Performance of fixed beam without interacting bars

Abstract

Increasing the bending capacity of reinforced concrete (RC) elements is one of important topics in structure engineering. The goal of this study is to develop a transferred stress system (TSS) on longitudinal reinforcement bars for increasing the bending capacity of RC frames. The study is divided into two parts, i.e., experimental tests and nonlinear FE analysis. The experiments were carried out to determine the load-deflection curves and crack patterns of the ordinary and TSS fixed frame. The FE models were developed for simulating the fixed frames. The obtained load-deflection results and the observed cracks from the FE analysis and experimental tests are compared to evaluate the validation of the FE nonlinear models. Based on the validated FE models, the stress distribution on the ordinary and TSS bars were evaluated. We found the load carrying capacity and ductility of TSS fixed beam are 29.39% and 23.69% higher compared to those of the ordinary fixed beams. The crack expansion occurs on the ordinary fixed beam, although there are several crack openings at mid-span of the TSS fixed beam. The crack distribution was changed in the TSS fixed frame. The TSS fixed beam is proposed to employ in RC frame instead of ordinary RC beam for improving the performance of RC frame.

Keywords

transferred stress system / bending capacity / crack opening / crack propagation / FE nonlinear model / stress distribution

Introduction

Finding ways to increase the bending capacity and ductility of reinforced concrete (RC) beams has been of major interest in structural engineering. Several studies have focused on fiber-reinforced polymers (FRP), epoxy bonding the steel plates, and post and pre-tensioning. Several new materials and systems have been proposed to improve the bending capacity and ductility of RC beams. Some studies have focused on employing additional materials after building the RC beam to improve its performance [¹–⁷]. Although materials can be added after building the RC beam, methods before building the RC beam are important as they exhibit the possibility for designing frames with higher span [¹,⁸].

Elmessalami et al. [⁹] evaluated columns and beams consisting of FRP bars. The contribution in presented a numerical approach to evaluate the bending capacity of composite RC beams. Rabczuk and Eibl [¹⁰] proposed a numerical analysis method to evaluate the performance of prestressed concrete beam. The maintenance and implementation of post-tensioned FRP are easy. Moreover, using post-tensioned FRP increases the bending capacity to approximately 40% and simultaneously decreases the crack expansion [¹¹]. Ross et al. [¹²] demonstrated that the bending capacity is increased more than 30%, although it decreases ductility 40% approximately. Near surface mounting (NSM) is a system for increasing the bending capacity of RC beams, which can be implemented afterwards. This method improves the performance of the beam w.r.t. crack expansion [¹³]. Shishegaran et al. [¹⁴] evaluated a new reinforcement bar system (NRBS). The obtained compressive stress from steel bars are located above the neutral axis of the beam and can transfer tensile bars located at the bottom mid span of the RC beam. They used a bent-up bar to transfer stress from upper to bottom bars and named this approach transferring stress system (TSS). To avoid the interaction between bet-up bars and concrete, the bet-up bars were located in rubber tubes. Although the bending capacity was increased to 40%, the shear failure mode occurred in the rubber tube zones because the concrete area in these zones was less than other zones of the RC beam. Moreover, although the bending capacity was increased, the shear capacity was not; therefore, shear failure occurred. Moreover, the crack openings in NRBS were perpendicular with no interaction bet-up bars. In contrast with other systems for increasing the bending capacity, this method is a cheaper option, but an additional technique is needed to increase the shear capacity [¹⁴]. There is also a lack of research about the bending capacity and performance of beams, which are built by TSS.

There are numerous publications regarding the computational modeling of concrete and RC [¹⁵,¹⁶], in which evaluating cracks and failure modes was done. They can be considered to evaluate the crack opening and crack expansion on RC beam.

