1. Introduction

2. Segmental Analysis of Reinforced UHPC Shallow T-Shaped Beam

2.1. Tensile Constitutive Relationship of UHPC

The tensile response of UHPC material can be divided into the following stages [25,26]: (1) a linear elastic stage before initial cracking, characterized by a linear stress-strain relationship; (2) a strain-hardening stage, marked by the development of multiple micro-cracks; (3) a strain-softening stage, characterized by rapid crack propagation and stress reduction. In the linear elastic stage, UHPC remains uncracked. During the strain-hardening stage, numerous micro-cracks are dispersed throughout the UHPC matrix. In the strain-softening stage, tensile deformation rapidly develops at the visible main cracks, as shown in Figure 1.

Figure 1. Tensile constitutive responses of UHPC. (a) Stress-strain relationship; (b) Stress-crack width relationship.

Figure 1. Tensile constitutive responses of UHPC. (a) Stress-strain relationship; (b) Stress-crack width relationship.

The tensile stress σ_Ut of UHPC can be expressed as a piecewise linear function of strain ε_Ut or half crack width Δ(Δ=w/2):

$${\sigma }_{Ut}=E_U{\epsilon }_{Ut};{\epsilon }_{Ut}\le {\displaystyle \frac{f_{SH}}{E_U}}$$

(1a)

$${\sigma }_{Ut}=f_{SH}+E_{SH}\left({\epsilon }_{Ut}-{\displaystyle \frac{f_{SH}}{E_U}}\right);{\displaystyle \frac{f_{SH}}{E_U}}<{\epsilon }_{Ut}<{\displaystyle \frac{f_{Ut}}{E_U}}+{\epsilon }_{inel}$$

(1b)

$${\sigma }_{Ut}=f_i-m_i∆;{∆}_{i-1}<∆<{∆}_i$$

(1c)

where E_U is the elastic modulus of UHPC; f_SH is the initial cracking stress; E_SH is the modulus of strain hardening; f_Ut is the tensile strength of UHPC; ε_inel is the permanent plastic strain caused by multiple micro-cracks; m_i and f_i are the slope and y-intercept of segment i of the stress-half crack width segmental curve, respectively; Δ_i is the half crack width at the right endpoint of segment i of the stress-half crack width relationship σ(Δ). The parameters can be obtained either through direct tensile tests or via the secondary backwards method based on flexural tests.

To simplify calculations, the stress-half crack width relationship shown in Figure 1b is converted into an equivalent stress-strain relationship [27]. The equivalent tensile strain within the crack zone is therefore estimated using the following formula:

$${\epsilon }_{Ut}={\displaystyle \frac{∆}{\left(\frac{S_U}{2}\right)}}+{\displaystyle \frac{{\sigma }_{Ut}}{E_U}}+{\epsilon }_{inel}$$

(2)

where Δ/(S_U/2) is the strain caused by crack propagation; σ_Ut/E_U is the strain caused by elastic deformation of UHPC between cracks; ε_inel is the plastic strain caused by micro-cracks. By converting the stress-crack width relationship into a stress-strain relationship according to Equation (2), the stress-strain constitutive model shown in Figure 2 is obtained.

Figure 2. Equivalent stress-strain relationship for tensile UHPC.

Figure 2. Equivalent stress-strain relationship for tensile UHPC.

2.2. Determination of Internal Forces in Reinforced UHPC Shallow T-Shaped Beams

2.2.1. Internal Force of UHPC in Tensile Zone

As demonstrated by the reinforced UHPC shallow T-shaped beam shown in Figure 3, the axial force exerted in the tensile zone of UHPC can be quantitatively derived through the application of an integral, which is based on the stress-strain relationship of UHPC material as defined by Equation (1).

$$F_{Ut}={\int }_{hc}^{h-h_c}{\sigma }_{Ut}dA={{\overline{\sigma }}_{Ut}A}_{Ut}$$

(3)

where A_Ut is the area of the UHPC tensile zone; h−h_c is the height of the tensile zone; ${\overline{\sigma }}_{Ut}$ is the average stress of UHPC in the tensile zone and can be determined by

$${\overline{\sigma }}_{Ut}\approx {\displaystyle \frac{{\int }_0^{{\epsilon }_D}{\sigma }_{Ut}d\epsilon }{{\epsilon }_D}}$$

(4)

The arm of force of UHPC in tensile zone is defined as

$$l_{ct}=\eta \left(h-h_c\right)$$

(5)

$$\eta ={\displaystyle \frac{{\int }_0^{{\epsilon }_D}{\sigma }_{Ut}\epsilon d\epsilon }{{\epsilon }_D^2{\overline{\sigma }}_{Ut}}}$$

(6)

where η is the ratio of the distance between the centroid of the tensile stress distribution and the neutral axis to the height of the tensile zone.

Figure 3. Cross-section of reinforced UHPC shallow T-shaped beam.

Figure 3. Cross-section of reinforced UHPC shallow T-shaped beam.

