Flexural and direct tensile strength ratio for concrete unusual cross-sections

ABSTRACT The relationship between flexural and direct tensile strength (αfl ratio) has been explored in evaluations of the cracking moment for concrete structural elements. However, most results for αfl can be applied only for rectangular cross-sections. This manuscript addresses its obtaining for unusual cross-sections largely used in precast concrete elements. A theoretical analysis was performed in thirty-two different cross-sections regarding the compressive strength of concrete and the aggregate type used in the concrete composition. The results showed a smooth increase in αfl for higher strength concretes and lower elastic modulus aggregates. The theoretical procedure showed a good correlation with experimental data and prediction models and can be an interesting alternative for the obtaining of the αfl of unusual cross-sections.


INTRODUCTION
Concrete is a material of quasi-brittle behavior evaluated predominantly in compression due to its high compressive strength and limited tensile strength. Such low tensile strength property is, therefore, neglected in the design of reinforced concrete structures, and steel reinforcement is used to support tensile stresses [1]. On the other hand, the tensile strength of concrete is an important property in assessments of both cracking formation and deflections at the serviceability limit state [2], and cracking moment in prestressed elements, punching shear, concrete/steel bond strength, shrinkage, control of crack width in early-ages, and development of moment-curvature diagrams [3]. It can be obtained by three different test methods, namely direct tensile test, splitting tensile test, and flexural test.
Splitting and flexural tensile strengths have been widely used and defined from the indirect application of tensile stresses according to EN 12390-6 [4] and EN 12390-5 [5], respectively. However, studies on the determination of the direct tensile strength are limited, since this property is susceptible to testing techniques, such as boundary conditions, loading ratio, and size and shape of the specimens tested [6], [7]. According to Chen et al. [8], although uniaxial tensile tests are challenging, their results are easily interpreted. Contrarily, flexural tests show a nonuniform stress-strain distribution in the cross-section of the specimen, thus hampering the analysis of results. Both tensile strengths (direct, splitting, and flexural) are usually correlated by some standard codes. Although the direct tensile strength is the true tensile strength of concrete, the splitting tensile strength is useful and reliable to estimate the conventional strength due to its simplicity execution. On the other hand, flexural tensile strength can be used to obtain the tensile strength in structural elements subjected to bending. For example, the ABNT NBR 6118 [9] indicates values for the correlation between flexural and direct tensile strength to be used on the verification of the cracking moment for rectangular, I-, Tand inverted T sections.
In general, the direct tensile strength is acquired through correlations between other properties. Figure 1a displays the difference between the tensile behavior for both direct tensile and flexural tests. Direct tensile tests exhibit a linear hardening up to the direct tensile strength (fct) when a brittle failure occurs. Unlike direct tensile tests, flexural tests show nonlinear hardening after the tensile strength of concrete has been reached and a smooth failure when the flexural tensile strength (f ct,fl ) has been achieved (see Figure 1a). A typical nonlinear flexural behavior of plain concrete is shown by a moment-curvature relationship (see Figure 1b). Hillerborg et al. [10] proposed a plain concrete behavior under tensile loading based on a fictitious crack model, which considers the presence of a fracture process zone when the maximum stress reaches the tensile strength of concrete (Figures 1c-1d). Such a zone is characterized by a gradual softening of concrete due to micro-cracking and interlocking of the aggregates, cement, or fibers [11], [12]. A fictitious crack is formed in this region simultaneously with a tensile stress decrease in the bottom fiber. When the tensile stress is assumed zero, a real crack is installed, and its width increases according to the softening stress-strain relationship [13] (Figures 1d-1e).
where M cr is the cracking moment, f ct,fl is the flexural tensile strength, f ct is the direct tensile strength, I g is the moment of inertia of the gross concrete section, y t is the distance from the centroidal axis of the gross section, and α fl is the flexural and direct tensile strength ratio. Some researchers have addressed the flexural and direct tensile strength ratio (α fl ) due to differences between the flexural and direct tensile behaviors of concrete and the significance of their correlation. Maalej and Li [13] developed an analytical model to evaluate the flexural strength of fiber cementitious composites and observed the flexural and direct tensile strength ratio depends on the brittleness ratio and is affected by stress distribution in the fracture process zone. Ratio α fl is a function of the specimen geometry and should decrease as the specimen height increases [13], [14]. Sorelli et al. [15] performed bending and uniaxial tensile tests in hybrid fiber-reinforced concretes, and the results indicated both type and fiber geometry highly influence their post-cracking behavior. α fl was 1.46 for plain concrete and increased to 1.86 for macro fiber reinforced concrete.
Wu et al. [6] and Chen et al. [8] studied the effects of strain rate and testing method on the tensile strength of concrete and experimentally compared three methods, namely direct tensile, splitting tensile, and flexural tests for measuring it. The results confirmed the specimens tested under flexure showed higher tensile strength than those subjected to direct and splitting tension. The authors concluded the tensile strength increases and α fl decreases with a strain rate increment, reaching 2.1 to 2.5 values for plain concrete of 37 MPa compressive strength [6], [8]. Balbo [16] evaluated a relationship between splitting tensile strength and flexural strength for dry and plastic concretes used in pavements bases. The experimental data showed the flexural strength is usually 92% and 49% higher than the splitting tensile strength of dry and plastic concretes, respectively. Lin et al. [17] proposed a testing method with embedded steel bars that was considered suitable for assessing the direct tensile strength of normal strength concrete specimens. The results were approximately 50% lower than the flexural tensile strength. αfl varied between 1.92 and 2.02 in tensile tests performed in Ultra-High Performance Fiber Reinforced Cementitious Composites [18].
The studies addressed are limited and report results only for rectangular cross-sections. On the other hand, structural elements with unusual cross-sections have been largely used in several precast concrete industries due to their versatility, production speed, durability, and safety [19]. Besides, such studies usually disregard the influence of strength and aggregate type of the concrete, which are important factors in tensile behavior [20], [21]. This paper evaluates the flexural and direct tensile strength ratio for unusual cross-sections used mainly in precast concrete elements. A theoretical analysis was performed in thirty-two different cross-sections regarding the compressive strength of concrete and the aggregate type used in the mixture. A discussion on the influence of ultimate tensile strain and a comparison between prediction models are also addressed.

