Flexural Behaviour of Hybrid Fibre-Reinforced Ternary Blend Geopolymer Concrete Beams

Abstract

Geopolymer concrete is one of the innovative eco-friendly materials that has gained the attention of many researchers in the sustainable development of the construction industry. The primary objective of this experimental investigation is to study the flexural behaviour of the ternary blend geopolymer concrete (TGPC) with various proportions of hybrid fibres. In this study, 27 reinforced concrete beams were cast with a TGPC grade of M55 and tested under monotonic loading conditions. The specimens were beams of length 1200 mm, depth of 150 mm, and width of 100 mm. Crimped steel (metallic) fibres and polypropylene (non-metallic) fibres were used in hybrid form to study the effect on the TGPC beams under flexure. The volume fractions of steel fibres were varied up to 1% with an increment of 0.5%, and polypropylene fibres varied from 0.1% to 0.25% with an increment of 0.05%. The test results were analysed based on the first crack load, ultimate load, load-deflection behaviour, energy absorption capacity, moment-curvature relationship, and ductility behaviour and compared with TGPC specimens without fibres. The experimental study reveals that the TGPC is one of the best alternatives for conventional cement concrete. The addition of hybrid fibres potentially improves the flexural properties of TGPC to a great extent. The test results showcased that the HTGPC with 1% steel and 0.1% polypropylene fibres exhibited better flexural properties than the other combinations of hybrid fibres considered in this study. Additionally, an effort was made to develop a model to estimate the flexural strength of TGPC with hybrid fibres, and the predicted values were found satisfactorily with the test results.

1. Introduction

Geopolymer concrete (GPC) is one of the new generations of cementless concrete produced by the alkaline activation of aluminosilicate materials. Some of the materials that highly bear alumina and silica are fly ash, ground granulated blast furnace slag (GGBS), silica fume, metakaolin, and alccofine. These are widely used as the source material for the production of GPC. The silica and alumina in the source material were dissolved by the alkaline activator, resulting in a molecular chain to produce the binder for the concrete. The GPC produced with these kinds of source materials exhibits low carbon footprint and exceptional properties than conventional cement concrete [1,2,3]. Another advantage of the GPC is that no water is involved in the chemical reaction and curing of the concrete, such as the Portland cement concrete (PCC) [4].

The properties of the GPC are primarily susceptible to the type of source material, alkaline activator, and curing techniques. For instance, the source material with high calcium content will lead to the quick setting of the concrete mixture [5]. Hence, decent knowledge is required to select the materials for the manufacturing of GPC. Many researchers have performed investigations on blending two different source materials to enhance the strength and durability properties of GPC [6,7,8]. Jawahar and Mounika [9] studied the combined effect of fly ash and GGBS on the mechanical properties of GPC. They noticed that the mechanical properties were improved with the increase in GGBS under ambient curing conditions. Jianhe Xie et al. [10] noted that the blend of GGBS and fly ash produces a great synergetic effect on the mechanical and workability properties of GPC. Ternary blend geopolymer concrete (TGPC) is a novel method in the GPC technology to enhance the performance of concrete under various conditions [11,12]. Gharieb et al. [13] investigated the effect of the binary and ternary blend of source materials on the geopolymer pastes. They stated that the compressive strength, porosity, bulk density, and corrosion rate were significantly improved with binary blend and also concluded that ternary blend further enhances the respective properties.

Enormous research was performed to improve the properties of structural concrete by incorporating different types of admixtures [14,15,16]. The addition of fibres in dispersed form is a promising and widely used technology to improve concrete properties to a great extent. Several researchers investigated the effect of randomly distributed fibres in concrete and reported that the mechanical and durability properties were undoubtedly enhanced [17,18]. The flexural behaviour of fibre-reinforced lightweight concrete was examined by Alex and Arunachalam [19]. They observed that the peak load, post-yield deflection and ductility were effectively increased with the addition of fibres. A detailed review on the creep of FRC was performed by Tosic et al. [20], and they noted that the hybrid fibre-reinforced concrete (HFRC) could be the best practice to solve creep-related serviceability problems. They also stated that HFRC possesses economic and technical advantages over the FRC. The previous studies on the strength, durability, and tension stiffening effect of hybrid fibre-reinforced ternary blend geopolymer concrete (HTGPC) proved that the addition of mixed fibres improved the properties when compared with the specimens without fibres and with mono fibres [3,21,22].

