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Re: Foundations for a 3-pinned arch stadium

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In a rigid body world, with ideally rigid foundations, you do not have
instability until the pin approaches the level of the supports as you
suggest. However, in a non-linear elastic/plastic world with flexible
foundations, each side of the span takes on that funky deflection curve and
the stability of the arch is diminished in a way that doesn't occur with
straight members.

In your scenario, you are probably thinking of straight lines from pin to
base on each side to create a stable force triangle. More realistically,
consider if those lines were drawn from pin to point of Mmax and then to
base. This is potentially a 5 pin mechanism with asymmetric response due to
real imperfections in materials, load and geometry (This asymmetry will
probably NOT be considered by the analysis software, even using second-order

The point is that an instability can occur well before the mid-span pin
approaches the level of the supports.

Additionally, the stability is reduced as the supports spread apart,
causing the pin to move closer to the level of the supports. This occurs at
about a 1:1 ratio for your structure (1" increase between supports lowers
the pin about 1") before second order effects. As the foundations spread,
the horizontal reactions increase. Iterate.

In your case, the lateral foundation stiffness at each support must be
800 k/in +/- to hold a 1" foundation separation. That's 1/2" displacements
at each support. Load factoring not considered.
(quick calc tie rod: 330 in^2, 1131 plf, 20" dia solid round, 200 ft long at
each arch - increase for load factors - how much does the arch weigh after
the first design pass?)

It is the sensitivity of the arch to base displacements that may be an
issue. The flatter the aspect ratio of the arch, the more sensitive it will
be. This can be checked easily.

Paul Ransom, P.Eng.
ph 905 639-9628
cell 905 802-3707
fax 905 639-3866

> From: "Kevin Below" <kbofoz(--nospam--at)>

> Paul, the arch is a continuous curve, so it is perfectly flat at the apex.
> I would like to understand your concern.  I can't see a mechanism forming
> until the top pin deflects to the level of the supports.  The geometry is a
> 200 ft span, with the central pin some 50 ft higher.
> On 10/9/07, Paul Ransom <ad026(--nospam--at)> wrote:
>>> From: "Kevin Below" <kbofoz(--nospam--at)>
>>> I will ask potential arch suppliers what they expect for the allowable
>>> movement.  They may reply that it doesn't affect their structure, because of
>>> the central pin.  It may become an architectural constraint.
>> Is the arch configured to be nearly flat at the apex (central pin)? If so,
>> you have potential catastrophic problems with base movement. Recall those
>> structures classes where stability and mechanisms were discussed. Run a
>> quick model and look at the first order deflected shape.

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