8.4 Principle of Mutuality of Works
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gradually load the system with load 2. This load will cause additional deformation
of the system. Let us call the work of load 2 on displacements caused by additional
deformation after applying load 2 as A 22 .
However, in the case of additional deformation, the work will be made not only
by load 2 but also by load 2 since the points of application of forces of system 1 in
the case of additional deformation will get additional displacements. Load 1 on these
displacements remains constant and will make the work that we will designate as
A 12 . As a result, the work of external forces is expressed by the sum A 11 +A 22 +A 12 .
In these addends, the first index indicates what load does the work and the second
one shows the forces that caused displacements of this load.
Let us change the order of loading. At first, let us apply load 2 to the non-loaded
system that will do some work A 22 . on the displacements caused by it. Let us keep
load 2 and will gradually apply load 1. As in the first loading case, load 1 will cause
additional deformation of the system and will do work A 11 . on the displacements
caused by it. During this time, load 2 remaining constant will do additional work
A 21 on displacements caused by the application of load 1. As a result, the work of
external forces in the second method of loading is expressed by the sum A 22 +A 11 +
A 21 .
The total work of external forces in each of these cases of loading equals the
work of internal forces taken with a reverse sign. This work is defined by the final
state of the system and does not depend on the loading sequence. Since the final
states of the system in the two considered cases of loading are the same, this means
that
A 12 = A 21 .
(8.11)
Thus, the following principle of mutuality of works (Betty) is proved [1].
Theorem 8.1 The work of forces of the first system on displacements caused by the
second system equals the work of the second system on displacements caused by the
first system of forces.
Let us substitute into the result (8.11) A 12 = P 1 δ 12 P 2 , where P 1 is the
generalized force [4] of the first system and δ 12 is the generalized displacement
of force P 1 in the direction of its action caused by P 2 = 1. In a similar fashion, let
us substitute A 21 = P 2 δ 21 P 1 , where δ 21 is the generalized displacement of force P 2
in the direction of its action caused by P 1 = 1. From the condition (8.11), it follows
that
δ 12 = δ 21 .
(8.12)
Equation (8.12) expresses (Maxwell) the principle of mutuality of displacements
that we formulate as follows.
Theorem 8.2 The displacement of the point of application of the first force in the
direction of its action caused by the second single force equals the displacement
81
gradually load the system with load 2. This load will cause additional deformation
of the system. Let us call the work of load 2 on displacements caused by additional
deformation after applying load 2 as A 22 .
However, in the case of additional deformation, the work will be made not only
by load 2 but also by load 2 since the points of application of forces of system 1 in
the case of additional deformation will get additional displacements. Load 1 on these
displacements remains constant and will make the work that we will designate as
A 12 . As a result, the work of external forces is expressed by the sum A 11 +A 22 +A 12 .
In these addends, the first index indicates what load does the work and the second
one shows the forces that caused displacements of this load.
Let us change the order of loading. At first, let us apply load 2 to the non-loaded
system that will do some work A 22 . on the displacements caused by it. Let us keep
load 2 and will gradually apply load 1. As in the first loading case, load 1 will cause
additional deformation of the system and will do work A 11 . on the displacements
caused by it. During this time, load 2 remaining constant will do additional work
A 21 on displacements caused by the application of load 1. As a result, the work of
external forces in the second method of loading is expressed by the sum A 22 +A 11 +
A 21 .
The total work of external forces in each of these cases of loading equals the
work of internal forces taken with a reverse sign. This work is defined by the final
state of the system and does not depend on the loading sequence. Since the final
states of the system in the two considered cases of loading are the same, this means
that
A 12 = A 21 .
(8.11)
Thus, the following principle of mutuality of works (Betty) is proved [1].
Theorem 8.1 The work of forces of the first system on displacements caused by the
second system equals the work of the second system on displacements caused by the
first system of forces.
Let us substitute into the result (8.11) A 12 = P 1 δ 12 P 2 , where P 1 is the
generalized force [4] of the first system and δ 12 is the generalized displacement
of force P 1 in the direction of its action caused by P 2 = 1. In a similar fashion, let
us substitute A 21 = P 2 δ 21 P 1 , where δ 21 is the generalized displacement of force P 2
in the direction of its action caused by P 1 = 1. From the condition (8.11), it follows
that
δ 12 = δ 21 .
(8.12)
Equation (8.12) expresses (Maxwell) the principle of mutuality of displacements
that we formulate as follows.
Theorem 8.2 The displacement of the point of application of the first force in the
direction of its action caused by the second single force equals the displacement