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2.12.8 Second-Order Refined Plastic Hinge Analysis

The main limitation of the conventional elastic-plastic hinge approach is that it overpredicts the strength of columns that fail by inelastic flexural buckling. The key reason for this limitation is the modeling of the member by a perfect elastic element between the plastic hinge locations. Furthermore, the elastic-plastic hinge model assumes that material behavior changes abruptly from the elastic state to the fully yielded state. The element under consideration exhibits a sudden stiffness reduction upon formation of a plastic hinge. This approach, therefore, overestimates the stiffness of a member loaded into the inelastic range (White et al. 1991, 1993; Liew et al. 1993). This leads to further research and development of an alternative method called the refined plastic hinge approach. This approach is based on the following improvements to the elastic-plastic hinge model:

1. A column tangent modulus model Et is used in place of the elastic modulus E to represent the distributed plasticity due to axial force effects along the length of a member. The member inelastic stiffness, represented by the member axial and bending rigidities EtA and EtI, is assumed to be the function of axial load only. In other words, EtA and EtI can be thought of as the properties of an effective core of the section, considering column action only. The tangent modulus captures the effect of early yielding in the cross-section due to residual stresses, which was believed to be the cause for the low strength of inelastic column buckling. The tangent modulus approach also has been utilized in previous work by Orbison (1982), Liew (1992), and White et al. (1993) to improve the accuracy of the elastic-plastic hinge approach for structures in which members are subjected to large axial forces.

2. Distributed plasticity effects associated with flexure are captured by gradually degrading the member stiffness at the plastic hinge locations as yielding progresses under increasing load as the cross-section strength is approached. Several models of this type have been proposed in recent literature based on extensions to the elastic-plastic hinge approach (Powell and Chen 1986) as well as the tangent modulus inelastic hinge approach (Liew et al. 1993; White et al. 1994). The rationale of modeling stiffness degradation associated with both axial and flexural actions is that the tangent modulus model represents the column strength behavior in the limit of pure axial compression, and the plastic hinge stiffness degradation model represents the beam behavior in pure bending, thus the combined effects of these two approaches should also satisfy the cases in which the member is subjected to combined axial compression and bending.

It has been shown that with the above two improvements, the refined plastic hinge model can be used with sufficient accuracy to provide a quantitative assessment of a member's performance up to failure. Detailed descriptions of the method and discussion of results generated by the method are given in White et al. (1993) and Chen et al. (1996). Significant works have been done to implement the refined plastic hinge methods for the design of three-dimensional real-size structures (Al-Bermani 1995; Liew et al. 2000).

2.12.9 Second-Order Spread of Plasticity Analysis

Spread of plasticity analyses can be classified into two main types, namely three-dimensional shell element and two-dimensional beam-column approaches. In the three-dimensional spread of plasticity analysis, the structure is modeled using a large number of finite three-dimensional shell elements, and the elastic constitutive matrix, in the usual incremental stress-strain relations, is replaced by an elastic-plastic constitutive matrix once yielding is detected. This analysis approach typically requires numerical integration for the evaluation of the stiffness matrix. Based on a deformation theory of plasticity, the combined effects of normal and shear stresses may be accounted for. The three-dimensional spread-of-plasticity analysis is computational intensive and best suited for analyzing small-scale structures.

The second approach for plastic-zone analysis is based on the use of beam-column theory, in which the member is discretized into many beam-column segments, and the cross-section of each segment is further subdivided into a number of fibers. Inelasticity is typically modeled by the consideration of normal stress only. When the computed stresses at the centroid of any fibers reach the uniaxial normal strength of the material, the fiber is considered as yielded. Compatibility is treated by assuming that full continuity is retained throughout the volume of the structure in the same manner as for elastic range calculations. Most of the plastic-zone analysis methods developed are meant for planar (2-D) analysis (Vogel 1985; White 1985; Chen and Toma 1994) Three-dimensional plastic-zone techniques are also available involving various degrees of refinements (Wang 1988; White 1988).

A plastic-zone analysis, which includes the spread of plasticity, residual stresses, initial geometric imperfections, and any other significant second-order behavioral effects, is often considered to be an exact analysis method. Therefore, when this type of analysis is employed, the checking of member interaction equations is not required. However, in reality, some significant behavioral effects such as the performances of joints and connections tend to defy precise numerical and analytical modeling. In such cases, a simpler method of analysis that adequately captures the inelastic behavior would be sufficient for engineering application. Second-order plastic hinge based analysis is still the preferred method for advanced analysis of large-scale steel frames.

2.12.10 Three-Dimensional Frame Element

The two-dimensional beam-column formulation can be extended to a three-dimensional space frame element by including additional terms due to shear force, bending moment, and torsion. The following

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FIGURE 2.123 Three-dimensional frame element: (a) nodal degrees of freedom and (b) nodal forces.

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