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This paper presents the results of structural reliability analysis of a structural element (beam) in building using First-order reliability method (FORM) to ascertain the level of safety. The natural stone (NS) which is the by-product of Precambrian deposits of the Bida trough was used as coarse aggregate: unwashed and washed aggregates were used. A total of 80 concrete cubes of 150 mm × 150 mm × 150 mm were cast and used for this study, sensitivity analysis was conducted by varying the span, depth, effective depth, area of shear reinforcement and dead load of the beam in bending. The result of the sensitiv - ity analysis revealed that the beam utilising unwashed and washed NS are both structurally safe at a span of 3000 mm with –5 –8 probabilities of failure of 9.20 × 10 and 2.06 × 10 and both safe at a depth of 600 mm with probabilities of failure of –4 –4 4.19 × 10 and 2.602 × 10 , respectively, in bending. Keywords Beam · Bending · Natural stones · Concrete · Structural reliability Introduction In any structure, there exist uncertainties or variability in loading and material properties. These uncertainties lead Concrete is the most widely used construction material in to variability in structural response during the life cycle of the world, second to water as the most utilised substance a structure (Jalayer et al. 2011). In order to design struc- on earth (Alhaji 2016). It is obtained by mixing cement, tures that can perform the intended purpose with desired water and aggregates in right quantities (with admixtures). confidence, these uncertainties involved must be taken into Aggregates ideally constitute 75% of concrete as they are account. The traditional way to tackle these uncertainties is extremely important in the quality of concrete produced, to use the extreme values of the uncertain quantities and/or this makes it important that they meet certain standards in safety factors in the framework of deterministic design. How- order to achieve a strong, durable and economical concrete. ever, a more accurate and precise way of treating the uncer- Concrete compressive, tensile and flexural strengths are tainties is by utilising probability-based design methods that measures used in determining the amount of resistance a have been evolving and gaining widespread acceptability for structural element can offer to deformation, they remain the the past few decades (Ayyub et al. 1995; Wen 2000). most important properties of concrete. Several researches In order to ensure durability, functionality (serviceability and have been conducted and many are still being conducted limit states) and safety (ultimate limit state), the evaluation of the on the relationship between the composition of structural structural behaviour of the reinforced concrete (RC) systems in concrete and its mechanical properties. Despite the seem- different loading conditions is a primary requirement. Advanced ingly huge interest in the study of concrete, cases of poor tools like non-linear finite element analyses (NLFEA ) is one construction and structural failure still exist (Alhaji 2016). of the solutions being utilised for the design of complex new RC structures. Reliability assessment of reinforced concrete structures always involves two different types of uncertainties namely: aleatory and epistemic uncertainties (Castaldo et al. * Daniel Ndakuta Kolo 2020). The aleatory uncertainties affect material properties firstname.lastname@example.org and geometry, while the epistemic uncertainties are related to Department of Civil Engineering, Federal University assumptions performed within the NLFEA . of Technology, Minna, Nigeria Vol.:(0123456789) 1 3 486 Asian Journal of Civil Engineering (2021) 22:485–491 The natural stone (NS) utilised is mostly found in Bida structure is built. As long as the strength exceeds the stress basin (Trough), it is a by-product of the Precambrian decom- on a component, safety is anticipated (Barambu et al. 2017). position, transportation and deposition of rocks in this Basin. It RC structures usually undergo degradation, this could mani- is an extension of the Iullemmeden Basin which runs through fest in the form of changes to strength and stiffness beyond Niger Republic and Mali in West Africa. Bida basin is found the baseline conditions assumed during structural design in Northern Nigeria and is delimited to the North East and (Castaldo et al. 2017). These ageing effects could result in South West by Basement Complex. The Precambrian rocks in degradation of structural components most especially when Northern Nigeria can generally be classified into four groups RC is exposed to aggressive environment, increasing the namely: Basement Complex, Older Granite series, Younger probability of structural failure (Castaldo et al. 2017). The Metasediments and Volcanic Rocks. This research investigates reliability assessment performed on the NS in this study, the effects of utilising washed and unwashed NS in