## 3 Numerical Example
It is emphasized that the second elevation pattern (P-2) for the floor forces, which includes an additional force at the top of tall frames with more than four floors, should always be considered in pushover analysis. This consideration is crucial because, in the deep nonlinear range, the failure of several braces in compression causes the braced frame to become more flexible, making it more sensitive to the influence of higher modes and second-order effects. Finally, the stepping lateral stiffness matrix \(\mathbf{k}_{i}\) of the six-story steel-braced frame is calculated at the specific inelastic \(i\)-step where the top displacement \(u_{\text{top},i}=0.18\) m occurs, corresponding to a chord rotation profile angle approximately equal to \(\theta_{\text{pr},i}=0.01\)rad. This matrix reflects the reduction in lateral stiffness due to the progression of damage in the structure, particularly accounting for the failure of various braces in compression and the resulting changes in the frame's behavior. To facilitate the computation of the stepping lateral stiffness matrix, the stepping flexibility matrix of the six-story steel-braced frame is first determined at a top displacement \(u_{\text{top},i}=0.18\) m and then inverted, following the procedure outlined in phase (h) of the methodology. Decimal places within the stiffness matrix are omitted for brevity. For the P-1 pushover analysis with positive floor forces, the damage stiffness matrix \(\mathbf{k}_{i}\) at a roof displacement \(u_{\text{top},i}=0.18\) m is as follows:  
\[\mathbf{k}_{i}= \begin{bmatrix}389635&-288907&113882&-23766&8882&-4922\\ -288907&433530&-316476&116213&-23348&4029\\ 113882&-316476&426223&-294602&101353&-13897\\ -23766&116213&-294602&409167&-269417&59041\\ 8882&-23348&101353&-269417&437511&-251167\\ -4922&4029&-13897&59041&-251167&203664\\ \end{bmatrix}  \tag{10}\]  
Figure 19: Axial force-axial deformation behavior of X-braces of the steel frame at roof displacement \(u_{\text{top},i}=0.18\) m in pushover analyses (**a**) P-1 and (**b**) P-2 along the negative direction. The lateral stiffness matrix of the six-story steel-braced frame in the health (undamaged) state is calculated using the same procedure outlined in phase (h) of the methodology:  
\[\mathbf{k}_{o}= \begin{bmatrix}1703412&-994416&129131&-19167&-17371&10389\\ -994416&1489715&-681819&119026&11485&-32439\\ 129131&-681819&1021146&-568196&109771&-12825\\ -19167&119026&-568196&871983&-482707&78014\\ -17371&11485&109771&-482707&717627&-327892\\ 10389&-32439&-12825&78014&-327892&279033\end{bmatrix}  \tag{11}\]  
Therefore, the stepping damage stiffness matrix of the steel-braced frame at the same inelastic \(i\)-step is calculated by the general relation \(\mathbf{\Delta k}_{i}=\mathbf{k}_{o}-\mathbf{k}_{i}\). The resulting matrix is as follows:  
\[\mathbf{\Delta k}_{i}= \begin{bmatrix}1313778&705509&15249&\text{N.E.}&\text{N.E. }&\text{N.E.}\\ 705509&1056185&365344&2812&\text{N.E.}&\text{N.E.}\\ 15249&365344&594923&273594&8418&\text{N.E.}\\ \text{N.E.}&2812&273594&462816&213289&18973\\ \text{N.E.}&\text{N.E.}&8418&213289&280115&76725\\ \text{N.E.}&\text{N.E.}&\text{N.E.}&18973&76725&75369\end{bmatrix}  \tag{12}\]  
We observe that all the aforementioned matrices are symmetrical, as expected. The notation \"N.E.\" within the damage stiffness matrix denotes that the coupled terms \(k_{14},\ k_{15},\ k_{16},\ k_{25},\ k_{26},\ k_{36}\) are not evaluated due to their high sensitivity. Below is the stepping damage stiffness matrix of the six-story steel-braced frame for the P-2 pushover analysis case with positive floor forces at the specific inelastic \(i\)-step where \(u_{\text{top},i}=0.18\) m occurs:  
\[\mathbf{\Delta k}_{i}= \begin{bmatrix}1161639&753503&57773&2958&\text{N.E.}& \text{N.E.}\\ 753503&1155753&438387&17049&\text{N.E.}&\text{N.E.}\\ 57773&438387&639968&287343&10022&\text{N.E.}\\ 2958&17049&287343&502948&234327&5673\\ \text{N.E.}&\text{N.E.}&10022&234327&397925&173901\\ \text{N.E.}&\text{N.E.}&\text{N.E.}&5673&173901&182399\end{bmatrix}