Materials and Structures

, 51:139 | Cite as

Analytical bond strength of deformed bars in concrete due to splitting failure

  • Feng XuEmail author
  • Zhimin Wu
  • Weiwei Li
  • Weiqing Liu
  • Shuguang Wang
  • Dongsheng Du
Original Article


The bond behavior of deformed bar in concrete is important for the design and analysis of concrete structures, especially with insufficient concrete cover or confinement. This paper presents an analytical solution for the bond strength of deformed bar in concrete due to the splitting failure of concrete cover. A hypothesis of the equivalent elastic deformation field is proposed to approximate the deformation within the inner cracked concrete, while a linear crack profile is assumed to formulate the width of fictitious cracks. In conjunction with a realistic softening relationship, the analytical solutions of the internal pressure and bond strength are then derived for the splitting failure. The comparisons between the analytical predictions and the extensive experimental results demonstrate that the presented analytical model is capable of predicting the bond capacity with good accuracy. Finally, the influences of geometric and material parameters on the prediction results are examined, and an expression for the required bonded or splice length is suggested for design applications.


Reinforced concrete Deformed bars Bond strength Splitting failure Analytical solution 

List of symbols


Cross-section area of reinforcing bar


Thickness of concrete cover

c1, c2

Material constants for softening relationship


Diameter of reinforcing bar


Radial distance from the fictitious crack front to the circle of rebar


Elastic modulus of concrete


Elastic modulus of reinforcing bar


Tensile strength of concrete


Yielding strength of reinforcing bar


Critical bonded length for short bonded specimen


Bonded/splice length


Number of fictitious cracks


Radial internal pressure


Maximum radial internal pressure


Radial internal pressure at boundary between cracked and uncracked parts


Random radial distance to the circle of rebar


Outer radius of concrete cover


Inner radius of concrete cover (radius of reinforcing bar)


Radial deformation of inner cracked concrete


Calculated radial deformation of inner cracked concrete


Radial deformation at boundary between cracked and uncracked parts


Width of fictitious crack


Crack width when the cohesive stress is zero


The crack width at the rebar-concrete interface


Inclination angle of bond action


Reduction coefficient considering the effect of bonded length


Circumferential deformation of the inner cracked concrete

\(\Delta_{{_{\theta } }}^{{R_{s} }}\)

Circumferential deformation of the cracked concrete at the bar-concrete surface


Strain of concrete at tensile strength


Radial strain of concrete


Circumferential strain of concrete


Poisson’s ratio of concrete


Radial stress in the outer elastic concrete


Cohesive stress across the fictitious crack in the inner cracked concrete


Circumferential stress in the outer elastic concrete

\(\sigma_{\theta }^{e}\)

Circumferential stress at boundary between cracked and uncracked parts


Longitudinal bond stress


Maximum bond stress



This study is funded by the National Basic Research Program of China (973 Program) with Grant No. 2011CB013801 and the National Natural Science Foundation of China with Grant No. 51308285. The financial supports from the Natural Science Foundation of Jiangsu Province with Grant No. BK20130943, Natural Science Foundation for College and University in Jiangsu Province with Grant No. 13KJB560004 and the China Scholarship Council (CSC) with Grant No. 201508320041 are also greatly acknowledged.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© RILEM 2018

Authors and Affiliations

  1. 1.College of Civil EngineeringNanjing Tech UniversityNanjingPeople’s Republic of China
  2. 2.State Key Laboratory of Coastal and Offshore EngineeringDalian University of TechnologyDalianPeople’s Republic of China

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