1 Introduction

The enormous growth of human population worldwide along with rising economic challenges, are in lockstep exacerbating the lack of affordable housing associated with the high cost of building construction, especially in developing countries [31]. As such, the construction industry is continually seeking measures that could alleviate building costs without compromising engineering safety. For example, utilization of plastic aggregates in concrete has been suggested to be a potential cost – effective and energy saving measure [22]. But an interesting emerging development on issues of basic housing construction, is use of rectangular beams and lintels made with the triangular cage of steel (TRC) layout or triangular reinforcing, hereafter referred to as TRC beams.

Expectedly, TRC beams are more cost – effective than the conventional beams reinforced with the rectangular cage (RRC) of steel layout or rectangular reinforcing, owing to less amount of steel used in the former. Hence use of TRC beams could contribute towards affordability of housing. Indeed in some developing countries, TRC beams are already being informally employed in basic housing construction, owing to cost effectiveness of the triangular reinforcing layout used. However, use of the TRC steel layout in beams, has not been widely investigated to fully understand its effects on structural performance. A quick scan of the literature shows that nearly all researches done on structural performance of reinforced concrete (RC) beams, have been based on the conventional RRC layout of steel. However, few papers have began to emerge on TRC layout of steel reinforcement, as later discussed (Section 1.2).

1.1 Ring beams in building construction

1.1.1 Ties for retrofit

Buildings that are not designed and built as rigid frame structures, are typically constructed in a manner that incorporates a ring beam(s). In housing construction, RC ring beams are built at the lintel level crossing at the top of window and door openings. Various types of low to medium storey buildings such as ordinary houses, apartments, schools, offices, hospitals, shopping malls etc., incorporate RC ring beams to partly serve as a horizontal structural tie system. Ring beams are particularly essential in masonry structures wherein bricks and blocks are used as walling units. In such structures, the ring beam supports the upper wall section, roof or floor, and transfers these loads uniformly to walls beneath it, while bridging across door or window openings.

As already mentioned, ring beams partly serve as a structural tie system making the building formidable and sturdy to resist horizontal forces. Earthquake events are among the most destructive combination of external vertical and horizontal natural forces that can severely damage the structural joint and tie system of buildings. Earthquake damage of buildings is often associated with joint failures and lack of major structural components. Satyarno [30] identified the lack of ring beams in buildings, to be one of the main reasons responsible for the damage or collapse of 200,000 houses during the earthquake disaster in Indonesia. In fact, incorporation of ring beams is one of the important recommendations in earthquake guidelines for resistance of buildings to seismic damage [4]. Finite element analysis was done by Rao et al. [27] to evaluate seismic behaviour of masonry buildings built with or without ring beams. They reported that the wall sections enclosed by ring beams, also collectively referred to as horizontal bands, provided remarkable resistance to earthquake damage of buildings. Horizontal bands are wall sections typically enclosed between three RC ring beams located at the plinth level, lintel and roof—eve level of the same building [12, 32]. Indeed, Bothara et al. [8] reported that school buildings constructed with horizontal bands of ring beams along with other fittings, exhibited effective seismic resistance, incurring no damage during the 2015 Gorkha epic earthquake that occurred in Nepal destroying entire community houses.

1.1.2 Structural role

In conventional housing construction wherein seismic retrofit is not incorporated or not of concern, one RC ring beam is typically installed usually only at the lintel level, for normal structural purposes. As such, RC ring beams are also flexural elements that must possess adequate structural capacity. This aspect was the main focus of the present study, aimed at evaluating structural performance of TRC beams. Although the construction practice of employing TRC in lieu of the conventional RRC ring beams, emanated from the need to reduce building costs for affordability of housing, it is essential that structural performance of the former is fully understood. As such, the present investigation was conducted to evaluate effects of the TRC steel layout or triangular reinforcing, on structural behaviour of concrete beams.

1.2 Concrete beams reinforced with the triangular steel layout

Al-Ansari [2] conducted a limited investigation comprising experimental testing and finite element analysis of two (2) triangular beams and one (1) T – beam of reinforced concrete. All the beams were 2 m long. The triangular beams were of two different cross—section sizes 250 mm base × 217 mm height or 300 mm base × 260 mm height, while the T – beam was of cross—Sect. 150 mm web width × 200 mm overall height × 350 mm wide flange. Both beam types were reinforced with 2T12 mm tension steel, along with stirrups spaced at 200 mm centre to centre (c/c). Compared to T – beams, it was reported that the triangular beams exhibited greater ductility along with less cracking.

