11  Exam Questions

Exercise 1

A rectangular concrete beam is subjected to a uniformly distributed load along its length. The stress distribution along the length of the beam shall be analyzed. The beam has a length of 10 meters, a width of 0.3 meters, and a height of 0.5 meters. The uniformly distributed load is 5000 N/m.

  1. Create a NumPy array x with 100 evenly spaced points along the length of the beam from 0 to 10 meters.

  2. Calculate the bending moments \(M(x)\) along the length of the beam using the formula: \[ M(x) = \frac{w \cdot x \cdot (L - x)}{2} \] where \(w\) is the distributed load (in N/m), \(x\) is the position along the beam (in m), and \(L\) is the length of the beam (in m).

  3. Compute the maximum bending stress \(\sigma_{\text{max}}(x)\) at each point along the beam using the formula: \[ \sigma_{\text{max}}(x) = \frac{M(x) \cdot c}{I} \] where \(c\) is the distance from the neutral axis to the outermost fiber of the beam (in m), and \(I\) is the moment of inertia.
    The moment of inertia of a rectangular cross-section is given by: \[ I = \frac{b \cdot h^3}{12} \] where \(b\) is the width (in m) and \(h\) is the height of the beam (in m).

  4. Determine the maximum bending stress.

  5. Plot the stress distribution \(\sigma_{\text{max}}(x)\) along the length of the beam.