Investigation of Deep Beams with Various Load Configurations

Investigation of Deep Beams with Various Load Configurations

Brown, Michael D

A series of tests on deep beams was performed to examine the impact of load distribution and shear reinforcement on the behavior of the beams. The specimens were subjected to single or double concentrated loads or uniformly distributed loads. Test results indicate differences in behavior among specimens subjected to different load distributions. The differences were apparent in the cracking patterns, failure modes, ultimate strengths, and strain distributions within the beams. Those differences indicate that distributed loads are a much less severe loading type than concentrated loads. The nominal strengths of the specimens were determined using strut-and-tie modeling provisions of both AASHTO LRFD and ACI 318-05 and then compared with the measured strengths.

Keywords: load; shear; strut-and-tie model.

(ProQuest: … denotes formulae omitted.)

INTRODUCTION

The use of strut-and-tie modeling (STM) has become increasingly popular in U.S. design practice. ACI Committee 3181 added provisions for the use of STM in the 2002 version of the building code. The AASHTO LRFD Bridge Design Specifications2 included provisions for the use of STM in the first edition in 1994.

Concurrent with the new STM provisions in ACI 318, a special publication3 was developed to present examples regarding the use of strut-and-tie models in concrete structures. Similarly, the Portland Cement Association (PCA) issued a set of example problems4 in which the AASHTO LRFD STM provisions were used to design typical bridge structures. Each of the examples in the ACI special publication3 is a structure, or portion thereof, subjected to concentrated loads. Four of the five examples presented in the PCA publication4 are structures subjected to concentrated loads. The fifth example is a footing subjected to a concentrated load on top of the footing and distributed soil pressure on the bottom of the footing.

In an actual building, beams are often loaded through a slab, which distributes the load. Very few examples regarding the use of STM for distributed loading are available. Both Marti5 and Muttoni et al.6 present conceptual examples of applying STM to structures with distributed loads but present no detailed calculations.

A series of tests was performed to investigate experimentally the application of STM to beams with uniform loads. These tests consisted of deep beams loaded with one or two concentrated loads, or with a uniform load. The tests were used to observe behavioral differences, if any, between the various load configurations. Based on the observed behavior, recommendations can be made about modeling a distributed load with concentrated loads for application of STM.

RESEARCH SIGNIFICANCE

A series of beam tests were performed to examine the behavior of struts in beams. The beam test specimens consisted of deep beams subjected to various uniform or concentrated loads. The loading on these members was also placed asymmetrically within the span. The asymmetrically applied distributed loading makes these specimens a unique addition to the technical literature. These beams contained horizontal and vertical shear reinforcement. Horizontal shear reinforcement consists of shear reinforcement parallel to the flexural tension reinforcement. Strain gauges were used to measure strain distributions within the beams to discern behavioral differences between beams subjected to uniform and concentrated loads. Recommendations for the application of STM to structures with distributed loads are also offered.

TEST SETUP

The tests were conducted at the Phil M. Ferguson Structural Engineering Laboratory. The reaction frame is shown in Fig. 1 with a specimen installed for testing. The frame consisted of four columns connected with heavy steel W-shapes to form the crosshead. Each of the columns was attached to the strong floor. The maximum load the frame could withstand was 480 kip (2140 kN). The maximum load was limited to the capacity of the anchors that connected the columns to the strong floor. The reaction frame was post-tensioned to the strong floor to prevent movements of the entire frame during testing.

The reaction frame was designed to allow for different load applications. In Fig. 1, the frame is configured to apply a single concentrated load to the specimen. Note the single hydraulic ram at the center of the frame. The concentrated loads were produced using hydraulic rams capable of applying 200 kip (890 kN).

In the cases where specimens were subjected to uniform loads, the load was produced by 30 identical hydraulic rams (10 kip [44.5 kN] each). A photograph of the uniform load apparatus is shown in Fig. 2. Each of the rams was connected to the same hydraulic manifold with an identical hose and coupler to ensure that the pressure supplied to each ram was identical. During early stages of the research study, pressure transducers were placed on randomly selected rams to verify constant pressure throughout the hydraulic system. The rams were arranged such that a pair of rams acted on a single bearing plate along the top surface of the beam. There was a small gap between adjacent bearing plates on top of the specimen.

EXPERIMENTAL PROGRAM

The basic details of the test specimens are shown in Fig. 3 and Table 1. The beams had a cross section that was 6 x 30 in. (152 x 762 mm) with an effective depth of 27 in. (686 mm). To prevent anchorage failures, the longitudinal bars were anchored with standard hooks as per ACI 318-05. A standard hook, as described by Section 7.1 of ACI 318-05, consists of a 90-degree bend with an extension of 12 times the diameter of the bar beyond the bend. The hook was positioned such that the point at which the bars could be fully developed was outside the exterior edge of the bearing plate. In terms of STM, the tie (longitudinal tension reinforcement) could be fully developed at the vertical face of the CCT node (outer face of the bearing plate) at each support.