structural however, does not incorporate the time-dependent and envi- elements. ronmental effects on the concrete. The use of natural stone (NS) sourced from Bida for The aim of this study is to conduct structural reliability concrete production is gaining wide acceptance especially studies on reinforced concrete beam subjected to bending among the dwellers of the Bida basin because the production forces with natural stone as coarse aggregate. The objectives of crushed granite is more labour intensive and expensive. are to (i) determine the physical properties of fine and coarse However, despite the wide acceptance of its use, the body of aggregates; (ii) determine the 28-day compressive strength literature is still not robust with research related to the use of of concrete cubes produced using unwashed and washed the NS as coarse aggregate for concrete production. The NS is natural stone and (iii) determine the reliability indices for used conveniently for mass concrete production and not com- the structural element. monly used for suspended reinforced concrete elements such as beams, columns or slabs because the structural reliability in that regard has not been evaluated. Jalayer et al. (2011) used a constant water–cement ratio of 0.65 to produce concrete with Materials and methods the NS as coarse aggregates, this research determined just the compressive strength of the concrete at 3, 7, 14, 21 and 28 days Ordinary portland cement (OPC) of curing. The flexural strength, splitting tensile strength and elastic modulus were not considered. Furthermore, Alhaji The Cement used was obtained from the Building Materi- (2016) developed statistical models for predicting the com- als Market Minna, Niger State. The bags of Cement were pressive strength, flexural strength, splitting tensile strength stored on a raised platform where adequate protection from and elastic modulus of concrete produced from the NS. Cur- external effect was guaranteed. The OPC conforms with BS rently, no research exists on structural reliability studies of 12 (1996). concrete produced utilising this coarse aggregate. Based on this premise, reliability studies on the NS concrete become Fine aggregate timely and justifiable. Reliability in structural engineering is a means through The sand was collected from Chanchaga, Minna, Niger which the performance or functionality of a structural system is assessed. It is a measure of the safety of the structural com- State. It was ensured that the sand was clean, sharp, free from clay and dirt. Fine aggregates generally refer to aggre- ponents and subsequently that of the whole system. Reliability analysis is a tool to predict to a certain acceptable degree, the gates passing through sieve size 4.75 mm BS 882 (1992). ability of the system or components to fulfil the design func- tion under given conditions within the structural design life Water (Abubakar 2015). The objective of any structural design is to ensure safety Water fit for drinking was sourced from the Civil Engineer - and economy of the structure operating under a given envi- ronment. For this purpose, designers always check whether ing Laboratory, Federal University of Technology Minna and used in casting the cubes. BS EN 1008 (2002) stipulates the capacity of the structure exceeds the demand as shown in Eq. (1): that water to be used for concrete production must be clean, drinkable and free from deleterious materials. Capacity(C) > Demand(D) (1) As far as the condition in (1) is satisfied, the safety of the structure is ensured for the intended purpose for which the 1 3 Asian Journal of Civil Engineering (2021) 22:485–491 487 the compressive strengths were calculated using Eq. (2). The Coarse aggregate test was conducted in accordance with BS 1881:116. The Coarse aggregate used for this work was sourced from Average load Bida, Niger State, Nigeria. Bida is located in the Middle belt F = N∕mm (2) cu Area region of Nigeria within Latitude N 9° 55′ E and Longitude N 5° 52′ E. The coarse aggregate conforms to specifications for natural aggregates as in BS 882 (1992). Reliability assessment Casting of concrete cubes for compressive strength First-Order Reliability Method (FORM) was utilised in test assessing the reliability of the structural element under consideration. This method is a simplified reliability model, The mix design method employed in this study is the Brit- it uses only the mean values and standard deviations for ish Standard mix design (DoE) method. The Concrete load and resistance values in a particular limit state in order specimens tested were cast in 150 mm × 150 mm × 150 mm to obtain the reliability index. The knowledge of type of moulds for compressive strength test. The samples were distribution for the random variables is not needed for this thoroughly mixed with the aid of a concrete mixer until analysis, hence all variables were assumed to be normally the desired homogeneity of the mixture was achieved. The distributed. standard iron moulds of 150 mm × 150 mm × 150 mm were used, it was ensured that the moulds were well lubricated with oil in order to reduce friction and enhance the removal of cubes from the mould. Each mould was then filled with Bending failure concrete in three layers each tampered 25 times. 