Reinforcing assemblies for use in structural concrete members, as well as reinforced concrete structures, are provided that are uniquely suited to deal with punching shear failure in structural concrete members such as slabs, beams footings and flat foundations. The reinforcing assemblies include a support base, comprising two elongate rebar members disposed substantially parallel to each other, and several shear-resisting rebar elements having two opposing ends and cast, or bent, in hairpin shape on a common plane in generally parallel orientation to each other and secured to the base. The reinforced concrete structures include a concrete member, having a first face and a generally opposed second face, and at least four reinforcing assemblies that are embedded and retained at prescribed locations within the concrete member by their base and by the opposed faces of the concrete member, and which comprise a support base and multiple hairpin-shaped rebar elements secured to the support base.

FIELD OF THE INVENTION 
       [0001]    This invention relates to reinforcing assemblies for use in structural concrete members. In particular, this invention is concerned with a reinforcing system that is uniquely suited to deal with shear failure and with punching shear failure in structural concrete members such as slabs, beams, footings and flat foundations. More particularly, the invention relates to a structural concrete reinforcing assembly that may be made entirely of rebar. Specifically, the invention relates to a novel technique for eliminating or minimizing shear failure and punching shear failure in structural concrete members such as slabs, beams, footings and flat foundations by means of a unique reinforcing assembly made up of hairpin-shaped rebars, which novel technique allows the user to fabricate reinforced concrete structures with enhanced punching-shear-resistant capabilities. 
       BACKGROUND OF THE INVENTION 
       [0002]    Commercial concrete is a mixture of cement, sand and stone aggregate held together as rigid structures by the action of small amounts of water. While concrete made in this fashion, usually referred to as “unreinforced concrete”, has fairly good resistance to compressive stresses, any significant tension will tend to break the rigid structure and cause undesirable cracking and separation of the concrete. For practical reasons, most commercial and industrial structural concrete members are made of “reinforced concrete”, that is, from a mixture of cement, sand and stone aggregate where a solid member made of a material with high strength in tension, such as steel, is placed and made to remain embedded. Reinforced concrete sections where the concrete is capable of resisting the compression and the steel placed inside is designed to resist the tension are often made into many shapes and sizes for the construction industry. Commercial and industrial structural concrete members such as slabs, beams, footings and flat foundations, even when made with reinforced concrete, are susceptible to shear forces that create tensile forces along them and often result in structural failure, that is, undesirable cracking and/or breaking of the structural concrete members. The type of failure caused by these shear forces and the attendant incline cracks that tend to propagate throughout the concrete, usually from the area under tension towards the area under compression, are not always easy to detect and correct since they are often not visible when they occur. One-way shear failure, often referred to as “shear failure”, usually occurs in beams and, occasionally, in walls, slabs, footings and other vertical members. Two-way shear failure, often referred to as “punching shear failure”, tends to occur in horizontal concrete members such as slabs, footings and flat foundations. Various methods and techniques exist for reinforcing structural concrete members so as to prevent or minimize the undesirable cracking and/or breaking of concrete structures caused by shear forces. Some of the methods and techniques that are commonly employed with various degrees of success include the use of open or closed stirrups that are strategically placed in designated areas of the concrete structures. A particular type of situation that is often encountered, for example when a load-bearing concrete slab is supported by a column, is what is commonly referred to as “punching shear” failure. Shear strength of flat slabs in the vicinity of columns or concentrated loads is often controlled by the two-way shear known as “punching shear”. Shear considerations can therefore be the controlling factor in determining the required slab thickness or increasing column size, especially of post-tensioned (“PT”) flat-plates. Punching shear failure is usually a substantial inclined cracking that occurs at about a 19-to-34-degree angle with respect to the top surface of a PT slab and extends from the edge of the load being applied. Post-tensioned slabs and slabs that make use of high-strength concrete are particularly susceptible to punching shear failure. Conventional solutions to shear problems are not always satisfactory when dealing with punching shear problems. To prevent punching shear failure at slab-column connections, conventional solutions usually provide for the use of stirrups, structural steel shearheads or studrail reinforcements. Each of these techniques, while adequate in many cases, also has its share of disadvantages. 