In the present study, fixed RC frames employing the TSS and ordinary steel bars are tested, and their results are compared to specify and evaluate the performance of ordinary and TSS fixed frames. A new test setup for testing the fixed frame is designed and conducted. Because of various stress distributions in TSS beams, there are different crack expansions for the TSS fixed frame in comparison to the ordinary fixed frame. We also propose nonlinear FE models to simulate the TSS and ordinary fixed frames. The FE results are compared with the experimental tests to evaluate the validation of the FE nonlinear models. The proposed FE models are subsequently used to assess the stress distribution of steel bars for two specimens.

TSS beam and ordinary beam

Figure 1(a) shows a fixed RC frame, which the static load is applied on the mid-span of the beam. The obtained moment diagram for column and beam and the deformed shape of neutral axis of this frame are shown in Fig. 1(b). To avoid buckling the column, the beam with 0.2 L of column is divided and tested in the present study, as shown in Figs. 1(c) and 1(d). This test setup is proposed for the first time and in the following, the theory of this test setup is explained clearly.

The fixed RC frame include two short columns and a beam, which is defined as the fixed beam. In this study, there are two kinds of fixed beams, including the ordinary and TSS fixed beam. The fixed ordinary beam includes six longitudinal steel reinforcement bars with diameter 14 mm, which three of them are located on the upper neutral axis of beam, and other three bars are located on the bottom neutral of beam, as shown in Fig. 2(a). The shear reinforcement bars with diameter 10 mm are employed and their distance is considered as 6.7 cm, as shown in Fig. 2. There are four steel reinforcement bars with diameter 14 mm in the divided columns, which two shear reinforcement bars with diameter 10 mm are used. Figure 2(b) shows the sections of beam and columns.

Figure 3 shows the fixed TSS frame, where there are six longitudinal steel reinforcement bars with diameter 14 mm in its beam like the fixed ordinary beam, and three of them are prepared by TSS method. To prepare TSS for longitudinal bar, two rubber tubes were used, which are located on 0.2 of length of bars, where the positive moment is changed to negative moment that is named the flection point, as shown in Fig. 3. As above-mentioned, three of longitudinal bars are located on the upper neutral axis of beam, which the second bar is prepared by TSS approach. Moreover, three longitudinal bars are located on the bottom neutral of beam, which the first and third bars are prepared by TSS method. The shear reinforcement bars with diameter 10 mm are employed and their distance is considered as 6.7 cm, as shown in Fig. 3. According to this figure, there are four steel reinforcement bars with diameter 14 mm in the divided columns, which two shear reinforcement bars with diameter 10 mm are used.

For explaining the theory of TSS approach, first, the compressive and tensile stresses on reinforcement bars, which are created due to the concentrated load, are specified. The stress of each zone of longitudinal reinforcement bar should be specified by applying the static load at mid-span of beam. According to the applied static load, two inflection points on beam occurs, in which moment is zero. In other words, there is the negative moment before the inflection point, which is changed to positive moment after the flection point, as shown in Fig. 4(a). If two rubber tubes are located on the flection points, there is no interaction between concrete and two parts of reinforcement bar, which are located into rubber tube. This part is named rubber tube part (RTP). As a result, there is tensile stress in the first side of the RTP, and in contrast, there is compressive stress in the last side of the RTP, and also no stress is created in the bar, which is located into rubber tube. As a result, the compressive stress are subjected against the tensile stress. By contrast, the compressive and tensile stresses are transferred to RTP, and are subjected against together; therefore, the values of these stresses are subtracted, and also the obtained value is considered as the final stress in this zone. The final stress, which is created in the rubber tube, is applied to two side of RTP. The created stress in RTP changes the stress distribution and pattern in length of longitudinal bar, which is prepared by TSS approach. As a result, the beam deformation and stress distribution on concrete of beam will change.

The superposition method is used to explain and justify how the bending capacity of a beam can be increased when TSS longitudinal steel bars are used. Figure 4(b) shows the compressive and tensile stresses that are created in the fixed ordinary beam due to the concentrated static load. The created stresses in the fixed TSS beam can be divided into the stresses, which are distributed due to positive and negative moments separately. After distributing stress due to positive and negative moments separately, the distributed stress in length of the longitudinal bar in two situations can be summed based on superposition method, as shown in Fig. 4(c).

