In the 1930s, with the advent of the prestressed concrete industry by French engineer Eugene Freyssinet

Deeper girders with larger prestressing forces tend to have more, longer, and broader cracks

Hollow girders are incredibly vulnerable to horizontal end cracking amongst the PC-girders

In the pretensioning method, first, the tension applies to PC-strand, and the strand becomes slightly thinner than the original thickness; after that, concrete to be poured. The prestress application starts once concrete gets its 70% of strength by releasing strands. At this time, the strand attempts to expand outward to get its original thickness, but concrete does not allow the strand to expand

This study's main objective was to determine that whether only placing end-zone reinforcements or debonding some strands can successfully eliminate horizontal cracks at the ends of BS18 hollow PC-girder or requires the combination of both methods. BS represents the B-live load slab girder, and the number illustrates the span length of the girder. Considering the long span of BS18 girder, a large amount of prestressing need to be applied at the girder ends, this results in the crack formation and increase the number of crack occurring points. The most common method to reduce principal stresses at the girder ends is designing end-zone reinforcements

For simulating the BS18 girder, author used Midas FEA software as the Finite Element Analysis (FEA) software. It is challenging to create a representative finite element model of a prestressed concrete girder through any FEA software

BS18 girder is 18 000 mm (18 m) long with 700 mm height and 700 mm width. This girder uses 16 PC-strands organized into three horizontal rows in the cross-section. The girder is entirely hollow except having six transverse diaphragms to form stiffened the girder. According to JIS A 5373 standards

Nonlinear FEA of concrete cracking simulation of a girder with 18 m length can be computationally expensive in time and storage space

The nonlinear analysis was carried out using the structural analysis software Midas FEA. This software is an analysis tool with standard FEM analysis functions in the construction field and can perform detailed analysis to analyze the concrete cracking.

This study used the smeared crack approach in which the stresses and strains of concrete and steel were evaluated by average or smeared values crossing several cracks. Total Strain Crack Model (Rotating Crack Model) was utilized to model the concrete behavior. Hordijk and Parabolic functions were used to model the concrete in tension and compression, respectively. The Von Mises model was used to model PC-strands and steel reinforcement bars. Based on the pre-analysis results ^{2} and 0.2 mm, respectively. Constitutive laws of some materials used in this analysis are shown in

In _{c} and f_{t} show the concrete compressive and tensile strength, respectively, G_{fc} and G_{ft} represent compressive and tensile fracture energy, respectively, ε_{cu} and ε_{tu} denote the ultimate strain of concrete in compression and in tension, respectively, h is the characteristic element length, d_{max} is the maximum size of coarse aggregate, and f_{ud} is the design tensile strength of PC-strands. The values selected for h and d_{max} in this study are 40.34 mm and 20 mm, respectively.

The compressive strength of concrete at the prestressing time was selected 35 N/mm^{2} as per JIS A 5373

Material Properties Analysis Type Compressive Strength (N/mm2) Tensile strength (N/mm2) Modulus of Elasticity (GPA) Compressive fracture energy (N/mm) Tensile fracture energy (N/mm) Prestressing 35 2.46 29.5 52.1 0.1

This study's strand type is SWPR7BL ^{2}, diameter and cross-section area of each strand is 15.2 mm and 138.7 mm^{2}, respectively. The Japan Road Association (JRA) Specifications for concrete bridges^{2}, some relaxations occur after prestressing ^{2}. After detensioning, axial force transfers to concrete, and pretensioning stress decreases to 1110 N/mm^{2} in strands.

It is stated by AASHTO LRFD Bridge Design Specifications ^{2}. The primary finite element models also specify this, stresses which the reinforcement bars carry are far below their yielding stress. The constitutive model of steel reinforcements used in this study is shown in

SD295 type reinforcement bars used in this study; their modulus of elasticity is 200 000 N/mm^{2}. The reinforcement bars and concrete interaction is considered the reinforcements' stiffness adds to the continuum elements' stiffness (Concrete) in which the reinforcements are located. The added ductility of the concrete after cracking is provided by the reinforcement bars. Once concrete elements reach their cracking limit, their load-carrying capacity drops with increasing deformations, larger loads are then transferred to the steel bars. The degrees of freedom of the bar and concrete elements are the same, and the stresses will be distributed to the surrounding bar elements once the concrete elements soften. Since the concrete material's tension stiffness is too small, even small tension can cause local cracking failure, leading to temporary instability and eventually to convergence problems.

Three-dimensional hexahedral elements were used to model concrete, loading plate, and supports in the nonlinear analysis. These elements are an appropriate type of solid elements that are quite practical in modeling arbitrary shapes while an arbitrary mesh is required. Steel reinforcement bars were created as "Embedded Reinforcement Bar Element." In this concept, the reinforcements' stiffness adds to the continuum elements' stiffness in which the reinforcements are located. PC-strands were created as 2-node linear three-dimensional truss elements. A truss element transmits only axial forces and may combine with tension-only/compression-only functions

In pretensioned girders, total prestressing forces transfer to concrete by the bond between the prestressing strand and the surrounding concrete ^{2} and 0.2 mm, respectably. The constitutive model of bond-slip is illustrated in