         [0003]    Stirrups, with longitudinal bars or with vertical bars, are difficult to place in the concrete structures and often present anchorage slip problems. Increasing the strength of concrete slabs with conventional stirrups is common, but the anchorage of stirrups is difficult to provide in thin slabs (e.g., less than about 6 inches in height) and therefore should be used only if the stirrups are closed and contain a longitudinal bar at each corner. The use of the stirrups as shear reinforcements in slabs is practicable only if the effective depth of the slab is greater than about 6 inches, but not less than 16 times the shear reinforcement bar diameter. So-called “shearheads” are structural steel shapes such as “I” shapes or channel sections. Shearheads are rarely used because their installation is very expensive and because they often interfere with the placement of flexural reinforcing bars and post-tensioned cables. So-called “studrails” consist of headed studs in the form of vertical bars mechanically anchored at each end by a plate or head capable of developing the yield strength of the bars. Studrails are often used to increase the shear strength in flat slabs, but in order to develop the full yield strength of the studs the area of the anchor head must be a minimum of 10 times the cross sectional area of the stud stem. Also, because of their anchor heads, studrails do not always provide adequate confinement of the concrete where the punching shear failure tends to occur. 
         [0004]    It is apparent that a need exists to provide a solution to the problems associated with punching shear failure that does not suffer from the shortcomings attendant the use of stirrups, shearheads, studrails and similar conventional devices currently in commercial use. The present invention provides one such solution. 
       SUMMARY OF THE INVENTION 
       [0005]    It is an object of the present invention to provide a practicable and economical solution to the problems associated with shear failure and punching shear failure in concrete structures. Another object of the present invention is to provide a safe and cost-effective reinforcing assembly for use in structural concrete members that is uniquely suited to deal with shear failure and punching shear failure in structural concrete members such as slabs, beams, footings and flat foundations. It is also an object of this invention to provide a structural concrete reinforcing assembly that may be made entirely of rebar members. An additional object of the invention is to provide a safe and low-cost reinforced concrete structure that is particularly resistant to punching shear forces and that is uniquely suited for commercial use with structural concrete members such as slabs, footings and flat foundations. A further object of the invention is to provide a structural concrete reinforcing assembly that is particularly effective at eliminating or minimizing shear failure in structural concrete members while affording better anchorage than the conventional devices used in preventing or minimizing shear failures. A particular object of the invention is to facilitate an efficient technique for eliminating or minimizing shear failure in structural concrete members such as slabs, footings and flat foundations by means of a unique reinforcing assembly that may be made up entirely of hairpin-shaped rebars, which novel technique permits the fabrication of reinforced concrete structures with enhanced punching-shear-resistant capabilities. These and other objects of the invention will be apparent from the disclosures that follow. 
         [0006]    The reinforcing assembly of the present invention involves a support base that comprises two elongate rebar members disposed substantially parallel to each other, and several shear-resisting rebar elements having two opposing ends and a hairpin shape on a common plane in generally parallel orientation to each other and secured to the base. The rebar elements may be cast in the hairpin shape, or they may be reformed by bending; and they may have a smooth surface, which affords cost savings, or be ribbed or corrugated in order to improve cohesion between the steel and the concrete. The reinforcing assemblies of the present invention are best used as sets of four units that are strategically placed at or near the areas of high shear failure propensity. The reinforced concrete structure of the invention includes a concrete member, having a first face and a generally opposed second face, and at least four reinforcing assemblies that are embedded and mechanically retained in a prescribed location within the concrete member by their base and by the opposed faces of the concrete member. Each of the four reinforcing assemblies of the reinforced concrete structure comprises a support base and multiple hairpin-shaped rebar elements secured to the support base. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]    A clear understanding of the key features of the invention summarized above may be had by reference to the appended drawings, which illustrate the system of the invention, although it will be understood that such drawings depict preferred embodiments of the invention and, therefore, are not to be construed as limiting its scope with regard to other embodiments which the invention intends and is capable of contemplating. 
           [0008]    Accordingly,  FIG. 1  represents a cross-sectional view of a concrete slab structure at the area surrounding the intersection of the slab structure with a supporting column, and showing undesirable cracking caused by punching shear forces. 
