Development of a Novel Apparatus to Determine Multiaxial Tensile Failure Criteria of Bridge Repair Materials

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This paper details the development and proof of concept testing for a novel apparatus capable of testing brittle materials in multiaxial tensile states, allowing the development of more robust failure models.

Abstract

Aging infrastructure is increasingly costing taxpayers due to increased repair and replacement costs. Ultra-high performance concrete (UHPC) has recently been recognized as a viable material for both the repair of concrete and steel infrastructure as well as a replacement material for new structures due to its enhanced mechanical and durability properties. Such uses require a much better understanding of the multiaxial tensile properties of UHPC to utilize the material more efficiently. This study focused on developing a novel apparatus capable of subjecting specimens to tensile forces in each of the three principal directions simultaneously. Such an apparatus could collect data for a portion of the failure surface that currently only has a small dataset to establish trends. The “Looney Bin” was designed to test 50-mm cube specimens in triaxial tension, biaxial tension, tension-compression, and tension-tension-compression stress states. Once the apparatus and fixtures were designed and fabricated, trial tests were conducted on a non-proprietary UHPC without steel fibers to establish a test method for each of the stress states evaluated. Data were then collected for different stress states using the established procedures and plotted against previously published failure models for UHPC to verify that the collected data were reasonable.

1. Introduction

Ultra-high-performance concrete (UHPC) has been shown to have far superior mechanical and durability properties relative to conventional concretes [1,2]. Due to this performance, the research community has been increasingly interested in understanding its behavior to more efficiently use UHPC in conventional construction. The increased cost of proprietary UHPC mix designs has limited its use to applications requiring relatively small quantities of the material, such as joint material for bridge elements [3,4] and overlays to existing bridge decks [5]. However, the development of non-proprietary mix designs with comparable performance to proprietary mix designs [6,7] has made it more economical to use UHPC as a direct replacement for conventional concrete in structural applications.

A requirement for using UHPC as a full structural element is accurately modeling its response in the structure for design purposes. Accurate analysis of structures requires failure criteria to predict when a material will fail in the analysis model. Numerous studies have been conducted to determine the failure criteria of UHPC in biaxial [8,9] and triaxial compression stress conditions [10,11]. Ritter and Curbach have also collected comprehensive failure data for UHPC in multiple stress states [12] and used that data to create a series of equations to better form the concrete’s failure surface [13].

Noticeably absent is data representing portions of the failure surface with two or more principal stresses in tension. In the case of conventional concrete, the absence of this data did not hinder design and analysis efforts since the tension strength is typically ignored due to its relatively low value when compared to its compressive strength [14]. However, UHPC has a much higher tensile strength, especially when incorporating steel fibers. Ignoring the tensile behavior of UHPC could be considered overly conservative, may not accurately depict behavior, and would ignore the contribution of the most expensive component of UHPC—the steel fibers. Therefore, a new test method is necessary to fill the gap in tensile data that would better inform failure models. With multiaxial tensile data, a complete failure surface could be developed to help improve the understanding of UHPC behavior as well as the analysis and design of UHPC structural elements. This study focuses on designing, fabricating, and evaluating a new test setup able to apply tension to materials (specifically concrete) in all three principal directions simultaneously to provide additional data for the material’s failure surface.

2. Concept Development

2.1. General Analysis of the Problem

Before the development of the test setup, a specimen geometry had to be chosen. The majority of triaxial testing that has been conducted on concrete has used cylindrical specimens. However, applying tensile forces in more than one direction on a cylindrical specimen is extremely challenging without altering the specimen geometry in ways that create stress concentrations. Due to this difficulty and the need to reduce stress concentrations, cube specimens were chosen for this study. Cubes are ideal for triaxial testing since they provide flat faces in three orthogonal directions for load application.

The next step was determining the specimen size. Ritter and Curbach selected 100 mm cubic specimens [12]. However, the use of a non-standard cube size would create issues with the repeatability of testing and with form construction. Therefore, the specimen size was set to 50 mm × 50 mm × 50 mm due to it being a standard size for compression testing of grouts and mortars. Moreover, since most UHPC mix designs use fibers that are 13 mm long, the 50 mm dimension was still at least three times the size of the largest constituent in the mix, thus meeting typical requirements for fiber-reinforced concrete test specimens. However, using this specimen size limits the fibers to a maximum length of 17 mm per ASTM C1609 [15]. Therefore, an attempt was made during the design of the triaxial test setup to provide the option of testing larger cube specimens. This consideration will create the opportunity for triaxial tension testing of specimens with longer fiber lengths.

