The frame structure shown in Figure $1$ is the support structure for a hoist located at the point of

application of load $W.$ The supports at A and $B$ are completely fixed. Other connections are welded.

Assuming the structure to be modeled using the minimum number of beam$-$axial $($frame$)$ elements:

a$)$ Develop the FEM model, showing nodes, elements, and degrees of freedom of the structure.

b$)$ Indicate the nodal connectivity of the FEM model.

c$)$ Develop the stiffness matrix of one the element.

d$)$ What is the size of the assembled global stiffness matrix?

e$)$ Indicate the nodal boundary conditions.

f$)$ What is the size of the reduced global stiffness matrix after application of the constraint

conditions?

g$)$ Assuming the reduced matrix is determined, how do to compute the nodal displacement?

Develop the matrix equation which needed to determine the reactions at the supports?

h$)$ To reduce the number element of the model, we may introduce an element containing three

nodes. For such element, how many shape functions do we need? Write an expression to

compute the deformation of the element with three nodes.

i$)$ Where do you recommend to have element with three nodes?