Impact Loading (Falling Mass, Energy Method)

dynamicsstress

Catch a dropped weight on the end of a bar and the bar feels far more than the weight itself. The falling mass has to be brought to rest, and all of its kinetic energy has to go somewhere — into the member, as strain energy. The faster it is moving when it lands, the more energy the member must swallow, and the higher the stress climbs above the quiet static value $W/A$. This page is the classic energy method for that: no stopwatch, no equation of motion, just a before-and-after energy balance that turns a drop height into a stress.

One number does all the work: the impact factor

Lay the weight $W = mg$ on gently and the member deflects by its static deflection $\delta_{st}$. Now instead drop it from height $h$. It falls that $h$ plus the extra $\delta_{impact}$ the member gives way, so it loses potential energy $W(h + \delta_{impact})$; the member, behaving as a linear spring of stiffness $k = W/\delta_{st}$, stores $\tfrac{1}{2} k\,\delta_{impact}^2$. Setting the two equal and writing $\delta_{impact} = n\,\delta_{st}$ collapses everything to a single quadratic in the dimensionless impact factor $n$:

$n^2 - 2n - \frac{2h}{\delta_{st}} = 0 \quad\Longrightarrow\quad n = 1 + \sqrt{1 + \frac{2h}{\delta_{st}}}$

Then the peak deflection and the peak stress are both just $n$ times their static values: $\delta_{impact} = n\,\delta_{st}$ and $\sigma_{impact} = n\,\sigma_{st}$. Two limits are worth committing to memory:

• A suddenly applied load already doubles the stress. Set $h = 0$ — the load is not dropped, just released onto the member with no gap — and the formula gives $n = 2$ exactly. Placing a load suddenly is twice as severe as easing it on. Drag the drop height to zero in the widget and watch $n$ settle on 2.
• A real drop multiplies it many times over. For $h \gg \delta_{st}$, $n \approx \sqrt{2h/\delta_{st}}$, which for a stiff member with a tiny static deflection can be in the hundreds.

Stiffer takes it harder — the counter-intuitive part

Look where the material enters. The static stress $\sigma_{st} = W/A$ for the rod is material-blind — it does not know $E$. But the impact factor does: $\delta_{st} = WL/EA$ shrinks as $E$ grows, so a stiffer member has a smaller $\delta_{st}$ and therefore a larger $n$. Multiply a fixed static stress by a bigger factor and the impact stress goes up. Swap the default aluminium for steel and watch $\sigma_{impact}$ climb even though $\sigma_{st}$ never moves. This is the exact opposite of the static intuition, and it is why a rigid mounting can be worse than a compliant one — the same reason a helical spring is put under a falling load to absorb the energy over a long, soft stroke. The lesson the Euler column keeps making — stiffness is not strength — returns here with the sign flipped: under impact, stiffness actively hurts.

Rod versus cantilever: the same drop, wildly different answers

The configuration selector switches the struck member, and only the static response changes — the impact factor formula is identical for both:

• Axial rod ($\sigma_{st} = W/A$, $\delta_{st} = WL/EA$). An axial member barely deflects, so $\delta_{st}$ is minute and $n$ is enormous — often 100 or more. Yet the rod usually survives, because its static stress is so low that even a hundredfold amplification stays modest. Impact factor alone is a trap: what fails a member is $\sigma_{impact} = n\,\sigma_{st}$, not $n$.
• Cantilever tip strike ($\sigma_{st} = Mc/I$ with $M = WL$, $\delta_{st} = WL^3/3EI$, reusing the cantilever beam verbatim). Bending is far more compliant, so $\delta_{st}$ is large and $n$ is modest — but the static bending stress is already high, so the product is what bites. Switch to steel here and the cantilever crosses yield while the rod is still comfortable.

Volume, not section modulus, is the impact capacity

Push the drop height up and the axial impact stress approaches a clean limit:

$\sigma_{impact} \;\to\; \sqrt{\frac{2\,m g h\,E}{V}}$

with $V = A L$ the member’s volume. The whole volume shares the strain energy — every cubic millimetre stores $\sigma^2/2E$ — so what governs how much drop energy a member can take is its volume, not its section modulus. A short stubby bar and a long slender one of the same volume absorb the same energy at the same peak stress. To make something impact-tough you give it material to strain, which is why energy-absorbing parts are bulky, not just strong. This is the inverse of the flywheel problem, where the trick is to store kinetic energy rather than survive a delivery of it.

g is a defined constant, not a knob

The weight $W = mg$ pulls in the standard acceleration of gravity, 9.80665 m/s² — a value fixed by definition (3rd CGPM, 1901), not something to dial in. As on the critical-speed page, it appears here as a labeled, cited constant with a source, never a slider; the knob is the mass $m$, and the weight is computed from it. This is the second page to draw on that constants mechanism.