The paper by Mahzuz [21] was a theoretical analysis of triangular RC beams relative to the conventional rectangular beams. In the analysis, an attempt was made to modify the standard theory equations that are conventionally used for design of rectangular beams, then the modified formulae were employed to analyse triangular beams. The working stress design philosophy was employed, but their study focused mainly on the resulting cost saving of materials owing to use of triangular RC beams. Compared to rectangular beams, use of triangular beams reportedly reduced the concrete section area by 42.8% to 45.1%, while steel area reduced by 8.9% to 12.5%. Mahzuz [21] also reported that triangular beams performed better than rectangular beams, with the former giving higher moment capacity and higher shear resistance. However, this claim was flawed since no thorough theoretical analysis was done to show superior performance of triangular beams, nor were experiments conducted to validate the proposed modifications made to the standard design equations.

Ambroziak et al. [3] conducted a study wherein triangular – shaped precast lintels, were evaluated for shear resistance. The lintels were of size 120 mm base × 180 mm height × 1200 mm long, reinforced with 3Y6 mm tension steel at the bottom and 1R6 mm compression steel at the top. The triangular links were spaced at 75 mm c/c at support for a distance of 300 mm from the beam end, while a wider spacing of 150 mm was used along midspan length of the beam. The 25 MPa concrete was used, along with longitudinal and shear reinforcement bars of yield strengths 500 MPa and 235 MPa, respectively. The test arrangement specified in Eurocode EN 845–2 [17] was employed in the experiment. Accordingly, the test was done by applying a single point load, located at a distance = 75 mm + height (mm) of the lintel, from its end. It was reported [3] that standard theory gave accurate prediction of measured shear capacity values of the triangular lintels. Their findings showed that the general performance of triangular lintels didn’t seem to significantly differ from that of conventional rectangular lintels.

The experiment by Bhakare and Pise [7] was an investigation into the shear behaviour of rectangular concrete beams reinforced with spiral—shaped reinforcement cage (SRC) of steel layout, RRC or TRC steel layout. A total of 27 rectangular beams of size 150 × 150 × 700 mm were made, comprising nine (9) RRC, nine (9) TRC and nine (9) SRC beams. The spacing of stirrups was varied to 70, 90 and 110 mm c/c. The beams were reinforced with tension or compression steel of 3Y10 bars, while stirrups were made using R6 mm steel. The Y10 and R6 mm bars were of Fe 500 and Fe 250 steel grades. The 27 beams were cast using 20 MPa concrete, then cured in water for 28 days prior to testing using the two – point loading arrangement.

As expected, their results [7] showed that spacing of stirrups significantly influenced shear and flexural responses of the different beams. At 70 mm spacing of stirrups, the load carrying capacity of TRC beams, was similar to that of their counterpart RRC beams. But at the wider link spacings of 90 and 110 mm, the load carrying capacity levels of TRC beams were respectively 7.1% and 17.9% lower. For the RRC beams, theoretical shear capacities were compared with experimental results. At the link spacing of 70 mm, the theoretical /experimental shear capacity values were 37.32 /44.18 kN, compared to 32.33 /39.25 kN and 29.34 /30.99 kN at 90 mm and 110 mm spacings, respectively. It can be seen that theoretical predictions and experimental results were generally in agreement at the higher link spacing of 110 mm wherein shear became the prevailing failure mechanism. But at the smaller spacings of 70 and 90 mm, the experimental results were about 20% higher than theoretical predictions, indicating that the mechanism of failure in these beams may have changed to flexure or shear – flexure interaction.

Interestingly, however, TRC beams consistently showed significantly reduced maximum deflection results relative to those of their counterpart RRC beams. At link spacings of 70, 90 and 110 mm, TRC beams gave deflection values that were 42.0%, 21.2% and 23.6% lower than those of the control RRC beams, respectively. The foregoing experimental results showed that while TRC beams tended to become increasingly vulnerable to flexural failure as link spacing increased, they exhibited significantly lower deflections relative to those of the corresponding conventional RRC beams. It may be noted that although different steel cage layouts were used in the study [7], all the beams were reinforced with the same amount of tension and compression steel areas, hence no cost saving of materials was associated with the TRC beams investigated.

Sathya [29] conducted an experiment to investigate shear behaviour of triangular concrete beams. Five (5) triangular beams of size 230 mm base × 400 mm height × 1200 mm length were cast, along with one (1) RRC beam to serve as the control. Two (2) triangular beams were made with triangular stirrups that were equally spaced at 85 mm c/c. The other three (3) triangular beams were made with triangular stirrups that were spaced at 85 mm c/c at beam supports, then spacing was gradually increased towards midspan of the beam length. The concrete beams were reinforced with the tension and compression steel bars 3Y16 and 3Y12 mm, respectively. After casting, the concrete beams were cured in water for 28 days, following which they were tested using the two – point loading arrangement. It was reported that triangular beams showed 12.71% higher load carrying capacity relative to that of the control. However, it is quite notable that only one (1) control RRC beam was cast and tested, which is a major flaw since the single value reported may not be representative of the true result. Typically, testing of at least two beams would be necessary to obtain a representative average result, especially for the control.

