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An analysis model of the shear capacity of prestressed ultrahigh performance concrete (UHPC) beams under the combined action of bending and shearing was established in this paper based on the modified compression field theory and by considering the unique material constitutive relation of UHPC. Shear tests were performed using three prestressed UHPC-T beams with different shear-span ratios to verify the correctness of the model. The results showed that the shear-span ratio greatly influenced the shear capacity and failure modes of UHPC-T beams. Upon increasing the shear-span ratio, the failure modes of the three beams were inclined compression failure, shear compression failure, and diagonal tension failure, successively. When the shear-span ratio changed from 1.04 to 2.12, the shear bearing capacity decreased greatly; however, when the shear-span ratio changed from 2.12 to 3.19, the decrease of the shear bearing capacity was very small. In addition, the MCFT analysis model was used to analyze the experimental data, and the predicted results were in good agreement, which proved the applicability of the model. Finally, according to the existing shear test results of UHPC beams and based on the main influencing factors, a simplified formula for predicting the shear capacity of UHPC beams was obtained by fitting. Comparing the MCFT model with the results of other pieces of literature, this formula accurately predicted the shear capacity of UHPC beams. The MCFT model and the simplified formula presented in this paper provide a powerful tool for predicting the shear performance of UHPC-T beams, which will contribute to the design and analysis of UHPC-T beams.

Ultrahigh performance concrete (UHPC) is a new type of steel fiber-reinforced cement-based composite material with many excellent properties [

With increasing in-depth research and applications of UHPC in engineering, scholars have carried out a series of studies on the shear performance of UHPC structures. Voo et al. [

Due to the deficiency of the above theoretical model, Vecchio and Collins proposed the modified compression field theory [

This paper aims to establish an analysis model for the shear capacity of UHPC beams based on the modified compression field theory. Considering the unique constitutive relation of UHPC materials, an improved analysis model was established and verified by experiments. A comparison of the results from codes and experiments showed that the existing shear strength formula is not suitable for predicting the shear capacity of UHPC-T beams; therefore, according to the MCFT model and the existing test results, a simplified prediction formula for the shear capacity of UHPC beams was obtained by fitting. The research results provide a practical calculation method for the shear design of UHPC beams and also provide a deeper understanding of the shear performance of UHPC-T beams.

The modified compression field theory was used to establish the strain compatibility equation and stress balance equation based on material mechanics and then combined with the UHPC constitutive relations to obtain the calculation method of the shear bearing capacity according to three convergence conditions.

The basic assumptions of the MCFT are as follows:

The shear stress and normal stress of the concrete microelement body are uniformly distributed when it is under load

The stress-strain of concrete is the average stress-strain of the cracked area

The principal stress direction of concrete is the same as the principal strain direction

Reinforced concrete does not experience bond-slip between each other, and there is an ideal bond between them

The axial stress of reinforcement is the only considered factor, and the reinforcement shear stress is not considered

According to the basic assumption, assuming that the UHPC and the reinforcing bar are in an ideal bonding state, and without considering the shear stress of the reinforcing bar, equations (

The average strain of the reinforced concrete microelements conforms to Mohr’s circle for strain. According to the geometric conditions in Figure

Mohr’s circle for average strain.

Combining equations (

According to the basic assumption, the average stress and average strain of concrete after cracking conform to Mohr’s circle for stresses. Equation (

Mohr’s circle for average stresses.

Figure

Analysis model of a UHPC beam based on MCFT. (a) Principal stress of UHPC; (b) equilibrium of UHPC element forces.

Considering the balance of forces in the vertical direction, equation (

From equations (

Considering the balance of forces in the horizontal direction and combining with equations (

Based on the strain coordination equation and stress balance equation, the constitutive models of ordinary steel bars, prestressed steel bars, and UHPC are introduced to establish the equation group.

It is assumed that the stress-strain relationship of the ordinary steel bars constitutive model is an ideal elastoplastic relationship as follows:

The constitutive relationship of prestressed steel bars is expressed in two stages, as follows:

Concrete is in the biaxial stress state and the uniaxial stress state, respectively, when considering the pure shear and pure bending stress states; therefore, the constitutive relationship for these two different stress states should be selected separately.