2.2.2. Internal Force of UHPC in Compressive Zone

The UHPC in the compressive zone is considered to be linear elastic as defined by the modulus of elasticity E_U. The axial force in the UHPC compressive zone is thus given by

$$F_{Uc}={\int }_0^{h_c}{\sigma }_{Uc}dA={\displaystyle \frac{1}{2}}{b}_f{h}_c^2{E}_U\phi$$

(7)

where φ is the curvature, b_f is the flange width, and the height of the arm of force in the compression zone relative to the neutral axis is 2/3h_c.

2.2.3. Internal Force of Reinforcement in Tensile Zone

The elevation of tension-stiffening zone between two cracks for a reinforced UHPC element is shown in Figure 4. The axial force in tension reinforcement within the T-shaped beam is given by:

$$F_{rt}=\gamma {\alpha }_E{A}_{rt}{E}_U\phi \left(h_{rt}-h_c\right)+F_{rt0}$$

(8)

$$F_{rt0}=-E_r{A}_{rt}{\epsilon }_{sh}$$

(9)

where α_E is the elasticity modulus ratio of reinforcement to UHPC; φ is the curvature; h_c is the height of the compressive zone; γ is the increased stiffness coefficient due to tension stiffening effect. Before the formation of macro-cracks (ε_D < f_Ut/E_U+ε_inel), γ is taken as 1, and after the formation of a macro-crack (ε_D ≥ f_Ut/E_U+ε_inel), γ is taken by the following expression:

$$\gamma ={\displaystyle \frac{\xi -n_f}{1-n_f+\frac{\xi -1}{\left(\frac{E_U{A}_{U-r}}{E_r{A}_{rt}}+1\right)}}}$$

(10)

$$n_f={\displaystyle \frac{m_i}{E_U}}{\displaystyle \frac{S_U}{2}}$$

(11)

$$\xi ={\displaystyle \frac{{\lambda }_1\frac{S_U}{2}}{tanh\left({\lambda }_1\frac{S_U}{2}\right)}}$$

(12)

$${\lambda }_1=\sqrt{k{u}_s\left({\displaystyle \frac{1}{E_r{A}_{rt}}}+{\displaystyle \frac{1}{E_U{A}_{U-r}}}\right)}$$

(13)

Figure 4. Elevation of tension stiffening region between cracks of reinforced UHPC beam.

Figure 4. Elevation of tension stiffening region between cracks of reinforced UHPC beam.

2.2.4. Internal Force of Reinforcement in Compressive Zone

The reinforcement in compressive zone is as assumed to be linear elastic. Hence, the axial force of reinforcement is derived as

$$F_{rc}={\alpha }_E{A}_{rc}{E}_U\phi \left(h_{rc}-h_c\right)+F_{rc0}$$

(14)

where, the force attributed to shrinkage strain is given by

$$F_{rc0}=-E_r{A}_{rc}{\epsilon }_{sh}$$

(15)

2.4. Segmental Analysis of Reinforced UHPC Shallow T-Shaped Beams

To ascertain the moment-rotation behavior of a beam, the uncracked segment as shown in Figure 6a is initially considered. It is assumed that the stress transfer length L_tr is equal to half the crack spacing (S_U/2), while the A-A section represents the initial position of the segment. As the shrinkage strain develops in the segment, if the reinforcement is not bonded, the resulting contraction is manifested as Section B-B. Due to the bond nature between UHPC and reinforcement, the contraction results in compressive forces being exerted on the reinforcement, while tensile forces are generated in the concrete to maintain equilibrium. This leads to the formation of Section C-C at a rotation angle θ_sh. The application of an external moment results in an increase in rotation θ to achieve equilibrium between force and moment, leading to the formation of Section D-D. For analytical purposes, the shrinkage strain ε_sh can be determined by shrinkage-relevant tests of UHPC material. The influence of creep can also be incorporated through the modification of the elastic modulus of UHPC using the age-adjusted effective modulus method [28,29]. A division of the deformation profile in Figure 6a by the transfer length L_tr produces the strain profile illustrated in Figure 6b, which represents the strain subsequent to the imposition of shrinkage strain and external moment. The sectional strain distribution of both UHPC and reinforcement is defined in Figure 6b, where the difference in their strains representing the shrinkage strain.

Figure 6. Detailed analysis of reinforced UHPC shallow T-shaped beam.

Figure 6. Detailed analysis of reinforced UHPC shallow T-shaped beam.

