# Shock loading stresses due to falling: what strength to use for acceptance criteria?

We are evaluating a design in which a fall-arresting safeguard would be subject to a shock load in the event of a catastrophic failure of a different component, which is required to meet compliance with our design standard. We believe that the equation below (found in Machinery's Handbook, p.279 of the 29th Ed.) is appropriate for approximating the stresses due to this type of loading:

$$p = p_s\left(1+\sqrt{1+\frac{2h}{y}}\right)$$

Where:

• $$p$$ = stress (psi) due to shock caused by impact of a moving load
• $$p_s$$ = stress (psi) resulting when moving load is applied statically
• $$h$$ = distance (in) that load falls before striking member
• $$y$$ = deflection (in) resulting from static load

Our problem is that in sample calculations of real-world scenarios, we are finding that the resultant stress is far higher than the ultimate tensile strength of the material. My intuition is telling me that the transient nature of the loading means that tensile strength alone is not an appropriate constraint but I'm unable to find any literature to back up that hypothesis. Additionally, our design standards (CMAA 70 and 74) don't seem to indicate any special allowed stress specifically for shock loading.

To be clear, since this is a safe-guard in case of catastrophic failure, we are not concerned with investigating fatigue rupture. If this safeguard is ever activated once, the product is rendered unable to function and replacement of the entire assembly is necessary.

So my question is, what strength is appropriate to use (if not ultimate tensile) as a measure of acceptable dynamic stresses due to shock load as a result of falling and what reputable sources are there to cite this information? Otherwise, are we incorrect to be using the above formula to estimate stresses due to shock load as a result of falling?

Here is a sample calculation:

Inputs:

• $$p_s = \frac{Wl}{Z} = \frac{1102.5lbf\times1in}{\frac{4in\times(0.375in)^{2}}{6}} = 11760 psi$$
• $$h = 0.5 in$$
• $$y = \frac{Wl^3}{3EI} = \frac{1102.5lbf\times(1in)^3}{3\times(2.90\times10^7psi)\times\frac{4in\times(0.375in)^{3}}{12}} \approx 7.21x10^{-4} in$$

Result: $$p \approx 4.50x10^5 psi$$. Without any special factor on strength, comparing this to $$UTS=65000psi$$ for the steel we are using, this would lead one to believe that this shock would cause this safe-guard to fail (to put it lightly).

• Are the numbers correct? ps is awfully high and y is really low. Is the mechanism actually so stiff that it doesnt defelct by 10^-4 in under the static pressure already? – mart Sep 9 '20 at 15:01
• I'm afraid the numbers appear to be correct for this sample case. I guess it's a matter of perspective, but that static stress doesn't seem unusually high for me for your average steel. Keep in mind that this specific member's sole purpose is to be very stiff as it's only function is to engage in the event of a failure to prevent dangerous load drop. I'll try and whip up a diagram showing the member and loading if you want to verify. – Luke Sep 9 '20 at 15:11
• That would be good and might help to answer the question. – mart Sep 9 '20 at 15:15
• @mart Image is added, if it's difficult to read please see full res here: i.imgur.com/ZDIziJ9.png – Luke Sep 9 '20 at 15:26
• w is the load that causes this static stress? where does the value for y given in the question come from? – mart Sep 9 '20 at 15:39

Your static deflection is way too low.

let's consider a 1-inch length of your angle for a basic check. Assuming steel E=29000ksi

$$I=1*0.38^3/12=0.00457 in^4$$ $$\delta = \frac{wL^3}{(3EI)} = \frac{11760*1^3}{3EI}=0.0295 in$$

Therefore

$$p = p_s\left(1+\sqrt{1+\frac{2h}{y}}\right) =11760(1+\sqrt{1+\frac{2*0.5}{0.0295}}=11760(1+6.899)=92786 psi \ \text{no good}$$

Let's look at it from the energy angle.

The stiffness of each one-inch length of the angle is $$K=[\frac{Ebh^3}{4L^3}]=\frac{29000000*1*0.38^3}{4*1}=397.822 ksi$$

$$mgh_ \text{potential e}= 1/2 Kx^2_\text{strain e} \rightarrow 11760*9.8*0.5=1/2*397822*x^2 \\x_{deflection}^2=\frac{11760*9.8}{397822}=0.29in\ \text{no good, large angle}$$

The angle is not okay.