In each test, the load was placed asymmetrically within the 10 ft (3048 mm) span. By placing the load asymmetrically, the load in one reaction was increased relative to the other. The location of failure could then be predetermined with some confidence. Therefore, instrumentation could be used more efficiently. If the specimens were symmetric, twice the amount of instrumentation would be needed for essentially the same amount of measured data because only one side of the specimen would fail.

The various load configurations are shown in Fig. 4. For each of the load configurations, failure was expected in the north portion of the beam due to the higher shear force at that section. To further ensure that the desired mode of failure occurred, the bearing plate used at the north support was smaller than the bearing plate at the south support. The smaller bearing plate at the north reaction was 6 x 6 in. (152 x 152 mm) and the larger plate at the south reaction was 6 x 8 in. (152 x 203 mm). The smaller bearing plate on the north end of the specimen resulted in a smaller CCT node adjacent to the support and, consequently, smaller areas at the ends of the struts framing into that node.

Two modes of failure are possible in concrete struts: crushing of the strut and splitting of the strut due to transverse tension. Crushing of a strut is primarily affected by concrete strength and size of the bearing area that, in turn, affects the minimum cross-sectional area of a strut. Splitting of the strut is affected primarily by the transverse reinforcement within the strut. To observe experimentally both modes of failure, the likelihood of both failure types had to be balanced within the specimens. If a large bearing plate, which was more representative of field practice, had been used, transverse strut splitting would have been the only observable mode because the crushing strength would be substantially greater than the splitting strength. Conversely, if only very large amounts of transverse reinforcement were used, only strut crushing would have been observed. Based on previous tests of isolated struts,7,8 the likelihood of a failure within a node is unlikely due to the beneficial effects of confinement within a nodal zone.

Additionally, it was desirable that specimens subjected to various load configurations had similar modes of failure so that the results of the experiments could be compared. Reducing the bearing size at the north reaction limited the possibility that crushing could occur directly beneath the applied load for specimens subjected to a single concentrated load. Crushing directly beneath the applied load is an unlikely mode of failure for a specimen subjected to a distributed loading. Those two modes of failure are quite different and little would be gained by comparing such results.

Each of the 10 specimens can be identified by a unique code. The codes are explained in Fig. 5. The first set of characters indicates the type of loading that was applied to the beam. The loading configurations are shown in Fig. 4. The second and third numbers are the spacing of the vertical and horizontal shear reinforcement in inches, respectively. A value of zero in either placeholder indicates that no such reinforcement was used. Vertical shear reinforcement consisted of No. 3 closed stirrups (f^sub y^ = 73 ksi [503 MPa]; Abar = 0.11 in.2 [71 mm2]), and horizontal shear reinforcement consisted of pairs of straight No. 3 bars (f^sub y^ = 73 ksi [503 MPa]). The horizontal shear reinforcement was distributed along the vertical faces of the members, similar to skin reinforcement that is often used for deep beams. One bar was placed near each face, and the bars spacing is the vertical distance between those bars. Within the 10 tests, there were two nominally identical specimens. Those two specimens are distinguished by the letters a and b. Details of the specimens are given in Fig. 3 and Table 1.

TEST RESULTS

During the tests, a computerized data acquisition system was used to gather and record the data. In each test, both of the support reactions were measured with load cells, and the applied load was monitored using a pressure transducer. The beams were tested such that the reaction at the north end of the beam was the greatest in magnitude. In Fig. 4, the north support reaction is indicated by the arrow at the top of the figure. At the time of testing, the age of the concrete for all test specimens was between 50 and 150 days.

Diagonal cracking and ultimate loads

Periodically during the tests, the load was paused to observe cracks. The load at which cracks occurred was determined by visual inspection then later confirmed using the data from the internal, and when necessary, external, strain gauges. For all tests, the visual observations regarding the loads at which cracks appeared were in good agreement with the electronically gathered data. The diagonal cracking and ultimate loads are listed in Table 1. The diagonal cracking loads listed in Table 1 are not the loads at which the first crack formed but are the loads at which a diagonal shear crack formed in the north portion of the beam. Often, small flexural cracks occurred near the point of maximum moment before the formation of the diagonal crack. The shear forces reported in Table 1 include the shear due to the self weight of the test specimen up to the face of the support. Furthermore, the values of shear listed in Table 1 represent the peak shear carried by the specimen. For design using traditional sectional methods, the shear would be calculated at a distance d away from the face of the support for uniformly loaded specimens. For STM, however, the peak shear is used and the critical section is taken at the location of peak shear.

