Dynamic and Cyclic Loading

D'Alembert's principle formally reduces a motion problem to equilibrium equations by adding inertia forces to the active forces:

$$\mathbf F_i=-m\mathbf a.$$

For translational motion of a particle:

$$\sum \mathbf F+\mathbf F_i=0.$$

For rotation of a rigid body about a fixed axis, the corresponding inertia moment is:

$$M_i=-J\varepsilon.$$

• $\mathbf F_i$ — inertia force;
• $m$ — mass;
• $\mathbf a$ — acceleration;
• $M_i$ — inertia moment;
• $J$ — mass moment of inertia about the rotation axis;
• $\varepsilon$ — angular acceleration.

Inertia forces are calculation quantities directed opposite to the corresponding accelerations. After introducing them, internal forces and stresses can be determined by methods analogous to static analysis.

For an idealized linearly elastic system without energy losses, when a weight falls from height $h$ and then deforms the structure, the maximum displacement can be expressed through the static displacement $\delta_{st}$ caused by the same weight $P$:

$$\delta_{max}=K_d\delta_{st},$$

where the dynamic factor is:

$$K_d=1+\sqrt{1+\frac{2h}{\delta_{st}}}.$$

If stresses in the linear system are proportional to load:

$$\sigma_{dyn}=K_d\sigma_{st}.$$

• $P$ — weight acting as the reference static force;
• $h$ — drop height;
• $\delta_{st}$ — static displacement under $P$;
• $\delta_{max}$ — maximum displacement during impact;
• $K_d$ — dynamic factor;
• $\sigma_{st}$, $\sigma_{dyn}$ — static and maximum dynamic stresses.

For suddenly applied loading with zero initial velocity, $h=0$ and therefore $K_d=2$. The formula assumes elastic behavior, negligible energy loss, and a correctly determined static compliance in the impact direction.

Because $K_d$ is dimensionless and has no separate quantity in the registry, the calculator uses the equivalent displacement relation without treating $K_d$ as a variable.

A variable normal-stress cycle is defined by its maximum and minimum values $\sigma_{max}$ and $\sigma_{min}$. The main parameters are:

$$\sigma_m=\frac{\sigma_{max}+\sigma_{min}}{2},$$

$$\sigma_a=\frac{\sigma_{max}-\sigma_{min}}{2},$$

$$R=\frac{\sigma_{min}}{\sigma_{max}}.$$

• $\sigma_{max}$ — maximum cycle stress;
• $\sigma_{min}$ — minimum cycle stress;
• $\sigma_m$ — mean stress;
• $\sigma_a$ — stress amplitude;
• $R$ — stress ratio.

For a fully reversed cycle, $R=-1$ and $\sigma_m=0$. For a pulsating cycle from zero to a positive maximum, $R=0$.