Strain energy is the elastic energy stored in a structure when loads deform it. Engineers use it to find deflections, rotations, and load sharing in beams, frames, trusses, and shafts. This cheat sheet helps students connect internal force diagrams to energy integrals and displacement results.
It is especially useful when direct deflection methods are long or when a displacement is needed at one specific point.
Key Facts
- For a linearly elastic member under axial force, strain energy is U = integral of N^2/(2EA) dx.
- For a beam in bending, strain energy is U = integral of M^2/(2EI) dx.
- For a circular shaft in torsion, strain energy is U = integral of T^2/(2GJ) dx.
- For transverse shear deformation, strain energy is commonly written U = integral of V^2/(2 k G A) dx, where k is the shear correction factor.
- Castigliano's second theorem states that the displacement in the direction of load P is delta = partial derivative of U with respect to P.
- The rotation in the direction of an applied moment M0 is theta = partial derivative of U with respect to M0.
- If no real load acts at the point where displacement is needed, add a dummy load Q, compute delta = partial derivative of U with respect to Q, then set Q = 0.
- For linear elastic structures, total strain energy can be found by adding the energy from axial, bending, torsion, and shear effects as U_total = U_axial + U_bending + U_torsion + U_shear.
Vocabulary
- Strain energy
- The recoverable elastic energy stored in a body due to deformation under load.
- Castigliano's theorem
- An energy method stating that the derivative of strain energy with respect to a load gives the displacement in that load's direction.
- Dummy load
- An artificial force or moment added at a point so Castigliano's theorem can be used to find a desired displacement or rotation.
- Flexural rigidity
- The product EI, which measures a beam's resistance to bending deformation.
- Torsional rigidity
- The product GJ, which measures a shaft's resistance to twisting deformation.
- Complementary energy
- An energy quantity equal to strain energy for linear elastic behavior and useful in force based energy methods.
Common Mistakes to Avoid
- Using external load instead of internal force in the energy integral is wrong because U depends on N, V, M, or T inside each member, not just the support reaction or applied load.
- Forgetting to split the member where the internal force expression changes is wrong because a single M(x) or N(x) expression may not apply across loads, supports, or hinges.
- Setting the dummy load to zero before differentiating is wrong because the derivative must be taken while the dummy load is still present in the internal force expressions.
- Ignoring the direction of the load or moment is wrong because Castigliano's result is positive only in the assumed direction of that generalized force.
- Mixing units such as N, mm, MPa, and m without conversion is wrong because strain energy and its derivatives require consistent units to produce correct displacement or rotation.
Practice Questions
- 1 A prismatic steel bar has length 2.0 m, area 400 mm^2, modulus E = 200 GPa, and axial load P = 50 kN. Find the axial strain energy U = P^2 L/(2EA).
- 2 A cantilever beam of length 3.0 m has a tip load P = 4 kN and constant EI = 18 MN m^2. Using U = integral of M^2/(2EI) dx, find the vertical tip deflection from Castigliano's theorem.
- 3 A solid circular shaft has torque T = 800 N m, length L = 1.5 m, shear modulus G = 80 GPa, and polar moment J = 2.5 x 10^-6 m^4. Find the angle of twist using theta = T L/(GJ).
- 4 When using Castigliano's theorem, why must a dummy load be added at the exact point and in the exact direction of the desired displacement?