In sectional design methods, the effects of shear are completely separated from the effects of flexure. Experience has shown that some of the load in a member that is applied near the support flows directly into the support by compression in the web above the inclined shear crack, as described in Section R11.1.3.1 of ACI 318-05. Because this small amount of load has a direct load-path into the support, it is not considered in the shear design of the member. STM explicitly considers the flow of forces near supports and the additional strength associated with the direct load-paths. Additionally, the size of the bearing area directly affects the strength calculated using a strut-and-tie model. Thus, all force acting on that bearing area must be considered. In sectional shear design methods, only the shear crossing the inclined crack emanating from the face of the support is considered. Hence, when using STM, the load near the support can not be neglected as in sectional design procedures.

In the south portion of the specimens, there is a region of the beam that is not within a distance h from a geometric or force discontinuity. Hence there is a B-region in the south portion of the specimens. Conversely, the entire north portion of the beams lies within a distance h of an applied load or reaction, making it a D-region. It was expected that the strength of the south portion of the specimen would be controlled by sectional shear behavior whereas the strength of the north portion would be governed by strut-and-tie action. Before testing, it was believed that the 3-to-1 ratio of the support reactions combined with the larger bearing plate at the south reaction would be adequate to force shear failure to occur in the north portion. This failure location did occur in nine of 10 tests. It should be noted that Specimen UL-17-17 failed in shear in the south portion whereas all other specimens failed in the north portion.

Table 1 indicates the amount of shear reinforcement within the specimen. The specimens contained various combinations of horizontal and vertical reinforcement. To compare the various combinations, the equation presented in ACI 318-05, Appendix A, (Eq. (1)) was adopted. That equation is based on the components of the vertical and horizontal shear reinforcement perpendicular to the strut axis. If the value of is not less than 0.003, the strut is considered reinforced as per ACI 318-05, Section A.3.3.1, provisions.

… (1)

The ratios of shear at which diagonal cracking occurred to shear at the ultimate load as a function of the shear reinforcement in the beam are plotted in Fig. 6. There is not a strong correlation between the diagonal cracking load and the amount of reinforcement crossing the splitting crack. However, the average ratio of shear at which diagonal cracking occurred to shear at the ultimate load for specimens without shear reinforcement appears to be greater than for specimens with large amounts of shear reinforcement. There is a possible explanation for this observation. Reinforcement has little effect on diagonal cracking loads but can increase the ultimate strength. Based on this reasoning, it is possible that the ratio of shear at which diagonal cracking occurred to shear at the ultimate load decreases with increasing vertical shear reinforcement. The load-versus-deflection plots for all 10 specimens are shown in Fig. 7.

Effect of load distribution

Specimens subjected to uniform loads failed at higher shear forces than the beams with one or two concentrated loads, whereas the diagonal cracking loads remained similar (25 to 66% of the ultimate load) between load configurations. There does not appear to be a trend between the amount or type of shear reinforcement and the first diagonal cracking load. The initial crack forms due to elastic stresses. Reinforcement has little effect of the stress distribution in a reinforced concrete structure before cracking. After cracks form, however, the reinforcement significantly affects the stress distributions and strength.

There were two sets of nominally identical specimens tested. The first set of beams was UL-0-0, 2C-0-0, and CL-0-0 (shown in Fig. 8). These beams had no shear reinforcement. Of these three beams, the specimen tested under uniform load carried the greatest shear, while the remaining two specimens failed at nearly the same shear force. It should be noted that Specimens UL-0-0 and 2C-0-0 had approximately the same concrete strength, while the concrete strength for CL-0-0 was somewhat less (Table 1).

For Specimen UL-0-0 (at bottom in Fig. 8), the crack along which shear failure occurred did not propagate toward the centroid of the uniform load. Instead, the crack propagated toward the point at which the shear was zero. For the specimens that were subjected to a uniform load, the point of zero shear was located 45 in. (1143 mm) from the north reaction. That point is marked in the bottom photograph in Fig. 8.

In the tests where two concentrated loads were applied, one of the loads (the load nearer to midspan) was applied at 45 in. (1143 mm) from the north support. That location coincided with the point of zero shear for the specimens subjected to uniform load. In Specimen 2C-0-0, as was the case with Specimen UL-0-0, the failure crack propagated toward the point where the applied shear was zero. The shape of the failure cracks in Specimens UL-0-0 and 2C-0-0 are very similar, whereas the cracking pattern of Specimen CL-0-0 is quite different. The similar cracking patterns for the specimens with uniform or two concentrated loads suggest that these two load distributions are inducing comparable levels of distress in the specimens.

