Mechanical model for the shear strength of steel fiber reinforced concrete (SFRC) beams without stirrups


Numerous studies have shown that adding steel fibers to the concrete mixture improves both the shear strength and ductility of reinforced concrete beams. Most current shear design formulations for steel fiber reinforced concrete beams are empirically based and their predictions provide acceptable results when compared with tests results, but only in limited ranges of the parameters involved. On the other hand, many shear theoretical models suggested in the literature are derived by extending previous formulations for conventional reinforced concrete beams, just introducing the stresses transferred across the critical shear crack. Since the effects of steel fibers on the others shear resisting mechanisms are not accounted for, “adjusting” empirical factors must be used to fit the experimental results. There is, therefore, the need for developing mechanical models capable to rationally account for the effects of steel fibers on the global shear strength and on each shear resisting mechanisms. In this paper, the previously derived and validated Multi Action Shear Model for RC elements is extended to steel fiber reinforced concrete beams without stirrups. The effects of steel fibers on each resisting mechanism have been identified and incorporated in the formulation of each shear component and in the Multi-Action Shear Model equilibrium equations. The residual tensile stresses of fibrous concrete, obtained through a simple formulation, has been used to: capture the enhancement of the compression chord contribution; the shear transferred by bridging effect of the fibres along the critical shear crack and by dowel action. The proposed model is shown to properly estimate the shear resistance of a large set of available test data, being able to account for most influencing parameters, like fibers types and amounts, concrete strength, longitudinal reinforcement ratios and beam geometry.

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As, Asw :

Longitudinal rebar and stirrups area

C :

Resultant of compression force

E c :

Young modulus of plain concrete

F τ :

Fiber factor

G f :

Energy fracture of plain concrete

L f :

Fiber length

M, Mcr :

Bending moment and cracking bending moment

R τ :

Reduction factor

T f :

Horizontal component of fiber resultant force

T l :

Resultant force of longitudinal rebar

VExp, VPre :

Experimental and predicted shear strength

V f :

Volumetric percentage of fibers

Vc, Vcw, Vs, Vl, Vcf :

Shear strength contribution: concrete, concrete web, stirrups, dowel action, and fibers

V u :

Ultimate shear strength

a :

Shear span length

b, d, h :

Width, effective depth and total depth of beam cross-section

d f :

Fiber diameter

d g :

Maximum aggregate size

d 0 :

Effective depth of the cross-section (d), but not less than 100 mm

fck, fcm :

Characteristic and average value of the plain concrete compressive strength

f ct :

Tensile strength under direct tension of plain concrete

fFts, fFtu :

Tensile stress level representative of SLS and ULS

fR1, fR3 :

Residual flexural tensile strength corresponding to CMOD = 0.5 and 2.5 mm

fy, fyw :

Steel rebar and steel stirrups yield strength

k :

Size effect parameter

k dg :

Average crack specimen factor

k v :

Strain-aggregate size parameter

l c :

Critical fiber length

l cr :

Distance from shear critical crack to the support

n :

Homogenization factor

vc, vcw, vs, vl, vcf :

Non-dimensional shear strength contribution: concrete, concrete web, stirrups, dowel action, and fibers

v u :

Non-dimensional ultimate shear strength

x, xf :

Neutral axis depth of plain concrete and SFRC

y :

Vertical abscissa from neutral axis

w u :

Critical crack width

z :

Inner lever arm

β :

Reduction factor

β τ :

Fiber bond factor

εx, εsb :

Average normal strain of concrete and of longitudinal reinforcement

η0, ηl :

Fiber orientation factor and length efficiency factor for fiber

μ cr :

Non-dimensional cracking moment

θ :

Shear crack inclination angle

ρ :

Geometrical percentage of longitudinal reinforcement

σ1, σ2 :

Principal tensile stresses

σ c,crit :

Normal stress at the critical point

σ c,max :

Normal stress at the top fiber of cross-section

σ sy :

Steel fiber yield strength

σ tu :

Average residual tensile stress of SFRC

σx, σy :

Tensile stresses along x and y direction

τ, τu :

Shear stress and ultimate shear stress

τ f :

Mean fiber-matrix shear stress

ξ :

Size and slenderness factor

ζ :

Abscissa along the longitudinal axis beam


  1. 1.

    Marí A, Bairán J, Cladera A et al (2015) Shear-flexural strength mechanical model for the design and assessment of reinforced concrete beams. Struct Infrastruct Eng 11:1399–1419.

    Article  Google Scholar 

  2. 2.

