DOB Findings and Recommendations for WTC, Feb 2003

NYC 9/11 Public Portal Document

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The Alternate Path Approach assumes a hypothetical damage state that ignores all other damage to the structure that may accompany the removal of critical column support. It assumes a girder spanning a single bay is transformed into a girder spanning two bays. The transition from the original structural configuration to the damaged state is assumed to be instantaneous, exposing the structure to a dynamic effect. Because it is not reasonable to require a structure to respond elastically to the effects of an instantaneous column removal, structures are permitted to develop plastic hinges and sustain significant inelastic deformations when subjected to these extreme-loading conditions. This enables the structure to dissipate significant amounts of energy that would otherwise impose much greater dynamic loadings to the individual members. The inclusion of geometric non-linearity resulting from large deformations can account for the redistribution of loads as a column is removed and the structure attempts to re-equilibrate to the larger spans through a change in behavior from a flexural response to a membrane response. The members that originally spanned a single bay must now span two bays and the center span will be at the location of the damaged column, where the connection details may have limited capacity to develop positive moments. ■ The inclusion of geometric non-linearity will enable the designer to account for the tension-membrane stiffening of the slabs and spandrel beams as they sag and develop catenary resistance. These membrane forces must be compared to the tensile capacity of the members and their connections to make sure they are capable of developing the axial forces. The effect of this dynamic phenomenon is depicted in the attached figure. This figure shows the response characteristics of the standard elastic-plastic single-degree-of-freedom spring mass system typically used to represent the dynamic behavior of a structural element. The loading represented in this figure is a suddenly applied step pulse, which corresponds to the instantaneous removal of a column and corresponding transfer of the gravity load to the double-bay span. Because the transition is assumed to occur instantaneously and its duration is prolonged, the behavior to the step pulse is represented by large ratios of T/Tn (duration of loading relative to the fundamental period of the structure) along the horizontal axis. The extent of inelastic deformation relative to its elastic limit is represented by the ductility, Xm/Xe, along the vertical axis. As the ductility increases, the structure sustains larger inelastic deformations that dissipate more energy. Both steel and reinforced concrete structures may be detailed to sustain a ductility ranging from 10 to 20 in response to an extreme loading condition. The third scale shown next to the curves represents the required capacity (ultimate resistance) of the structural element (prior to developing the plastic hinge) as a ratio of the applied step load (Ru/P). For a ductility of 1.0, which corresponds to an elastic response, the required capacity must be twice the applied step load in order to account for the dynamic behavior (this is represented by the bottom curve that asymptotes to a unit value of ductility at large ratios of T/Tn). This is the most conservative dynamic amplification factor, which ignores the considerable amount of inelastic' deformation that will accompany the redistribution of loads. A less conservative idealization of the redistribution of loads is represented by the thick black line drawn at a ductility, of 2.5 (a modest amount of inelastic deformations by most accounts). This line is intermediate. between the 1.2 and 1.3 Ru/P curves and represents the dynamic effect of the step load of infinite duration that is suddenly applied to the inelastic structure, which requires the structure to support 1.25 times the . magnitude of the step load. This more physically meaningfol amplification factor accounts for the dynamic redistribution of loads to the undamaged portions of the structure without introducing unwarranted conservatism into the analysis. This corresponds to the increase from yield to ultimate strength that is not reflected when using the Plastic Design or Ultimate Strength methods.

February 2003 WTC Building Code Task Force Page 37 Findings and Recommendations

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