## 5 Adaptive Load Vectors
### Adaptive Upper-Bound (AUB) Pushover Analysis  
The adaptive upper-bound (AUB) pushover analysis procedure was developed by [PERSON] et al. [72] to assess the performance of tall buildings. It considers the effects of higher modes and the dynamic changes in structural characteristics during inelastic responses. The accuracy of the proposed procedure is tested using four steel moment-reasisting frame buildings. The procedure can be summarized as follows:  
1. Initialization: Begin with the model. Set the step to \(k=0\). Initialize the load factor \(\Delta\lambda\) and conduct an eigenvalue analysis. The upper-bound (UB) load vector is determined using the given Equation (54):\[f_{UB}=\omega_{1}^{2}M\varphi_{1}+\omega_{2}^{2}M\varphi_{2}\bigg{(}\frac{q_{2}}{q _{1}}\bigg{)}_{UB}, \tag{54}\]
2. Target displacement calculation: using Equation (55), compute the predefined target displacement: \[u_{r}=u_{rTLP}\bigg{[}1+\bigg{(}\frac{q_{2}}{q_{1}}\bigg{)}_{UB}\bigg{]},\] (55)
3. Load vector application: implement the newly determined load vector. 4. Displacement check: Examine if the roof displacement has reached the predefined target. If it has, the procedure ends; if not, proceed to the next step. 5. Frequency update: Check if there is a need to update the frequency due to yielding. If required, apply the UB lateral load pattern as per Equation (54). 6. Eigenvalue analysis: conduct an eigenvalue analysis using the updated stiffness (K) and mass (M) matrices. 7. Inelastic response spectrum construction: construct the inelastic response spectrum considering the current ductility ratio using Equation (56): \[\mu_{k}=\frac{d_{k}}{d_{y}},\] (56)
8. Determine values: Identify the values of \(S_{dIk}\) and \(S_{d2k}\). Then, calculate the \((q_{2}/q_{1})\) (\(k\)) ratio using Equation (57): \[\bigg{(}\frac{q_{2}}{q_{1}}\bigg{)}_{UB}=\bigg{|}\frac{\Gamma_{2}S_{d2}}{ \Gamma_{1}S_{d1}}\bigg{|},\] (57)
9. Lateral load pattern calculation: determine the new lateral load pattern using Equation (58): \[f_{i}^{(k)}=\max\Big{(}f_{i,UB}^{(k)},F_{i,Unif}^{(k)}\Big{)},\] (58)
10. AUB lateral load pattern application: apply the AUB lateral load pattern as per Equation (59): \[P^{(k)}=P^{(k-1)}+\Delta P^{(k)}=P^{(k-1)}+\Delta\lambda^{(k)}\overline{f}^{(k )}{}_{\circ}P_{0},\] (59)
11. Iteration: return to Step 3 and repeat the process until the roof displacement meets the predefined target. A Multi-Mode Adaptive Displacement-Based Pushover Analysis Procedure for Estimating the Seismic Demands of RC Moment-Resisting Frames (MADP)  
The multi-mode adaptive displacement-based pushover (MADP) method, cited in [56], combines'multi-modal' and 'adaptive' pushover approaches. It starts with a load pattern based on the structure's elastic mode shape and updates it as the analysis progresses. The method uses the structure's 'first-mode' inelastic mode shape to estimate target displacement accurately and is employed to assess the seismic performance of RC moment-resisting frames. Key advantages include overcoming traditional NSP limitations by considering higher-mode effects and structural changes [73], enabling adaptive and detailed seismic demand evaluation, and utilizing a simple, computationally efficient target-displacement calculation method [44]. The procedure is summarized in the following steps:  
1. Create a structural mathematical model: develop a mathematical model of the structure that integrates the monotonic nonlinear behavior and degradation features of the structural components. Eigenvalue analysis and mode shapes: Conduct an eigenvalue analysis of the elastic building model to compute the natural frequencies \(\omega_{n}\) and the associated mode shapes \(\Phi n\). Normalize the mode shapes so that the roof component of \(\Phi n\) equals unity (\(\Phi nr=1\)). 3. Compute the lateral force distribution: calculate \(Sn=m\Phi n\), where \(Sn\) represents the distribution of lateral forces across the structure's height for the first stage of the multistage pushover analysis. 4.