The bond between PC-strand and concrete can be attributed to adhesion, mechanical interlock, and friction, depending upon the stage of bond development and the nature of the strand's surface. It is essential to evaluate the bonding mechanism simulation with a 3D strand. The strand geometry was modeled as a circular rod, and contact interaction definitions were used to simulate bond behavior. In reality, a strand expands in the radial direction due to Poisson's ratio once prestress is released. This expansion is thought to be one of the possible reasons for end cracking in concrete girders. Evaluating the 3D bond behavior of PC strands thus requires simulation of the strand as a three-dimensional object to simulate its radial interaction with the surrounding concrete. The strand's actual shape is a bundle of seven twisted wires, as shown in

Horizontal cracking occurs immediately after prestressing release when there is no service load on the girder. Therefore, the primary loadings considered for finite element analyses were prestressing forces. Time-dependent effects, such as thermal contraction and creep, were irrelevant during the short interval when cracking occurs. Prestressing loads transfer to girder via bond and friction at the girder ends over a "transfer length" distance. At the girder ends where the strands' slip occurs, stresses in concrete are zero, but concrete carries full effective prestress from strands at the end of the transfer length. As PC-strands were included in the finite element model as truss elements, prestressing forces were directly applied to the strands. When concrete got 70% of its compressive strength (35 N/mm^{2}), 187.2 KN prestressing force was applied to each strand.

Because of material nonlinearity, the model simulation needs considerable computation time. Finer meshes lead to further accurate results, but the time duration for completing analysis and computations is also a factor in selecting the mesh size. At the ends of the girder, which are the main regions of interest, a finer mesh is expected to provide a much more accurate solution for the inelastic material so that a finer mesh was generated and used for the whole model. The mesh size was selected as 50 mm for this model, as shown in

In this study, the Iteration method was used as the solution method. FEA solves a set of linear equations at each iteration in the nonlinear analysis. The nonlinear analysis used the “Newton Raphson” method as the solution technique. The relationship between a force vector and displacement vector is no longer linear in the nonlinear analysis. There are many reasons; for example, in the case of material nonlinearity, the relationship becomes nonlinear, and the displacements often depend on the displacements in the earlier stages, e.g., in the case of plastic material behavior

In Case #3 and #4, end-zone reinforcements were placed in both ends of the models. Researchers proposed a design procedure for designing end-zone reinforcements. According to their design procedure

A_{ps} shows the total prestressing steel area, which is equal to 2496.6 mm^{2} for the BS18 girder. The total steel area computed from Eq. (4) was equal to 988.64.56 mm^{2}. The used steel bars in this study were with 10 mm diameter (D10), and the cross-section area of one bar was calculated as 78.5 mm^{2}. The number of stirrups to be placed at each end of the girder was 13 bars. The reference

However, placing 6 or 7 stirrups within 87.5 mm length is difficult. Therefore, the designed steel bars were arranged as a mesh pattern, and it is referred to as the end-mesh, as shown in

Four cases were analyzed for the girder to investigate horizontal end crack occurrences under different conditions, as shown in

Additionally, time-based changes of the maximum principal stresses and strains were investigated at the time of prestressing for each case to check that horizontal cracks occur in which percentage of prestressing at the ends of the BS18 girder. Cracking occurs when the maximum principal stresses exceed the tensile strength of concrete. In this work, variations of the principal stresses and strains were studied in four prestressing stages with 5%, 40%, 70%, and 100% prestressing percentages.

Case |
Conditions |

1 |
Fully bonded all PC-strands |

2 |
Debonding two PC-strands |

3 |
Case 1 condition + End-zone reinforcement |

4 |
Case 2 condition + End-zone reinforcement |

Temporal changes of the principal stress and strain contours at the ends of the BS18 girder during prestressing for all four cases are shown in

Many researchers studied horizontal cracking at the ends of the ordinary PC-girders. But the methods to prevent horizontal cracks at the ends of the hollow girders are not available in the literature, while hollow beams are incredibly vulnerable to the horizontal end crack occurrence. Thus, in this study, a detailed analytical examination was carried out to propose measures to suppress horizontal cracks at the hollow PC-girder's ends during prestressing. First, prestressing test analysis was performed for four cases with the same material, geometric and other model properties. It was confirmed that horizontal cracks at the ends of the BS18 hollow PC-girder could be suppressed by debonding four PC-strands at the ends of the girder for a distance equal to the transfer length and placing one end-mesh made of D10 bars at each end of the girder.

Followings are the conclusions of this work:

Numerical analysis illustrated that the magnitude of principal stresses is more than the tensile strength of concrete at the ends of ordinary BS18 girder and horizontal cracks are likely to occur.

Debonding four PC-strands at the ends of the girder for a distance equal to the transfer length can reduce principal stresses to the level to become less than the tensile strength of concrete, consequently, horizontal end cracks can be eliminated.

When all PC-strands are fully bonded, placing one end-mesh made of D10 bars at each end of the girder cannot reduce principal stresses to the desired level, and horizontal cracks are likely to occur.

Combining both methods (Strand-debonding and placing end-zone reinforcements) can further reduce the magnitude of principal stresses to eliminate horizontal end cracks completely.

Debonding four PC-strands and placing one end-mesh made of D10 bars at each end of the BS18 girder can further reduce principal stresses to a level to be significantly less than the tensile strength of concrete even in the stage of 100% prestressing, and horizontal cracks can be entirely suppressed.

In this study, horizontal cracks are eliminated at the ends of the BS18 hollow girder with numerical analysis. In the future, it is needed to re-examine these methods through experiments.