Lastly, the tensile force transfer method had to be determined. The application of tensile forces to brittle materials is inherently complicated by the required load transfer mechanisms. For example, in uniaxial testing, tension is applied to steel specimens through textured wedge grips that are designed to compress into the specimen as tension is applied [16]. This load transfer method is feasible due to the general malleability and ductility of steel. If this method were attempted with concrete, its brittle nature would not allow the wedge grip teeth to mold the surface without causing the concrete surface to fracture, thus losing grip strength.

There have been numerous studies evaluating the best method for applying tensile forces to concrete specimens for uniaxial testing [17,18,19,20,21]. The most common method among these studies was applying tension through collars on dogbone specimens. The use of collars to apply the tensile forces would require casting projections with angled faces on each side of the cube specimen for the collar to react against. This method would cause complications in the specimen fabrication since it would require specialized formwork to cast the bell shapes on each face. Moreover, the projections from the cube face would inevitably create stress concentrations on the cube specimen, which are best avoided.

Lepissier was successful in transferring tensile forces through aluminum end caps that were directly epoxied to the ends of dogbone specimens [20]. While the specimen shape provided a larger area for epoxying the end caps than the center area, the epoxy used in that study was not the strongest available (tensile strength of 26.9 MPa), and using a stronger epoxy had the potential to mitigate the bond strength issues. The tapered design not only provided a larger area for epoxy; however, the distance between the epoxy surface and the test surface was far enough to avoid issues associated with confinement by the epoxy. While this is beneficial for determining tensile strength, the bevels required for changes in diameter will inherently create areas of stress concentration, which could potentially cause pre-mature failure. Ritter and Curbach attempted to reduce the effects of confinement by using brush heads for load transfer. However, the researchers were not able to bond directly to the cube face and had to resort to embedding screws to provide better anchorage [12]. The embedded screws potentially created additional stress states near the surface of the test specimen, which could alter the resulting data. Again, the cube specimen could be formed to provide projections on each cube face that create a larger area for epoxy; however, this route was not the most advantageous. Therefore, the method for load transfer would be through metal plates that were epoxied directly to the cube faces using a stronger epoxy than that used by Lepissier [20].

2.2. Design of the “Looney Bin”

After the initial analysis of the problem, the following requirements were set prior to designing the Looney Bin apparatus:

• It must be small enough to be easily transported by one person.
• Since the load application will occur in three orthogonal directions simultaneously, it must have a cube shape.
• It must be easily deconstructed to ensure that attaching and removing test specimens takes a reasonably short amount of time.
• It must be large enough to provide space for the test fixtures needed for tensile force applications.
• It must be possible to adjust the test setup for larger specimen sizes to facilitate future testing of longer fibers.
• It must allow for the application of compressive loads along one of the three axes.

These requirements led to the test setup being constructed out of bolted ASTM B209 [22], Grade 6061 (yield strength of 240 MPa) aluminum plates. Aluminum plates were chosen to reduce the overall weight of the test setup while also making fabrication easier. Using bolts to connect the plates ensured ease of construction and deconstruction. The plate thickness would need to be large enough to accommodate bolts that are tapped through the plate’s thickness, large enough to withstand the anticipated failure loads of the test specimens, and thin enough to keep the weight manageable for one person to maneuver. As a starting point for design, the bolt size was set to 6 mm in diameter. With this bolt size chosen, an initial plate thickness of 13 mm was chosen.

