Shear Mechanism of Precast Segmental Concrete Beam Prestressed with Unbonded Tendons

Abstract

The shear tests are conducted on six precast segmental concrete beams (PSCBs) in this paper. A new specimen design scheme is presented to compare the effects of segmental joints on the shear performance of PSCBs. The failure modes, shear strength, structural deflection, stirrup strain, and tendon stress are recorded. The factors of shear span ratio, the position of segmental joints, and hybrid tendon ratio are focused on, and their effects on the shear behaviors are compared. Based on the measured responses, the shear contribution proportions of concrete segments, prestressed tendons, and stirrups are decomposed and quantified. With the observed failure modes, the truss–arch model is employed to clarify the shear mechanism of PSCBs, and simplified equations are further developed for predicting the shear strength. Using the collected test results of 30 specimens, the validity of the proposed equations is verified with a mean ratio of calculated-to-test values of 0.96 and a standard deviation of 0.11. Furthermore, the influence mechanism of shear span ratio, segmental joints, prestressing force, and hybrid tendon ratio on the shear strength is clarified. The increasing shear span ratio decreases the inclined angle of the arch ribs, thereby reducing the shear resistance contribution of the arch action. The open joints reduce the number of stirrups passing through the diagonal cracks, lowering the shear contribution of the truss action. The prestressing force can reduce the inclination of diagonal cracks, improving the contribution of truss action. The external unbonded tendon will decrease the height of the arch rib due to the second-order effects, causing lower shear strength than PSCBs with internal tendons.

1. Introduction

Precast segmental concrete beams (PSCBs) have become popular construction options in transportation infrastructure, including highways, railways, and urban bridges [1,2]. In industrialized manufacturing, the PSCBs are prefabricated in segments and then post-tensioned with prestressed tendons during on-site assembly. Therefore, they show significant advantages, including ease of construction, faster project completion, and environmental sustainability. Due to their replaceable and non-grouting properties, unbonded prestressing tendons are encouraged for use in PSCBs. However, the unbonded phenomenon is not conducive to controlling the joint opening width once the joint cross-section is decompressed [3,4,5]. Then, the shear force transmission between segments can only be achieved through the limited compression zones, consequently altering the shear transfer mechanism from that of monolithic beams, which causes a decrease in shear strength [6].

To date, extensive research has been conducted on the flexural properties of PSCBs. Yuan et al. conducted three flexural tests of precast segmental box girders with internal and external hybrid tendons [7]. They found that the hybrid ratio of internal and external tendons had a significant influence on bending deformation, flexural strength, and tendon stress increments. When the ratio varied from 3:1 to 1:3, the flexural capacity decreased by 47.8%, and the tendon stress increments reached 30% to 50% of the effective tendon stress. Jiang et al. tested the flexural behaviors of four segmental beams and one monolithic beam [8]. The tests revealed that the failure occurred at the joint cross-section adjacent to the loading point, characterized by obvious concentrated plastic deformation. The flexural capacity of the segmental beam decreased by 41.0% compared to the monolithic ones. Yang et al. [9] and Tan et al. [10] proposed the use of FRP tendons as external or internal unbonded prestressing reinforcements for PSCBs, demonstrating improvements in structural ductility. The initial stiffness was found to be generally the same for PSCBs with different types of strands. In contrast, the secondary stiffness after decompression and joint opening is less for those with lower elastic moduli in FRP tendons. Zhu et al. proposed the application of ultra-high-performance concrete (UHPC) and novel segmental joint forms to enhance the flexural performance of PSCBs [11]. In addition, numerical models, such as solid element models [12,13] and beam truss element models [14,15], were also studied to clarify the effects of unbonded tendons and segmental joints on the flexural behaviors. Overall, the flexural performance of PSCBs has been thoroughly understood with the previous efforts. Generally, the flexural failure of PSCBs occurred at the joint position with the maximum bending moment, and the structural ultimate deformations were larger than those of monolithic ones, causing larger stress increments in unbonded strands. Based on the tested failure modes, simplified formulas have been proposed to predict flexural capacity, guiding structural design [16,17].

The direct shear performance of the shear-keyed joint is another mechanical property of PSCBs that has received extensive attention [18,19]. Jiang et al. have conducted extensive research in this field [20,21,22], including on the direct shear behaviors of dry joints, steel fiber-reinforced concrete joints, and RPC joints. The tests revealed two distinct direct shear failure modes under varying confining stress conditions, and a sequential failure phenomenon of multi-keyed dry joints was observed, resulting in a decrease in shear strength. They proposed that when the AASHTO specification’s direct shear strength calculation formula [23] was applied to multi-key dry joints, a reduction factor of 0.7 should be adopted. Similar conclusions were also obtained by Ahmed et al. [24,25], who put forward that the AASHTO specification’s formula underestimated the direct shear bearing capacity of multi-key epoxied joints by approximately 11.3% and overestimated that of multi-key dry joints by about 19.3%. Further, Pan et al. [26,27] conducted direct shear tests of single and multiple-keyed joints of UHPC to explore improvements in high-performance concrete on the direct shear performance. The test results showed that the shear resistance increased by 23.8% as the steel fiber volume content increased from 1% to 2%. The AASHTO specification’s formula was relatively conservative, and a new prediction formula was proposed based on the mechanism analysis and fitting of test results. Extensive previous research has clarified the failure mechanism of single- and multiple-keyed joints, with a broad consensus. Direct shear strength prediction formulas are also available for single-, dry multiple-, or epoxied keyed joints [28,29].

