Shear design of high strength concrete prestressed girders

Abstract

Normal strength prestressed concrete I-girders are commonly used as the primary superstructure components in highway bridges. However, shear design guidelines for high strength PC girders are not available in the current structural codes. Recently, ten 7.62 m (25 feet) long girders made with high strength concrete were designed, cast, and tested at the University of Houston (UH) to study the ultimate shear strength and the shear concrete contribution (V_c) as a function of concrete strength ({f^{\prime }}_{\mathrm{c}}). A simple semi-empirical set of equations was developed based on the test results to predict the ultimate shear strength of prestressed concrete I-girders. The UH-developed set of equations is a function of concrete strength (\sqrt{{f^{\prime }}_{\mathrm{c}}}), web area (b_wd), shear span to effective depth ratio (a/d), and percentage of transverse steel (ρ_t). The proposed UH-Method was found to accurately predict the ultimate shear strength of PC girders with concrete strength up to 117 MPa (17000 psi) ensuring satisfactory ductility. The UH-Method was found to be not as overly conservative as the ACI-318 (2011) code provisions, and also not to overestimate the ultimate shear strength of high strength PC girders as the AASHTO LRFD (2010) code provisions. Moreover, the proposed UH-Method was found fairly accurate and not exceedingly conservative in predicting the concrete contribution to shear for concrete strength up to 117 MPa (17000 psi).

Graphical abstract

Keywords

shear design / high strength concrete / prestressed girders / full-scale tests

1 Introduction

Normal strength prestressed concrete (PC) I-girders are commonly used as the primary superstructure components in highway bridges. The developments in new materials and technology in recent years have made it possible to construct and assemble more commercially attractive long-span high strength PC girders. Thus, the use of high strength concrete (i.e., compressive strengths {f^{\prime }}_{\mathrm{c}}>70 MPa or 10,000 psi) has gained wide acceptance in the PC industry. Standardization in the design and manufacturing of the precast bridge components has optimized bridge design. However, design guidelines for high strength PC girders are not available in the current structural codes.

In PC girders design, an adequate margin of safety ensuring gradual or ductile failures must be provided considering various modes of failure. Shear failures, the subject of this paper, are generally considered hazardous because the failures could be brittle and give little or no warning. Moreover, it is known that the brittleness of concrete increases with increasing concrete strength. Hence, the ductility of the shear failure in PC girders made with high strength concrete needs to be ensured. Ductility can be achieved by limiting the transverse steel thus reaching the yielding stress in steel before concrete crushes. But, maximum transverse steel is needed to achieve larger ultimate shear strength of the cross-section. Hence, the optimum amount of transverse reinforcement is required to be provided that ensures higher ultimate ductile shear strength for a given cross-section. The optimum amount of transverse steel can be provided only if the concrete contribution to the shear strength could be well predicted. Generally, shear failures are encountered at two locations: 1) web shear failure near the supports where shear is large and bending moment is small, and 2) flexural-shear failure near the one-third or quarter points of the span where both shear and bending are large or where the bridge girders are continuous over supports which are generally critical in flexural-shear.

The guidelines for shear design in the current structural codes, such as American Concrete Institute (ACI) and American Association of State Highway Transportation Officials (AASHTO) Specifications, are complicated and vary drastically, especially for high strength concrete. For a concrete strength {f^{\prime }}_{\mathrm{c}} of 100 MPa (14,500 psi), AASHTO Specifications allows a V_n,max value to be 2.5 times as large as that in the ACI 318 code. Moreover, there is a dearth of data points (i.e., girder tests) for shear with concrete strength greater than 55 MPa (8,000 psi). Recently, a rational approach (UH-Method) was developed at the University of Houston to estimate the maximum shear strength based on the extensive studies of two-dimensional (2D) membrane elements using the Universal Panel tester and full-scale PC girder tests at the University of Houston [1]. The proposed UH-Method ensured satisfactory shear-ductility for strength of concrete up to 77 MPa (11,000 psi), and were found to be safe and not overly conservative than the current codal provisions.

Nonetheless, full-scale shear girder tests with high strength concrete (i.e., compressive strengths {f^{\prime }}_{\mathrm{c}}>70 MPa or 10,000 psi) are necessary to be investigated so as to establish reliable design guidelines for high strength PC girders and possibly revise the structural codes. This paper presents the experimental program of testing ten girder specimens, compromising of three high strength-groups, to validate the UH-Method for shear of high strength concrete and to review the shear design provisions in current codes.

2 Research significance

The UH-Method [1] for shear design of PC girders, ensured satisfactory shear-ductility for strength of concrete up to 77 MPa (11000 psi), and was found to be safe and not overly conservative than the current codal provisions. However, these guidelines are required to be validated for concrete strengths greater than 77 MPa (11,000 psi). Therefore, large-scale shear tests of PC girders made with high strength concrete (i.e., compressive strengths {f^{\prime }}_{\mathrm{c}}>70 MPa or 10,000 psi) are necessary to be investigated so as to establish reliable design guidelines and possibly revise the structural codes. The main objective of this work was to validate the proposed UH-Method for different strengths of concrete up to 117 MPa (17,000psi). It is believed that the results of this work will be greatly appreciated by the bridge design engineers as it would enable them to design safely, optimally and commercially viable long-span high strength PC girders.

