# Effective Strength of Concrete Struts in 2-D Strut-Tie Model : Numerical Method

As an attempt to improve a design of structural concrete, a strut-tie model approach has been widely used since the 1980’s. The approach has been proved, from many research results, to be very effective for the design of structural concrete having geometric and/or static discontinuities. However, to apply the approach in practice, a proper way of estimating the effective strength of concrete strut must be established.

Many experimental and analytical studies concerning the effective strengths of concrete struts have been conducted, and various equations and values for the effective stress levels of concrete struts have been suggested. However, the conventional methods employing equations and values are only applicable to the limited strut-tie models for structural concrete with similar loading and geometrical conditions used in experimental and analytical studies.

In this post, a method by which the effective strengths of concrete struts can be determined consistently regardless of the types of structural members and strut-tie models is introduced. In this method, the effective strengths of the concrete struts are initially determined using the biaxial failure envelope of plain concrete, the deviation angles between the struts and compressive principal stress trajectories, and an algorithm implementing the degree of confinement in relation to the reinforcement details. Above figureô shows the relationship between the principal stresses of a two-dimensional finite element and the effective strength of a concrete strut. To determine the initial effective strength of the strut, the principal stressesô of a two-dimensional finite element located along the longitudinal axis of the strut should be obtained from the two-dimensional finite element analysis.ô By extending the line between the two points, (0, 0) and (ü1,ô ü2), to the failure envelope of plain concrete, ü2fô which equals to the effective strength of the finite elementô can be determined.ô Similarly, the effective strength of the other finite elements located along the longitudinal axis of the strut should be determined. Then, the effective strength of the strut is obtained by averaging the effective strengths of all finite elements that are in the range of standard deviation.ô If the concrete strut deviates from the compressive principal stress flows with deviation angle, the effective strength of the finite element should be reduced by transforming the stress to the strut axis or by using Mohr’s circle, as shown in below figure. Reference

1. Yun, Y.M. (2000)ô Nonlinear Strut-Tie Model Approach for Structural Concrete, ACI Structural Journal, Vol. 97, No. 4, pp. 581-590.
2. Yun, Y.M. (2000) Computer Graphics for Nonlinear Strut-Tie Model Approach, Journal of Computing in Civil Engineering, ASCE,ô Vol. 14, No. 2, pp. 127-133.