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Stress–strain curve for concrete in circular columns based on elastoplastic analysis

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Abstract

This paper presents a stress–strain curve that describes the axial behavior of concrete in circular columns with lateral reinforcement. As opposed to most studies on this subject, which used semiempirical methods, the proposed stress–strain relations are based on a theoretical derivation. They have been derived from analysis of the problem’s full range according to the theories of elasticity and plasticity. Based on these theories, the current study analytically examines the influence of the main variables, such as the volumetric lateral reinforcement ratio and the material properties, on the behavior of circular confined concrete columns and proposes theoretical expressions that describe their stress–strain relations. Application of the proposed curve shows good agreement with published test results. Since these expressions were derived from a theoretical analysis, they can be considered as an analytical verification of existing empirical curves, yet they are also simple enough for practical applications.

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Abbreviations

a :

Cylinder radius

a 1, a 2, a 3, a 4 :

Functions that define the descending stress-strain curve

A :

Constant of the failure and softening surfaces

b :

Width of lateral steel reinforcement (b 1 = b/a)

B :

Constant of the failure and softening surfaces

D :

Gross diameter of RC column and a constant of the failure and softening surfaces

E c :

Cylinder elastic modulus

E s :

Ring elastic modulus

f c0 :

Unconfined concrete compressive strength

f cc :

Confined concrete compressive strength

f sp :

Lateral steel stress at concrete peak stress

f y :

Yield strength of the transverse steel reinforcement

f yℓ :

Yield strength of the longitudinal steel reinforcement

f fail , f soft :

Functions of the failure and softening surfaces

f 3 :

p/q ratio

F, F 1 :

Functions that define the stresses

I 1 :

First invariant of the stress tensor

\(I^{trans}_{1}\) :

Coefficient of the softening surface

J 2 :

Second invariant of the deviatoric stress tensor

k :

Constant with a dimension of length−1 (k 1 = ka)

K :

Function that depends on the confined concrete properties

l 1, l 2, l 3, l 4 :

Functions that depend on the confined concrete properties

m :

Steel-to-concrete modulus ratio

n A :

Parameter that defines the ascending stress-strain curve

p :

Lateral pressure

q :

Axial pressure

s :

Clear spacing of the transverse steel reinforcement (s 1 = s/a)

s b :

Center-to-center tie spacing

t :

Tie thickness (t 1 = t/a)

t eq :

Tie equivalent thickness (t eq1 = t eq /a)

V s :

Original tie volume

V s,eq :

Equivalent tube volume

\(\epsilon_{sf}\) :

Fracture strain of lateral reinforcement

\(\epsilon_{p}\) :

Accumulated plastic strain

\(\epsilon_{pult}\) :

Ultimate accumulated plastic strain at which the stress state reaches the residual strength envelope

\(\epsilon_{z}\) :

Axial strain

\(\epsilon^{0}_{z}\) :

Axial strain at zero volumetric strain

\(\epsilon^{lim}_{z}\) :

Axial strain at which cracking occurs in the lateral direction

\(\epsilon_{c0}\) :

Concrete compressive strain corresponding to unconfined concrete strength

\(\epsilon_{cc}\) :

Concrete compressive strain corresponding to confined concrete strength

\(\epsilon_{c85}\) :

Strain at 85% of the confined concrete strength

\(\epsilon_{v,max}^{u}\) :

Maximum volumetric contraction strain experienced by the confined concrete

ν :

Poisson’s ratio

ψ :

Softening parameter

ρ v :

Volumetric ratio of transverse steel reinforcement

ρ vf :

\(\rho_{v}f_{sp}/f_{c0}\)

ρ vfy :

\(\rho_{v}f_{y}/f_{c0}\)

σ lat :

Lateral (radial) stress

ϕ t :

Cross-section diameter of transverse steel reinforcement

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Author information

Authors and Affiliations

  1. Halcrow Yolles, 207 Queen’s Quay West, Suite 550, P.O. Box 132, Toronto, ON, M5J 1A7, Canada

    Rami Eid

  2. Faculty of Civil and Environmental Engineering, National Building Research Institute, Technion – Israel Institute of Technology, Haifa, 32000, Israel

    Avraham N. Dancygier

  3. Department of Civil Engineering, Faculty of Engineering, University of Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada

    Patrick Paultre

Authors
  1. Rami Eid
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  2. Avraham N. Dancygier
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  3. Patrick Paultre
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Corresponding author

Correspondence to Rami Eid.

Appendix A

Appendix A

The constants of the Imran and Pantazopoulou concrete constitutive model [20] are:

$$A=2.877; \quad B=0.546; \quad C=9.315; \quad D=0.274$$
(37)

For the axisymmetric cylinder under the axial stress (σ 3) and the equivalent uniform lateral pressure (\(\sigma_{1}=\sigma_{2}\ge\sigma_{3}\)—compression negative), the first invariant of the stress tensor, I 1; the second invariant of the deviatoric stress tensor, J 2; and the major principal stress, σ 1, are:

$$ \begin{array}{ll} I_{1}=\sigma_{3}+2\sigma_{1} &\quad J_{2}=\frac {1} {3} (\sigma_{3}-\sigma_{1})^{2} \\ \sigma_{3}=-f_{cc} &\quad \sigma_{1}=-t_{eq}f_{y}=-\sigma_{lat} \end{array}$$
(38)

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Eid, R., Dancygier, A.N. & Paultre, P. Stress–strain curve for concrete in circular columns based on elastoplastic analysis. Mater Struct 43, 63–79 (2010). https://doi.org/10.1617/s11527-009-9470-6

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