Apparatuses for constructing displacement aggregate piers

Abstract

Apparatuses for constructing displacement aggregate piers are disclosed. In one example, a mandrel is provided comprising a tamper head that has cutting teeth on the leading edge thereof. In another example, hydrojet nozzles are provided within one or more of the cutting teeth of the tamper head. In yet another example, the mandrel comprises grout tubes (or grout injection lines) and/or grout inspection lines. In yet another example, the mandrel and/or tamper head can comprise cutting teeth, hydrojet nozzles, grout tubes (or grout injection lines), grout inspection lines, and any combinations thereof.

Claims

1. A system for constructing aggregate piers, comprising: a mandrel, the mandrel comprising an upper feed tube portion; a tamper head; at least one hydrojet nozzle on the lower edge of the tamper head, wherein each hydrojet nozzle is entirely disposed at an angle ranging between 10 and 80 degrees from horizontal; and, a passage extending from the feed tube to the tamper head therethrough for feeding aggregate through the feed tube to the tamper head, wherein the tamper head comprises a plurality of unbiased cutting teeth on a lower edge of the tamper head opposite the feed tube and surrounding a perimeter of the tamper head edge, and wherein the cutting teeth have a flat bottom that ranges in width from 0.5 inches (1.2 cm) to 6 inches (15.2 cm) and ranges in depth from 0.25 inches (0.6 cm) to 6 inches (15.2 cm).

2. The system of claim 1 wherein the cutting teeth cover 20% to 80% of the cross-sectional area of the tamper head edge.

3. The system of claim 1 wherein a distance between cutting teeth of the plurality of cutting teeth is equal to about the width of the cutting teeth.

4. The system of claim 1 wherein a distance between cutting teeth of the plurality of cutting teeth is greater than or less than the width of the cutting teeth.

5. The system of claim 1 wherein the cutting teeth increase driving stresses at the tamper head edge by a factor of 1.25 to 5.

6. The system of claim 1 wherein the at least one hydrojet nozzle is installed in at least one of the cutting teeth.

7. The system of claim 1 wherein the at least one hydrojet nozzle comprises a stream, where the stream ranges in diameter from between 1/1000 of an inch (0.0254 mm) to 0.25 inches (0.6 cm).

8. The system of claim 1 wherein the at least one hydrojet nozzle is fluidly connected to an interior manifold that connects to one or more jet tubes extending internally or externally down the feed tube.

9. The system of claim 1 wherein the at least one hydrojet nozzle comprises a stream that ranges in pressure from 10 psi (68.9 kPa) to 4,000 psi (27,579 kPa).

10. The system of claim 1 wherein the mandrel further comprises one or more diametric restriction elements.

11. A method of constructing aggregate piers, the method comprising the steps of: a) providing a mandrel, the mandrel comprising an upper feed tube portion, a tamper head, at least one hydrojet nozzle on the lower edge of the tamper head, wherein each hydrojet nozzle is entirely disposed at an angle ranging between 10 and 80 degrees from horizontal and a passage extending therethrough for feeding aggregate through the feed tube to the tamper head, wherein the tamper head comprises a plurality of unbiased cutting teeth on a lower edge of the tamper head opposite the feed tube and surrounding a perimeter of the tamper head edge, and wherein the cutting teeth have a flat bottom that ranges in width from 0.5 inches (1.2 cm) to 6 inches (15.2 cm) and ranges in depth from 0.25 inches (0.6 cm) to 6 inches (15.2 cm); b) driving the mandrel in a vertical motion into free-field soils to a specified depth; c) lifting the mandrel a specified distance; d) repeating the driving and lifting of the mandrel; and e) adding aggregate to form the pier.

12. The method of claim 11 wherein the mandrel further comprises one or more diametric restriction elements.

13. The method of claim 12 wherein the one or more diametric restriction elements expand radially forming a compaction surface within the tamper head during driving of the mandrel.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Having thus described the presently disclosed subject matter in general terms, reference will now be made to the accompanying Drawings, which are not necessarily drawn to scale, and wherein:

(2) FIG. 1A and FIG. 1B illustrate a side view and a bottom end view, respectively, of an example of a mandrel that has a tamper head with cutting teeth on the leading edge thereof;

(3) FIG. 2 illustrates a side view showing more details of the cutting teeth of the tamper head shown in FIG. 1A and FIG. 1B;

