1 Introduction

Shallow and deep beams in numerous concrete structures may subject to torsion, frequently in combination with bending and shear. Provisions for the design of Ultra-High Performance Fiber Reinforced Concrete (UHPFRC) beams were not included in the available codes such as ACI 318-2019 [1], Eurocode 2 (EC-2) [2] and the Egyptian code (ECP-203-2020) [3]. For the torsional design of normal reinforced concrete beams, these codes adopted the space truss model. According to ACI 318-2019, a beam with the tested whose clear span to total height ratio (L/h) does not exceed 4.0 is considered deep, while that with L/h ratio more than 4.0 is considered shallow beam. Recently, the use of UHPFRC becomes attractive for long span bridges. During the past two decades, extensive researches studied the different properties of UHPFRC, including the mechanical strengths, fiber distribution characteristics, shrinkage behavior and bond performance [4,5,6,7]. Previous studies indicated that using steel fibers with volume fraction of 1.5–2.0% will result in practical UHPFRC mixes with cylinder comprssive strength approaching 200 MPa [8,9,10]. The first design guidelines for UHPFRC structures were published in 2002 by the French Association of Civil Engineers (AFGC) [11] and updated in 2013 [12]. It depends on the provisions of EC-2 and considers the role of steel fibers. Another design guidelines for UHPFRC structures were published [13, 14].

Many experimental and analytical investigations on the shear performance of UHPFRC shallow and deep beams have been published [15,16,17,18,19,20,21,22,23]. In addition, experimental studies of fibre reinforced concrete beams in torsion have been reported [24,25,26]. However, a small number of researches studied the torsional behavior of UHPFRC beams with and without web reinforcement [27,28,29,30,31,32]. Very recently, Zhou et al. [32] investigated experimentally and theoreticaly the effect of steel fiber property on the torsional behavior of ultra-high performance concrete (UHPC) rectangular beams without steel reinforcement. They tested 8 UHPC rectangular specimens under pure torsion until failure. The variables in the specimens included the volume fraction, type, dimension and hybrid effect of steel fibers. Their results show that the addition of steel fibers significantly increased the cracking and ultimate torques of UHPC beams, with maximum increases of 79% and 159%, respectively. So, many questions have been raised about the web reinforcement suitable for design of UHPFRC shallow and deep beams subjected to torsion.

In this study, the torsional response of UHPFRC shallow and deep beams with web reinforcement has been investigated. In addition, the effect of steel fibers and web reinforcement is included in a proposed 3-D numerical finite element model to simulate the actual properties of UHPFRC beams under pure torsion.

2 Experimental Program

2.1 Details of Tested Beams

The test specimens were composed of five shallow beams and five deep beams from the same UHPFRC mix and subjected to pure torsion. The tested span for the beams was changed to give the required (L/h) ratio. The main studied parameters are the web reinforcement ratio ρv %, the longitudinal bars, and the (L/h) ratio. Details of tested beams are given in Table 1 and the reinforcement is shown in Fig. 1. The supports of the beams were arranged to be fixed at one end and the other end was free to twist. Figure 2 shows the test setup of UHPFRC beams under pure torsion. The longitudinal bars and stirrups at the middle of the tested span were bonded with electrical strain gauges. The load was applied up to failure at a distance (e mm) from the longitudinal centerline of the tested beams (e = 290 mm for shallow beams and e = 250 mm for deep beams). Torsional rotations were recorded and the crack widths were measured and their propagation was marked on the surface of the tested beams.

