(Solved): The plane structure shown in Figure \( 2.21 \) consists of a rigid, weightless bar and linear spri ...

The plane structure shown in Figure \( 2.21 \) consists of a rigid, weightless bar and linear springs of stiffness \( k^{(1)} \) and \( k^{(2)} \). Only small vertical displacements are permitted. The reduced stiffness matrix \( \widehat{\mathbf{K}} \) of this structure is \( 2 \times 2 \) but can have various forms, depending on the choice of global displacement matrix. Determine \( \widehat{\mathbf{K}} \) for each of the following choices of lateral translations: a. \( u_{1 y} \) at \( x=0 \) and \( u_{2 y} \) at \( x=L \) (see Figure 2.21, right). b. \( u_{1 y} \) at \( x=0 \) and \( u_{A y} \) at \( x=L / 2 \). c. \( u_{2 y} \) at \( x=L \) and \( u_{B y} \) at \( x=2 L \). lint: consider the number of modes for the rigid bar, and follow the process from Q2a. The oints for questions a. b. c. are the locations where external force/displacement is applied, and ou need to find the relationship between concentrated forces applied there and the internal orces due to the deforming springs relative to the rigid bar. Degrees-of-freedom for Part (a)