Analysis of Fixed Beam Using Flexibility and Stiffness Methods Assignment

Assignment Task

1. 1. The structure shown in Fig. 1 is fixed at support A and is restrained with a roller at support D.

• Find the reaction at D for the beam using the flexibility method of analysis.
• Draw the shear force diagram (SFD) and bending moment diagram (BMD) showing all the peak values. 

2. The structure shown in Fig. 2 is constructed from the assembly of beam elements 1 and 2. It is fixed at both end nodes 1 and 3 and is subjected to a point load at node 2.

• Develop the stiffness matrix of beam elements 1 and 2.
• Assemble the stiffness matrix for the full structure.
• Determine the node displacements and the member end forces for the beams using the matrix stiffness method.
• Draw the shear force diagram (SFD) and bending moment diagram (BMD) showing all the peak values. 

2. 1. The structure shown in the below figure is fixed at Node C and constrained with roller support at Node A. The structure is subjected to concentrated moment at Node B. Assume El is constant.

i. Determine the degree of static indeterminacy (DSI), i.e., the excess of unknown actions than the available number of equations of static equilibrium. 

ii. Find the reaction at support A using the flexibility (force) method of analysis.

iii. Draw the shear force diagram (SFD) and bending moment diagram (BMD) showing all the peak values. 

2. For the same structure shown in the figure:

i. Determine the degree of kinematic indeterminacy (DKI), i.e., the number of independent translations and rotations (the unknown joint displacements).

ii. Develop the stiffness matrix of Beam Elements 1 (connecting A to B) and Beam Element 2 (connecting B to C) and assemble the global stiffness matrix.

iii. Determine the nodal displacements as well as the element end-forces using the stiffness (displacement) method of analysis. Draw the shear force diagram (SFD) and bending moment diagram (BMD) showing all the peak values.

(Refer to online lecture slides for stiffness matrix of a beam element. You do not need to show working for the inverse of matrices (use your calculator). Feel free to rotate the structure 90° if you are more comfortable to analyse it horizontally) 2 m 2 m B24 kN.m C

3. 1. The structure shown in Fig. 1 is fixed at Node A and constrained with roller support at Node B. The structure is subjected to concentrated moments at Node C. Assume El is constant.

i. Determine the degree of static indeterminacy (DSI), i.e., the excess of unknown actions than the available number of equations of static equilibrium.

ii. Find the reaction at support B using the flexibility (force) method of analysis.

iii. Draw the shear force diagram (SFD) and bending moment diagram (BMD) showing all the peak values. (Refer to online lecture slides for the integration table) '

2. For the same structure shown in Fig. 1

i. Determine the degree of kinematic indeterminacy (DKI), i.e., the number of independent translations and rotations (the unknown joint displacements).

ii. Develop the stiffness matrix of Elements 1 and 2 and assemble the global stiffness matrix.

iii. Determine the nodal displacements and the element end-forces using the stiffness (displacement) method of analysis.

iv. Draw the shear force diagram (SFD) and bending moment diagram (BMD) showing all the peak values. (Refer to online lecture slides for stiffness matrix of a beam element. You do not need to show working for the inverse of matrices. Use your calculator) 

4. 1. In the structure shown in Fig. 1 assume that the supports B and C are rollers and A is pinned. Assume El is constant.

• Find the reaction at support B using the flexibility (or force) method of analysis.
• Draw the shear force diagram (SFD) and bending moment diagram (BMD) showing all the peak values. 

2. The structure shown in Fig. 2 is fixed at Node 1 and constrained with roller supports at Nodes 2 and 3. The structure is subjected to concentrated moments at Nodes 2 and 3. Assume El is constant.

• Develop the stiffness matrix of Element 1 (connecting Nodes 1 and 2) and Element 2 (connecting Nodes 2 and 3).
• Assemble the stiffness matrices and determine the nodal displacements and the elements end forces using the stiffness (or displacement) method of analysis.
• Draw the shear force diagram (SFD) and bending moment diagram (BMD) showing all the peak values. 

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