Table 1 Test specimens details
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Fig. 1
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Reinforcement details

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Fig. 2
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Test setup of UHPFRC beams under pure torsion

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2.2 Provided Torsional Reinforcement for the Tested Beams

For normal reinforced concrete beams, according to ACI 318-2019, the diameter of the longitudinal bars is limited to the smaller of sv/24 or 10 mm, while EC-2 and ECP-203-2020 limited the diameter of these bars to the smaller of sv/15 or 12 mm. The provided longitudinal torsional reinforcement was chosen 4 bars with a diameter of 12 mmfor all the tested beams except beams BTU3, BTU4 and DBTU3 which were provided with 4 bars with diameter 16 mm (in addition to 2 skin bars of diameter 10 mm for beam DBTU3). The maximum stirrups spacing sv,max adopted by ACI 318-19 for shallow beams is the smaller of (ph/8 = 85 mm for beams BTU1, BTU2, BTU3 and BTU4 and 60 mm for beam BTU5 or 300 mm), while EC-2 and ECP-203-2020 require sv,max to be the smaller of (ph/8) or 200 mm. These stirrups were arranged with sv = 75 mm which is approximately the maximum allowed or with sv = 200 mm which is 235% of the maximum adopted by the studied codes. The maximum spacing of web reinforcement sv,max given by ACI 318-2019 for deep beams, is the lesser of (d/5 = 80 mm or 300 mm), while according to ECP-203-2020, sv,max should not exceed 200 mm). According to EC-2, sv,max is the lesser of (2b = 250 mm or 300 mm). The minimum torsional web reinforcement ratio ρv % for normal reinforced concrete deep beams adopted by ACI 318-2019 is equal to 0.25%, while that adopted by ECP-203-2020 is equal to 0.30% for steel 240 and 0.25% for high grade steel. The tested spacing between stirrups were sv = 100 mm and 200 mm. It is expected that the existence of steel fibers in the concrete mix will cause a reduction of the quantity of web reinforcement required for UHPFRC beams compared with similarly constructed from normal reinforced concrete. Table 1 shows that the provided stirrups spacing sv is considerably greater than sv,max of ACI 318-2019 (sv provided = 250% or 125% from sv,max of ACI 318-2019). Consequently, the provided torsional web reinforcement ratio ρv % is considerably less than ρv,min adopted by the studied codes. The provided ρv of beams DBSU1, DBSU3 and DBSU4 is only about 40% of ρv,min of ACI 318-2019, while the tested ρv of beams DBSU2 and DBSU5 is about 80% of ρv,min of ACI 318-2019. The provided web and longitudinal reinforcement of the tested beams with diameters 6.0 mm, 10.0 mm, 12.0 mm and 16.0 mm have yield strength fy equal to 336.2 MPa, 408.0 MPa, 480.1 MPa and 497.5 MPa, respectively.

2.3 Details of UHPFRC Mix

Portland cement grade of 52.5 N according to BS EN 197/1 with specific surface area of 355 m2/kg was utilized. The natural with a specific density of 2.65 was used. In addition to a crushed quartz powder of a mean diameter of about 20 µm, a silica fume with a diameter of 1.0 μm to 0.1 μm with a fineness of 19,000 m2/kg was used. In all the mixtures a new generation of polycarboxlic ether high range water reducer was used. Table 2 provides a summary of the mixing proportions for one cubic meter. The volume fraction of steel fibers used for all the tested beams has been kept constant of about 1.5%. A hooked-ended straight steel fiber with aspect ratio equal to 25 was used in the concrete mix. The fibers have yield strength equals to 552 MPa. The mixture’s ingredients were mixed with a high speed-mixer for 10 min. Concrete specimens were cured within the first 24 h at room temperature, 21 ± 2 °C. After demolding, the specimens were moist cured until the day of testing. The characteristic compressive strength fcu and the cylinder concrete compressive strength fc' based on an average of three specimens of the UHPFRC mix are 190.2 MPa and 175.7 MPa, respectively. while the splitting tensile strength fsp and the flexural strength fr are 12.5 MPa and 40.5 MPa, respectively.