Compression constitutive model of UHPC in the biaxial stress state.

The constitutive relationship of UHPC compression under biaxial stress is described by

The tensile constitutive model of UHPC under the biaxial stress state is shown in Figure

Tensile constitutive model of UHPC in the biaxial stress state.

The uniaxial tensile constitutive model of UHPC is based on the uniaxial tensile test results of UHPC provided by Zhang et al. [

Uniaxial tensile constitutive model of UHPC.

The tensile constitutive relationship of UHPC under uniaxial stress state is shown in the following equation:

In the section analysis of bending-shearing composite action, the test beam is analyzed separately according to pure bending and pure shear stress forms, and the corresponding UHPC constitutive models are used in the calculation.

In the shear analysis and calculations of the test beam, it is assumed that the shear force is uniformly distributed at the section height. Since the shear capacity of a T-shaped beam is affected by the flange width, the contribution of the flange plate to the shear bearing capacity is considered. In the calculation, the effective width of a T-shaped beam is taken as _{i}, and the UHPC constitutive model used equations (

Shear section of the T-shaped beam.

In the flexural analysis of beams, since ordinary concrete does not consider the strength of cracked concrete, the contribution of this part is not considered in the modified compression field theory; however, the residual tensile strength of UHPC beams after cracking cannot be ignored. Therefore, this paper introduces the improved UHPC tensile component, which is calculated by equations (

Figure

Stress-strain under the pure bending moment: (a) strain distribution; (b) stress distribution.

The flexural capacity

The main variables calculated in this paper are the section failure angle

Because stirrup stress is equal to the assumed value, the discriminant conditional expression (

The discriminant conditional expression (

The discriminant conditional expression (

Combining the above deformation compatibility equation, stress balance equation, material constitutive relation, and three convergence discriminants, the MCFT-based calculation procedure of the shear bearing capacity of UHPC beam was compiled by MATLAB, as shown in Figure

Block diagram of the calculation program.

UHPC adopts UA type with reference to the Swiss specification, with a designed compressive strength of 120 MPa and an elastic tensile strength of 7 MPa [

Mix proportion of the UHPC matrix.

Component | Cement | Silica fume | Quartz flour | Quartz powder | Superplasticizer | Water |
---|---|---|---|---|---|---|

Mass ratio | 1.000 | 0.250 | 1.100 | 0.300 | 0.019 | 0.2 |

Mechanical properties of UHPC.

42.2 GPa | 126 MPa | 21.7 MPa |

The calculated spans of the three constructed prestressed UHPC-T beams were 1.4 m, 2.8 m, and 4.2 m, respectively, and the shear-span radios were

(a) Test beam section. (b) Longitudinal reinforcement layout (mm).

Mechanical properties of reinforcement.

Mechanical property | Diameter | |||
---|---|---|---|---|

Structural reinforcement | 384 | 542 | 210 | |

Stirrup | Φ8 | 345 | 500 | 210 |

Longitudinal bar | 345 | 500 | 210 | |

Prestressed reinforcement | Φ15.2–4 | 1820 | 1940 | 195 |

After pouring UHPC, the test beams were covered with plastic film for 2 days and then demolded, followed by steam curing for 3 days (temperature ≥ 90°C; relative humidity > 90%). After steam curing, the beams were stored in a room for 28 days, then tensioned and grouted, and finally naturally cured until the test began.

The device and instruments used in this test are shown in Figure

Load test: this was read by the pressure sensor arranged between the Jack and the reaction frame.

Displacement test: all three test beams were equipped with five displacement sensors (the numbers are 0–4 from left to right).

Strain test: the distribution of the strain gauges of the full beam is illustrated in Figure

Crack test: the width and length of the crack were measured by a crack observation instrument and ruler.

Experimental device and instruments.

Positions of strain gauges. (a) L1, (b) L2, and (c) L3.

The test results are shown in Table

Shear-resistance test results.