In another experimental study, Chavan and Joshi [11] tested a total of nine (9) square beams of size 200 × 200 × 1000 mm long, reinforced with the square steel cage layout, lattice—shaped steel cage layout or TRC steel layout. For the beams reinforced with the square – shaped steel cage, two bars were embedded as tension or compression steel. But for the TRC beams, a single bar was embedded as compression steel along with two bars of tension reinforcement. The bar size(s) and spacing of stirrups used, were not mentioned in the paper. The beams were cast using 25 MPa concrete and cured in water for 28 days, then subjected to flexural testing. Results showed that TRC beams gave 19.2% lower peak load, but also exhibited a significant 26.8% lower deflection. It is possible or even likely that the link spacing used in the study by Chavan and Joshi [11], may have been fairly wide to the extent of adversely affecting the peak load values of TRC beams [7]. Meanwhile the observation by Chavan and Joshi [11] showing that TRC beams gave a deflection reduction of 26.8%, is consistent with findings by Bhakare and Pise [7] who also reported deflection reductions of 21% to 42%, depending on the link spacing.

In summary, the literatures reviewed in the foregoing, give contradictory findings regarding the effects of TRC steel layout or triangular reinforcing, on load bearing response of RC beams, with some studies reporting increase while others showed decrease in peak load. Two aspects are, however, evident:- Firstly, link spacing influences the load bearing responses of beams. Studies [7] have shown that peak load decreases with increase in link spacing, and vice—versa. Secondly, deflection values of TRC beams are consistently and significantly lower than those of their counterpart RRC beams.

1.3 Objectives

The objectives of the present study were to evaluate flexural and deformation responses of rectangular TRC beams, relative to those of the conventional RRC beams. Use of TRC beams is emerging, as they are increasingly being employed informally for housing construction in developing countries. But very little research presently exists on structural tendencies of the rectangular beams reinforced with triangular steel cage layout i.e. TRC beams.

In the present study, a set of 16 beams were made using 30 MPa and 60 MPa concretes, comprising eight (8) TRC beams and eight (8) RRC beams reinforced with Y10 or Y 12 mm longitudinal steel bars, while stirrups were made using Y6 mm steel bars. The link spacing used was 100 or 200 mm c/c. The beams were under—reinforced or over—reinforced to exhibit the different structural responses. Performance behaviours of TRC beams were compared with those of the corresponding RRC beams. The limit state theory as given in SANS 10100–1 and BS 8110 design codes, was employed to predict theoretical failure loads and deformation behaviours of the beams. Actual results were compared with the predicted values obtained based on standard theory.

2 Experimental study

2.1 Beam samples

All the beams were of rectangular shape and were divided into three (3) groups of samples. Each group represented samples made with different reinforcement details embedded for the various beams to exhibit different failure modes comprising flexure failure (Group F), shear failure (Group S) or crushing failure (Group C). Two (2) TRC or RRC beams of 30 MPa or 60 MPa concrete strength, were cast for testing under each failure mode.

RRC beams of Groups F and S were under—reinforced with 2Y12 mm tension (bottom) and 2Y10 mm compression (top) steel bars, while those of Group C were over—reinforced with 3Y12 mm tension (bottom) reinforcement along with 2Y10 mm compression steel. The tension reinforcement of TRC beams was the same as that of RRC beams, while the compression (top) steel of the former was a single bar comprising 1Y10 mm. High yield Y6 mm stirrups were used for all the beams. The link spacing used for the Group F and Group C beams was 100 mm c/c, while that for Group S beams was 200 mm c/c, as shown in Table 1. The reinforcement bars had a high yield strength of 543.65 MPa along with the elastic modulus of 207.9 GPa.

Table 1 Beams reinforced with the rectangular cage (RRC) or triangular cage (TRC) of steel layout
Full size table

All the beams were of size 110 × 130 × 2000 mm length, as shown in Fig. 1. Eight (8) TRC and eight (8) RRC beams were cast, of which twelve (12) of them were made with 30 MPa concrete while four (4) were made with 60 MPa concrete (Table 1).