Table 2 Mixture proportions for 1.0 m3 (kg)
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3 Experimental Results

3.1 Torsional Strength and Behavior

The cracking and ultimate torsional moments and corresponding recorded angles of twist are given in Table 3, while photographs of the crack patterns of the tested specimens are shown in Figs. 3 and 4. For the tested beams, spiral cracks formed at angles with respect to horizontal plane of the beam between 30 and 50 degrees. Just before reaching the ultimate torsional moment of the beam, the width of main crack rapidly increases and the corner longitudinal bars pullout from the concrete, causing failure of the beam. Plots of the recorded maximum crack width versus the torsional moments are shown in Fig. 5. The spiral crack width reduces with increasing the area of the corner bars of beams with the same (L/h) ratio and stirrups. The development of the spiral cracks of beams BTU3 and DBTU3 (with longitudinal bars 4D16) were less than that of beams BTU2 and DBTU1 with the same stirrups but with 4D12, respectively. For the beams with the same corner bars, increasing the stirrups spacing or (L/h) ratio increases the crack width. These results are validated by previous tests [29,30,31]. In Fig. 5a, the width of the major crack in beam BTU2 and beam BTU3 with sv equals to 200 mm, was slightly more than BTU1 and BTU4 with sv equals to 75 mm, respectively. Table 3 shows that increasing sv resulted in a little reduction of the cracking and ultimate torsional moment. The cracking experimental torsional moment of beams BTU2 (with sv equal to 235% of the maximum allowed by the codes) was about 94.6%, of that of the similar beams BTU1 (with sv = 75 mm which is approximately the maximum allowed). Although ρv of beam BTU2 is 0.11% which is only 36.7% of BTU1 (with the minimum adopted by the studied codes), the reduction in the experimental ultimate torsional strength was only about 6.7%. This was because the ultimate torsional strength of UHPFRC beam is obtained as the summation of four parts; torsional strength resisted by cement matrix, torsional strength resisted by steel fibers, torsional strength resisted by horizontal web reinforcement and torsional strength resisted by stirrups; and the contribution of the steel fibers in the ultimate torsional strength comprised the greatest part and consequently, the contribution of the stirrups became relatively small.

Table 3 Experimental results
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Fig. 3
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Crack pattern for the tested UHPFRC shallow beams (showing the load in tons applied at 290 mm from the beam longitudinal axis)

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Fig. 4
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Photographs showing pattern of cracks for UHPFRC deep beams specimens

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Fig. 5
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Experimental torsional moment-maximum crack width relationships

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3.2 Steel Strains Relationships

Figures 6a and b show the recorded torsional moment with the strains in vertical leg of the middle stirrup for the tested beams. Before cracking of the concrete, the recorded strains were very small and a high increase occurs after the formation of the first spiral crack. For the beams of the same (L/h) ratio and the same corner bars, the strains in the stirrup bar were reduced by reducing the stirrups’ spacing. The strains recorded in the stirrup of beam BTU1 (sv = 75 mm), DBTU2 and DBTU5 (sv = 100 mm) were considerably less than that of beams BTU2 (sv = 200 mm), DBTU1 and DBTU4 (sv = 200 mm), respectively. Figures 6c and d show the strains recorded in the corner longitudinal bars of the beams. Reducing the area of corner bars increases the strains for the beams of the same stirrups and (L/h) ratio. The strains measured in the corner bars of beams BTU2 and DBTU1 (with 4 D12) were less than that of beams BTU3 and DBTU3 (with 4D16), respectively. The recorded strains in the corner bars of all the tested beams increases gradually with increasing the torsional moments causing an increase in the main spiral crack width. A torsional hinge forms at the location of the main spiral crack after yielding of the corner bars whereas the crushing of concrete occurs at a torsional moment more than that at yielding of corner bars. This indicates that the failure modes of the tested UHPFRC shallow and deep beams are controlled by the cross-sectional area of longitudinal corner bars without considerable effect of web reinforcement.

Fig. 6
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Experimental torsional moment-steel strains relations

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3.3 Experimental Angle of Twist Relationships

The recorded angle of twist relationships for the tested UHPFRC beams are given in Fig. 7. The area under the torsional moment–angle of twist curves is defined as the torsional fracture energy which can be used to compare the ductility of the beams. Increasing the area of longitudinal bars reduces the angle of twist for the same web reinforcement ratio and (L/h) ratio as can be seen from the comparison between beams BTU3 with BTU2 and DBTU3 with DBTU1, respectively. Increasing the stirrups spacing had little effect on the recorded angle of twist. The torsional post cracking stiffness reduces considerably with increasing (L/h) ratio from 4 to 6 as can be seen for beams BTU1 and BTU5 in Fig. 7a, while the same conclusion can be observed with increasing (L/h) ratio from 1 to 2 as can be seen for deep beams DBTU4 and DBTU1 in Fig. 7b.

