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Minimum steel ratios in reinforced concrete beams made of concrete with different strengths—Theoretical approach

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Abstract

Minimum reinforcement is provided in concrete beams in order to improve their behaviour towards cracking and ductility at failure.

Generally, codes of practice equations for the minimum steel ratios, longitudinal and transversal, are mainly empirical and do not include all the influential parameters in them. For this reason and due to the fact that they do lack of a theoretical background, different codes can give values for the minimum steel ratios that greatly differs from one another. Also the validity of these equations may be questioned particularly in the case of high strength concrete beams and prestressed concrete beams for which limited test data are available.

In this work, a theoretical approach for the minimum steel ratios that are required for the ductile behaviour at failure in bending, shear and torsion, in concrete beams made of concrete with different strengths is presented. Comparisons are also made between the proposed expressions, the codes expressions and available test results.

Résumé

Un renforcement minimum des poutres en béton armé est prévu pour améliorer leur comportement en cas de fissuration et leur ductilité dans la rupture.

En général les formules des normes techniques pour les pourcentages minimum des armatures longitudinales et transversales sont simplement empiriques et ne contiennent pas tous les paramètres influents. Pour cette raison et comme elles ne prennent pas en considération des fondements théoriques, les différents règlements donnent des valeurs très différentes pour les pourcentages minimum d'acier. La validité de ces formules peut être mise en doute particulièrement dans les cas de poutres en béton à haute résistance et des poutres précontraintes pour lesquelles seuls des résultats d'essais très limités sont disponibles.

Dans cette communication une approche théorique pour les pourcentages minimum d'acier requis pour le comportement ductile dans la rupture par flexion, effort tranchant et torsion, des poutres en béton armé exécutées avec des bétons de différentes résistances est presentée. Des comparaisons entre les expressions proposées et celles des règlements sont présentées.

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Abbreviations

A c :

Area of a concrete section

A eff :

Area confined by the centre lines of the walls of a hollow section

A s :

Area of longitudinal steel

A s, min :

Minimum area of longitudinal steel

A sw :

Area of transversal steel

A sw, min :

Minimum area of transversal steel

F s, l :

Force in longitudinal steel

M :

Bending moment

M cr :

Cracking bending moment

M p :

Plastic flexural moment

M t :

Torsional moment

N :

Normal force

V cr :

Cracking shear force

V p :

Shear force due to the inclination of prestressing cables

b :

Width of a rectangular section

d :

Effective depth of a rectangular section

h :

Section height

t ef :

Effective thickness of a thin walled section

x :

Neutral axis depth

z :

Lever arm

f c :

Compressive strength of concrete

f ck :

Characteristic compressive strength of concrete

f ct :

Tensile strength of concrete

f ct, f :

Flexural tensile strength of concrete

f y :

Yield strength of steel

f yw :

Yield strength of web steel

n :

Number

α:

Coefficient

β:

Coefficient

θ u :

Truss angle

ε co :

Concrete strain at peak stress in compression

ε cu :

Ultimate concrete strain in compression

ε su :

Ultimate steel strain

ε tu :

Ultimate concrete strain in tension

ε o :

Concrete strain at peak stress in tension

σ cp :

Average stress in concrete section due to normal force

τ max :

Maximum shear stress

μ cr :

Normalized cracking moment (Mcr/fc.b.h2)

ρ s,bal :

Balanced steel ratio

ρ s,min :

Minimum longitudinal steel ratio

ρ sw,min :

Minimum transversal steel ratio

ν:

Effectiveness factor

References

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Authors and Affiliations

  1. COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    I. A. E. M. Shehata, L. C. D. Shehata & S. L. G. Garcia

  2. Universidade Federal Fluminense, Rio de Janeiro, Brazil

    L. C. D. Shehata

Authors
  1. I. A. E. M. Shehata
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  2. L. C. D. Shehata
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  3. S. L. G. Garcia
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Shehata, I.A.E.M., Shehata, L.C.D. & Garcia, S.L.G. Minimum steel ratios in reinforced concrete beams made of concrete with different strengths—Theoretical approach. Mat. Struct. 36, 3–11 (2003). https://doi.org/10.1007/BF02481565

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