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Stress block parameters for concrete flexural members reinforced with superelastic shape memory alloys

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Abstract

The unique properties of superelastic shape memory alloys (SMAs) have motivated researchers to explore their use as reinforcing bars. The capacity of a steel reinforced concrete (RC) section is calculated by assuming a maximum concrete strain ε c-max and utilizing stress block parameters, α 1 and β 1, to simplify the non-linear stress–strain curve of concrete. Recommended values for ε c-max, α 1, and β 1 are given in different design standards. However, these values are expected to be different for SMA RC sections. In this paper, the suitability of using sectional analysis to evaluate the monotonic moment–curvature relationship for SMA RC sections is investigated. A parametric study is then conducted to identify the characteristics of this relationship for steel and SMA RC sections. Specific mechanical properties are assumed for both steel and SMA. Results were used to evaluate ε c-max, α 1, and β 1 values given in the Canadian standards and to propose equations to estimate their recommended values for steel and SMA RC sections.

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Abbreviations

\( \overline{Y} \) :

Distance between point of action of the concrete compressive force and the extreme compression fibre of the concrete section

\( A_{\text{s}}^{\prime} \) :

Compressive reinforcement area

A g :

Gross area of concrete section

ALI:

Axial load index which represents the ratio between the applied axial load to the axial capacity of the cross-section

A s :

Tensile reinforcement area

b :

Cross-section width

C :

Compression zone height

C c :

Compressive force in concrete

cc:

Point at which concrete reaches its crushing strain

E cr-SMA :

SMA modulus of elasticity before the start of martensite variant reorientation (austenite phase)

E p1 :

SMA modulus of elasticity before the start of the stress induced martensite phase

E p2 :

SMA modulus of elasticity after the start of the stress induced martensite phase (martensite phase)

E u-s :

Steel plastic modulus of elasticity

E u-SMA :

SMA post-yielding modulus of elasticity

E y-s :

Steel elastic modulus of elasticity

\( f_{\text{c}}^{\prime} \) :

Concrete compressive strength

f c :

Concrete compressive stress

f cr-SMA :

SMA critical stress (start of martensite variant reorientation)

f p1 :

Martensite stress induced stress

f s :

Steel stress

f u-s :

Steel ultimate stress

f u-SMA :

SMA ultimate stress

f y-s :

Steel yielding stress

f y-SMA :

SMA yielding stress

h :

Cross-section height

H :

Point at which strain in the SMA bars exceeds ε p1

M :

Moment

M code :

Moment obtained using A23.3 recommended values (Eq. 1)

M f :

The failure moment

M r :

Moment obtained using the proposed equations for α 1 and β 1

M u :

Ultimate moment

M y :

Yielding moment

NSC:

Normal strength concrete

P :

Axial load

R :

Coefficient of determination

r :

Point at which rupture of reinforcing bars occurs

T s :

Tensile force in the reinforcing bars

y :

Point at which bars reach f y-s for steel or for f cr-SMA SMA

Z :

Slope of compressive strain softening branch

α 1, β 1 :

Stress block parameters

ε c :

Concrete compressive strain

ε c-max :

Concrete maximum strain corresponding to the peak moment

ε cr-SMA :

SMA critical strain

ε cu :

Ultimate concrete compressive strain

ε end :

End part of the bar strain

ε mid :

Middle part of the bar strain

ε p1 :

Martensite stress induced strain

ε SMA :

SMA strain

ε SMA-avg :

SMA average bar strain

ε top :

Concrete top compressive strain

ε u-s :

Steel strain at failure

ε u-SMA :

SMA strain at failure

ε y-s :

Steel yielding strain

ε y-SMA :

SMA yielding strain

ρ :

Tensile reinforcement ratios

ρ′:

Compressive reinforcement ratios

Φ:

Curvature

Φcr-SMA :

Curvature corresponding to the SMA critical stress

Φmax :

Curvature corresponding to the peak moment

Φu :

Ultimate curvature

Φy-s :

Curvature corresponding to the steel yielding stress

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario, Canada, N6A 5B9

    Y. I. Elbahy, M. A. Youssef & M. Nehdi

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  1. Y. I. Elbahy
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  2. M. A. Youssef
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  3. M. Nehdi
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Correspondence to M. A. Youssef.

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Elbahy, Y.I., Youssef, M.A. & Nehdi, M. Stress block parameters for concrete flexural members reinforced with superelastic shape memory alloys. Mater Struct 42, 1335–1351 (2009). https://doi.org/10.1617/s11527-008-9453-z

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