The other set of nominally identical beams (UL-8.5-0a, UL-8.5-0b, 2C-8.5-0, and CL-8.5-0) show a similar trend (Fig. 9). Specimen CL-8.5-0 failed at the lowest shear force of the 10 specimens tested in this series, and Specimen UL-8.5-0a carried the greatest shear force. As the load distribution became more uniform, the shear strength of the beams increased. It is interesting to note that this trend could be observed in beams without shear reinforcement (Fig. 8) and with shear reinforcement (Fig. 9). Of these four specimens, three of them had very similar concrete strengths, but Specimens 2C-8.5-0 had a somewhat greater measured concrete compressive strength.

In Specimens UL-8.5-0a and UL-8.5-0b, all of the cracks propagated toward the point of zero shear as in the companion specimen without shear reinforcement (UL-0-0). Also, the diagonal cracks that formed near the failure load of Specimen CL-8.5-0 had similar location and direction to its companion specimen that contained no shear reinforcement (Specimen CL-0-0).

Specimen CL-8.5-0 (which contained shear reinforcement) failed at a load less than Specimen CL-0-0 (which had no shear reinforcement). Specimen CL-8.5-0 also had stronger concrete than CL-0-0. It was expected that CL-8.5-0 would resist more shear force; however, this was not the case. This counterintuitive result is likely due to the large amounts of scatter associated with the shear strength of reinforced concrete beams. A discussion of the general scatter involved with shear strength testing can be found in Reference 9.

Effect of reinforcement

Photographs of all the specimens subjected to uniform loads can be seen in Fig. 10 and 11. These photographs show the specimens after failure. The specimens shown in Fig. 10 are identical except for the shear reinforcement. The concrete strength within those members was nominally identical. The details of the specimens are given in Table 1 and Fig. 3.

Of the six specimens loaded with uniform load, only two contained no vertical shear reinforcement. Those two specimens (UL-0-0 and UL-0-8.5) carried the least shear of the six beams, although the reduction in shear strength was not large. The specimen with only horizontal shear reinforcement actually carried less shear than the specimen with no shear reinforcement of any kind. These two tests (UL-0-0 and UL-0-8.5) indicate the horizontal shear reinforcement did not positively affect the shear strength of the specimen. Specimen UL-0-0 had a greater concrete compressive strength than did Specimen UL-0-8.5. The difference in concrete strength could account for the lack of large variation in ultimate strength between these two specimens. Specimen UL-17-0, however, should also be considered, as it related to these two specimens. Specimen UL-17-0 had vertical reinforcement that was 1/2 the amount of horizontal reinforcement in Specimen UL-0-8.5 for approximately the same concrete strength. Specimen UL-17-0 carried approximately 15% more load than UL-0-8.5. In these two specimens, a small amount of vertical reinforcement increased strength much more effectively than a large amount of horizontal reinforcement.

Unlike horizontal shear reinforcement, small amounts of vertical shear reinforcement had a significant, positive effect on shear strength of the specimens tested in the study. Specimens UL-17-0, UL-8.5-0a, and UL-8.5-0b all had similar modes of failure. That mode involved concrete crushing at the north reaction. Those same three specimens also carried the three greatest shear forces. Specimen UL-17-0 had only half as much vertical shear reinforcement as Specimen UL-8.5-0, and therefore half of that required by ACI 318-05, Section A.3.3.1, provisions (Eq. (1)), but the same mode of failure resulted. These tests indicate that only a small amount of vertical shear reinforcement may be necessary to change the mode of failure from diagonal tension (Specimen UL-0-0) to concrete crushing adjacent to the node (Specimens UL-17-0). Additional increases in shear reinforcement do not seem to produce any additional shear strength. Specimens UL-8.5-0a and UL-8.5-0b carried peak shear forces of 162 and 143 kip (719 and 634 kN), respectively. Specimen UL-17-0 carried a peak shear force of 143 kip (638 kN). Reducing the spacing of the shear reinforcement from 17 in. (432 mm) (Specimen UL-17-0) to 8.5 in. (216 mm) (Specimens UL-8.5-0a and UL-8.5-0b) did not result in significant increases in strength.

The reason the additional reinforcement present in UL-8.5-0a and UL-8.5-0b compared with UL-17-0 provided little additional strength has to do with the observed failure mode. As mentioned previously, there are two possible failure mechanisms for a concrete strut: crushing or splitting in the transverse direction. Splitting in the transverse direction is controlled by the amount of reinforcement that crosses the splitting crack. There is a critical amount of reinforcement that is needed to maintain equilibrium in a strut after cracking.10 Once the critical amount of reinforcement is provided, the mode of failure changes from splitting in the transverse direction to crushing of the concrete along the strut axis. The strut crushing strength is not affected by transverse reinforcement. These results indicate that for the specimens tested, the critical amount of reinforcement is not greater than the amount of reinforcement provided in UL-17-0.