The next step was to design the test fixtures and determine the overall size of the test setup. An anticipated peak tensile load of 22 kN was chosen for the design of the test setup. This load would represent a failure stress of 8.6 MPa for the cube specimens, which is near the failure stresses of the UHPC test by Graybeal and Baby [17]. The load application method chosen involved tightening a threaded rod that would react off the plate wall and be attached to a clevis that was epoxied to the cube. The other side of the cube would then need to be attached to the wall of the test setup to complete the load path. Epoxying directly to the plate wall would not be feasible since the process of running multiple tests would require replacing the plates for each test. Therefore, separate individual blocks of ASTM B221 [23] Grade 6061 aluminum bars that would bolt directly to the plate wall would be epoxied to the cubes to act as plinths. This would allow for easy attachment and removal of test specimens without the need to disassemble the entire apparatus.

The plinths and clevis plates that would be epoxied to two side cube faces were fabricated to have a cross-section that was 0.4 mm shorter in each direction to ensure that there were no interferences at the corners of the cube specimens. The plinth and clevis that would be epoxied to the top and bottom faces of the cube were fabricated to have a cross-section that was 0.4 mm longer in each direction to ensure they were slightly larger than the cube face since this would be the loading direction taken to failure. The direct attachment of flat, metal faces to the cube specimen does inherently create confinement near the surface. However, the forces required for the anticipated failure loads would require as much area for attachment as possible to ensure a bond failure does not occur. The authors decided to allow this confinement to occur since anticipated strains at failure are traditionally small for brittle materials, thus minimizing Poisson’s effect. Moreover, Poisson’s effect states that, when applying tensile forces, the materials will contract in the transverse direction. Confining the specimen from contracting counters a potential strengthening effect from the compression created by this contraction.

No pre-manufactured clevises were found that had the required dimensions; therefore, the clevises were fabricated out of ASTM A36 [24] steel plate stock. The clevises were designed to be attached to a 13 mm ball joint using a steel shank, and the ball joint would attach to the threaded rod. The ball joint size was chosen for its availability and had a tensile strength of 44 kN, which exceeds the anticipated peak load. ASTM A193 [25], Grade B7 threaded rods with a diameter of 13 mm were chosen to simplify the connection to the ball joint. The plinth height was set to 38 mm to allow for the bolts to thread 25 mm into the plinth, and two offset bolts were used to reduce the risk of rotating the plinths when tightening the connection bolts. A total of nine plinths and nine clevises were constructed to allow for the gluing and testing of three cubes during a single session. The American Institute of Steel Construction (AISC) Steel Construction Manual [26] was used to determine the capacities of the steel fixtures, and the yield stress was used to determine the axial capacity of the aluminum plinths to ensure they were adequately sized for the design force. The capacities are shown in

Table 1. Capacities of load application fixtures.

Fixture	Material	Controlling Case	Capacity [kN]
Clevis Plate	ASTM A36	Bolt Bearing (Equ. J3.6b [26])	73
Clevis Plate Weld	ASTM A36	Base Metal (Equ. J2-2 [26])	145
Clevis Pin	ASTM A36	Shear (J3-1 [26])	57
Plinth	Aluminum, Grade 6061	Axial Tension	547
Threaded Rod	ASTM A193, B7	Bolt Tension (J3-1 [26])	56

.

With the plinth sizes chosen, the specimen would have to be attached to a corner of the cube test setup, which provided the benefit of increasing the stiffness of the overall test since the load would be applied close to support (plate wall). The test apparatus was designed to have two plates that are inset on all four sides (labeled left and right plates), two plates that are inset on the top and bottom and directly supported by the left and right plates on the other two edges (labeled side plates), and the top and bottom plates would be directly supported on all four edges by the other four plates. The plates were attached together with three bolts on each edge to provide stability and stiffness. All edges that were inset had 25-mm-deep threaded holes that were centered in the plate thickness. All edges that were supported by a plate edge had 7 mm holes for the 6 mm-diameter bolts.

