Work and Strain Energy

Work and strain energy provide the energy basis for many structural-analysis methods. In an elastic structure, work done by external actions is stored as strain energy in the members.

Work of a gradually applied force

If a force $P$ increases from zero to its final value and the structure is linear elastic, with final displacement $\delta$ in the force direction,

$$A=\frac12 P\delta.$$

The factor $1/2$ appears because force and displacement increase proportionally during gradual loading.

Member strain energy

For an elastic member, strain energy can be expressed through internal force effects. Typical contributions are

$$U_N=\int\frac{N^2}{2EA}\,dx,\qquad U_M=\int\frac{M^2}{2EI}\,dx.$$

Torsional and shear strain-energy terms can be added when they are included in the structural model.

Energy balance

For quasistatic elastic loading without energy loss, external work equals the stored strain energy. This provides a way to relate loads and displacements without directly integrating the differential equation of the elastic curve.

Virtual and reciprocal work

For linear-elastic systems, reciprocal-work relations provide the basis of the unit-load method: a real internal-force state is combined with an auxiliary unit state, and products of corresponding internal forces are integrated along the members.

Applications

Energy relations are used to calculate deflections and rotations, coefficients of force-method compatibility equations, and independent checks of structural calculations.