2.5. Total energy stored in the beam

Try these first

1. 1.

   Define Castigliano’s theorem.

   Hint

   Think about displacement in relation to energy.

2. 2.

   What is axial energy in beams?

   Hint

   Consider how force affects length.

3. 3.

   What principle does Castigliano’s theorem describe?

   • Displacement in structures
   • Stress concentration
   • Fatigue analysis

   Hint

   Consider how external loads relate to internal deformations.

4. 4.

   True or False: Shear energy only pertains to loading perpendicular to the longitudinal axis.

   • True
   • False

   Hint

   Think about stress direction relative to beam length.

5. 5.

   A simply supported beam with a span of 5m is subjected to a central load of 30 kN. Calculate the total energy stored in the beam if the bending moment at the center is known to be 75 Nm, with E = 210 GPa and I = 0.0001 m^4.

   Hint

   Apply the bending energy formula carefully.

6. 6.

   If a beam is fixed at one end with a torsional force applied, find the energy stored due to torsion and shear combined, given T = 100 Nm, L = 4 m, E = 200 GPa, and G = 80 GPa.

   Hint

   Remember the formulas for each energy type mentioned!