Fig. 7
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Experimental torsional moment–angle of twist relatios

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4 Modification for ACI 318-2019 Equation for Predicting

4.1 Cracking Torsional Strength of UHPFRC Beams

The cracking torsional strength Tcr of reinforced concrete beams according to ACI 318-2019 and ECP-203-2020 is calculated as follows:

$$ T_{{{\text{cr}}}} = 0.33 \, \left( {\frac{{A_{cp}^{2} }}{{P_{cp} }}} \right)\sqrt {f_{{\text{c}}}^{\prime } } $$
(1)

where Acp is the concrete section gross area, pcp is the outer perimeter of the concrete section and fc' is the design cylinder compressive strength.

The experimental cracking torsional strength Tcr,exp of UHPFRC specimens is given in Table 3 with the calculated values Tcr,cal of the previous equation. The average of the ratio (Tcr,exp/Tcr,cal) for the tested shallow and deep beams are 3.821 and 2.223, respectively. This indicates that the equation used by ACI 318-2019 and ECP-203-2020 highly underestimates the cracking torsional strength of UHPFRC specimems because this equation does not include the effet of steel fibers ratio (Vf %) and (L/h) ratio. In order to propose a modification for Eq. (1) so as to be suitable (safe and conservative) for UHPFRC shallow and deep beams, many trials were examined to include L/h ratio and Vf %. The proposed modified equation can be expressed as follows:

$$ T_{{\text{cr,pro}}} = \left( {0.20\frac{l}{h} + \, V_{{\text{f}}} \% } \right)*0.33 \, \left( {\frac{{A_{cp}^{2} }}{{P_{cp} }}} \right)\sqrt {f_{{\text{c}}}^{\prime } } $$
(2)

The predicted cracking torsional strength of UHPFRC beams Tcr,pro using the proposed equation is given in Table 3. The average of the ratio (Tcr,exp/Tcr,pro) for the shallow and deep beams are 1.646 and 1.260, respectively. This shows that the proposed equation is conservative and safe when used for UHPFRC shallow and deep beams.

5 Proposed Numerical Model for Prediction of Trsional Behavior of UHPFRC Shallow and Deep Beams

A three-dimensional nonlinear finite element model is proposed to predict the torsional response of UHPFRC shallow and deep beams. For this model, the computer program ABAQUS [33] is used. Concrete is modeled using the SOLID C3D8R element, which is defined by eight nodes and capable of cracking in tension and crushing in compression. A 25 mm * 25 mm * 25 mm element was the mesh element size. The main and web reinforcements are modeled using a bar element T2D3. The concrete damage plasticity (CDP) model has been adopted in order to simulate the concrete behavior with degradation under both compression and tension, within a specified range. The damage property lowers the elastic stiffness when the element plasticizes. Subsequently, it cannot recover to its initial strength. This model assumes that the two main failure mechanisms of concrete are the compressive crushing and tensile cracking. The main parameters in this model are the elastic modulus, Poison's ratio, and stress–strain curve of concrete in compression and tension. CDP needs some important parameters to be inputs in the software ABAQUS to make a realistic simulation for the concrete and these parameters stand for the triaxial compressive test of concrete which are five parameters and defined in Drucker-Prager plastic flow function and yield function [33]. A bar element T2D3 is used for modeling the main and web reinforcements. Elastic-perfectly plastic material in both tension and compression is used for modeling the steel reinforcement. A perfect bond between concrete and reinforcement was assumed. The elasticity modulus of reinforcing steel Es is assumed 200 GPa and the Poisson’s ratio is taken as 0.30. The effect of steel fibers is included in the proposed model through the material properties of UHPFRC. The Poisson's ratio of UHPFRC is assumed to equal 0.20 and the tensile strength is taken equal to 8 MPa. The elasticity modulus of UHPFRC is calculated from the following equation [21]:

$$ {\text{E}}_{{\text{c}}} = { 3737}\sqrt {f_{cu} }\; \left( {{\text{MPa}}} \right) $$
(3)

The adopted value of Ec is equal to 51,538 MPa for fcu equal to 190.2 MPa.

