# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 3 Development of the MINLP Model Formulation for Truss Superstructures
### The MINLP Model Formulation for Truss Superstructures
#### 3.2.4 Structural Analysis Constraints  
Non-linear and linear parameter constraints (Equations (2)-(17)) as well as the lower and upper bounds on continuous variables represent a rigorous system of functions taken from structural analysis. In the proposed MINLP formulation, structural analysis constraints are divided into boundary constraints, finite element equations, design constraints and deflection constraints.  
The boundary constraints, Equations (2)-(5), define the supporting and loading conditions of the structure. The supporting conditions are established through coefficients \(\mathbf{s}_{\chi,i}\) and \(\mathbf{s}_{y;i}\) defined for each node \(i\). The coefficient \(\mathbf{s}_{\chi,i}\) is valued 1 if node \(i\) is supported in the direction of axis \(X\); otherwise, it is valued 0. Similarly, \(\mathbf{s}_{y;i}\) is 1 when node \(i\) is supported in \(Y\) direction. Consequently, the corresponding deflections of the supported nodes are defined to be zero by Equations (2) and (3). The load on a truss is given by the nodal forces, which are defined by the loading coefficients \(\mathbf{p}_{\chi,k,i}\) and \(\mathbf{p}_{y,k,i}\) in Equations (4) and (5). Each parameter \(\mathbf{p}_{\chi,k,i}\) contains the value of the nodal force, which acts on node \(i\) in the direction of axis \(X\) and corresponds to the loading case \(k\), \(k\)\(\in\)\(K\). In the same way, the nodal forces in the direction of axis \(Y\) are defined by coefficients \(\mathbf{p}_{y,k,i}\). In the proposed MINLP-TS formulation, the supporting and loading coefficients are arranged into a set of corresponding vectors \(\mathbf{s}_{\chi}=[\mathbf{s}_{\chi,i}]\), \(\mathbf{s}_{y}=\{\mathbf{s}_{y;i}\}\), \(\mathbf{p}_{\chi,k}=[\mathbf{p}_{\chi,k,i}]\) and \(\mathbf{p}_{y,k}=\{\mathbf{p}_{y,k,i}\}\).  
The finite element equations, Equations (6)-(8), are used to calculate the reactions, axial forces and displacements of the structure. Equations (6) and (7) are defined for each node \(i\). As node \(i\) can, in general, be either the starting or the ending joint of a bar, the contributions of elements \(i\)\(\rightarrow\)\(j\) (\(i<j\); \(i\) is the starting joint) as well as \(j\)\(\rightarrow\)\(i\) (\(j>i\); \(i\) is the ending joint) have to be considered. Parameters \(l_{i,j}^{\text{sf}}\) and \(m_{i,j}^{\text{sf}}\) are the cosines of the angles between the local longitudinal axis of bar \(i\)\(\rightarrow\)\(j\) and the global axes \(X\) and \(Y\), respectively. When the shape of the structure is also optimized, parameters \(l_{i,j}^{\text{sf}}\) and \(m_{i,j}^{\text{sf}}\) become substituted functions of the joint coordinates; see Equations (11) and (12).  
The design constraints, Equations (13)-(15), represent the calculation of the stresses and buckling of bars, where \(\sigma^{\chi,\text{max}}\) and \(\sigma^{\chi,\text{max}}\) are the tensile and compressive strength of the material, while \(\sigma_{i,j}^{\text{cr}}\) is the limit buckling stress of bar \(i\)\(\rightarrow\)\(j\). It should be noted that, according to Equation (8), the positive sign of the axial force \(F_{k,i,j}\) represents a tensile force. The deflection constraints, Equations (16) and (17), ensure that the deformations of joints remain within the allowed limits.