In particular, the shear failure of PSCBs usually occurred under the combined bending and shear-loading states, so their shear resistance performance was comprehensively affected by both bending and shear effects [30]. The shear failure mechanism of PSCBs was more complex and differed from that of monolithic ones due to the presence of unbonded effects and segmental joints. Li et al. conducted six tests of externally prestressed PSCBs under combined shear and bending loading [31]. In their tests, the loading points were adjacent to the joint cross-sections. Under ultimate states, the joints were first opened, and then the top flange failed under the combined shear and compressive stress. No noticeable diagonal cracks were observed in the test specimens. Addressing this failure mode, they proposed a shear strength prediction formula based on the concrete shear compression strength model. Yuan et al. [32] conducted a study on three hybrid tendon prestressed PSCBs under shear-bending loading, and observed shear compression failure, as well as vertical sliding of the top flange. The failure mode was similar to the flexural ones, and no obvious diagonal cracks could be observed. Jiang et al. [33,34] finished the shear tests of 11 PSCBs with external tendons and 6 PSCBs with hybrid tendons to study the failure modes and shear strength. The shear span ratio and loading locations were considered as experimental variables, and different failure modes observed by Li et al. [31] were noted. The joint adjacent to the loading point was first opened, and then diagonal cracks were formed from the joint to the loading point. Finally, diagonal compression and shear compression failure modes were shown. Due to the discontinuity of the open joints, the shear-carrying capacities of the PSCBs decreased by 11% compared to the monolithic beam. Moreover, the formulas in the design codes [23] were found to be conservative for predicting shear strength, and the strut-and-tie model was recommended as an alternative option. Recently, the shear behaviors of precast UHPC segmental beams were tested by Jiang et al. [35,36]. The results showed that the shear strength can be improved by the application of UHPC and the web stirrups can be replaced, providing convenience for lightweight PSCBs.

Compared with the flexural and direct shear behaviors of PSCBs, the shear behaviors of PSCBs were more complicated. Although some experiments have been conducted to explore failure modes, additional issues remain to be addressed. First, the unfavorable effects of segmental joints on the shear-carrying capacity should be clarified. In a previous study, the impact of segmental joints was examined by comparing monolithic and segmental beams, which may differ in terms of materials, loading, and prestressing force. These differences may compromise the comparative conclusions, as observed in cases where the shear capacity of monolithic beams was lower than that of segmental ones with the same configuration in a previous study. Second, the shear resistance contributions of different components were still unclear, and the shear-carrying mechanism of PSCBs requires further clarification. As shear failure is a brittle mode, it should be avoided as much as possible in structural design. Especially for the shear performance of PSCBs with unbonded tendons, further research is still needed to explore their failure mechanism and clarify the composition of their shear-carrying capacity.

To address these issues, six PSCB specimens are subjected to shear-loading tests in this study. The shear behaviors are tested, and the shear-carrying mechanism is discussed based on the test results. The highlights of this research can be concluded from three aspects. First, a new experimental design is developed to address the effects of segmental joints. The control group is designed at two shear spans of the same beam to specifically examine the impact of the joint on the shear resistance performance. This design can directly reflect the effects of segmental joints and reduce the variability effects of different specimens, as seen in previous studies. In addition, the segmental joint positions, the hybrid tendon ratio, and the shear span ratio are taken as the test variables, aiming to investigate their influences on the shear behavior of PSCBs. Second, the contribution ratio of each component, such as concrete segments, stirrups, and prestressing tendons, in the shear resistance capacity is calculated based on the test results. Furthermore, based on the observed failure modes and analysis of shear resistance contributions, the combined truss and arch model is employed to elucidate the shear mechanism of PSCBs, and simplified shear strength prediction equations are proposed.

2. Experimental Project

2.1. Design of Specimens

A total of six scaled specimens were tested to evaluate the shear behaviors of PSCBs, comprising one monolithic beam and five segmental beams. The structural design, as outlined in the AASHTO-PCI-ASBI Segmental Box Girder Standards [37], is taken as the prototype beam. The cross-section of the specimen is determined by the scale ratio of 6:1. Meanwhile, for the convenience of specimen preparation, the box cross-section is transformed to a T-shape one according to the principle of invariable cross-sectional neutral axis relative position and similarity of the height and web thickness. Figure 1 shows the dimension details of the specimens. The key parameter values of the specimens are shown in

Table 1. The key parameter values of specimens.