Fig. 1
figure 1

Details of beams reinforced with the rectangular cage (RRC) or triangular cage (TRC) of steel layout: UR is under – reinforced and OR is over – reinforced beams

Full size image

2.2 Casting and curing

Concrete mixtures were prepared using 19 mm quartzite stone and crusher quartzite sand, along with CEM II 42.5 N cement supplied by Pretoria Portland Cement (PPC) (Pty) Ltd. Table 2 shows mixture designs of the 30 MPa and 60 MPa concretes used. To cast each beam, the fabricated reinforcement steel cage along with cover depth spacers were placed inside the mould, as shown in Fig. 2a. Subsequently, concrete was mixed using the 50 L flat pan mixer, then poured into each mould in four (4) layers of equal thickness. Each layer of fresh concrete was compacted by tamping using the standard steel rod, followed by knocking along the outside surface of the beam mould, using a rubber mallet hammer. After completing the compaction of all four (4) layers of fresh concrete, surface finishing was done, then the beam samples were covered for 24 h using plastic sheeting. Afterwards, the beams were demoulded then sprinkled with water and again covered fully with plastic sheeting, to prevent moisture loss. This curing procedure of sprinkling the beams with water, was repeated daily to maintain adequate moisture presence until testing of the beams at the age of 28 days. During casting of the beams, 100 mm cubes were also made and cured for 28 days in water. The cubes were used to determine 28 – day compressive strength values of the concrete mixtures (Table 2).

Table 2 Concrete mixtures used
Full size table
Fig. 2
figure 2

Beam casting and test set – up

Full size image

2.3 Testing and instrumentation

The samples were set – up on the beam machine for testing using the two – point loading arrangement, as shown in Fig. 2b. The set – up involved using a manual dial gauge placed at midspan underneath the beam, to record and measure deflection readings. Also simultaneously used was the digital deflection measurement device, which was the machine’s in-built data acquisition system. During testing, monotonic loading was applied until beam failure occurred. The load and deflection values at which the first crack appeared, were also observed and recorded. Cracks that formed were marked on both sides of the beam, as they occurred during monotonic loading to failure. The results obtained were used to evaluate structural responses of the different beams.

3 Test results and discussion

3.1 Results

Subsequently discussed are the test results obtained for beams of the different Groups F, S and C. The peak loads and maximum deflections measured during testing, are shown in Table 3. Also included in the table are values of the lever arm (x/d). It can be seen that Group F and S beams had small x/d values below 0.5 confirming that they were under – reinforced (Section 2.1), although F-TRC-30 and S-TRC-60 beams seemed to be balanced sections with x/d values of 0.62 and 0.68, respectively. The Group C beams had high x/d values ≥ 0.78, thereby confirming that they were over – reinforced, as designed (Section 2.1).

Table 3 Measured peak loads and maximum deflections of beams
Full size table

3.1.1 Group F beams

It may be recalled that four (4) sets of Group F beams were made comprising:- (a) RRC beams made with 30 MPa concrete (F-RRC-30) and TRC beams made with 30 MPa concrete (F-TRC-30), (b) RRC beams made with 60 MPa concrete (F-RRC-60) and TRC beams made with 60 MPa concrete (F-TRC-60). The load—deflection behaviours of the different Group F beams, are shown in Fig. 3a, b. The beams were under—reinforced as per the details given in Fig. 2, which would expectedly allow the structural element to preferentially exhibit the flexural failure mode. The observed crack patterns shown in Fig. 4, show that indeed RRC beams exhibited the flexural failure mode, as characterized by progressive development of numerous, closely—spaced short cracks formed directly beneath the point or region of loading. But TRC beams showed the tendency towards combined shear – flexural failure, exhibiting extended diagonal cracking, following the development of flexural short cracks at the bottom edge below the point of loading. However, the flexural failure mode seemed dominant in both the RRC and TRC beam types of Group F, being under – reinforced as confirmed by their small x/d values (Section 3.1).

Fig. 3
figure 3

Load – deflection graphs of Group F beams made with (a) 30 MPa concrete and (b) 60 MPa concrete

