Materials and Structures

, 51:35 | Cite as

Interaction diagrams for design of hybrid fiber-reinforced tunnel segments

  • Yiming Yao
  • Mehdi Bakhshi
  • Verya Nasri
  • Barzin Mobasher
Original Article


Fiber reinforcement has emerged as an alternative to traditional reinforcing bars and welded wire mesh reinforcement for precast concrete tunnel segments. This is mainly due to improved postcracking behavior and crack control characteristics of fiber-reinforced concrete (FRC) segments. A hybrid solution of fibers and reinforcing bars is adopted when FRC is not adequate as the sole reinforcing system. Often times, this is the case in large-diameter tunnels with large curved length segments in order to achieve required strength for embedment loads in shallow cover, TBM thrust jack forces, and loading from imperfect construction and irregularities. P–M interaction diagrams are used as one of the main design tools since segment cross section, under most of governing load cases, is subjected to a combined axial force and bending moment. Standard FRC constitutive laws recently allows for a significant residual strength in tension zone below the neutral axis. However, design capacity of hybrid fiber-reinforced concrete (HRC) segment is significantly underestimated using conventional Whitney’s rectangular stress block method. Methods that currently incorporate contribution of fibers on P–M diagrams are based on numerical and finite-element analyses. However, closed-form solutions offer important advantages. This paper presents material models, derivations and for the first time closed-form solutions to construct P–M interaction diagram of HRC segments. Parametric studies are conducted and validity of the model is verified by simulating experimental results of HRC columns and model-predicted results of precast and cast-in-place concrete linings. Results show that using appropriate material models for fiber and reinforcing bar, engineers can use the proposed methodology to obtain P–M interaction diagrams for HRC tunnel segments.


Analytical method Fiber Hybrid fiber-reinforced concrete Interaction diagram Tunnel Lining Residual strength Segment TBM 

List of symbols


Core dimensions to centerlines of lateral bars


Area of longitudinal reinforcement


Area of transverse reinforcement in y-direction of the cross section (x axis coincides with the longitudinal direction of the beam that is perpendicular to y–z plane)


Area of transverse reinforcement in z-direction of the cross section


Beam width


Coefficients for normalized moment in Table 2


Effective depth at location of steel rebar


Diameter of steel rebar


Elastic tensile modulus of concrete


Elastic compressive modulus of concrete


Elastic modulus of steel


Stress components in stress diagram


Cylindrical ultimate compressive strength of concrete


Design strength of concrete


Characteristic compressive strength of concrete


Value of fck of confined concrete


Yield strength of longitudinal steel


Yield strength of lateral steel


Force components in stress diagram


Full height of a beam section or height of each compression and tension zone in stress diagram


Neutral axis depth ratio


Neutral axis depth ratio at balanced failure


Bending moment


Bending moment at balanced failure


Bending moment at first cracking


Nominal bending moment capacity


Ultimate bending moment


Normalized bending moment


Modulus ratio (Es/E)


Axial force


Axial force at balanced failure


Nominal axial force


Ultimate axial force


Normalized axial force


Tie spacing


Moment arm from force component to neutral axis


Normalized depth of steel reinforcement


Normalized tensile strain (εtcr)


Normalized concrete compressive modulus (Ec/E)




Concrete compressive strain


First cracking tensile strain


Concrete compressive yield strain


Confined concrete compressive yield strain


Ultimate concrete compressive strain


Confined ultimate concrete compressive strain


Concrete tensile strain


Ultimate tensile strain


Compressive strain at top fiber


Tensile strain at bottom fiber


Steel yield strain


Strength reduction factor


Normalized steel yield strain (εsycr)


Normalized compressive strain (εccr)


Normalized ultimate compressive strain (εcucr)


Normalized ultimate compressive yield strain due to confinement (εcu,ccr)


Normalized residual tensile strength (σpcr)


The critical normalized residual tensile strength that change deflection-softening to deflection-hardening


Steel reinforcement ratio per effective area


Steel reinforcement ratio per gross area


Concrete compressive stress


Confined concrete compressive stress


Concrete tensile stress


Cracking tensile strength


Compressive yield strength


Confined compressive yield strength


Residual tensile strength


Normalized concrete compressive yield strain (εcycr)


Normalized concrete compressive yield strain due to confinement (εcy,ccr)


Normalized steel strain (εscr)



Elastic compression zone 1 in stress diagram


Plastic compression zone 2 in stress diagram


At first cracking


At ultimate concrete compressive strain


At concrete compressive yielding


At stage i of normalized concrete compressive strain and tensile steel condition


Refer to steel in tension side


Refer to steel in compression side


At steel yielding


Elastic tension zone 1 in stress diagram


Residual tension zone 2 in stress diagram


Extreme top fiber of cross section


Extreme bottom fiber of cross section


At concrete ultimate tensile stain


At ultimate bending moment/axial load


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

11527_2018_1159_MOESM1_ESM.docx (1.3 mb)
Supplementary data, associated program and the model user guide related to this article can be found at and (DOCX 1312 kb)


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Copyright information

© RILEM 2018

Authors and Affiliations

  1. 1.Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of EducationSoutheast UniversityNan JingChina
  2. 2.AECOMNew YorkUSA
  3. 3.School of Sustainable Engineering and the Built EnvironmentArizona State UniversityTempeUSA

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