A rectangular Mindlin plate is pinned along its edges at \(x=0\) and \(x=a\), and is constrained by elastic supports possessing torsional stiffness \(k_{T}\) along its edges at \(y=0\) and \(y=b\). Establish the boundary conditions for the structure.

For the pinned edges, the deflection is zero. The pinned edges are located at \(x=0\) and \(x=a\), so for these edges, we have: 1. At \(x=0\): \(w(0,y)=0\) 2. At \(x=a\): \(w(a,y)=0\) These conditions represent the deflection restrictions at these edges.

For the edges with elastic constraints, we have torsional stiffness \(k_{T}\). These are located at \(y=0\) and \(y=b\). Here, the conditions are related to the bending moment \(M_{x}\) and the torsional moment \(M_{xy}\): 1. At \(y=0\): Bending moment \(M_{x}(x,0) + k_{T}w(x,0) = 0\) 2. At \(y=b\): Bending moment \(M_{x}(x,b) + k_{T}w(x,b) = 0\) 3. At \(y=0\): Torsional moment \(M_{xy}(x,0) = 0\) 4. At \(y=b\): Torsional moment \(M_{xy}(x,b) = 0\) These conditions represent the balance between the plate's internal constraints and the external torsional stiffness constraints. To sum up, the boundary conditions for the rectangular Mindlin plate are: 1. Pinned edges: * At \(x=0\): \(w(0,y)=0\) * At \(x=a\): \(w(a,y)=0\) 2. Edges with torsional stiffness: * At \(y=0\): \(M_{x}(x,0) + k_{T}w(x,0) = 0\) and \(M_{xy}(x,0) = 0\) * At \(y=b\): \(M_{x}(x,b) + k_{T}w(x,b) = 0\) and \(M_{xy}(x,b) = 0\)