Full size image
Fig. 4
figure 4

Group F beams exhibiting the flexural or shear – flexural failure mode

Full size image

It is evident in Fig. 3a that in each case, the first and second samples gave similar results throughout loading. For the F-RRC-30 beams, the difference in curves as seen at higher loading, may be related to the observed cracking behaviour, as beam RRC2 showed earlier cracking at 12.9 kN which is much lower than the 31.1 kN cracking load of beam RRC1. Meanwhile, beams TRC1 and TRC2 of F-TRC-30 type gave similar cracking loads of 19.8 kN and 22.3 kN, along with their similar ultimate failure loads of 37.4 kN and 40.0 kN, respectively. The F-TRC-30 beams gave an average peak load of 38.7 kN which is somehow higher but close to the corresponding 34.6 kN of F-RRC-30 beams (Fig. 3a). Also, the average maximum deflection of the F-TRC-30 beams was 14.0% lower, giving 15.10 mm compared to 17.56 mm of the corresponding F-RRC-30 beams. Similar observations can be seen for the high strength F-TRC-60 and F-RRC-60 beams with the former giving the maximum deflection of 12.36 mm which is 49.0% lower than the corresponding 24.25 mm of the latter, while their peak loads were generally similar, being 47.3 kN and 50.9 kN respectively (Fig. 3b, Table 3). As expected, peak loads of the 60 MPa concrete beams F-RRC-60 and F-TRC-60, were higher than those of their counterpart F-RRC-30 and F-TRC-30 beams made at moderate 30 MPa strength.

3.1.2 Group S beams

The link spacing for Group S beams was relatively larger, being 200 mm c/c (Table 3) to preferentially cause shear failure as the predominant mode. Two sets of beams were cast comprising RRC beams made with 30 MPa concrete (S-RRC-30) and TRC beams made with 30 MPa concrete (S-TRC-30). Figure 5 gives the load—deflection graphs of Group S beams, along with the shear failure mode exhibited by both beam types, as seen in Fig. 6.

Fig. 5
figure 5

Load – deflection graphs of Group S beams

Full size image
Fig. 6
figure 6

Group S beams exhibiting the shear failure mode

Full size image

The ultimate or peak load of S-TRC-30 beams was 34.0 kN, which is close to 31.5 kN of S-RRC-30 beams. The S-TRC-30 beams, however, gave a maximum deflection of 13.75 mm which is higher than 12.10 mm of S-RRC-30 beams. During testing, it was observed that the S-TRC-30 beams exhibited severe shear failure tendency, characteristically emitting a sudden and loud bang sound at breaking point (Fig. 6b). In contrast, the S-RRC-30 beams showed a controlled less drastic manner of shear failure (Fig. 6a).

3.1.3 Group C beams

As discussed in Section 3.1, the high x/d values of Group C beams confirm that they were over – reinforced (Fig. 2), thereby causing them to preferentially exhibit failure by concrete crushing. Two sets of beams of Group C category were made comprising RRC beams made with 30 MPa concrete (C-RRC-30) and TRC beams made with 30 MPa concrete (C-TRC-30). The link spacing used for Group C beams was 100 mm c/c (Table 1). Figure 7 shows the load—deflection graphs of Group C beams.

Fig. 7
figure 7

Load – deflection graphs of Group C beams

Full size image

The peak load /maximum deflection values of C-TRC-30 and C-RRC-30 beams were 37.1 kN /15.83 mm and 42.0 kN /15.61 mm, respectively. It can be seen that C-RRC-30 beams gave a higher peak load which may be attributed to the higher compression (top) steel comprising 2Y10 mm used, compared to the single bar 1Y10 mm employed as compression steel in C-TRC-30 beams. However, both beam types gave similar values of maximum deflection, in contrast with those of Groups F and S, wherein TRC beams clearly exhibited lower deflection values relative to those of the corresponding RRC beams (Sections 3.1.1 and 3.1.2).

3.2 Discussion of load – deflection results

It is evident that when links were closely – spaced sufficiently to prevent shear failure, TRC beams exhibited relatively lower deflection values while maintaining similar levels of load carrying capacity as RCC beams. It was found that TRC beams of Group F gave 14% to 49% lower deflections relative to their counterpart RRC beams (Section 3.1). These findings are consistent with those presented in the literatures, wherein use of TRC beams reportedly led to deflection reduction levels of 21% to 42% [7, 11]. In the present study, TRC beams made with the high strength 60 MPa concrete gave the maximum deflection of 12.36 mm, being significantly lower than 15.10 mm of the same beam type made with the normal strength 30 MPa concrete (Tables 3 and 4).

Table 4 Measured and predicted values of ultimate load and maximum deflection
Full size table

In the case of Group S beams that exhibited shear failure as the dominant mechanism, the peak load /maximum deflection values of TRC beams were lower /higher relative to those of corresponding RCC beams (Section 3.2). For the same link spacing, TRC beams of Group S seemed to be more vulnerable to shear damage than their counterpart RRC beams. It is evident from the foregoing observations, that employment of TRC beams should be accompanied with careful consideration of shear reinforcement.