(4) FIG. 3A and FIG. 3B illustrate a side view and a bottom end view, respectively, showing more details of the tamper head shown in FIG. 1A and FIG. 1B comprising hydrojet nozzles;

(5) FIG. 4A and FIG. 4B illustrate a top view and a cross-sectional view, respectively, of an example of a mandrel that comprises one or more grout tubes for adding grout to aggregate piers;

(6) FIG. 5A and FIG. 5B illustrate a side view and a bottom end view, respectively, of another example of a mandrel that comprises one or more grout tubes for adding grout to aggregate piers;

(7) FIG. 6A and FIG. 6B illustrate a top view and a cross-sectional view, respectively, of an example of a mandrel that comprises one or more grout tubes for adding grout to aggregate piers, according to another embodiment; and

(8) FIG. 7 shows a plot of the grouted pier modulus test results for a 24-inch (61-cm) diameter pier formed using, for example, the mandrel shown in FIG. 5A and FIG. 5B.

DETAILED DESCRIPTION

(9) The presently disclosed subject matter now will be described more fully hereinafter with reference to the accompanying Drawings, in which some, but not all embodiments of the presently disclosed subject matter are shown. Like numbers refer to like elements throughout. The presently disclosed subject matter may be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will satisfy applicable legal requirements. Indeed, many modifications and other embodiments of the presently disclosed subject matter set forth herein will come to mind to one skilled in the art to which the presently disclosed subject matter pertains having the benefit of the teachings presented in the foregoing descriptions and the associated Drawings. Therefore, it is to be understood that the presently disclosed subject matter is not to be limited to the specific embodiments disclosed and that modifications and other embodiments are intended to be included within the scope of the appended claims.

(10) In some embodiments, the presently disclosed subject matter provides apparatuses for efficiently constructing displacement aggregate piers in difficult driving conditions and/or in soils requiring that the pier be grouted to achieve structural support. In one example, the aggregate piers constructed using the presently disclosed apparatuses are used to support structures, such as buildings, foundations, floor slabs, walls, embankments, pavements and other improvements.

(11) An aspect of the presently disclosed apparatuses for efficiently constructing displacement aggregate piers is that they provide improvements to the tamper head driven mandrel method of efficiently constructing displacement aggregate piers.

(12) Another aspect of the present disclosure is that it provides improved apparatuses and methods for constructing grouted aggregate piers.

(13) In some embodiments, a mandrel is provided that has a tamper head with cutting teeth on the leading edge thereof, wherein the cutting teeth provide a more efficient means of penetrating the mandrel into hard or dense materials during driving.

(14) In other embodiments, hydrojet nozzles are provided within one or more of the cutting teeth for delivering liquid under pressure, which can be used for the loosening of dense, stiff, and/or cemented materials that may be encountered during driving.

(15) In yet other embodiments, a mandrel is provided that has grout tubes (or grout injection lines) and/or grout inspection lines, wherein the grout tubes are used to facilitate adding grout more accurately to piers constructed in very soft and weak soil.

(16) In still other embodiments, the presently disclosed mandrel and/or tamper head can comprise cutting teeth, hydrojet nozzles, grout tubes (or grout injection lines), grout inspection lines, and any combinations thereof.

(17) Referring now to FIG. 1A and FIG. 1B, a side view and a bottom end view, respectively, are provided showing an example of a mandrel 100 that has a tamper head with cutting teeth 114 on a leading edge thereof. FIG. 1A and FIG. 1B show an exemplary embodiment of the present embodiment for conditions characterized by difficult tamper head driving. For example, mandrel 100 may comprise a feed tube 110 and a tamper head 112, wherein feed tube 110 has a passage 116 running therethrough for feeding aggregate (not shown) to tamper head 112. Additionally, tamper head 112 may comprise a plurality of cutting teeth 114 on the leading edge thereof; namely, cutting teeth 114 may be on the edge of tamper head 112 opposite feed tube 110. Feed tube 110 and tamper head 112 can be formed of, for example, metallic materials such as steel, cast iron, and/or aluminum.

