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Design shear strength formula for high strength concrete beams

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Abstract

After a brief review on the concrete shear strength mechanisms, two very reliable expressions for predicting the shear strength of beams without transverse reinforcement are reported: the one proposed by Bažant and Kim [7], which is valid for Normal strength Concrete (NSC) beams, and the other recently proposed by the authors, which is valid for High Strength Concrete (HSC) beams. Hence a summary of a shear strength model for beams with stirrups is provided, which was derived [27] on the basis of the Bažant and Kim expression and therefore is adequate only for NSC beams.

On the basis of the expression obtained for HSC without stirrups and of the model already proposed for NSC with stirrups, a shear strength expression for HSC beams with stirrups is derived. The obtained expression is applied to an experimental program of 116 HSC beams with stirrups, and is found to predict the test results better than ACI Code [1], Eurocode [12] and CEB/FIP Model Code [9].

A design formula is hence proposed, which is adequately conservative and accurate.

A design example of a HSC beam with stirrups is carried out, and the various design expressions previously considered are compared.

Résumé

Après une brève revue des mécanismes de résistance au cisaillement, deux expressions très fiables prévoyant la résistance au cisaillement de poutres sans armure transversale sont rapportées: l'une proposée par Bažant et Kim [7], valable pour les poutres en béton ordinaire, l'autre récement proposée par les auteurs, valable pour les poutres en béton à haute performance. Le résumé d'un modèle de résistance au cisaillement est fourni pour les poutres avec arçons, déduit [27] à partir de l'expression de Bažant et Kim et donc adéquat seulement pour les poutres en béton ordinaire avec arçons.

En se basant sur l'expression obtenue pour les poutres en béton à hautes performances (BHP) sans arçons et sur le modèle déjà proposé pour les poutres en béton ordinaire avec arçons, on traouve une expression pour les poutres en BHP avec arçons. L'expression ainsi obtenue est appliquée à un échantillon expérimental de 116 poutres en BHP avec arçons, et s'avère plus apte à prévoir les résultats expérimentaux que le Code ACI [1], l'Eurocode [12], ou le code de référence CEB/FIP [9].

Il est donc proposé pour ce projet une expression à la fois traditionnelle et précise. Un exemple d'étude de poutre en BHP avec arçons est exposé et les différentes expressions déjà considérées pour le projet sont comparées.

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Abbreviations

A s :

area of the longitudinal reinforcement

A v :

area of stirrups

C :

internal compression force

I b :

beam action index

I bc :

beam action index in critical conditions

M b :

beam action contribution to the ultimate bending moment

M c :

resisting bending moment

M FL :

pure flexure bending moment

M u :

ultimate bending moment in beams with stirrups

M uc :

ultimate bending moment in beams without stirrups

T :

tensile force in the longitudinal reinforcement

V a :

resultant of the aggregate interlocking stresses at the crack interface

V c :

shear force due to the concrete resisting contribution

V d :

shear force due to the dowel action

a :

shear span

(a/d) c :

critical value of the shear span to effective depth ratio

b :

beam width

c :

concrete cover

d :

beam effective depth

d a :

maximum aggregate size

f c :

concrete compressive cylindrical strength

f yl :

yielding strength of the longitudinal reinforcement

f yv :

yielding strength of the transverse reinforcement

h :

beam depth

j :

ratio between the internal lever arm and the beam effective depth

k, q :

coefficients

r :

coefficient which is the 0.05 fractile of the corresponding statistical distribution

s :

stirrups spacing

v s :

shear strength provided by stirrup

v si :

shear strength due to stirrups inclusion

v u :

shear strength of a beam with transverse reinforcement

v uc :

shear strength provided by concrete

x :

distance between the support and the transverse section at the ending point of the crack

ϕ:

angle between the diagonal compression strut and the beam longitudinal axis

ζ:

function taking into account the size effect

ρ:

longitudinal reinforcement percentage

ρ v :

geometrical percentage of the transverse reinforcement

ψ:

stirrup effectiveness function

meas :

measured

calc :

calculated

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Authors and Affiliations

  1. Department of Civil Engineering, University of Udine, Udine, Italy

    G. Russo, G. Somma & P. Angeli

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  3. P. Angeli
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Russo, G., Somma, G. & Angeli, P. Design shear strength formula for high strength concrete beams. Mat. Struct. 37, 680–688 (2004). https://doi.org/10.1007/BF02480513

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