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Studies on ductility and evaluation of minimum flexural reinforcement in RC beams


Some experimental investigations on ductility and prediction of minimum flexural reinforcement in reinforced concrete (RC) beams are reported. The minimum flexural reinforcement was evaluated using optimum ductility in RC beams. Beams of size 100 mm, 200 mm and 400 mm were tested, which were designed with varying percentages of flexural reinforcement i.e. 0.15, 0.30, 0.60 and 1.0. The beams were tested under four-point loading to study the flexural behaviour under uniform bending moment. The experimentally obtained average compressive strength of concrete was 30 MPa. The influence of beam size (depth) on cracking and normalised ultimate flexural strength, ductility and overall average rotation has been studied. The cracking in RC beams is complex phenomenon in small size beams, while the cracking strength decreases as the depth increases beyond 200 mm. The flexural strength of RC beams, from the present study, appears to decrease as the depth increases. The ductility of RC beams increases as the percentage of flexural reinforcement increases. The ductility number has been derived from dimensional analysis using fracture mechanics principles. The ductility of RC beams decreases as the depth of beams increases. An optimum percentage of flexural reinforcement has been established using optimum ductility number, Np, which is equal to 0.20. The minimum flexural reinforcement was found to decrease as the beam depth increases, and decreases as the yield strength of reinforcement increases.

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A constant obtained from the analysis


Area of c/s of beam

Ag :

Gross area of cross-section

As :

Area of steel reinforcement


A constant obtained from the analysis


A constant obtained from the analysis


Size of the member

Ec :

Modulus of elasticity of concrete

Es :

Modulus of elasticity of steel reinforcement

fy :

Yield strength steel reinforcement

ft :

Tensile strength of concrete

GF :

Fracture energy of concrete


Beam depth

KIc :

Concrete fracture toughness

wc :

Crack opening displacement

ρmin :

Minimum flexural reinforcement


  1. ACI 318 (2005) Building code requirements for structural concrete and commentary. American Concrete Institute 1995, Farmington Hills, Michigan

  2. IS 456 (2000) Code of practice for design of plain and reinforced concrete structures. Bureau of Indian Standards, New Delhi

  3. EN 1992 (2003) Design of concrete structures-part 1.1: general rules and rules for buildings. European Standards

  4. Shehata IAEM, Shehata LCD, Garcia SLG (2003) Minimum steel ratio in RC beams made of concrete with different strength–theoretical approach. Mater Struct 36:03–11

    Article  Google Scholar 

  5. Carpinteri A (1984) Stability of fracturing process in RC beams. ASCE Struct J 110(3): 544–558

    Article  Google Scholar 

  6. Bosco C, Carpinteri A, Debernardi PG (1991) Use of brittleness number as a rational approach to minimum reinforcement design. RILEM workshop on Analysis of Concrete Structures by Fracture Mechanics, Sweden, pp 133–151

  7. Hillerborg A (1990) Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforced concrete beams. Eng Fract Mech 35(1/2/3):233–240

    Google Scholar 

  8. Bosco C, Carpinteri A, Debernardi PG (1990) Minimum reinforcement in high strength concrete. J Struct Eng—ASCE 116(02): 427–437

    Article  Google Scholar 

  9. Bosco C, Carpenteri A, Debernardi PG (1990) Fracture of reinforced concrete; scale effects and snap–back instability. Eng Fract Mech 35(4/5): 665–677

    Article  Google Scholar 

  10. Baluch MH, Azad AK, Ashmawi W (1992) Fracture mechanics application to reinforced concrete members in flexure. In: International Workshop on application of fracture mechanics to reinforced concrete, Italy, pp 413–436

  11. Gerstle WH, Dey PP, Prasad NNV, Rahulkumar P, Xie M (1992) Crack growth in flexural members—a fracture mechanics approach. ACI Struct J, 89(6):617–62

    Google Scholar 

  12. IS 4031-1968: Test methods for ordinary portland cement. Bureau of Indian Standards, New Delhi

  13. IS 2386-1963: Methods of test for aggregates for concrete. Bureau of Indian Standards, New Delhi

  14. IS 10262-1999: Recommended guidelines for concrete mix design. Bureau of Indian Standards, New Delhi

  15. IS 1786-1985: Specification for high strength deformed steel bars and wires for concrete reinforcement. Bureau of Indian standards, New Delhi

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Correspondence to Gangolu Appa Rao.

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Appa Rao, G., Vijayanand, I. & Eligehausen, R. Studies on ductility and evaluation of minimum flexural reinforcement in RC beams. Mater Struct 41, 759–771 (2008).

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  • Ductililty
  • Minimum reinforcement
  • Average rotation
  • Size effect
  • Flexural strength