Energy Methods

For a linearly elastic system loaded statically from zero to the final force values, the work of external forces equals the stored strain energy:

$$A=U=\frac{1}{2}\sum_i P_i\delta_i+\frac{1}{2}\sum_j M_j\varphi_j.$$

For one generalized force $P$ and its corresponding displacement $\delta$:

$$U=\frac{1}{2}P\delta.$$

• $A$ — work of external forces;
• $U$ — strain energy;
• $P_i$ — generalized force;
• $\delta_i$ — corresponding linear displacement;
• $M_j$ — generalized moment;
• $\varphi_j$ — corresponding angular displacement.

The factor $1/2$ appears because, in a linear system, the force and its corresponding displacement increase proportionally from zero to their final values.

For a linearly elastic member, strain energy can be calculated by integrating contributions from internal force resultants along its length:

$$U_N=\int_0^L\frac{N^2(x)}{2EA}\,dx,$$

$$U_M=\int_0^L\frac{M^2(x)}{2EI}\,dx,$$

$$U_T=\int_0^L\frac{T^2(x)}{2GJ_p}\,dx.$$

If shear deformation is included, its contribution is written consistently with the adopted beam model, for example:

$$U_Q=\int_0^L\frac{Q^2(x)}{2\kappa GA}\,dx,$$

• $U_N$, $U_M$, $U_T$, $U_Q$ — strain-energy contributions;
• $N(x)$ — axial force;
• $M(x)$ — bending moment;
• $T(x)$ — torque;
• $Q(x)$ — shear force;
• $E$, $G$ — Young's and shear moduli;
• $A$ — cross-sectional area;
• $I$ — second moment of area;
• $J_p$ — polar second moment of area;
• $\kappa$ — shear correction factor.

In a linear system, the total strain energy is the sum of the applicable independent quadratic contributions.

For a linearly elastic system, the work of one set of forces through the displacements caused by another set equals the reciprocal work:

$$\sum_i P_i^{(1)}\delta_i^{(2)}=\sum_i P_i^{(2)}\delta_i^{(1)}.$$

As a special case for two unit forces, Maxwell's reciprocal-displacement theorem gives:

$$\delta_{ij}=\delta_{ji}.$$

• $P_i^{(1)}$, $P_i^{(2)}$ — generalized forces in loading states 1 and 2;
• $\delta_i^{(1)}$, $\delta_i^{(2)}$ — corresponding displacements;
• $\delta_{ij}$ — displacement at coordinate $i$ caused by a unit force at coordinate $j$.

Thus the displacement at point $i$ in the direction of force $i$ caused by a unit force at point $j$ equals the corresponding displacement at point $j$ caused by a unit force at point $i$, provided the system is linearly elastic and reciprocity conditions are satisfied.

For a linear-elastic system, the displacement in the direction of a generalized force is obtained by differentiating the strain energy with respect to that force:

$$\delta_i=\frac{\partial U}{\partial P_i}.$$

Similarly, the rotation at a generalized applied moment is:

$$\varphi_i=\frac{\partial U}{\partial M_i}.$$

• $U$ — strain energy;
• $P_i$ — generalized force;
• $\delta_i$ — corresponding linear displacement;
• $M_i$ — generalized moment;
• $\varphi_i$ — corresponding rotation.

If the required load is absent from the actual system, an auxiliary force or moment may be introduced in the required direction, $U$ written as a function of that parameter, differentiated, and the auxiliary parameter then set to zero.

The displacement at a specified point and direction can be determined by the unit-load method. For bending of a beam or frame:

$$\delta=\int_0^L\frac{M(x)\,\bar M(x)}{EI}\,dx,$$

where $M(x)$ is the bending moment from the real loading and $\bar M(x)$ is the moment from a unit force applied at the point and in the direction of the required linear displacement.

For a required rotation, apply a unit moment.

For a general member, applicable contributions may be summed:

$$\delta=\int\frac{N\bar N}{EA}\,dx+\int\frac{M\bar M}{EI}\,dx+\int\frac{T\bar T}{GJ_p}\,dx+\int\frac{Q\bar Q}{\kappa GA}\,dx,$$

• $N$, $M$, $T$, $Q$ — internal force resultants from the real loading;
• $\bar N$, $\bar M$, $\bar T$, $\bar Q$ — corresponding resultants from the unit-load state;
• $E$, $G$, $A$, $I$, $J_p$, $\kappa$ — stiffness and section parameters.

The sign of $\delta$ indicates whether the actual displacement agrees with the direction of the introduced unit load.