3.3 Cracking patterns and failure modes

As mentioned earlier (Section 2.3), cracks were marked on the beam samples as the fractures formed during testing. Figure 8 shows the cracking patterns observed upon ultimate failure of the RRC and TRC beams of all three categories comprising Group F, Group S and Group C (Table 1). The patterns of beams F-RRC-30 and F-TRC-30, both showed progressive cracking, leading to development of numerous closely—spaced short cracks underneath the loaded region, which depicts the flexural failure mode. However, the F-TRC-30 beam type showed a tendency towards shear – flexural failure. The observed cracking patterns and flexural or shear – flexural failure mode, highlight the significant effects of link spacing. It is well—established that links in beams not only provide shear resistance but they also enhance beam action. Stirrups restrain the wobbling of longitudinal bars and maintain the geometric shape of the steel reinforcement cage under load, thereby allowing effective stress transfer throughout the network of connected bars. Accordingly, provision of adequate link spacing is necessary to sufficiently aid beam action, which increases ductility of the flexural member, in turn leading to controlled progression towards failure rather than sudden collapse or rapid failure. Indeed, the beams of Groups F and C made with 100 mm link spacing exhibited load – deflection curves showing greater ductility relative to those of Group S made with the wider 200 mm link spacing (Figs. 3, 4 and 5).

Fig. 8
figure 8

Crack patterns of Group F, S and C beams upon loading to failure

Full size image

It can be seen in Fig. 8 that for both beam types S-RRC-30 and S-TRC-30 (Fig. 8) wherein the wider 200 mm link spacing was used, large diagonal shear cracks occurred emanating from one or both supports, a pattern characteristic of severe shear failure (Fig. 6). Typically in RC beams, external shear force is initially resisted by the combined action of concrete in compression, aggregate interlock and dowel action of the tension steel reinforcement, all of which collectively give concrete shear resistance, commonly referred to as ‘critical shear’ (Vc). Once the external shear force is sufficiently large to exceed Vc, the initial formation of diagonal cracks causes the aggregate interlocking effect to diminish, forcing the dowel action resistance to rapidly increase, which in turn transfers the internal shear force to links tied to the steel reinforcement. The higher shear force transferred to the links, tends to pull them apart, a mechanism which may lead to controlled yielding or disentangling and even rupturing of stirrups, depending on link spacing. It was discussed earlier in Section 3.2, that the observed shear failure of S-TRC-30 beams was characterized by a loud bang sound which depicted disentangling of the triangular links, leading to formation of large diagonal crack(s) upon loading of the flexural member to the point of collapse (Fig. 6).

In addition to rapid increase in dowel action force once diagonal cracks initially form, the concrete compression force also rapidly increases at the point of loading, which can lead to crushing failure, depending on concrete strength. This failure mechanism can be seen in crack patterns of the over – reinforced beams C-RRC-30 and C-TRC-30 (Fig. 8), showing crushing of concrete in the compression zone to develop combined shear – flexural cracks, otherwise not seen in patterns of the under – reinforced beams F-RRC-30.

4 Theoretical prediction

4.1 Theory

The limit state design philosophy as employed in SANS 10100–1 [28] and BS 8110 [9] design codes, was used to predict structural responses of the concrete beams subjected to flexural loading. The philosophy comprises the ultimate limit state theory used to design flexural elements for load and shear resistance, while the serviceability limit state is the basis for checking flexural elements against deflection and cracking.

The ultimate moment of resistance for rectangular RC beams, is given by Eq. (1). The neutral axis of the section is obtained by equating compression forces of concrete plus those of compression steel, with the tension force of steel reinforcement acting on the beam section. In the present study, Eq. (1) was employed in accordance with the assumptions specified for application of the design theory [9, 28].

$$\text M={\text C}_{\text c}\left(\text d-0.45\text x\right)+{\text C}_{\text s}\left(\text d-{\text d}^{\prime}\right)$$
(1)

where M—ultimate moment of resistance, d – effective depth for tension (bottom) reinforcement, d’ – effective depth for compression (top) reinforcement, x – neutral axis, Cc – concrete compression force, Cs – compression force of (top) steel reinforcement.

Since the two – point loading arrangement (Fig. 2) was employed for testing of the beams to failure, the ultimate load carrying capacity, is given by Eq. (2).

$$\text{P}=\text{M}/\text{a}$$
(2)

where P – ultimate or peak load (kN), a – distance from support to the loading point (a = 480 mm, Fig. 1).

Prediction of beam deflection was based on the partially – cracked section, for which its values of neutral axis and stiffness, were employed in the analysis. For a partially – cracked section, the moment resisted by concrete in tension is determined and deducted from the total moment [19], hence the net moment (Mnet) is given by Eq. (3).

$${\text{M}}_{\text{net}}=\text{M}-\frac{1}{3}\frac{\text{b}{(\text{h}-\text{x})}^{3}}{(\text{d}-\text{x})}{\text{f}}_{\text{ct}}$$
(3)

where M, x, d – as defined for Eq. (1), h – overall depth of section, b – width of section, and fct – tensile strength of concrete. In the present study, tensile strength was taken to be 10% of compressive strength. Equation (4) gives the maximum deflection (δ) at midspan.

$$\updelta =\frac{\text{P}.\text{a}}{24\text{ E}.\text{I}}(3{\text{L}}^{2}-4{\text{a}}^{2})$$
(4)

where E – Modulus of Elasticity, I – Moment of Inertia of the partially—cracked section, and L – distance between supports, P and a – as defined for Eq. (2).