(18) Cutting teeth 114 are typically installed or machined at the bottom edge or leading edge of tamper head 112. The purpose of cutting teeth 114 is to provide a more efficient means of penetrating mandrel 100 into hard or dense materials during driving. It is well known by those skilled in the art that the penetration of an object into the ground depends on, among other things, the characteristics of the subsurface materials, the presence or lack of ground water, the driving energy applied, and the cross-sectional area of the object being driven into the subsurface materials. The presently disclosed mandrel 100 provides for the installation or machining of driving teeth or cutting teeth that advantageously reduce the cross-sectional area of tamper head 112 at the point of penetration into the ground.

(19) The driving stress that is applied to the leading edge of tamper head 112 may be computed as the ratio of the driving force applied by the driving hammer to the cross-sectional area of the driving surface of tamper head 112. The greater the driving stress, the more rapid the penetration of tamper head 112 into the subsurface materials. In one example, cutting teeth 114 cover from about 20% to about 80% of the cross-sectional area of the cross-sectional driving area of tamper head 112.

(20) Referring now to FIG. 2, a side view of the tamper head 112 shown in FIG. 1A and FIG. 1B is provided showing more details of cutting teeth 114. Cutting teeth 114 have a width w and a length l. Further, there is a space s between adjacent cutting teeth 114. In one example, the width w of cutting teeth 114 can be from about 0.5 inches (1.3 cm) to about 6 inches (15.2 cm). In one example, the length l of cutting teeth 114 can be from about 0.25 inches (0.6 cm) to about 6 inches (15.2 cm). The width w of cutting teeth 114 and space s between cutting teeth 114 can be the same or can be different.

(21) The provision of cutting teeth 114 increases the driving stresses at the leading edge of tamper head 112 by a factor of from about 1.25 to about 5 depending on the configuration and geometry of cutting teeth 114. The magnification of the mandrel-bottom driving stresses allows tamper head 112 to cut into hard-driving materials more rapidly and at reduced wear and tear to the driving hammer as compared with conventional tamper heads. Examples of hard-driving materials include dense or cemented sand, stiff and hard clay, and subsurface obstructions, such as buried concrete pieces and bricks,

(22) FIG. 3A and FIG. 3B show another embodiment of mandrel 100 for difficult driving conditions. FIG. 3A and FIG. 3B illustrate a side view and a bottom end view, respectively, showing tamper head 112 equipped with both cutting teeth 114 and one or more hydrojet nozzles 118 that may be installed or machined near the bottom of tamper head 112. The purpose and configuration of cutting teeth 114 is as described above with reference to FIG. 1A, FIG. 1B, and FIG. 2. Hydrojet nozzles 118 may be installed within cutting teeth 114 and inclined at angles ranging from about 10 degrees to about 80 degrees from horizontal. Hydrojet nozzles 118 may be installed in one cutting tooth 114, in two cutting teeth 114, or in many cutting teeth 114 depending on the required effectiveness of the jetting operations. Hydrojet nozzles 118 may generate streams of pressurized water or other liquids (e.g., liquid 120) having a diameter ranging from about 1/1000 of an inch (0.0254 mm) to about 0.25 inches (0.6 cm) depending on the mandrel design and driving requirements. Hydrojet nozzles 118 may be hydraulically connected to an interior manifold (not shown) that may be in turn connected to one or more supply lines (not shown) that extend down along mandrel 100 (or within mandrel 100) from the top of mandrel 100. The purpose of hydrojet nozzles 118 and the interior manifold is to distribute pressurized water or other liquids (e.g., liquid 120) from working grades downward through mandrel 100 and out of hydrojet nozzles 118.

(23) Water jet pressures that can range from about 10 psi (68.9 kPa) to about 4000 psi (27,579 kPa) may be applied through hydrojet nozzles 118 during downward driving. The provision of hydrojet nozzles 118 allows for the loosening of dense, stiff, and/or cemented materials that may be encountered during driving. The application of the high pressure water loosens these materials and allows for more effective driving. Hydrojet nozzles 118 are typically installed at angles that are inclined from vertical so as to prevent clogging the nozzles during mandrel driving and extraction. Hydrojet nozzles 118 that are installed at steep angles provide for easiest driving but are also more easily clogged during mandrel penetration into soil materials. By inclining hydrojet nozzles 118 from vertical, hydrojet nozzles 118 have the advantage that they still angle downwards to loosen the subsurface materials yet are not as easily clogged by soil particles. The provision of cutting teeth 114 allows hydrojet nozzles 118 to be inclined at angles greater than zero degrees from vertical thus allowing for the inclined configuration of hydrojet nozzles 118.