According to Fig. 4(c), the fixed TSS beam is analyzed under positive and negative moment, respectively. First, beam is analyzed under positive moment. The red and blue arrows show the tensile and compressive stresses, which are created due to positive and negative moments. The green arrow demonstrates the stress that are distributed because of exist compressive and tensile stresses, which are created in the zones, which the positive or negative moment is applied to them, which these stresses are shown by red and blue in Fig. 4(c).

Methodology

There are two parts in this study. First, the results of experimental tests of ordinary and TSS frames are evaluated and compared, and then crack expansion is evaluated in each model. In the second part, FE nonlinear model is developed for simulating the TSS and ordinary frames. In this part, the results of FE models are compared with the results of the experimental tests. Moreover, the obtained crack expansion in each model from FE nonlinear model is compared with the actual cracks in the experimental test. Based on the validated FE nonlinear models, the stress distribution of reinforcement bars is evaluated. Finally, the theory of TSS is evaluated based on the obtained results. Figure 5 shows a flowchart that describes the steps of the present study.

Experimental test

Experimental Program is carried out within two main stages: first, the mechanical behavior of steel bar and concrete is tested and determined based on conventional test procedures [¹⁷–²¹]. The steel bar is tested by extend-meter test because the strength of this member plays important role in the ordinary and the fixed TSS frames. Figure 6 shows the extend-meter test of the prepared steel bar. The steel bars should be prepared like a dumbbell for carrying out the extend-meter test, as shown Fig. 6.

In the second stage, four samples of the divided and fixed frames consist of two ordinary beams and two novel TSS beams with 0.2 length of their column are prepared. Each beam is of 15 cm width (w_b), 15 cm depth (d_b) and 130 cm length (L_b) and its column is of 15 cm width (w_c), 15 cm depth (d_c), and 15 cm length (L_c). For the purpose of this study, the new set-up is considered, which is schematically described in Fig. 7. As shown, the samples can be regarded as a frame, where the columns are cut into the length to eliminate the effect of buckling. Furthermore, in this new setup, effect of monotonic loading on the behavior of the column-beam joints can be investigated throughout the test. Figure 7 shows the details of devised the fixed TSS frame versus ordinary frame.

Figure 7(a) shows the reinforcement bars of the ordinary fixed frame. There are three longitudinal bars in top of beam, and also there are three longitudinal bars in bottom of beam. The distance between shear bars is 6.7 cm. Figure 7(b) shows the reinforcement bars of the TSS fixed frame. A rubber tube of 15 cm long is encased the top reinforcement bar and two provide for bottom reinforcement bars at the calculated distance (200 mm from the beam end) to ensure desired performance of the tubes at inflection point. Figure 7(c) shows the molding and preparing the fixed frame. All samples were prepared in a specific day and the curing condition of all specimens were the same.

Test setup and the location of measuring devices are depicted in Figs. 7(d) and 7(e). The displacements are recorded and monitored during the tests by using a deflector meter at the mix span of beam. Samples are examined by applying a monotonic concentrated load at the mid-span of beam (three points bending test). A hydraulic jack of 30 tons capacity is placed on a 100 mm diameter plate at the mid-span of the sample to apply the load until the sample failed. The short columns of each sample are located on two solid steel plates.

Simulating the ordinary and TSS fixed frame

The principle aim of this subsection is related to develop the FE nonlinear model for simulating the behavior of samples. After verifying the developed FE nonlinear models, the obtained crack expansion, the performance of the samples, and bending capacity of the samples from ABAQUS and experimental tests are compared. Based on the validated sample, the stress distribution on steel bars in the fixed ordinary and TSS frames can be evaluated to justify the hypothesis of this study. In the following, first, the considered behavior for concrete and steel is presented, and then the detail of the developed FE nonlinear models for simulating both the ordinary and TSS fixed beam are proposed.