The data available on the flexural behaviour of concrete beams were limited to lightweight concrete, high-performance concrete, GPC, FRC, HFRC, etc. [23,24,25,26]. The flexural behaviour of HTGPC is not yet reported. Hence, this experimental study deals mainly with the effect of the steel and polypropylene fibres in a hybrid form on the flexural behaviour of the TGPC beams. A method was proposed to predict the flexural strength of TGPC, and the results were found to be convincing to the actual test results.

2. Experimental Programme

2.1. Materials

The following materials were used in this present investigation.

2.1.1. Ternary Blend Source Material

The primary source material used for this study was low calcium Class F fly ash complying with IS 3812:2003 [27]. It is a by-product of coal combustion procured from Mettur Thermal Power Station, Tamil Nadu, India. It appears dark grey with an average spherical particle size and a specific gravity of 75 microns and 2.30.

Table 1. Chemical components of fly ash.

Al2O3	SiO2	Fe2O3	TiO2	K2O	CaO
27.75%	55.36%	9.74%	3.54%	2.55%	1.07%

shows the chemical components of the used fly ash.

GGBS, a by-product of the iron and steel manufacturing industry, was obtained from a local supplier conforming to BS 6699:1992 [28] used as a partial replacement for fly ash. It has an off-white coloured appearance with an irregular shape and a mean particle size of 30 microns. The specific gravity of the used GGBS is 2.88. The chemical components of the GGBS are given in

Table 2. Chemical components of GGBS.

CaO	SiO2	Al2O3	MgO	S	FeO	Mn	Cl
37.04%	32.49%	20.86%	7.82%	0.98%	0.68%	0.11%	0.012%

.

Metakaolin sourced from dehydroxylation of the kaolinite mineral was also added as the source material to form the ternary blend of the binder for the TGPC. It has an appearance of creamy-ivory coloured powder with an irregular shape. The average particle size and the specific gravity of the metakaolin are 2–3 microns and 2.56, respectively. The chemical components of the metakaolin used in this study are given in

Table 3. Chemical components of metakaolin.

SiO2	Al2O3	Fe2O3	Na2O	K2O	MgO	TiO2	CaO
56.64%	42.38%	0.42%	0.11%	0.04%	0.2%	0.1%	0.1%

.

2.1.2. Fine and Coarse Aggregate

The fine aggregate used in the study was manufactured sand conforming to IS 383:1970 (reaffirmed 2002) [29]. It was Zone II fine sand with 4.75 mm of maximum size. The fine aggregate’s specific gravity and fineness modulus were 2.40 and 2.92, respectively. Ordinary crushed blue metal was used as a coarse aggregate with a maximum size of 12.5 mm. The specific gravity and fineness modulus of the coarse aggregate were 2.92 and 6.79, respectively.

2.1.3. Alkaline Activator

The alumina and silica in the ternary blend source material were activated by sodium silicate and sodium hydroxide mixture. The sodium silicate used in this study was composed of 64% water, 28% SiO₂, and 8% Na₂O by weight. 99% purity of sodium hydroxide in pellets form was used for this investigation.

2.1.4. Superplasticiser and Water

To improve the workability of the TGPC mixtures, Conplast SP 430 was used as a chemical admixture. Potable water from the laboratory was used for preparing the alkaline activator solution and mixing the TGPC mixtures.

2.1.5. Polypropylene and Steel Fibres

This study used polypropylene fibre with an aspect ratio of 300 and crimped steel fibres with an aspect ratio of 66 in a hybrid form. Figure 1 shows the image of the fibres used in this study. The properties of the polypropylene and steel fibres are given in

Table 4. Properties of fibres.

