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Interaction diagrams for design of hybrid fiber-reinforced tunnel segments

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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.

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Abbreviations

a c/b c :

Core dimensions to centerlines of lateral bars

A s :

Area of longitudinal reinforcement

A sy :

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)

A sz :

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

b :

Beam width

C 1–12 :

Coefficients for normalized moment in Table 2

d :

Effective depth at location of steel rebar

d s :

Diameter of steel rebar

E :

Elastic tensile modulus of concrete

E c :

Elastic compressive modulus of concrete

E s :

Elastic modulus of steel

f :

Stress components in stress diagram

f′ c :

Cylindrical ultimate compressive strength of concrete

f cd :

Design strength of concrete

f ck :

Characteristic compressive strength of concrete

f ck,c :

Value of fck of confined concrete

f sy :

Yield strength of longitudinal steel

f yd :

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

k b :

Neutral axis depth ratio at balanced failure

M :

Bending moment

M b :

Bending moment at balanced failure

M cr :

Bending moment at first cracking

M n :

Nominal bending moment capacity

M u :

Ultimate bending moment

M′:

Normalized bending moment

n :

Modulus ratio (Es/E)

P :

Axial force

P b :

Axial force at balanced failure

P n :

Nominal axial force

P u :

Ultimate axial force

P′:

Normalized axial force

s c :

Tie spacing

y :

Moment arm from force component to neutral axis

α :

Normalized depth of steel reinforcement

β :

Normalized tensile strain (εt/εcr)

γ :

Normalized concrete compressive modulus (Ec/E)

ε :

Strain

ε c :

Concrete compressive strain

ε cr :

First cracking tensile strain

ε cy :

Concrete compressive yield strain

ε cy,c :

Confined concrete compressive yield strain

ε cu :

Ultimate concrete compressive strain

ε cu,c :

Confined ultimate concrete compressive strain

ε t :

Concrete tensile strain

ε tu :

Ultimate tensile strain

ε top :

Compressive strain at top fiber

ε bot :

Tensile strain at bottom fiber

ε sy :

Steel yield strain

ϕ :

Strength reduction factor

κ :

Normalized steel yield strain (εsy/εcr)

λ :

Normalized compressive strain (εc/εcr)

λ cu :

Normalized ultimate compressive strain (εcu/εcr)

λ cu,c :

Normalized ultimate compressive yield strain due to confinement (εcu,c/εcr)

μ :

Normalized residual tensile strength (σp/σcr)

μ crit :

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

ρ :

Steel reinforcement ratio per effective area

ρ g :

Steel reinforcement ratio per gross area

σ c :

Concrete compressive stress

σ c,c :

Confined concrete compressive stress

σ t :

Concrete tensile stress

σ cr :

Cracking tensile strength

σ cy :

Compressive yield strength

σ cy,c :

Confined compressive yield strength

σ p :

Residual tensile strength

ω :

Normalized concrete compressive yield strain (εcy/εcr)

ω c :

Normalized concrete compressive yield strain due to confinement (εcy,c/εcr)

χ :

Normalized steel strain (εs/εcr)

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

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Authors and Affiliations

  1. Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education, Southeast University, Nan Jing, 210096, China

    Yiming Yao

  2. AECOM, New York, 10004, NY, USA

    Mehdi Bakhshi & Verya Nasri

  3. School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, 85281, AZ, USA

    Barzin Mobasher

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  1. Yiming Yao
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  2. Mehdi Bakhshi
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Correspondence to Barzin Mobasher.

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The associated excel spreadsheet program is available at https://doi.org/10.13140/RG.2.2.14437.09440.

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11527_2018_1159_MOESM1_ESM.docx

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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Yao, Y., Bakhshi, M., Nasri, V. et al. Interaction diagrams for design of hybrid fiber-reinforced tunnel segments. Mater Struct 51, 35 (2018). https://doi.org/10.1617/s11527-018-1159-2

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