Page 174 - Shigley's Mechanical Engineering Design

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Deflection and Stiffness 149
as shown by the graph. Any mechanical element having such a characteristic is called a
nonlinear softening spring.
If we designate the general relationship between force and deflection by the equation

F = F(y) (a)
then spring rate is defined as
F dF
k(y) = lim = (4–1)
y→0 y dy
where y must be measured in the direction of F and at the point of application of F. Most
of the force-deflection problems encountered in this book are linear, as in Fig. 4–1a. For
these, k is a constant, also called the spring constant; consequently Eq. (4–1) is written
F
k = (4–2)
y
We might note that Eqs. (4–1) and (4–2) are quite general and apply equally well for
torques and moments, provided angular measurements are used for y. For linear dis-
placements, the units of k are often pounds per inch or newtons per meter, and for
angular displacements, pound-inches per radian or newton-meters per radian.

4–2 Tension, Compression, and Torsion

The total extension or contraction of a uniform bar in pure tension or compression,
respectively, is given by
Fl
δ = (4–3)
AE
This equation does not apply to a long bar loaded in compression if there is a possibil-
ity of buckling (see Secs. 4–11 to 4–15). Using Eqs. (4–2) and (4–3) with δ = y, we see
that the spring constant of an axially loaded bar is
AE
k = (4–4)
l
The angular deflection of a uniform solid or hollow round bar subjected to a twist-
ing moment T was given in Eq. (3–35), and is
Tl
θ = (4–5)
GJ
4
where θ is in radians. If we multiply Eq. (4–5) by 180/π and substitute J = πd /32 for
a solid round bar, we obtain
583.6Tl
θ = (4–6)
Gd 4
where θ is in degrees.
Equation (4–5) can be rearranged to give the torsional spring rate as
T GJ
k = = (4–7)
θ l
Equations (4–5), (4–6), and (4–7) apply only to circular cross sections. Torsional load-
ing for bars with noncircular cross sections is discussed in Sec. 3–12 (p. 101). For the
angular twist of rectangular cross sections, closed thin-walled tubes, and open thin-
walled sections, refer to Eqs. (3–41), (3–46), and (3–47), respectively.