Strengthening a tube connected building group by base isolators and story isolators

3.1. Tube model

Elastic tube can be modeled by beam elements [11]. The classical beam model can simulate bending moment effects on deformations without transverse shear effect. The Timoshenko model developed the classical beam theory with first-order shear deformation effects with the assumption that cross sections remain plane and rotate about the same neutral axis as in the classical beam model, but do not remain normal to the deformed longitudinal axis [12]. The tube bears both flexural deformation and shear deformation when subjected to lateral forces. The Timoshenko beam elements are therefore chosen to simulate the tube in this study. As shown in Fig. 3, at each floor level the tube is divided into a number of segments which are connected at nodal points.

Fig. 3Tube and simulation model

3.1.1. Timoshenko beam element

Suppose the Timoshenko beam length is $l$, transverse displacement is $u$, bending rotation is $\theta$ at $x$ place (as shown in Fig. 4(a)), simple linear shape functions $N_1=1-x/l$ and $N_2=x/l$ are used, then the Timoshenko element’s consistent mass matrix [12] is:

1

${\mathit{M}}^e=\rho \left\{\begin{array}{cc}\begin{array}{cc}lA/3& 0\\ 0& lI/3\end{array}& \begin{array}{cc}lA/6& 0\\ 0& lI/6\end{array}\\ \begin{array}{cc}lA/6& 0\\ 0& lI/6\end{array}& \begin{array}{cc}lA/3& 0\\ 0& lI/3\end{array}\end{array}\right\},$

where $\rho$ is the density of the tube, $A$ is the section area and $I$ is the moment of inertia [12, 13]. Element stiffness matrix is:

2

${\mathit{K}}^e={\mathit{K}}_s^e+{\mathit{K}}_b^e=\frac{EI}{l}\left\{\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& -1\\ 0& 0& 0& 0\\ 0& -1& 0& 1\end{array}\right\}+\frac{\mu GA}{l}\left\{\begin{array}{cccc}1& l/2& -1& l/2\\ l/2& l^2/4& -l/2& l^2/4\\ -1& -l/2& 1& -l/2\\ l/2& l^2/4& -l/2& l^2/4\end{array}\right\},$

where $E$ is Young’s modulus, $G$ is shear modulus and $\mu$ is the shear coefficient of the Timoshenko beam. For the thin walled hollow circular section, the value of shear coefficient is $\mu =\frac{6(1+\vartheta ){(1+m^2)}^2}{\left(7+6\vartheta \right){\left(1+m^2\right)}^2+(20+12\vartheta )m^2}$ in which $\vartheta$ is Poisson ratio [12, 13]. The coefficient $m$ equals to $\frac{d-t}{d+t}$ in which$d$, $t$ are section size as shown in Fig. 4(b).

Fig. 4Timoshenko plane beam element

a) Timoshenko beam element

b) Tube section

3.2. Model of connection isolators

Lead rubber bearings are installed in the structural system acting as story isolators. Lead rubber bearings can be modeled by Bouc–Wen models [1, 20]. The properties of the lead rubber bearings are defined by 3 parameters: total lead rubber bearings yield force $f_y$, total initial shear stiffness $k_i$ and post-yield shear stiffness $k_y$. Total initial shear stiffness of the lead rubber bearings is supposed to be 10 times the total post-yield stiffness (i. e. $k_i=10k_y$) [1]. At each floor level, two LRB300 isolators are chosen to connect tubes and frames. The lateral post-elastic stiffness of a LRB300 is 0.435 MN/m and its yield force is 25.48 kN.

3.3. Isolated frame model

3.3.2. Floor shear model

Multi-degree-of-freedom shear models can be used to simulate the isolated frames. The masses are assumed to be lumped at each floor level. Table 3 shows the masses and the lateral stiffnesses of the two frames shown in Fig. 5. The total masses of the left frame and the right frame are respectively 7.667×10³ tons and 6.835×10³ tons. When these frames are fixed to the ground, their fundamental periods are 1.1326 s and 1.1917 s, respectively.