3 Background and literature review

The shear behavior and strength of PC I-girders depend mainly on the web. The web can be considered as 2-D membrane element. For many years, the shear research group at the UH has been working on understanding the behavior and strength of the 2-D membrane element subjected to shear using the Universal Panel Tester.

Using the Universal Panel Tester, a Rotating-Angle Softened-Truss Model (RA-STM) was developed [2,3], which truly treated the cracked reinforced concrete as a smeared, continuous material. This model, like other models that are based on the rotating-angle, could not logically produce the concrete contribution because shear stresses could not exist along the rotating-angle cracks. To predict the “concrete contribution,” Hsu and his colleagues [4–6] proposed the Fixed-Angle Softened-Truss Model (FA-STM). In FA-STM the direction of cracks is assumed to be perpendicular to the principal applied tensile stresses at initial cracking rather than following the rotating cracks. The constitutive laws of concrete are set in the principal coordinates of the applied stresses at initial cracking. The only shortcoming of FA-STM is that it is more complicated than RA-STM because of the complexity in the stress-strain relation of concrete in shear.

Recently, the research group at UH tried to utilize their previous experience and clear understanding of the behavior of 2D element in shear to better estimate the shear capacity of PC girders. Laskar [1,7] used the softened truss model [6] to express the maximum shear strength V_n,max as:

(1)V_{\mathrm{n},\max }={\sigma }_{\mathrm{2}}^{\mathrm{c}}{b}_{\mathrm{w}}\left(\mathrm{0.9}d\right)\sin a_{\mathrm{1}}\cos a_{\mathrm{1}},

where, {\sigma }_{\mathrm{2}}^{\mathrm{c}} is the compression strength of the concrete struts. (0.9d) is the height of the truss measured from the centroid of the steel to the centroid of the compression zone; and a₁ is the angle of the normal to failure surface with respect to the longitudinal axis of the girder (a₁ = 45° when an element is subjected to pure shear).

To develop an expression for V_n,max that is applicable to the whole range of concrete strengths from 20 MPa (3,000 psi) to 100 MPa (14,500 psi), Zhang and Hsu [6] tested full-sized reinforced concrete (RC) panel elements (1.4 m × 1.4 m × 0.178 m) with concrete strengths up to 100 MPa. These extensive panel tests showed that the strength of the concrete struts in the principal compressive direction is “softened” by the perpendicular principal tensile strain \overline{{\epsilon }_{\mathrm{1}}}. Thus, the resulting strength of the concrete struts is function of the concrete compressive strength {f^{\prime }}_{\mathrm{c}} and the principal tensile strain \overline{{\epsilon }_{\mathrm{1}}}. Thus the maximum shear strength, V_n,max, can be given by:

(2)V_{\mathrm{n},\max }=\mathrm{5.8}\sqrt{{f^{\prime }}_{\mathrm{c}}\left(\mathrm{M}\mathrm{P}\mathrm{a}\right)}f\left(\overline{{\epsilon }_{\mathrm{1}}}\right)b_{\mathrm{w}}\left(\mathrm{0.9}d\right)\cdot \mathrm{0.5}.

Equation (2) shows that V_n,max is a function of \sqrt{{f^{\prime }}_{\mathrm{c}}} for concrete compressive strength {f^{\prime }}_{\mathrm{c}} up to 100 MPa (14,500 psi). Equation (2) is simplified as:

(3)V_{\mathrm{n},\max }=C_{\mathrm{1}}\sqrt{{f^{\prime }}_{\mathrm{c}}\left(\mathrm{M}\mathrm{P}\mathrm{a}\right)}\cdot b_{\mathrm{w}}\cdot d,

where C₁ is a constant which was determined by calibrating the balance condition constant C_b based on the shear tests of prestressed girders available in literature and was found to be taken 1.5 [1].

If the beam cross section is designed to be at balanced condition, this means that the steel yields at the same time the concrete crushes giving the limit for design. This mode of failure is ductile and is desirable by engineers. If the concrete crushes before the steel yields, the cross-section is said to be designed as over-reinforced, and the mode of failure is brittle — undesirable by engineers. On the other hand if the steel yields before the concrete crushes, the cross-section is designed to be under-reinforced, and the resulting mode of failure is ductile — desirable by engineers. Thus, in order to achieve a ductile failure mode, the girder cross-section must be designed to be under-reinforced or balanced, and the constant C₁ has to be less than or equal to the balance condition constant C_b Thus, constant C₁ was decided to be taken equal to 1.33 [1]. Therefore from Eq. (3), the maximum shear strength V_n,max can be given by:

(4)V_{\mathrm{n},\max }=\mathrm{1.33}\sqrt{{f^{\prime }}_{\mathrm{c}}\left(\mathrm{M}\mathrm{P}\mathrm{a}\right)}\cdot b_{\mathrm{w}}\cdot d.