With the plinth and clevis sizes determined, the interior dimensions required for the specimen and test fixtures could be determined. The length of each piece of the test setup in one direction was as follows: 38 mm plinth, 50 mm specimen, 54 mm clevis, and 38 mm shank thread length for the ball joint. The combined length (excluding epoxy thickness) was 180 mm, which would set the minimum interior dimension of the test setup to 205 mm to provide additional space for a load measuring device. However, fibers with up to 30 mm lengths have been tested in UHPC [18]. This fiber size would set the minimum cubic specimen dimension at 90 mm. Accounting for this new size, the fixture length would increase to 220 mm, with a minimum interior dimension of 245 mm. Aluminum plate stock with a 305 mm width was readily available at the time of fabrication and, using a 13 mm thickness would allow for an interior dimension of 280 mm with inset plate construction. Therefore, the interior dimension was set to 280 mm with an outside dimension of 305 mm. This width would provide a large enough interior section to house cubes with dimensions up to 100 mm without being overly bulky. A plate thickness was determined assuming the plate was simply supported on two edges, the 22 kN design force was applied directly in the center of the plate, and the material was only stressed to its yield stress, which were conservative assumptions. With these assumptions, a minimum plate thickness of 11 mm was determined, which is smaller than the 13 mm thickness assumed when determining the interior clearances.

The final step in the Looney Bin design process was to ensure the plates and bolts were adequate to resist the design force. The bolts chosen were SAE-J429 [27] Grade 5 bolts, having a minimum tensile strength of 827 MPa. The shear strength of the bolts was taken to be 45% of the tensile strength (nominal shear strength in bearing-type connections) per Table J3.3 of the ASCE Steel Construction Manual [25], or 372 MPa. A 6 mm-diameter bolt with 20 threads per inch has a solid cross-sectional area of 20 mm². Therefore, the shear strength per bolt was 7.5 kN and a minimum of three bolts would be required to withstand the 22 kN design load. Each plate that was bolted had a minimum of six bolts; therefore, the bolting pattern was sufficient.

2.3. Fabricated and Assembled Looney Bin

The fabricated and assembled Looney Bin is shown in Figure 1. Examples of the fabricated clevises and plinths are shown in Figure 2 and Figure 3, respectively.

The Looney Bin can be configured to test cube specimens in multiple stress states. This study focused on evaluating the Looney Bin when used to apply uniaxial tension, triaxial tension (TTT), biaxial tension (TT), and tension-tension-compression (TTC) stress states. The TTT stress state is generally the weakest stress state of concrete, and the TTT strength represents the closure (apex) point of the failure surface. The TT stress state is a common stress state in plate bending, and it represents the weakest stress state in the biaxial failure surface. The TTC stress state represents the most damaging stress state for brittle materials due to Poisson’s effect. The drastic reduction in compressive stress as tensile stresses are applied in the two other principal directions can be quantified more clearly using this test setup. Another point about each of these stress states is that they are not well understood, and very little data on these stress states exists for UHPC [8,9,10,11,12,13]. The uniaxial tension stress state was applied by attaching the appropriate fixtures to opposite faces of the cube specimen. The TTT stress state was applied by attaching the appropriate fixtures on all sides of a cube specimen. The TT stress state was applied by attaching the appropriate fixtures to two adjacent sides of the cube specimen. The TTC stress state was applied by attached fixtures in the same manner as the TT test: placing a plinth to support the bottom face of the cube and applying compression on the top face of the cube using a separate compression test machine. It is also possible to apply the biaxial tension-compression stress state with the Looney Bin.

2.4. Data Collection Method

With the apparatus fabricated, the tensile load application method was designed. Earlier, it was stated that the tensile force would be developed by tightening a nut on the threaded rod that would react with the Looney Bin plate wall. A load cell would then need to be constructed to measure the load applied from tightening the nut and connecting the ball joint to the threaded rod. This was facilitated by constructing a load cell using a standard 50-mm-long coupling nut with strain gauges attached to measure the tensile force. Two strain gauges were attached on diametrically opposite sides of the coupling nuts along their centerline to provide an average strain occurring at the coupling nut. The strain gauges and their exposed wires were covered with RTV silicone to help protect them from accidental contact. The strain gauge wires were zip-tied to the coupling nut approximately 13 mm from the strain gauge for strain relief.

The strain gauge readings were collected using National Instruments™ (NI) data collection equipment coupled with the program, LabVIEW™. The strain gauge output was measured by attaching the wires to an NI 9236 quarter-bridge, 350 Ohm strain gauge module. The module was then inserted into an NI CompactDAQ eight-slot USB chassis, which was then directly connected to a computer using a USB cable. Since the Looney Bin was designed to be a closed system, a hole was required through a plate wall to allow the wires to be fed out of the apparatus for data collection.

