Modeling of Strain Penetration Effects in Fiber-Based Analysis of Reinforced Concrete Structures

ACI Structural Journal, Mar/Apr 2007 by Zhao, Jian, Sritharan, Sri

PROPOSED METHOD

A zero-length section element at the end of a beam-column element, as shown in Fig. 3, can incorporate the fixed-end rotation caused by strain penetration to the beam-column element. This is because a zero-length section element is assumed to have a unit length such that the element deformation (for example, rotation) is equal to the section deformation (for example, curvature). Because of the fiber representation of the section at the member interface, the proposed approach models the bond slip of the longitudinal bars individually during the state determination of the zero-length section element. Hence, this approach is amenable to the fiber analysis concept and allows the strain penetration effects to be captured during flexural analysis of concrete members regardless of the cross-sectional shape and direction of the lateral load. The concept of using a zero-length section element to capture strain penetration effects is equally applicable to beam bars anchored into interior buildings joints. Such application of the proposed concept, however, requires further research and is beyond the scope of this paper.

The unit length assumption also implies that the material model for the steel fibers in the section element would represent the bar slip instead of strain for a given bar stress. Focusing on capturing the bond slip due to strain penetration along fully anchored bars into concrete footings and bridge joints, suitable material models for the zero-length section element are as follows.

Material model for steel fibers

For the selected anchorage condition, the material model for the steel fibers in the zero-length section element must accurately represent the bond slip of fully anchored bars loaded only at one end. To minimize the error in the material model for the steel fibers, the previously discussed approaches involving local bond-slip and steel stress-strain models were not preferred to establish the bar stress versus loaded-end slip relationship. Instead, a generic model based on measured bar stress and loaded end slip from testing of steel reinforcing bars that were anchored in concrete with sufficient embedment length is advocated in this paper.

Monotonic curve-It is proposed that the monotonic bar stress σ versus loaded-end slip s relationship can be described using a straight line for the elastic region and a curvilinear portion for the post-yield region, as shown in Fig. 4. The slope of the straight line was taken as K, whereas the curvilinear portion was represented by

... (1)

where ... is the normalized bar stress; ... is the normalized bar slip; ... is the ductility coefficient; b is the stiffness reduction factor, which represents the ratio of the initial slope of the curvilinear portion at the onset of yielding to the slope in the elastic region (K); f^sub y^ and f^sub u^ are, respectively, the yield and ultimate strengths of the steel reinforcing bar; and s^sub y^ and s^sub u^ are, respectively, the loaded-end slips when bar stresses are f^sub y^ and f^sub u^.