###### Abstract  
The double friction pendulum (DFP) bearing is adapted from the well-known single friction pendulum (SFP) bearing. This type of bearings has been widely used for structural vibration controls. The main advantage of the DFP is its capacity to accommodate larger displacements as compared with the SFP one. This paper aims to assess the effect of the vertical earthquake component on the seismic behaviour of a base-isolated high-rise building. In this respect, the mathematical model of the building subjected to earthquake excitations with an implementation of a DFP bearing system is established. The model presented herein considers earthquake excitations in horizontal (X and Y) and vertical (Z) directions. A series model of two friction elements is presented for the bearing, where the friction load of the bearing surface is governed by a modified Bouc-Wen model, which is dependent on the sliding velocity and the contact pressure. The numerical results of an example of a base-isolated 9-story steel building subjected to near-source and far-field earthquakes show the high effectiveness of the bearing system in reduction of the seismic response of the building, especially in the near-source region, as well as exhibit considerable effectiveness of the vertical earthquake component on the bearing and structural behaviour. seismic isolation; double friction pendulum; high-rise building; vertical earthquake component 202123809911380911380911380911380911380913809113809113809138091138091380913809138  
surfaces and unequal friction coefficients of the two sliding interfaces. Also, in their work, the effects of the height of the articulated slider and the friction in the rotational part of the articulated slider on the lateral force-displacement relation were presented. An advanced DFP bearing model with tri-linear behaviour was later studied by [PERSON] and [PERSON] [9]. The effects of the DFP bearing system with different friction values and restoring properties on a simple bridge are investigated under different earthquakes. These studies have focused on the planar behaviour of the DFB bearing. The effect of the vertical excitation is assumed to be small and is neglected. Besides, recent studies have indicated that the rotational component of earthquakes also affects the overall response of buildings [10]; however, this effect on the DFP bearing has not been investigated so far. Regarding this issue, [PERSON] and [PERSON] [11] studied the effect of the vertical ground acceleration component on the horizontal response of structures isolated with DFP bearings. In which, a mathematical model of the DFP bearing is presented. The model, which represents the force-displacement relationship of the bearing with identical sliding surfaces under bi-directional excitations, is developed based on governing equations of the FP bearing undergoing the unidirectional excitation. The model is simplified by considering equal curvature radii and friction coefficients of the concave surfaces, and hence the sliding velocities on each surface are equal. Recently, [PERSON] et al. [12] presented a theoretical model of the DFP system under variable vertical loadings and unequal friction coefficients between two sliding surfaces. Both the studies assume a constant friction coefficient of the sliding surface, which is independent of the sliding velocity and the contact pressure. Recently, [PERSON] and [PERSON] [13] developed a three-dimensional DFP bearing model including uplift and impact behaviour; the model facilitates investigations of the extreme behaviour of FP bearing systems. Experimental studies have also been conducted to investigate the effect of vertical excitation on the horizontal response of buildings. The earlier test was conducted by [PERSON] and [PERSON] [14] for a 6-story building isolated with TP bearings; the authors concluded that the effect of the vertical excitation is minor. [PERSON] and [PERSON] [15] presented a full-scale shake table experiment of a 5-story frame building isolated with TFP bearings.