Simulating behavior of concrete and steel

Among the well-known fracture modeling methods [²²–²⁶], the concrete damaged plasticity (CDP) presents and develops the constitutive behavior of concrete in ABAQUS environment by presenting the scalar damage variables for both tensile and compressive response, which are shown in Fig. 8. The damage variables in compressive and tensile behaviors are defined as d_c and d_t, respectively. The amount of them is changed from zero to one, which zero and one are referred to undamaged material and completely damaged material, respectively [²⁷,²⁸]. Based on the CDP method, there are two failure mechanisms including, the compressive crushing of concrete and tensile cracking. Two hardening variables, including {\epsilon }_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}and {\epsilon }_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}, which are related to the failure mechanisms under compressive stress and tensile stress, respectively, govern the yield surfaces. The tensile and compressive damage parameters are determined based on the below equation, which were presented by Birtel and Mark [²⁹].

(1)d_{\mathrm{c}}=\mathrm{1}-{\displaystyle \frac{{\sigma }_{\mathrm{c}}{E}_{\mathrm{c}}^{-\mathrm{1}}}{{ϵ}_{\mathrm{c}}^{\mathrm{pl}}(\mathrm{1}/b_{\mathrm{c}}-\mathrm{1})+{\sigma }_{\mathrm{c}}{E}_{\mathrm{c}}^{-\mathrm{1}}}},

(2)d_{\mathrm{t}}=\mathrm{1}-{\displaystyle \frac{{\sigma }_{\mathrm{t}}{E}_{\mathrm{c}}^{-\mathrm{1}}}{{ϵ}_{\mathrm{t}}^{\mathrm{pl}}(\mathrm{1}/b_{\mathrm{t}}-\mathrm{1})+{\sigma }_{\mathrm{t}}{E}_{\mathrm{c}}^{-\mathrm{1}}}},

where d_t and d_c are defined as tensile and compressive damage parameters, {\sigma }_{\mathrm{t}} and {\sigma }_{\mathrm{c}} are defined as tensile and compressive stresses of concrete. E_{\mathrm{c}} is defined as the Young’s modulus of concrete, {\epsilon }_{\mathrm{t}}^{\mathrm{p}\mathrm{l}} and {\epsilon }_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}are denoted as plastic strains corresponding to tensile and compressive strengths of concrete. b_{\mathrm{t}}andb_{\mathrm{c}} are constant parameters, which the amount of them should be considered between zero and one based on the previous studies [¹⁴–¹⁶].

In this study, the nonlinear performance is considered for steel. Isotropic hardening model with Mises yielding criterion is applied for steel [³⁰].

Fracture energy of concrete

Since a local damage (or damage-plasticity) model results in an ill-posed boundary value problem and associated mesh dependency, the stress-strain curve is “scaled” such that the correct energy dissipation at postlocalization is ensured, which finally yields mesh independent results, see e.g. the overview in Ref [³¹]. This requires an additional material parameter, the fracture energy of concrete (G_f) [³²–³⁶]. This parameter can be obtained experimentally [³⁴,³⁵]. However, in case of lack of experimental data [³⁴,³⁵], provides an estimation of G_f given as:

(3)G_{\mathrm{F}}=\mathrm{73}\cdot {\sigma }_{\mathrm{c}\mathrm{m}}^{\mathrm{0.18}},

where {\sigma }_{\mathrm{c}\mathrm{m}} is the mean compressive strength. The scaling for a linear softening in order to avoid mesh dependency follows the simple relationship [³⁵,³⁶]:

(4){\epsilon }_{\max }={\displaystyle \frac{G_{\mathrm{f}}}{\mathrm{0.5}a{\sigma }_{\mathrm{t}.\max }}},

where a and {\sigma }_{\mathrm{t}.\max }are the characteristic size of a finite element and the maximum tensile stress, respectively.