The maximum shear strength V_n,max in the case of prestressed concrete girders with only straight strands is the summation of concrete contribution V_c and steel contribution V_s,

(5)V_{\mathrm{n},\max }=V_{\mathrm{c}}+V_{\mathrm{s}}.

Laskar et al. [1] used the shear friction concept put forth by Loov [8] and the available tests on prestressed girders in literature to develop a rational equation for the estimation of the concrete contribution, V_c. From the test results of girders B1 to B5 by Laskar et al. [1,7], the concrete contribution V_c was observed to be a strong function of the shear span to depth ratio (a/d). Based on the analysis of tested girders and those which were in the literature, Laskar et al. [1] presented a new and simple equation to predict the concrete contribution V_c as follows in US Customary units:

(6)V_{\mathrm{c}}={\displaystyle \frac{\mathrm{1.17}}{{\left(a/d\right)}^{\mathrm{0.7}}}}\sqrt{{f^{\prime }}_{\mathrm{c}}\left(\mathrm{M}\mathrm{P}\mathrm{a}\right)}{b}_{\mathrm{w}}d\le \mathrm{10}\sqrt{{f^{\prime }}_{\mathrm{c}}}{b}_{\mathrm{w}}d,

where b_w = width of the web of girder, mm; a = shear span, mm, and d = effective depth from the centroid of the strands to the top compression fiber of the girder, mm.

The value of d is not taken to be less than 80% of the total girder depth.

The steel contribution V_{s^{\prime }} must be based on the observed failure shear crack as it intersects the transverse reinforcement i.e., stirrups. For design, the failure crack can simply be assumed to be inclined at an angle of 45°, similar to the ACI 318 Code. The assumption of a 45° failure shear crack has also been supported by a study of shear energy dissipations in the failure zone [9]. Also, this assumption was supported by the test observation of the flexure shear failure of girder B4 tested by Laskar et al. [1]. In Laskar et al. [1,7], a more realistic concept of seeking a path of minimum shear resistance among a series of individual stirrups was used, as shown in Fig. 1(a) and (b). The minimum number of stirrups intersecting the minimum shear resistance line at 45° is taken as [d/s−1]. This differs from the ACI Code procedure, which is based on the concept of smearing the stirrups, resulting in an average number of stirrups, d/s, crossing the 45° shear crack. Hence, the steel contribution in shear can be calculated as follows:

(7)V_{\mathrm{s}}=A_{\mathrm{v}}{f}_{\mathrm{y}}\left(d/s-1\right).

UH-Method developed at the University of Houston [1] depicted in Eqs. (1) through (7) was used to design the high strength PC girders in the present research. Large-scale shear tests of the designed girders with different class of high strength concrete and different shear span to depth ratio were carried out at UH to validate the proposed equations.

4 Experimental program

4.1 Test girders

The Texas Department of Transportation (TxDOT) currently uses Tx-series of PC girders for highway bridge construction. These girders typically represent the ones used by other Department of Transportation in the USA. The Tx-series girders have a web thickness of 178 mm (7 in.) and depths ranging from 711 to 1829 mm (28 to 72 in.) [14,19]. The Tx28 girder has a wide web compared to its shallow depth, making the web relatively stouter and more compact. The Tx70 girder has a relative thin web compared to its larger depth, making the web very slender and susceptible to end zone failure.

Therefore, to ensure shear failure without local failure in the end regions, Tx54 was studied for this research. Typically Tx-series girders have a top slab with a thickness 203 mm (8 in.) and the minimum spacing between girders is 2032 mm (80 in.), as shown in Fig. 2. In this research an internal Tx54 was considered with a top slab width of 2032 mm (80 in.) as shown in Fig. 3 (a). The resulting girder cross section was scaled down to 43% to form the modified Tx28 girder. The cross section of the modified Tx28 girder is shown in Fig. 3(b).

Ten full-scale modified Tx28 PC girders with a length of 7.62 m (25 ft.) were tested at UH under this research work. Two “Group A” girders (A1 and A2) with concrete compressive strength {f^{\prime }}_{\mathrm{c}} of 49 MPa (7,100 psi); four “Group F” girders (F1 to F4) with concrete compressive strength {f^{\prime }}_{\mathrm{c}} of 91 MPa (13,200 psi); and four “Group C” girders (C1 to C4) with concrete compressive strength {f^{\prime }}_{\mathrm{c}} of 108 MPa (15,660 psi) were investigated in this research. Table 1 shows the concrete mix proportions used for casting the girders in these groups. All the PC girders were cast using local materials typically employed in manufacturing of TxDOT girders, in the state of Texas at two precast plants i.e., Texas Concrete Company, Victoria-TX and Flexicore of Texas, Houston-TX. Texas Concrete Company manufactured the Groups A and F PC girders while Flexicore of Texas produced the Group C PC girders.