ID	Prestressed Tendons		b (mm)	a (mm)	Shear Span Ratio a/h	Effective Prestress (MPa)
	Internal	External				Internal	External
M	1	2	—	600	1.5	641.1	631.1
S-1	1	2	420	600	1.5	618.9	675.9
S-2	1	2	550	600	1.5	602.6	608.3
S-3	1	2	420	800	2.0	678.9	659.4
S-4	1	2	620	800	2.0	663.7	669.4
S-5	0	4	420	600	1.5	—	351.8 (Straight) 654.9 (Draped)

, in which “M” represents the monolithic beam and “S” represents the segmental beam. The test variables primarily concern the shear span ratios (a/h), hybrid ratios of internal to external tendons, and joint locations (b).

The total length of the specimen is 2.7 m, with a simply supported span of 2.5 m. The specimens are designed with the same T-shape cross-section of 0.4 m in height, 0.5 m in top flange width, and 0.1 m in web width. The external tendons are designed as draped profiles with an inclination angle of 6.2 deg, and the internal tendons are designed as straight profiles. For segmental beams, only one segmental joint is placed at the right shear span, and the left shear span is kept continuous to compare the differences in shear force transmission across the joint under the same loading states. The concrete segments are connected with multi-keyed epoxy joints. Every segmental joint has four shear keys with a root height of 50 mm and a wedge angle of 36.9 deg.

The internal and external tendons in the test beam are all A^s 15.2 steel strands with a nominal diameter of 15.2 mm. The specimens have consistent reinforcement layouts and are cut off at the joint cross-section. The web reinforcement configuration comprises two 20 mm-diameter longitudinal bars at the bottom, complemented by two 10 mm bars at mid-height and the top. Four additional 10 mm longitudinal bars are arranged along the top flange. Transverse reinforcement utilizes 6 mm-diameter double-legged stirrups, spaced at 140 mm intervals within shear-critical regions and 75 mm within the pure bending section.

The summaries of the experimental comparison groups are as follows:

(1)

The comparisons between specimens M and S-1 are employed to study the effects of joints on the shear performance.

(2)

The comparisons among specimens S-1, S-2, S-3, and S-4 will show the influences of the relative position between loading points and joints on the structural shear behaviors.

(3)

The experimental control group of specimens S-1 and S-3 is set to compare the shear span ratio on the shear resistance capacity.

(4)

Specimens S-1 and S-5 are compared to study the hybrid tendon ratio (area ratio of internal tendons to external tendons) on the shear behaviors.

(5)

In addition, for each segmental beam, the behaviors of the left and right shear spans are compared to show the effects of segmental joints on the shearing capacity.

2.2. Fabrication of Specimens

The segmental beams are cast in two stages by the short-line matching method. The wooden formwork is used for supporting concrete pouring, and a PVC pipe with a diameter of 20 mm is pre-embedded inside for the subsequent tensioning of the internal tendon. At the joint position, the high-density trapezoidal foams are adhered to and positioned on the web of the T-shaped wooden formwork for pouring and shaping the shear-keyed joints, as shown in Figure 2.

The left concrete segment is first poured, and then the joint formwork is removed after three days of curing. After that, the surfaces of the formed shear-keyed joint are roughened and oiled, and the right segment is poured in to match. Prestress tensioning is carried out after the second batch of concrete has cured for 28 days.

2.3. Material Properties

The concrete strength of the specimens is designed as C50 grade. Two batches of commercial concrete with the same mix proportion are used. The concrete mix proportions are shown in

Table 2. Mix proportion of concrete.

Cement (kg/m3)	Water (kg/m3)	Sand (kg/m3)	Coarse Aggregate (kg/m3)	Sand Percentage	Fly Ash (kg/m3)
326	163	713	1025	41%	86

. Ordinary Portland cement with a strength grade of 42.5 is used, and the maximum particle size of the aggregate is 20 mm. During the pouring of the test beams, specimens of 150 mm cubic, 100 mm × 100 mm × 300 mm prism, and 100 mm cubic are made to test the material properties of compressive strength (f_cu), elastic modulus (E_c), and splitting tensile strength (f_tr). The testing method follows the specifications outlined in the Chinese code [38]. The mechanical properties of non-standard specimens are modified in accordance with the code [38]. The results after modification are shown in

Table 3. Mechanical properties of concrete.

Batch	fcu (MPa)	fc (MPa)	ftr (MPa)	Ec (MPa)
Left segment	52.0	42.7	3.63	32,600
Right segment	48.5	41.4	3.58	32,500

. The material properties of the reinforcements and steel strands used in the specimens are tested according to the Chinese code [39,40], and the test results are shown in

Table 4. Mechanical properties of reinforcements and steel strand.

Bar Size	Diameter (mm)	Area (mm2)	Yield Strength fy (MPa)	Ultimate Strength fu (MPa)	Elastic Modulus Es (GPa)
A6	6.0	28.3	356.2	636.6	206
C10	10.0	78.5	435.5	674.8	208
C20	20.0	314.2	450.6	587.8	208
As15.2	15.2	139.0	1754.6	1955.4	195

.

