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Theoretical Formula of Ultimate Shear Strength for RC Beams without Transverse Reinforcement by Using External Vertical Prestressing Rebars

  • Structural Engineering
  • Published:
KSCE Journal of Civil Engineering Aims and scope

Abstract

External vertical prestressing rebars (EVPRs) can effectively improve shear strength of concrete structures by the way of providing vertical compressive stress to the concrete and acting as stirrups. The objective of this study is to obtain a shear strength prediction model of EVPRs strengthening concrete beam without transverse reinforcement enhanced by modifying the ACI-318 and Critical Shear Crack Theory (CSCT). At the same time, this article analyzes the contribution of the strength of the shear strength, including the EVPRs as stirrups, compression zone of concrete, dowelling action, aggregate interlock and residual tensile strength. The results indicate that the shear strength prediction model can fit well with the experimental data and EVPRs can effectively enhance shear strength.

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Abbreviations

a (mm):

Shear span of the segment without transverse reinforcement

A s (mm2):

Area of longitudinal bars intersects at point A of the critical shear crack

A sv (mm2):

Area of stirrups within spacing sP

A svP (mm2):

Area of the EVPRs within spacing sP

b (mm):

Width of the beam

b ef (mm):

Effective width of concrete in tension

c (mm):

Depth of the compression zone

c 1 c 2 c 3 :

Related parameters in the CSC

c b (mm):

Concrete cover

d (mm):

Effective height of beam cross-section

d b (mm):

Diameter of the longitudinal bar

d F (mm):

Vertical height of the CSC

d g (mm):

Maximum aggregate size

E c (MPa):

Modulus of elasticity of conrete

E s (MPa):

Modulus of elasticity of the longitudinal bar

E svP (MPa):

Modulus of elasticity of the EVPR

f c (MPa):

Concrete compressive strength in cylinder

f ct,ef (MPa):

Effective tensile strength of concrete

F P (N):

Pulling force in the EVPRs

f t (MPa):

Concrete tensile strength in cylinder

f yv (MPa):

Yield strength of stirrups

f y vp (MPa):

Yield strength of the EVPRs

G F (N/mm):

Fracture energy of concrete

h (mm):

Height of beam cross-section

h F (mm):

Vertical distance from the point F of the CSC to the beam top

k b :

Reduction factor tensile strength of concrete

l (mm):

Span length

l 1 l 2 l 3 l F1 (mm):

Integration limits

l B (mm):

Length of segment A-B of the CSC

l ef (mm):

Effective length of concrete in tension

l F (mm):

Length of segment B-F of the CSC

m:

Number of longitudinal bars activated in dowel action

n P :

The arranged number of the EVPRs along the beam width

r F (mm):

Horizontal distance from the point F of the CSC to the loading

s (mm):

The arranged spacing of stirrups along the beam length

s P (mm):

The arranged spacing of the EVPRs along the beam length

u A (mm):

Horizontal opening of the CSC at point A

v A (mm):

Vertical opening of the CSC at point A

V Agg (N):

Shear capacity contribution of aggregate interlock of concrete in the beam without transverse reinforcement

V c(N):

Shear capacity contribution of the concrete in the beam without transverse reinforcement

V C ompr (N):

Shear capacity contribution of compression zone of concrete in the beam without transverse reinforcement

V comprP (N) :

Shear capacity contribution of compression zone of concrete in the beam without transverse reinforcement but enhanced by the EVPRs

V cP (N):

Shear capacity contribution of the concrete in the beam without transverse reinforcement but enhanced by the EVPRs

V Dowel (N):

Shear capacity contribution of dowel action of longitudinal bars in the beam without transverse reinforcement

V DowelP (N):

Shear capacity contribution of dowel action of longitudinal bars in the beam without transverse reinforcement but enhanced by the EVPRs

V Res (N):

Shear capacity contribution of residual tensile strength of concrete in the beam without transverse reinforcement

V ResAB (N):

Shear capacity contribution of residual tensile strength of concrete at the segment A-B of the CSC in the beam without transverse reinforcement

V ResABP (N):

Shear capacity contribution of residual tensile strength of concrete at the segment A-B of the CSC in the beam without transverse reinforcement but enhanced by the EVPRs

V ResBF (N):

Shear capacity contribution of residual tensile strength of concrete at the segment B-F of the CSC in the beam without transverse reinforcement

V ResBFP (N):

Shear capacity contribution of residual tensile strength of concrete at the segment B-F of the CSC in the beam without transverse reinforcement but enhanced by the EVPRs

V ResP (N):

Shear capacity contribution of residual tensile strength of concrete in the beam without transverse reinforcement but enhanced by the EVPRs

V s (N):

Shear capacity of the ordinary stirrups

V sP (N):

Shear capacity contribution of the EVPRs as stirrups

V u (N):

Total shear capacity of the beam without transverse reinforcement

V uP (N):

Total shear capacity of the beam without transverse reinforcement but enhanced by the EVPRs

w C (º):

Maximum opening of concrete allowing to transfer stress in the CSC

x A (mm):

Horizontal distance from the support to point A of the CSC

x F (mm):

Horizontal distance from the support to point F of the CSC

β AB (º):

Angle of the segment A-B of the CSC

β BF (º):

Angle of the segment B-F of the CSC

ρ :

Reinforcement ratio of longitudinal bars

δ A (mm):

Crack sliding of the CSC at point A

ε s :

Strain of the longitudinal bar

σ zP0 (MPa):

Vertical compression stress in the concrete provided by the initial pulling force of the EVPRs

σ zP1 (MPa):

Vertical compression stress in the concrete provided by the EVPRs during the loading process of the beam

σ zPy (MPa):

Vertical compression stress in the concrete provided by the EVPRs when the EVPRs at yielding strength

ψ (º):

Rotating angle of the CSC

ω A (º):

Opening at point A of the CSC

k c :

Coefficient compression zone

ω B(º):

Opening at point B of the CSC

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Acknowledgements

This research was funded by the Scientific Research Project of Educational Department of Liaoning Province (LNZD202005) and the financial support from the project of the MOE Key Lab of Disaster Forecast and Control in Engineering of Jinan University grant number (20200904005). We would like to thank Editage (https://www.editage.cn) for English language editing.

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Authors and Affiliations

  1. School of Traffic Engineering, Shenyang Jianzhu University, Shenyang, 110168, China

    Xingwei Xue & Meizhong Wu

  2. Shenyang Jianzhu University, Shenyang, 110168, China

    Peng Zhou

  3. Dalian University of Technology, Dalian, 116024, China

    Junlong Zhou

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  1. Xingwei Xue
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  2. Meizhong Wu
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  3. Peng Zhou
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Correspondence to Junlong Zhou.

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Xue, X., Wu, M., Zhou, P. et al. Theoretical Formula of Ultimate Shear Strength for RC Beams without Transverse Reinforcement by Using External Vertical Prestressing Rebars. KSCE J Civ Eng 25, 2522–2533 (2021). https://doi.org/10.1007/s12205-021-0911-2

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  • DOI: https://doi.org/10.1007/s12205-021-0911-2

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