Reading the widget

The readouts give the static and impact deflections and stresses, the impact factor $n$, and the safety factor against yield. Two warnings guard the model’s assumptions. Increase the section or pick a dense material and the member-mass warning fires once the struck member’s own inertia stops being negligible beside the falling mass — the energy balance quietly assumes a massless member. Push the drop height or the stiffness far enough and $\sigma_{impact}$ reaches yield: past that point the linear-elastic energy balance is void, and the number is a failure, not a margin.

Try it

Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

Impact failures are the ones that surprise people, because the load that causes them looks harmless. A part sized comfortably for a weight sitting on it can shatter when the same weight is dropped a few centimetres onto it — the impact factor turned a safe static stress into one well past ultimate. The effect is worst for brittle materials (cast iron, hardened steel, ceramics, cold glass), which have almost no capacity to absorb energy plastically: they take the full elastic spike and crack. It is also why the same drop that dents a ductile bracket fractures a hard one, and why toughness — energy absorbed before fracture — is a different property from strength.

The honest caveat: the energy method can under-predict

It is tempting to read $\sigma_{impact} = n\,\sigma_{st}$ as a safe worst case. It is not always. The energy balance is a lumped, quasi-static model: it assumes the member deflects in its ordinary static shape, the whole thing responding at once, and it distributes the strain energy over the entire volume. Two real effects it cannot see can push the local peak stress above what it predicts:

• Stress waves. A sharp impact does not load the member all at once. A compression wave travels from the point of contact at the speed of sound in the material, and until it reflects and settles, the material near the strike sees a stress governed by the wave — for a bar struck axially, on the order of $\sigma \sim \rho\,c\,v$ (density × wave speed × impact velocity), independent of the length the lumped model leans on. For very stiff, short, or hard-struck members the wave peak can exceed the energy-balance value before the member ever reaches its static-shape deflection.
• Contact stresses. Right at the point of contact the two bodies press through a tiny area, and the local Hertzian contact pressure can be enormous and completely local — the source of impact brinelling, spalling, and surface cracking that a whole-member stress number knows nothing about.

Juvinall and Marshek carry this caution explicitly: the energy method is a good design estimate for members that are long and compliant compared with the contact, and it grows optimistic as the member gets stiffer and the contact harder. This widget models only the lumped energy balance; stress-wave propagation, Hertzian contact, and strain-rate effects on the material properties are named here and never computed — treat the number as an estimate, not a proof of safety.

What else the balance leaves out

The same assumptions that make the algebra clean are each a real member’s escape or trap:

• A massless member. All the drop energy is assumed to go into strain, none into accelerating the member itself. When the struck member’s own mass is not negligible beside the falling mass, some of the energy goes into moving it, and the true impact factor is lower than shown — the widget warns when this crosses 10%.
• No losses and no rebound. Real impacts lose energy to sound, heat, local plasticity, and a bounce. Every one of those reduces the energy the member has to store, so on this axis the model is conservative — it hands the member the entire drop energy.
• A single event. Repeated impacts are a fatigue problem, not a static-yield one; a member that survives one blow at $\sigma_{impact}$ can still fail after thousands. This page models one clean strike against first yield only.

So the number here is neither a guaranteed ceiling nor a guaranteed floor: it over-counts by ignoring losses and under-counts by ignoring waves and contact. It is a teaching estimate of the right order, and the real lesson is qualitative and robust — suddenly-applied is already 2×, a drop is many times more, and the way to survive it is volume of material to strain, not raw strength. That is exactly why a helical spring under a falling load is the classic energy-absorber (a long soft stroke stores the same energy at a fraction of the stress), and why the compliant mounting the static intuition of the Euler column would never suggest is often the safe one.

Sources

• Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §2.8 (impact loading: the falling-collar energy balance, the impact factor, and the suddenly-applied-load n = 2 result), §9.8 (strain energy of bending), §9.10 (deflections produced by impact); the end-loaded cantilever tip deflection WL³/3EI is Ch. 9 material (worked examples / Appendix H).
• Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — impact and energy loads (ch. 7): the same impact-factor result, the strain-energy-density design view (energy absorbed per unit volume, σ²/2E), and the caution that the energy method is unconservative against stress-wave and contact effects.
• BIPM, The International System of Units (SI Brochure), 9th ed., 2019, and NIST Special Publication 330 (2019): the standard acceleration of gravity gₙ = 9.80665 m/s², a conventional value fixed BY DEFINITION (3rd CGPM, 1901) — not a measurement.