2. Figure 1 shows a stepped beam ABC of length 2L fixed to rigid walls at...

 

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2. Figure 1 shows a stepped beam ABC of length 2L fixed to rigid walls at A and C. The flexural rigidity of portion AB is 2EI and that of BC is EI. The beam is subjected to two distributed loads as shown. It is intended to model each of portions AB and BC using a single beam element. A www. w₁ (N/m) L B W₂ (N/m) L Figure 1 C y X (a) Draw a finite element model of the beam showing element numbers, node numbers and symbols for nodal displacements/rotations and nodal forces/ moments. List all the boundary conditions. (5 marks) (b) Convert the distributed loads into equivalent lumped nodal loads using the formula {}=[qL/2_qL²/12_qL/2 -q1²/12] where q is the transverse load intensity (N/m) along y direction and show the nodal loads pictorially on a neat sketch of beam elements. (5 marks) (c) Write all element matrices and load vectors and assemble them to obtain the global equilibrium equations. Apply the boundary conditions to obtain a reduced system of equations. (5 marks) = (d) For L = 1 m, EI = 10000 Nm², wi 10000 N/m and w2 = 5000 N/m, solve the reduced system of equations for the vertical deflection and cross sectional rotation at B. (5 marks) 2. Figure 1 shows a stepped beam ABC of length 2L fixed to rigid walls at A and C. The flexural rigidity of portion AB is 2EI and that of BC is EI. The beam is subjected to two distributed loads as shown. It is intended to model each of portions AB and BC using a single beam element. A www. w₁ (N/m) L B W₂ (N/m) L Figure 1 C y X (a) Draw a finite element model of the beam showing element numbers, node numbers and symbols for nodal displacements/rotations and nodal forces/ moments. List all the boundary conditions. (5 marks) (b) Convert the distributed loads into equivalent lumped nodal loads using the formula {}=[qL/2_qL²/12_qL/2 -q1²/12] where q is the transverse load intensity (N/m) along y direction and show the nodal loads pictorially on a neat sketch of beam elements. (5 marks) (c) Write all element matrices and load vectors and assemble them to obtain the global equilibrium equations. Apply the boundary conditions to obtain a reduced system of equations. (5 marks) = (d) For L = 1 m, EI = 10000 Nm², wi 10000 N/m and w2 = 5000 N/m, solve the reduced system of equations for the vertical deflection and cross sectional rotation at B. (5 marks)