Force-flowconcepts22are useful in visualizing the paths taken by lines of force as they pass through a machine or structure from points of load application to points of structural sup- port. If the structure is simple and statically determinate, the equations of static equilib- rium given in (4-1) are sufficient to calculate all the reaction forces. If, however, there are redundantsupports, that is, supports in addition to those required to satisfy the conditions of static equilibrium, equations (4-1) are no longer sufficient to explicitly calculate the magnitudes of anyof the support reactions. Mathematically, this happens because there are more unknowns than equilibrium equations. Physically, it happens because each support behaves as a separate “spring,” deflecting under load in proportion to its stiffness, so that the reactions are shared among the supports in an unknown way.
22Discussed in 4.3.
F
Spring 2 (soft) Spring 1
(stiff)
k1 k2
y1 = y2 = y
Figure 4.19
Parallel spring configuration.
F
Spring 2 (soft)
Spring 1 (stiff)
y = y1 + y2 L2
L1 y1 = ΔL1 y2 = ΔL2
Figure 4.20
Series spring configuration.
From a mathematical point of view, additional deflection equations must be written and combined with the equilibrium equations of (4-1) so that the number of unknowns matches the number of independent equations.
From a physical point of view it is important to appreciate how load sharing relates to relative stiffness. If a stiff spring, or stiff load path, is in parallelwith a soft spring or soft load path, as illustrated in Figure 4.19, the stiff path carries more of the load.23 If a stiff spring or stiff load path is in serieswith a soft spring or load path, as illustrated in Figure 4.20, the loads carried are equal but the stiff spring deflection is smaller than the soft spring deflection.24The importance of these simple concepts cannot be overemphasized because all real machines and structures are combinations of springs in series and/or parallel.
Using these concepts, a designer may quantitatively determine the overall spring rate for any member or combination of members in a machine or structure. Consequently, load car- rying and load sharing may be quantitatively evaluated at an early design stage.
Examples of machine and structural elements for which load-sharing concepts may be useful include bearings, gears, splines, bolted joints, screw threads, chain and sprocket drives, timing belt drives, multiple V-belt drives, filamentary composite parts, machine frames, housings, and welded structures. In certain applications, such as machine grips, hub-and-shaft assemblies, or stacked elements subjected to cyclic loads, load-sharing con- cepts may be used to design for local strain-matchingat critical interfaces.25
Machine Elements as Springs
Although the rigid body assumption is a useful tool in force analysis, no real machine el- ement (or material) is actually rigid. All real machine parts have finite stiffnesses, that is, finite spring rates. A uniform prismatic bar in axial tension as analyzed in 2.4, leads to (2-7), repeated here as
(4-82) kax = AE
L
23May be verified using (4-87). 24May be verified using (4-90).
25If small-amplitude cyclic relative motions are induced between two contacting surfaces because of differential stiffness qualities of the two mating bodies, fretting fatigue or fretting wear may become potential failure modes.
Reducing the amplitude of the cyclic sliding motion by strain-matching may reduce or eliminate fretting.
See 2.9.
Load Sharing in Redundant Assemblies and Structures 181
where is axial spring rate, Ais cross-sectional area, Eis modulus of elasticity, and Lis bar length. Figure 2.2 depicts the linear force-deflection curve characterizing an axially loaded uniform bar in tension. Many, but not all, machine elements exhibit linearforce- deflection curves. Some exhibit nonlinearcurves, as sketched in Figure 4.21. For example, nonlinearstiffeningcurves are typified by Hertz contact configurations such as bearings, cams, and gear teeth, and nonlinear softeningcurves are associated with arches, shells, and conical-washer (Belleville) springs. In addition to axial loading, other configurations ex- hibiting linear spring rates include elements in torsion, bending, or direct shear. From ear- lier equations for deflection, given in 4.4, spring rate (force or torque per unit displacement) expressions may be developed.
For torsionally loaded members, from (4-44),
(4-83) where is torsionalspring rate, Tis applied torque, is angular displacement, Gis shear modulus,Lis length, and Kis a cross-sectional shape constant from Table 4.5.
For members in bending the concept is the same but details are more involved, re- quiring that members in bending be handled on a case-by-case basis. For example, case 1 of Table 4.1 would give
(4-84) where is bendingspring rate for case 1 of Table 4.1, Pis concentrated center load, is deflection at the loading point, Eis modulus of elasticity, and Lis length of the sim- ply supported beam.
For simple members in direct shear,26the spring rate is
(4-85) where is the direct shearspring rate, is displacement of the loaded edge, and Gis shear modulus of elasticity.
For nonlinear springs such as bearings or gear teeth (Hertzian contacts), the spring rate is not constant and cannot be simply described. A linearizedspring rate approxima- tion is often defined about the operating point, especially if vibrational characteristics are of interest.
dP
kdir-sh
kdir-sh = P dP
= AG L yload
kbend-1
kbend-1 = P
yload = 48EI L3
u ktor
ktor = T u = KG
L kax
Force Force
Displacement
Force
Displacement Displacement
(a) Linear (b) Stiffening (c) Softening
Figure 4.21
Various force-deflection characteristics exhibited by springs or machine elements when loaded.
26For example, a short block of “length” Land cross-sectional “shear” area A, if fixed at one “end” and loaded at the other by a force P, illustrates the case of direct shear.