For flexural resistance, in girders of Groups A and F, 14 13 mm (0.5 in.) diameter, seven-wire, low-relaxation prestressed straight were used, while in Group C, a 13 mm (0.5 in.) diameter oversize, seven-wire, low-relaxation straight were used to increase the bending moment capacity so as to ensure shear failure of the girders. The prestressing strands had an ultimate tensile strength of 1862 MPa (270 ksi). The locations of prestressing strands and different types of reinforcing steel in the tested girder specimens are shown in Figs. 4 and 5.

Both web-shear failure and flexure-shear failure were investigated in each group of the girders with different concrete strength. The test results of girders made with different concrete strength were then used to validate the UH-Method [1] which determined the concrete and steel shear contributions at different shear span to depth (a/d) ratios of a girder. Girders A1, F1, F3, C1, and C3 were designed to study the behavior of girders in web-shear with shear span to effective depth ratio of 1.77. Girders A2, F2, F4, C2, and C4 were designed to study the flexural-shear behavior of girders with greater shear span to effective depth ratio. The locations of the applied actuator loads and support reactions are shown in Fig. 6.

In this study, structural behavior of PC girders under web-shear and flexure-shear failure modes was investigated. Girder cross-sections were designed to ensure ultimate shear failure mode. Girders A1, A2, F1, F2, C1, and C2 were designed at balanced condition while girders F3, F4, C3, and C4 were designed to have over-reinforced cross-sections. The web of all girders was reinforced in the transverse direction with one legged stirrups namely S-rebars which was fabricated using 16 mm diameter (#5) mild steel bars with ultimate tensile strength of 413 MPa (60,000 psi).

In addition to the transverse direction reinforcement, 16mm diameter (#5) bars were used for U rebars which were designed to resist the end zone bearing, spalling, and bursting stresses. 13 mm diameter (#4) bar was used for C, A, and T rebars; where C rebars were designed to confine concrete and act as secondary reinforcements in the bottom flange, while A and T rebars were designed to be the lateral and longitudinal flexural reinforcement in the top flange, respectively.

Table 2 presents the reinforcement details for all the tested girders.

4.2 Test set-up

The girders were subjected to vertical loading up to their maximum shear capacity to failure in a specially built steel loading frames (Fig. 7). Two of the four actuator frames were each installed on the north and south ends of the PC girder, respectively. These two actuator frames were placed on top of two WF18 × 97 steel girders, bolted securely to the strong floor. The two WF18 × 97 steel girders were 6.1 m (20 ft.) long and spaced at 2210 mm (87 in.) center to center. Two of the four hydraulic actuators attached to the vertical steel frame were used to apply vertical loads on the girders. Each of the two actuators had a capacity of 1423 KN (320 kips) in compression.

PC girder specimen was longitudinally positioned in the middle of the loading frames on top of two load cells placed at the north and south ends (Fig. 7). The load cells of 2224 kN (500 kips) capacity were placed on top of steel pedestals fixed to the strong floor. The load cell readings were used to check force-equilibrium of the loading system throughout the tests. On top of the load cells, bearing plates to support the girders were placed with a roller on the north end and a hinge on the south end, thus allowing the girder to rotate freely at the supports and to expand freely along its length. The actuators were provided with steel bracings for their lateral stability. The position of vertical loads and load-cell supports for all girders of Group A, F, and C are shown in Fig. 6.

Actuator loads were applied through a roller assembly consisting of two 152 mm × 305 mm × 51 mm (6 in. × 12 in. × 2 in.) thick hardened steel bearing plates and two hardened steel rollers of 51 mm (2 in.) diameter and 305 mm (12 in.) in length. This ensured uniform and frictionless load transfer from actuators on to the girder surface. The bearing plates and rollers were heat-treated to maximum possible hardness, to minimize local deformations. Lead sheets were used between the load bearing plates and girder surface to aid in uniform loading.

MTS “MultiFlex” System precisely controlled the applied loads and displacements through the servo-controlled actuators. Each girder was first loaded using the two actuators under a load-control mode at a rate of 9 KN/min (2 kips/min). As soon as the slope of load versus displacement curve for girder being tested dropped, the actuator control-mode was switched to a displacement-control mode at a rate of 5 mm/hour (0.2 inch/hour) until shear failure occurred at either end of the girder. The displacement-control mode was essential in capturing the ductility or brittleness behavior of the girder failing in shear.

4.3 Instrumentations

Strains in both the transverse steel and concrete were measured as the load was applied on the girders. Electrical-resistance foil-type strain gauges were pasted on the surface of transverse steel rebars (i.e., shear steel reinforcement) to measure local maximum strains at critical locations in the girders during the load tests. The strain gauge data obtained during the load tests were used to ascertain the number of transverse steel rebars that may have yielded at the failure shear load.

The girders designed to fail in web-shear had strain gauges installed on transverse rebars along the line joining the points of applied load and the load-cell support. Girders that were designed to fail in flexure-shear had the strain gauges installed on transverse rebars along two lines representing anticipated shear crack direction in the web. The first line was at 45° to the horizontal as per ACI-318 [12] code recommendations, and the second line at an angle “θ” based on the AASHTO LRFD [13] code specification.

