area, called the
shearing (or tangential) stress and designated by the letter
(tau), we obtain
,
.
The
total stress is expressed in terms of the normal and shearing
stresses by the following formula
.
The total stress is not considered to be a convenient measure of internal forces in a body as materials resist normal and shearing stresses in different ways. Normal stresses tend to bring closer together or separate individual particles of a body in the direction of the normal to the plane of the section. Shearing stresses tend to move particles of a body with respect to each other on the plane of the section.
In determining the stress at any point of a body, it is possible to pass an infinite number of differently oriented planes through this point. To fully characterize the state of stress at a given point, we have to know not only the magnitude and direction of the stress but also the inclination of the plane. In the following we shall see how the stress at a given point varies with the inclination of a plane passed through this point. The concepts of strain and stress are the fundamental concepts in strength of materials.