6
1 Summary of Elasticity Theory: Basic Concepts
(A) Straight lines connecting each pair of points are parallel to an arbitrarily defined
direction.
(B) The distance between these points is low as compared to the body dimensions.
A strain is called homogeneous when the distance (l) between these points is
increased by the value ((l), proportional to l. The relation of
ε =
l
l
is referred to as relative elongation in the specified direction. Negative relative
elongation is defined as shortening.
1.4 Internal Forces: Method of Sections
Interaction forces of solid body particles form a system of internal forces. Some of
them act upon the Newton law of gravitation and represent interest only in the case
of large astronomic bodies. Such forces between individual parts of engineering
structures are negligibly low and we will not take them into account. Other internal
forces of a solid body for its two elementary particles have a significant value
only if the considered particles are spaced by no more than the distance between
molecules. Therefore, by mentally separating one part of the body from the other,
we will represent the interaction of these parts by forces distributed over the section
surface.
For any dissected parts of the body, the system of internal forces replaces the
action of the discarded part on the considered part. Thus, the method of imagined
sections, firstly, allows detecting internal forces and, secondly, transfers internal
forces to the category of external loads applied to each of the dissected parts of
the body. Using the interrelation principle of action and counter-action, these loads
applied to each of the sides of the made section are equal in value and opposite in
direction.
Let us explain the above using Fig. 1.1. Assume an arbitrary solid body loaded
by internal forces. Some of these forces (or even all of them) can be reactive. Let us
designate the aggregate of external forces with (P e ).
Let us mentally dissect the body by any surface (for example, by the plane α,
Fig. 1.1a) into two parts—right (R) and left (L). Let us detach these parts (Fig. 1.1b).
Let us replace the action of each part with the other one with internal forces
applied to each of the section sides (see Fig. 1.1b). Internal forces are continuously
distributed over the section surface in a complicated manner. Let us designate the
aggregate of these forces by (P i ).
For any distribution over the section surface, internal forces must be such that
they satisfy the equilibrium conditions for the left and right parts of the body
individually. Symbolically, this can be written as follows:
1 Summary of Elasticity Theory: Basic Concepts
(A) Straight lines connecting each pair of points are parallel to an arbitrarily defined
direction.
(B) The distance between these points is low as compared to the body dimensions.
A strain is called homogeneous when the distance (l) between these points is
increased by the value ((l), proportional to l. The relation of
ε =
l
l
is referred to as relative elongation in the specified direction. Negative relative
elongation is defined as shortening.
1.4 Internal Forces: Method of Sections
Interaction forces of solid body particles form a system of internal forces. Some of
them act upon the Newton law of gravitation and represent interest only in the case
of large astronomic bodies. Such forces between individual parts of engineering
structures are negligibly low and we will not take them into account. Other internal
forces of a solid body for its two elementary particles have a significant value
only if the considered particles are spaced by no more than the distance between
molecules. Therefore, by mentally separating one part of the body from the other,
we will represent the interaction of these parts by forces distributed over the section
surface.
For any dissected parts of the body, the system of internal forces replaces the
action of the discarded part on the considered part. Thus, the method of imagined
sections, firstly, allows detecting internal forces and, secondly, transfers internal
forces to the category of external loads applied to each of the dissected parts of
the body. Using the interrelation principle of action and counter-action, these loads
applied to each of the sides of the made section are equal in value and opposite in
direction.
Let us explain the above using Fig. 1.1. Assume an arbitrary solid body loaded
by internal forces. Some of these forces (or even all of them) can be reactive. Let us
designate the aggregate of external forces with (P e ).
Let us mentally dissect the body by any surface (for example, by the plane α,
Fig. 1.1a) into two parts—right (R) and left (L). Let us detach these parts (Fig. 1.1b).
Let us replace the action of each part with the other one with internal forces
applied to each of the section sides (see Fig. 1.1b). Internal forces are continuously
distributed over the section surface in a complicated manner. Let us designate the
aggregate of these forces by (P i ).
For any distribution over the section surface, internal forces must be such that
they satisfy the equilibrium conditions for the left and right parts of the body
individually. Symbolically, this can be written as follows: