Final answer:
When manually solving for the unknown degrees of freedom (DOFs) by using symmetry in the given beam, there are several steps to follow:
- A. Set up a free-body diagram for the beam, including all forces acting on it and labeling them with their components in the chosen reference frame.
- B. Choose a pivot point and indicate its location on the free-body diagram to compute torques of the forces.
- C. Write the equilibrium equations for force components in the x-direction and y-direction, as well as the torque equation for rotation about the chosen pivot point.
- D. Solve the system of equations for equilibrium to find the unknown quantities.
Step-by-step explanation:
To manually solve for the unknown degrees of freedom (DOFs) of the beam using symmetry, you can follow these steps:
A. Apply symmetry considerations to identify redundant supports:
- - Identify the symmetrical axis of the beam. In this case, it seems to be the vertical dashed line passing through the center.
- - Determine if any supports or loads are symmetrically placed with respect to the axis. If so, those supports or loads can be considered redundant because they will not affect the unknown DOFs.
B. Calculate the reactions at the supports:
- - Since we have identified redundant supports, we can ignore them for now and focus on the non-redundant supports.
- - Calculate the reactions at these non-redundant supports using the equilibrium equations. These equations are based on the sum of forces and moments being equal to zero.
- - Make sure to consider any known loads or moments applied to the beam.
C. Determine the unknown displacements:
- - With the reactions at the supports known, you can solve for the unknown displacements.
- - To do this, you will need to apply the principles of structural analysis, such as the equations of compatibility and equilibrium.
- - These equations will help you determine the displacements at the supports based on the reactions and geometry of the beam.
D. Analyze the equilibrium of the beam under symmetrical loading:
- - Consider the symmetrical loading on the beam. Determine if the loading is symmetric with respect to the vertical axis.
- - Use the principles of equilibrium to analyze the forces and moments acting on the beam.
- - Check if the sum of forces and moments acting on the beam is equal to zero.