Bond-slip Model to Capture Strain Penetration Effects

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    In fiber-based analysis, the flexural member is represented by unidirectional steel and concrete fibers. The member stiffness and forces are obtained by numerically integrating the stiffness and forces of sections along the member length. The section deformation (e.g., displacement or rotation) is used to obtain the strain in each fiber using the plane sections remain plane assumption. The fiber stress and stiffness are updated according to the material models, followed by upgrading of the section force resultant and the corresponding stiffness.

    A zero-length section element is a fiber discretization of the cross section of a structural member. Zero-length section elements have been generally used for section analyses to calculate moment–curvature responses. Described below is a method that uses a zero-length section element to capture the member end rotation resulting from the strain penetration effects.

    The zero-length section element in OpenSees is  assumed to have a unit length such that the element deformations (i.e., elongation and rotation) are equal to the section deformations (i.e., axial strain and curvature). Hence, the zero length section element can be used to calculate rotation at beam-column end under a moment. To incorporate a zero-length section element in analysis, a duplicate node is required (i.e., the distance between node i and j is zero). In addition, the translational degree-of-freedom of the nodes should be constrained to each other to prevent sliding of the beam-column element (between nodes j and k) under lateral loads because the shear resistance is not included in the zero-length section element.

    Because of the unit-length assumption, the material model for the steel fibers in the section element represents the bar slip instead of strain for a given bar stress.  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. The major difference is the material model for the steel fibers in the zero-length section elements due to the different anchorage conditions. Focusing on column/wall bars fully anchored  into concrete footings and bridge joints, suitable bar stress vs. slip models are developed as follows.

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    For the longitudinal bars anchored in footings and bridge joints, the material model for the steel fibers in the zero-length section element must accurately represent the bond slip behavior of fully anchored bars loaded at one end. The existing approaches involving local bond-slip relationships and steel stress-strain models are not used to establish the bar stress vs. loaded-end slip relationship. Instead, a generic model is established below based on measured bar stress and loaded end slip from testing of steel reinforcing bars that were anchored in concrete with sufficient embedment length.

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