Type of Fibre	Length	Diameter	Tensile Strength	Density
Polypropylene	12 mm	0.04 mm	550–600 MPa	950 kg/m3
Crimped steel	30 mm	0.45 mm	800 MPa	7950 kg/m3

.

2.2. Mixture Proportions for TGPC

In this present study, the mixture proportions for the TGPC were calculated by trial and error method in accordance with the guidelines proposed by Rangan [30], as there is no standard procedure for GPC mix design yet. The detailed studies on the mix design of TGPC by the authors are published elsewhere [12,31]. The TGPC with a volume fraction of 60% fly ash, 25% GGBS, and 15% metakaolin was picked in this work as a ternary blend. The alkaline activator to binder ratio was constant at 0.3 with a 14 M sodium hydroxide solution. The sodium silicate to sodium hydroxide ratio was fixed at 2.5 for all the mix proportions. For better workability of TGPC, additional water to binder ratio of 0.2 with a superplasticier content of 1.5% of the weight of the source material was considered. Eight HTGPC mix proportions with different volume fractions of steel fibres (0.5% and 1%) and polypropylene fibres (0.1%, 0.15%, 0.2% and 0.25%) were considered and compared with TGPC (without fibres). The mixture proportions of all the specimens are tabulated in

Table 5. Mixture proportions of ternary blend geopolymer concrete (TGPC) and hybrid fibre-reinforced ternary blend geopolymer concrete (HTGPC).

Materials, kg/m3	TGPC	HTGPC1	HTGPC2	HTGPC3	HTGPC4	HTGPC5	HTGPC6	HTGPC7	HTGPC8
Fly ash	237.47	237.47	237.47	237.47	237.47	237.47	237.47	237.47	237.47
GGBS	122.62	122.62	122.62	122.62	122.62	122.62	122.62	122.62	122.62
Metakaolin	64.52	64.52	64.52	64.52	64.52	64.52	64.52	64.52	64.52
Fine aggregate	554.40	554.40	554.40	554.40	554.40	554.40	554.40	554.40	554.40
Coarse aggregate	1293.60	1293.60	1293.60	1293.60	1293.60	1293.60	1293.60	1293.60	1293.60
Na2SiO3	90.99	90.99	90.99	90.99	90.99	90.99	90.99	90.99	90.99
NaOH	36.40	36.40	36.40	36.40	36.40	36.40	36.40	36.40	36.40
Water	84.92	84.92	84.92	84.92	84.92	84.92	84.92	84.92	84.92
Superplasticiser	6.37	6.37	6.37	6.37	6.37	6.37	6.37	6.37	6.37
Steel fibre	-	39.25 (0.5%)	39.25 (0.5%)	39.25 (0.5%)	39.25 (0.5%)	78.50 (1%)	78.50 (1%)	78.50 (1%)	78.50 (1%)
Polypropylene fibre	-	0.95 (0.1%)	1.425 (0.15%)	1.90 (0.2%)	2.375 (0.25%)	0.95 (0.1%)	1.425 (0.15%)	1.90 (0.2%)	2.375 (0.25%)

.

2.3. Details of the Specimen

Nine beams of length 1200 mm, depth of 150 mm, and width of 100 mm were cast to study the effect of hybrid fibres on the flexural behaviour of TGPC. For each mixture proportion, three beams were tested, and the average test result was considered for this investigation. The measurements and reinforcement details of the beam are shown in Figure 2. Two high yield strength deformed (HYSD) bars of 10 mm diameter were used as a tension reinforcement, and two number of 6 mm diameter HYSD bars were used as a compression reinforcement, as shown in Figure 2. Two leg stirrups with 6 mm diameter bars were provided at 80 mm c/c spacing as shear reinforcement to ensure the flexural failure of the beam. The mechanical properties of the bars are provided in

Table 6. Properties of steel bars.

