Learning topic
Impact Loading Calculations
Calculate impact response with energy balance and the dynamic factor Kd: falling weights, suddenly applied loads, maximum stress, deflection, and model limits.
Impact loading occurs when a body with nonzero velocity contacts a structure and transfers energy to it over a short time. Maximum forces, stresses, and displacements during impact may be several times larger than the static values produced by the same weight.
Energy model
In the simplest elastic model, the loss of potential energy of a falling weight is converted into strain energy of the structure. If a weight $P$ falls through height $h$ and the maximum additional deformation after contact is $\delta$, the work of the weight through the distance $h+\delta$ is equated to the maximum elastic strain energy.
For a linear system whose static displacement under $P$ is $\delta_{st}$:
$$P(h+\delta)=\frac{1}{2}\frac{P}{\delta_{st}}\delta^2.$$
Solving this quadratic relation gives the dynamic factor $K_d$.
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.
Static displacement
$\delta_{st}$ is the displacement of the impact point in the load direction that would be produced by statically applying force $P$. It is calculated by ordinary strength-of-materials methods for an axial bar, beam, spring, or another linearly elastic system.
Suddenly applied load
If force $P$ is applied instantaneously without a drop height and without initial velocity, the idealized model has $h=0$ and $K_d=2$. Thus the maximum elastic displacement and stress are twice their corresponding static values.
Example
Suppose a weight falls through $h=20\ \text{mm}$ and the static displacement under its weight is $\delta_{st}=0.5\ \text{mm}$. Then:
$$K_d=1+\sqrt{1+\frac{2\cdot20}{0.5}}=1+\sqrt{81}=10.$$
If the static stress from the weight is $\sigma_{st}=25\ \text{MPa}$ and the system remains linear, the estimated maximum impact stress is $\sigma_{dyn}=250\ \text{MPa}$.
Horizontal impact and specified velocity
If the initial energy is specified as kinetic energy $mv^2/2$ rather than by a drop height, write the energy balance directly using that kinetic energy. The resulting response depends on impactor mass $m$, velocity $v$, structural compliance, and the adopted contact model.
Calculation procedure
- Determine the weight or initial kinetic energy of the impactor.
- Calculate the static displacement $\delta_{st}$ of the impact point under the corresponding static force.
- Write the energy balance for the adopted model.
- Determine the maximum displacement $\delta_{max}$ or dynamic factor $K_d$.
- Use linear proportionality to determine maximum internal forces and stresses.
- Verify that the stresses remain within the range where the elastic model is acceptable.
Limitations
Real impacts may involve plastic deformation, local contact crushing, rebound, damping, and stress-wave propagation. In such cases, the simple energy formula may overestimate or underestimate the actual maximum response and a more detailed dynamic model is required.
About this topic
Under impact loading, kinetic energy from falling masses transforms instantaneously into strain energy. This section presents procedures for calculating dynamic impact factors under vertical and horizontal collisions, alongside dynamic stress and deflection evaluation formulas.