           [0009]      FIG. 2  is a top view of the concrete slab structure depicted in  FIG. 1 . 
           [0010]      FIG. 3  is a schematic diagram of a preferred embodiment of the reinforcing assembly system of the invention, and also shows several shapes that may be used for the hairpin-shaped rebar elements of the reinforcing assembly. 
           [0011]      FIG. 4  is a schematic diagram of another preferred embodiment of the invention, illustrating one manner in which four reinforcing assembly systems may be used to provide reinforcement to a critical area surrounding the intersection of a slab and a supporting column. 
           [0012]      FIG. 5  is a cross-sectional plan view of a reinforcing arrangement that uses eight of the reinforcing assembly system depicted in  FIG. 3 , showing the immediate vicinity of the intersection of a post-tensioned slab and a supporting column. 
           [0013]      FIG. 6  is a sectional view of a specific section identified in  FIG. 5 , showing details of the reinforcing arrangement depicted therein. 
           [0014]      FIG. 7  is a graph depicting and comparing load deflection relationship test results obtained by means of the reinforcing assembly system of the present invention. 
           [0015]      FIG. 8  is a graph depicting and comparing load deflection relationship test results obtained by means of a reinforcing assembly system used in the prior art. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0016]    The invention will now be described with reference to its application to a post-tensioned slab structure that is supported by a column, which may be a steel supporting column or a concrete supporting column.  FIG. 1  represents a cross-sectional view of a concrete slab structure  101  at the “loaded” area  102  (also referred to as the “reaction” area) surrounding the intersection of the slab structure with concrete supporting column  103 , and showing undesirable cracking  104  caused by punching shear forces. This is a typical two-way shear situation.  FIG. 2  is a top view of the concrete slab structure depicted in  FIG. 1  and shows post-tensioned concrete slab  201  (slab  101  in  FIG. 1 ) intersected by concrete supporting column  203  (column  103  in  FIG. 1 ) and creating loaded, or reaction, area  202  delineated by shear critical section perimeter  204 . The maximum distance  205  at which the American Concrete Institute (“ACI”) requires any reinforcing assembly to be located inside a concrete slab supported by a column is usually expressed as “d/2”, with “d∞ being the effective depth of the slab, that is, the distance from the extreme compression fiber to the centroid of longitudinal tension reinforcement. This distance is shown in  FIG. 2  as distance  205  from outer face  206  of support column  203  to the shear critical section perimeter  204 . 
         [0017]      FIG. 3  shows a schematic diagram of a preferred embodiment of this invention illustrating one manner in which the reinforcing assembly system of the invention may be designed to provide reinforcement to a critical area of interconnection between a slab and a cross member concrete structure such as a support column. The reinforcing assembly system of this invention is sometimes referred to as the “PTE Strand punching shear reinforcement”, or simply the “PTE-PSR”. The system involves a support base, comprising two elongate rebar members, or reinforcement bars, disposed substantially parallel to each other, and several hairpin-shaped shear-resisting rebar elements that are attached to the base. As shown in  FIG. 3 , base  301  comprises first elongate rebar member  302  and second elongate rebar member  303  disposed parallel to each other. Elongate rebar members  302  and  303  (sometimes referred to as “anchorage bars” or “rail bars”) are made of steel, but they may also be made of some other metal or hard material. The cross-sectional area of rebar members  302  and  303  is round, but it may also be rectangular or some other form. The length and thickness of rebar members  302  and  303  may vary with the specifications of the concrete structure for which they are designed. In the case of a standard flat slab, for example, they typically may be anywhere from about 1 to 5 feet long and from about ⅜ to ¾ inch in diameter (if round), and ¼ to ¾ inch in thickness (if square), depending on the applied loads and the effective depth of the slab. Also by way of illustration, if rectangular, the thickness of the rebar members may be anywhere between about ½ and 2 inches by between about ⅛ and ⅜ inch in cross-sectional area. Shear-resisting rebar elements  304 - 309  have a hairpin shape and are welded to base  301  at each of their ends, e.g., shear-resisting rebar element  309  is welded to rebar member  302  at end  310  and to rebar member  303  at end  311 , while shear-resisting rebar element  308  is welded to rebar member  302  at end  312  and to rebar member  303  at end  313 , and so on. The cross-sectional areas of shear-resisting rebar elements  304 - 309  may be substantially round or rectangular. The PTE-PSR reinforcing assembly depicted in  FIG. 3  has six hairpin-shaped rebar elements, but it will be understood that more or less rebar elements may be used depending on the dimensions and other specifications of the concrete structure. Hairpin-shaped rebar elements  304 - 309  are made of steel, but they may also