2.4. Loading and Measurement Scheme

(1)

Segment assembly

First, the epoxy resin adhesive is evenly applied to the joint cross-section of the left segment, with a thickness controlled to no more than 3 mm. The prestressed strands are temporarily tensioned by jacks to maintain a compressive stress of approximately 1 MPa during the curing of the epoxy resin adhesive. The tendons are post-tensioned with a controlled tension stress of 900 MPa (for S-5, the controlled tension stress for the two straight steel strands is 450 MPa). During the post-tensioning process, the tension forces are monitored by the anchor cable force gauge at the tensioning end. Finally, the assembled beam is placed onto the loading frame, and the anchor cable force gauges are reread to record the effective prestress of each tendon.

(2)

Loading scheme

A 100 T hydraulic jack is employed to apply static loading to the test beam, as illustrated in Figure 3. A load-controlled loading scheme is implemented, and the loading test is conducted in accordance with the specifications of the Chinese code [41]. Before the formation of evident shear cracks in the test beam, each loading increment is set at 10 kN. Following the appearance of visible diagonal cracks, the loading increments are reduced to 5 kN until the test beam reaches failure. After each loading stage, a 5 min pause is introduced to record the test data, including load, displacement, and strain, while also observing the crack morphology of the test beam.

(3)

Measurement scheme

The schematic diagram of the measurement scheme is shown in Figure 4, which includes structural deformations, bearing capacity, concrete compressive strain, stirrup strain, and tendon force. The concrete compression strain of the top flange and the strain of the bottom rebars are used to identify whether bending failure occurs. The stirrup strain is used to analyze the shear contributions that occur after diagonal cracks form.

3. Test Results

3.1. The Main Test Results

The main test results are shown in

Table 5. Summary of the test results.

ID	Cracking Load (kN)	Ultimate Load (kN)	Tendon Stress Increment (MPa)		Calculated Flexural Strength P (kN)	Maximum Deflection (mm)	Failure Mode
			Internal	External
M	360	696.1	487.8	430.7	1112.6	11.79	DC
S-1	350	666.7	787.1	643.5	895.1	14.94	SC
S-2	310	643.2	890.6	730.5	725.6	19.11	SC
S-3	340	522.6	627.8	544.0	897.2	19.88	SC
S-4	240	457.1	905.5	675.8	665.2	15.34	SC
S-5	400	546.3	-	323.4 (S) 298.1 (D)	942.3	10.83	SC

, in which “DC” denotes diagonal compression failure mode, and “SC” denotes shear compression failure mode. For the segmental beams, the small diagonal cracks first appear at the web of the shear span, but their development is slight. Instead, due to the cutoff steel reinforcements and unavoidable matching defects, the segmental joint sequentially opens and forms a master crack, ultimately leading to structural failure. Therefore, the cracking loads in Table 5 refer to the joint opening loads for segmental beams and the diagonal cracking loads for monolithic beams. Employing the built numerical model of flexural analysis by Yan et al. [15], the flexural strength of each specimen is calculated without shear behaviors considered. It can be seen that the ultimate loads of all specimens are lower than their theoretical flexural strength, and all specimens fail in shear.

The effects of shear span ratio, joints, and tendon arrangements on the shear strength can be concluded as follows:

(1)

Effects of the segmental joints

The results comparisons of specimens M, S-1, and S-2 show that the shear-bearing capacity of S-1 and S-2 decrease by 4.2% and 7.6%, respectively, compared with that of M. The segmental joints reduce the structural shear strength. Moreover, all of the segmental beams fail at the right shear span with the segmental joint, indicating the adverse effect of the joint on the shear strength directly.

Meanwhile, the relative position between the joint and the loading points will also influence the shear capacity. The distance between the loading point and the joint of S-1 is greater than that of S-2 (180 mm for S-1 and 50 mm for S-2). That of S-3 is greater than that of S-4 (380 mm for S-3 and 220 mm for S-4). The test results show that the shear-bearing capacity of S-2 is 3.5% lower than that of S-1, and that of S-4 is 12.5% lower than that of S-3. As the distance between the loading point and the joint decreases, the shear-bearing capacity of the structure decreases, and the degree of decline increases with the increase in the shear span ratio.

(2)

Effects of the shear span ratio

Specimens S-1 and S-4 have the same tendon arrangements and relative position of the joint to the loading point, but the shear span ratio is different (the shear span ratio of S-1 is 1.5 and that of S-4 is 2.0). The shear-bearing capacity of the S-4 test beam is 31.4% lower than that of S-1. As the shear span ratio increases, the shear-bearing capacity of the structure decreases significantly. The comparison between S-2 and S-3 shows a similar conclusion. The shear-bearing capacity of S-3 is 18.8% lower than that of S-2.

