**4. Tests and results**

not necessarily mean more effective reinforcement. After the initial tendon force exceeded a threshold (53% in this study), the external tendon will yield and fracture in advance and

**2.** Effect of tendon cross-sectional area. When the initial tendon force and tendon sag stay unchanged, there is a positive relationship between tendon cross-sectional area and the structural performance of a balanced-reinforced beam. As the tendon cross-sectional area increased, the structure improved both in yield strength and ultimate strength, and the duration of the hardening stage was extended. **Figure 6b** shows the load-deflection curves for different tendon cross-sectional areas and constant initial tendon force (20 kN) and tendon sag (200 mm). When the tendon cross-sectional area was 48, 86, and 134 mm2

RC beam's yield strength increased by 15, 22, and 30%, respectively, compared with that of the unreinforced beam. The corresponding increases in the beam's ultimate strength were 16, 37, and 48%. These results demonstrate significant improvement in structural ductility. According to the concept of an over-reinforced beam, there should be an upper limit on

**3.** Effect of tendon sag. When the initial tendon force and cross-sectional area remain constant, the structural performance of the RC beam tends to vary positively with tendon sag. As the tendon sag increased, the structure showed increases in both yield strength and ultimate strength. **Figure 6c** illustrates the load-deflection curves for a tendon specimen with

200, 300, and 400 mm, the beam's yield strength increased 15, 29, and 45%, respectively, after the reinforcement. The corresponding increases in its ultimate strength were 16, 33, and 48%, respectively. These results suggest great effect of sag on the beam's bearing capacity. When the tendon sag exceeded a threshold (300 mm in this case, equal to beam height), the duration of hardening experienced by the beam was shortened. Further research is needed

The analysis above shows that the application of DWM external prestressing not only improved the bearing capacity of the SSB but also altered the plastic zone developed in the beam. **Figure 7** shows the contours of stress in the plastic zone throughout the deformation processes of the unreinforced beam and the RC beam. When the unreinforced beam was subjected to an external load, a plastic zone arose first in the beam segment in the stage of pure bending. As the load increased, the plastic zone tended to expand toward the two ends symmetrically about the midspan position. The height of the plastic zone at midspan gradually increased and always peaked around the midspan. The plastic zone within the segment in

In the RC beam, the plastic zone in the region corresponding to the pure-bending segment of the unreinforced beam expanded at a slower rate due to the presence of web members. The plastic zone's height decreased compared to that in the unreinforced beam. Along the beam bottom, it was symmetrically distributed about the midspan web member. As the load

tendon cross-sectional area. This needs to be discussed in future research.

initial tendon force of 20 kN and cross-sectional area of 48 mm2

to verify if this threshold equals beam height in all cases.

shear bending gradually expanded from the loading point to the supports.

*3.3.4. Characteristics of plastic zone development*

, the

. When the tendon sag was

the RC beam becomes more likely to fail by brittle fracture.

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#### **4.1. Experimental design**

**Table 1** shows the number of test specimens and their materials. Two parts were prepared for each of the specimens indicated by \*. The specimens were divided into three groups. Then "1-" and "2-" were added to the original specimen numbers.

**Figure 8** shows the design of the specimens (reinforcement ratio 1.27%). The properties of the rebars, steel wire ropes, and other materials used in the RC beam are presented in **Table 3**. The concrete strength, at 28 MPa, was measured using rebound hammer.

Loading scheme: A three-point bend test was performed on the specimens using a hydraulic servo jack (**Figures 9** and **10**). Each test process was first controlled by load, which increased 10 kN for each stage. After the load reached 100 kN, displacement control was applied, and the displacement increased 5 mm each stage. The loading time was 3 min and the period of sustained load was 30 min.

Observation scheme: (1) observed variables: load, midspan displacement, beam-end displacement, and stress in concrete, wire ropes, rebars, and web members; (2) observation method: measurements by load transducer, displacement meter, and resistance strain gauge and manual measurement record using coordinate paper (rope length change was measured as strain in rope) and calculation using the Hooke's law; and (3) test devices: static strain gauge, ruler, and so on [12].