Strain distributions

For five of the 10 tests, strain gauges were placed on the surface of the beams. The strain gauges were distributed along an axis that was transverse to a line that connected the north reaction and the centroid of the applied loads. The locations of the strain gauges can be seen in the top portion of Fig. 8. For the specimens subjected to a single concentrated load, a strut was expected to form between load point and the support. Strain gauges were placed to measure the distribution of stress across the width of that strut. For the specimens with uniform or pairs of concentrated loads, strain gauges were placed in the same positions to compare the distribution of strains among the specimens.

Strain distributions from Specimens CL-0-0, 2C-0-0, and UL-0-0 are shown in Fig. 12. In the figure, the negative values of distance from the strut axis correspond with strain gauges below and to the left of the strut axis, that is, strain gauges closer to the extreme tension fiber. The strain distributions shown represent strains measured at the ultimate load. For the specimens with single concentrated loads, the peak strain was measured on the strut axis and the magnitude of the strain decreased with distance from the axis. This strain distribution suggests the presence of a single strut between the load point and the support reaction.

The strain distributions from beams loaded with two concentrated loads or a uniform load were very different from the distributions produced by single loads (Fig. 12). There was no sharply defined peak in the distributions from beams with double or uniform loads. Rather, the distributions were largely uniform with the exception of the large tensile strains recorded by the surface strain gauges nearest the tension face of the beam. The lack of a distinct peak in the strain distributions of Specimens 2C-0-0 and UL-0-0 shown in Fig. 12 indicates that a single strut did not form between the centroid of the applied loads and the support. If such a strut had formed, the strain distributions for those specimens would have been much more similar to that of Specimen CL-0-0. These data suggest different loadcarrying mechanisms, or strut-and-tie models, for beams loaded with single concentrated loads and beams with multiple loads, for example, two loads or a uniform load, are different. Only the distributions for specimens without shear reinforcement are shown in Fig. 12, but these distributions are typical of all specimens regardless of shear reinforcement.

The strain distributions from the specimens loaded with a pair of concentrated loads or a uniform load indicate that a larger portion of the beam is carrying load than the specimens with only a single concentrated load. The uniform load case may be a less punishing load case because the strains are distributed to a larger portion of the beam. The uniformly distributed load is likely generating a compression fan between the point where the loads are applied and the support. This compression fan causes reduced strain over a larger area as compared with a single strut, as in the case of a concentrated load. This distribution of strain along with the reduction in strain mobilizes a larger portion of the specimen to resist the applied loads. Based on the experimental data presented herein, it is difficult to determine precisely where the border between multiple diagonal struts and a compression fan lies. To make such a determination, more tests of specimens with additional load configurations are necessary.

As shown in Fig. 8, the main diagonal crack in the north portion of the specimens passed between the stain gauge in the center of the pattern and the gauge just below the center gauge. In Specimen 2C-0-0 only, there was a crack near the strain gauge nearest the tension fiber of the specimens. For Specimens UL-0-0 and 1C-0-0, the region near the surface-mounted strain gauge nearest the extreme tension fiber remain uncracked.

Observations from tests

The failure of several of the test specimens involved concrete crushing adjacent to the support. Before failure occurred, however, two nearly parallel shear cracks were observed in many of the tests. The formation of the parallel shear cracks occurred between 60 and 80% of the ultimate load. This type of parallel diagonal cracking was indicative of impending shear failure in the test specimens regardless of the amount of shear reinforcement present. Strain gauges were placed on the longitudinal reinforcement at the location of maximum moment within the test specimen. Readings from those gauges indicate that the longitudinal bars did not yield during any of the tests.

The strain distribution in a deep beam subjected to a single concentrated load showed a distinct peak on the axis of the strut and reduced strain away from that axis. As the load distributions became more uniform, the strain distribution also became more uniform, indicating the presence of a different load-carrying mechanism for beams with multiple or uniform loads.

During testing, two modes of failure were observed among the test specimens: 1) crushing of the diagonal strut abutting the north reaction; and 2) splitting of the diagonal strut abutting the north reaction with the exception of Specimen UL-17-17, which failed in the south portion. Crushing of the strut, when it occurred, happened at the interface between the CCT node above the north reaction and the diagonal strut. Crushing of the strut occurred primarily in specimens with distributed loading, heavy shear reinforcement, or both. Strut splitting is characterized by a large diagonal crack that approximately follows the axis of the diagonal strut. When this mode of failure occurred, there was relatively little additional damage to the specimen. This mode occurred primarily in specimens with concentrated load, little shear reinforcement, or both.

APPLICATION OF CODE STM PROVISIONS TO TEST SPECIMENS

A truss model is any mathematical model of a structural element that uses truss behavior as its basis. A strut-and-tie model is a truss model that is applied to a fully cracked concrete structure that is capable of undergoing plastic deformation. All of the specimens tested fall under the STM provisions of both ACI 318-05 and AASHTO LRFD Bridge Design Specifications. Each of those codes was used to determine the nominal capacity of the specimens, that is, without strength reduction factors f. This nominal capacity was then compared with the measured capacity.