Nominal Diameter, mm	Actual Diameter, mm	Yield Strength, MPa	Ultimate Strength, MPa	Modulus of Elasticity, GPa
10	9.94	532	580	235
6	6.10	525	575	230

.

2.4. Casting and Curing Procedure

At first, the dry components, which include the source materials, fine aggregate and coarse aggregate, were mixed thoroughly in a horizontal tilting drum concrete mixer. Sodium silicate and sodium hydroxide (14 M) solution were mixed before 24 h of the TGPC mixture as an alkaline activator [12,32]. The alkaline activator mixed with water and superplasticiser was then added to the dry mix. The fibres were added according to the mixture designation. It is advisable to add the polypropylene fibres before adding the liquid components to achieve a uniform distribution of fibres. The steel fibres were finally added prior to 5–10 rotations of the mixer to control the distortion of fibres. Then, the mixture was filled and compacted with a needle vibrator in a steel mould. Three specimens were cast for each mix proportion. The specimens were then wrapped using a plastic film to prevent moisture loss at the time of steam curing. After 24 h, The beam specimens were cured inside the steam curing chamber [3] at 60 °C for an extra 24 h. The samples were then removed from the curing chamber, unmoulded, and stored in normal room condition till testing.

2.5. Testing Procedure

All the beams were whitewashed before testing to view the crack patterns clearly. The beams were tested under four-point loading in compression and bending testing machine of capacity 3000 kN, as shown in Figure 3. The test was performed in load-controlled loading conditions, and the rate of loading was kept constant at 4 kN/min till the beam failed. Due to the inherent limitations of the testing machine, a complete cure beyond the peak load could not be obtained. Hence, full attention was given to recording the deformation of up to 80% of the post-peak load. The results obtained up to 80% of the peak load could be considered accurate in view of the above. For calculation purposes, only these recorded values, up to 80% of the post-peak load, were utilised. The concrete surface strains at the midspan were noted with the help of two linear variable differential transformers (LVDT) located at 20 mm below the top and 20 mm above the bottom. A dial gauge with a least count of 10 microns was positioned at the bottom centre of the beam to monitor the deflection at every stage of loading. The crack pattern at each stage of loading was viewed using a crack detection microscope having the least count of 20 microns. The test readings were noted for every 2 kN increase in applied load.

3. Results and Discussions

Table 7. Test results.

Beam ID	First Crack Load, kN	Ultimate Load, Pu, kN	Deflection at Pu, mm	Energy Absorption Capacity, kNm	Deflection at 0.8 Pu, mm	Deflection at Yield Load, mm	Ductility Factor
TGPC	16	46	4.48	0.155	8.60	2.92	2.94
HTGPC1	18	50	5.10	0.222	18.35	3.13	5.87
HTGPC2	19	52	5.2	0.257	18.17	2.94	6.17
HTGPC3	20	53	5.25	0.261	18.55	3.07	6.04
HTGPC4	25	55	6.26	0.275	21.06	3.07	6.86
HTGPC5	28	59	6.5	0.324	21.42	2.76	7.76
HTGPC6	26	57	6.37	0.301	20.42	2.99	6.83
HTGPC7	25	56	6.12	0.297	19.96	2.93	6.81
HTGPC8	21	54	5.9	0.291	19.89	2.92	6.81

presents the flexural test results of the beam specimens. The tabulated test results are the mean value of three specimens with the same volume fraction of hybrid fibres.

3.1. Load-Deflection Response

Load versus mid-span deflection curves for all the beams were plotted, and the observation is given in Figure 4. Each curve in Figure 4 is obtained from the mean test result of 3 specimens considered for each mix proportion. The statistical significance threshold considered for the present study was 5%. It may be noted from Figure that all the curves are linear up to the first crack load. The additional applied load developed multiple cracks, and the curve deviated from linearity to a nonlinear region. After the multiple cracking stages, the slope of the curve decreases for the specimen reinforced with hybrid fibre [33]. A sudden drop in the load was observed beyond the peak load for the specimens without fibres and a more or less flat curve for the specimens with hybrid fibres [34]. The specimens with higher fibre content showed less deformation for the same magnitude of load due to the higher stiffness of steel fibres. The maximum deflection is obtained from HTGPC specimens with 1% steel fibres. This may be due to the fact that as the steel fibre content increases, the ability to bridge wide cracks at high strain levels increases. The specimens with hybrid fibres produced an increase in peak load and a more ductile softening behaviour. The specimens with 1% of steel and 0.1% of polypropylene fibre exhibited better performance than the other specimens.

