*3.2. P-*Δ *Effects*

In this study, the P-Δ effects are taken into account with the lean-on column, which is subjected to the gravity of each floor. The lean-on column is simulated by several bar elements and connected to the moment-resisting frame by a rigid diaphragm at each floor. The discretization of the cross-section, i.e., fiber section, is not used to simulate the bar elements of the lean-on column, which is assumed to behave elastically. The second-order restoring force is the product of the structural geometric stiffness **<sup>K</sup>***g* and the structural displacement **X***i*+1. The structural geometric stiffness **<sup>K</sup>***g* is the assembly of the geometric stiffness *kg* of the bar elements, which is expressed as:

$$k\_{\mathcal{S}} = \frac{P}{L\_{\mathfrak{e}}} \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix} \tag{15}$$

where *P* is the gravity subjected to the lean-on column. It should be noted that the geometric stiffness is a symmetric matrix.

### *3.3. Time History Analysis of a Four-Story Frame Subjected to Earthquake*

The primary structure is a four-story steel frame, as shown in Figure 4. The stiffnessbased beam-column elements with fiber sections are used to simulate the beams and columns of the moment resisting frame, which is completely symmetric, and the bar elements are adopted to model the lean-on column. There are 24 fibers for each section of the beam-column elements. The elastic modulus and yield strength of the steel are 200 GPa and 345 MPa, respectively. The elastic–perfectly plastic constitutive relationship is adopted for steel. A consistent mass is used to build the mass matrix. The formulation of the initial stiffness matrix follows the procedure in Section 3.1. Shear deformation is not considered in the finite element model. According to the modal analysis, the first and second natural periods of the frame are 1.02 and 0.32 s, respectively. The Rayleigh damping assumption with a 2% damping ratio for the first and second order modes is applied to generate the damping matrix.

**Figure 4.** Four-story steel frame.

The structure is subjected to the unscaled 1940 El Centro NS ground motion. GCR algorithms with *κ*1 = 1/2, *κ*2 = 1/4 and two time steps of 0.01 and 0.001 s are used to conduct the time history analysis. The Newmark CAA algorithm with a time step of 0.001 s is taken for comparison. It should be noted that the GCR algorithms are explicit, whereas the CAA is implicit, so the computing efficiency of the GCR algorithms far exceeds that of the CAA algorithm with the same time step. If a larger time step is adopted, the efficiency of the GCR algorithm can be further improved. Figure 5 compares the lateral displacements of different stories using variant algorithm schemes.

**Figure 5.** *Cont.*

**Figure 5.** Time history curves of lateral displacements. (**a**) First story; (**b**) second story; (**c**) third story; (**d**) fourth story.

Figure 5 shows that the numerical results of the GCR algorithms match well with those of the CAA algorithm. Furthermore, two error indices are adopted to measure the errors between the reference model (CAA) and computing models (GCR):

$$\text{NEE} = \left| \frac{\sum\_{i=1}^{N} \mathbf{x}\_{RM,i}^{2} - \sum\_{i=1}^{N} \mathbf{x}\_{CM,i}^{2}}{\sum\_{i=1}^{N} \mathbf{x}\_{CM,i}^{2}} \right| \tag{16}$$

$$\text{NRMSE} = \frac{\sqrt{\sum\_{i=1}^{N} \frac{\left(\mathbf{x}\_{RM,i} - \mathbf{x}\_{CM,i}\right)^2}{N}}}{\max(\mathbf{x}\_{CM}) - \min(\mathbf{x}\_{CM})} \tag{17}$$

where *xRM* and *xCM* are the structural responses of the reference model and computing model, respectively; *N* is the sampling number. NEE and NRMSE are sensitive to the amplitude and frequency errors, respectively. Table 2 provides the corresponding error indices for Figure 5. It can be concluded from Table 2 that the differences between the GCR algorithms and the CAA algorithm with the same time step of 0.001 s are extremely small; even for the GCR algorithm with a larger time step of 0.01 s, the maximum NEE and NRMSE are less than 4% and 1%, respectively, which are acceptable in engineering practice. This indicates that GCR algorithms are viable for solving nonlinear dynamic problems with superior computational efficiency and accuracy.

**Table 2.** Error indices of lateral displacements using different integration algorithms (unit: %).

