Rolling and rocking of rigid uplifting structures

Abstract

Summary Allowing structures to uplift modifies their seismic response; uplifting works as a mechanical fuse and limits the forces transmitted to the superstructure. However, engineers are generally reluctant to construct an unanchored structure because the system could overturn due to lacking redundancy. Using a safety factor for the design of a flat rocking foundation, ie, designing it wider, goes against the main idea of this seismic modification method as the force demand for the structure increases. We propose to extend the flat base of a rocking block with curved extensions to better protect the block from overturning, yet not prevent its uplifting. After investigating the seismic response of such rocking blocks, we extend the study to investigate the seismic response of rolling and rocking frames comprising columns with curved base extensions. The equations of motion are derived, time history analyses are performed, and rocking spectra are constructed. We draw two important conclusions: (a) the response of a class of rocking oscillators with curved base extensions is equivalent to the response of a flat‐base rocking oscillators of the same slenderness, yet larger size; (b) the rotation demand on two negative stiffness rocking and rolling oscillators with the same uplifting acceleration and the same size is roughly the same as long as the rocking oscillators are not close to overturning. The above findings can serve as a basis for the rational seismic design of structures supported on rocking columns with curved bases, a system that has been used since the 1960s.

Shake table testing of a rocking podium: Results of a blind prediction contest

“…[22][23][24][25][26] Motivated by the increasing stability of larger rocking blocks with identical aspect ratios, researchers have suggested using rocking as a seismic design strategy because the uplift of the block acts as a mechanical fuse and limits the design forces of both the superstructure and the foundation. The idea is applicable to both buildings ( Figure 2, left) [27][28][29][30][31][32] and bridges (Figure 2, right). [33][34][35][36][37] However, seismic analysis of rocking structures is not straightforward because, during the uplift, such structures have negative stiffness (in terms of the relation between the lateral force F and displacement u, shown in Figure 1).…”

“…Motivated by the increasing stability of larger rocking blocks with identical aspect ratios, researchers have suggested using rocking as a seismic design strategy because the uplift of the block acts as a mechanical fuse and limits the design forces of both the superstructure and the foundation. The idea is applicable to both buildings (Figure 2, left) 27–32 and bridges (Figure 2, right) 33–37 …”

“…Rocking isolation of a building 27–32 (left); rocking frame that can model a rocking bridge 33 (right). Both figures are adapted from Ref.…”

Simplified analysis of bilinear elastic systems exhibiting negative stiffness behavior

“…This increases the displacement capacity while keeping the uplifting acceleration constant ( Figure 5C). 17 The size and curvature of the extensions control the shape of the post-uplift part of the pushover curve, essentially creating a bilinear or trilinear force-deformation loop, as discussed in detail in Bachmann et al 17 The post-uplift stiffness can be positive or negative depending on the curvature of the extension. A similar behavior can be obtained by using a flexible restraining system 3,5 ( Figure 6).…”

“…Rocking has been proposed as a seismic isolation method for both bridges [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and buildings, [17][18][19][20][21] because uplift works as a mechanical fuse and limits the design forces of both the superstructure and the foundation. Unlike structures designed to yield, the free rocking rigid block 22 of Figure 1A,B exhibits negative post-uplift stiffness ( Figure 1C).…”