First, the SSB's mechanical properties before reinforcement were measured, and the results were plotted as load-deflection curves. Stop loading when the maximum fracture width in the concrete in the tensioned region reached 0.2 mm and then unload the specimen. Later, the initial prestressing force was applied to each specimen based on the reinforcement design

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**Figure 11** compares the load-deflection curves for four specimens before and after reinforcement. When the beam deflection reached 4 mm, specimens 1JGL-1-1, 2JGL-1-1, 1JGL-5-2, and 2JGL-5-2, respectively, showed 29, 30, 41, and 43% increases in load compared to those before reinforcement. For a deflection of 7 mm, the load increased 26, 28, 35, and 37%, respectively, compared with those before reinforcement. The experimental results demonstrate that the

**Figure 12** compares the load-deflection curves for six specimens before and after reinforcement. The fracture strength of specimens 1JGL-1-1, 2JGL-1-1, 1JGL-5-2, and 2JGL-5-2 increased by 21, 26, 38, and 42%, compared to the fracture strength of specimen 2JGL-5-2. Their ultimate strengths were up 13%, 15%, 41%, and 43%, respectively, compared with specimen 2JGL-5-2. The experimental results show that the reinforcement improved the stiffness of the specimens and resulted in 24 and 40% increases in their fracture strength, 24 and 37% increases in yield strength, and 15 and 42% increases in ultimate strength on average. Specimens with larger tendon cross-sectional areas and sags exhibited better structural performance, consistent with

The failure characteristics of the six specimens are presented in **Table 4**. The unreinforced beams were balanced-reinforced. As they failed when the rebars began yielding, their failure mode was ductile failure. The RC beams were divided into two groups: one group with small tendon cross-sectional areas and small tendon sag, and the group with larger tendon cross-sectional areas and larger tendon sags. The rebars all yielded as the tendons failed. The structure ductility was relatively high in both groups. The former group failed before the external tendons fractured, while the latter group failed before the concrete was crushed. This difference demonstrates that large cross-sectional areas and tendon sags in reinforcement design can deliver

1-FJGL is an unreinforced beam. When the external load applied to it reached 40 kN, the first fracture arose at midspan. Later, new fractures developed in the pure-bending beam segment at intervals of about 130 mm. A diagonal fracture developed in the shear-bending segment when the external load was 60 kN and continuously propagated upward as the load increased. After the external load exceeded 100 kN, a number of vertical fractures occurred at midspan, resulting in a sharp increase in beam deformation. The load-deflection curve had

requirements until the specimen failed.

specimens became stiffer after the reinforcement.

**4.2. Experimental results**

*4.2.1. Load-deflection curve*

the numerical results.

better results.

*4.2.3. Fracture distribution*

*4.2.2. Failure characteristics*

**Figure 8.** Sectional properties of tested beam (the unit used in the figure is mm).


**Table 3.** Material properties table.

**Figure 9.** Schematic diagram for loading setup (the unit used in the figure is mm). Notes: (1) fixed hinged support; (2) buttress; (3) test beams; (4) rolling hinged support; (5) backing plate; (6) rolling hinged support; (7) distributive girder; (8) jack; (9) pressure sensor; (10) reaction beam; (11) rolling hinged support; (12) wire rope; and (13) web member.

**Figure 10.** Load diagram of testing beam.

First, the SSB's mechanical properties before reinforcement were measured, and the results were plotted as load-deflection curves. Stop loading when the maximum fracture width in the concrete in the tensioned region reached 0.2 mm and then unload the specimen. Later, the initial prestressing force was applied to each specimen based on the reinforcement design requirements until the specimen failed.

#### **4.2. Experimental results**

#### *4.2.1. Load-deflection curve*

**Name Diameter Φ** 

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**Table 3.** Material properties table.

**Figure 10.** Load diagram of testing beam.

**(mm)**

**Density ρ (kg/m3 )**

**Figure 8.** Sectional properties of tested beam (the unit used in the figure is mm).

*Es*  **(N/mm2**

**Figure 9.** Schematic diagram for loading setup (the unit used in the figure is mm). Notes: (1) fixed hinged support; (2) buttress; (3) test beams; (4) rolling hinged support; (5) backing plate; (6) rolling hinged support; (7) distributive girder; (8) jack;

(9) pressure sensor; (10) reaction beam; (11) rolling hinged support; (12) wire rope; and (13) web member.