The strain gauge readings were calibrated to display load using a universal testing machine. Threaded rods were attached to loading platforms using standard nuts, with the coupling nut used to attach the top and bottom threaded rods. The standard nut was then tightened to at least five loads, and the corresponding strains from the coupling nut were recorded. A linear calibration was then fitted to the strains, and the slope of that line was used as a multiplier for the strain data to convert it to load. This process was then replicated to check the calibration factor and verify accuracy.

The coupling nut-load cell was attached to the ball joint and threaded rod. Each fixture was only threaded 13 mm into the coupling nut load cell to ensure the threaded regions did not overlap with the attached strain gauges. A smaller nut on each of the fixtures was used as a guide to ensure the appropriate amount of thread was inserted into each coupling nut load cell. The guide nut was located 13 mm from the end of the respective fixtures, and then the fixture was threaded into the coupling nut load cell until it reached the nut. Once both fixtures were attached, the guide nuts were tightened on the coupling nut load cell (applying slight compression) to ensure the threaded rod and ball joint did not come loose during testing. Once the coupling nut load cell was calibrated, the Looney Bin fabrication was complete, and trial runs could commence.

3. Evaluation of the Fabricated Looney Bin

3.1. Establishing Specimen Gluing Procedure

Since this test was designed to achieve at least 22 kN in tension, the adhesive used to attach the fixtures to the specimen must have a direct tensile strength of at least 8.6 MPa. To achieve this bond strength, a commercially available, high-modulus epoxy was chosen to attach the test fixtures to the specimens. While a low-modulus epoxy would have been preferable, no readily available product could achieve the desired bond strength. Moreover, a low-viscosity epoxy was chosen to ensure no epoxy leaks down the specimen prior to curing. The chosen epoxy is advertised to have a tensile strength of 34.6 MPa, which is stronger than the epoxy used by Lepissier [20].

Prior to applying any epoxy, the cube specimens were measured in all three orthogonal directions for accurate stress calculations. Initially, each specimen was glued to the looney bin. The epoxy procedure started with applying the epoxy to the three faces of the cube specimen that would be in contact with the plinths (the plinth configuration just prior to cube placement is shown in Figure 3). Painter’s tape was placed on the top of the plinths and side clevises to eliminate the overlapping of epoxy from the top clevis with the side clevises and plinths. Once those three sides were covered with epoxy, the cube was carefully placed on the plinths. Then, the two clevises that would go on the sides of the specimen were coated in epoxy. Pieces of wood were cut to act as stands for the side clevises while the epoxy was curing. Once the two side clevises were put in place, the top clevis was coated in epoxy and carefully placed on top of the cube. The Looney Bin was carefully leveled to ensure the top clevis stayed in place while the epoxy cured. After placement of the top clevis, a scraper was used to remove epoxy from the adjacent fixtures not protected by painter’s tape. Painter’s tape was attached to both the Looney Bin plate wall and the top clevis to help hold the top clevis in place. The specimens were then allowed to cure for 24 h per the epoxy manufacturer’s recommendations. A glued specimen is shown in Figure 4.

One aspect of UHPC that drastically improves its durability is its very low permeability. While low permeability is great for durability, it drastically reduces the epoxy bond strength. The first test was conducted using a cube with no surface preparation on either the cube or fixture faces. This test only reached a stress of approximately 2.8 MPa before an epoxy bond failure occurred. This issue led to the idea of sandblasting the cube and fixture faces prior to gluing. The sandblasting removes any laitance on the concrete faces and helps to create a roughened surface while not doing any internal damage to the specimen. The specimens were then measured after sandblasting. A comparison of a sandblasted cube to the cube as it appears after curing is shown in Figure 5.

3.2. Finalizing the Test Setup

Once the specimens were glued and allowed to cure, they were bolted into the Looney Bin, and the loading assembly consisting of a ball joint, coupling nut load cell, and threaded rod was attached to the clevises using steel pins. A glued specimen with fixtures attached for a TTT test, just prior to sealing the Looney Bin with the top plate, is shown in Figure 6.