80 cubes were cast in total, 40 cubes for the unwashed NS and 40 for Table 1 presents a summary of the input parameters uti- the washed NS. The cubes were cured for 28 days using lised for the reliability analysis of the reinforced concrete ponding method of curing. Cube casting was performed in beam subjected to bending forces. The input parameters are accordance to BS 1881 (1983). normally distributed because first-order reliability method (FORM) was used for analysis. All dimensional properties Compressive strength test are treated as normal distribution. Figures 1 and 2 present Probability Density Plots (PDF) Compressive strength test on concrete cubes (80 Cubes) for the compressive strength of concrete produced using was determined using the compressive testing machine. The unwashed and washed natural stones, respectively. The PDF weight of each cube was taken before crushing, this is, how- plots are in complete agreement with normal distribution ever, a destructive method of testing cubes. After crushing, used in the reliability analysis. Table 1 Input data for bending Input parameter X Mean C.O.V Std. Distribution failure Dev Slab thickness, hs (mm) X 150 0.07 10.50 Normal Beam height, h (mm) X 400 0.05 20.00 Normal Beam width, b (mm) X 225 0.05 11.25 Normal Effective depth of beam, d (mm) X 359 0.05 17.95 Normal Unit weight of concrete, X 24 0.04 0.96 Normal Diameter of tension bar, ϕ (mm) X 16 0.04 0.64 Normal Area of tension reinforcement, As (mm ) X 402 0.04 16.08 Normal Area of shear reinforcement, Asv (mm ) X 101 0.04 4.04 Normal Yield strength of steel, f (N/mm ) X 460 0.05 23.00 Normal y 9 Compressive strength of concrete, f (N/ mm ) X 22.52 0.25 5.63 Normal cu 10 Dead load intensity, DL (kN/m) X 17.70 0.10 1.77 Normal Live load intensity, LL (kN/m) X 3.33 0.18 0.60 Normal Span of beam, L (mm) X 4000 0.22 880 Normal C.O.V coefficient of variation, Std. dev standard deviation 1 3 488 Asian Journal of Civil Engineering (2021) 22:485–491 The maximum bending moment at the mid-span of a simply supported beam is represented by Eq. (3): M = wl (3) In order to satisfy flexural strength requirements, the maximum bending moment must conform to the inequal- ity M ≤ M , where M = Maximum bending Moment. u R u M = Ultimate Moment of resistance. Performance Function Equation (4) presents the equation for beam performance Fig. 1 Probability density function for compressive strengths of function unwashed natural stone (NS) Z = R− S (4) (R Resistance, S Load effect) K = (5) f bd cu M = Kf bd cu wL g (X)= 0.156 f bd − 1 cu For a balanced reinforced concrete beam section, the per- formance function is enumerated as shown in Eq. (6): 0.87f A y s g (X) = g (X) = 0.87f A d − − wl 1 1 y s 0.9f b cu Fig. 2 Probability density function for compressive strengths of 0.87f A y s (1.4DL + 1.6LL)L washed natural stone (NS) g (X) = 0.87f A d − − , 1 y s 0.9f b 8 cu (6) where f yield strength of steel, A area of tension reinforce- y s ment, d effective depth of beam, F Compressive strength of cu concrete, DL Dead load, LL live load. 0.03 0.3 3 Sieve size Fig. 3 Sieve analysis result for fine aggregate 1 3 % Cumulative Passing Asian Journal of Civil Engineering (2021) 22:485–491 489 6.5 Unwashed Aggregate Washed Aggregate 5.5 4.5 3.5 2.5 1.5 0.5 10 2400 2600 2800 3000 3200 3400 3600 Span L, (mm) Seive size Fig. 5 Relationship between safety index and span of beam in bend- ing for washed and unwashed N Fig. 4 Sieve analysis result for coarse aggregate (NS) Table 2 Specific gravity result for Fine Aggregate Trials 1 2 3 Washed Aggregate Unwashed Aggregate Weight of cylinder 115.0 116.5 16.6 Weight of cylinder + sample 207.2 240.6 248.9 Weight of cylinder + sample + water 87.3 512.0 484.9 Weight cylinder + water 428.0 435.7 402.2 Specific gravity of sample (S.G) 2.80 2.60 2.70 500 600 700 800 Average specific gravity (S.G) 2.70 Depth d, (mm) Fig. 6 Relationship between safety index and depth of beam in bend- ing for washed and unwashed NS aggregate Table 3 Specific gravity result for Coarse Aggregate (NS) Trials 1 2 3 Weight of cylinder 115.0 116.5 116.6 obtained. In order to classify a soil as well graded, the Weight of cylinder + sample 307.2 328.4 325.9 C > 6 (Arora 2003), hence it is concluded that both aggre- Weight of cylinder + sample + water 553.2 566.5 533.3 gates are not well graded. Weight cylinder + water 438.7 34.6 402.6 Specific gravity of sample (S.G) 2.50 2.65 2.66 Average specific gravity (S.G) 2.60 Aggregate specific gravity test Table 2 presents the results for specific gravity test per - Performance function linearisation formed on fine aggregate, an average specific gravity of 2.7 was obtained and is within the natural aggregates range From Eq. (6), substituting f = X f = X , DL = X y 9 cu 10 