To measure the average or smeared strain in concrete within the expected failure region of the girder web, a set of 10 Linear Voltage Differential Transformers (LVDTs) were used in a rosette formation on the east and west faces and either ends of the girder (Fig. 8). For more details the readers are referred to the research report by Labib et al. [19].

A maximum of six LVDTs were placed under the girder to measure the vertical displacement i.e., deflection of the girder during the test. Two of the LVDTs were positioned under the applied load while others were located at the sides of each support, to measure the total and net displacements of the girder, respectively. On average, each girder was instrumented with 44 LVDTs and several strain gauges. Data from all the above discussed sensors were monitored continuously and stored by the HBM “Spider-8” Data Acquisition System (Fig. 7). Shear and flexure cracks formed on the girder during the load test were regularly marked on a grid and crack widths were measured at different load intervals using a hand-held microscope having a 0.0254 mm (0.001 in.) measuring precision.

5 Test results

5.1 Shear force vs. deflection curves

Figures 9–11 show plots of shear force applied on the girder specimen versus the girder net deflection. The shear force plotted in these figures was obtained from the load cell placed under the end supports of the girders. For each end, net deflection was obtained by subtracting the measured deflection by the LVDT placed underneath the girder at the locations of the actuators and the support settlement measured by the LVDT placed beside the support during test.

5.1.1 Group A girders

Two “Group A” girders (A1 and A2) with average concrete compressive strength {f^{\prime }}_{\mathrm{c}} of 49 MPa (7,100 psi) were tested. Figure 9 shows the shear force-deflection curves for Girder A1 and A2 at North and South ends. Although the south end was stiffer than the north end a web-shear failure was induced first at the south end at 502.51 KN (112.97 kips). Because the failure of the south end did not affect the stability of the girder, testing the north end was continued until it had a web-shear failure at a shear load of 614.79 kN (138.21 kips; curves No. 1 and 2 in Fig. 9).

Thereafter, to study the effect of the shear span to depth ratio on the shear behavior and the ultimate shear capacity, girder A2 was loaded with a shear span to depth ratio a/d equal to 3.00. Figure 9 shows the shear force versus net deflection curves for both the north and the south ends, which reflects the identical behavior of both ends starting from the appearance of the first shear crack until web-shear failure at the south end at a shear load of 534.43 KN (120.14 kips). After the south end had failed, the south load cell was moved toward the north end thus reducing the girder span to 5.18 m (17 ft.). Then the north end was reloaded until failure occurred at a shear load of 558.69 KN (125.60 kips).

5.1.2 Group F Girders

Four Group F girders with average concrete compressive strength {f^{\prime }}_{\mathrm{c}} of 91 MPa (13,200 psi) were tested. Girders F1 and F2 were designed with transverse steel close to balanced condition, while F3 and F4 were designed to be over-reinforced in shear. The four girders were designed with different shear span to effective depth ratio a/d. Figure 10 shows the shear force versus net deflection curves for the four girders.

Girder F1 was loaded simultaneously at both ends with shear span to effective depth ratio a/d = 1.77. It was noted that as the applied loads at both ends increased, the south end had more shear cracks than the north end. Hence, it was believed that this would have softened the south end more than the north, which resulted in to a web-shear failure of the south end before the north end under a shear load of 879.37 KN (197.69 kips) (curve 2 in Fig. 10). After the south end failure, the south support was moved to get a net girder span of 6.40 m (21 ft.). The north end was retested and failed finally in web-shear mode at 865.40 KN (194.55 kips) shear force (curve 3 in Fig. 10).

Girder F2 was tested to fail in flexure-shear mode with shear span to effective depth ratio of a/d = 3.00. Both ends behaved identically as shown in curve 4 of Fig. 10. During loading, flexure-shear cracks initiated at both the ends of the girder. Because of the noteworthy increase in the concrete strength for this group, the shear strength increased more significantly than its flexure capacity. Therefore, to avoid the undesired pure flexure failure, it was decided to hold the test and to reduce the shear span resulting into a smaller shear-span-to-depth ratio so as to ensure a flexure-shear failure. Thus, the shear span to depth ratio a/d was then reduced to 2.25. The girder was reloaded simultaneously at both ends having a net span of 6.40 m (21 ft.). The concrete struts formed on the web in the previous run were crushed at the south end indicating web-shear failure at a shear force of 824.48 KN (185.35 kips; curve 6 in Fig. 10). After the failure of the south end, the south load cell was moved to get a net girder span of 4.57 m (15 ft.), and the north end was retested with the same shear span to depth ratio of 2.25. The north end had also a web-shear failure at 868.47 KN (195.24 kips) shear force, (curve 7 in Fig. 10).

Girder F3 was designed to fail in web-shear mode with shear span to the effective depth ratio a/d = 1.77. The south end failed first in web-shear under a shear load of 887.64 KN (199.55 kips; curve 9 in Fig. 10). Consequently, after the failure of the south end, the girder was reloaded after moving the support toward the interior at the failed south end thus resulting in a net girder span of 6.40 m (21 ft.). The north end of the girder failed at a shear force of 858.20 KN (192.93 kips; curve 10 in Fig. 10).

