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

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## 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 *ε*_{sh} 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 *ε*_{sh} 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

*A*_{c}Cross sectional area of concrete in prism

*A*_{r}Cross section area of reinforcement

*A*_{rc}Total

*A*_{r}in compression*A*_{rt}Total

*A*_{r}in tension- B
Bond force at reinforcement/interface

*B*_{n}*B*in*n*th prism segment*b*Width of beam

*C*Moisture diffusion coefficient

*C*_{1}Maximum value of

*C**C*_{n}Force in concrete on left hand side of

*n*th segment*D*Displacement of reinforcement relative to position at unstressed state

*D*_{b-long}Long term deflection of beam at mid-span

*D*_{b-short}Short term deflection of beam at mid-span

*D*_{FI}*D*from full interaction analysis*D*_{PI}*D*from partial interaction analysis*d*Full depth of beam

*d*_{rt}Distance of tension reinforcement to tension face of beam; half depth of reinforced concrete prism

*d*_{NA}Depth of neutral axis from compression face

*d*_{NA-c}*d*_{NA}for concrete section*d*_{NA-cn}*d*_{NA}for concrete slice*n**d*_{NA-r}*d*_{NA}for all the longitudinal reinforcement- d
*Δ*/d*x* Slip strain

*E*_{c}Concrete modulus allowing for creep

*E*_{r}Reinforcement modulus

*E*_{( t)}Concrete elastic modulus at a given time

*E*_{( t0)}Initial elastic modulus of concrete

- EI
Flexural rigidity

*F*Force profile

*F*_{cc}Resultant force in concrete in compression

*F*_{ccn}*F*_{cc}in*n*th slice*F*_{ct}Resultant force in concrete in tension

*F*_{ctn}*F*_{ct}in*n*th slice*F*_{rc}Resultant force in compression reinforcement

*F*_{rt}Resultant force in tension reinforcement

- FI
Full interaction

*f*Surface factor

*f*_{c}Compressive strength of concrete

*f*_{ck}Characteristic compressive strength

*f*_{ct}Tensile strength of concrete

*g*_{s}(*t*)*E*_{(t0)}/*E*_{t}*H*Pore humidity

*H*_{c}*H*when*C*(*H*) = 0.5*C*_{1}*H*_{en}Environmental humidity

*H*_{s}Surface humidity

*k*_{sh}Shrinkage coefficient

*L*_{b}Length of beam

*L*_{def}Half length of symmetrically loaded segment; half length of concrete prism prior to straining

*L*_{per}Length of reinforcement perimeter

*L*_{s}Length of prism segment

*L*_{T}Total length of prism after straining

*m*Empirical constant for the diffusion coefficient

*M*Moment

*M*_{cr}Moment to cause cracking

*M*_{cr-in}*M*_{cr}to cause initial crack in uncracked beam*M*_{cr-pr}Moment to cause primary cracks

*M*_{cr-sec}Moment to cause secondary cracks

*n*Number of slices per half width

*P*Resultant longitudinal force in reinforcement

*P*_{cr}Resultant force in reinforcement at a flexural crack

*P*_{cr-pr}*P*_{cr}at the formation of primary cracks*P*_{cr-sec}*P*_{cr}to cause secondary cracks*P*_{cr}/*Δ*_{cr}Crack opening stiffness

*P*_{n}*P*on left side of*n*th element- PI
Partial interaction

- RC
Reinforced concrete

*S*_{cr}Crack spacing

*S*_{cr-pr}Primary crack spacing

*T*Temperature

*t*Time

*w*Width of crack; 2

*Δ*_{cr}*α*_{0}Empirical constant for the diffusion coefficient

*χ*Curvature

*Δ*Slip of reinforcement

*Δ*_{cr}*Δ*relative to crack face;*w*/2*Δ*_{n}Slip at

*n*th segment*δ*Deformation profile

*δ*_{b}Interface bond slip

*δ*_{c-n}Concrete deformation within

*n*th slice*δ*_{rc}Contraction of compression reinforcement

*δ*_{rt}Extension of tension reinforcement

*δΔ*_{n}change in slip in

*n*th segment*ε*Strain profile; strain

*ε*_{c}Concrete strain distribution

*ε*_{c-sh}Strain in concrete due to shrinkage

*ε*_{cn}*ε*_{c}in*n*th slice; mean concrete strain in*n*th prism segment*ε*_{ct}Fracture strain of concrete;

*f*_{ct}/*E*_{c}*ε*_{pk}Peak strain

*ε*_{r}Reinforcement strain distribution

*ε*_{rc}Strain in compression reinforcement

- \(\varepsilon_{\text{s}}^{0}\)
Magnitude of the ultimate shrinkage strain

*ε*_{rt}Strain in tension reinforcement

*ε*_{rn}Mean reinforcement strain in

*n*th prism segment*ε*_{r-sh}Strain in reinforcement due to shrinkage

- \(\varepsilon_{\text{s}}^{0}\)
ultimate concrete shrinkage strain

*ε*_{sh}Shrinkage strain

*ε*_{sh n}Shrinkage strain distribution in

*n*th slice of concrete or*n*th prism*Φ*Creep coefficient

*θ*Rotation

*σ*Stress profile

*σ*_{c}Concrete stress distribution

*σ*_{cc}Stress in concrete at level of compression reinforcement

*σ*_{ct}Stress in concrete at level of tension reinforcement

*σ*_{r}Reinforcement stress distribution

*σ*_{rc}Stress in compression reinforcement

*σ*_{rt}Stress in tension reinforcement

*τ*_{b}Interface bond stress

*τ*_{bn}*τ*_{b}in*n*th 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

## References

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