ANALYTICAL SOLUTION
Ananthan et al. [22] investigated the fracture behavior of plain concrete slender beams subjected to flexural loading using equilibrium equations, and proposed a one-dimensional model, called softening beam model, which accurately predicts the maximum load of rectangular concrete specimens under bending. The model was developed from uncracked ligament equilibrium and use of the strain softening modulus, calculated by Equation 4: where E T is the strain softening modulus (MPa), and Ɛ ut and Ɛ pt are the ultimate and peak strains of concrete in tension, respectively. The relationship between strain softening modulus and elastic modulus is given by where E* is the relation between strain softening modulus and initial modulus and E is the elastic modulus of concrete (MPa). According to Ananthan et al. [22], ratio E* features the failure mechanism in concrete specimens through the slope of the post-peak softening branch of the tensile stress-strain diagram. The material displays a perfectly brittle behavior when E* = ∞ and perfectly ductile behavior for E* = 0 (see Figure 2a). Figures 2b and 2c show the stress distribution for both perfectly brittle and perfectly ductile behaviors, respectively. The ultimate moment capacity can be obtained from the moment equilibrium of cross-section for both cases, and αfl assumes values of 1.0 and 3.0 for brittle and ductile materials, respectively. Such results indicate the limit range where α fl can be considered [22], [23]. Since a strain softening in tension characterizes the concrete, the idealized stress-strain relationships ( Figure 2a) do not apply to an uncracked-ligament real behavior, whose description considers a slope of strain-softening modulus with 0 < E* < ∞ ( Figure 3a). The softening beam model assumes the stress-strain relationship of concrete in tension can be indicated by a bilinear diagram (Figure 3a). The plane section remains plane after deformation, and the compression behavior simulated is linearly proportional. The equilibrium conditions should be satisfied up to the fracture onset, represented in the stress-strain distribution diagrams in Figure 3b [22]. According to the stress-strain relationship in Figure 3a, the stress in the post-peak softening branch is given by where σ t is the tensile stress (MPa), and Ɛ t is the corresponding tensile strain.
Ɛ t can be obtained by Equation 7, derived from the relationships depicted in Figure 3b.
where λ and δ are variable factors from 0 to 1 that characterize the stress-strain distribution diagrams.