    Zararis PD, Papadakis GC (2001) Diagonal shear failure and size effect in RC beams without web reinforcement. J Struct Eng 127:733–742.

    Article  Google Scholar 

  3. 3.

    Minelli F (2005) Plain and fiber reinforced concrete beams under shear loading: structural behavior and design aspects. University of Brescia, Italy

    Google Scholar 

  4. 4.

    Lantsoght EOL (2019) Database of shear experiments on steel fiber reinforced concrete beams without stirrups. Materials (Basel) 12:917.

    Article  Google Scholar 

  5. 5.

    Dinh HH, Parra-Montesinos GJ, Wight JK (2011) Shear strength model for steel fiber reinforced concrete beams without stirrup reinforcement. J Struct Eng 137:1039–1051.

    Article  Google Scholar 

  6. 6.

    Minelli F, Conforti A, Cuenca E, Plizzari G (2014) Are steel fibres able to mitigate or eliminate size effect in shear? Mater Struct 47:459–473.

    Article  Google Scholar 

  7. 7.

    Shoaib A, Lubell AS, Bindiganavile VS (2014) Size effect in shear for steel fiber reinforced concrete members without stirrups. ACI Struct J 111:1081–1090.

    Article  Google Scholar 

  8. 8.

    Zarrinpour MR, Chao S-H (2017) Shear strength enhancement mechanisms of steel fiber-reinforced concrete slender beams. ACI Struct J 114:729–742.

    Article  Google Scholar 

  9. 9.

    Qi J, Ma ZJ, Wang J (2017) Shear strength of UHPFRC beams: mesoscale fiber-matrix discrete model. J Struct Eng 143:04016209.

    Article  Google Scholar 

  10. 10.

    Casanova P, Rossi P (1997) Analysis and design of steel fiber reinforced concrete beams. ACI Struct J 94:595–602.

    Article  Google Scholar 

  11. 11.

    RILEM TC-162-TDF (2003) Test and design methods for steel fibre reinforced concrete: s–e design method. Final recommendation. Mater Struct 36:560–567

    Article  Google Scholar 

  12. 12.

    German Committee for Structural Concrete (DAfStb) (2010) Guideline—fiber reinforced concrete. Berlin

  13. 13.

    Foster S (2010) Design of FRC beams for shear using the VEM and the draft Model Code approach. In: fib Bulletin No. 57: shear and punching shear in RC and FRC elements. International Federation for Structural Concrete (fib), Salò, Lake Garda, Italy, pp 195–210

  14. 14.

    Aoude H, Belghiti M, Cook WD, Mitchell D (2012) Response of steel fiber-reinforced concrete beams with and without stirrups. ACI Struct J 109:359–368.

    Article  Google Scholar 

  15. 15.

    Spinella N, Colajanni P, La Mendola L (2012) Nonlinear analysis of beams reinforced in shear with stirrups and steel fibers. ACI Struct J 109:53–64

    Google Scholar 

  16. 16.

    Colajanni P, Recupero A, Spinella N (2012) Generalization of shear truss model to the case of SFRC beams with stirrups. Comput Concr 9:227–244.

    Article  Google Scholar 

  17. 17.

    Spinella N, Colajanni P, Recupero A (2010) Simple plastic model for shear critical SFRC beams. J Struct Eng 136:390–400.

    Article  Google Scholar 

  18. 18.

    Spinella N (2013) Shear strength of full-scale steel fibre-reinforced concrete beams without stirrups. Comput Concr 11:365–382.

    Article  Google Scholar 

  19. 19.

    Voo YL, Foster SJ, Gilbert RI (2006) Shear strength of fiber reinforced reactive powder concrete prestressed girders without stirrups. J Adv Concr Technol 4:123–132.

    Article  Google Scholar 

  20. 20.

    Tung ND, Tue NV (2018) Shear resistance of steel fiber-reinforced concrete beams without conventional shear reinforcement on the basis of the critical shear band concept. Eng Struct 168:698–707.

    Article  Google Scholar 

  21. 21.

    Kupfer HB, Gerstle KH (1973) Behavior of concrete under biaxial stresses. J Eng Mech Div 99:853–866

    Google Scholar 

  22. 22.

    Marì A, Cladera A, Ribas C et al (2018) Simplified Multi-action Shear Model for plain or steel fibre reinforced concrete beams longitudinally reinforced with steel or FRP bars. In: fib bulletin 85—towards a rational understanding of shear in beams and slabs. Fédération internationale du béton (fib), pp 260–273

  23. 23.

