Higher concrete grade or more reinforcement? - A data driven approach for optimal shear strength

1. Introduction

Many a time the structural engineer has to choose between higher concrete grade and higher percentage of reinforcement. Some believe that increasing the concrete grade can result in savings and some intuitively view a high concrete grade to be very uneconomical for a small project.

The shear strength of concrete section varies based on the % of reinforcement and the grade of concrete.

In this article, let us examine whether higher concrete grade or higher reinforcement % makes significant difference for the shear strength of the section, by referring to the Indian concrete code IS 456:2000.

2.0 Data collection

2.1 Shear strength:

The below table (table 1) from IS 456:2000 is our data to be examined.

The parameters A_{s ,} b and d in table 1 are defined as below:

A_s=area of tension reinforcement. 
b=width of the concrete member
d=effective depth of the concrete member (overall depth minus distance between reinforcement centroid and extreme concrete fiber on tension side).

Table 1

2.2 Material price

The below material prices have been assumed.

Table 2  

                                                     

Grade of Conc.	Price (Rs/cum)	Shuttering (Rs/Sqm)
M15	4500	500
M20	4800	500
M25	5000	500
M30	5500	500
M35	5800	500
M40	6000	500

Reinforcement cost (Fe500): Rs. 60000/- per ton

2.3 Data clean-up

In the absence of an excel file for the shear strength table, the above picture was converted into Excel by using the below online character recognition tool.

https://www.onlineocr.net/

The data converted into excel had some decimal points missing, and some numbers not properly recognized. The data table being small, this was cleaned up by visual inspection.

3.0 Case study

Let us consider a one way slab of 200 mm thickness. Assume 20mm clear cover and 16mm dia rebar. Then effective depth of the section d=200-20-16/2=172 mm.

Our shear strength studies will be based on this slab.

4.0 Assumptions

The following assumptions have been made.

1. Maximum concrete grade is M40.
2. Minimum % of reinforcement is 0.15.
3. Maximum % of reinforcement is 3.0
4. The depth of the section is constrained to be constant.
5. Costing data is assumed.
6. The slab requires no shear links.

5.0 Exclusion

The below is the exclusions

• Distribution reinforcement is not considered in estimation and costing.

6.0 Data generation

For the slab under consideration, the following data is further generated from the shear strength and material pricing data collected.

6.1 Material price of reinforced concrete

The following material prices have been arrived at per cu m.

Table 3  

		Cost of reinforced concrete in Rs. per cu m
		M15	M20	M25	M30	M35	M40
100As/bd	0.15	7608	7908	8108	8608	8908	9108
	0.25	8013	8313	8513	9013	9313	9513
	0.5	9025	9325	9525	10025	10325	10525
	0.75	10038	10338	10538	11038	11338	11538
	1	11051	11351	11551	12051	12351	12551
	1.25	12063	12363	12563	13063	13363	13563
	1.5	13076	13376	13576	14076	14376	14576
	1.75	14089	14389	14589	15089	15389	15589
	2	15101	15401	15601	16101	16401	16601
	2.25	16114	16414	16614	17114	17414	17614
	2.5	17127	17427	17627	18127	18427	18627
	2.75	18139	18439	18639	19139	19439	19639
	3	19152	19452	19652	20152	20452	20652

6.2 Normalized material price of reinforced concrete

In table 3 above, M15 @ 0.15% reinforcement costs the least. Let us normalize the reinforced concrete costs with respect to the same, presented in the below table.

Table 4

		Cost of reinforced concrete per cu m (normalized w.r.t. M15 @ 0.15% As)
		M15	M20	M25	M30	M35	M40
100As/bd	0.15	1.00	1.04	1.07	1.13	1.17	1.20
	0.25	1.05	1.09	1.12	1.18	1.22	1.25
	0.5	1.19	1.23	1.25	1.32	1.36	1.38
	0.75	1.32	1.36	1.39	1.45	1.49	1.52
	1	1.45	1.49	1.52	1.58	1.62	1.65
	1.25	1.59	1.63	1.65	1.72	1.76	1.78
	1.5	1.72	1.76	1.78	1.85	1.89	1.92
	1.75	1.85	1.89	1.92	1.98	2.02	2.05
	2	1.99	2.02	2.05	2.12	2.16	2.18
	2.25	2.12	2.16	2.18	2.25	2.29	2.32
	2.5	2.25	2.29	2.32	2.38	2.42	2.45
	2.75	2.38	2.42	2.45	2.52	2.56	2.58
	3	2.52	2.56	2.58	2.65	2.69	2.71

