Effectiveness of external prestressing in enhancing the non-ductile hanger failure mechanism in reinforced concrete inverted T-beams

Abstract

Recently, inverted T-beams have been used in reinforced concrete (RC) bridges to support transverse precast stringers. Inverted T-beams, contrary to practice with conventional beams, are loaded on the flanges upper surface. This loading configuration causes hanger failure due to the generation of vertical tensile stresses near the bottom of the web. The key purpose of this study is to investigate the efficiency of vertical external prestressing stainless-steel bars in mitigating non-ductile hanger failure in reinforced concrete inverted T-beams. An experimental study on six inverted-T beams, including two un-strengthened specimens, was carried out. The study showed that the value of the prestressing level had a considerable impact on the performance of hanger mechanism in relation to crack pattern, ultimate loads, cracking behavior, load–deflection, strains, and ductility. The experimental results indicated that the suggested method for strengthening inverted T-beams had efficacy in reducing the seriousness of the non-ductile hanger failure and resulted in a strength increase of up to 53% when compared to that of the un-strengthened specimen. Additionally, two analytical models for estimating the hanger capacity and the average crack width of the strengthened RC inverted T-beams were proposed. The models that were proposed exhibited a high degree of agreement with the experimental results.

Graphical abstract

Keywords

RC bridges / inverted T-beams / strengthening / hanger failure / external prestressing

1 Introduction

Reinforced concrete (RC) inverted T-beams are a common construction solution for many existing RC bridges, as shown in Fig.1. Clearance restrictions may affect the overall height of the bridge when it is built over a roadway. In conventional rectangular beams, the precast stringer sits on the top of the rectangular section. This method leads to an increase in the overall bridge height, which increases the overall cost. The utilization of RC inverted T-beam involves positioning the precast stringer atop the upper surface of the ledges (bottom flange). This configuration results in a reduction in the overall depth of the bridges and the potential clearance height beneath them.

Precast stringers bearing on the top face of an inverted T-beam’s flange create a more complex behavior than that of regular rectangular beams [1–6]. Failures of RC rectangular-section beams may be separated into two distinct types, namely flexural and shear failures. Shear failure can occur even if the beam’s flexural capacity is not exceeded, resulting in a non-ductile failure that is marked by its suddenness. This type of failure is considered to be more catastrophic than flexural failure. In contrast, there are six different failure modes that can occur as a result of the position of loads on RC inverted T-beams. These failure modes include flexural failure caused by tension reinforcement yielding, flexural failure caused by top concrete crushing, and overall shear failure in the web near supports. Additionally, there are three other failure modes that may occur, related to the bottom flange: shear friction failure at the interface between the flange and the web, punching shear failure of the flange at the point of loading, and failure of the hanger reinforcement responsible for supporting the load up to the compression chord. The six failure modes in RC inverted T-beams are illustrated in schematic form (Fig.2). When an external load is applied to the bottom flange the most common type of failure that can occur in this setting is known as hanger failure. This particular type of failure is characterized by the development of a vertical breakdown between the web and the flange.

Mirza and Furlong [7] investigated the behavior of one-third scale models of typical inverted T-beams used for bridges in terms of strength and serviceability. The specimens had various reinforcement details and different amounts of hanger reinforcement for the same cross section and approximately the same concrete strength. Of the 27 observed failures, four resulted in pure punching shear failure of the flange, six tests resulted in failures due to pure hanger failure between the web and flange connection, and four tests resulted in failures due to combined hanger failure and punching shear failure. Based on these findings, Raths [8] highlighted some concerns about the safety level provided by the PCI equations for the ledge’s carrying capacity. The purpose of that research was to review common issues related to inverted L-beams and to develop design recommendations. Raths developed the hanger reinforcement requirement that is addressed in the PCI Handbook [9] and the research recommended using the following equation (Eq. (1)) to calculate the hanger reinforcement (A_sh):

(1)A_{\mathrm{s}\mathrm{h}}=\frac{V_{\mathrm{u}}(b_{\mathrm{w}}+e-0.5)}{\varphi f_{\mathrm{y}}(b_{\mathrm{w}}+d_{\mathrm{s}}^{\prime }-0.5)},

where V_u is the ultimate applied load, f_y is the reinforcement yield strength, b_w is the width of the web, d'_s is the distance from centroid of hanger reinforcement to the inner web face, e is the distance from the applied load to the inner web face, and ϕ is the strength reduction factor.