be made of some other metal or hard material. The length and the thickness of hairpin-shaped rebar elements  304 - 309  may vary with the specifications of the concrete structure for which they are designed. In the case of a standard 8-inch-thick flat slab, for example, they may be about 15 inches long (from end to end) and from about ¼ to ⅝ inches thick, depending on the applied loads and the effective depth of the slab. The hairpin-shaped rebar elements are sometimes referred to herein as “hairpins”. Various alternative shapes, e.g., shapes  314 - 319 , of hairpin-shaped rebar elements  304 - 309  may be used, but the shapes should be substantially symmetrical, that is, the length of the legs, such as vertical legs  320  and  321  in the illustration of  FIG. 3 , should be essentially the same. Likewise, for any given assembly, the height of each hairpin, that is, the distance from its highest vertical point to the base of the assembly, should also be constant. In the illustration of  FIG. 3  the height of each hairpin attached to base  301  is 4 inches, and the distance between the two legs of each hairpin is 2 inches, while the length of base  301  is 1 1/2  feet. Rebar members  302  and  303  and hairpin-shaped rebar elements  304 - 309  may be made of steel reinforcing bars having the same thickness, or they may be made of other metals and have different thicknesses. In the illustration of  FIG. 3 , rebar members  302  and  303  and hairpin-shaped rebar elements  304 - 309  are made of ⅜-inch-diameter round steel. The elongate rebar members may be cast or cut to size, while the hairpin-shaped rebar elements may be shaped by bending, by hand or by special equipment, or they may be preformed by casting to the desired shapes and sizes. The elongate rebar members and the hairpin-shaped rebar elements may have a smooth surface, when cost savings constitute an important consideration, or they may be ribbed or corrugated in order to improve cohesion between the steel and the concrete, thereby providing better anchorage. 
         [0018]    In using the reinforcing assembly system of the invention it is often best to place several, e.g., four, of these assemblies, or “PTE-PSRs”, right next to the loaded or reaction area, arranged in the manner depicted in  FIG. 4 , which is a schematic diagram illustrating one manner in which the assembly system of the invention may be used to provide reinforcement to a shear critical area surrounding the intersection of a post-tensioned slab and a concrete supporting column. Thus assemblies  401 ,  402 ,  403  and  404  are placed in cross-like fashion around loaded or reaction area  405  within critical section  406  that may occur, surrounding said intersection on the same plane, each disposed at an approximately right angle with the other. 
         [0019]      FIG. 5  is a top cross-sectional view of a reinforcing arrangement using eight PTE-PSR reinforcing assemblies, showing the immediate vicinity of the intersection of a post-tensioned slab and a concrete supporting column. As stated above, the maximum distance at which the American Concrete Institute requires any reinforcing assembly to be located inside a concrete slab supported by a column is usually expressed as “d/2”, with “d” being the effective depth of the slab, that is, the distance from the extreme compression fiber to the centroid of longitudinal tension reinforcement. This critical distance is shown in  FIG. 5  as distance  501  from outer face  502  of supporting column  503  to the first hairpin  504  of PTE-PSR  505 . PTE-PSR  505  has six hairpins and is similar to the PTE-PSR depicted in  FIG. 3 . The placement of PTE-PSR  505  is such that the location of first hairpin  504  is made to coincide with shear critical section perimeter  514  of loaded or reaction area  513 . PTE-PSRs  506 ,  507 ,  508 ,  509 ,  510 ,  511  and  512  are similarly configured and similarly placed around supporting column  503  on the same spatial plane as PTE-PSR  505  in cross-like fashion as illustrated in  FIG. 5 . When configured in sets of two hairpin assemblies, as illustrated in  FIG. 5 , improved shear resistance capabilities are also obtained if the distance between each assembly in a set is maintained at 2 d or less, that is, if the distance between each assembly is made to be equal to or less than twice the effective depth of the slab or concrete member. Thus, in the illustration of  FIG. 5 , distance  515  between PTE-PSR  505  and PTE-PSR  506  is set at less than twice the effective depth d of the slab; distance  516  between PTE-PSR  507  and PTE-PSR  508  is also set at less than twice the effective depth d of the slab; distance  517  between PTE-PSR  509  and PTE-PSR  510  is also set at less than 2 d; and distance  518  between PTE-PSR  511  and PTE-PSR  512  is likewise set at less than 2 d. 