Based on the strain distribution and constitutive relationship of UHPC material, the sectional stress distribution can be obtained as shown in Figure 6c, and its integration then produces the internal force profile in Figure 6d. By applying the force and moment equilibrium equation, the relationship between the external moment M and curvature can then be solved. As the growing of the external moment, the strain ε_D of UHPC bottom face will eventually reach the multi-cracking strain. Thereafter, the segment in Figure 6a is replaced by the one ass shown in Figure 6e. The emergence of microcracks leads to the stress hardening of UHPC, as illustrated in Figure 6g. Upon arriving of strain ε_D at the macrocracking strain, f_Ut/E_U+ε_inel, macrocracks are initiated on the bottom face of beam, as represented in Figure 6i. The presence of macrocracks indicates the stress softening of tensile response, as shown in Figure 6k. In this situation, the effective stiffness of the tensile reinforcement increases due to the tensile stiffening effect. It is represented by the multiplication of the axial rigidity of the reinforcement by the tension-stiffening parameter defined by Equation (10). By employing the moment-rotation approach, this enables the derivation of the relationship between moment-curvature and moment-crack width, and further determines the deflections and crack widths in the section.

The strain at the bottom of UHPC beam ε_D is applied, and then the curvature is

$$\phi ={\displaystyle \frac{{\epsilon }_D}{h-h_c}}$$

(18)

To determine the neutral axis depth, the force equilibrium is referred as

$$F_{Ut}+F_{rt}+F_{Uc}+F_{rc}=0$$

(19)

Substituting Equations (1), (5), (8), (14), and (18) into Equation (19), for a T-shaped section of UHPC shallow beam the following is obtained:

$$a_0+a_1{h}_c+a_2{h}_c^{2 }=0$$

(20)

where

$$a_0=E_U{\epsilon }_D\left(\gamma {\alpha }_E{A}_{rt}{h}_{rt}+{\alpha }_E{A}_{rc}{h}_{rc}\right)+\left(F_{rt0}+F_{rc0}\right)h+{\overline{\sigma }}_{Ut}{b}_w{h}^2$$

(21a)

$$a_1={-E}_U{\epsilon }_D\left(\gamma {\alpha }_E{A}_{rt}+{\alpha }_E{A}_{rc}\right)-\left(F_{rt0}+F_{rc0}\right)-2{\overline{\sigma }}_{Ut}{b}_{w}h$$

(21b)

$$a_2={\overline{\sigma }}_{Ut}{b}_w-{\displaystyle \frac{1}{2}}{b}_f{E}_U{\epsilon }_D$$

(21c)

After determining the height of neutral axis from Equations (20) and (21), the curvature can be calculated using Equation (18), and then internal forces can be evaluated using Equations (3), (7), (8), and (14), thereby determining the external moment M of the section. By rearranging Equation (2), it arrives

$$w=S_U\left[\phi \left(y-h_c\right)-{\displaystyle \frac{{\sigma }_{Ut}}{E_U}}-{\epsilon }_{inel}\right]\ge 0$$

(22)

By applying aforementioned process, it allows for the evaluation of the moment-curvature and moment-crack width relationships in a parametric manner across a range of bottom strains ε_D. It is noteworthy that the maximum crack width of the section can be obtained from Equation (22) under the assumptions made during the derivation of crack spacing.

3. Simplified Analysis Model for Reinforced UHPC Shallow T-Shaped Beams

Figure 7. Simplified analysis model for moment-curvature and moment-crack width relationships: (a) Moment-curvature relationship; (b) Moment-crack width relationship.

Figure 7. Simplified analysis model for moment-curvature and moment-crack width relationships: (a) Moment-curvature relationship; (b) Moment-crack width relationship.

4. Four-Point Bending Test of Reinforced UHPC Shallow T-Shaped Beams

Figure 8. Geometric dimensions of T-shaped beam specimen (unit: mm).

Figure 8. Geometric dimensions of T-shaped beam specimen (unit: mm).

Figure 9. Cross-sectional reinforcement configuration of T-shaped specimen (unit: mm).

Figure 9. Cross-sectional reinforcement configuration of T-shaped specimen (unit: mm).

Table 1. Characteristics of steel fibers used within UHPC.

Table 2. Design parameters and measured material properties of various specimens.

Figure 10. Photos of four-point bending tests for reinforced UHPC specimens: (a) test setup and loading scheme; (b) electro-hydraulic automatic control testing machine.

Figure 10. Photos of four-point bending tests for reinforced UHPC specimens: (a) test setup and loading scheme; (b) electro-hydraulic automatic control testing machine.

5. Result Analysis and Discussion

Figure 11. Load-deflection curves of reinforced UHPC shallow T-shaped specimens. (a) Measured load-deflection curves from static four-point bending tests; (b) Comparison between theoretical and measured load-deflection curves.

Figure 11. Load-deflection curves of reinforced UHPC shallow T-shaped specimens. (a) Measured load-deflection curves from static four-point bending tests; (b) Comparison between theoretical and measured load-deflection curves.

Figure 12. Load-crack width curves of reinforced UHPC shallow T-shaped specimens in static flexural tests. (a) Load-crack width curves of the whole response; (b) Load-crack width curves prior to reinforcement yielding.

Figure 12. Load-crack width curves of reinforced UHPC shallow T-shaped specimens in static flexural tests. (a) Load-crack width curves of the whole response; (b) Load-crack width curves prior to reinforcement yielding.

6. Conclusions

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