Girder F4 was tested to fail in flexure-shear mode with shear span to an effective depth ratio of a/d = 2.25. The girder test showed that its south end was stiffer than north end. The north end failed first in web-shear mode with ultimate shear load of 706.38 KN (158.80 kips; curve 11 in Fig. 10). Consequently, the girder was reloaded after moving the support toward the interior at the failed north end. At the peak shear load, the south end also demonstrated a web-shear mode of failure, having a maximum shear load of 768.79 kN (172.83 kips; curve 13 in Fig. 10).

5.1.3 Group C girders

Four “Group C” Girders with average concrete compressive strength {f^{\prime }}_{\mathrm{c}} of 108 MPa (15,660 psi) were tested. Girders C1 and C3 were designed to be tested to study web-shear failure similar to Girders F1 and F3 in the previous group. Girders C2 and C4 had a different cross section with a reduced width of top flange and were designed to have a flexural-shear failure mode. Girders C1 and C2 were designed to have transverse steel at the balanced condition, while girders C3 and C4 were designed to be over reinforced shear reinforcement. The four girders in this group were designed with a different shear span to an effective depth ratio a/d.

Curves 1 to 3 in Fig. 11 show the shear load versus net deflection curves at the north and the south ends during the first test and at the north end during the second run. Girder C1 failed first in web-shear mode at the south end under a shear force of 749.53 KN (168.50 kips; curve 2 in Fig. 11). After the failure of the south end, the south support was moved to get a net girder span of 6.40 m (21 ft.). The north end was retested and failed at 834.04 KN (187.50 kips; curve 3 in Fig. 11).

Girder C2 was tested to fail in the flexure-shear mode with a shear span to effective depth ratio a/d = 3.00. The girder failed in flexural-shear mode first at the south end at a shear load of 732.18 KN (164.60 kips; curve 5 in Fig. 11). After the failure of the south end, the south load cell was moved to produce a net girder span of 5.18 m (17 ft.). The north end was retested and failed also in flexural-shear mode at a shear force of 845.61 KN (190.10 kips; curve 6 in Fig. 11).

Girder C3 was tested to fail in web-shear mode with a shear span to effective depth ratio a/d = 1.77. The north end failed first at a shear force of 953.88 KN (214.44 kips; curve 7 in Fig. 11). After the failure of the south end, the south support was moved to achieve a net girder span of 6.40 m (21 ft.). The south end was retested and failed finally in web-shear at shear force of 1015.26 KN (228.24 kips; curve 9 in Fig. 11).

Then girder C4 was tested similarly to girder C2 by simultaneously loading it at both ends using a shear-span-to-depth ratio a/d = 3.00. The first test ended with a flexure-shear failure at the south end under a shear load of 762.87 KN (171.50 kips; curve 11 in Fig. 11). After the south end failure, the girder was reloaded after moving the south support toward the interior at the failed end to yield a test span of 5.18 m (17 ft.). The north end was retested and failed at 862.96 KN (194 kips; curve 12 in Fig. 11).

5.2 Ultimate shear strength

Table 3 summarizes the experimental ultimate shear strength, the test variables and mode of failure for each end of the girders tested. The ultimate strength of each end shown in this table includes half the girder self-weight. The balanced steel contribution toward shear strength (V_s) is determined by Eq. (7). The average strain in transverse steel shown in Table 3 was measured strain during the test using the strain gauges. It was observed that these measured strains in the transverse steel rebars intersecting the failure shear crack were close to the yielding value, thus conforming the assumption of “path of minimum shear resistance” as shown in Fig. 1(a), employed in UH-Method. 2. The experimental results show that UH-Method ensures adequate ductility at the estimated ultimate shear strength, which is required to provide warning before shear failure. The experimental concrete contribution to shear strength (V_c) was calculated by Eq. (5).

Figure 12 shows the variation of the normalized ultimate shear strength versus the concrete strength for girders tested in the present study and other data available in the literature. The figure shows that the ultimate shear strength of the prestressed girders made with normal strength concrete, namely of Bennett and Balasooriya [15], Rangan [16], Ma et al. [17], and the two girders of Group A of this research work, can be closely predicted by all the available code provisions (ACI 318, AASHTO LRFD, NCHRP 549) as well as by the proposed UH-Method [1]. Furthermore, it should be noted that AASHTO and NCHRP 549 equations shown in Fig. 12 are function of d_v instead of d, as it is previously known. To make them comparable to the ACI and Laskar’s equations, it was assumed that d_v = 0.9d.