Klein [10] conducted an experimental full-scale load test on inverted L-beams. The failure loads were observed to be less than the loads predicted by the current PCI equations. Furthermore, Klein stated that when designing the hanger reinforcement for ledge and web, it is important to consider the eccentricity of the applied load to the centroid of the resisting section of the L-shaped beam. Recently, Garber et al. [11] studied several factors that affect the strength of RC inverted T-beams, including the hanger reinforcement ratio and the thickness of the ledge. That study found that the hanger reinforcement in the web was the key factor for strength. Rizkalla et al. [12] calibrated the specimens tested by Klein [10] using the finite element method (FEM) to confirm its capability relative to the measured behavior. The calibrated models’ outputs were in good accordance with the observed behavior. The hanger carrying capacities predicted by the FEM analysis were 90% of the experimental capacities. Also, the calibrated models confirmed that the PCI equations were not conservative in estimating the hanger capacity.

Despite the use of RC inverted T-beams in some existing bridges, which were designed based on older codes [13,14], there has been little study on assessing their strengthening techniques. Externally prestressing techniques were studied in research [15–24] as an alternative to traditional strengthening techniques such as concrete jackets [25–27], steel plates [28–30], or Fiber Reinforced Polymers (FRPs) [31–46]. However, one of the types of failures that may occur when using FRPs in the strengthening is de-bonding, which is an undesirable failure mode. In this case, the FRP materials delaminate from the concrete surface prior to reaching the rupture strain. Therefore, the FRPs are unable to utilize their full tensile capacity. Using a concrete jacket for strengthening the girders in RC bridges leads to an increase in the cross-sectional dimensions, hence an increase in the self-weight. Strengthening using steel plates is widely used for various structural elements. However, steel plates do not provide a long-term solution due to corrosion issues, especially in applications exposed to severe environmental conditions.

For the purpose of prestressing, a technique known as external post-tensioning involves positioning tendons externally to the concrete element. Because of its many benefits, the use of external prestressing as a technique for strengthening reinforced concrete structures has become more common. The benefits include affordable building practices, uncomplicated monitoring and maintenance, and ease of tendon application. This research presents experimental and analytical studies on the application of external vertical prestressing stainless-steel bars for strengthening RC inverted T-beams to enhance their hanger failure mechanism under symmetric loading.

2 Experimental work

2.1 Test specimens

As presented in Tab.1, the experimental work program consists of study of six RC inverted T-beams. There were two reference specimens; B1 without vertical stirrups (hanger reinforcement) at the mid-span; B2 with minimum vertical stirrups at the mid-span acting as hanger reinforcements. There were four strengthened inverted T-beams using external vertical prestressing technique with different levels of prestressing (0%, 30%, 45%, and 60% ε_py). The prestressing levels were taken to be similar to earlier studies that involved the strengthening of RC structural elements, which varied from 30% to 60% ε_py [47–52].

The specimens were designed based on the formula presented by Mirza and Furlong [7] with the objective of avoiding all types of failure except for hanger failure. The specimens were subjected to simple support with a center-to-center distance of 3000 mm, and had a total length of 3200 mm. It was decided that for all the specimens, the width of the web would be 200 mm, and the total height would be 600 mm. As a preventative step against the possibility of punching shear failure occurring, a flange thickness of 200 mm and a width of 600 mm were decided upon. As can be seen in Fig.3, both the upper and lower steel reinforcements were the same for all specimens. Fifteen bars with a diameter of 18 mm were used in the T-beam construction process. Nine bars were employed as lower reinforcement, while the remaining six were utilized to fix the stirrups. To prevent shear failure within the web, a series of 10 mm-diameter vertical stirrups were employed, with a spacing of one stirrup per 100 mm and distributed at intervals of 1200 mm from both ends of the specimen. It is noteworthy that specimen B1 was deliberately constructed without the presence of hanger stirrups at the mid-span section. This was done with the aim of investigating the extent to which the existing hanger reinforcement contributes to the load-carrying capacity. Nevertheless, in accordance with ACI 318-19 [53], other specimens were provided with minimal stirrups at the mid-span (8 mm in diameter and placed at 200-mm intervals).

