## Appendix B Determination of Range for Parameter \(\lambda\)  
\(\lambda\) is defined in Section 2 as the ratio between the equivalent stiffness of beams \(EI_{b,eq,n}\) and columns \(EI_{c,eq,n}\) in the surrogate frames used in the study. This Appendix explains how a practical range of values for \(\lambda\) (between 0.1 to 10 centered around 1) was determined. Figure 11 shows a regular steel frame, with identical span length \(L\) at each bay and height \(h\) at each story, where all the beam elements at story \(n\) are similar and present a stiffness \(El_{b,n}\), where \(E\) is the steel Young modulus and \(I_{\chi,n}\) the relevant second moment of inertia of element \(x\) at story \(n\) for in-plane deformation. Likewise, all columns at story \(n\) are assumed identical, with stiffness \(El_{c,n}\). Shear and axial flexibility are neglected. The corresponding surrogate frame is shown in Figure 11; the corresponding beam and column at level \(n\) present stiffnesses \(El_{b,\alpha,n}\) and \(El_{c,\alpha,n}\), defined as  
\[El_{b,\alpha,n}=SEI_{b,n}, \tag{12}\]  
\[El_{c,\alpha,n}=(S+1)El_{c,n}/2, \tag{13}\]  
where \(S\) is the number of bays of the original frame. This conversion assumes that the inflection points of all beams are located exactly at midspan, resulting in identical rotations for all joints at the same level; this is the same assumption used to transform regular frames to so-called fish-bone models [45], resulting in a stiffness matrix identical to that of the surrogate frame in this study. Typical moment-resisting frames are designed imposing capacity conditions at every joint in story \(n\), such as:  
\[M_{\mathbb{R},c,n}+M_{\mathbb{R},c,n+1}\geq\gamma\sum M_{\mathbb{R},b,n}, \tag{14}\]  
where \(M_{\mathbb{R},c,n}\) and \(M_{\mathbb{R},c,n+1}\) represent the bending strength of columns at story \(n\) and \(n+1\), respectively, \(\sum\ M_{\mathbb{R},b,n}\) is the sum of bending moment strength of beams concurring to the joint considered (\(M_{\mathbb{R},b,n}\) at the edge, \(2M_{\mathbb{R},b,n}\) at the inner joint, see Figure 11), and \(\gamma>1\) is a factor that in Eurocode 8 takes a value of 1.3 [42] (clause 4.4.2.3(4)); this condition must be applied at all joints for \(n<N\). For the frame represented in Figure 11, the condition imposed in the inner joint is more restrictive to columns. Two different limiting strength scenarios are assumed:  
1. \(M_{\mathbb{R},c,n}=M_{\mathbb{R},c,n+1}\) leading to the condition \(M_{\mathbb{R},c,n}\geq 1.3M_{\mathbb{R},b,n}\);
2. \(M_{\mathbb{R},c,n}>>M_{\mathbb{R},c,n+1}\) leading to the condition \(M_{\mathbb{R},c,n}\geq 2.6M_{\mathbb{R},b,n}\). Assuming IPE sections for beams and HEB sections for columns in the original frame, \(E=210\) GPa, and the same yield strength in both beams and columns, once the beam section is fixed, the values of \(El_{b,n}\) and \(M_{\mathbb{R},b,n}\) are known, and the flexural strength of the columns can be found according to strength scenario #1 or #2; the minimum HEB section providing this strength is determined and its stiffness \(El_{c,n}\) obtained; then Equations (12), (13) and (7) are used to find values of \(\lambda\) for different numbers of bays \(S\) (1, 2, 3, 4, and 10). The process is repeated for all possible IPE sections from IPE200 to IPE600. Relevant statistics (median \(\mu\), standard deviation \(\sigma\) and coefficient of variation \(c_{\bar{\sigma}}\)) of the results obtained for \(\lambda\) across the full range of cross-sections are shown in Table 11; actual values are expected to lie between both strength scenarios; for the class of frames analyzed, \(\lambda\) falls between 0.35 and 1.64 and \(\lambda=1\) is adopted as central value. If the column yield strength adopted is 1.5 times the beam yield strength (for instance, for S235 beams and S355 columns), \(\lambda\) lies between 0.58 and 2.75. The range of values finally adopted is expanded considerably [between 0.1 and 10] and includes \(\lambda=\infty\), corresponding to a pure shear frame.