Wire rope 9 / 1.4 75 1550 Wire rope 15 / 1.4 208 1550 Reinforcement 14 7800 1.95 / 400

**) × 105 Breaking force(kN)** **Tensile strength (N/mm2**

**)**

**Figure 11** compares the load-deflection curves for four specimens before and after reinforcement. When the beam deflection reached 4 mm, specimens 1JGL-1-1, 2JGL-1-1, 1JGL-5-2, and 2JGL-5-2, respectively, showed 29, 30, 41, and 43% increases in load compared to those before reinforcement. For a deflection of 7 mm, the load increased 26, 28, 35, and 37%, respectively, compared with those before reinforcement. The experimental results demonstrate that the specimens became stiffer after the reinforcement.

**Figure 12** compares the load-deflection curves for six specimens before and after reinforcement. The fracture strength of specimens 1JGL-1-1, 2JGL-1-1, 1JGL-5-2, and 2JGL-5-2 increased by 21, 26, 38, and 42%, compared to the fracture strength of specimen 2JGL-5-2. Their ultimate strengths were up 13%, 15%, 41%, and 43%, respectively, compared with specimen 2JGL-5-2.

The experimental results show that the reinforcement improved the stiffness of the specimens and resulted in 24 and 40% increases in their fracture strength, 24 and 37% increases in yield strength, and 15 and 42% increases in ultimate strength on average. Specimens with larger tendon cross-sectional areas and sags exhibited better structural performance, consistent with the numerical results.

#### *4.2.2. Failure characteristics*

The failure characteristics of the six specimens are presented in **Table 4**. The unreinforced beams were balanced-reinforced. As they failed when the rebars began yielding, their failure mode was ductile failure. The RC beams were divided into two groups: one group with small tendon cross-sectional areas and small tendon sag, and the group with larger tendon cross-sectional areas and larger tendon sags. The rebars all yielded as the tendons failed. The structure ductility was relatively high in both groups. The former group failed before the external tendons fractured, while the latter group failed before the concrete was crushed. This difference demonstrates that large cross-sectional areas and tendon sags in reinforcement design can deliver better results.

#### *4.2.3. Fracture distribution*

1-FJGL is an unreinforced beam. When the external load applied to it reached 40 kN, the first fracture arose at midspan. Later, new fractures developed in the pure-bending beam segment at intervals of about 130 mm. A diagonal fracture developed in the shear-bending segment when the external load was 60 kN and continuously propagated upward as the load increased. After the external load exceeded 100 kN, a number of vertical fractures occurred at midspan, resulting in a sharp increase in beam deformation. The load-deflection curve had

**Beam number** **Cable diameter (mm)**

**Table 4.** Characteristic load and failure of beam.

**Figure 13.** Beam crack mapping. 1-FJGL, 2JGL-1-1, 2JGL-5-2.

**Displacement (mm)**

1FJGL 94 13.5 113 93.9 6.9 2FJGL 99 13.2 116 96.2 7.1 1JGL-1-1 117 14.8 125 146.4 9.9 2JGL-1-1 122 14.1 128 136.2 9.6 1JGL-5-2 133 11.3 152 100.2 8.9 2JGL-5-2 132 10.6 162 85.9 8.1

**Load values (kN)**

**Table 5.** Ductility coefficient of beam.

**Sag of cable (mm)**

**Internal force of cable (kN)**

1FJGL — / / 42 124 Ductility 2FJGL / / / 40 115 Ductility 1JGL-1-1 9 200 20 50 136 Broken of cable 2JGL-1-1 9 200 20 53 141 Broken of cable

1JGL-5-2 15 250 30 58 175 Crush 2JGL-5-2 15 250 30 60 180 Crush

**Beam number Yield step Ultimate load 90% Displacement ductility ratio (mm)**

**Load values (kN)**

**Displacement (mm)**

**Cracking load (kN)**

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**Ultimate load (kN)**

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**Failure characteristics**

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**Figure 11.** Comparison of bearing capacity between, before, and after reinforcement.