2.2 Material properties

The specimens in the plywood formwork were cast concurrently and subjected to curing with moist burlap. The present study employed ready-mixed concrete comprising 1040 kg/m³ of coarse aggregate with a maximum size of 20 mm, 760 kg/m³ of fine aggregate, 400 kg/m³ of cement, and 160 kg/m³ of water. Six standard cylinders of concrete, with an average diameter of 150 mm and an average height of 300 mm, were cast and allowed to cure for quality control assessments. The mechanical properties of the concrete, including the compressive strength of the cylinders, modulus of elasticity, and Poisson’s ratio, were determined in accordance with ASTM standards [54,55]. In accordance with the previously stated standards, the absolute values for the aforementioned properties were determined to be 29.70 MPa, 26435 MPa, and 0.240, respectively, as presented in Tab.2.

The values shown in Tab.3 were obtained by tensile testing of the steel bars to estimate the yield strength, ultimate strength, and modulus of elasticity, in accordance with the guidelines prescribed by the ASTM A615/A615M standard [56]. Stainless steel bars with a high strength and a diameter of 12 mm were utilized in order to provide the prestressing force. These stainless-steel bars were used in conjunction with mild steel plates that had standard dimensions of 150 mm × 500 mm × 30 mm. To guarantee that the suggested technique of strengthening would be successful, one of the most important steps was to choose steel plates with an appropriate level of flexural stiffness. As a result, the steel plates were selected with great care to ensure that they had an adequate level of stiffness and to avoid any potential bending deflections. In addition to this, the distance of the clear span between the two stainless-steel bars was deliberately designed to be relatively close to the web formed by the inverted T-beams. The yield strength of the stainless-steel bars was measured at 600 MPa, while their ultimate strength was measured at 740 MPa, and their modulus of elasticity was measured at 200 GPa. The steel plates had a modulus of elasticity of 210 GPa, a yield strength of 255 MPa, and an ultimate strength of 375 MPa.

2.3 Prestressing technique

In this paper, the details of the external prestressing developed by the author are shown in Fig.4. The external prestressing technique uses six stainless-steel bars, six steel plates, and 12 nuts for each specimen. Two 12 mm diameter stainless steel bars were fastened from the top and bottom of the used steel plates by 14 mm nuts and steel plates with a thickness of 30 mm. The prestressing force was generated using a torque key by tightening the nuts at the bars’ ends. A strain gauge was installed on each bar, to control the prestressing level before the test, and to measure the increase in strain during the test. The losses caused by nuts’ movements and elastic deformations of steel plates were not considered because the test was performed immediately after the prestressing process.

One of the important aspects that should be considered to simulate the reality of the application of the proposed solution for an existing RC bridge (e.g., the one shown in Fig.1) is that, before the implementation of the suggested strengthening system, the area of the slab above the strengthened inverted T-beams in-between the two precast stringers should be removed first. After applying the prestressing technique, all removed parts should be cast using a special category of high-performance fiber reinforced cementitious composites (HPFRCC), such as strain-hardening cementitious composites (SHCC), to avoid corrosion attack.

2.4 Test setup and instruments

As shown in Fig.5, the experimental tests consisted of applying a symmetric four-point loading protocol to all inverted T-beams that only had simple supports. The specimen’s tested span was measured to be 3000 mm, while the distance between the loading points was 800 mm. For all experimental work, a specially manufactured steel frame with high stiffness was designed to uniformly distribute the applied load. The deflection under each loading point was obtained using 100 mm linear variable differential transducers (LVDTs). In addition to this, a pair of LVDTs were fastened atop the supporting points so that they could measure the total net deflection of the specimens that were being evaluated. In order measure the relative displacements between the upper and lower plates, to confirm the stiffness of the steel plates that were used, two LVDTs were mounted beneath them. The results of this measurement were that the relative displacements between the plates were almost negligible. A microscope with a resolution of 0.02 mm was used to measure the width of the cracks at each of the different loading steps. Electrical strain gauges were attached to the stirrup that was placed in the mid-zone, the stirrup that was positioned in the shear span, and the flexural reinforcement that was located in the mid-span. During the period of the test, a π-shaped displacement transducer that was linked to the mid-span concrete surface was responsible for recording the maximum compressive strain that the concrete experienced. During each stage of the test, a data logger device (TDS-150) collected automated recordings of the strains, as well as the loads and vertical displacements.

