# Time-Dependent Behavior of Reinforced Concrete Beams under High Sustained Loads

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## Abstract

**:**

## 1. Introduction

_{c}). Creep under this level of stress has been the focus of most past research [13,17,18]. The stress level (σ

_{c}) falls within the elastic domain of concrete response and creep deformations are caused by the breaking and reformation of atomic bonds at various highly stressed sites within the colloidal microstructure of the calcium silicate hydrate gels in the hardened cement paste [19]. At sustained stress levels ≤ ~0.4 f’

_{c}, there is a nearly linear relationship between the short-term strains (strain under short-term loading) and delayed creep strains (strains developed with time) and the rate of creep strain development decreases with time [16].

_{c}), the concrete undergoes additional damage processes as a result the propagation of microcracks initiated during loading [20,21,22]. Additional nonlinear creep strains are developed as the relationship between the linear creep strain and elastic strain is lost [16].

_{c}. During this stage, concrete experiences coalescence of microcracks, which ultimately results in concrete failure [16]. The tertiary creep stage is signified by a rapid increase in the creep strain rate.

_{c}, microcracks grow slowly [20], creep deformations are limited under infinite loading, and concrete does not fail [1]. When the sustained stress is greater than 0.80 f’

_{c}, microcracks grow rapidly [20] and concrete strains increase rapidly so that concrete fails at stresses lower than f’

_{c}. Iravani and MacGregor [12] demonstrated that as concrete strength increased, the ratio of sustained load strength to ultimate strength increased.

## 2. Materials and Methods

#### 2.1. Specimen Design

#### 2.2. Test Setup

#### 2.3. Instrumentation

^{−6}strain.

#### 2.4. Material Properties

#### 2.5. Testing Procedure

## 3. Results

## 4. Discussion

#### 4.1. Deflection under Sustained Loading

_{s}) divided by the sustained load deflection at 24 days (δ

_{[email protected] days}) for series I and series II. Despite some differences, all specimens show a similar rate of deflection increase with time. Most of the deflection increase took place in the first few days. In the first 24 h for beam series I, 44%, 61%, and 65% of the deflection increase occurred in specimens B2-SL, B3-SL, and B4-SL, respectively. After 7 days, 67%, 72%, and 80% of the deflection increase occurred in these specimens. Moreover, higher sustained load induced greater deflections in the first few days. In the first 24 h for beam series II during the first stage of sustained loading, 35%, 72%, and 71% of the deflection increase occurred in specimens B6-SL, B7-SL, and B9-SL, respectively. Specimen B6-SL experienced the slowest gain in deflection, most likely due to the much lower load level compared to the other specimens (84% vs. >91%). After 7 days, 71%, 85%, and 84% of the 24-day deflection had occurred in the specimens. Again, the results show that most of the deflection increase occurred in the first days of sustained loading. After about 2 days the rate of deflection increase was nearly constant at about 0.025 mm/day (0.001 in./day) for the specimens as expected in the secondary stage of creep. Only specimen B10-SL (series II) reached the third stage (tertiary) and failed under sustained load, as shown in Figure 18. The two earlier increases in deflection were due to load adjustment, while the last increase occurred during tertiary creep with no adjustment in the loading. This stage happened within 2 min before failure.

#### 4.2. Peak Load

#### 4.3. Strain

#### 4.4. Deflection vs. Strain

## 5. Conclusions

- Even if plain concrete can experience tertiary creep and eventual failure under load levels near 75–80% of its compressive strength, the sustained load level for RC beams to cause failure is very close to their short-term capacity. Only one beam (B10-SL) failed under sustained load at a load level of 98%. Another beam (B9-SL) was able to carry a load level of 95% for 30 days without failure.
- Deflections increased during sustained loading for all specimens. At 24 days, the increase in deflection was on average 24% of the initial deflection. The increase in deflection was higher for specimens with higher levels of sustained load.
- On average 55% of the increase in deflection under sustained load took place in the first 24 h. The sharper increase in deflection early in the loading coincided with the primary stage of creep deformations and time-dependent bond slip. The secondary stage exhibited nearly a linear increase in deflection with time. Only specimen B10-SL experienced a tertiary stage that showed a sharp increase in deflection with time just 2 min before failure.
- Sustained load increased the deflection at shear failure for all specimens. For beams series I, the increase in deflection at shear failure compared to the control specimen was 230% while for beams series II, the increase in deflection was 150%. The large increase in deflection shows that sustained loading can significantly affect the failure behavior of a beam. This large increase in deflection would allow for load redistribution in redundant systems or provide warning signs of impending failure.
- Shear-controlled beams tested under sustained load showed a different failure behavior from the control specimen that was tested under monotonically increasing load to failure. Unlike the control specimen, which failed brittlely in shear, specimens tested under sustained loads experienced significant increases in deflection and flexural cracking before ultimate shear failure.
- Both the tensile and compressive strains increased under sustained load. The increase in strain shows that the reinforcement took more of the loading as the concrete softened under the sustained load. The increase in the compression strain was higher than in the tension strain.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Effect of sustained stress level and duration on the concrete strain (adapted from Rüsch [1]).

