through arching against its neighbors; the second and third spheres 

 collapse at progressively lower capacities because less support is 

 given them through arching. Thus, a foundation underlain by crushable 

 spheres will not exert uniform pressures on the bearing surface and may 

 still have a tendency to rock. 



A system of crushable, thin-walled, open-ended cylinders, illus- 

 trated in Figure 6, requires three and possibly four elements along any 

 one line. Both axially and diametrically collapsing cylinders were 

 analytically examined; axial collapse was predicted by the empirical 

 thin-wall buckling formulas of reference 24, and diametric collapse was 

 predicted via structural plastic design theory. Two materials were 

 considered: mild steel and rigid polyvinyl chloride, PVC. The errors 

 involved in applying the empirical buckling formula, which was developed 

 for metals, to the PVC are expected to be small; however, test data are 

 needed to verify this hypothesis. A preliminary design was carried 

 through for a 40-kip , low-profile installation. Foundation size was 

 governed by the cylinder size and configuration. 



Figure 6a illustrates the configuration assumed for the axially- 

 loaded cylinders, and Table I presents some preliminary analysis 

 results. Reasonable cylinder diameters and wall thicknesses were 

 selected, and allowable stresses were calculated from the buckling 

 formula. The load capacity per steel cylinder, even for very thin- 

 walled cylinders, is very high; therefore, steel cylinders of reason- 

 able dimensions supporting the assumed 40-kip load would not properly 

 collapse at points of load concentration to achieve conformity with the 

 rock surface. Thus, axially-loaded steel cylinders are not suitable 

 for eliminating foundation rocking. Cylinders made out of PVC perform 

 much better with 1/16-inch wall cylinders having an ultimate of 4.8 

 kips versus 3.75 kips required ultimate. Again, these results are 

 theoretical, and other failure modes, such as splitting, may control. 



Figures 6b, 6c, and 6d illustrate the configurations which were 

 assumed for diametrically loaded cylinders supporting a structure, that 

 is, for cylinders lying on their side. The individual collapsible pod 

 is made up of a number of progressively smaller cylinders to give the 

 pod a somewhat constant load-carrying capacity through a considerable 

 deflection. Thus, if the outermost 48-inch diameter cylinder were to 

 develop plastic hinges, it would deform until the 36-inch diameter 

 cylinder picked up the load, and so forth. For the analysis, the param- 

 eters selected were the foundation loading, the number of pods support- 

 ing each foundation, and the diameter of the outermost cylinder in the 

 pods. From these parameters the outermost-cylinder wall thickness re- 

 quired by plastic design theory was calculated. An impact loading of 

 20 kips was added to the 40-kip foundation loading, for a total loading 

 on the foundation of 60 kips. Two configurations having eight and six- 

 teen pods respectively were selected. Table II indicates the wall 

 thickness required to support the ultimate load for the two configura- 

 tions: steel cylinders seem a viable contender for the eight-pod con- 

 figuration; PVC cylinders appear to serve well for both eight- and 



16 



