Work-Energy Method 1

[Mrennui]

I will try and keep this brief

I'm trying to calculate the force on impact of a tile falling onto a glass roof using the work-energy method. I understand that there many many unknown variables and I have expalined this to the client and he requested simplified results anyhow.

I know the kinetic energy at impact 1/2mv². Then using work-energy the average force can be calculated from F=1/2mv²/d (later d = v). Where d is the displacement or deflection of the glass.

I then supposed that the maximum displacemt could be found using the engineers bending equation (1) M/I = E/R = -s/y

Where s is the stress, in this case the limiting stress(the example attached is for a steel beam). I supposed using max stress would give max deflection.

Knowning that (2) 1/R = d²v/dx² where v is the vertical displacement (page 324 Megson, I know this may not be strictly valid as glass can be subeject large deflections, but ignoring this)

Putting (2) into (1) Ey d²v/dx² = s

Integrating twice (I have attached my example)

Ey v = sx²/2 + s L/2 x

which gives a deflection of 41.9mm

I checked this supposing Mmax = smax Z

Then Mmax = WL/4 giving Wmax = 4Mmax/L

Finally v = WmaxL³/48EI

But this gives v as -27.9mm

I freely admit I'm not the best engineer or mathematician, can someone explain to me what I'm doing wrong? Is it flawed methodology, have I just got the number wrong? In the sheets attached I was troubled that v came out as positive. Any idea's are very much appreicated.

 

[rb1957]

i haven't gone over the entire story in detail, but i'd delete the -ve sign in equation (1) ... s = My/I.

also a -ve displacement probably means displacement in the -ve direction (ie down), which would make sense.

but 1" displacement sounds large, maybe it's a very heavy tile, maybe its a very big panel.

the typical problem with calculating the force due to an impact is determining the time interval that the force acts over ... KE = Impulse = F*t ... short times make for large forces !

 

[Mrennui]

Thanks rb1957

I had investigated calculating the using F = mass x acceleration. But I considered estimating the time of deceleration too inaccurate. That is why I decided to use the work energy-method. I thought I would be able to calcualte max deflection based on max allowable stress. And then put this value into the work-energy equation to obtain the mean force.

But, as stated, the two answers are inconsistent. But the equation used to check deflection (WL³/48EI) is also derived from the Engineers bending equation using a similar (parallel) double integration. I therefore think the two results should be consistent. Hmmm...

I don't have time to do more work now, but I will give it another go tonight.

 

[Mrennui]

Although I have just thought about the -ve sign, you are correct. I have been inconsistent with my force system the forces acts in a negative direction, therefore

v = - (-W)L/48EI

which means the sign convention issue is resolved.

 

[rb1957]

that's good. but somewhere in your analysis, by proxy, you have assumed a time interval. it'd be something like the change in velocity of the tile as it deforms the panel ... constant deceleration, at t = 0, x = 0, v = [your KE result]; t = t [the time of the impact], x = 27mm, v = 0.

now the important question is what happens to the panel of glass ?

and what if the tile falls on it's edge, concentrating the impact ??

 

[Mrennui]

You may be right about the time, I haven't worked it through. But I don't need to know a time for the equations for work-energy just the energy and the distance.

I did explain to the client that the smaller the contact area the higher the stress, he was happy with a simplified approach using a given area.