## 3 Methods for the Structural Analysis of Diagrid Tall Buildings
[18] carry out a static condensation procedure, where the contributions of vertical displacements and out-of-plane rotations are condensed. Specifically, Equation (13) is re-written in the following form:  
\[\left\{ \begin{array}{c}\{F_{H}\}\\ \{F_{V}\}\end{array} \right\}=\left[ \begin{array}{cc}[K_{HH}]&[K_{HV}\\ [K_{VH}]&[K_{VV}]\end{array} \right]\left\{ \begin{array}{c}\{\delta_{H}\}\\ \{\delta_{V}\}\end{array} \right\}, \tag{14}\]  
where \(\{\delta_{H}\}\) (\(\{F_{H}\}\)) include the contributions of horizontal displacements (forces) and torsional rotations (moments), and \(\{\delta_{V}\}\) (\(\{F_{V}\}\)) include the contribution of vertical displacements (forces) and out-of-plane rotations (moments). The condensation procedure yields the following relation, where only the horizontal DOFs are considered through the \(3N\times 3N\) condensed stiffness matrix \([K_{HH}]^{*}\):  
\[\{F_{H}\}=\left([K_{HH}]-[K_{HV}][K_{VV}]^{-1}[K_{VH}]\right)[\delta_{H}]=[K_ {HH}]^{*}\{\delta_{H}\}. \tag{15}\]  
The MBM has been used within the GA framework to investigate the stiffness interaction between an external steel diagrid and an internal concrete core. In particular, a square, 18-story tall building is considered in [18], and the coupled behavior is analyzed under lateral forces and torque moments. Although the torsional behavior is obviously governed by the external diagrid tube, the distribution of shear forces at the various floor levels is not trivial, and gives rise to an oscillating trend along the height of the building, due to the shear-bending coupling of the two structural systems (Figure 7). In a more recent paper, [PERSON] et al. [28] investigate the effect of the diagonal inclination on the diagrid-core coupled system. As already pointed out by [PERSON] et al. [10], when the diagrid angle is in the optimal range, the diagrid's lateral stiffness prevails over the internal core's one. The influence of the type of internal core, i.e., closed- or open-section shear wall, is also investigated in [28]. Although the diagrid-core coupling mechanism is almost the same in terms of lateral deformability, a remarkable difference between the two types of internal cores (open- and closed-section shear wall) is observed in terms of torsional behavior. In the case of internal open-section shear walls and steep diagrid angles, a clear inflection point in the torsional deformation curve is obtained due to the warping effect of the internal shear wall [28]. Although simplified, the methods presented in this section for the structural analysis of diagrid systems, integrated with the stiffness- and strength-based methodologies for the preliminary design shown in Section 1, can provide a valid and efficient alternative to FE calculation in the preliminary stages. As a matter of fact, they allow the quick investigation of the overall structural behavior, while capturing the fundamental parameters governing diagrid performance. It has to be noted that, due to the increasing power of modern computing technologies, simplifying the structural model, i.e., reducing the DOFs of the system, is not often an imperative need. Current FE software in modern computers are able to perform the structural analysis of buildings with very large numbers of DOFs, and the trend nowadays is to consider even more detailed models for the sake of general analysis. However, although using detailed models is necessary in the ultimate design stages, it can make one lose sight of the overall structural behavior during the preliminary stages. As a matter of fact, in these stages, the correct comprehension of the overall structural behavior has important implications in the definition of the optimal resisting elements. These choices in the preliminary phases are in turn known to have a strong influence on the cost and efficiency of the final solution, especially in tall buildings.