We will obtain bending moment diagrams for three beams and calculate stresses inside the beams. For each beam, take the -axis along the neutral axis of the beam, oriented from left to right, with origin at .
To obtain the bending moment diagrams for the three beams and calculate stresses inside the beams, you will need to follow these steps:
Step 1: Determine the type of loading and supports for each beam.
- Identify the forces and moments acting on each beam.
- Determine the supports and their types (e.g., simply supported, fixed, cantilevered, etc.).
Step 2: Draw the beam and establish coordinate axes.
- Sketch each beam and label the dimensions (length, height, width, etc.) as well as supports and loads.
- Establish the coordinate axes with the x-axis along the neutral axis of the beam, oriented from left to right, and select an appropriate origin.
Step 3: Analyze the equilibrium equations.
- Apply the equilibrium equations (sum of forces and sum of moments) to each segment of the beam.
- Consider both vertical and horizontal equilibrium conditions.
Step 4: Calculate the reactions at the supports.
- Determine the reactions at the supports using the equilibrium calculations.
- Take into account both vertical and horizontal forces.
Step 5: Integrate the bending moment equation.
- Use the equilibrium equations and the loading conditions to determine the bending moment equation along the length of the beam.
- Integrate the equation to obtain the bending moment diagram.
Step 6: Determine the maximum and minimum bending moments.
- Determine the peak values of the bending moment from the moment diagram.
- Identify locations where the bending moments change signs.
Step 7: Calculate the stresses inside the beam.
- Apply the stress formula (Stress = Moment × Distance / Section Modulus) at critical locations along the beam.
- Calculate the stresses using the maximum and minimum bending moments and the appropriate distances from the neutral axis.
- Use the section modulus, which depends on the beam's geometry, to calculate the stress.
Step 8: Plot the bending moment diagrams.
- Plot the bending moment diagrams for each beam according to the results obtained in the previous steps.
- Label the maximum and minimum bending moments and their respective locations.
By following these steps, you will be able to obtain the bending moment diagrams for the three beams and calculate the stresses inside the beams along the neutral axis.