Figure 12 depicts the following observations as follows; although the AASHTO LRFD [13] formula has the minimum margin of safety in predicting the maximum shear strength for PC girders with low-normal strength concrete, it is unsafe for high strength concrete. NCHRP 549 [18] reduced the maximum shear strength by 28% to be safe in the higher range of concrete strength. This had been done to avoid the high possibility of a local failure in the end zone (D-region) before reaching the ultimate shear strength in the B-region in the case of high strength PC girders. Nonetheless, NCHRP 549 [18] is overly conservative for low-normal strength concrete. Additionally, ACI 318 [12] is noted to be overly conservative in determining the maximum shear strength of PC girders made with low-normal and high strength concretes. The new equations i.e. UH-Method, proposed by Laskar et al. [1] can closely predict the ultimate shear strength of girders for different concrete strength from 30 to 117 MPa (17,000 psi). The UH-Method is less conservative than the ACI-318 [12] equations as well as reliable than the AASHTO LRFD [13] specifications.

Figure 12 shows that the equation provided by NCHRP 549 [18] can well predict the ultimate shear strength up to concrete strength 100 MPa (14,500 psi). However it is able to predict only 93% of the test data in the range of high strength concrete. This is different from Laskar’s Equation (2010) which has almost the same and uniform margin of safety for different concrete strength.

5.3 Strains at failure

Smeared strains during the entire loading process were measured using the LVDT rosette shown in Fig. 8. These strains reflect the entire behavior of the web during the loading process including the first shear crack and the maximum compression strain in concrete struts at failure. Reaching the maximum compression strain in concrete struts ensures reaching the shear capacity of girders at the end of the loading process.

The maximum average LVDT strain was beyond the yield at both ends in girders A2, F1, F2, F3, and at one end of girders A1 and C2. The transverse steel at both ends of girders F4, C1, C3, C4, and at one end of girder C2 did not reach the yield. On the other hand, the average local strain measured using strain gauges at both ends of each girder is presented in Table 4 and was used to predict the steel contribution and concrete contribution, as discussed previously. Thus, the smeared strains measured by LVDT in this work were used to cross-verify the rebar strains locally measured by strain gauges and pinpoint the formation and crushing of concrete struts in the web-shear zone. For more details, readers are directed to Labib et al. [19] and Labib [14] publications.

5.4 Contribution from steel and concrete

The steel contribution V_s to the ultimate shear strength can be calculated based on the method proposed by Laskar et al. [1], by simply assuming the failure shear crack to be inclined at an angle of a 45°, similar to the ACI-318 [12] Code. Assuming the yielding of all stirrups, the steel contribution can be calculated as:

(8)V_{\mathrm{s}}=A_{\mathrm{v}}{f}_{\mathrm{y}}\left(d/s-1\right),\hspace{0.17em}\mathrm{if}\hspace{0.17em}{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{g}}\ge {\epsilon }_{\mathrm{y}}.

In the case of over-reinforced girders in the transverse direction, it is expected that some of the transverse bars cutting the 45° crack might not reach the yielding point. From the strain gauges readings, the local strain in these bars can be known and the average strain can be calculated. Hence, if the average rebar strain is less than the yield strain, the steel contribution, V_S, to the ultimate shear strength can be calculated as:

(9)

The calculation of the experimental concrete contribution can be seen in Table 4. It is based on calculating the steel contribution, V_s assuming the yield stress of the mild steel of 413 MPa (60,000 psi) and the modulus of elasticity of 200,000 MPa (30,000 ksi). By subtracting the steel contribution from the ultimate shear strength of the tested girders V_test, the concrete contribution V_c, can be calculated.

The experimental normalized concrete contribution values for the ten girders tested in this research work with different shear span to depth ratio lie above the conservative curve proposed by Eq. (6) [1] as seen in Fig. 13. Comparison of UH-Method with other published works is also presented in Fig. 13. It is clear that the proposed UH-Method [1] for predicting the concrete contribution of PC girders is conservatively valid for concrete strength up to 117 MPa (17,000 psi).

6 Conclusions

1) The shear behavior of ten PC Modified Tx28 I-girders with different concrete strength was critically examined with web-shear or flexure-shear failure. From their experimental results, the UH-Method [1] can reliably predict the ultimate shear strength of PC girders having concrete strength up to 117 MPa (17,000 psi). Shear design provisions of AASHTO LRFD [13] code overestimate the ultimate shear strength for PC girders with high strength concrete. The proposed UH-Method is not as overly conservative as the ACI-318 [12] code provisions nor is it unsafe as the AASHTO LRFD [13] code for high strength concrete.

2) The experimental results show that UH-Method ensures adequate ductility at the estimated ultimate shear strength, which is required to provide warning before shear failure.

3) The experimental results of PC I-girders presented in this work and others in literature shows that the UH-Method for predicting the concrete shear contribution (V_c) in a PC girder remains valid for high strength concrete up to 117 MPa (17,000 psi) with different shear span to depth ratio and different ratios of transverse steel.