3.2. First Crack and Ultimate Crack Load

The values of the test results obtained from the flexural test on the beams are given in Table 7. It can be observed from Table that the first crack load increased with the increase in volume fraction of fibres. This may be associated with the improvement in the tensile strain carrying capacity of HTGPC in the region of fibres [35]. Ultimate load also gradually increased as the fibre content increased. The increase in the ultimate load ranges from 8% to 28% for the hybrid fibres. The first crack load increased by about 75% for the HTGPC with a volume fraction of 1% steel fibre and 0.1% polypropylene fibre. This may be due to polypropylene fibres’ ability to arrest the microcracks, which delayed the formation of macrocracks [36]. During the development of microcracks in the matrix, the polypropylene fibres in the neighbourhood of such microcracks will try to control these cracks and avoid further development. The steel fibres arrest as the cracks develop, and the bridging effect of fibres minimise the cracks from widening. Additionally, due to the interception of steel fibres, the cracks have to take a longer path, which results in the need for further energy; this consecutively results in the increase in ultimate load [37]. However, it has been noticed as the volume fraction of polypropylene fibres increases; the strength is reduced due to the balling effect of fibre, which results in reduced workability of concrete.

3.3. Energy Absorption Capacity and Ductility

The energy absorption capacity of the specimens can be expressed from the area under the load-deflection plots. It may to observed that the energy absorption capacity ranges from 0.155 kNm to 0.324 kNm for the specimens. In addition, the HTGPC beams with 1% steel and 0.15% polypropylene fibres exhibited the highest energy absorption capacity compared with other specimens considered in this investigation. Hence, it may be analysed that the hybrid fibre combination of more than 1% steel and 0.15% polypropylene cannot enhance the energy absorption capacity. This may be due to the poor binding force of the fibres with concrete resulting from the balling effect of the fibres [11]. The increase in the energy absorption capacity may be associated with the higher load-carrying capacity of the beam that can withstand larger deformation due to the effect of hybrid fibres [33].

The ductility factor is defined as the ratio of the deflection corresponding to 80% of the ultimate load (P_u) in the post-peak load region (δ_u) to deflection at first yield (δ_y) [38]. This takes into account the softening part of the load-deflection curve. Softening ductility consideration is important in seismic design and in cases where large deformations are expected [39]. The method followed to calculate the deflection at first yield load was explained elsewhere [40]. The results for all the tested specimens are given in Table 7. It can be seen that the addition of hybrid fibres resulted in a considerable increase in ductility. The improved ductility signifies the improved inelastic energy absorption due to the incorporation of fibres in the hybrid form [41]. The increase in the values of ductility factor was about 2.64 times for the HTGPC beams with 1% steel and 0.1% polypropylene fibre than the TGPC specimens.

3.4. Cracking Behaviour

As expected, flexure cracks were initiated in the pure bending zone, but due to the constant moment applied in the middle span of the beam, the sequence of crack formation was random within this region. As the load increased, new cracks were developed and merged with the existing cracks along the span [42]. Near the peak load, the beam deflected significantly, thus indicating that tensile steel must have yielded at failure. The cracks at the mid-span of the beams opened widely when the beam reached the ultimate load and near failure. Figure 5 shows the crack patterns of the tested beams. In TGPC specimens, compression concrete spalling occurred, whereas the HTGPC beams were intact, and the hybrid fibres prevented the spalling of concrete [43]. In HTGPC beams, crack initiation and propagation were delayed, and a large number of finer cracks developed in the flexure span and started widening when the load was increased. At higher loads, the crack pattern was not uniform. This may be due to the interception of fibres. The bridging action of the steel and polypropylene fibres was visible in the tensile part of the members. This justifies the control of cracks at macro and micro levels, respectively [36]. The addition of hybrid fibres improved the first crack load, ultimate load and deflection at ultimate load.