3 Experiment results

3.1 Crack pattern and ultimate loads

Fig.6 depicts the final crack patterns for all tested specimens. For the control specimen B1, without hanger reinforcement, no cracks were observed in the early stages of loading. At the point that the tensile strain in concrete reached its maximum cracking value, vertical flexural cracks at the tension region appeared. Shortly after the flexural crack formed, the first crack initiated at the intersection between the web and flange at about 254 kN (65%P_u) and then moved up at an angle of approximately 40° to create arch cracks in the web. As the load increased, the horizontal crack width between the web and the flange increased until the occurrence of non-ductile hanger failure at a load of about 392 kN. The implementation of mid-zone hanger reinforcement, as shown in specimen B2, was able to successfully postpone the appearance of the first hanger cracks (horizontal cracks at the connection between the web and the flange) until a load limit of about 262 kN. In addition to this, the failure mode that was displayed by specimen B2 was somewhat more ductile than that displayed by the control specimen B1. Additionally, when compared to specimen B1, the ultimate load of the tested specimen B2 showed a 23% increase.

Conversely, the prestressing technique had a significant impact on increasing the initial cracking loads and the ultimate loads for all tested specimens, as indicated in Tab.4. Furthermore, it was observed that the strengthened specimens exhibited a mode of failure that was predominantly ductile, as opposed to the un-strengthened specimens B1 and B2. The initial cracking loads of the strengthened specimens B3, B4, B5, and B6 exhibited increments of 6%, 23%, 35%, and 48%, respectively, in comparison to B2. The prestressing technique resulted in a rise in the ultimate loads of the strengthened specimens. The specimens B3, B4, B5, and B6 exhibited ultimate loads of 599, 681, 713, and 739 kN, respectively. These values were found to be 24%, 41%, 48%, and 53% higher than the ultimate load of specimen B2.

3.2 Experimental hanger capacity versus analytical estimation

Fig.7 depicts a schematic illustration of the resistance forces that develop throughout the strengthening of inverted T-beams as a result of the prestressing approach. Equation (2) demonstrates that the hanger capacity of RC inverted T-beams that have been strengthened with external prestressing bars is comprised of three distinct contributions. According to Eq. (3) [57], the first contribution \left(V_{\mathrm{c}}\right), refers to the resistance that is produced by the concrete section. The resistance is primarily contingent upon the effective area of the cross section (B_f× d_f) and the average concrete compressive strength (f_{\mathrm{c}}^{\prime }) of standard cylinders.

(2)V_{\mathrm{u}}= {2V}_{\mathrm{c}}+V_{\mathrm{s}\mathrm{w}}+V_{\mathrm{p}\mathrm{s}},

(3)V_{\mathrm{c}}= 0.17\sqrt{f_{\mathrm{c}}^{\prime }}{B}_{\mathrm{f}}{d}_{\mathrm{f}},

where d_f is the flange’s effective depth; and B_f is the flange’s width. The second contribution of web reinforcement, namely hanger resistance \left(V_{\mathrm{s}\mathrm{w}}\right), is shown in Eq. (4). This contribution is dependent upon the effective depth of the flange (d_f) and the width of the bearing steel plate (B_s). The third term is the contribution of the prestressing strengthening technique \left(V_{\mathrm{p}\mathrm{s}}\right). This contribution depends on the prestressing force in each bar. According to the experimental outcomes, all recorded strain values of the external prestressing bars reached the yield strength and can be calculated by Eq. (5).