11, of 2.6–2.7 (Neville and Brooks 2008). LL = X and linearising yields Eq. (7) which is a linearised Table 3 presents the results for specific gravity test per - version of the beam performance function formed on the coarse aggregate, an average specific gravity 2 −1 g (X) = 125556.66X − 604.03984X X of 2.6 was obtained and is within the natural aggregates 1 9 9 10 (7) range of 2.6–2.7 (Neville and Brooks 2008). This implies − 1575000X − 1800000X 11 12 that the aggregate is suitable for construction work. Results and discussion Figures 3 and 4 present the sieve analysis result obtained for fine and coarse aggregates, respectively, a Coefficient of uniformity (C ) of 3.5 was obtained for the fine aggre- gate, while for the coarse aggregate C value of 1.22 was 1 3 % Cummulative Passing Safety Index (β) Safety Index (β) 490 Asian Journal of Civil Engineering (2021) 22:485–491 Unwashed Aggregate Washed Aggregate of Tension reinforcement (A ). A general increase in safety index (β) was obtained as the Area of steel reinforcement was increased. As seen on Fig. 7, the unwashed aggregates recorded higher reliability indices values when compared with the washed aggregates for bending, varying the A . Figure 8 presents the result of sensitivity analysis con- 500 600700 800 900 1000 1100 1200 1300 ducted in bending for unwashed and washed NS varying the Area of Steel, As (mm ) beam dead load (g ). A general increase in safety index (β) was obtained as the design dead load was reduced (unwashed Fig. 7 Relationship between safety index and area of tension rein- forcement in Bending for washed and unwashed NS aggregate and washed aggregate). Washed Aggregate Unwashed Aggregate Conclusion 4.5 The following conclusion is drawn from the study: the rein- 3.5 forced concrete beam utilising the natural stone (NS) as 2.5 coarse aggregate is structurally safe at a span of 3000 mm 1.5 utilising unwashed and washed NS with Probabilities of fail- 0.5 –5 –8 ure of 9.20 × 10 and 2.06 × 10 , respectively, and depth of 6789 10 11 12 13 14 15 600 mm for both unwashed and washed NS with probabili- Dead Load gk (kN/m) –4 –4 ties of failure of 4.19 × 10 and 2.602 × 10 , respectively, in bending. The safety index is very sensitive to the span and Fig. 8 Relationship between safety index and dead load of beam in depth of the beam, hence they are the critical factors to be bending for washed and unwashed aggregate put into consideration when designing reinforced concrete beam utilising the NS as coarse aggregate. Sensitivity analysis Compliance with ethical standards Figure 5 presents the result of sensitivity analysis con- ducted in bending for both unwashed and washed NS vary- Conflict of interest On behalf of all authors, the corresponding au- ing the beam span (L). As seen from Fig. 5, increasing the thor states that there is no conflict of interest. beam span reduces the safety level with reliability indices lower than the 3.0 target reliability (β ), whereas decrease T Open Access This article is licensed under a Creative Commons Attri- in the span increases the safety level for the concrete under bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long consideration. The beam is reliable at span of 3000 mm as you give appropriate credit to the original author(s) and the source, with reliability index of 3.74 and 4.08 for unwashed and provide a link to the Creative Commons licence, and indicate if changes washed aggregates, respectively. Increasing the span of were made. The images or other third party material in this article are beams increases the bending moment experienced which included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in contributes to the bending of beams. The washed aggre- the article’s Creative Commons licence and your intended use is not gates gave higher reliability indices when compared with permitted by statutory regulation or exceeds the permitted use, you will the unwashed aggregates. This result is in line with the need to obtain permission directly from the copyright holder. To view a findings of research by Ogunbiyi et al. (2017) and Ode copy of this licence, visit http://creativ ecommons .or g/licenses/b y/4.0/. and Eluozo (2016) where washed aggregates performed better than unwashed aggregates in concrete production. Figure 6 presents the result of sensitivity analysis conducted in bending for both unwashed and washed NS References varying the beam depth (d). A general increase in safety Abubakar, M. (2015). Reliability Based Design of Concrete Mixes index (β) was observed as the depth was increased from Admixed with Cow Bone Ash, Unpublished Master of Engineer- 400 to 800 mm. As seen the washed aggregates recorded ing (MEng) Thesis, Department of Civil Engineering, Federal higher reliability indices when compared with unwashed University of Technology, Minna, Niger State, Nigeria. aggregates. Alhaji, B. (2016). 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Published: Nov 22, 2020
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