Keywords: high strength concrete, flexural ductility, balanced strain reinforcement ratio, empirical relations, theoretical model.

Ductility is the essential structure property that is responsible for post-yield behavior during severe shaking. Structure ductility is also important for static load cases as it provides warning of imminent failure and leads to a more rational bending moment distributions between sagging and hogging moments in statically indeterminate reinforced concrete structures [1,2].

The ductility of a section is normally expressed in terms of its curvature ductility factor where and are the curvatures and the first yield of tension reinforcement and at ultimate, respectively. Ductility is a basic requirement in the structural design of high strength concrete (HSC), particularly due to its brittleness. Therefore, adequate evaluation of ductility and rotational capacity of high strength concrete sections is of an important practical interest.

This paper investigates the curvature ductility for high strength reinforced concrete sections and provides a simple, yet efficient, empirical formula to calculate section curvature ductility.

First, a computer program is developed for numerical calculations of section moment-curvature relationship. This program follows an iterative approach and utilizes a recently published stress-strain relationship for high-strength concrete [3].

The established moment-curvature relationship is then utilized to predict the yield and ultimate curvature values. Calculation of is based on the first yield of tensile reinforcement. However, the maximum curvature is controlled by the allowable maximum concrete compression strain at the extreme fiber, , since the limiting steel strain is typically high. This curvature may be expressed as where is the neutral axis depth corresponding to the maximum compression strain. In our computations, the maximum strain at the extreme compression fiber is taken equal to .

The program is then implemented in an extensive parametric study to evaluate the effects of various design parameters on section curvature ductility. Finally, the obtained numerical results are used to establish an empirical formula for curvature ductility of high strength concrete sections through nonlinear regression analysis.

The arguments of the proposed formula are the section ratios of tension, , and compression, , reinforcement and the concrete characteristic strength, . It is found that the effects of these three variables on ductility can be represented by a single dimensionless parameter where is the section balanced reinforcement ratio in the form, .

This formulas' constant parameters ( and ) depend on the yield strength of steel, the ratio between compression and tension reinforcement , and the ratio where and are the depth from the extreme compression fibers to the compression and tension reinforcement, respectively. It is valid for rectangular sections with MPa, , and .

Empirical formulas for as functions of are also developed for milled as well as high-grade steel. For milled steel with MPa, or, . And, for high-grade steel with MPa, or, .

The proposed formulas should be useful to design engineers. For instance, a design engineer can use them find out and quantify different ways to achieve the desired ductility level, e.g. by using a certain amount of compression reinforcement, by adjusting the depth to compression reinforcement d', or by increasing the concrete compressive strength to a specific level. He can then decide which way is more proper (technical and economical) for his specific application.

1

T. Pauly, M.J.N. Priestley "Seismic Design of Reinforced Concrete and Masonry Buildings" John Wiley & Sons, New York, 12-144, 1992.

2

R. Pendyala, P. Mendis, L. Patnaikuni, "Full Range Behavior of High-Strength Reinforced Concrete Flexural Concrete Members" ACI Structural Journal, 93 (1), 1996.

3

M. Attard, M. Stewart, "A Two Parameter Stress Block for High-Strength Concrete" ACI Structural Journal, 95(3), 305-317,1998.

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