(4)V_{\mathrm{s}\mathrm{w}}=\frac{B_{\mathrm{s}}+2d_{\mathrm{f}}}{S}{f}_{\mathrm{y}\mathrm{w}}{A}_{\mathrm{s}\mathrm{w}},

(5)V_{\mathrm{p}\mathrm{s}}=f_{\mathrm{p}\mathrm{s}}{A}_{\mathrm{p}\mathrm{s}},

where f_yw is the hanger reinforcement yield strength, A_sw is the amount of the hanger reinforcement, f_ps is the external prestressing bar yield strength, A_ps is the amount of the external prestressing bars, and S is the hanger reinforcement’s longitudinal spacing. An illustrative representation of the hanger resistance pertaining to the inverted T-beam B6 is provided in Appendix A.

3.3 Load–deflection behavior

Fig.8 illustrates the vertical mid-span deflection and the corresponding applied load for all specimens at each load step. Furthermore, the outcomes of the initial stiffness are presented in Tab.4 and were derived from the gradient of the load–deflection graph immediately preceding the occurrence of cracking [58]. It is important to keep in mind that, to track post-peak behavior after reaching the ultimate load, the jacking load was gradually released. A bilinear relationship can be used to roughly describe the load–deflection relationship of specimens. The first stage started at the beginning of the test and ended when the initial hanger crack appeared at the connection between the flange and the web. The specimens behaved linearly at this stage, and the initial stiffness depended on the value of prestressing. The second stage began with the when the cracking began and ended with failure. Around this stage, the stiffness began to degrade more quickly, and the load–deflection relationship was affected by the strengthening technique. In particular, at the same load level (480 kN), the implementation of the external prestressing technique resulted in a noteworthy reduction of mid-span deflection, amounting to 75.1% of that for the un-strengthened reference specimen B2. Additionally, as indicated in Tab.4, the stiffness of specimens B3, B4, B5, and B6 exhibited a rise of 11.2%, 30.4%, 52.2%, and 96.1%, correspondingly, in comparison to the un-strengthened reference specimen B2.

3.4 Ductility

In comparison to brittle materials, ductile materials possess the ability to undergo inelastic deformation beyond their initial yield points while still preserving their load resistance. This study selectively used two ductility indices, namely displacement-based and energy-based ductility, to assess the ductility effectiveness of the pre-stressing technique. The displacement ductility index \left({\mu }_{\mathrm{\Delta }}\right) can be defined as the ratio of the maximum displacement ({\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}) and the yielding displacement ({\mathrm{\Delta }}_{\mathrm{y}}), as established by prior research [59]. The energy ductility index ({\mu }_{\mathrm{E}}) is a metric that quantifies the relationship between the energies of a specimen at the point of failure (E_{\mathrm{m}\mathrm{a}\mathrm{x}}) and yielding (E_{\mathrm{y}}) loads. To obtain additional details, the theoretical computation of the maximum deflection ({\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}) and yielding deflection ({\mathrm{\Delta }}_{\mathrm{y}}) was conducted using the methodology recommended by Baraghith et al. [60], as illustrated in Fig.9. As shown in Tab.5, the specimens that were strengthened with external vertical pre-stressed stainless-steel bars showed low ductility. In detail, the strengthened specimens had a loss in ductility of approximately 18% compared to the un-strengthened specimen B2. The main cause of the reduction in ductility was low vertical deflection due to the generated external prestressing force.

The enhancement of load capacity Is a crucial consideration in the strengthening of RC inverted T-beams. The present study involves the computation of the performance factor to ascertain the efficacy of the employed strengthening method. According to Afefy et al. [59], the performance factor (PF) can be expressed as the result of multiplying the displacement ductility factor (DF) and the strength factor (SF). The ductility factor (DF) can be defined as the value obtained by dividing the displacement ductility index of a given specimen by the displacement ductility index of the control specimen B1. The strength factor (SF) is defined as the ratio of the ultimate load of a given specimen and the ultimate load of the control specimen B1. Specimens that were strengthened with external vertical prestressing stainless-steel bars exhibited greater performance factor when compared to the control specimen B1.

