Inertia Forces

When a structural member moves with acceleration, its mass produces an inertia effect that must be considered when determining internal forces and stresses. D'Alembert's principle provides a convenient way to formulate such problems.

Translational motion

For a body of mass $m$ with translational acceleration $a$, the magnitude of the inertia force is $ma$ and its direction in the D'Alembert calculation scheme is opposite to the acceleration. If mass is distributed along a member, the inertia loading may also be distributed.

For a bar with constant mass per unit length $m_l$ and uniform acceleration $a$ along its length, the magnitude of the distributed inertia load is:

$$q_i=m_l a.$$

Stress in an accelerating bar

If a straight uniform bar of length $L$ and area $A$ accelerates along its own axis, the axial force at a section must accelerate the portion of mass on one side of that section. Therefore $N$ varies along the length and normal stress is $\sigma=N/A$.

For example, if a bar of density $\rho$ is pulled at one end and the entire bar has axial acceleration $a$, then for a section at distance $x$ from the free end:

$$N(x)=\rho Aax,\qquad \sigma(x)=\rho ax.$$

The largest stress occurs near the end through which the accelerating force is transmitted.

Rotational motion

For angular acceleration $\varepsilon$, an inertia moment $J\varepsilon$ is introduced. During steady rotation with angular velocity $\omega$, material points have centripetal acceleration $a_n=\omega^2r$, so rotating components develop stresses associated with distributed mass forces.

Example: thin ring

For an idealized thin ring of material density $\rho$, radius $r$, and angular velocity $\omega$, the circumferential stress in a simple membrane model is:

$$\sigma_\theta=\rho\omega^2r^2.$$

Thick disks, complex rims, and nonuniform rotating components require more detailed stress models.

Calculation procedure

1. Determine the law of motion and accelerations of the masses.
2. Determine concentrated or distributed inertia forces.
3. Add them to the calculation scheme according to D'Alembert's principle.
4. Determine internal force resultants.
5. Calculate stresses and displacements using the usual strength-of-materials relations within the adopted model.

Limitation

If acceleration changes rapidly, vibration develops, or deformation waves propagate through the member, a quasi-static inertia-force representation may be insufficient. A full dynamic analysis is then required.