FE nonlinear models

In the present study, a 3D FE nonlinear model is developed by using ABAQUS Explicit to simulate the fixed ordinary and TSS frames, which are subjected to the concentrated static load. Concrete and reinforcement bars are considered as 3D solid element (C3D8R) and B31 element, respectively. The elastic modulus and Poisson's ratio of steel are considered as 200 GPa and 0.3 like the experimental results. Moreover, the nonlinear behavior of steel is considered, and isotropic hardening model with Mises yielding criterion is applied for steel. As a result, the plastic model in ABAQUS is used to simulate the nonlinear material behavior of steel as a ductile material.

Table 1 shows yield parameters for mechanical behavior of steel bar. The stress-strain of reinforcement bars simplified with bilinear performances, including a linear elastic and linear strain hardening zones for compressive and tensile behavior that are shown in Fig. 9. The proposed stress-stain for simulating steel bar is fitted to the obtained results of tensile behavior of the steel bar from experimental test.

Concrete with compressive strength 30 MPa is considered in FE nonlinear model like the test results; therefore, the elastic modulus and Poisson's ratio of concrete are determined and considered as 25.7 GPa and 0.18. As above-mentioned, the CDP model is employed to simulate the behavior of concrete. The considered value of plasticity parameters for the CDP model are demonstrated in Tables 2–4. These values can be obtained from the same experimental results or calibration process and sensitivity analysis in the previous studies, which reported experimental data about the concrete with compressive strength 30 MPa [⁸,¹⁷].

Figure 10 shows the TSS and ordinary fixed beam. The dimensional geometry of the experiment and simulation samples are the same. The dimensions of short columns are considered as 15 cm * 15 cm * 15 cm. The dimensions of beam are considered as 130, 15, and 15 cm for length, width, and depth of beam, respectively. The reinforcement bars are used in both TSS and ordinary models with the same length, diameter, and bar distance with the experimental specimens. The distance between shear reinforcement bars and their diameters for both models are selected as 67, 10 mm, respectively. Six longitudinal reinforcement bars are modeled in both models, which three of them are located on above the neutral axis and other bars are located on the bottom neutral axis. Figure 10 shows a schematic FE nonlinear model.

There are two no interaction parts on TSS steel bar like the experimental situation. In TSS fixed beam, the second upper reinforcement bar is prepared by TSS steel bar, and also the first and third bottom reinforcement bars are prepared by TSS steel bar. For Providing the TSS steel reinforcement bars in FE nonlinear model, the elements of reinforcement bars along the defined length at the inflection points of beam are considered as solid elements because of ability of this element to assign frictionless contact. Hence, using element-coupling modeling strategy, the solid elements are connected to beam element part of reinforcement bars, as shown in Fig. 11.

The same boundary condition with experimental test for FE models is considered. The load is applied at the mid-span of beam on the circle plate with diameter 10 cm. The short columns are subjected to two solid steel beams. The mesh sensitivity for both samples was carried out to select the mesh size for concrete and steel bars. As a result, the mesh size of the concrete and steel bars was selected as 15 and 10 mm, respectively.

Result and discussion

The obtained load-deflection of ordinary and TSS fixed samples from experimental tests and FE models are shown in Fig. 12. According to the results of experimental tests, the load carrying capacity in yield stress status and corresponding deflection to this status for the first and second ordinary fixed beams were obtained 134.45 kN and 4.03 mm, and 147.79 kN and 3.88 mm, respectively. The load carrying capacity of yield stress status and corresponding strain to this status for the first and second TSS fixed beams were obtained 179.90 kN and 4.01 mm, and 185.27 kN and 3.53 mm, respectively. As a result, the mean load carrying capacity of the TSS fixed beam is 29.39% larger than the mean load carrying capacity of the ordinary fixed beam. Moreover, the deflection in yield stress status of the ordinary fixed beams 4.67% are larger than those of the TSS fixed beam. The load carrying capacity of the first and second ordinary fixed beams in ultimate limit status were measured as 126.85 and 140.94 kN, respectively. In addition, the load carrying capacity of the first and second TSS beams in ultimate stress status were determined as 169.73 and 160.65 kN, respectively. The maximum deflection of the first and second ordinary fixed beam were determined as 24.20 and 28.29 mm, respectively. The maximum deflection of the first and second TSS fixed beam were measured as 34.94 and 31.99 mm, respectively. Although, the deflection of the TSS fixed beam in yield stress status is less than those of the ordinary fixed beam, the deflection of TSS beam in ultimate limit status is larger than those of the ordinary fixed beam. This phenomenon shows that the reinforcement bars of TSS fixed beam with more forces are yielded because of the transferred compressive stress to exist tensile stress. Because of changes in distribution stress on longitudinal reinforcement bars, the deflection of the TSS fixed beam in yield stress status was obtained less than those of the ordinary fixed beam in yield stress status. According to comparison of the obtained results from Figs. 12(a) and 12(b), the hypothesis and theory of the transferred stress can be justified because the load carrying capacity of TSS fixed beam is larger than those of the ordinary fixed beam is. In other words, the results shows that by increasing static load, the longitudinal reinforcement bars were yielded because the compressive stress was applied to the tensile zone by using rubber tube; therefore, the more load is needed to yield the longitudinal reinforcement bars at mid-span of the TSS fixed beam in comparison to the ordinary fixed beam.

