On the Modeling of Elastic and Inelastic, Critical- and Post-Buckling Behavior of Slender Columns and Bracing Members

Abstract

Analyzing tall braced frame buildings with thousands of degrees of freedom in three dimensions subject to strong earthquake ground motion requires an efficient brace element that can capture the overall features of its elastic and inelastic response under axial cyclic loading without unduly heavy discretization. This report details the theory of a modified elastofiber (MEF) element developed to model braces and buckling-sensitive slender columns in such structures. The MEF element consists of three fiber segments, two at the member ends and one at mid-span, with two elastic segments sandwiched in between. The segments are demarcated by two exterior nodes and four interior nodes. The fiber segments are divided into 20 fibers in the crosssection that run the length of the segment. The fibers exhibit nonlinear axial stress-strain behavior akin to that observed in a standard tension test in the laboratory, with a linear elastic portion, a yield plateau, and a strain hardening portion consisting of a segment of an ellipse. All the control points on the stress-strain law are user-defined. The elastic buckling of a member is tracked by updating both exterior and interior nodal coordinates at each iteration of a time step, and checking force equilibrium in the updated configuration. Inelastic post-buckling response is captured by fiber yielding in the nonlinear segments. A user-defined probability distribution for the fracture strain of a fiber in a nonlinear segment enables the modeling of premature fracture, observed routinely in cyclic tests of braces. If the probabilistically determined fracture strain of a fiber exceeds the rupture strain, then the fiber will rupture rather than fracturing. While a fractured fiber can take compression, it is assumed that a ruptured fiber cannot. Handling geometric and material nonlinearity in such a manner allows the accurate simulation of member-end yielding, mid-span elastic buckling and inelastic post-buckling behavior, with fracture or rupture of fibers leading to complete severing of the brace. The element is integrated into the nonlinear analysis framework for the 3-D analysis of steel buildings, FRAME3D. A series of simple example problems with analytical solutions, in conjunction with data from a variety of cyclic load tests, is used to calibrate and validate the element. Using a fiber segment length of 2% of the element length ensures that the elastic critical buckling load predicted by the MEF element is within 5% of the Euler buckling load for box and I-sections with a wide range of slenderness ratios (L/r = 40, 80, 120, 160, and 200) and support conditions (pinned-pinned, pinned-fixed, and fixed-fixed). Elastic post-buckling of the Koiter-Roorda L-frame (tubes and I-sections) with various member slenderness ratios (L/r = 40, 80, 120, 160, and 200) is simulated and shown to compare well against second-order analytical approximations to the solution. The inelastic behavior of struts under cyclic loading observed in the Black et al. and the Fell et al. experiments is numerically simulated using MEF elements. Certain parameters of the model (e.g., fracture strain, initial imperfection, support conditions, etc.) that are not controllable and/or unmeasured during the tests are tuned to realize the best possible fit between the numerical results and the experimental data. A similar comparison is made between numerical results using the MEF element and the experimental data by Tremblay et al. collected from cyclic testing of single-bay braced frames. Finally, a FRAME3D model of a full-scale 6-story braced frame structure that was pseudodynamically tested by the Building Research Institute of Japan subjected to the 1978 Miyagi-Ken-Oki earthquake record, is analyzed iv and shown to closely mimic the experimentally observed behavior. To summarize, the MEF element is able to incorporate all the characteristic features of slender columns and braces that significantly affect their elastic and inelastic, critical and post-buckling behavior, and is remarkably effective in capturing the essence of said behavior, even with the vast uncertainty associated with the buckling phenomenon. To aid in the evaluation of the collapse-prediction capability of competing methodologies, a benchmark problem of a water-tank subjected to the Takatori near-source record from the 1995 Kobe earthquake, scaled down by a factor of 0.32, is proposed. The water-tank is so configured as to have a unique collapse mechanism (under all forms of ground motion), of overturning due to P - instability resulting from column and brace buckling at the base. A FRAME3D model of the tank reveals severe buckling in the bottom megacolumns on the west face of the tower, followed almost instantaneously by compression brace buckling on the north and south faces, when the structure is hit by the Takatori near-source pulse, resulting a tilt in the structure. Subsequent shaking induces P - instability resulting in complete collapse of the tank.

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