The bolts for the top plate were tightened to snugness. Before attempting any tests, concerns were raised about friction development between the nut, washer, and the Looney Bin plate wall during load application. To help reduce friction development, needle-roller thrust bearings sandwiched by equal-diameter thin washers were placed between the plate wall and larger-diameter washers. These washers were designed for slowly rotating equipment and provided a large reduction in friction. A larger washer was used between the tightening nut and the thrust bearing washer to prevent damage to the thrust bearing washers. Moreover, the threaded rod was greased where it contacted the nut to reduce friction between the two sets of threads. The addition of grease was sufficient to allow for smoother tightening without also rotating the threaded rod.

Once the nuts were in place and ready to test, the entire test setup was clamped to a steel table to keep the apparatus from rotating when each nut was tightened. Moreover, aluminum blocks were clamped to either side of the apparatus to provide further resistance to rotation. The completed test setup prior to load application is shown in Figure 7.

The load directions were labeled σ₁ for the load applied through the top plate, σ₂ for the load applied through the side wall, and σ₃ for the load applied through the right wall. Since this test setup would not allow for easy proportional loading, the load was applied in the σ₂ and σ₃ directions (labeled side stresses) first. Load application was started in the σ₂ direction up to the specified stress first, then the σ₃ direction load was applied, and lastly, the load was applied in the σ1 direction until failure, all accomplished by tightening the nut, which reacted off of the plate wall and applied tension to the threaded rod. After each test was completed, the principal stresses were ordered in the conventional manner (σ₁ > σ₂ > σ₃) for reporting. A sample of the collected raw data for a cube with side stresses set to 3.5 MPa (9 kN) and a σ₁ failure load of 18.7 MPa is shown in Figure 8. An example of a failed specimen is shown in Figure 9.

In the initial test attempts, the load in the σ₁ direction was applied quickly until failure. However, to reduce the effect of large stress rates on the failure loads, the load in the σ1 direction was applied in a relatively slow manner. Care was taken to apply the load at a rate less than 0.24 MPa/s. However, since the load application method was manual, the load rates did vary between 0.1 and 0.24 MPa/s. The loads in the σ₂ and σ₃, in three directions, were applied relatively quickly without regard for the load rate. A flowchart showing the steps for the TTT test is shown in Figure 10.

The TT test setup was the same as the TTT test setup. However, loading was applied in two ways. First, the load was applied in a non-proportional manner, similar to the TTT test. A stress was set for one direction, then the load was gradually applied with approximately the same stress rate until failure. Then, a test was conducted where the load was applied to both faces simultaneously, i.e., proportional loading. The same load rate range was kept for the proportional loading case. The TT test setup is shown in Figure 11. A flowchart showing the steps for the TTT test is shown in Figure 12.

The TTC test begins with setting the two tensile side stresses, and then applying the compressive load to failure. Care was taken to keep the stress rate to less than 1.03 MPa/s using a different machine designed for compression load applications. A loading bar was used to ensure the compression tester loading platen did not compress the Looney Bin plate walls. A neoprene pad was placed between the loading bar and the top of the specimen to help apply the compression load more evenly. The tensile side stresses were applied without regard to load rate. The TTC test setup is shown in Figure 13. A flowchart showing the steps for the TTC test is shown in Figure 14.

3.3. Evaluation of the Fabricated Looney Bin

Initial trials for the Looney Bin were conducted on a non-proprietary UHPC without steel fibers. The collected data are shown in

Table 2. Looney Bin Data.

	Side Stresses	No. of Tests Averaged	Mpa
			σ1	σ2	σ3	fc	θ°
TTT	0–0	3	8.50	0.00	0.00	111.70	0.0
	1.38–1.38	2	6.86	1.50	1.47	121.18	0.3
	2.07–2.07	3	6.26	2.15	2.11	111.10	0.5
	2.76–2.76	2	5.28	2.81	2.71	115.74	1.9
	3.45–1.72	3	5.93	3.33	1.84	112.52	21.0
	3.45–3.45	2	8.63	3.42	3.40	107.71	0.1
	3.45–3.45	2	7.07	3.42	3.36	107.71	0.9
	3.45–3.45	2	7.37	3.52	3.43	126.35	1.2
	4.41–2.07	2	5.73	4.07	2.11	99.11	32.7
	6.41–4.41	3	6.74	4.10	4.09	126.35	0.1
TT	Proportional	2	4.94	4.94	0.00	110.59	59.7
	2.41	2	4.62	2.53	0.00	110.59	33.2
TTC	1.38–1.38	2	1.35	1.33	−16.26	120.13	59.9
	1.38–1.38	3	1.37	1.36	−22.85	113.79	60.0
	2.07–2.07	2	2.04	2.00	−28.36	111.84	59.9
	2.07–2.07	2	2.11	2.03	−20.95	113.79	59.8
	2.76–1.38	2	2.70	1.37	−19.44	111.84	56.9
	2.76–2.76	2	2.78	2.67	−14.68	120.13	59.7

