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?
Understanding Strain Energy and Castigliano's Theorem
Energy methods work because a load does work while a member changes shape. In a linear elastic structure, that work is stored and can be recovered if the load is removed. The amount stored depends strongly on where the internal forces are largest.
A short region with a large bending moment can contribute more energy than a long region with a small moment. This is why a careful bending moment diagram matters so much.
The diagram is not only used to find stress. It shows which parts of a member control its flexibility.
Castigliano's theorem turns this stored energy into a targeted displacement calculation. First, choose the load whose displacement is required. Express every internal force in terms of that load.
For a beam, this usually means writing the bending moment as a function of position and the chosen load. Substitute that expression into the energy integral, integrate over each section, then differentiate the final energy expression with respect to the chosen load. The result is the displacement in the same direction as that load.
A chosen applied moment gives a rotation rather than a straight line displacement. Sign conventions still matter. A negative result means the actual movement is opposite to the positive direction chosen at the start.
A dummy load is useful when the needed point has no actual force. Place a small imagined force at that point in the direction of the wanted movement. Include it in the reactions and internal force expressions.
Differentiate the total energy with respect to the dummy force. Only after differentiating should the dummy force be set equal to zero. Removing it earlier makes its effect disappear from the calculation.
This method is often called the unit load approach when the imagined force has a value of one, but the energy logic is the same. It is especially helpful for the sideways movement of a point on a frame or the vertical movement of a truss joint.
Students should split members wherever the loading, cross section, material, or internal force equation changes. Each region needs its own integral limits and expression. For example, a point load changes the bending moment equation at its location.
A stepped shaft has different torsional stiffness in each diameter region. Use consistent units throughout. If force is in newtons and length is in millimetres, elastic modulus and section properties must match those units.
Bending deformation is often the main effect in slender beams, while axial deformation can matter in trusses, torsion in drive shafts, and shear in short deep beams. Adding every effect without checking its size can waste time. Leaving out an important effect can give a misleading deflection.
These calculations appear in real design whenever excessive movement causes trouble before a part breaks. A floor beam may be strong enough yet feel bouncy. A machine shaft may survive the torque yet twist enough to misalign gears.
A bridge truss may carry its loads safely while one joint moves enough to affect connected components. Castigliano's theorem is valuable because it focuses on one movement that matters.
When learning it, draw the free body diagram first, label the coordinate direction, state the positive displacement direction, and check whether the final units reduce to length or radians. Those checks catch many errors before the result is used.