5 Stiffness and Strain Considerations
5.3 THE JOINT .1 B ASIC C ONCEPTS.1 BASICCONCEPTS
5.3.2 C OMPUTING J OINT S TIFFNESS
(5:14)
and
KJ¼ F
DTJ (5:15)
where
1 K ¼ 1
K1
þ 1 K2
¼ T2 EA1
þ T2 EA2
(5:16)
Theoretically, the relationship between applied compressive force and deflection for our pair of blocks should be linear as long as the force stays within the elastic limit of the material. In practice, however, we will often find that the stiffness of a joint is not linear and may not be fully elastic. Some report the preload–compression relationship shown in Figure 5.8 [1].
Others report a variety of nonlinear effects. We’ll look at some of these in Chapter 13. Before we consider these complexities, however, it is useful to review the ‘‘classical’’ theories, which have been used to evaluate joint behavior in the past. Although simplified, they are often used as a basis for more complex theories. They’re also good enough for many applications. So let’s take a look at them now.
5.3.2 COMPUTING JOINTSTIFFNESS
We assumed a simplified, equivalent body shape for a bolt, to make routine calculations of stiffness and deflection less complicated. We have to do the same sort of thing for the joint.
That portion of the joint which is put in compressive stress by the bolt can be described as a barrel with a hole through the middle, as suggested in Figure 3.3. Some workers, therefore,
Compression (TJ)
Preload (FP)
FIGURE 5.8 The deflection (TJ) of joint members can be nonlinear levels (low FP).
have substituted an ‘‘equivalent barrel’’ for the joint [4], but more common substitutions are hollow cylinders [1] or a pair of frustum cones [5] as in Figure 5.9.
5.3.2.1 Stiffness of Concentric Joints
A discussion of eight proposed ways to estimate the stiffness of a hard (non-gasketed) joint is given by Motosh [4]. The equivalent cylinder approach is described at length by Meyer and Strelow [1]. Unfortunately, each of these techniques assumes that
1. Joint behavior will be linear and fully elastic.
2. There is only one bolt in the joint and it passes through the center of the members being clamped together (this is called a ‘‘concentric’’ joint).
3. The external load applied to the joint is a tension load and it is applied along a line that’s concentric with the bolt axis.
Your own experience, I’m sure, will tell you that limitations 2 and 3 are substantial ones and mean that these equations and recommendations may not apply—at least not very accurately—to many of the joints with which we will be dealing. They’re our only choice at the present state of the art, however, except as noted below. At least they’re our main
‘‘theoretical’’ choice. If the approximations they give us aren’t good enough, we have to determine joint stiffness experimentally.
We will use the equivalent cylinder approach, in this book, to estimate stiffness. This involves the general equation
KJ¼EAC
T (5:17)
where
KJ ¼ stiffness of joint (lb=in., N=mm) E ¼ modulus of elasticity (psi, MPa)
AC¼ cross-sectional area of the equivalent cylinder used to represent the joint in stiffness calculations (in.2, mm2)
T ¼ total thickness of joint or grip length (in., mm)
Note the similarity of this equation to Equation 5.12. The big difficulty here is AC, the cross-sectional area of the equivalent cylinder. The equations we’ll use for ACare summarized in Figure 5.10. Note that there are three different equations, depending on the diameter of contact (DB) between the bolt head (or washer) and the joint, and its relationship to the FIGURE 5.9 Equivalent shapes, substituted for joint members in calculating joint stiffness and deformation.
outside diameter of the joint (DJ) [1,7]. If the joint has a square or rectangular cross section, its diameter is the length of one side (or of the shortest side of the rectangle). DH is the diameter of the hole.
5.3.2.2 Stiffness of Eccentric Joints
Most bolts don’t run through the centerline of the joint or external tension loads don’t align themselves with bolt axes. If the bolt, load, or both lie away from the joint centerline, the joint is called ‘‘eccentric’’ and our choice of stiffness equations is diminished still further. The German engineering society, Verein Deutscher Ingenieure (VDI), however, has published equations that can be used to estimate the stiffness of eccentric joints as long as the cross-sectional area of that portion of a joint which is loaded by one bolt is not much larger than the contact area between bolt (or nut or washer) and joint [7]. With reference to Figure 5.11, the area we assume to be loaded by the bolt is AJ. The stiffness equations which follow assume that
FIGURE 5.10 Equations used to compute the stiffness of concentric joints using the equivalent cylinder method. We’ll call this stiffness Kic.