**Figure 12.** Comparison of load-deflection curves between reinforced beam and unreinforced beam.

only one peak, which corresponded to midspan point in the fractured region on the sides of the specimen (**Figure 13**).

Specimens 2JGL-1-1 and 2JGL-5-2 were loaded until the maximum fracture width reached 0.2 mm. Then they were unloaded and reinforced by a prestressing force. At this point, all fractures in them were closed and the beams formed inverted arches with vertical displacements of 1 mm and 2 mm, respectively.

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**Table 4.** Characteristic load and failure of beam.

**Figure 13.** Beam crack mapping. 1-FJGL, 2JGL-1-1, 2JGL-5-2.


**Table 5.** Ductility coefficient of beam.

only one peak, which corresponded to midspan point in the fractured region on the sides of

**Figure 12.** Comparison of load-deflection curves between reinforced beam and unreinforced beam.

**Figure 11.** Comparison of bearing capacity between, before, and after reinforcement.

Specimens 2JGL-1-1 and 2JGL-5-2 were loaded until the maximum fracture width reached 0.2 mm. Then they were unloaded and reinforced by a prestressing force. At this point, all fractures in them were closed and the beams formed inverted arches with vertical displace-

the specimen (**Figure 13**).

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ments of 1 mm and 2 mm, respectively.

Specimen 2JGL-5-2. These fracture characteristics of 2JGL-5-2 were similar to those of the previous specimen. Their fracture characteristics differed in two ways: (1) the magnitudes of load at characteristic points were higher than those observed for 2JGL-1-1. For example, the primary fractures opened again when the load was 60 kN and new fractures arose extensively in the shear-bending segment as the load was reached and (2) the four peaks were more obvious in the load-deflection curve for 2JGL-5-2 (See 2JGL-5-2 in **Figure 12**).

The phenomena described above are consistent with the stress contour plots (**Figure 7**), demonstrating the reliability of the analytic method.

#### *4.2.4. Ductility of RC beam*

Ductility is an important indicator considered in seismic design for beams. It is usually measured by displacement-based ductility coefficient, *μ*, [13]:

$$
\mu = \omega \mu / \omega y \tag{14}
$$

**5. Comparison of the theoretical, numerical, and experimental** 

**Figure 14** compares the theoretical, numerical, and experimental load-deflection curves for

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Due to the fundamental assumptions mentioned earlier, the theoretical values for the stage of elastic deformation were slightly smaller than corresponding experimental and numerical values, thus ensuring the safety of the specimens. This demonstrates that the theoretical results can accurately describe the mechanical behavior of the specimens and the calculation method is reliable. For the stage of plastic deformation, the theoretical values were significantly greater than the experimental and numerical values, indicating that the theoretical cal-

The experimental data were highly consistent with the numerical data for both elastic deformation and plastic deformation stages, demonstrating the validity of the numerical method proposed.

**Figure 15** compares the structural performance extracted from all experimental curves with the structural performance of the beam models observed in numerical analysis. The findings

**1.** The specimens reinforced by prestressing had better structural performance than unrein-

**2.** The structural performance curves fit the numerical and experimental data and roughly

move in the same manner, thus confirming the numerical results.

**Figure 14.** Comparison of theoretical, simulated, and experimental values.

**results**

are as follows:

forced beams.

specimens JGL-1-1 and JGL-5-2.

culation cannot provide reliable guidance.

where △*y* is the displacement when the longitudinal rebars in the beam begin yielding and △*u* is the displacement when the load is decreased to 90% of the maximum load. The ductility coefficients of the test beams are presented in **Table 5**.

**Table 5** reveals that the RC beams had much higher ductility than the unreinforced beams. The specimens with small tendon cross-sectional areas and small initial tendon forces exhibited slightly higher ductility than the specimens with larger tendon cross-sectional areas and greater initial tendon forces.

#### **4.3. Summary**

The analysis performed earlier suggests that after being reinforced by DWM external prestressing, the SSB exhibited slightly increased stiffness, improved fracture strength, yield strength, and ultimate strength and significantly increased ductility. This is because the mechanical behavior of the RC beam was constrained by the external prestressing force.