The same strut-and-tie models were used for calculations based on ACI 318-05 and AASHTO LRFD. Three different strut-and-tie models were used for the three different load distributions. The trusses upon which the strut-and-tie models were based are shown in Fig. 13. In each case, the critical element of the various strut-and-tie models was the interface between the CCT node above the north reaction and the abutting diagonal strut. That particular CCT node and strut were, based on design calculations, critical regardless of the load distributions. Based on the calculations performed in accordance with code provisions, all of the specimens were expected to have the same mode of failure: crushing in the strut adjacent to the CCT node above to the north reaction.

For these test specimens, 3/4 of the applied load must flow to the north support reaction based on static equilibrium. As can be calculated from the data presented in Table 1, the ratio of applied load to load measured at the north reaction ranged between 0.73 and 0.76. Consequently, the ratio of the north reaction to the south reaction varied from 2.70 to 3.16. The node formed above that reaction was subjected to the maximum force in the test specimens, and that force was confined to the minimum area. The strut framing into the node above the north reaction is the strength-limiting portion of the strut-and-tie model. A node at its abutting strut(s) must share a common face with a single numerical value of area for that face. Using the provisions of ACI 318-05 or AASHTO LRFD, it is rare that the effective stress in a nodal region is less than the effective stress in the abutting strut. Hence, the strut that carries the load toward the north reaction is critical regardless of the strut-and-tie model chosen.

The simplest of the three trusses is that which was used to model the specimens subjected to a single concentrated load (Fig. 13(b)). For that model, the load was carried to the north reaction by a direct strut that connected the loading point with the north reaction. Between the loading point and the south reaction, a more complex truss was used. That truss included two vertical ties that allowed for the formation of a multiple panel truss rather than a single strut at a very shallow inclination, connecting the loading point with the south reaction. Because the direct strut abutting the north reaction was inclined at approximately 45 degrees, it was unlikely that vertical ties would be needed.

For the specimens subjected to two concentrated loads, a slightly more complex truss was needed (Fig. 13(c)). The north reaction supported 3/4 of the total applied load. Some of the load applied at the inner load point must, therefore, flow to the north reaction along with the entire load applied at the outer loading point. This load distribution gives rise to the funicular arch that is shown to carry the load toward the north reaction. For the portion of the truss nearest the South reaction, the number of vertical ties was reduced from two to one so that the struts would have an inclination of approximately 45 degrees.

Some explanation of the truss model used for uniformly loaded specimens (Fig. 13(a)) is warranted. Based on the strain distributions presented in the preceding section, two concentrated loads were chosen to model the uniformly distributed load. To maintain a simple model, the two loads that model the uniform load were not equal to one another. The ratio of the two loads was made equal to the ratio of the two support reactions. As such, the right-hand point load was set equal to 3/4 of the total applied load, and the left-hand load was equal to 1/4 of the applied load. The concentrated loads were then placed at the centroids of the portions of the uniform load they were intended to model. By dividing the uniform load in this manner, there was no need for the funicular arch that was used for the specimens subjected to two concentrated loads. The strut supporting 3/4 of the load can be thought of as a strut acting at the centroid of a compression fan.

Application of ACI 318-05 STM provisions

The STM provisions of ACI 318-05 allow for the method to be used on specimens without shear or crack control reinforcement. If such reinforcement is present, the effective stress in struts is increased compared with struts without reinforcement. Strut-and-tie modeling is a plastic design method and, as such, it requires the ability for plastic redistribution within a structure. The ability of members with little or no reinforcement to undergo plastic redistribution is questionable. Because the ACI 318-05 provisions allow the application of STM to specimens without shear reinforcement, however, all 10 specimens will be modeled using those provisions.

To determine the capacity of the test specimens as per ACI 318-05 recommendations, only the inclined strut that frames into the north reaction must be checked. All of the specimens were designed such that the inclined strut was the critical element in the STM.

As per ACI 318-05, the effective compressive strength in a strut is determined by

f^sub ce^ = 0.85β^sub s^ f^sub c^sub ‘ (2)

The effective compressive stress in a strut, f^sub ce^, was then multiplied by the area of the end of the strut. The area of the end of the strut is equal to the area of the inclined face of the CCT node. A schematic representation of the CCT node can be seen in Fig. 14. For all of the specimens tested, l^sub b^ = 6 in. (152 mm) and ^t^ = 6 in. (152 mm). The angle of inclination θ varied based on the load distribution on the specimen in question (Fig. 13). The node shown in Fig. 14 is based on provisions of both AASHTO LRFD and ACI 318-05. These provisions produce the minimum possible area over which stresses can act for given values of l^sub b^ and w^sub t^. This minimum area is also perpendicular to the axis of the strut abutting the node such that it is assumed that all stress are perpendicular to the face of the node. The node was assumed to act over the full width of the cross section (b = 6 in. [152 mm]). Based on the concrete strength, nodal area, and efficiency factor, which was a function of the reinforcement in the strut, the strength of the critical element of the STM can be determined. The total loads allowed on the specimens as per ACI 318-05 are shown in Table 2.