         [0020]      FIG. 6  is a front sectional view of Section A-A′ identified in  FIG. 5  and showing frontal views of PTE-PSRs  505  and  510  (identified in  FIG. 6  as PTE-PSRs  605  and  610 ) and showing the locations of PTE-PSRs  511  and  512  of  FIG. 5  (identified in  FIG. 6  as PTE-PSRs  611  and  612 ), all embedded within shear critical section perimeter  603  of loaded or reaction area  606  surrounding supporting column  602  in post-tensioned slab  601 . First hairpin  604  of PTE-PSR  605  is located at distance  613  from supporting column face  607 . First hairpin  604  of PTE-PSR  605  is also spaced from adjacent hairpin  608  of PTE-PSR  605  at a distance which is the same as distance  613 ; adjacent hairpin  608  is spaced from further adjacent hairpin  609  at a distance which is the same as distance  613 , and so on. Likewise, first hairpin  614  of PTE-PSR  610  is located at distance  615  from supporting column face  616  which is the same as distance  613  between hairpin  604  and column face  607 ; while adjacent second hairpin  617  of PTE-PSR  610  is spaced from hairpin  614  at a distance which is the same as distance  615 ; and adjacent third hairpin  618  is separated from hairpin  617  at a distance which is the same as distance  615 , and so on. In the illustration of  FIG. 6 , as well as in the illustration of  FIG. 5 , the slab effective depth (“d”) is 8 inches, and critical distances  501 ,  613  and  615  are 4 inches each. 
         [0021]    Following is a detailed description of tests conducted in order to confirm the efficiency of the present invention and compare it with prior art techniques for reinforcing concrete members.  FIG. 7  is a graph depicting and comparing load deflection relationship test results obtained by means of the reinforcing assembly system of the present invention (PTE-PSRs), as discussed below, while  FIG. 8  is a graph depicting and comparing load deflection relationship test results obtained by means of a reinforcing assembly system used in the prior art (studrails), also discussed below. The specifications of the compared reinforcing assembly systems and the test parameters are shown in Table 1 below. The PTE-PSRs were made from reformed rebars (as shown in  FIG. 3 ) PTE-PSRs were tested and compared using the same spacing and strength criteria as studrails. An important goal of these tests was to conduct an experimental investigation of the punching shear behavior of post-tensioned slabs (“PT slabs”) with hairpin assembly reinforcements. The slabs were designed to ensure punching shear failure and the tests were conducted by applying a central concentrated load on post-tensioned slabs. The test results obtained by using hairpin assembly reinforcements were compared to results obtained by using headed-stud reinforcements, also known as “headed-studs” or “studrails”, which had been previously used to increase punching shear capacity in flat slabs. The punching shear failure of the PT slabs was ensured by the following approach: The dimensions and thickness of the slab were chosen to yield a nominal punching shear stress of about 25 tons using the ACI equation for headed studs (8√{square root over (f′ c )} at d/2 distance from the column face) which was about 40% of the capacity of the loading jack used for the tests. The slab was then analyzed using the so-called “Adapt floor-pro program”. The tests were conducted at the facilities of P.T.E. Strand Co., Inc., in Florida, which also provided materials and assistance in casting, stressing the tendons and conducting the tests. The maximum compressive stress in the concrete at the critical sections were below 75% of the concrete compressive strength, which would ensure a punching shear failure mode of the post-tensioned slab. The numbers and the profiles of the prestressing cables were then determined using the Adapt floor-pro program. Four post-tensioned concrete slabs with the same dimensions and post-tensioning were tested. Tests conducted with the hairpin assembly reinforcements (the PTE-PSRs) were designated PT- 1  and PT- 2 , while tests conducted with the headed-stud reinforcements (the studrails) were designated PT- 3  and PT- 4 . The specifications and parameters of the reinforcements are shown in Table 1. The test specimens were 6′-0″(1.83 m)×6′-0″(1.83 m)×4″(102 mm). The shear reinforcements of both specimens were located in the central region of the slab, around the loaded or reaction area. The concrete used for the slabs had a design strength of 3,500 psi (23.3 Mpa) and was supplied by a local ready-mix plant. The coarse aggregate used was #16 crushed limestone, and the water-cement ratio was 0.42. All specimens were cast from the same batch. Five standard cylinders of 6 in. (152.5 mm) radius×12 in. (305 mm) height were cast from each batch and kept in the same environment as the slab. The compressive strength after 7 days was 2,710 psi (18.7 Mpa) and the strength at the time of testing was 4,350 psi (30 Mpa). 
         [0022]    The prestressing strands used were 0.5 in. (12.7 mm) in diameter, seven wire strands conforming to ASTM standard A421, with a specified ultimate strength of 270 ksi (1861 Mpa). The tendons were protected with a plastic sheathing to prevent the “cable-concrete” bond and to reduce friction at the time of stressing. #3 bars (10 mm) were used for fabricating hairpin assembly reinforcements conforming ASTM standard A421, with a specified yielding strength of 60 ksi (413.5 Mpa). The edge forms for the slabs were constructed using ¾ in. (19 mm) thick and 4 in. (102 mm) high plywood which were braced using 2 in. (51 mm)×6 in. (153 mm) wood studs. The edge forms were carefully aligned, leveled and nailed to the floor to help prevent movement during casting. The slabs were cast on the floor over plastic sheets. Prior to casting the slabs, the plastic sheets were oiled for ease of removal of the specimens. Chairs were used to ensure that the desired tendon profile was attained. Ready-mix concrete was delivered and pumped to the location of the specimens. The concrete was vibrated and the final slab finish achieved using steel trowels and wooden floats. A plastic cover was placed over the slabs for a 15-day curing period. 