Table 3Masses and lateral stiffnesses at each floor level

	Left frame		Right frame
	Mass (ton)	Lateral stiffness (MN/m)	Mass (ton)	Lateral stiffness (MN/m)
Ground floor	638.8	1.8462×101	569.6	1.5492×101
Other floor	638.8	1.1286×103	569.6	9.0694×102

The mass matrix of the isolated left frame is ${\mathit{M}}_{lf}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(m_{l,1},m_{l,2},\cdots ,m_{l,12})$, in which $m_{l,1},m_{l,2},\dots ,m_{l,12}$ are masses at each floor level as shown in Fig. 5.

Fig. 5Simulation model

The stiffness matrix of the isolated left frame is:

4

${\mathit{K}}_{lf}=\left\{\begin{array}{cccc}{k}_{l,iso}+k_{l,1}& -k_{l,1}& & \\ -k_{l,1}& k_{l,1}+k_{l,2}& \ddots & \\ & \ddots & \ddots & -k_{l,12}\\ & & -k_{l,12}& k_{l,12}\end{array}\right\},$

where $k_{l,iso},k_{l,1},\dots ,k_{l,12}$ are lateral stiffnesses at each floor level as shown in Fig. 5.

Rayleigh damping is adopted for the frame over isolation layer level with the first and second modal damping ratios ${\xi }_{lf}$ at 0.03. The damping of the isolated left frame at isolation layer is:

5

$c_{lf,iso}=\frac{2{\xi }_{lf,iso}{k}_{lf,iso}}{{\omega }_{lf,iso}},$

where ${\omega }_{lf,iso}$ is the first modal circular frequency of the left isolated frame. ${\xi }_{lf,iso}$ is the damping ratio of base isolation system and is supposed to be 3 %.

The mass matrix, stiffness matrix and damping matrix of the right isolated frame are similar to the above mass matrix, stiffness matrix and damping matrix of the left isolated frame.

3.4. Model of the new building group and equations of motion

Fig. 5 shows the simplified simualtion model of the building group system. Three tubes are simulated by three beams and two floor shear models represent the left frame and right frame. Bouc-Wen models are used to simulate the connection isolators between the tubes and frames. The stiffness of connection isolators $k$ equals to the initial stiffness of the connection isolators in elastic region and post-elastic stiffness of the connection isolators in plastic region.

Equations of motion [14] of the structural system shown in Fig. 5 are developed in the form of:

where mass matrix ${\mathit{M}}_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}}=diag\left({\mathit{M}}_t,{\mathit{M}}_{lf},{\mathit{M}}_t,{\mathit{M}}_{rf},{\mathit{M}}_t\right)$, in which ${\mathit{M}}_t$, ${\mathit{M}}_{lf}$ and ${\mathit{M}}_{rf}$ are respectively tube mass matrix, left frame mass matrix and right frame mass matrix. Similarly, damping matrix ${\mathit{C}}_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}}=diag({\mathit{C}}_t,{\mathit{C}}_{lf},{\mathit{C}}_t,{\mathit{C}}_{rf},{\mathit{C}}_t)$ and stiffness matrix ${\mathit{K}}_{group}=diag\left({\mathit{K}}_t,{\mathit{K}}_{lf},{\mathit{K}}_t,{\mathit{K}}_{rf},{\mathit{K}}_t\right)$. ${\mathit{I}}_{group}={\{{\mathit{I}}_t,\mathrm{}{\mathit{I}}_{lf},\mathrm{}{\mathit{I}}_t,\mathrm{}{\mathit{I}}_{rf},\mathrm{}{\mathit{I}}_t\}}^{\mathrm{T}}$, in which ${\mathit{I}}_{lf}$ and ${\mathit{I}}_{rf}$ are unit vectors indicating the position of the seismic forces applied on the left frame and the right frame.${\mathit{N}}_{LRB}={\{f_{ltlf,1},f_{ltlf,2},\dots ,f_{lfmt,1},f_{lfmt,2},\dots ,f_{mtrf,1},f_{mtrf,2},\dots {,f}_{rfrt,1},f_{rfrt,2},\dots \}}^T$ is the connection force provided by the story isolators. Subscripts lt, lf, mt, rf and rt respresent left tube, left frame, middle tube, right frame and right tube, respectively. The connection force is related to the relative displacement between the tubes and frames. For example at the first floor the story isolator force [1, 20] between the left tube and the left frame is:

where $k_y$ and $k_i$ are the respective yield stiffness and initial stiffness of LRBs.