(24) In yet another embodiment of the present invention, grout tubes (or grout injection lines) may be used to facilitate more accurately adding grout to piers constructed in very soft and weak soil. For example, FIG. 4A and FIG. 4B show a top view and a cross-sectional view, respectively, of an example of a mandrel 400 that comprises grout tubes for adding grout to aggregate piers. Mandrel 400 may include a feed tube 410 and a tamper head 412, wherein feed tube 410 has a passage 414 therethrough for feeding aggregate (not shown) to tamper head 412. Feed tube 410 and tamper head 412 can be formed of metallic materials, such as steel, cast iron, and aluminum. In one example, the length or height of tamper head 412 can be from about 6 inches (15.2 cm) to about 12 inches (30.5 cm).

(25) In some embodiments, a grout injection line 416 extends downward alongside feed tube 410 to discharge at a location above tamper head 412. In one example, grout injection line 416 has an inside diameter (ID) of about 2 inches (5 cm). The grout is injected near the bottom of mandrel 400 to provide greater confidence and accuracy in the provision of grout within the aggregate. Namely, in this embodiment, an injection port 418 is located above tamper head 412 to reduce the likelihood of grout injection line 416 clogging during compaction facilitated by tamper head 412. In one example, injection port 418 is located at least about 6 inches (15.2 cm) above the top edge of tamper head 412. Grout injection line 416 may be used to accurately inject known volumes of grout at known elevations of tamper head 412. This allows for greater confidence in the location and presence of the added grout within the aggregate pier. Because of this greater confidence, the total volume of grout added to the pier may be reduced, thereby providing cost efficiencies. Further, because grout injection line 416 allows for more accuracy, the pier may be constructed with grout extending to a lower top of grout elevation, thus reducing the potential for post-pier-construction grout chipping activities for piers that are constructed with grout above design elevations.

(26) Optionally, as shown in FIG. 4A and FIG. 4B, a grout inspection line 420 may be provided that also extends downward alongside feed tube 410. In one example, grout inspection line 420 has an ID of about 2 inches (5 cm). Grout inspection line 420 may be used for providing an independent verification of grout quantities. Grout inspection line 420 is optimally located above the discharge point (i.e., injection port 418) of grout injection line 416. Namely, grout inspection line 420 has an inspection port 422 that is typically located some distance above injection port 418 of grout injection line 416. Grout inspection line 420 may consist of a hardened pipe that is affixed or attached to the side of mandrel 400, or it may consist of a flexible hose, or a combination of the two. The purpose of grout inspection line 420 is to verify the elevation of the head of grout within mandrel 400. If grout is observed to emerge from grout inspection line 420 then the pressure head of grout at the elevation of inspection port 422 is known to be equal to or to exceed the elevation of injection port 418 of grout injection line 416.

(27) Further, optionally a set of diametric restriction elements 424 may be installed in tamper head 412 of mandrel 400. Diametric restriction elements 424 can be fabricated from individual chains, cables, or wire rope, or a lattice of vertically and horizontally connected chains, cables, or wire rope. In a specific example, the diametric restriction elements 424 are half-inch (1.3-cm), grade 100 alloy chains. In one example, after initial driving, mandrel 400 is raised and the diametric restriction elements 424 hang freely by gravity from the bottom of tamper head 412. As tamper head 412 is raised the aggregate/grout flows into the cavity left by tamper head 412. After raising tamper head 412 the prescribed distance, tamper head 412 is then re-driven downwardly to a depth preferably less than the initial driving depth into the underlying materials. This allows the diametric restriction elements 424 the opportunity to expand radially and bunch up forming a compaction surface within the tamper head 412 that substantially reduces or prevents aggregate from moving upward relative to the tamper head 412. It is further understood that the tamper head with teeth of FIG. 1A, 1B, 2, or 3A and 3B may also further comprise diametric restriction elements installed therein.