Application of AASHTO LRFD STM provisions

The STM provisions within AASHTO LRFD are more complex than those in ACI 318-05. The basic equations used in AASHTO LRFD STM provisions are repeated herein

… (3)

ε^sub 1^ = ε^sub s^ + (ε^sub s^ + 0.002) cot2 α^sub s^ (4)

The three inputs into the aforementioned equations are the inclination of the strut, concrete compressive strength, and the strain in the direction of the tie. The inclination of the strut was based on the strut-and-tie model used, and the tie strains were measured during the test. Tie strains were measured by electric resistance strain gauges that were placed on the longitudinal reinforcing bars at the center of the CCT node. The measured strain was somewhat different than that implied by AASHTO LRFD. The strains required for input into the AASHTO LRFD STM design procedures should be average strains across cracked concrete. Even though different measures of strain are used in this document and by the authors of the AASHTO LRFD strut-and-tie modeling provisions, comparisons can be made.

In design practice, the tensile strain ε^sub s^ should be calculated based on the factored forces in the reinforcement comprising the tie. For a laboratory specimen, however, there are no design loads that can be factored to calculate the appropriate value of strain. Furthermore, if the tie is designed such that it will yield under factored loads, the strain in that tie is incalculable based on the simple constitutive model that is used for conventional reinforcement. If the stress in the reinforcement is exactly equal to the yield stress, the strain in that bar can range between the strain at first yield (ε^sub y^ [asymptotically =] 0.002 for Grade 60 reinforcement) and the strain at the onset of strain hardening in the reinforcement (ε^sub shq^ [asymptotically =] 0.01).

The node geometry used by AASHTO LRFD is identical to that which is used by ACI 318-05 (Fig. 14). The results of the AASHTO LRFD STM-based design calculations are shown in Table 2.

The allowable loads calculated using AASHTO LRFD are slightly less conservative than their ACI 318-05 counterparts. However, there was still a wide margin between the predicted and measured capacities. The coefficients of variation of the ratio of measured to predicted strength for ACI 318-05 and AASHTO LRFD are only slightly different. The increased complexity used in AASHTO LRFD did not result in increased accuracy within these 10 specimens, but the AASHTO LRFD provisions do appear to be somewhat less conservative than the ACI 318-05 specifications.

It should be noted that none of the 10 specimens tested satisfied the required crack control reinforcement stipulated in Section 5.6.3.6 of AASHTO LRFD Bridge Design Specifications. That requirement specifies a reinforcement ratio of 0.003 in the horizontal and vertical directions.

SUMMARY AND CONCLUSIONS

This experimental program was used to examine the effects of load distribution and shear reinforcement on the strength of deep beams. Additionally, the strength of these specimens was calculated with a simple strut-and-tie model based on the provisions of two U.S. codes. The use of the STM provisions from both ACI 318-05 and AASHTO LRFD Bridge Design Specifications resulted in conservative estimates of strength for all 10 specimens. Strut-and-tie modeling conforms to the lower-bound theory of plasticity. Therefore, its application should result in conservative estimates of strength. Based on the test data developed in this study, the application of STM does produce conservative values of strength.

The results of these tests indicate that for modeling a uniform load with STM, two concentrated loads may be adequate, depending on the details of the member. The experimental results indicate that different load-carrying mechanisms are present for different load distributions. For a single concentrated load, a direct strut forms between the applied load and the nearest support reaction. In the case of a distributed loading, a compression fan likely forms. The compression fan focuses the distributed applied loads toward a reaction point. For these tests, two concentrated loads appeared to be sufficient to generate the same load-resisting mechanism present with a true uniform load, as witnessed by the similarity in measured strain distributions between the 2C and UL loading cases (Fig. 12).

The number of loads, however, cannot be the only criterion for determining an appropriate model for a uniform load. As the length of the beam increases, two loads become less similar to a uniform load. The spacing of the loads must also be considered. Certainly if the spacing of the applied loads is less than the depth of the member, the load is sufficient to be considered uniform. The upper limit on load spacing to generate a uniform load, however, needs to be investigated.