In each case
b¼ t if t (DBþ Tmin)
b¼ Dð Bþ TminÞ if t > Dð Bþ TminÞ where
W, t, b, and Tminare illustrated in Figure 5.11 (all in in., mm)
DB ¼ diameter of contact between bolt head (or washer) and the joint (in., mm) DH ¼ diameter of the bolt hole (in., mm)
If joint dimensions exceed the limits suggested above (for W ), the equations given in Figure 5.12 don’t apply. If the joint satisfies the limitations, then its stiffness may be estimated from the equations given in Figure 5.12,
W ≤ (DB +Tmin) T
where
and b(DB +Tmin)
or
if
if t >(DB +Tmin) b
t b t
W
LX
LX
CB
CJ
Tmin
AJ b W
t ≤ (DB+Tmin) LX
LX
AJb W if W(DBTmin)
AJb (DBTmin) if W (DBTmin) (5.18a) (5.18b)
FIGURE 5.11 Sketch of an eccentric joint. The shaded are, b W, can be considered that portion of the joint interface which is loaded by a single bolt. See text for the equations used to estimate this area, AJ.
where
CJ ¼ centerline of joint LX ¼ external load (lb, N)
A ¼ distance between external load and joint centerline (in., mm) s ¼ distance between bolt axis and joint centerline (in., mm)
AC ¼ cross-sectional area of equivalent concentric cylinder (see Figure 5.10) (in.2, mm2) kjc ¼ stiffness of equivalent concentric cylinder (see Figure 5.10) (lb=in., N=mm) KJ ¼ stiffness of eccentric joint (lb=in., N=mm)
r0t ¼ resilience of eccentric joint when load and bolt are coaxial (in.=lb, mm=N)
r’’t ¼ resilience of eccentric joint when load and bolt fall along different axes (in.=lb, mm=N) RG¼ radius of gyration of joint area AB(in., mm)
AJ ¼ cross-sectional area of joint (see Figure 5.11) (in.2, mm2) For reference, the radius of gyration for a square cross section is [8]
RG¼ 0:289d (5:24)
where d¼ length of one side.
LX
LX
CJ
LX CJ
CJ
a s
LX
a = s
(A)
(B)
where KJ = 1
rJ where
KJ = 1 rJ
rJ 1 KJc
+
= 1 s a AC RG2AJ
rJ 1 KJc 1+
= s2AC
RG2AJ
(5.22)
(5.23)
FIGURE 5.12 Equations used to compute the stiffness of eccentric joints when the line of action of the external load (LX) coincides with the bolt axis (A) and when it does not (B).
For a rectangular cross section it is
RG¼ 0:289d (5:25)
where d¼ length of the longer side.
For a circular cross section it is
RG¼ 0:25d (5:26)
where d¼ diameter of the circle.
5.3.3 STIFFNESS INPRACTICE
Experience shows that the stiffness of a ‘‘typical’’ joint (whatever that may be) is about five times the stiffness of the bolt that would be used in such a joint. Very thin joints—sheet metal and the like—will be substantially stiffer, although the stiffness of the bolt will also increase rapidly as it gets shorter, as suggested in Figure 5.13. In this figure, incidentally, we have used the equivalent cylinder approach to estimate the possible stiffness of a concentric, hard joint.
5.3.3.1 A Quick Way to Estimate the Stiffness of Non-Gasketed Steel Joints
Here’s another way to use Motosh and VDI data to estimate the stiffness of a non-gasketed steel joint. Both sources have published charts on which are plotted the joint-to-bolt stiffness ratio (KJ=KB) as a function of the bolt’s slenderness ratio L=D, where L ¼ the effective length of the bolt and D¼ nominal diameter.
Figure 5.14 shows a combined version of the published data for slenderness ratios varying from 1:1 to 16:1. The straight line represents the Motosh data; the curved line is from VDI. As you can see, they’re in good agreement above a slenderness ratio of about 4:1.
Figure 5.15 shows a similar plot for thinner joints, with L=D ratios of 1.2:1 or less [9].
Projections of the lower end of the VDI and Motosh curves are also shown in Figure 5.15, showing that the agreement in the data for thin joints is less than perfect. Nevertheless, for any slenderness ratio, this is the best information I’m aware of.
These curves can be used to estimate joint stiffness as follows:
1. Use Equation 5.10 or 5.12 or a version thereof to compute the stiffness of your bolts (KB).
2. Compute the L=D ratio of your bolt, using the effective length (Lbeþ Lse) for L.
3. Use Figures 5.14 or 5.15 with your L=D ratio and find the corresponding KJ=KB stiffness ratio.
4. Multiply the KBcomputed in step 1 by the KJ=KBratio to estimate KJ.
Note that the data in Figures 5.14 and 5.15 is good only for steel bolts used in non-gasketed steel joints. If your joint is made of something else, complete the above steps and then modify the estimate of stiffness as follows.
KJ0¼ KJ
Em
30 106 (5:27)
where
KJ ¼ stiffness of a steel joint as estimated from the procedure above (lb=in., N=mm) KJ0 ¼ stiffness of the same joint, but made from an alternate material (lb=in., N=mm) Em ¼ modulus of elasticity of the alternate material (psi, MPa)
Esteel¼ modulus of the steel joint material (psi, MPa)