(28) Referring now to FIG. 5A and FIG. 5B, a side view and a bottom end view, respectively, of another example of a mandrel 500 are provided. In this example, the mandrel 500 includes grout tubes (or grout injection lines) for adding grout to aggregate piers, wherein the grout tubes are used to facilitate more accurately adding grout to piers constructed in very soft and weak soil. For example, mandrel 500 includes a feed tube 510 and a tamper head 512, wherein feed tube 510 has a passage 514 running therethrough for feeding aggregate (not shown) to tamper head 512. Feed tube 510 and tamper head 512 can be formed of, for example, metallic materials such as steel, cast iron, and aluminum. Tamper head 512 has an upper end 516 and a lower end 518. In one example, feed tube 510 has an outside diameter (OD) of about 11 inches (27.9 cm) and an ID of about 9 inches (22.9 cm). In this example, the OD and ID of upper end 516 of tamper head 512 is smaller than the OD and ID of lower end 518 of tamper head 512. For example, upper end 516 of tamper head 512 has an OD of about 16 inches (40.6 cm) and an ID of about 14 inches (35.6 cm), while lower end 518 of tamper head 512 has an OD of about 18 inches (45.7 cm) and an ID of about 16 inches (40.6 cm).

(29) In some embodiments, a grout injection line 520 may extend downward alongside feed tube 510. In one example, grout injection line 520 is a 2-inch (5-cm) ID black pipe. A grout hose (not shown) may attach to the top of grout injection line 520. The bottom end of grout injection line 520 may be fluidly coupled to a splitter 522 that supplies two or more grout lines 524 (e.g., grout lines 524a, 524b). In one example, two grout lines 524 are integrated into the walls of tamper head 512 and wrap around the sides of tamper head 512 until they are opposite of each other. The two or more grout lines 524 can be, for example, hardened pipe, flexible hose, or a combination thereof. In certain other embodiments, the two or more grout lines 524 can be made of the same material as tamper head 512. It is understood that more than two grout lines may be provided (with resulting multiple splitting).

(30) Each of the two grout lines 524 may include a deflector plate 526. For example, a deflector plate 526a may be located below the end of grout line 524a and a deflector plate 526b may be located below the end of grout lines 524b. Deflector plates 526a, 526b help to direct the grout to the center of tamper head 512 as it is pumped and to keep soil/aggregate from plugging grout lines 524a, 524b during driving. In this embodiment, the grout is injected near the bottom of mandrel 500 to provide greater confidence and accuracy in the provision of grout within the aggregate.

(31) In other embodiments, the presently disclosed mandrel 500 and/or tamper head 512 may further include cutting teeth 114, hydrojet nozzles 118, grout tubes (or grout injection lines) (e.g., 416, 520), grout inspection lines (e.g., 420), and any combinations thereof.

(32) Referring now to FIG. 6A and FIG. 6B, a top view and a cross-sectional view, respectively, are shown of another example of a mandrel 400 that comprises one or more grout tubes for adding grout to aggregate piers. Unlike the embodiment shown in FIG. 4A and FIG. 4B, here the one or more grout lines 416 discharge directly into the tamper head 412 rather than discharging into the feed tube 410 above tamper head 412. Additionally, FIG. 6A and FIG. 6B show an embodiment of the mandrel 400 that optionally does not include one or more grout inspection lines, like grout inspection line 420 shown in FIG. 4A and FIG. 4B.

(33) Having generally described the presently disclosed apparatuses for constructing displacement aggregate piers, it is more specifically described by illustration in the following specific EXAMPLE.

EXAMPLE

(34) In one example of the present subject matter, a method of injecting grout into an aggregate pier within a targeted zone of very soft and weak soils using the grout injection tubes was demonstrated in full-scale field tests.

(35) The piers were installed with a Liebherr 125 base machine equipped with a grout pump and hopper. A pump hose ran from the pump to the top of the mandrel. The mandrel was equipped with a 2 inch ID grout pipe 520 similar to that shown in FIG. 5A and FIG. 5B that ran along the full length of the mandrel pipe 510. At the bottom of the mandrel, the pipe split into two pipes wrapped around the sides of the head until the discharge locations were opposite to each other. A deflector plate was located below the end of the grout discharge locations to help the grout move to the center during pumping.

(36) Several grout mixes were evaluated during the testing program resulting in a finalized grout mix that had the proper viscosity to allow for pumping but to not freely permeate through the voids within the aggregate pier. The final grout mix consisted of 242 lbs (110 kg) of water, 660 lbs (299 kg) of cement, 990 lbs (449 kg) of sand (playground), 500 mL of retarder (i.e., EUCON W.O.), and 1,650 mL of superplasticizer (i.e., EUCON 37 superplasticizer). The grout was mixed with a paddle mixer and tested with flow cone test per ASTM C939 to achieve a flow that ranged from 40 to 60 seconds.