3.5. Moment-Curvature Relationship

An attempt was made to plot the moment-curvature relationship for all the specimens. The curvature was computed from the strains on the concrete surface observed from the deformations measured using the two LVDTs placed at the top and bottom level of the beam specimen, as shown in Figure 6. The strain in compression (${\epsilon }_c$) and strain in tension (${\epsilon }_s$) were calculated using the following equations,

$${\epsilon }_c=\frac{{\Delta }_c}{GL}$$

(1)

$${\epsilon }_s=\frac{{\Delta }_s}{GL}$$

(2)

where,

• ${\Delta }_c$ is the deformation obtained from the top LVDT
• ${\Delta }_s$ is the deformation obtained from the bottom LVDT
• GL is the gauge length of LVDTs

Figure 6. Comparison of moment-curvature plots.

Figure 6. Comparison of moment-curvature plots.

The symbol “ϕ” denotes the curvature of the reinforced concrete member, and it is obtained from Equation given by,

$$\varphi =\frac{{\epsilon }_c+{\epsilon }_s}{d_l}$$

(3)

where, $d_l$ is the distance between the top and bottom LVDTs.

The moment was calculated using the equation (relevant to the particular loading configuration),

M = 0.195P

(4)

The moment-curvature plots for all the specimens obtained from the experimental results are shown in Figure 6. Figure shows that all the curves are linear up to the first crack moment, beyond which they exhibit nonlinear characteristics. This may be due to the following reasons. The material behaves similar to a homogeneous composite until the formation of the first cracking. After that, multiple micro and macro cracks develop, leading to an increase in deflection or curvature. Fibres bridges these cracks, so the cracks may have to deviate from their propagation path. The above mechanism of cracking results in the softening of the material, and this will continue till the yielding of steel [36]. The curve becomes more or less flat after the yielding of steel bars [44].

3.6. Prediction of Flexural Strength of HTGPC

3.6.1. Modification of Stress Block

As per IS 456:2000 (reaffirmed 2021) [45], the maximum strain under flexure is considered 0.0035, which is slightly conservative for HTGPC. The experimental investigation observed that maximum strain increases up to 0.005 and above for HTGPC specimens with different volume fractions of steel and polypropylene fibres. Therefore, to modify the stress block parameters, the maximum strain in compression is limited to 0.004 [46]. The following assumptions were also made:

• The strain diagram is linear.
• A parabolic cum rectangular stress block in the compression zone of the section.
• A rectangular stress distribution represents the contribution of fibres in the tension zone.
• The compressive strength in concrete shall be assumed to be 0.67 times the characteristic strength.

With the above assumptions, the strain and stress diagrams across the beam’s cross-section are shown in Figure 7. Figure 7a represents the cross-section of the beam, and the strain distribution diagram with modified strain in the compression is shown in Figure 7b. Figure 7c depicts the effect of fibres that were considered only below the neutral axis [47,48]. Referring to Figure, the total compressive force (C) and tensile force (T) are given by,

$$C=C_c+C_s$$

(5)

$$T=T_s+T_f$$

(6)

where,

• C_c = compressive force in concrete
• C_s = compressive force in compression steel
• T_s = tensile force in tension steel
• T_f = tensile force in concrete composite below neutral axis due to the tensile strength of fibres.

Figure 7. Modified stress-strain distribution across a cross-section: (a) cross section of the beam; (b) strain distribution; (c) stress distribution.

Figure 7. Modified stress-strain distribution across a cross-section: (a) cross section of the beam; (b) strain distribution; (c) stress distribution.