Substituting Equation 7 in Equation 6 yields
Because of the linear hardening of the stress-strain relationship, the peak strain of concrete in tension can be written as Replacing Equations 9 and 5 in Equation 8, the tensile stress in softening portion is given by According to the stress distribution diagram (Figure 3b), the compressive stress can be obtained by: where σ c ' is the compressive stress (MPa). The compressive and tensile horizontal forces acting on the uncracked ligament (Equations 12 and 13, respectively) are defined multiplying the tensile strength of concrete by the area of the stress distribution diagram: where F c is the horizontal compressive force (N), F t is the horizontal tensile force (N), h is the rectangular section height (mm), and b w is a rectangular section width (mm). The first equilibrium condition should be satisfied, since no external horizontal forces act on the section, thus: Substituting Equations 10 and 11 in Equation 15, the first equilibrium condition is defined as The solution to the quadratic equation is given by The moment equilibrium condition is accepted when the external bending moment is equal to the ultimate moment capacity generated by the horizontal tensile force on the compression center, and can be written as Finally, applying properties I g and y t for rectangular cross-section, and replacing Equation 18 in Equation 2, Ananthan et al. [22] defined α fl as Equation 19 represents α fl for a rectangular cross-section. It is noteworthy that the characterization of the stress distribution diagram and knowledge of the stress-strain relationship in tension are sufficient to obtain α fl .

SOLUTION FOR UNUSUAL CROSS-SECTIONS
The theoretical analysis was developed in two phases. The first involved the definition of the geometry of the crosssections and mechanical parameters employed, whereas in the second, the ultimate moment capacity of the crosssections was calculated by the moment-curvature diagram, and α fl was obtained for normal and high strength concretes of 20 to 90 MPa. Six different aggregate types, namely basalt, diabase, granite, gneiss, limestone, and sandstone were considered in each series.

Geometry of the cross-sections of precast concrete structures
Thirty-two cross-sections usually applied in precast concrete structures were employed. They were divided into four groups of eight and coined according to both structural element type and application position in situ. The BCS Group was comprised of one-dimensional structural elements frequently used in precast concrete buildings, such as beams, columns, and piles, and the FLS Group considered structural elements of one and two dimensions employed in buildings and bridge floors (e.g., slabs, rails, filler blocks, and double tees). Structural elements, such as U and Y-beams and tiles used in roofs of commercial and industrial buildings were inserted in the RFS group. Finally, the BRS Group was comprised of buried large structural elements employed in waterway and highway infrastructures (e.g., culverts and tunnels). Figure 4 illustrates the geometry of the cross-sections evaluated.

Mechanical parameters
The compressive behavior of concrete was described from a parabola-rectangle stress-strain relationship recommended by ABNT NBR 6118 [9], which shows an initial parabolic branch, and a constant branch between the strain at the maximum compressive strength and the ultimate compressive strain. [9]. The tensile behavior of concrete is represented by a bilinear stress-strain relationship proposed by Bažant and Oh [24]. This law considers a linear hardening characterized by the elastic modulus, and a linear softening after the tensile strength of concrete has been reached. Its ultimate tensile strain was 10 times greater than the peak tensile strain (Ɛ ut = 10Ɛ pt ) according to ACI 224.2R [25]. Safety factors β and γ c were considered in stress-strain diagrams and assumed values of 0.85 and 1.4, respectively, in accordance with ABNT NBR 6118 [9]. In this paper, the steel reinforcement contribution was not considered because only the portion of tensile strength of concrete is employed to assess the cracking moment of the structural elements. Figure 5 shows the compressive and tensile behaviors of concrete. The mechanical properties were obtained from the characteristic compressive strength of concrete (f ck ) using relationships indicated in ABNT NBR 6118 [9], leading to valid results. Table 1 displays the relationships employed for the mechanical properties of concrete. , α e is the correction factor of elastic modulus according to aggregate type, Ɛ c2 is the strain at the maximum compressive strength, Ɛ cu is the ultimate compressive strain, Ɛ ut is the ultimate tensile strain, Ɛ pt is the peak tensile strain, n is the exponent of compressive stress law, and β and γ c are safety factors.
The ultimate moment capacity was determined by the moment-curvature relations from a section analysis of the precast concrete elements. The geometry of the cross-sections, mechanical properties, stress-strain diagrams of concrete, force equilibrium, and strain compatibility were used for the obtaining of the moment-curvature relationships, assuming plane sections remained plane after bending. The neutral axis depth was adjusted for a given compressive strain of concrete, for satisfying the equilibrium of the internal forces, and the moment was calculated. The momentcurvature curves exhibited a linear branch up to the peak tensile strain of concrete, with a subsequent nonlinear behavior until the ultimate tensile strain had been achieved. The elastic modulus was multiplied by a correction factor (αe) that assumed values of 1.2, 1.0, 0.9 and 0.7 for mix compositions with basalt/diabase, granite/gneiss, limestone and sandstone, respectively, for consideration of the different aggregate types, thus changing the peak and ultimate tensile strain of concrete. Finally, Equation 20 determined α fl .

