Truss by the Finite Element Method Given the following structure made up of five Truss elements and four nodes. The loading is broken down into two horizontal and vertical loads, respectively Px and Py, both applied to node 4 . The boundary conditions are hinged supports (no displacements, only rotation allowed) at nodes 1 and 3 . We want to determine the displacements at nodes 2 and 4 , horizontal and vertical reactions at nodes 1 and 3 (resp. RHi and RVi at node i=1 and 3 ) and efforts ( Ni with i=1,,5) in this structure. Fig 1 : Truss mesh fig 2 : local coordinate systems remarks: We will use lowercase (resp. uppercase) for entities written in the local (resp. global) coordinate system Each node has 2 degrees of freedom noted ui,vi (resp Ui,Vi) for node i in the local (resp.global) coordinate system. Question 1: Fill the following connectivity table Q2: Give the stiffness matrix for each element in his local coordinate system. We will note ki this local stiffness matrix for element number i: k1= k2= k3= k4= k5= Q3: Give the stiffness matrix for each element in the global coordinate system. We will note Ki this global stiffness matrix for element number i: K1= K2= K3= K4= K5= Q4: Write the global stiffness matrix for the whole system, we note this matrix (which is the assembly of the previous global stiffness matrix for each element) K : Q5: Write the system to solve as [K]{Q}={F} where - [K] is the global stiffness matrix, - {Q} is the vector of the displacements in the global coordinate system - {F} is the vector of the efforts (reactions and loads) in the global coordinate system Q6: by applying the boundary conditions a) find the components of the displacement of each nodes. U1= V1= U2= V2= U3= V3= U4= V4= b) find the reactions forces at nodes 1 and 3 : RH1=RV1=RH3=RV3= Q7: Find the forces in each truss (we choose N>0 in tension) N1= N2= N3= N4=