         [0023]    The specimens were tested in an elevated position using a steel test frame. Application of an upward load allowed for the observation of the punching shear failure on the top surface of the slab. The reaction frame was designed and fabricated of structural steel. The central loading was applied upwards and the concrete slab was held down at four locations which were 3 ft. (915 mm) apart. The slab also had a 1.5 ft. (457 mm) overhang on each side. The tendons were stressed up to 33 kips (146.5 KN) after the slab was positioned on the test frame. The loading was accomplished by using a hydraulic jack. The hydraulic jack was previously calibrated to yield the applied load. A dial gauge was mounted at the center of the slab to measure the displacement of the slab. Load cells were also placed between the edge of the slab and the anchoring mechanism of the prestressing tendons to measure the force in the four central tendons. The strain-gage based load cells were manufactured by Honeywell Sensotec and had a capacity of 50 kips (222 KN) each. These load cells were connected to a 4-channel digital indicator which displayed the tendon force. 
         [0024]    The tests were carried out by increasing the pressure in the hydraulic jack and recording the central deflection and the tendon forces at each increment of the load. The data for one of the slabs with hairpin assembly reinforcements (PT- 1 ) are provided in Table 2. The first crack in PT- 1  was observed at a load of 30.5 kips (135.7 KN). The second perpendicular crack in PT- 1  appeared at a load of 43.2 kips (192.2 KN) in PT- 1 . The maximum load carried by PT- 1  was 63.8 kips (283.7 KN). Both specimens PT- 1  and PT- 2  failed in the flexural mode. The results presented in Table 2 indicate that the tendon force increases linearly with the load up to the point of failure. The maximum stress in the tendon at failure was 190 ksi (1308 Mpa). The use of the hairpin assembly reinforcements (the PTE-PSRs) caused flexural failure to occur before punching shear failure in both specimens PT- 1  and PT- 2 . The central load vs. deflection plots for PT- 1  and PT- 2  are provided in  FIG. 7 . The load deflection behavior appears to be essentially bi-linear with the change in slope occurring close to the first visible crack. 
         [0025]    The data obtained for one of the slabs with the studrails (PT- 3 ) is provided in Table 3. The first crack in PT- 3  was observed at a load of 30.5 kips (135.7 KN) and the second perpendicular crack appeared at a load of 39.9 kips (177.5 KN). The maximum load reached was 57.3 kips (255 KN). The central load vs. deflection plots for PT- 3  and PT- 4  are provided in  FIG. 8 . The load deflection behavior appears to be essentially bi-linear with the change in slope occurring close to the first visible crack. Both specimens PT- 3  and PT- 4  failed in punching shear mode. The shape of the failure surface was roughly square with a dimension of about 22 in. (559 mm) to 22 in. (559 mm) each side. 
         [0026]    The ACI approach for punching shear capacity with headed-stud reinforcements is based on the nominal punching shear stress of a critical section located at a distance d/2 from the column face and the length of the shear reinforcement as described earlier. The test results are compared with the values obtained from the ACI provision of headed-stud reinforcements in Table 4. The ACI-421.1 R recommendation for nominal punching shear stress using headed-studs is 8√{square root over (f′ c )} at a distance d/2 from the column face. For hairpin assembly reinforcements, or PTE-PSRs, (PT- 1  & PT- 2 ), the average punching shear strength was 62.2 Kips (276.7 KN) which is equal to a nominal punching shear stress of 10.2√{square root over (f′ c )}. For PT slabs reinforced with headed-studs (PT- 3  & PT- 4 ), the ACI predictions were found to be almost equal or slightly conservative. The average capacity was 50.2 Kips (223.3 KN) which corresponds to a nominal shear stress of 8.2√{square root over (f′ c )}. The results obtained for punching shear using the PTE-PSRs were about 20% higher than those obtained for punching shear using the headed-studs. 
         [0027]    The tests demonstrated that the PT slabs with headed-stud reinforcements failed in punching shear, while PT slabs with hairpin assembly reinforcements (PTE-PSRs) failed in flexure. The test results with the headed-stud reinforcements were almost equal to those predicted by the ACI equation. The tests with hairpin assembly reinforcements had the shear failure prevented, and almost complete redistribution of bending moments were achieved prior to collapse. The PT slab capacity with hairpin assembly reinforcements were about 20% higher than the PT slab capacity with the headed-stud reinforcements. The nominal shear stress for a PT slab with hairpin assembly reinforcements was equal to 10.2√{square root over (f′ c )} and could actually be higher as the slabs failed in flexure. The test results show that, in terms of shear reinforcement capability, concrete structures using hairpin assembly reinforcements in PT slabs provide superior confinement of the concrete where punching shear failures tend to occur and behave structurally better than concrete structures that use headed-stud reinforcements. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Specifications and Parameters of Load Deflection Relationship Test 
               