4.2 Comparison of test results with theoretical predictions

In this section, the measured ultimate or peak loads and maximum deflections obtained from testing of the beams, are compared with those predicted using the standard design theory (Section 4.1). It may be noted that the prediction was done for maximum deflection at the point of beam collapse, being the ultimate limit state.

Given in Table 4 are the predicted values presented along with actual results. It can be seen that the predicted ultimate loads are generally in agreement with actual results obtained from experimental testing, regardless of the failure mode. However, standard theory seemed to predict slightly lower ultimate loads for TRC beams compared to those predicted for RRC beams. Indeed, the ultimate load values predicted for the TRC beams were all between 30.5 to 37.5 MPa, which are somehow lower than the values comprising 32.9 to 43.8 MPa predicted for RRC beams.

The ratio of actual measured ultimate load (Pm) to predicted load (Pp), was calculated for each of the beams tested. It can be seen in Table 4 that the Pm /Pp values all fall within 0.96 to 1.47, giving an overall average of 1.14, which is reasonably close to the perfect 1.0. The ratio of measured maximum deflection (δm) to predicted maximum deflection (δp) was also determined for each of the various beams tested. Indeed, the average value of the ratio δm /δp was 1.08, which shows a nearly perfect agreement between the predicted values and actual deflection results.

Figure 9 shows the comparison of actual results against the corresponding predicted values, based on the Pm /Pp and δm /δp ratios. For beams cast using the normal strength 30 MPa concrete, it is evident in Fig. 9a that the Pm /Pp ratio is slightly higher for TRC beams compared to values of the corresponding RRC beams, implying that the predicted values of the former were somehow lower than those of latter, as earlier mentioned. For example, F-RRC-30 beams gave the Pm /Pp ratio of 1.04 which is lower than 1.24 of F-TRC-30 beams. This trend is evident for all the corresponding RRC and TRC beams made with the normal strength 30 MPa concrete, while those made using the high strength 60 MPa concrete seemed to exhibit an opposite trend. Meanwhile, the δm /δp values for TRC beams, were generally similar to those of the corresponding RRC beams. Interestingly, standard theory predicted the maximum deflection values of TRC beams more accurately while the corresponding predictions for RRC beams were less accurate, as indicated by the nearly perfect average δm /δp = 0.99 for the former, compared to 1.16 for the latter. Generally, however, the foregoing findings affirm soundness of the fundamental theory underlying the analytical method employed in the design codes. The findings also confirm applicability of the theoretical Eqs. (1) to (4) for use to design and analyse TRC beams.

Fig. 9
figure 9

Comparison of measured with predicted values for (a) ultimate load and (b) maximum deflection: Pm/Pp and δm/δp – as defined for Table 4

Full size image

4.3 Error statistics

Figure 10a shows the plot of measured ultimate load versus the corresponding predicted values, wherein the major factors affecting predictions are evident. It can be seen that the predicted ultimate load values for RRC beams made with the normal strength 30 MPa concrete, all fall on the line of equality, thereby confirming the veracity of the limit state theory. In contrast, the predicted ultimate or peak loads of TRC beams, generally fall below the equality line, indicating under—prediction. These observations seem to suggest the need to possibly modify the limit state equations of standard theory for employment to more accurately predict structural behaviour of TRC beams. Another important factor that was of major influence is concrete strength. It can be seen that upon employment of high strength 60 MPa concrete, structural capacity of the beams changed significantly and differed markedly from those of their counterparts cast using the normal strength 30 MPa concrete. It is evident that both beam types F-RRC-60 and F-TRC-60 made with the high strength 60 MPa concrete, exhibited significantly higher measured ultimate loads that were 32.1% and 25.6% greater than their predicted values, an observation that can be attributed to the difference in structural responses of normal strength and high strength concretes.