The ductility of the ordinary fixed beams was obtained 6.29 and 6.46 from tests 1 and 2, respectively. Moreover, the ductility of the TSS fixed beams were obtained 8.06 and 7.71 from test 1 and 2, respectively. As a result, the ductility of TSS fixed beams is larger than those of ordinary fixed frames.

Comparison of the results from FE nonlinear models and experimental tests is shown in Fig. 12. The load carrying capacity in yield stress status and corresponding deflection to this status for the ordinary fixed beams were obtained 138.22 kN and 4.11 mm from FE nonlinear analyses. As shown in Fig. 12(a), the results of FE model of the ordinary fixed frame are close to the obtained results from experimental tests. The mean obtained load carrying capacity, maximum deflection, and ductility of the ordinary fixed beams from experimental tests were measured 141.17 kN, 26.24 mm, and 6.37, respectively. Moreover, the values of these parameters were obtained 151.01 kN, 30.01 mm, and 6.30 from FE nonlinear model, respectively. As a result, there are 6.79%, 14.37%, and 1.1% errors in FE nonlinear model for simulating the behavior of the ordinary fixed frame that are related to the load carrying capacity, the maximum deflection, and ductility, respectively. As a result, the validation of the proposed FE nonlinear model for simulating the ordinary fixed frame is acceptable.

Figure 12(b) shows comparing the obtained results of the TSS fixed beam from FE nonlinear model and experimental tests. The load carrying capacity in yield stress status and corresponding deflection to this status for the TSS fixed beams were obtained 157.04 kN and 4.08 mm from FE nonlinear analyses. Figure 12(b) shows that the results of FE model of the TSS fixed frame are close to the obtained results from experimental tests. The mean obtained load carrying capacity, maximum deflection, and ductility of the TSS fixed frame from experimental tests were determined as 183.95 kN, 33.47 mm, and 7.89, respectively. Moreover, the values of these parameters were calculated as 157.04 kN, 34.15 mm, and 7.37 in FE nonlinear analyses, respectively. As a result, there are 17.13%, 2.03%, and 7.1% errors in FE nonlinear model for simulating the behavior of the TSS fixed frame that these errors are related to the load carrying capacity, the maximum deflection, and ductility, respectively. As a result, the proposed FE nonlinear model for simulating the TSS fixed frame is acceptable. According to the obtained results from FE nonlinear model and experimental tests, the TSS fixed frame is more strong than the ordinary fixed frame.