A finite element meshing for some of the tested UHPFRC beams is shown in Fig. 8, while the stress distribution for beams BTU1 and DBTU3 using the proposed model is shown in Fig. 9. The proposed numerical model were used to validate the experimental results of the tested UHPFRC beams in torsion. Table 4 compares the recorded experimental cracking torsional moment Tcr,exp and the ultimate torsional moment Tu,exp with the predicted values using the proposed model (Tcr,NUM and Tu,NUM). The average of the ratio (Tcr,exp/Tcr,NUM) for UHPFRC shallow beams is equal to 1.014 and the average of the ratio (Tu,exp/Tu,NUM) is equal to 1.077, while these ratios for the tested UHPFRC deep beams is equal to 0.782 and 0.984, respectively. This showed that the proposed numerical model gives good predictions of the torsional strength of UHPFRC beams. The experimental and numerical torsional moment–angle of twist relationships of the tested beams are given in Figs. 10 and 11. These comparisons showed that the proposed model gives good predictions for the twist relationships of UHPFRC shallow and deep beams with web reinforcement.

Fig. 8
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Meshing of UHPFRC beams used in the proposed numerical model

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Fig. 9
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Stress distribution of UHPFRC beams using the proposed numerical model

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Table 4 Experimental and numerical predictions of torsional strength
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Fig. 10
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Numerical and experimental torsional moment–angle of twist curves for shallow beams

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Fig. 11
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Numerical and experimental torsional moment–angle of twist curves for deep beams

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6 Conclusions

An experimental and numerical investigation to study the torsional behavior of UHPFRC shallow and deep beams with web reinforcement has been conducted. Ten UHPFRC beams with fcu equal to 190.2 MPa containing steel fibers volume fraction of 1.5% subjected to pure torsion were tested. The main studied parameters were the provided torsional web reinforcement ratio, the area of longitudinal bars and the tested span to total height ratio (L/h = 6.0, 4.0, 2.0 and 1.0). The provided longitudinal torsional reinforcement of the tested beams were chosen to cover the minimum required by the studied international codes, while the provided stirrups spacing sv is considerably greater than sv,max of ACI 318-2019. A numerical 3-D model has been proposed to predict the torsional response of the tested beams. The major findings are summarized as foollows:

  1. 1.

    The web torsional reinforcement has not considerable effect on the torsional strength of UHPFRC beams. Increasing the stirrups spacing to be 235% of the maximum adopted by the studied codes for normal reinforced concrete caused a reduction in the experimental ultimate torsional strength by only 6.7%.

  2. 2.

    The minimum torsional web reinforcement ratio of UHPFRC beams can be safely taken equal to 40% of that required by ACI 318-2019 for normal reinforced concrete. The maximum stirrups spacing can be safely taken the lesser of twice beam width or 300 mm. The minimum bar diameter of longitudinal torsional reinforcement can be safely taken the lesser of 12 mm or (1/15) web reinforcement spacing.

  3. 3.

    Increasing the area of corner bars and reducing (L/h) ratio restrict the development of torsional spiral cracks and enhance the torsional strength of UHPFRC beams. The strains in the corner bars control the failure of UHPFRC beams under pure torsion.

  4. 4.

    A modification to the equation used by ACI 318-2019 and ECP-203-2020 for calculating the cracking torsional strength of concrete beams is proposed to make this equation suitable for UHPFRC beams ( by including the ratio of steel fibers in the concrete mix and L/h ratio).

  5. 5.

    The proposed numerical model showed good estimations of the torsional strength and deformations for UHPFRC shallow and deep beams with web reinforcement.