Test beam | _{cr} (kN) | _{max} (kN) | Failure mode | |
---|---|---|---|---|

L1 | 1.06 | 945 | 2800 | Inclined compression failure |

L2 | 2.12 | 302 | 1690 | Shear compression failure |

L3 | 3.19 | 805 | 1622 | Diagonal tension failure |

As depicted in Figure

Failure modes of the test beams. (a) L1, (b) L2, and (c) L3.

The failure mode of beam L1 was inclined compression failure. When the load reached 945 kN, the first inclined crack appeared in the web of the right shear span. As the load increased, the crack increased continuously. At this time, the cracks on the left side of the beam appeared more densely, a series of approximately parallel oblique cracks formed at the connection line between the loading point and the support, and the inclination angles of these cracks were in the range of 37–57°. The concrete between the load point and the support was compressed into pieces when the load reached 2700 kN, some of the concrete peeled outward, and the beam was destroyed.

The failure mode of beam L2 was shear compression failure. The first diagonal crack appeared on the left side when loading to 302 kN. As the load increased, cracks appeared continuously on both sides of the webs. When the load reached 1000 kN, many dense diagonal cracks appeared on both sides, and the inclination angles of these cracks were in the range of 32–43°. When the load was increased to 1690 kN, a critical oblique crack between the support and the loading point appeared, and the inclination angle was about 41°. The sound of steel fiber tearing could be clearly heard, and the outer skin of the concrete was peeled off. The stirrup and the longitudinal bars successively yielded, and the concrete was crushed until the beam failed.

The failure mode of beam L3 was diagonal tension failure. When the load was increased to 805 kN, the first diagonal crack appeared on the right web, and the crack developed continuously upon increasing the load. When the load reached 1045 kN, the crack width reached 0.21 mm. As the load continued to increase, many diagonal cracks appeared on both sides of the webs. The inclination angles of these cracks were within the range of 26–35°, and bending cracks began to appear in the bottom slab. When the beam approached failure, critical diagonal cracks appeared on the right side of the beam. The inclination angle was about 28°, the bending cracks of the bottom plate extended and widened, and horizontal tearing occurred at the junction of the top plate and the web until the beam broke.

Upon increasing the shear-span ratio, the inclination angle of the inclined crack of the beam decreased, and the failure mode also changed. When the beam underwent inclined compression failure, the shear bearing capacity was controlled by the compressive strength of concrete, so there was no steel fiber tearing phenomenon. When the beam underwent shear compression failure, the shear bearing capacity was controlled by the bite force between the compressive zone at the top of the detached body and the steel fiber at the inclined crack. The sound of steel fiber tearing could be clearly heard during the failure process. When the beam underwent diagonal tension failure, the shear bearing capacity was controlled by the tensile strength of concrete; however, due to the bridge effect of steel fiber, the tensile strength of the beam was improved, so the ultimate bearing capacity of beams L2 and L3 was similar. The compressive strength of UHPC concrete is much larger than the tensile strength, so the ultimate bearing capacity of beam L1 was much larger than that of beams L2 and L3.

Figure

Load-displacement curves of the test beam.

Figure

Load-wed tensile strain relationship.

Figure

Load-bottom plate tensile strain relationship.

For the load-strain response, the data measured by strain gauges before cracking are reliable; however, after cracking, the distribution of cracks greatly influenced the strain test results. For example, the strain gauge across the cracks will record a larger strain, and the strain gauge not located across the cracks may even record a strain reduction due to cracking of the surrounding area; therefore, only the typical load-strain response is given here.

At present, there are few design specifications for UHPC. The French specification [

Comparison between the test values and code calculated values.

Test beam | _{exp}/_{cal} [ | _{exp}/_{cal} [ | _{exp}/_{cal} [ | _{exp}/_{cal} [ |
---|---|---|---|---|

L1 | 2.79 | 3.42 | 5.45 | 2.41 |

L2 | 1.69 | 2.06 | 3.29 | 1.58 |

L3 | 1.59 | 1.98 | 3.16 | 1.66 |

Average value | 2.02 | 2.49 | 3.97 | 1.88 |

Table

Most specifications are proposed for rectangular beams, and the contribution of the flange part of T-shaped beams to the shear bearing capacity is not considered; thus, the calculations of the specifications are conservative

The residual tensile strength after UHPC cracking contributes a larger proportion to the shear bearing capacity, and the values in the standard calculation are different from the actual values, resulting in calculation errors

The impact of the shear-span ratio is not sufficiently considered in the specifications

The MCFT models contain many calculation parameters and complex convergence conditions, and the shear capacity calculations in existing specifications are too conservative. To facilitate the engineering applications of UHPC beams, the test data from previous studies were collected for fitting [

To verify the correctness and applicability of the MCFT model and formula, the calculated results were compared and analyzed with previously published experimental data, as presented in Table

Comparison of experimental results and predicted values.