The parameters in Figure 7 were determined using the following method. In the characteristic stress-strain curve for concrete in flexural compression, assuming a parabola in the initial region up to a strain of 0.002, the depth of the rectangular portion will be 0.5x and below which it varies parabolically over a depth of 0.5x to zero at the neutral axis, as shown in Figure 7.

3.6.2. Determination of Total Compressive Force (C)

For the rectangular section of width b,

$$C_c=0.67f_{ck}b\left(\frac{x}{2}+\left(\frac{2}{3} \frac{x}{2}\right)\right)=0.56f_{ck}bx$$

(7)

The line of action of C_c is determined by the centroid of the stress block, located at a distance of k₂x from the extreme compression fibre. Considering the moments of compressive forces about the extreme compressive fibre, the total moment due to C_c will be equal to the moments due to C₁ and C₂.

$$C_c\left(k_{2}x\right)=C_1\left(\frac{1}{4}\right)x+C_2\left(x-\frac{5}{8}\frac{x}{2}\right)$$

(8)

$$\left(0.56f_{ck}bx\right)\left({\mathrm{k}}_2\mathrm{x}\right)=0.67f_{ck}bx\left(\left(\frac{1}{2}\frac{1}{4}x\right)+\left(\frac{2}{3}\frac{1}{2}\right)\left(x-\frac{5}{8}\frac{x}{2}\right)\right)$$

(9)

By solving,

$$k_2=0.425$$

(10)

C_s can be calculated by finding the strain in compression steel (ε_sc) using strain compatibility.

$${\epsilon }_{sc}=0.004 \left(1-\frac{d^{\prime }}{x}\right)$$

(11)

C_s = A_sc ε_sc E_sc

(12)

where,

• A_sc = area of steel in compression
• E_sc = modulus of elasticity of compression steel

Hence, the total compressive force (C) is given by,

$$C=A_{\mathit{sc}}{\epsilon }_{\mathit{sc}}{E}_{\mathit{sc}}+0.56f_{ck}bx$$

(13)

Point of action at C,

Let y_c be the distance from the extreme compression fibre to the point of action of C. Taking moment about the extreme top plane,

$$C{y}_c=C_s{d}^{\prime }+C_c{k}_{2}x$$

(14)

$$C{y}_c=A_{sc}\left(0.004\left(1-\frac{d^{\prime }}{x}\right)\right)E_{sc}{d}^{\prime }+0.56f_{ck}bx\left(0.425x\right)$$

(15)

$$y_c=\frac{A_{sc}\left(0.004\left(1-\frac{d^{\prime }}{x}\right)\right)E_{sc}{d}^{\prime }+0.56f_{ck}bx\left(0.425x\right)}{A_{sc}\left(\frac{d-d^{\prime }}{x}\right)0.004E_{sc}+0.56f_{ck}bx}$$

(16)

3.6.3. Determination of Total Tensile Force (T)

$$T_s=A_{st} f_y$$

(17)

The tensile strength of fibre-reinforced composite is given by the law of mixtures, as the sum of matrix strength and fibre strength [49] as follows:

$${\sigma }_t={\sigma }_m{V}_m+{\sigma }_f{V}_f$$

(18)

where,

• ${\sigma }_t$ = strength of fibre-reinforced composite
• ${\sigma }_f$ = strength of fibres
• ${\sigma }_m$ = strength of the matrix
• V_f = volume of fibres
• V_m = volume of matrix = 1 $-$ V_f

At the ultimate state, the strength contribution of the concrete matrix may be safely neglected after the cracking of the composite. Thus, the ultimate strength of the composite is given by,

$${\sigma }_t={\sigma }_f{V}_f$$

(19)

The tensile strength of the fibres depends on the length, bond efficiency factor, orientation and interfacial bond stress. The tensile strength of the fibre is given by [50],

$${\sigma }_f=\eta {\eta }_1 {\eta }_b 2\tau \frac{l_f}{d_f}$$

(20)

where $\eta$ is the orientation factor equal to 0.41 for the three-dimensional random orientation of fibres [50], ${\eta }_1$ is the length efficiency factor, which accounts for the varying fibre stress at the end portions of the fibres and is equal to 0.5 [46], ${\eta }_b$ is the bond efficiency factor, and its value varies from 1 to 1.2 depending upon fibre characteristics; 1.2 for crimped steel fibres and 1.0 for smooth straight fibres [50]. $\tau$ is the fibre-matrix interfacial bond stress, taken as 5.12 MPa for crimped steel fibres and 4.15 MPa for plain fibres [51].

