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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

Abstract

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.

Keywords

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

List of symbols

ac/bc

Core dimensions to centerlines of lateral bars

As

Area of longitudinal reinforcement

Asy

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)

Asz

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

b

Beam width

C1–12

Coefficients for normalized moment in Table 2

d

Effective depth at location of steel rebar

ds

Diameter of steel rebar

E

Elastic tensile modulus of concrete

Ec

Elastic compressive modulus of concrete

Es

Elastic modulus of steel

f

Stress components in stress diagram

f′c

Cylindrical ultimate compressive strength of concrete

fcd

Design strength of concrete

fck

Characteristic compressive strength of concrete

fck,c

Value of fck of confined concrete

fsy

Yield strength of longitudinal steel

fyd

Yield strength of lateral steel

F

Force components in stress diagram

h

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

k

Neutral axis depth ratio

kb

Neutral axis depth ratio at balanced failure

M

Bending moment

Mb

Bending moment at balanced failure

Mcr

Bending moment at first cracking

Mn

Nominal bending moment capacity

Mu

Ultimate bending moment

M

Normalized bending moment

n

Modulus ratio (Es/E)

P

Axial force

Pb

Axial force at balanced failure

Pn

Nominal axial force

Pu

Ultimate axial force

P

Normalized axial force

sc

Tie spacing

y

Moment arm from force component to neutral axis

α

Normalized depth of steel reinforcement

β

Normalized tensile strain (εtcr)

γ

Normalized concrete compressive modulus (Ec/E)

ε

Strain

εc

Concrete compressive strain

εcr

First cracking tensile strain

εcy

Concrete compressive yield strain

εcy,c

Confined concrete compressive yield strain

εcu

Ultimate concrete compressive strain

εcu,c

Confined ultimate concrete compressive strain

εt

Concrete tensile strain

εtu

Ultimate tensile strain

εtop

Compressive strain at top fiber

εbot

Tensile strain at bottom fiber

εsy

Steel yield strain

ϕ

Strength reduction factor

κ

Normalized steel yield strain (εsycr)

λ

Normalized compressive strain (εccr)

λcu

Normalized ultimate compressive strain (εcucr)

λcu,c

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

μ

Normalized residual tensile strength (σpcr)

μcrit

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

ρ

Steel reinforcement ratio per effective area

ρg

Steel reinforcement ratio per gross area

σc

Concrete compressive stress

σc,c

Confined concrete compressive stress

σt

Concrete tensile stress

σcr

Cracking tensile strength

σcy

Compressive yield strength

σcy,c

Confined compressive yield strength

σp

Residual tensile strength

ω

Normalized concrete compressive yield strain (εcycr)

ωc

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

χ

Normalized steel strain (εscr)

Subscripts

c1

Elastic compression zone 1 in stress diagram

c2

Plastic compression zone 2 in stress diagram

cr

At first cracking

cu

At ultimate concrete compressive strain

cy

At concrete compressive yielding

i

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

s

Refer to steel in tension side

s′

Refer to steel in compression side

sy

At steel yielding

t1

Elastic tension zone 1 in stress diagram

t2

Residual tension zone 2 in stress diagram

top

Extreme top fiber of cross section

bot

Extreme bottom fiber of cross section

tu

At concrete ultimate tensile stain

u

At ultimate bending moment/axial load

Notes

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  https://doi.org/10.13140/rg.2.2.19367.24482 and  https://doi.org/10.13140/rg.2.2.16011.80162 (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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