RESULTS AND DISCUSSIONS
Firstly, the theoretical model was compared with a combination of experimental results from flexural and uniaxial tensile tests conducted by Sorelli et al. [15], Lin et al. [17] and Wee et al. [26] in rectangular cross-section specimens. Different samples were tested under direct tensile and four-or three-point bending. The tensile strength of concrete was evaluated in models with 3 to 90-day curing time and 10 to 70 MPa compressive strength for distinct mix compositions.
The experimental and theoretical results of the comparison of α fl (Figure 6) show the theoretical model reasonably agreed with the experimental data. The higher differences were observed in tests performed at early ages, which showed small compressive strength. Numerous operations are performed on the specimens at this stage, and their properties are widely influenced by temperature, humidity, and curing conditions [1]. Besides, the drying shrinkage occurs by the imposition of tensile stress fields on concrete [16]. The difference between experimental and theoretical results was approximately 10%, considering normal and high strength concretes above 20 MPa. Therefore, the theoretical model showed a good fit for the strengths scope considered in this study. 9 Theoretical (R² = 0.883) Tests [15], [17], [26]  The results were also divided into two topics. Firstly, α fl was addressed in terms of compressive strength of concrete, aggregate type used in the mixture, and ultimate tensile strain, and in the second topic, it was compared according to different prediction models.

Influence of compressive strength of concrete and aggregate type
An extensive theoretical analysis evaluated the influence of the compressive strength of concrete and aggregate type on α fl . Table 2 shows α fl calculated for a typical concrete with f ck = 40 MPa and different types of aggregates. The results indicate the aggregate type used in the mix composition exerts a moderate influence on α fl . Low-stiffness aggregates provided greater deformability to the concrete [27], and compositions obtained higher values for α fl . The use of basaltic aggregates as reference promoted up to 12.9%, 6.9% and 4.3% increases for concretes that used sandstone, limestone, and granite aggregates, respectively, for all series analyzed. The difference decreased in function of the increase in the compressive strength of concrete.  Figure 7 more clearly shows the influence of the aggregate type on α fl . According to the correlations between the mechanical properties of concrete in Table 1, the elastic modulus reduction due to the aggregate type caused more deformability and improved the ultimate tensile strain of the concrete. Additionally, for the same tensile strength of concrete, the increase in the ultimate tensile strain reduced the softening branch slope and the strain softening modulus (E T ), increasing α fl . On the other hand, the α fl ratio of concretes with aggregates of lower elastic modulus showed a smaller increment than concretes with aggregates of larger elastic modulus (Figure 7). α fl can be sequentially higher in concretes with basalt, granite, limestone and sandstone, respectively, for the same ultimate tensile strain value. The increase in the compressive strength of concrete conduct to an increase in the tensile strength of the concrete reducing the neutral axis depth (λ) in structural elements subjected to bending (see Figure 4b), which smoothly increases the α fl ratio, according to Equation 19. The α fl value increased to 9.5% on average when the compressive strength of concrete improved from 20 MPa to 90 MPa. However, normal strength concretes (20 MPa to 50 MPa) showed an up to 7% increase against only 2.5% of high strength ones (60 MPa to 90 MPa).        The α fl variation in terms of compressive strength of concrete showed constant values for 50 MPa and 60 MPa compressive strengths due to the distinct mechanical parameters adopted for normal and high strength concretes (Table 1). Regarding mechanical properties, the linear compressive stress-strain relationship employed in the analytical solution proposed by Ananthan et al. [22] was different from the parabola-rectangle stress-strain one used in this study. However, the ultimate moment capacity produces small compressive stresses in the top fiber of the cross-section, and the stress-strain relationship in compression exerts a small influence on α fl .
According to the results, 75.1% of the calculated values of α fl remained between 1.20 and 1.60. Values above this range were mostly obtained by circular cross-section (BCS-2), rectangular cross-sections (BCS-1 and FLS-2), and U and Ybeams (RFS-2/3/6) used as roof structural elements. On the other hand, 14.6% of the results (α fl < 1.20) were associated with large structural elements, such as cross-sections for box culverts (BRS-1/3/4/5) and box girder bridges (FLS-8). The α fl decrease in such structural elements may be related to the size effect phenomenon. According to [28]- [30], the flexural tensile strength of specimens of large dimensions is reduced due to an increase in the cross-section height. In this study, the size effect was milder in elements with circular segments, such as cross-sections for tunnels (BRS-2/7/8).