3.5 Strains

The load–strain relationship of the prestressing bars is presented in Fig.10. The strain values were initially set at levels of 0%, 30%, 45%, and 60% ε_py, prior to loading. It is noteworthy that the strain values exhibited a slight alteration prior to the appearance of cracks at the intersection of the web and flange. The minimal strain change in the external prestressing bars can be attributed to the minor deformation assumption, prior to cracking, as predicted by classical elastic mechanics. Following the formation of cracks, the specimen experienced significant vertical deformation and a marked increase in strain due to the tensile stresses. As a result, the hanger resistance is composed of two factors: first, the resistance of concrete is increased by the external vertical compressive stress produced by the prestressing technique, and secondly, the external prestressing bars act as hanger reinforcement to carry the tensile force.

Fig.11 depicts the relationship between the load applied and the associated tensile strain experienced by the hanger stirrups in the studied zone. The load–strain graph for every specimen that was examined demonstrates an initial linear propagation from the origin point down the vertical axis until the occurrence of the first cracking. The behavior that was seen suggests that the employment of stirrups did not improve the ability of the specimen to resist hanger cracking before the initial development of hanger cracks. After the cracking stage, the second part of the relationship between load and strain grew in a way that was’t linear up to the stirrup’ yielding strain. In other words, a slight plateau was associated with the abrupt opening of cracks. Moreover, the strengthened specimens with externally prestressed stainless-steel bars showed lower tensile strain in the vertical stirrups than in the reference specimen B2 at any load level. In detail, the tensile strain of vertical stirrups at failure for specimen B2 was 2111 × 10⁻⁶, while this value for strengthened specimens was in the range of 974−1695 × 10⁻⁶.

3.6 Cracking behavior

Different standards set different restriction values for the maximum crack width for aggressive exposure of RC structures because crack width has an effect on the durability of the structure. The maximum crack width is limited to 0.2 mm by CEB-FIP [61] and ECP 203-2020 [62] standards, while 0.30 mm is the limit established by Euro-code [63]. Therefore, it is crucial to investigate the cracking behavior of the suggested strengthening system.

Fig.12 shows the development of the diagonal crack widths of the tested specimens against the applied load. For the specimens strengthened with external vertical pre-stressed bars, only thin visible cracks were observed up to almost 80% of the ultimate loads. In addition, providing external prestressing as a strengthening technique improved the cracking behavior. Thus, at the service stage, strengthened specimens B3, B4, B5, and B6 had smaller crack widths than those of the un-strengthened specimen B2. The reductions of the average crack widths were 22%, 46%, 54%, and 65% for specimens B3, B4, B5, and B6, respectively, compared to observations for specimen B2. In particular, the extreme crack width for the control specimens at the service load (67%P_u) was found to be 1.90–2.65 mm, but only 0.66–1.48 mm for the strengthened specimens. Finally, it was found that the crack width at the service load was significantly decreased in the strengthened specimens as a result of the compressive force provided by the prestressing technique.

3.7 Analytical evaluation of cracking characteristics for RC inverted T-beams

3.7.1 Hanger cracking capacity (P_cr)

Herein, a proposed equation is used to estimate the hanger cracking load for the reinforced inverted T-beams (refer to Eq. (6)). A relationship between the normalized hanger cracking capabilities and the prestressing level (µ _ps) is established as shown in Fig.13 to determine the impact of the prestressing level (µ _ps) on the hanger cracking load (P_cr) of the strengthened RC inverted T-beam. The relationshi’s trend line can be utilized to illustrate the impact of the prestressing level ratio via factor k_ps, as presented in Eq. (7). It is noteworthy that the capacities of the experimental hangers were normalized by dividing them by the product of the square root of the compressive strength of concrete (\sqrt{f_{\mathrm{c}}^{\prime }}), the width of the flange (B_{\mathrm{f}}), and the effective depth of the flange (d_{\mathrm{f}}).

(6)P_{\mathrm{c}\mathrm{r}}=k_{\mathrm{p}\mathrm{s}}\sqrt{f_{\mathrm{c}}^{\prime }}{B}_{\mathrm{f}}{d}_{\mathrm{f}}.