$x_{ltlf,1}=x_{lt,1}-x_{lf,1}$, in which $x_{lt,1}$ and $x_{lf,1}$ are the displacements of the left tube and the left frame. $\alpha =\frac{k_y}{k_{iy}}$ is stiffness ratio. $z_{ltlf,1}$ is hysteretic dimensionless quantity and satisfies the following relation [1]:

where $d_y$ is yield displacement and $A=1$, $\gamma =0.9$ and $\beta =0.1$ [1].

Eq. (6) may be rewritten in incremental form to be solved by Newmark-$\beta$ in combination with Newton-Raphson method [14] to obtain structural responses (e. g. acceleration, velocity and displacement) at any time $t$. In the analysis tubes and frames are assumed to behave linearly elastic throughout the loading history.

4.1. Site condition and earthquakes input

The building group is assumed to be located in a medium soft area with seismic intensity at the VIII degree in accordance with the Chinese code [18]. Fig. 6 shows the acceleration response spectra for the three earthquake records at 3 % damping.

Fig. 6Acceleration response spectra at damping ratio 3 %

4.2. Simulation results

In consideration of the nonlinear properties of the story isolators, small time interval $\mathrm{\Delta }t=$0.01/100 $=$ 1×10^-4 s is used. Fig. 7 shows the maximum drift of the tubes at each floor level when the system is under three earthquakes. Like a deformed cantilever beam subjected to transverse force, the maximum drifts of the tubes occure at the top floor level. The maximum drifts of the frame occur around at the second floor level. The maximum drifts of the building group and the corresponding story drift angles are shown in Table 4, which shows that the maximum drift angles are very small.

Table 5 indicates the reduction of maximum acceleration responses of the building group. The maximum accelerations of the tubes and the frames are reduced by at least 19.91 % and 64.66 %.

In addition, the maximum displacement can also be significantly reduced by the base isolators and story isolators. As an example the displacement histories of the left frame are shown in Fig. 8. It is observed that the retrofit strategy is effective in reducing the displacement responses of the left frame. Excited by three earthquakes before retrofit the root mean square displacement responses of the left frame are 37.92 mm, 36.16 mm and 14.60 mm, respectively. When base isolators and story isolators are installed, the root mean square displacement responses of the left frame are 14.78 mm, 19.04 mm and 9.59 mm, respectively. This indicates that the root mean square displacement responses can be reduced by more than 34.30 %. Similarly responses of the right frame can be observed.

Fig. 9 shows the maximum displacement envelopes of the frames. It is observed that the maximum displacement happens at the top floor level and the lateral displacement mainly occures at the isolation layer (the first floor level). These results show the two frames are well isolated from ground motions.

Fig. 7Maximum drifts of the tubes under three earthquakes

Fig. 8Displacement histories of the left frame under three earthquakes

Fig. 9Displacement envelope of the frames

Fig. 10Maximum relative displacement (mm)

Fig. 10 shows the maximum relative displacement between the tubes and frames at each floor level. The maximum relative displacement response between the tubes and the frames to the El Centro earthquake is larger than corresponding responses to Taft earthquake and Kobe earthquake. The maximum relative displacement responses between the left tube and the left frame, between the left frame and the middle tube, between the middle tube and the right frame, between the right tube and the right frame are 8.5 cm, 8.3 cm, 7.7 cm and 10.1 cm, respectively. These values are much less than the minimum design seismic joint width, which is 21 cm according to the Chinese Code for seismic design of buildings [16].

The aforementioned results show that under three suitable earthquakes the responses of the building group are small. The relative displacements between the tubes and frames comply with the Chinese seismic design code. When proper seismic joint width is designed, pounding within the building group can be prevented.