. Each result from the Looney Bin was a single data point representing the principal stresses at the failure of the cube. The side stresses at failure were determined by averaging the stress readings between the time of reaching the prescribed stress state and the time at failure. The σ₁ stress was determined by finding the maximum stress. The three principal stresses were then arranged in the appropriate order by magnitude to describe a single test. Three replicate tests were conducted for each stress state, and an outlier analysis was used to remove outliers. Each stress state described in Table 2 is the average of at least two tests of the stress condition listed. If the replicate tests were too dissimilar in results, the data were discarded and the stress state was retested. The tables list the principal stresses at failure, the compressive strength of the concrete at the time of testing determined from a 75 mm × 150 mm cylinder (fc), and the deviatoric polar angle where that data point falls on the failure surface (θ).

Ritter and Curbach developed a novel series of equations that were modified to fit concrete failure surface data in a variety of shapes [13]. After developing these equations, the authors used the UHPC failure data they collected to define parameters for two failure surface shapes. The data used to define the parameters was from fiber-reinforced UHPC and for the following stress conditions: triaxial compression, tension-compression-compression, and biaxial compression. Due to the absence of data for stress conditions with more than one tensile direction, the authors presented two versions of their failure surface with different shapes on the tension end of the failure surface. One shape was parabolic until the point that represents the theoretical triaxial tension strength, labeled iteration No. 1. The other shape creates reverse curvature in the tension region to the point of theoretical triaxial tension strength, labeled iteration No. 7 [13]. The Looney Bin data are plotted against both failure models presented by Ritter and Curbach [13] in Figure 15 and Figure 16. Each data point was divided by its uniaxial compressive strength to facilitate comparison to the failure models.

Overall, the Looney Bin data plots in the expected regions of the failure models, providing evidence that the collected data are reasonable. While reasonably close, neither model predicts the strengths well, which could be due to the models being calibrated using fiber-reinforcing UHPC. The parabolic failure surface (No. 1) appears to underestimate the strength in the tensile region and overestimate the TTC strengths. This implies that the Looney Bin data does not follow a parabolic pattern. The failure surface with reverse curvature in the tension region (No. 7) appears to follow the trend of the data more closely than the parabolic shape. The capacities in the tension region appear to be reasonably well estimated; however, the TTC strengths are also overestimated. However, the parameters of these equations could potentially be optimized to fit the data more closely and develop a failure model for UHPC without fibers. More data would need to be collected to develop such a failure model.

4. Conclusions and Recommendations

This study focused on developing an apparatus capable of testing UHPC in a triaxial tension stress state to further refine the shape of the failure surface in the tension region. Following specific rules to allow for a reasonable size, capacity, and functionality, the Looney Bin test apparatus was designed to test cube specimens in TTT, TT, and TTC stress states. Trial data were collected using UHPC without steel fibers and plotted against existing failure models for UHPC. This comparison showed that the data collected with the Looney Bin was reasonable and could be used to further develop the tension region of the failure surface for UHPC, thus improving failure models to be used during the analysis of full-scale UHPC elements. The following is a list of recommendations for the use of the Looney Bin in future work:

• The Looney Bin should be used to test other materials within the strength limitations of the test setup.
• Additional data should be collected with UHPC-containing fibers to assess test setup performance.
• The data collected should be evaluated against other published models for additional verification of the data and to refine those models for use in finite element analysis.

5. Patents

The Looney Bin is the subject of a pending U.S. patent application.