80
8 Linear Elastic Systems
Fig. 8.2 Deformation of a
helical spring
Let us calculate the normal force N as a projection of the force P onto the normal
line to the cross-section:
N = P sin α.
(8.8)
Let us assume that the curvature radius of the spring turn is large as compared to its
transverse dimension. While neglecting the transverse force, we can represent the
potential energy of spring deformation using formula (8.5), e.g.
A =
P l
2E
4 sin
2 α
πd 2 +
64R 2
πd 4 (1 + ν cos
2 α)
,
(8.9)
where l is the spring rod length that, in case of n turns, can be written as follows:
l = 2πRn sec α.
Formula (8.9) can be represented as follows in a different way:
A =
P 2 Rn
Ed 2
4 sin
2 α
cos α
+ 64
R
d
2
1
cos α
+ ν cos α
.
(8.10)
8.4 Principle of Mutuality of Works
Let us consider two various states of any linear elastic system loaded by two various
loads. Let us designate the loads and internal forces and system displacements in
these two states using indexes 1 and 2. Let us represent that the initially non-loaded
system is exposed to load 1. The system gets some deformations from load 1, and
the load has made work that we will designate as A 11 . Let us keep load 1 and