Hence, the ultimate tensile strength of hybrid fibre-reinforced composite can be taken as,

$${\sigma }_t= {\sigma }_{\mathit{fs}}{V}_s+ {\sigma }_{\mathit{fp}}{V}_p$$

(21)

where ${\sigma }_{fs}$ and ${\sigma }_{fp}$ are the tensile strengths of steel and polypropylene fibres, respectively.

Therefore, the tensile force in the concrete composite below the neutral axis due to the tensile strength of fibres is given by,

T_f = b(D x) σ_t

(22)

3.6.4. Depth of Neutral Axis

Equating the total compressive force (C) and tensile force (T),

$$C_c+C_s=T_s+T_f$$

(23)

A_sc ε_sc E_sc + 0.56 f_ck b x = A_st f_y + σ_t b (D x)

(24)

By solving Equation (24), the depth of the neutral axis ‘x’ can be determined.

3.6.5. Ultimate Flexural Strength

Ultimate flexural strength can be calculated as,

$$M_{u\left(pre\right)}=T_s\left(d-y_c\right)+T_f\left\{\left(\frac{D-x}{2}\right)+\left(x-y_c\right)\right\}$$

(25)

$$M_{u\left(pre\right)}=A_{st}{f}_y\left(d-y_c\right)+b\left(D-x\right){\sigma }_t\left\{\left(\frac{D-x}{2}\right)+\left(x-y_c\right)\right\}$$

(26)

The obtained results are tabulated in

Table 8. Comparison of experimental and predicted flexural strength.

Beam ID	Mu(exp), kNm	Mu(pre), kNm	Mu(exp)/Mu(pre)
TGPC	8.97	7.71	1.16
HTGPC1	9.75	8.79	1.11
HTGPC2	10.14	9.00	1.13
HTGPC3	10.14	9.21	1.10
HTGPC4	10.72	9.41	1.14
HTGPC5	11.51	9.47	1.21
HTGPC6	11.16	9.63	1.15
HTGPC7	10.92	9.84	1.11
HTGPC8	10.53	10.04	1.05
Average			1.13
Standard deviation			0.05
Coefficient of variation (%)			4.12

. It may be noted that the proposed method predicts flexural strength and compare satisfactorily with the experimental test results (M_u(exp)).

4. Conclusions

This study presented the experimental investigation of the flexural behaviour of HTGPC beams. Based on the test results, the following conclusions can be drawn:

• The experimental results revealed that the addition of fibres in TGPC enhances the post-peak performance, showing a softening behaviour of the material. The fibres in hybrid form limit the sudden failure and change to a soft form.
• The fibres in hybrid form impact the load at different levels and improve the deflection corresponding to the load.
• The addition of hybrid fibres improved the specimens’ first crack load and ultimate load. The first crack load was found to increase significantly by 75%, and the ultimate load was found to increase by 28% compared with the specimens without fibres.
• The displacement ductility factor and the energy absorption capacity were increased by 2.64 times and 2.09 times, respectively, for the HTGPC specimen with 1% steel fibres and 0.1% polypropylene fibres compared with the specimens without fibres.
• IS 456:2000 recommend a maximum strain of 0.0035 for the specimens under flexure. However, this value is found to be conservative based on the test results of HTGPC specimens, and a maximum strain of 0.004 could be considered in the stress block for HTGPC.
• The method proposed for estimating the flexural strength of HTGPC was compared satisfactorily with the test results. The effect of the addition of hybrid fibres in the tension zone is considered in this model.