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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