In most of the tests, shear failure was preceded by the formation of a shear crack that was nearly parallel to the first diagonal shear crack. The second of the parallel cracks formed between 60 and 80% of the ultimate capacity of the specimen. Failure of these specimens was then caused by concrete crushing between the two cracks. That type of failure is best illustrated by the photograph of Specimen UL-0-8.5 in Fig. 10. For specimens with small shear span-to-depth ratios, parallel diagonal cracking may be an indicator of impending shear failure. Further experimental observations must be made to examine the full meaning of the formation of parallel shear cracks in a variety of specimen types. The few tests described in this paper are not sufficient to make a firm conclusion regarding the importance of the formation of parallel shear cracks.

For the 10 specimens tested in this series, the addition of horizontal shear reinforcement did not seem to significantly affect the shear strength of the specimens. However, small amounts of vertical shear reinforcement (approximately 1/2 of that required using ACI 318-05 STM provisions and 1/3 of that required by AASHTO LRFD STM provisions) were enough to change the mode of failure from diagonal tension to strut-crushing of the strut adjacent to the support.

ACKNOWLEDGMENTS

The authors would like to thank the Texas Department of Transportation for providing financial support for this research program, and the guidance of Project Supervisor D. Van Landuyt is gratefully acknowledged. Opinions, findings, conclusions, and recommendations in this paper are those of the authors.

NOTATION

f’^sub c^ = specified compressive strength of concrete

f^sub ce^ = effective compressive strength in strut

f^sub cu^ = usable compressive strength

l^sub b^ = length of bearing area defining node

V^sub cr^ = shear force at which diagonal cracking in north portion of specimens occurred

V^sub max^ = maximum shear force carried by north portion of specimens

w^sub t^ = width of tie in strut-and-tie model

α^sub i^ = angle between axis of strut and i-th layer of reinforcement crossing that strut

α^sub s^ = smallest angle between compressive strut and adjoining tie

β^sub s^ = strut efficiency factor

= 0.75 for bottle-shaped struts satisfying Eq. (1)

= 0.60 for bottle-shaped struts not satisfying Eq. (1)

ε = tensile strain in concrete in direction of tension tie

θ = angle formed between axis of strut and adjoining tie

REFERENCES

1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05),” American Concrete Institute, Farmington Hills, Mich., 2005, 430 pp.

2. American Association of State Highway and Transportation Officials, “AASHTO LRFD Bridge Design Specifications,” 2nd Edition, Washington, D.C., 1998, 1253 pp.

3. Reineck, K., ed., Examples for the Design of Structural Concrete with Strut-and-Tie Models, SP-208, American Concrete Institute, Farmington Hills, Mich., 2002, 242 pp.

4. Mitchell, D.; Collins, M. P.; Bhide, S. B.; and Rabbat, B. G., “AASHTO LRFD Strut-and-Tie Model Design Examples,” Portland Cement Association, Skokie, Ill., 2004, 65 pp.

5. Marti, P., “Basic Tools in Reinforced Concrete Beam Design,” ACI JOURNAL, Proceedings V. 82, No. 1, Jan.-Feb. 1985, pp. 46-56.

6. Muttoni, A.; Schwartz, J.; and Thürlimann, B., Design of Concrete Structures with Stress Fields, Birkhäuser, 1997, 143 pp.

7. Brown, M. D.; Sankovich, C. L.; Bayrak, O.; Jirsa, J. O.; Breen, J. E.; and Wood, S. L., “Design for Shear in Reinforced Concrete Using Strut-and-Tie Models,” Report No. 0-4371-2, Center for Transportation Research, The University of Texas at Austin, Austin, Tex., Apr. 2006, 354 pp.

8. Brown, M. D.; Sankovich, C. L.; Bayrak, O.; and Jirsa, J. O., “Behavior and Efficiency of Bottle-Shaped Struts,” ACI Structural Journal, V. 103, No. 3, May-June 2006, pp. 348-355.

9. Brown, M. D.; Bayrak, O.; and Jirsa, J. O., “Design for Shear Based on Load Configurations,” ACI Structural Journal, V. 103, No. 4, July-Aug. 2006, pp. 541-550.

10. Brown, M. D., and Bayrak, O., “Minimum Reinforcement for Bottle-Shaped Struts,” ACI Structural Journal, V. 103., No. 6., Nov.-Dec. 2006, pp. 813-821.

ACI member Michael D. Brown is a Staff Engineer at Whitlock, Dalrymple, Poston, and Associates, Austin, Tex. He received his PhD, MSE, and BSCE from the University of Texas at Austin, Austin, Tex., in 2005, 2002, and 2000, respectively. He is a member of Joint ACI-ASCE Committee 445, Shear and Torsion.

ACI member Oguzhan Bayrak is an Associate Professor of Civil Engineering at the University of Texas at Austin. He is a member of ACI Committees 341, Earthquake Resistant Concrete Bridges; and E803, Faculty Network Coordinating Committee; and Joint ACI-ASCE Committees 441, Reinforced Concrete Columns; and 445, Shear and Torsion.

Copyright American Concrete Institute Sep/Oct 2007

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