6.3 Shear strength per unit normalized price of reinforced concrete

By dividing the shear strength in table 1 by the unit normalized price of reinforced concrete in table 4, let us arrive at the strength we are achieving per unit concrete price (normalized), as in table 5 below.

Table 5

		Shear strength per unit normalized price (MPa)
		M15	M20	M25	M30	M35	M40
100As/bd	0.15	0.28	0.27	0.27	0.26	0.25	0.25
	0.25	0.33	0.33	0.32	0.31	0.30	0.30
	0.5	0.39	0.39	0.39	0.38	0.37	0.37
	0.75	0.41	0.41	0.41	0.41	0.40	0.40
	1	0.41	0.42	0.42	0.42	0.41	0.41
	1.25	0.40	0.41	0.42	0.41	0.42	0.42
	1.5	0.40	0.41	0.41	0.41	0.41	0.41
	1.75	0.38	0.40	0.41	0.40	0.41	0.41
	2	0.36	0.39	0.40	0.40	0.40	0.40
	2.25	0.34	0.38	0.39	0.39	0.39	0.40
	2.5	0.32	0.36	0.38	0.38	0.38	0.39
	2.75	0.30	0.34	0.37	0.37	0.38	0.38
	3	0.28	0.32	0.36	0.36	0.37	0.37

7.0 Data visualization

From table 5, we see the value of shear strength we achieve per unit price increases as the % of reinforcement  increases initially and then starts decreasing. This happens for all values of concrete grade, from M15 through M40. On the other hand, the strength per unit price decreases as the concrete material grade increases at lower values of  100A_s/bd. This reverses at higher values of 100A_s/bd, with the strength per unit price increasing with concrete grade.

Since the strength per unit price increases with increasing 100A_s/bd for all concrete grades, let us plot the mean shear strength across all grades considered against 100A_s/bd as in the chart below.

8.0 Data analytics

8.1 Correlation

From the above data visualization, we understand there is a steep rise in the shear strength with reinforcement increasing up to 1% and then there is a relatively mild fall.

The correlation coefficient calculated in Excel works out to be around 0.208, which indicates mild positive correlation. 

8.2 ANOVA

Though ANOVA is ideally for categorical variables, it can also be used against continuous independent variables as in this case.

By performing single factor ANOVA across the rows and across the columns in Excel, below are the results obtained.

Table 6

Table 7

9.0 Interpretation of results

1. The mild positive correlation in the chart plotted is to initial steep increase in the value of strength followed by subsequent relatively mild decline.
2. From the ANOVA in Table 6, the P-value is very high across the columns i.e. for varying concrete grades. So we accept the null hypothesis and infer that the difference in the mean values of shear strength per unit price across varying concrete grades is not statistically significant.
3. From the ANOVA in Table 7, the P-value is very low and nearly zero across rows i.e. for varying % reinforcement. So we reject the null hypothesis and infer that the difference in the mean values of shear strength per unit price is statistically significant.

10.0 Conclusions

The below conclusions are drawn from this study.

1. It is the % of reinforcement that influences the shear strength achieved per every unit of currency, more than the grade of concrete.
2. From data visualization, it is be concluded that the influence of % reinforcement has a positive effect on the cost effectiveness of the cross-section with respect to shear strength, only up to certain % of reinforcement. 

11.0 Practical Applications

1. Reinforcement % increased up to some percentage (1% in the present study) in a flexural member will improve the cost efficiency of the section in terms of its shear strength.
2. Where the section design is governed by shear strength, it will be judicious to increase the reinforcement only up to some percentage. Beyond that, other options like revising the section dimensions/slab depth etc. need to be explored.