Keywords: prestressed concrete, crack, tendon, bond, slip, tension-stiffening.

A wide body of research has been conducted to consider the tension stiffening effect, and many numerical models have also been introduced [1]. Among these models, consideration of the strain softening branch in the tension region of the stress-strain relation of concrete is one of the generally adopted approaches [2]. Recently, modification of the stress-strain relation of steel has been emphasized, because reaching the yield strength of a bare bar at a cracked section does not necessarily indicate the complete yielding of steel embedded in a cracked element [3]. The average steel stress in a cracked element still maintains an elastic stress rather than the yield strength [4]. In spite of these efforts concentrated on the cracking behaviour of reinforced concrete (RC) structures, a numerical model that can simulate the tension stiffening effect of prestressed concrete (PSC) structures has not been introduced, primarily because of the different bond characteristics of tendons. Accordingly, to effectively simulate the post-cracking behaviour of PSC structures with bonded tendons, a modified stress-strain curve of tendon is introduced in this paper on the basis of the bond mechanism between a tendon and concrete, as was introduced at the reinforcing steel [5].

The tendon model could be expressed with the apparent yield stress, , considering the bond characteristics of embedded tendon bar. It means that the yield stress corresponding to the strain of 0.01 needs to be revised to . Then the yielding point of an embedded tendon bar can finally be calculated by

where is the nominal diameter of tendon, iss the tensile strength of the grouting material, and is the steel ratio, and is Young's modulus of concrete. Finally, the apparent yield stress calculated with above equation differs from the yield stress of bare tendon in this structure. Therefore, the bonded tendon model can be used with the multi-linear model which is based on the apparent yield stress.

The proposed model has been used to more exactly predict the ultimate resisting capacity and cracking behaviour of containment structures at various stages in its loading history. Especially, the proposed tendon model makes it possible to analyze PSC structures using commercialized software. Representative PSC beams were analyzed with the purpose of investigating the relative effects of bond-slip and tension stiffening, and finally numerical analysis of a CANDU containment structure was conducted. From the numerical analyses, in advance, the following conclusions can be drawn: (1) ignoring the tension stiffening effect clearly leads to underestimation of the stiffness and ultimate resisting capacity of PSC structures; (2) the plastic hinge length must be considered to exactly predict the ultimate resisting capacity of concrete structures where the plastic deformation is concentrated at any location with a narrow range; (3) for under-reinforced concrete structures, the use of modified stress-strain relations of reinforcing steel and tendon has the dominant influence on the cracking behaviour and ultimate resisting capacity of structure, while its effect has been decreased with an increase of the steel ratio; nevertheless (5) consideration of the modified stress-strain relations of reinforcing steel and tendon with the tension softening in concrete must be taken into account to yield a very satisfactory agreement of the numerical results with experimental data, regardless of the steel ration placed in the structure.

1

Comite Euro-International du Beton (CEB) task group 22, "RC Elements Under Cyclic Loading", Thomas Telford, 1996.

2

Maekawa, K., Pimanmas, A., and Okamura, H., "Nonlinear Mechanics of Reinforced Concrete", Spon Press, 2003.

3

Kwak, H.-G. and Kim, D.-Y., "Nonlinear Analysis of RC Shear Walls Considering Tension-Stiffening Effect", Computers and Structures, 75, 5, 499-517, 2001. doi:10.1016/S0045-7949(00)00157-7

4

Belarbi, A. and Hsu, T.T.C., "Constitutive Laws of Concrete in Tension and Reinforcing Bars Stiffened by Concrete", ACI Structural Journal, 91, 4, 465-474, 1994.

5

fib Task Group on Bond Models, "Bond of Reinforcement in Concrete", International Federation for Structural Concrete (fib), 2000.

6

Kwak, H.-G. and Kim, D.-Y., "Material Nonlinear Analysis of RC Shear Walls Subject to Monotonic Loadings", Engineering Structures, 26, 11, 1517-1533, 2004. doi:10.1016/j.engstruct.2004.05.013

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