**Figure 22.**Strain increase with time during the first stage of sustained loading for beams series II. (

**a**) Specimens B6-SL and B9-SL (

**b**) Specimen B10-SL.

Series | Specimen | Loading Age t_{o} (Days) | Duration t_{d} | Temperature °C (°F) | Relative Humidity % |
---|---|---|---|---|---|

Series I | BC1 | 64 | - | - | - |

B2-SL | 65 | 25 days | 15.4 ± 3.7 (59.7 ± 6.7) | NA | |

B3-SL | 91 | 42 days | 13.3 ± 2.7 (56.0 ± 4.9) | NA | |

B4-SL | 135 | 24 days | 16.5 ± 3.7 (61.7 ± 6.7) | NA | |

Series II | BC5 | 282 | - | - | - |

B6-SL | 284 | 32 days | 26.1 ± 2.5 (79.0 ± 4.5) | 73.5 ± 3.6 | |

B7-SL | 338 | 52 days | 26.4 ± 3.6 (79.5 ± 6.5) | 72 ± 7 | |

B9-SL | 492 | 34 days | 17.1 ± 5.9 (62.8 ± 10.7) | 28.1 ± 17.1 | |

B10-SL | 543 | 84.5 min. | 12.6 (54.6) | 22.6 |

Material | 3/8” Limestone kg (lb) | River Sand kg (lb) | Type I Cement kg (lb) | Air L (oz) | Retarder L (oz) | Water L (gal) |
---|---|---|---|---|---|---|

Amount | 1067.64 (2354) | 831.07 (1831) | 333.45 (738) | 0.135 (4.6) | 0.843 (28.5) | 133.68 (35.3) |

Specimen | BC1 | B2-SL | B3-SL | B4-SL |
---|---|---|---|---|

Sustained load kN (lb) | - | 16.01 (3600) | 17.13 (3850) | 18.24 (4100) |

Sustained load intensity (sustained load/peak load of BC1) | - | 0.82 | 0.87 | 0.93 |

Actual sustained load intensity (sustained load/measured peak load) | - | 0.79 | 0.86 | 0.88 |

Peak load kN (lb) | 18.36 (4352) | 20.28 (4559) | 20.01 (4498) | 20.68 (4659) |

Ratio of the peak load to the peak load of BC1 | 1 | 1.05 | 1.03 | 1.07 |

Deflection at the peak load mm (in.) | 12.37 (0.487) | 13.88 (0.547) | 20.47 (0.806) | 18.22 (0.717) |

Deflection at failure mm (in.) | 12.37 (0.487) | 40.39 (1.590) | 44.78 (1.763) | 37.41 (1.473) |

Initial deflection under short-term loading deflection (δ_{i}) mm (in.) | - | 5.31 (0.209) | 9.42 (0.371) | 10.66 (0.420) |

Deflection increase under sustained load (δ_{s}) mm (in.) | - | 1.11 (0.043) | 3.55 (0.140) | 2.18 (0.085) |

Creep coefficient—Ratio sustained to initial deflection (δ_{s}/δ_{i}) | - | 0.21 | 0.38 | 0.20 |

Rotation under sustained load | - | 0.0027 | 0.0087 | 0.0054 |

Sustained load deflection at 24 days (δ_{[email protected] days}) mm (in.) | - | 1.11 (0.043) | 3.11 (0.122) | 2.18 (0.085) |

Rotation at 24 days under sustained load | - | 0.0027 | 0.0075 | 0.0054 |

Specimen | BC5 | B6-SL | B7-SL | B9-SL | B10-SL |
---|---|---|---|---|---|

Sustained load kN (lb) | - | 15.35 (3750) | 16.68 (3450) | 17.48 (3750) | 18.02 (4050) |

The reinforcement depth mm (in.) | 116 (4.573) | 112 (4.412) | 112 (4.423) | 111 (4.357) | 114 (4.473) |

Sustained load intensity (sustained load/peak load of BC1) | - | 0.84 | 0.91 | 0.95 | 0.98 |

Actual sustained load intensity (sustained load/measured peak load) | - | 0.71 | 0.89 | 0.91 | 1.00 |

Peak load kN (lb) | 18.35 (4216) | 21.75 (4889) | 18.70 (4205) | 19.26 (4330) | 18.02 (4050) |

Ratio of the peak load to the peak load of BC5 | 1 | 1.18 | 1.02 | 1.05 | 0.98 |

Deflection at the peak and failure mm (in.) | 5.64 (0.222) | 18.45 (0.727) | 11.76 (0.463) | 12.52 (0.493) | 6.13 (0.241) |

Initial deflection under short-term loading deflection (δ_{i}) mm (in.) | - | 4.98 (0.196) | 6.45 (0.254) | 6.84 (0.269) | 5.87 (0.231) |

Deflection under sustained load (δ_{s}) mm (in.) | - | 0.91 (0.036) | 1.60 (0.063) | 1.95 (0.077) | 0.26 (0.010) |

Creep coefficient—Ratio sustained to initial deflection (δ_{s}/δ_{i}) | - | 0.19 | 0.25 | 0.29 | 0.05 |

Rotation under sustained load | - | 0.0015 | 0.0026 | 0.0032 | 0.00043 |

Sustained load deflection at 24 days (δ_{[email protected] days}) mm (in.) | - | 0.90 (0.035) | 1.58 (0.062) | 1.92 (0.076) | - |

Rotation at 24 days under sustained load | - | 0.0015 | 0.0026 | 0.0032 | - |

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**MDPI and ACS Style**

Shubaili, M.; Elawadi, A.; Orton, S.; Tian, Y. Time-Dependent Behavior of Reinforced Concrete Beams under High Sustained Loads. *Appl. Sci.* **2022**, *12*, 4015.
https://doi.org/10.3390/app12084015

**AMA Style**

Shubaili M, Elawadi A, Orton S, Tian Y. Time-Dependent Behavior of Reinforced Concrete Beams under High Sustained Loads. *Applied Sciences*. 2022; 12(8):4015.
https://doi.org/10.3390/app12084015

**Chicago/Turabian Style**

Shubaili, Mohammed, Ali Elawadi, Sarah Orton, and Ying Tian. 2022. "Time-Dependent Behavior of Reinforced Concrete Beams under High Sustained Loads" *Applied Sciences* 12, no. 8: 4015.
https://doi.org/10.3390/app12084015