(3)

Effects of the tendon arrangement

The shear span ratio, joint position, and prestressing degree of S-1 and S-5 are the same, except for the tendon arrangement. The test results show that the shear strength of S-5 with fully externally prestressed tendons is about 18.1% lower than that of S-1 with hybrid tendons, indicating that increasing the proportion of internal tendons has a positive effect on the shear strength.

3.2. Load–Deflection Curves

The load–displacement curves and the structural deformation curves at different loading levels of each test beam are shown in Figure 5. The summaries are as follows:

(1)

Load–displacement curves

The load–displacement curves of each test beam present a two-broken-lines feature. Before the joint cracks, the test beam shows a linear elastic deformation. After the joint opens, the deformation increases significantly due to the reduction in structural stiffness.

(2)

Structural deformation characteristics

The monolithic beam exhibits symmetrical deformation in each loading section, with a deformation shape that is approximately parabolic. When close to the ultimate load, the vertical deformation at the loading point on the N side is greater than that on the S side, showing an asymmetrical structural deformation, and a shear brittle failure occurs on the N side. The segment beam specimens show a similar pattern. Before the joint opens, the deformation of the segment beam is consistent with that of the monolithic beam, with the maximum deformation occurring at the mid-span position. After the joint opening, the deformation at the joint position (N side) develops rapidly, and the deformation becomes significantly asymmetrical, with the maximum deformation occurring at the joint position.

(3)

Maximum deformation

The deformation of the monolithic beam is significantly lower than that of the segment beams, and there is no apparent portent of failure before it occurs, indicating a brittle failure. For the segmental beam, deformation at the joint is significant before failure, indicating better ductility compared to the monolithic beam. As the shear span ratio increases, the maximum structural deformation also increases. For test beams with the same shear span ratio, the smaller the distance of the joint position from the loading point, the greater the structural deformation.

3.3. Failure Modes

The failure modes of each test beam are presented in Figure 6.

(1)

Specimen M

Under a loading level of 310 kN, the monolithic beam M initially develops diagonal cracks in the shear span regions at both ends, with the crack direction extending from the loading points toward the supports. With the increasing load, the width of the main cracks increases, while new noticeable shear cracks do not appear. The shear crack width on the N side is slightly larger than that on the S side. During the loading process at P = 700 kN, the test beam experiences sudden shear compression failure on the N side, as evidenced by fractured, exposed stirrups. No flexural cracks are observed in the test beam, and no yielding occurs in the tendons or bottom reinforcements.

(2)

Segmental beams

For specimen S-1, small diagonal cracks first appear in both shear span regions under a loading level of P = 300 kN. At P = 360 kN, slight cracking is initiated at the bottom edge of the joint cross-section, occurring at the interface between the concrete and the adhesive layer. At P = 370 kN, the segmental joint opens and then extends upward to approximately 110 mm from the bottom edge before propagating toward the loading point, forming diagonal shear cracks. As the load increases, the concrete in the shear compression zone is segmented into a series of parallel “short inclined columns” by these cracks, ultimately leading to shear compression failure of the compression zone. Prior to the failure, S-1 exhibits significant joint opening, with a clear indication of impending failure. The failure modes of the S-2 to S-5 test beams are consistent with those of the S-1 test beam.

3.4. Analysis of Shear-Bearing Contribution

The shear capacity of the test beam can be decomposed according to Equation (1), as shown in Figure 7. V represents the total shear force of the test beam; V_c represents the shear contribution by the concrete, including the shear compression zone and the interlocking effect of the aggregate between the crack surfaces; V_d represents the shear contribution of the anchorage effect of the longitudinal steel bars and stirrups inside the beam; V_s represents the shear contribution produced by the tensile force of the stirrups passing through the inclined cracks; and V_p represents the shear contribution of the tendon forces.

$$V=V_c+V_d+V_s+V_p$$

(1)

Based on the test data of stirrups crossing the cracks, the shear contribution of stirrups V_s can be quantified. The shear resistance of prestressed tendons, V_p, can be calculated based on the tension force and structural deformation at different loading levels. Therefore, the shear resistances of the specimens can be decomposed into three parts: V_s, V_p, and V_c + V_d. Then, the variation in the contribution of these three components to the shear strength at different loading stages can be analyzed.

According to the measured responses, the shear resistance contribution ratio of each component can be calculated, as shown in Figure 8.

(1)

Throughout the entire loading process until failure, the shear resistance (V_c + V_d) contributed by concrete and the dowel action plays the most significant role in the structural shear capacity.

(2)

During the initial loading phase (before the appearance of noticeable flexural and shear cracks), the shear force is predominantly borne by V_c. After the formation of diagonal cracks, the proportion of shear force carried by stirrups crossing these cracks increases significantly. Once the stirrups at the diagonal cracks yield, they can no longer bear additional shear force, and the subsequent increase in shear force is distributed among other components.

(3)

The shear force contribution of internal straight tendons is lower than that of draped external strands. Notably, their contribution ratios increase substantially after joint opening.