Specimen number | Test value (kN) | MCFT (kN) | Equation ( | Theoretical/experimental | Formula/test |
---|---|---|---|---|---|

L1 | 2800 | 2470.0 | 2324 | 0.88 | 0.83 |

L2 | 1690 | 1811.4 | 1825.2 | 1.07 | 1.08 |

L3 | 1622 | 1811.4 | 1411.1 | 1.12 | 0.87 |

Ji et al. [ | |||||

L-2-100-a | 556.0 | 552.5 | 487 | 0.99 | 0.88 |

L-3-100-a | 410.0 | 415.6 | 290.8 | 1.01 | 0.71 |

L-4-100-a | 360.0 | 363.4 | 301.8 | 1.01 | 0.84 |

L-1-100-a | 755.0 | 756.5 | 695.5 | 1.00 | 0.92 |

L-2-200-a | 500.0 | 515.5 | 457.5 | 1.03 | 0.92 |

L-2-150-a | 507.0 | 517.9 | 493.8 | 1.02 | 0.97 |

Chen [ | |||||

B-2-60-90 | 320.0 | 330.5 | 385.2 | 1.03 | 1.20 |

B-2-60-180 | 285.0 | 308.7 | 368.7 | 1.08 | 1.29 |

B-2-30-90 | 314.0 | 310.0 | 338.6 | 0.99 | 1.08 |

B-2-30-180 | 246.0 | 251.0 | 308.9 | 1.02 | 1.26 |

B-2-0-90 | 287.5 | 246.5 | 310.3 | 0.86 | 1.08 |

B-2-0-180 | 240.0 | 239.8 | 276.3 | 1.00 | 1.15 |

B-3-60-90 | 285.5 | 291.8 | 291.6 | 1.02 | 1.02 |

B-1-60-90 | 370.0 | 359.4 | 494.6 | 0.97 | 1.34 |

1.01 | 1.03 | ||||

0.061 | 0.17 |

Table

In the MCFT analysis model proposed in this paper, to simplify the calculation,

The following conclusions can be drawn according to the above research:

In this paper, the shear analysis model of UHPC-T beams based on the modified compression field theory fully considered changes in the shear performance of UHPC structures under the combined action of bending and shearing. The comparison between the calculated and experimental results proved the rationality of the assumptions in the theory, indicating that the established model was suitable for predicting the shear bearing capacity of UHPC beams.

The shear capacity of the UHPC beam decreased upon increasing the hear-span ratio, but the extent of the decrease gradually slowed. In this paper, when the shear-span ratio changed from 1.04 to 2.12, the shear bearing capacity decreased greatly, whereas the decrease was very small when the shear-span ratio changed from 2.12 to 3.19.

The shear-span ratio affects the failure mode of beams. When the shear-span ratio is 1.06, the failure mode is inclined compression failure. When the shear-span ratio is 2.12, the failure mode is shear compression failure. When the shear-span ratio is 3.19, the failure mode is diagonal tension failure.

The shear capacity of the UHPC beam calculated by existing specifications led to conservative results; the smaller the shear-span ratio, the more obvious the difference. The simplified prediction formula proposed in this paper based on the MCFT model and existing test data fully considered the influence of the shear-span ratio. The calculation process was simple and was verified by previously published experimental results, which can provide a reference for the shear design of UHPC beams in practical engineering applications.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was funded by the National Natural Science Foundation of China (no. 51278183), scientific research project of Shaanxi Provincial Department of Transportation (no. 14-18k), and special funds for Basic Scientific Research Fees of Central Universities (no. 310821161102).