(37) The piers were constructed by driving the mandrel through the fill, peat, and clean sand to a depth of about 30 feet (9.1 m). Stone was wetted and added to the mandrel hopper. An ungrouted pier was constructed using a 5 ft/4 ft (1.5 m/1.2 m) stroke pattern over the lower 17 feet (5.2 m) in the clean sand. At a depth of 1 foot (0.3 m) below the peat layer, the mandrel was held stationary and grout was introduced into the mandrel through the grout pipes. After a specific volume of grout was introduced, a single lift was constructed with a 3 ft/3 ft (0.9 m/0.9 m) stroke pattern at a depth of 13 feet (4.0 m), and then the upper portion of the pier through the peat and fill from a depth of 13 feet (4.0 m) to 4 feet (1.2 m) was constructed with grouted stone using a 3 ft/2 ft (0.9 m/0.6 m) stroke pattern. At a depth of 4 feet (1.2 m) grouting was stopped and the upper 4 feet (1.2 m) of the pier was constructed with ungrouted stone using a 3 ft/2 ft (0.9 m/0.6 m) stroke pattern.

(38) A modulus test was performed on a constructed pier. The results shown in plot 700 of FIG. 7 indicate that the constructed piers confirmed the design and were sufficient to support the structure.

(39) Several hundred piers were installed at this site with the technique described above. Traditional aggregate pier grouting methods with low viscosity grout were not feasible at this site because the grout would permeate through the pier and into the clean sand matrix soil along the lower 17 feet (5.2 m) of the pier. Additionally, traditional grouting methods do not allow for accurately starting and stopping the grouting process at the targeted depth of the peat soils. The advantage of introducing grout within a targeted zone rather than grouting the entire pier length as with traditional aggregate pier grouting methods resulted in a significant reduction in the volume of grout required for each pier and in the overall cost of the project.

(40) Following long-standing patent law convention, the terms a, an, and the refer to one or more when used in this application, including the claims. Thus, for example, reference to a subject includes a plurality of subjects, unless the context clearly is to the contrary (e.g., a plurality of subjects), and so forth.

(41) Throughout this specification and the claims, the terms comprise, comprises, and comprising are used in a non-exclusive sense, except where the context requires otherwise. Likewise, the term include and its grammatical variants are intended to be non-limiting, such that recitation of items in a list is not to the exclusion of other like items that can be substituted or added to the listed items.

(42) For the purposes of this specification and appended claims, unless otherwise indicated, all numbers expressing amounts, sizes, dimensions, proportions, shapes, formulations, parameters, percentages, parameters, quantities, characteristics, and other numerical values used in the specification and claims, are to be understood as being modified in all instances by the term about even though the term about may not expressly appear with the value, amount or range. Accordingly, unless indicated to the contrary, the numerical parameters set forth in the following specification and attached claims are not and need not be exact, but may be approximate and/or larger or smaller as desired, reflecting tolerances, conversion factors, rounding off, measurement error and the like, and other factors known to those of skill in the art depending on the desired properties sought to be obtained by the presently disclosed subject matter. For example, the term about, when referring to a value can be meant to encompass variations of, in some embodiments, 100% in some embodiments 50%, in some embodiments 20%, in some embodiments 10%, in some embodiments 5%, in some embodiments 1%, in some embodiments 0.5%, and in some embodiments 0.1% from the specified amount, as such variations are appropriate to perform the disclosed methods or employ the disclosed compositions.

(43) Further, the term about when used in connection with one or more numbers or numerical ranges, should be understood to refer to all such numbers, including all numbers in a range and modifies that range by extending the boundaries above and below the numerical values set forth. The recitation of numerical ranges by endpoints includes all numbers, e.g., whole integers, including fractions thereof, subsumed within that range (for example, the recitation of 1 to 5 includes 1, 2, 3, 4, and 5, as well as fractions thereof, e.g., 1.5, 2.25, 3.75, 4.1, and the like) and any range within that range.

(44) Although the foregoing subject matter has been described in some detail by way of illustration and example for purposes of clarity of understanding, it will be understood by those skilled in the art that certain changes and modifications can be practiced within the scope of the appended claims.