    Lim TY, Paramisivam P, Lee SL (1987) Bending behavior of steel-fiber concrete beams. ACI Struct J 84:524–536.

    Article  Google Scholar 

  24. 24.

    Cuenca E, Conforti A, Minelli F et al (2018) A material-performance-based database for FRC and RC elements under shear loading. Mater Struct 51:11.

    Article  Google Scholar 

  25. 25.

    Barros JAO, Foster SJ (2018) An integrated approach for predicting the shear capacity of fibre reinforced concrete beams. Eng Struct 174:346–357.

    Article  Google Scholar 

  26. 26.

    Fib (2010) Fib model code for concrete structures 2010. Lausanne, Switzerland

  27. 27.

    American Society for Testing and Materials (2006) ASTM C 1609/C 1609M-05: Standard test method for flexural performance of fiber reinforced concrete (using beam with third-point loading). In: Annual book of ASTM Standards. West Conshohocken, PA

  28. 28.

    Voo JYL, Foster SJ (2003) Variable engagement model for fibre reinforced concrete in tension. University of New South Wales, School of Civil and Environmental Engineering, Sydney

    Google Scholar 

  29. 29.

    Lee S-C, Cho J-Y, Vecchio FJ (2011) Diverse embedment model for steel fiber-reinforced concrete in tension: model development. ACI Mater J 108:516–525.

    Article  Google Scholar 

  30. 30.

    Lee S-C, Cho J-Y, Vecchio FJ (2013) Simplified diverse embedment model for steel fiber-reinforced concrete elements in tension. ACI Mater J 110:403–412.

    Article  Google Scholar 

  31. 31.

    Bentz E (2000) Sectional analysis of reinforced concrete members. National Library of Canada = Bibliothèque nationale du Canada

  32. 32.

    Mohr S, Bairán JM, Marí AR (2010) A frame element model for the analysis of reinforced concrete structures under shear and bending. Eng Struct 32:3936–3954.

    Article  Google Scholar 

  33. 33.

    Cladera A, Marí A, Bairán JM et al (2016) The compression chord capacity model for the shear design and assessment of reinforced and prestressed concrete beams. Struct Concr 17:1017–1032.

    Article  Google Scholar 

  34. 34.

    Cladera A, Marí A, Ribas C et al (2015) Predicting the shear–flexural strength of slender reinforced concrete T and I shaped beams. Eng Struct 101:386–398.

    Article  Google Scholar 

  35. 35.

    CEN (2005) Eurocode 2—design of concrete structures: part 1–1. General rules and rules for buildings. CEN 1992-1-1

  36. 36.

    Bažant ZP, Yu Q, Gerstle W et al (2007) Justification of ACI 446 proposal for updating ACI code provisions for shear design of reinforced concrete beams. ACI Struct J 104:601–610.

    Article  Google Scholar 

  37. 37.

    Cladera A, Marì A, Ribas C et al (2019) A simplified model for the shear strength in RC and PC beams, and for punching shear in slabs, without or with shear reinforcement, including steel, FRP and SMA. In: SMAR 2019—5th international conference on smart monitoring, assessment and rehabilitation of civil structures. Potsdam, Germany

  38. 38.

    Bentz EC, Vecchio FJ, Collins MP (2006) Simplified modified compression field theory for calculating shear strength of reinforced concrete elements. ACI Struct J 103:614–624.

    Article  Google Scholar 

  39. 39.

    Yin WS, Su ECM, Mansur MA, Hsu TTC (1989) Biaxial tests of plain and fiber concrete. ACI Mater J 86:236–243.

    Article  Google Scholar 

  40. 40.

    Lantsoght EOL (2019) Database of experiments on SFRC beams without stirrups failing in shear. Accessed 27 Aug 2019

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The financial support provided by the University of Messina (Italy), through the scholarship granted for a 2-months research and teaching stage of the first author, is acknowledged.


This work is part of the Research Projects BIA2015-64672-C4-1-R, funded by the Spanish Ministry of Economy and competitiveness, and RTI2018-097314-B-C21, funded by The Spanish Ministry of Science and Innovation.

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Correspondence to Nino Spinella.

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Marì Bernat, A., Spinella, N., Recupero, A. et al. Mechanical model for the shear strength of steel fiber reinforced concrete (SFRC) beams without stirrups. Mater Struct 53, 28 (2020).

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  • Steel fiber reinforced concrete (SFRC)
  • Shear strength
  • Multi Action Shear Model
  • Shear resisting mechanisms
  • Mechanical model