According to the results of Fig. 12, the deflection of the TSS fixed beam is larger than the deflection of ordinary fixed beam. As a result, the ductility of TSS fixed beam is larger than the ductility of ordinary fixed beam. Figure 13 is presented to justify this phenomenon. Figures 13(b) and 13(c) show the TSS fixed samples, which their statuses were ultimate stress. As shown in these figures, crack opening occurred at mid-span of these samples, which is main reason for increasing the deflection of TSS fixed beams. On the other hand, only the crack opening at mid-span of the TSS fixed beams occurred, as shown in Figs. 13(b) and 13(c) although crack expansion observed on the ordinary fixed beams and there is no crack opening in these samples, as shown in Fig. 13(a). As a result, the maximum deflection at mid-span of TSS fixed beam is larger than those of the ordinary fixed beam. The cracks can be specified the zones, which there is the tensile in; therefore, the cracks can be specified the negative or positive moment on length of beam. The several cracks were created on beam-to-column connection and mid-span of the ordinary fixed beams, which the cracks on beam-to-column connection occurred on upper corners because of the negative moments in two corners of ordinary fixed beam; therefore, there is the significant tensile stress on the upper longitudinal reinforcement bars at the first and end of the ordinary fixed beam. Moreover, the several cracks occurred at bottom mid-span of the ordinary fixed beam because of the positive moment. As a result, there is the significant tensile stress at bottom mid-span reinforcement bars. In contrast, only the crack expansions and crack openings occurred at bottom mid-span of the TSS beams, which this phenomenon shows that the significant tensile stress did not occur on upper longitudinal reinforcement bars at beam-to- column connection zones. Instead of the significant tensile stress in these zones, high tensile stress occurred on the bottom reinforcement bars, which subjected to bottom mid-span of the TSS fixed beam. Based on comparison of the cracks of the TSS and ordinary fixed samples, the theory of the transferred stress can be justified and explained because there is no crack on upper beam-to column connection zones. Also, there is significant crack opening at mid-span of the TSS fixed beam, which happened because of high level tensile stress at bottom mid-span of beam. The obtained cracks in the ordinary and TSS fixed frames show that the TSS fixed frame is stronger than the ordinary fixed frame.

Figure 14 shows the obtained cracks from the FE nonlinear analyses. Based on the Fig. 14(a), several cracks occurred on the bottom mid-span and upper corners of the ordinary fixed beam, which are related to tensile cracks which happened because of positive and negative moments in these zones, respectively. Moreover, the diagonal cracks occurred on the ordinary fixed beam due to shear stress. There are two kinds of crack in short columns, which are referred to the tensile cracks which happened because of negative moment of columns. There are the same cracks in FE nonlinear model and experimental tests of ordinary fixed frame, as shown in Figs. 13(a) and 14(a). Based on comparison the obtained cracks from experimental tests and FE nonlinear model, the FE nonlinear model for simulating the ordinary fixed frame is valided.

Figure 14(b) shows the cracks, which occurred on the TSS fixed beam. The obtained cracks of the TSS fixed frame from the FE nonlinear model and experimental tests is the same. There are several cracks on bottom mid-span of the TSS fixed beam, which distance between them are larger than the cracks, which occurred on bottom mid-span of the ordinary fixed beam. These cracks occurred because of tensile stress corresponding to the positive moment in this zone. Moreover, according to Figs. 14(a) and 14(b), the width of crack on the bottom mid-span of the TSS fixed beam is larger than those of the ordinary fixed beam. The obtained wide cracks from FE nonlinear model show the crack opening, which occurred in experimental tests, as shown in Figs. 13(b) and 13(c). There is a crack on upper corners of the TSS fixed beam due to tensile stress corresponding to the negative moment in these zones that Fig. 13(b) shows the same crack in experimental tests of the TSS fixed beam. In addition, Fig. 14(b) shows a diagonal damage on the short columns of the TSS fixed frame due to shear stress. The same diagonal crack occurred on the short columns of the TSS fixed frame, as shown in Figs. 13(b) and 13(c). Based on Figs. 13 and 14, there is no the diagonal crack in the TSS fixed beam in both experimental tests and FE nonlinear analysis, although there are several diagonal cracks on the ordinary fixed beam in both experimental tests and FE nonlinear analysis that are related to shear stress. As a result, TSS method can change the stress distribution on concrete and the reinforcement bars of TSS sample.