Comparison with prediction models
The theoretical results of α fl were compared with different prediction models from the literature. Codes for the design of concrete structures have shown fixed values or simple expressions for α fl . According to Model Code [14], α fl depends only on the cross-section height and is reduced with its increase. In contrast, ABNT NBR 6118 [9] recommends the use of fixed values for α fl . Both models disregard the mechanical characteristics of the structural element.
Based on nonlinear fracture mechanics, Buchaim [23], Müller and Hilsdorf [31] and Rokugo et al. [32] proposed analytical models considering the influence of the characteristic length (l ch ) on the flexural behavior, defined by Hillerborg et al. [10] according to both fracture energy and mechanical properties of concrete. Although this parameter has no direct physical meaning, it is a property that determines the fracture process zone size [13]. Table 3 shows the summarized expressions of the codes and authors for the prediction of α fl .  Figure 12 shows α fl for each prediction model compared to theoretical results for rectangular cross-section (BCS-1). The energy fracture was obtained according to the Model Code [14], and a 0.10 h/L ratio was considered. Predictions of design codes do not compute the concrete composition and show constant values for α fl . The results ranged between 1.22 and 1.50 for Model Code [14] and ABNT NBR 6118 [9], respectively, whereas in the other prediction models, they varied up to 10% due to an increase in the compressive strength of concrete. The largest variations between the prediction models evaluated ranged between 36.8% and 26.5% for normal and high strength concretes, respectively.
Although most prediction models are defined only for rectangular cross-sections, ABNT NBR 6118 [9] establishes α fl for I and T beams -see Figure 13 for a comparison of α fl for rectangular, and I and T beams.  In general, the theoretical procedure used in this study showed a good agreement with the prediction models described, except for Model Code [14], which was more conservative. A greater disparity was observed for low strength concretes, which subsequently balanced α fl with the increase in the compressive strength.

CONCLUSIONS
This study reported a theoretical analysis of ratio α fl for unusual cross-sections widely used in precast concrete structures. Thirty-two different cross-sections were evaluated and divided into four groups of elements commonly 30 60 90 Ratio α fl Theoretical ABNT NBR 6118 [9] Ratio α fl Theoretical ABNT NBR 6118 [9] employed in beams, columns, floors, roofs, and buried structures. Fictitious crack model considerations were used in the theoretical analysis for the obtaining of the ultimate moment capacity of precast concrete elements. Parametric studies investigated the effects of the compressive strength of concrete and aggregate type of the mix composition on α fl . Normal and high strength concretes of 20 MPa to 90 MPa compressive strength and six aggregate types were considered in the analysis. An increment in the compressive strength of concrete smoothly increased α fl . Similarly, lower elastic modulus aggregates caused a greater deformability in the concrete and increased α fl . Such an increment in α fl due to the compressive strength and aggregate type was higher in normal strength concretes than in high strength ones. The analyses revealed 75.1% of ratio α fl results ranged between 1.20 and 1.60, highlighting its higher values for circular, rectangular, and U and Y beams. On the other hand, buried large cross-sections showed a significant decrement in α fl due to the size effect.
The proposed methodology was compared with experimental results, and prediction models from the literature showed a reasonable agreement, with more significant differences observed concerning the Model Code [14]. According to the results, the theoretical procedure has proven a viable alternative and can be a consistent way for assessing the α fl of precast concrete elements with unusual cross-sections.