References

[1] Laskar A, Hsu T T C, Mo Y L. Shear strength of prestressed concrete girders part 1: Experiments and shear design equations. ACI Structural Journal, 2010, 107(3): 330–339

[2] Belarbi A, Hsu T T C. Constitutive laws of concrete in tension and reinforcing bars stiffened by concrete. ACI Structural Journal, 1994, 91(4): 465–474

[3] Belarbi A, Hsu T T C. Constitutive Laws of Softened Concrete in Biaxial Tension-Compression. ACI Structural Journal, 1995, 92(5): 562–573

[4] Pang X B, Hsu T T C. Fixed-angle softened-truss model for reinforced concrete. ACI Structural Journal, 1996, 93(2): 197–207

[5] Hsu T T C, Zhang L X. Tension stiffening in reinforced concrete membrane elements. ACI Structural Journal, 1996, 93(1): 108–115

[6] Zhang L X, Hsu T T C. Behavior and analysis of 100MPa concrete membrane elements. Journal of Structural Engineering, 1998, 124(1): 24–34

[7] Laskar A, Wang J, Hsu T T C, Mo Y L. Rational Shear Provisions for AASHTO LRFD Specifications. Technical Report 0−4759−1 to Texas Department of Transportation, Department of Civil and Environmental Engineering, University of Houston, Houston, TX, 2007, 216

[8] Loov R E. Shear Design of Uniformly Loaded Girders. Presented at the Sixth International Conference on Short and Medium Span Bridges, Vancouver, Canada, 2002-July-31

[9] Laskar A. Shear behavior and design of prestressed concrete members. Dissertation for the Doctoral Degree. Houston: University of Houston, 2009, 322

[10] TxDOT 0−4759−1. Rational Shear Provisions for AASHTO LRFD specifications. Technical Report 0−4759−1, by Laskar A, Wang J, Hsu T T C, Mo Y L. Texas Department of Transportation, TX, University of Houston, TX, 2007

[11] ASTM C 494/C 494M−99. Standard Specification for Chemical Admixtures for Concrete. ASTM International, West Conshohocken, P A, 1999, 10

[12] ACI-318. Building Code Requirements for Structural Concrete Commentary. American Concrete Institute, Farmington Hills, Michigan, 2011

[13] AASHTO. AASHTO LRFD Bridge Design Specifications. 4th ed. American Association of State Highway and Transportation Officials (AASHTO), Washington, D C, 2010

[14] Labib E L. Shear behavior and design of high strength concrete prestressed bridge girders. Dissertation for the Doctoral Degree. Houston: University of Houston, 2012, 592

[15] Bennett E W, Balasooriya B M A. Shear strength of prestressed girders with thin webs failing in inclined compression. ACI Journal Proceedings, 1971, 69(3): 204–212

[16] Rangan B V. Web crushing strength of reinforced and prestressed concrete girders. ACI Structural Journal, 1991, 88(1): 12–16

[17] Ma Z J, Tadros M K, Baishya M. Shear behavior of pretensioned high-strength concrete bridge I-girders. ACI Structural Journal, 2000, 97(1): 185–193

[18] NCHRP 549. Simplified Shear Design of Structural Concrete Members. NCHRP Report 549, Transportation Research, 2005

[19] TxDOT 0-6152-2. Shear in High Strength Concrete Bridge Girders. Technical Report 0−6152−2, by Labib E L, Dhonde B H, Howser R, Mo Y L, Hsu T T C, Ayoub A. Texas Department of Transportation, Department of Civil and Environmental Engineering, University of Houston, Houston, TX, 2013, 303

[20] Hernandez G. Strength of Prestressed Concrete Girders with Web Reinforcement. Report, the Engineering Experiment Station, University of Illinois, Urbana, Illinois, 1958, 135

[21] Mattock A H, Kaar P H. Precast-prestressed concrete bridges – 4: Shear tests of continuous girders. Journal of the PCA Research Development Laboratories, 1961, 19–47

[22] Hanson J M, Hulsbos C L. Overload Behavior of Pretensioned Prestressed Concrete I-Girders with Web Reinforcement. Highway Research Record 76, Highway Research Board, 1965, 1–31

[23] Elzanaty A H, Nilson A H, Slate F O. Shear capacity of prestressed concrete girders using high-strength concrete. ACI Journal, 1986, 83(3): 359–368

[24] Kaufman M K, Ramirez J A. Re-evaluation of ultimate shear behavior of high-strength concrete prestressed I-girders. ACI Structural Journal, 1988, 85(3): 295–303

[25] MacGregor J G, Sozen M A, Siess C P. Strength and Behavior of Prestressed Concrete Girders with Web Reinforcement. Research Report, The Engineering Experiment Station, University of Illinois, Urbana, Illinois, 1960

[26] Bruce R N. An experimental study of the action of web reinforcement in prestressed concrete girders. Dissertation for the Doctoral Degree. Champaign: University of Illinois Urbana, 1962

[27] Lyngberg B S. Ultimate shear resistance of partially prestressed reinforced concrete I-girders. ACI Journal, 1976, 73(4): 214–222

[28] Robertson I N, Durrani A J. Shear strength of prestressed concrete T girders with welded wire fabric as shear reinforcement. PCI Journal, 1987, 32(2): 46–61

[29] Shahawy M A, Batchelor B. Shear behavior of full-scale prestressed concrete girders: Comparison between AASTHO specifications and LRFD code. PCI Journal, Precast/Prestressed Concrete Institute, 1996, 41(3): 48–62

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