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The 3D interaction (yield) surface of P-M2-M3 hinges may be defined explicitly, or automatically through AISC-LRFD eqn. H1-1a and H1-1b (Φ=1) or FEMA-356 eqn. 5-4 for steel, or ACI 318-02 (Φ=1) for concrete. Post-yield behavior is interpolated from one or more user-defined P-θ curves, in which θ represents the relationship between M2 and M3. During analysis, an energy-equivalent moment-rotation curve is generated relative to the input P-θ curve(s) and the interaction-surface yield point.

Moment-rotation curve

The moment-rotation curve of a P-M2-M3 hinge is a monotonic backbone relationship used to describe the post-yield behavior of a beam-column element subjected to combined axial and biaxial-bending conditions. The 3D interaction surface of a P-M2-M3 hinge indicates the envelope of yield points. Performance beyond this limit state must be interpolated from one or more moment-rotation curves. Because P-M2-M3 response extends linearly through 3D coordinates to the yield surface, then beyond in a manner that will not exactly resemble the input moment-rotation curve, an energy-equivalent curve is created by holding the area under the user-defined curve constant. Deformation capacity is reduced or increased to maintain equivalency, based on yield-point distance from the M2-M3 plane. This energy-equivalent curve then extends from the interaction surface in a nonlinear manner.

P-M2-M3 parameters

A moment-rotation curve is defined by the relationship between a series of resultant moments M and projected plastic rotations R_p. As described in the CSI Analysis Reference Manual (Moment-Rotation Curves, page 137), these coordinates are obtained through an iterative and qualitative experiment in which the element is modeled in SAP2000 and subjected to a constant axial load P and moments M₂ and M₃ which increase according to a fixed ratio (cosθ, sinθ) which corresponds to their moment angle θ, shown in Figure 1.

Resultant moment M is then given as M = M₂ cosθ + M₃ sinθ, and projected plastic rotation R_p is given as R_p = R_p2 cosθ + R_p3 sinθ.

These relationships indicate that the moment and rotation values of a P-M2-M3 moment-rotation curve are obtained through basic geometric relationships between components projected along the M2 and M3 axes, as shown in Figure 1:

Figure 1 - Moment and rotation components

Notes

Design forces (Pu, M2, M3) must be within the interaction surface for the section to pass design. This can be checked visually by plotting the M2-M3 interaction diagram for a constant axial load P (design force), then by making sure that the design point (M2, M3) falls within this domain.

See Also

Fiber hinge article

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