             
          
           
               
                   
                   
                   
                   
                   
                 Number of 
                 Number of 
               
               
                   
                 Number of 
                 Studrail/ 
                 Studrail/ 
                 Studrail/ 
                 assembly rails/ 
                 studrails/ 
               
               
                 Assembly 
                 test 
                 hairpin 
                 hairpin 
                 hairpin 
                 elongated rebars 
                 hairpins per 
               
               
                 system 
                 specimens 
                 diameter 
                 height 
                 spacing 
                 per column 
                 assembly 
               
               
                   
               
               
                 Studrails 
                 2 
                 ⅜ in 
                 4 in 
                 1.5 in 
                 8 
                 6 
               
               
                 PTE-PSRs 
                 2 
                 ⅜ in 
                 4 in 
                 1.5 in 
                 8 
                 6 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Test results for PT-1 
               
             
          
           
               
                   
                   
                 Tendon forces measured by 
                   
               
               
                 Load 
                 Deflection 
                 load cells 
               
               
                 (Kips) 
                 (in) 
                 (lbs) 
                 Remarks 
               
               
                   
               
             
          
           
               
                 0.00 
                 0.000 
                 21111 
                 22670 
                 23217 
                 25750 
                   
               
               
                 5.2 
                 0.009 
                 21110 
                 22676 
                 23223 
                 25723 
               
               
                 10.5 
                 0.017 
                 21152 
                 22602 
                 23237 
                 25731 
               
               
                 16.4 
                 0.027 
                 21204 
                 22645 
                 23259 
                 25748 
               
               
                 22.3 
                 0.038 
                 21269 
                 22707 
                 23288 
                 25794 
               
               
                 25.0 
                 0.046 
                 21306 
                 22745 
                 23317 
                 25798 
               
               
                 30.5 
                 0.069 
                 21387 
                 22828 
                 23422 
                 25902 
                 1st visible 
               
               
                   
                   
                   
                   
                   
                   
                 crack 
               
               
                 36.6 
                 0.116 
                 21500 
                 22924 
                 23748 
                 26230 
               
               
                 43.2 
                 0.199 
                 21808 
                 23135 
                 24282 
                 26772 
                 2 nd  visible 
               
               
                   
                   
                   
                   
                   
                   
                 crack 
               
               
                 49.2 
                 0.338 
                 22500 
                 23746 
                 25067 
                 27603 
               
               
                 54.6 
                 0.446 
                 23167 
                 24391 
                 25736 
                 27323 
               
               
                 60.6 
                 0.629 
                 24114 
                 25305 
                 26545 
                 29182 
               
               
                 63.8 
                 0.801 
                 24562 
                 25743 
                 26848 
                 19465 
                 punching 
               
               
                   
                   
                   
                   
                   
                   
                 shear 
               
               
                   
                   
                   
                   
                   
                   
                 failure 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Test results for PT-3 
               
             
          
           
               
                   
                   
                 Tendon forces measured 
                   
               
               
                 Load 
                 Deflection 
                 by load cells 
               
               
                 (Kips) 
                 (in) 
                 (lbs) 
                 remarks 
               
               
                   
               
             
          
           
               
                 0.00 
                 0.000 
                 24188.0 
                 21328.0 
                 23519.0 
                 17841.0 
                   