Fig. 10
figure 10

Measured versus predicted values of (a) ultimate load and (b) maximum deflection: RRC – reinforced with the rectangular steel cage layout, TRC – reinforced with the triangular steel cage layout

Full size image

It is recognized that the rectangular stress block standardized for structural design of normal strength 20 to 50 MPa concretes, is not representative of high strength concrete behaviour. As such, attempts have been made to propose different stress block types for design of high strength concretes [24,25,26]. However, the various stress – block types that have been suggested in the literatures, for design of high strength concretes, are still experimental propositions that have not been standardized. The unique behaviour of high strength concretes, which requires employment of a stress block different from that of normal strength concretes, seems to be responsible for the observed disparity between predicted and measured ultimate loads of the beams (F-RCC-60, F-TRC-60) made with the 60 MPa concrete (Fig. 10a).

A chart comparing the predicted and measured values of maximum deflection, is shown in Fig. 10b. Again, data values generally fall along the line of equality, indicating a strong agreement between the predicted and measured results. Notably, data points of TRC beams are clustered more closely to the equality line, while those of RCC beams are widely scattered further away from the line, which confirms earlier observations discussed in Section 4.2, showing that standard theory gave better deflection predictions for the former than for the latter.

Figure 11a, b shows residuals between the predicted and measured values of ultimate loads or maximum deflections. It can be seen that data points are clustered around the zero—line, indicating that the results generally depict a normal distribution. Fanning out heteroscedasticity can be seen in both graphs, as commonly exhibited by different performance properties and various natural phenomena that typically occur in concrete [23].

Fig. 11
figure 11

Residuals of (a) ultimate load and (b) maximum deflection

Full size image

Statistical error analysis was done comprising determination of the root mean square of error (RMS) and coefficient of variation of error (CV). The strong agreement observed between measured and predicted results (Section 4.2) is depicted by the low CV values for ultimate load and for maximum deflection, which were determined to be 22.4% and 12.6% respectively, along with their associated low RMS values of 7.8 and 4.4. The observed low CV values show good to excellent prediction accuracy level similar or better than those of standard code – type models, being typically 20% to 50% [1, 5, 6, 10, 13,14,15,16, 18, 20, 23].

5 Cost comparisons

It may be recalled that the RRC and TRC beams were reinforced as per the details given in Fig. 1 and Table 1. In this section, cost estimates of producing the Group F, S, and C beams, were determined taking into consideration the amount of main steel reinforcement as the only variable, while keeping constant the quantities of all other materials including cement, aggregates and Y6 mm bars used to make stirrups. Commercial unit prices of the Y12 mm and Y10 mm steel bars were obtained from the local industry, being R17 per kg and R20 per kg, respectively. Table 5 shows cost comparison based on commercial prices for the amount of steel reinforcement used in the different RRC and TRC beams of Groups F, S and C. As expected, less amount of steel was used to fabricate TRC beams of the different groups, leading to their lower cost estimates, relative to those of their counterpart RRC beams. Interestingly, the under – reinforced beams of Groups F and S showed a significant cost saving of 10% to 10.5%, which could beneficially contribute to affordability among sectors such as housing construction in developing countries, for example.

Table 5 Cost saving resulting from use of the triangular steel cage (TRC) relative to employment of the conventional rectangular steel cage (RRC) layout
Full size table

6 Conclusions

In the foregone experimental study, rectangular beams reinforced with the triangular steel cage (TRC) layout i.e. TRC beams, were evaluated for structural performance. The structural responses of TRC beams, were compared with those of the corresponding conventional beams reinforced with the rectangular steel cage (RRC). The parameters measured were load carrying capacity, deformation and failure mode. The actual results obtained were compared with theoretical predictions calculated based on the standard limit state theory employed in the design codes SANS 10100–1 and BS 8110. Relative cost estimates of producing the different beam types, were also compared. The following findings have been drawn.

  1. 1.

    TRC beams made with normal strength concrete consistently gave lower deflections, being 14% to 49% less than those of the conventional RRC beams, while the load carrying capacity values of both beam types, were similar.

  2. 2.

    The maximum deflections predicted were in excellent agreement with actual measured results, for both the TRC and RRC beams. Interestingly, the theoretical maximum deflections predicted for TRC beams, were more accurate than those predicted for RRC beams.

  3. 3.

    Theoretical predictions gave realistic ultimate load values for both the TRC and RRC beam types. However, the predicted ultimate load values of TRC beams were marginally lower compared to those predicted for RRC beams, perhaps highlighting the need to modify the standard theoretical equations for use to design the former. Further research is needed on this aspect.

  4. 4.

    Compared to the conventional RRC beams, the TRC beams seemed to be more vulnerable to shear failure as link spacing increased. Further research is needed to more comprehensively evaluate shear performance behaviour of TRC beams.

  5. 5.

    It was found that production of TRC beams resulted in cost saving of up to 10.5% relative to the cost of producing RRC beams. Such a level of cost saving can be of significant benefit towards affordability in some sectors such as housing construction in developing countries.