(4)

At the ultimate state, the contribution ratio of V_c + V_d ranges from 48.3% to 78.3%, the contribution ratio of V_s ranges from 18.4% to 31.6%, and the contribution ratio of V_p ranges from 3.3% to 20.1%.

(5)

The contribution ratio of shear resistance for each component at the ultimate loading state correlates with the shear span ratio. In test beams with a shear span ratio of 2.0 (S-4 and S-5), the contribution ratios of V_s and V_p are higher than those in beams with a shear span ratio of 1.5. Conversely, the contribution ratio of V_c + V_d is lower in beams with a shear span ratio of 2.0 compared to those with a shear span ratio of 1.5.

4. Shear Mechanism Analysis Based on the Combined Truss and Arch Model

(1)

Shear strength prediction based on combined truss and arch models

Based on the experimental observations, the typical failure mode of the segmental beam at the ultimate state is as follows: a diagonal crack first forms between the joint and the loading point, leading to the eventual failure of the concrete in the shear compression zone. According to the combined truss and arch model, the shear-bearing diagram for the segmental beam can be illustrated as in Figure 9. The shear strength can be decomposed into three components, as expressed in Equation (2). V_arch denotes the shear contribution by the arch effect, corresponding to the V_c + V_d terms in Equation (1). V_truss denotes the shear contribution by the truss effect, corresponding to the V_s terms in Equation (1).

$$V=V_{arch}+V_{truss}+V_p$$

(2)

From the perspective of simplifying the calculation, this study quantitatively evaluates the shear resistance contributions of the truss effect, the arch effect, and the vertical component of external prestressing tendons based on the following assumptions:

(1)

The arch axis of the arch effect is simplified as a straight line connecting the loading point to the support. According to research by He et al. [42] and Qi et al. [43], this simplified straight–arch axis form can provide sufficient accuracy for simplified design purposes.

(2)

The rise of the arch rib in the arch effect is defined as the distance from the centroid of the prestressed tendon to the top of the beam.

(3)

In the truss effect, diagonal cracks are assumed to develop between the loading point and the adjacent joint within the shear span region. The shear resistance contribution of the truss effect considers only the contribution of stirrups, excluding the shear resistance provided by concrete.

(4)

The shear resistance contribution of the prestressed tendon is determined by the vertical component of the effective prestress within the tendon.

Based on these assumptions, the shear-carrying contribution by the arch effect, V_arch, can be denoted as Equation (3). f_c represents the concrete’s axial compressive strength (MPa); b_v stands for the width of the beam web (mm); a is the shear span length (mm); d_v is the rise of the arch rib (mm), which can be taken as the distance from the center of the prestressed tendons to the top of the beam; and w_arch represents the width of the arch rib (mm). According to the suggestions by He et al. [42] and Qi et al. [43], w_arch = 0.5 d_v. In particular, v is introduced here to consider the effects of cracking on compression, as presented in the modified compression field theory [44]. Following the recommendation of Maekawa et al. [45], a lower limit value of v = 0.6 is adopted in this paper.

$$\begin{array}{cc}{V}_{arch}& =v{f}_c{w}_{arch}{b}_v\sin \beta \hfill \\ & ={\displaystyle \frac{v{f}_c{w}_{arch}{b}_v}{\sqrt{{\left(a/d_v\right)}^2+1}}}\hfill \end{array}$$

(3)

The shear-carrying contribution by the truss effect, V_truss, can be denoted as Equation (4) by ignoring the contribution of concrete tension between cracks. Among them, V_truss1 represents the shear capacity component of the stirrups passing through the failure inclined section, n is the number of stirrups, f_y is the stirrups’ yield strength, and A_s is the area of a single stirrup (mm²). The purpose of V_truss2 is to prevent the crushing failure of the web concrete before the stirrups yield. The number of stirrups, n, for the truss effect in the monolithic beam can be calculated according to the inclined crack angle, θ, and the effective height, d_v, as suggested by the AASHTO specification [23].

$$\begin{array}{l}{V}_{truss1}=n{f}_y{A}_s,V_{truss2}=0.25f_c{b}_v{d}_v\\ V_{truss}=\min \left\{V_{truss1},V_{truss2}\right\}\end{array}$$

(4)

The shear-carrying contribution, V_p, of the prestressed tendons can be calculated according to Equation (5), where A_p is the area of the draped prestressed tendons (mm²); f_ps is the tendon stress, which is simplified to be the effective prestress; and α is the angle between the draped prestressed tendons and the axis of the beam.

$$V_p=A_p{f}_{ps}\sin \left(\alpha \right)$$

(5)

Equations (2) to (5) are the proposed shear strength prediction formulas for PSCBs based on the combined truss and arch model. Using these equations, the shear strength of specimens in this research and of those in Jiang et al.’s research [33,34] are predicted and compared with the test results to verify the effectiveness. The ratio of the calculated shear strength to the test results is shown in Figure 10 and

Table A1. Comparison table of the calculated and tested shear-carrying capacity.