(7)k_{\mathrm{p}\mathrm{s}}=0.50 +\frac{{\mu }_{\mathrm{p}\mathrm{s}}}{3}.

3.7.2 Average hanger crack width (w_{\mathrm{a}\mathrm{v}\mathrm{g}})

Tab.6 presents a comprehensive analysis of previous analytical models [64–66] utilized to calculate the average crack width. Tab.7 presents the computed average crack width values based on the previously established models, in comparison to the experimental outcomes of the average crack widths for all the tested RC inverted T-beams. It is noteworthy that the experimental outcomes concerning the average crack widths of every specimen were evaluated at the service load level (0.67P_u). Evidently, as depicted in Fig.14, the previously utilized models demonstrate a tendency toward conservative outcomes and have neglected to account for the influence of prestressing levels. Thus, it is imperative to develop a new model that considers such influence. It is important to recognize that the symbols referenced in previous analytical models [64–66] have been clarified in the notation.

(1) Derivation of average hanger crack width formula

The equation proposed for predicting the average hanger crack width of reinforced concrete inverted T-beams was formulated on the basis of strain compatibility and the theory of nearly linear correlations between the mean crack width (w_{\mathrm{a}\mathrm{v}\mathrm{g}}), concrete’s tensile strain ({\epsilon }_{\mathrm{c}\mathrm{t}\mathrm{r}}), the hanger reinforcement’s tensile strain ({\epsilon }_{\mathrm{s}\mathrm{w}}), and the tensile strain of external prestressing bars ({\epsilon }_{\mathrm{p}\mathrm{s}}), as presented in Eq. (8). The symbol \theta represents the angle of cracking of the crack and can be assumed to be close to zero, as per experimental findings (specifically, the initial crack located between the flange and web, where \mathrm{c}\mathrm{o}\mathrm{s} \theta =1.0). Additionally, the symbol l_{\mathrm{w}} denotes the horizontal projection length of the crack between the flange and web, which is equivalent to the clear distance between the bearing steel plates.

(8)w_{\mathrm{a}\mathrm{v}\mathrm{g}}=\left[{\int }_0^{l_{\mathrm{w}}}\left({\epsilon }_{\mathrm{s}\mathrm{w}}-{\epsilon }_{\mathrm{p}\mathrm{s}}{-\epsilon }_{\mathrm{c}\mathrm{t}\mathrm{r}}\right)\mathrm{d}x\right]\mathrm{c}\mathrm{o}\mathrm{s}\theta .

Equation (9) presents a simplified expression for the average crack width, wherein the influence of tension stiffening is ignored. This is due to the fact that the tensile strain exhibited by the concrete material ({\epsilon }_{\mathrm{c}\mathrm{t}\mathrm{r}}) is notably lower than that of the hanger reinforcement ({\epsilon }_{\mathrm{s}\mathrm{w}}). Thus, it is possible to derive the mean crack width by integrating Eq. (9), resulting in the expression shown in Eq. (10).

(9)w_{\mathrm{a}\mathrm{v}\mathrm{g}}=\left[{\int }_0^{l_{\mathrm{w}}}\left({\epsilon }_{\mathrm{s}\mathrm{w}}-{\epsilon }_{\mathrm{p}\mathrm{s}}\right)\mathrm{d}x\right],

(10)w_{\mathrm{a}\mathrm{v}\mathrm{g}}=\left({\epsilon }_{\mathrm{s}\mathrm{w}}-{\epsilon }_{\mathrm{p}\mathrm{s}}\right)l_{\mathrm{w}}.

The tensile strain of the internal hanger stirrups ({\epsilon }_{\mathrm{s}\mathrm{w}}) is calculated by Eq. (11) where A_sw is the total area of internal stirrups, E_sw is the internal steel stirrup’s modulus of elasticity, and P_cr is the hanger cracking capacity load as depicted in Eq. (6).

(11){\epsilon }_{\mathrm{s}\mathrm{w}}=\frac{P_{\mathrm{c}\mathrm{r}}}{2{A_{\mathrm{s}\mathrm{w}}E}_{\mathrm{s}\mathrm{w}}}.

