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Time dependent deflection of RC beams allowing for partial interaction and non-linear shrinkage

  • Noor Md. Sadiqul HasanEmail author
  • Phillip Visintin
  • Deric John Oehlers
  • Terry Bennett
  • Md. Habibur Rahman Sobuz
Original Article
  • 22 Downloads

Abstract

Existing analysis and design techniques to quantify serviceability member deflection of reinforced concrete (RC) beams are generally built on two major premises: (1) full interaction through the use of moment curvature approaches; and (2) a uniform longitudinal shrinkage strain εsh within the member to simplify the analysis. Both premises are gross approximations. With regard to the first premise, RC beams are subject to flexural cracking and the associated partial interaction (PI) behaviour of slip between the reinforcement and adjacent concrete. Furthermore with regard to the second premise, numerous tests have shown that εsh varies along both the width and depth of the beam that is, it is far from uniform. Hence the quantification of the serviceability deflections of RC beams for design is subjected to two major sources of error: that due to the PI mechanisms which occur in practice; and that due to the time dependent material properties of shrinkage and creep. This paper deals with the development of PI numerical mechanics models with non-linear shrinkage strain variations required to simulate the PI behaviour of RC beams in order to considerably reduce the source of error due to the mechanics model. This new mechanics model will allow: the development of appropriate design mechanics rules for serviceability deflection; and also assist in the better quantification of creep and shrinkage by removing the existing mechanics source of error. Specifically, this paper describes a numerical approach for quantifying the deflection of RC beams that not only allows for the PI behaviours of flexural cracking, crack spacing and reinforcement slip but also allows for the variation in the longitudinal shrinkage strains along both the depth and width of the member. The analysis is also compared with time dependent test results of six RC beams tested by Gilbert and Nejadi (An experimental study of flexural cracking in reinforced concrete members under sustained loads. UNICIV Report No. R-435, School of Civil and Environmental Engineering, University of New South Wales, Sydney, Australia, 2004) under sustained loading conditions for the period of 400 days and the predicted deflection of RC beams have shown conservative estimates with the measured deflections.

Keywords

Reinforced concrete beams Time dependent deflection Concrete shrinkage Concrete creep Partial interaction mechanics Segmental approach 

List of symbols

Ac

Cross sectional area of concrete in prism

Ar

Cross section area of reinforcement

Arc

Total Ar in compression

Art

Total Ar in tension

B

Bond force at reinforcement/interface

Bn

B in nth prism segment

b

Width of beam

C

Moisture diffusion coefficient

C1

Maximum value of C

Cn

Force in concrete on left hand side of nth segment

D

Displacement of reinforcement relative to position at unstressed state

Db-long

Long term deflection of beam at mid-span

Db-short

Short term deflection of beam at mid-span

DFI

D from full interaction analysis

DPI

D from partial interaction analysis

d

Full depth of beam

drt

Distance of tension reinforcement to tension face of beam; half depth of reinforced concrete prism

dNA

Depth of neutral axis from compression face

dNA-c

dNA for concrete section

dNA-cn

dNA for concrete slice n

dNA-r

dNA for all the longitudinal reinforcement

dΔ/dx

Slip strain

Ec

Concrete modulus allowing for creep

Er

Reinforcement modulus

E( t)

Concrete elastic modulus at a given time

E( t0)

Initial elastic modulus of concrete

EI

Flexural rigidity

F

Force profile

Fcc

Resultant force in concrete in compression

Fccn

Fcc in nth slice

Fct

Resultant force in concrete in tension

Fctn

Fct in nth slice

Frc

Resultant force in compression reinforcement

Frt

Resultant force in tension reinforcement

FI

Full interaction

f

Surface factor

fc

Compressive strength of concrete

fck

Characteristic compressive strength

fct

Tensile strength of concrete

gs(t)

E(t0)/Et

H

Pore humidity

Hc

H when C(H) = 0.5C1

Hen

Environmental humidity

Hs

Surface humidity

ksh

Shrinkage coefficient

Lb

Length of beam

Ldef

Half length of symmetrically loaded segment; half length of concrete prism prior to straining

Lper

Length of reinforcement perimeter

Ls

Length of prism segment

LT

Total length of prism after straining

m

Empirical constant for the diffusion coefficient

M

Moment

Mcr

Moment to cause cracking

Mcr-in

Mcr to cause initial crack in uncracked beam

Mcr-pr

Moment to cause primary cracks

Mcr-sec

Moment to cause secondary cracks

n

Number of slices per half width

P

Resultant longitudinal force in reinforcement

Pcr

Resultant force in reinforcement at a flexural crack

Pcr-pr

Pcr at the formation of primary cracks

Pcr-sec

Pcr to cause secondary cracks

Pcr/Δcr

Crack opening stiffness

Pn

P on left side of nth element

PI

Partial interaction

RC

Reinforced concrete

Scr

Crack spacing

Scr-pr

Primary crack spacing

T

Temperature

t

Time

w

Width of crack; 2Δcr

α0

Empirical constant for the diffusion coefficient

χ

Curvature

Δ

Slip of reinforcement

Δcr

Δ relative to crack face; w/2

Δn

Slip at nth segment

δ

Deformation profile

δb

Interface bond slip

δc-n

Concrete deformation within nth slice

δrc

Contraction of compression reinforcement

δrt

Extension of tension reinforcement

δΔn

change in slip in nth segment

ε

Strain profile; strain

εc

Concrete strain distribution

εc-sh

Strain in concrete due to shrinkage

εcn

εc in nth slice; mean concrete strain in nth prism segment

εct

Fracture strain of concrete; fct/Ec

εpk

Peak strain

εr

Reinforcement strain distribution

εrc

Strain in compression reinforcement

\(\varepsilon_{\text{s}}^{0}\)

Magnitude of the ultimate shrinkage strain

εrt

Strain in tension reinforcement

εrn

Mean reinforcement strain in nth prism segment

εr-sh

Strain in reinforcement due to shrinkage

\(\varepsilon_{\text{s}}^{0}\)

ultimate concrete shrinkage strain

εsh

Shrinkage strain

εsh n

Shrinkage strain distribution in nth slice of concrete or nth prism

Φ

Creep coefficient

θ

Rotation

σ

Stress profile

σc

Concrete stress distribution

σcc

Stress in concrete at level of compression reinforcement

σct

Stress in concrete at level of tension reinforcement

σr

Reinforcement stress distribution

σrc

Stress in compression reinforcement

σrt

Stress in tension reinforcement

τb

Interface bond stress

τbn

τb in nth segment

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Research involving human participants and/or animals

The authors declare that there is no involvement of any human participants and/or animals to conduct this research.

Informed consent

The authors also declare that all authors are aware about this manuscript to be submitted in this journal.

Supplementary material

11527_2019_1350_MOESM1_ESM.docx (92 kb)
Supplementary material 1 (DOCX 91 kb)

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Copyright information

© RILEM 2019

Authors and Affiliations

  • Noor Md. Sadiqul Hasan
    • 1
    Email author
  • Phillip Visintin
    • 1
  • Deric John Oehlers
    • 1
  • Terry Bennett
    • 1
  • Md. Habibur Rahman Sobuz
    • 1
    • 2
  1. 1.School of Civil, Environmental and Mining EngineeringThe University of AdelaideAdelaideAustralia
  2. 2.Department of Building Engineering and Construction ManagementKhulna University of Engineering and TechnologyKhulnaBangladesh

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