Source	ID	fc	dv	bv	a	fy	As	fpe	Ap	Vcal	Vtest	Vcal/Vtest
This paper	M	43	346	100	600	356	28.3	631.1	278	602	696.1	0.86
	S-1	43	346	100	600	356	28.3	675.9	278	564	666.7	0.85
	S-2	43	346	100	600	356	28.3	608.3	278	520	643.2	0.81
	S-3	43	346	100	800	356	28.3	659.4	278	431	522.6	0.83
	S-4	43	346	100	800	356	28.3	669.4	278	391	457.1	0.86
	S-5	43	346	100	600	356	28.3	654.9	278	523	546.3	0.96
Jiang et al. [33,34]	FM1.6-N-N-e	45	270	70	450	345	28.3	983	109.6	354	389	0.91
	FD1.6-40-N-e	45	270	70	450	345	28.3	937	109.6	316	335	0.94
	FM2.2-N-N-e	47	270	70	600	345	28.3	851.3	109.6	311	320	0.97
	FD2.2-40-N-e	45	270	70	600	345	28.3	1016	109.6	304	285	1.07
	FE2.2-40-N-e	45	270	70	600	345	28.3	988	109.6	304	275	1.10
	FD2.2-20-50-e	50	270	70	600	345	28.3	815.3	109.6	323	340.2	0.95
	FE2.2-20-50-e	50	270	70	600	345	28.3	731.3	109.6	322	345	0.93
	FM2.7-N-N-e	48	270	70	750	345	28.3	799	109.6	274	284.1	0.97
	FD2.7-40-N-e	54	270	70	750	345	28.3	778.5	109.6	336	307.4	1.09
	FE2.7-40-N-e	54	270	70	750	345	28.3	739.8	109.6	336	330.2	1.02
	FD2.7-70-N-e	37	270	70	750	345	28.3	592	109.6	190	280	0.68
	FD2.7-40-70-e	54	270	70	750	345	28.3	719	109.6	257	304.1	0.85
	FE2.7-40-70-e	54	270	70	750	345	28.3	753.3	109.6	258	275.2	0.94
	FD2.7-20-50-e	51	270	70	750	345	28.3	761.3	109.6	326	329	0.99
	FE2.7-20-50-e	51	270	70	750	345	28.3	732.8	109.6	325	259.7	1.25
	SM1.6-N-N-h	35	270	110	450	345	28.3	651.7	109.6	405	425.2	0.95
	SD1.6-20-50-h	35	270	110	450	345	28.3	655	109.6	405	399.5	1.01
	SD1.6-20-50-e	35	270	110	450	345	28.3	630.8	109.6	404	366.7	1.10
	SM2.2-N-N-h	35	270	110	600	345	28.3	649.3	109.6	340	359.5	0.95
	SD2.2-20-50-h	35	270	110	600	345	28.3	622.3	109.6	301	315	0.96
	SD2.2-20-50-e	35	270	110	600	345	28.3	608	109.6	300.7	338.6	0.89
	SM2.7-N-N-h	35	270	110	750	345	28.3	645.3	109.6	296.3	305.4	0.97
	SD2.7-20-50-h	35	270	110	750	345	28.3	643	109.6	296.3	298.8	0.99
	SD2.7-20-50-e	35	270	110	750	345	28.3	625.8	109.6	296.0	267.7	1.11

. The comparison results indicate that the simplified model presented in this section can estimate the ultimate bearing capacity of each test beam with relative accuracy. The mean value of the ratio of the calculated value to the test value is 0.96, and the standard deviation is 0.11. However, due to the introduction of certain assumption conditions, the simplified calculation formula still leads to a certain degree of dispersion in the calculation results.

(1)

Influence mechanism of factors

Based on the mechanism analysis of the truss–arch model and the simplified prediction formulas for shear strength, the main influencing factors and their corresponding influence laws on the shear capacity of PSCBs can be concluded as follows:

(2)

Influence of shear span ratio

It is worth noting that the term a/d_v in Equation (3) is analogous to the shear span ratio. According to Equation (3), as the shear span ratio increases, the inclination angle β of the arch rib gradually decreases, thereby reducing the shear contribution of the arch effect. This observation aligns well with the experimental results, indicating that the shear span ratio primarily affects the shear capacity of PSCBs by altering the load transfer path associated with the arch effect.

(3)

Influence of segmental joints

In segmental beams, the web stress flow is “interrupted” when the segmental joints open, limiting its propagation across the joints. The effects of segmental joints on the shear behaviors can be concluded from two aspects, as shown in Figure 11. (a) Due to the open joint, the shear cracks mainly form between the loading point and the joint cross-section, reducing the number of stirrups crossing the web shear cracks. This reduction further diminishes the shear contribution of the stirrups within the truss effect. The closer the loading point is to the joint, the fewer stirrups participate in the shear resistance, leading to a more significant reduction in the shear strength. (b) The open joints also change the width of the arch rib to be smaller. When stress flows across the open joint, it can only be transmitted from the compressive zone, and the arch effect will be reduced. Therefore, compared with a monolithic beam, the shear-carrying capacity of PSCBs is mainly reduced by weakening the arch and truss effects.

