Cantilever Deflection Under "Sudden" loading...
Cantilever Deflection Under "Sudden" loading...
(OP)
There’s an equation from a text that is giving me something of a fit…
A vessel strikes the tip of a cantilevered finger pier at some oblique angle.
1. The initial condition finds a mass (m) moving along at a constant velocity (v) resulting in a kinetic energy (KE = ½ mv^2); while the cantilever is as yet undeflected, resulting in a stored energy (strain energy, potential energy, spring energy, etc.) equal to zero.
2. Upon contact of the moving load with the cantilever tip, the cantilever deflects until the kinetic energy of the mass is dissipated, leaving the system (briefly) at rest; the deflection having (presumably) converted the kinetic energy into stored energy in the cantilever (i.e. conservation).
Can anyone produce the derivation for the following relationship: Δ = (KE/k)^1/2 ...?
The absorbed energy side of the equation appears to be k x Δ^2, but this just does not look right for the strain energy of a cantilever...
Thanks,
walterbrennan
A vessel strikes the tip of a cantilevered finger pier at some oblique angle.
1. The initial condition finds a mass (m) moving along at a constant velocity (v) resulting in a kinetic energy (KE = ½ mv^2); while the cantilever is as yet undeflected, resulting in a stored energy (strain energy, potential energy, spring energy, etc.) equal to zero.
2. Upon contact of the moving load with the cantilever tip, the cantilever deflects until the kinetic energy of the mass is dissipated, leaving the system (briefly) at rest; the deflection having (presumably) converted the kinetic energy into stored energy in the cantilever (i.e. conservation).
Can anyone produce the derivation for the following relationship: Δ = (KE/k)^1/2 ...?
The absorbed energy side of the equation appears to be k x Δ^2, but this just does not look right for the strain energy of a cantilever...
Thanks,
walterbrennan






RE: Cantilever Deflection Under "Sudden" loading...
Cheers
Greg Locock
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RE: Cantilever Deflection Under "Sudden" loading...
Strange stuff...
RE: Cantilever Deflection Under "Sudden" loading...
I found stuff online that gets close to the requested derivation, but has that pesky extra 2 in it.
Strain Energy in a cantilever due to bending (U) = (P^2 * L^3)/(6EI) Page 6
k = (3EI)/L^3 Link
Δ = (PL^3)/(3EI)
U = (P^2)/2k [plugging k into the U formula]
Δ = P/k [plugging k into the Δ formula] ---> P = k*Δ
U = (k^2)(Δ^2)/2k = k * (Δ^2)/2 ----> Δ = (2U/k)^.5
I don't suppose by chance that the book you referenced was looking at two posts? :)
Are there any other energies being neglected? Are we allowed to assume that all the energy is being conserved and this is a perfectly elastic collision? Or is there a typo in the book?
RE: Cantilever Deflection Under "Sudden" loading...
No, unfortunately the author was not considering two posts in the design example... it's a cantilevered finger pier. And he isn't considering other energy losses.
The publication is Marinas and Small Craft Harbors,first edition, by Tobiasson; and the Δ = (KE/k)^1/2 reference is found on page 375 about halfway down the page.
At first I though he might be accounting for the presence of two stringer beams; but a unit load deflection calculation earlier in the example demonstrates that he is checking the finger as a whole; considering the stringer beams as a composite cross-section, for the purposes of resisting the entire transverse load.
If anyone has a second edition of Tobiasson, with the same example problem, I would be interested in knowing if the same Δ = (KE/k)^1/2 relation is indicated there.
Thanks,
PBW
RE: Cantilever Deflection Under "Sudden" loading...
TTFN

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RE: Cantilever Deflection Under "Sudden" loading...
The real "boat hits pier" problem has the boat impacting horizontally into a long pier regularly supported (at 10-15 foot intervals) by flexible piers mounted into mud and dirt supporting a horizonal "board" that is trying to flex against the strong side of the member.
RE: Cantilever Deflection Under "Sudden" loading...
Mike McCann, PE, SE
RE: Cantilever Deflection Under "Sudden" loading...
One comparison is the ASCE7 loads for car impact into a bollard. The forces are way lower than you would calculate for an elastic cantilever.
ps: I am not a pier expert!
RE: Cantilever Deflection Under "Sudden" loading...
Cheers
Greg Locock
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RE: Cantilever Deflection Under "Sudden" loading...
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin
RE: Cantilever Deflection Under "Sudden" loading...
On further investigation, I noted that Mr. Tobiasson’s own work referenced an older book, Design and Consruction of Ports and Marine Structures (Alonzo DeF. Quinn, 2nd Ed. 1972).
According to Mr. Quinn, the rule of thumb is as follows:
“The energy to be absorbed by the fender system and dock is usually taken to be 1/2E, as the remaining one-half is assumed to be absorbed by the ship and the water, because of the rotation of the center of mass of the ship around the point of contact of the bow with the fender, which is assumed to be at the one-fourth point of the length of the ship.”
Mr. Quinn goes on to elucidate that, for berthing points other than the one-quarter-point, the energy imparted to the fender/dock is necessarily higher than the fifty percent presumed above. For a direct strike in the line of the ship’s travel, for instance, or a side-strike amidships (either of which would orient the reaction vector toward the center of the ship mass), the full energy would be presumed to be imparted to the fender/dock.
Mr. Tobiasson’s example problem appears to invoke the 1/2E rule on the basis that the his model boat, while striking directly in the line of the boat’s travel, is impacting the end of a cantilevered finger pier at an oblique angle. In essence, he appears to presume that some sidesway/rotation of the boat will follow the impact, as the finger deflects laterally; gobbling up some of the energy, along the way. Right, wrong or indifferent, it appears that Mr. Tobiasson was, in fact at least employing a known rationale when he trimmed his design energy in half.
Being in the realm of what I would consider to be a learning exercise (or, at least, an historical study
I've attached excerpts from both references to this post.
Thanks,
PBW