The strain of external prestressing bars ({\epsilon }_{\mathrm{s}\mathrm{p}}) is calculated by Eq. (12) where f_ps is the yield strength of prestressing bars, E_ps is the modulus of elasticity of external prestressing bars and {\mu }_{\mathrm{p}\mathrm{s}} is the prestressing level.

(12){\epsilon }_{\mathrm{p}\mathrm{s}}={\mu }_{\mathrm{p}\mathrm{s}}\frac{f_{\mathrm{p}\mathrm{s}}}{E_{\mathrm{p}\mathrm{s}}}.

By substitution into Eq. (10), the calculation of the average width of hanger cracks can be determined using Eq. (13).

(13)\begin{array}{rl}{w}_{\mathrm{a}\mathrm{v}\mathrm{g}}=& \left({\epsilon }_{\mathrm{s}\mathrm{w}}-{\epsilon }_{\mathrm{p}\mathrm{s}}\right)l_{\mathrm{w}}\\ =& \left[\frac{\left(0.50+{\displaystyle \frac{{\mu }_{\mathrm{p}\mathrm{s}}}{3}}\right)\left(\sqrt{f_{\mathrm{c}}^{\prime }}{B}_{\mathrm{f}}{d}_{\mathrm{f}}\right)}{2A_{\mathrm{s}\mathrm{w}}{E}_{\mathrm{S}\mathrm{w}}}-{\mu }_{\mathrm{p}\mathrm{s}}\frac{f_{\mathrm{p}\mathrm{s}}}{E_{\mathrm{p}\mathrm{s}}}\right]l_{\mathrm{w}}.\end{array}

(2) Experimental crack width versus proposed crack width

Tab.7 presents a comparison between the experimental outcomes for the average crack width of RC inverted T-beams and the predicted values derived from the proposed theoretical model. Furthermore, an illustration of the anticipated average crack width of the strengthened specimen B6 is provided in Appendix B. The mean ratio between the data from experiments and the anticipated results is 1.06, with a corresponding standard deviation of 0.036. Evidently, the aforementioned formula (Eq. (13)) exhibits a high level of precision in estimating the average width of hanger cracks (w_{\mathrm{a}\mathrm{v}\mathrm{g}}).

4 Conclusions

The current study investigated the suitability of adopting the vertical prestressing technique in order to improve the hanger failure mechanism for existing RC inverted T-beams under symmetric loading. Six tests were conducted, and the following conclusions were obtained.

1) Strengthening of a RC inverted T-beam that had inadequate hanger reinforcement using external prestressing technique in the hanger zone enhanced resistance to the non-ductile failure mechanism and showed adequate results in terms of both serviceability and ultimate capacity.

2) Using external prestressing technique in strengthening inverted T-beams delayed the appearance of the hanger cracks by about 6% to 48% and reduced the crack width at the service load by about 22% to 65% compared to the values for the same phenomena in the un-strengthened beam B2 based on the value of the prestressing force.

3) The vertical compressive stress generated by external pre-stressed stainless-steel bars can decrease the tensile stress affecting reinforced concrete elements by preventing early-stage cracking. However, they served as a stirrup in the ultimate stage and enhanced the hanger capacity by up to 53% compared to capacity for the un-strengthened beam B2.

4) The proposed strengthening technique showed adequate results from both ultimate capacity and ductility viewpoints; the application of external pre-stressed stainless-steel bars showed an outstanding performance factor that was increased by more than 55% above that for the un-strengthened beam B2.

5) The proposed analytical expressions used to evaluate the capacity of RC inverted T-beams that were strengthened with an external prestressing technique were shown to be satisfactory and conservative; the variation between the experimental capacities and the predicted values calculated from the proposed analytical model was about 1%.

6) The enhanced cracking characteristics of the strengthened RC inverted T-beams (the hanger cracking capacity and the average hanger crack width) were analytically estimated. The proposed models were in good accordance with the experimental results.

It should be clear that, while the results obtained from testing the strengthened RC inverted T-beams are promising, there are a significant number of factors that need to be investigated and confirmed experimentally and analytically to create design guidelines for strengthening the existing RC inverted T-beams by vertical prestressing technique.

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