Additionally, the joint types, such as epoxy and dry joints, will also impact the shear behavior of PSCBs. For PSCBs with dry joints, the joint will open once decompressed, and the height of the compressive zone is smaller than that of PSCBs with epoxy joints. The shear-carrying contribution of arch effects of PSCBs with dry joints will be reduced. In the current work, this effect is not considered in the proposed equations from the perspective of simplifying calculations.

(4)

Influences of tendon stress

First, the vertical component of draped prestressed tendons contributes to bearing part of the shear force. Meanwhile, the application of prestress reduces the inclination angle of the beam’s shear cracks at ultimate load, enabling more stirrups to traverse these cracks and enhancing the overall shear capacity of the structure. This suggests that the tendon stress will influence the shear strength of PSCBs through its effects on the vertical components and the truss action.

(5)

Influence of internal/external prestressing type

The prestressed tendons of the beam serve as tension bars to balance the axial force of the arch. At ultimate load, the distance from the centroid of the prestressed tendons to the top of the beam represents the rise of the arch effect. For PSCBs with all external prestressing, the absence of the eccentricity effect induced by second-order behavior leads to a reduction in the rise of the arch effect, which in turn decreases the inclination angle of the arch ribs and consequently lowers the shear capacity of the structure.

(6)

Discussion of the proposed equations

The proposed equations are established based on the tests and observations of scaled T-shape cross-section specimens. For beams with other diameters, such as the full-scale girder in practice, the size effect inevitably exists, and further studies are required to evaluate the applicability of these equations. In future work, the shear strength of full-scale PSCBs can be calculated by finite element models and then compared with the predictions to verify the applicability.

In AASHTO specifications [23], the equations for calculating the shear strength of prestressed concrete beams are provided without distinguishing between segmental and monolithic beams. For beams without inclined transverse reinforcement, the relevant equations in AASHTO are presented in Equation (6), where β represents the factor indicating the ability of diagonally cracked concrete to resist tension and shear, and θ denotes the angle of inclination of diagonal compressive stresses. The variables β and θ are calculated according to the modified compression field theory. It is observed that Equation (6) shares a similar form with the proposed equations derived from the mechanical behavior analysis. Specifically, the term V_c corresponds to the arch effect, while the term V_s corresponds to the truss effect. Following current design guidelines and based on the clarified influence mechanism of segmental joints on shear performance, the term V_s can be modified as expressed in Equation (4) to account for the effects introduced by segmentation.

$$\begin{array}{l}V=V_c+V_s+V_p\\ V_c=0.0632\beta \sqrt{f_c}{b}_v{d}_v\\ V_s={\displaystyle \frac{A_s{f}_y{d}_v\cot \theta }{s}}\end{array}$$

(6)

It should be emphasized that the bearing mechanism and shear-carrying capacity calculation formula proposed in this paper are specifically applicable to PSCBs constructed with ordinary concrete and steel strand prestressed tendons. For PSCBs incorporating new materials such as CFRP tendons and UHPC segments, the shear-bearing mechanisms may be influenced by additional factors. These aspects fall beyond the scope of the current study and warrant further investigation.

5. Conclusions

This paper conducts static loading tests on the shear performance of one monolithic beam and five PSCBs. The effects of the shear span ratio, joints, and prestressing tendons on the failure modes and shear strength are analyzed and compared. Based on the combined truss and arch model for PSCBs, the shear mechanism is clarified, and simplified shear strength prediction equations are proposed. The main conclusions are as follows:

(1)

The discontinuity of the open segmental joints alters the failure modes of PSCBs, differing from the monolithic beam, and the maximum reduction of shear-carrying capacity is 7.6% in the tests. The closer the distance between the loading point and the joint, the more significant the adverse effect of the joint.

(2)

Under extreme test conditions, the proportion of shear resistance contributed by concrete segments is the highest, ranging from 48.3% to 78.3%; the contribution from stirrups ranges between 18.4% and 31.6%, while that from prestressed tendons falls within 3.3% to 20.1%. The contribution ratio of each component is found to be correlated with the shear span ratio, with an increasing shear span ratio leading to a decrease in the contribution from concrete segments.

(3)

The truss–arch model can reasonably explain the shear-bearing mechanism of PSCBs and the action mechanism of each influencing factor. The proposed simplified calculation formula reasonably estimates the shear capacity of thirty segmental beams, with a mean ratio of calculated-to-test values of 0.96 and a standard deviation of 0.11.

(4)

This research focuses on the shear behavior of PSCBs constructed with ordinary concrete and steel strand prestressed tendons. For PSCBs incorporating new materials such as CFRP tendons and UHPC segments, further studies are still needed.