               
               
                 5.2 
                 0.008 
                 24191 
                 21337 
                 23528.0 
                 17852 
               
               
                 10.5 
                 0.018 
                 24196.0 
                 21357 
                 23588 
                 17903 
               
               
                 16.4 
                 0.031 
                 24218.0 
                 21388 
                 23640 
                 17987 
               
               
                 22.3 
                 0.043 
                 24241.0 
                 21416 
                 23738 
                 18075 
               
               
                 27.8 
                 0.064 
                 24305.0 
                 21502 
                 23888 
                 18229 
               
               
                 30.5 
                 0.082 
                 24421.0 
                 21618 
                 23970 
                 18303 
                 1st visible 
               
               
                   
                   
                   
                   
                   
                   
                 crack 
               
               
                 36.6 
                 0.154 
                 24919.0 
                 22156 
                 24340 
                 18675 
               
               
                 39.9 
                 0.203 
                 25217.0 
                 22493 
                 24590 
                 18941 
                 2 nd  visible 
               
               
                   
                   
                   
                   
                   
                   
                 crack 
               
               
                 43.2 
                 0.251 
                 25544.0 
                 22824 
                 24841 
                 19238 
               
               
                 49.2 
                 0.369 
                 26270.0 
                 23622 
                 25496 
                 20030 
               
               
                 54.6 
                 0.540 
                 27147.0 
                 24645 
                 26329 
                 21057 
               
               
                 57.4 
                 0.653 
                 27660.0 
                 25272 
                 26857 
                 21770 
                 punching 
               
               
                   
                   
                   
                   
                   
                   
                 shear 
               
               
                   
                   
                   
                   
                   
                   
                 failure 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Comparison of punching shear test results with ACI provision 
               
             
          
           
               
                   
                 Test results 
                 ACI Equation 
               
             
          
           
               
                   
                   
                   
                 Shear 
                 Nominal shear 
                 Nominal shear 
               
               
                 Specimen 
                 Failure loads 
                 Average failure 
                 perimeter, b 0   
                 stress. 
                 stress. 
               
               
                 designation 
                 Kips/KN 
                 load, Kips/KN 
                 in/mm 
                 psi/Mpa 
                 psi/Mpa 
               
               
                   
               
               
                 PT-1 
                 63.82/283.8 
                 62.21/276.65 
                 28.80/731.5 
                 10.2{square root over (ƒ′ c )}/0.85{square root over (ƒ′ c )} 
                 N.A 
               
               
                 PT-2 
                 60.60/269.5 
               
               
                 PT-3 
                 57.36/255.0 
                 50.30/223.60 
                 28.80/731.5 
                 8.2{square root over (ƒ′ c )}/0.68{square root over (ƒ′ c )} 
                 8{square root over (ƒ′ c )}/0.66{square root over (ƒ′ c )} 
               
               
                 PT-4 
                 43.21/192.2 
               
               
                   
               
               
                 Conversion Factors and Notations: 1 in. = 25.4 mm; 1 kip = 4.448 N; 1 ksi = 6.895 Mpa 
               
               
                 A c  = area of critical section = b 0 d 
               
               
                 b 0  = perimeter of critical shear section at a distance d/2 from the column face. 
               
               
                 d = distance from the extreme compression fiber to the centroid of tension reinforcement 
               
               
                 ƒ′ c  = concrete compressive strength 
               
               
                 N.A = not applicable 
               
             
          
         
       
     
         [0028]    The hairpin assembly reinforcements described herein may be set in the concrete structure by placing them in a light-weight plastic support base, such as a stool, or “chair”, and setting the support base and the hairpin assembly reinforcements on wood forms of the type commonly used commercially to set concrete and concrete reinforcements. Normally, the assembly can be secured to the support base by means of wires and the support based nailed or otherwise fastened to the wood forms before pouring the concrete. The techniques for embedding reinforcing devices in concrete are well known to those skilled in art of concrete pouring and handling, and any of a number of such techniques may be used for this purpose. The reinforcing assemblies of the present invention may also be placed in the post-tensioned anchorage zone of the reinforced concrete structure. 
         [0029]    While the present invention has been described in terms of particular embodiments and applications, in both summarized and detailed forms, it is not intended that these descriptions in any way limit its scope to any such embodiments and applications, and it will be understood that many substitutions, changes and variations in the described embodiments, applications and details of the method and system illustrated herein and of their operation can be made by those skilled in the art without departing from the spirit of this invention.