Figure 15 shows the stress distribution on reinforcement bars of the ordinary and TSS fixed frames. The tensile and compressive stresses are shown negative and positive values, respectively. According to Fig. 15(a), there is no stress at the inflection points in reinforcement bars of the ordinary fixed frame. In addition, there is tensile stress at bottom mid-span bars and upper corner bars of the ordinary fixed beam, which the value of them is larger than the yielded stress, which was considered for reinforcement bar. As a result, failure occurred due to tensile stress, which happened because of positive moment at mid-span of the beam. Moreover, the stress of several shear bars at mid-span of beam is more the yield stress, which causes the shear cracks; thus, it is considered as another failure mode of the ordinary fixed beam.

There are two kinds longitudinal steel bars in the TSS fixed beam including, TSS and ordinary bars. According to the theory of TSS bars, which was mentioned in Section 2, there are two parts in TSS bars, which do not interact with concrete. Figure 15(a) shows the stress of the ordinary, TSS, and shear steel bars of the TSS fixed frame. Although there is no stress in the ordinary steel bars of the TSS fixed beam at the inflection points, there are tensile stress in the bottom TSS bars of this beam at the inflection points; therefore, the tensile stress was transferred to corners bars. Moreover, the compressive stress was transferred from corner steel bars to bottom mid-span bars. As a result, the length of bottom mid-span bars of the TSS fixed beam was subjected to tensile stress is less than the bottom mid-span bars of the ordinary fixed beam. This phenomenon occurred for the TSS steel bar, which was located at upper neutral axis. On the other hand, the tensile stress of the upper corner bars was decreased because of the compressive stress, which was transferred from the upper mid-span bars to upper corner bars. As a result, there is no yielded bars at corner bars of the TSS fixed beam, although the corner bars of the ordinary fixed beam were yielded, as shown in Figs. 15(a) and 15(b). The changing stress distribution in the TSS fixed beam causes changing stress distribution on the shear bars. The stress distribution on steel bars, the cracks of specimens in both FE model and experimental tests justify and confirm the transferred stress theory.

Conclusions

We developed a theory of the transferred stress system (TSS) for fixed beams. This system is employed for transferring compressive stresses to tensile stresses and vice versa in order to increase the bending capacity of the beam. The principle aim of this study was to evaluate the load carrying capacity of the beam of RC frame under static loading. Therefore, we experimentally and computationally studied the performance of ordinary and TSS fixed beams. To simplify the test setup, the RC frames with 0.2 length of their columns are considered avoiding buckling. Two ordinary and TSS fixed beams with 0.2 column length were prepared. We can summarize the key finding of our study as follows.

1) The load carrying capacity of the TSS fixed frame is higher compared to those of the ordinary fixed frame. Moreover, the ductility and maximum deflection of the TSS fixed beam is higher than those of the ordinary fixed frame.

2) Several types of cracks, such as diagonal cracks, tensile cracks at bottom mid-span and upper sides of the ordinary beam are observed for the ordinary fixed frame. In contrast, there is only one crack on the side of the TSS fixed frame and several cracks at the bottom mid-span of the TSS fixed frame caused by tensile stresses due to the negative moments.

3) The number of the micro cracks in the ordinary fixed frame is higher compared to the TSS fixed frame. Furthermore, crack expansions occurred on the ordinary frame, while only crack opening was observed in the TSS fixed beam. This is probably the reason why the maximum deflection and ductility of the TSS fixed frame is higher compared to the ordinary fixed beam.

4) The TSS can change the stress distribution on the steel longitudinal reinforcement bars and concrete justifying the theory of transferred stresses.

5) The developed FE nonlinear models demonstrated that the stress distribution changed in the TSS fixed frame.

6) The obtained stresses in longitudinal reinforcement bars of the ordinary fixed frame are related to the moment, which is applied to the beam. In contrast, the obtained stresses in longitudinal bars, which do not have interaction with concrete, do not depend on the moment, which is applied to beam, and they depend on the stresses, which are obtained in the reinforcement bars interacting with concrete after and before the rubber tube.

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