Graphical Analysis of a Sandstone Arch

Webpage developed by David Aynardi

Sandstone Arch Introduction Part 1 - Stability Analysis Part 2 - Rainbow Bridge References
Appendix 1. Analysis Step by Step 2. Finding the Centerline 3. Distributed Load 4. Previewing Thrust Line Contact Form
Other Projects
Sandstone Arch Introduction Part 1 - Stability Analysis Part 2 - Rainbow Bridge References
Appendix 1. Analysis Step by Step 2. Finding the Centerline 3. Distributed Load 4. Previewing Thrust Line Contact Form
Other Projects

Appendix 4

Previewing Points on a Thrust Line

If two funicular polygons are constructed for the same system of parallel forces, the ratio between the lengths of each ordinate in one of the polygons to the length of the corresponding ordinate in the other, is the same 1. This provides a convenient method for previewing part of a funicular polygon without constructing all of it. Such previews are invaluable where successive trials are necessary to fit a funicular to the shape of an arch.

Ordinates
Figure 21: Ordinates Prior to Construction of a Thrust Line

Example

It is proposed to construct a thrust-line which passes through the two points marked by red dots in Figure 21. The dashed red line connecting these dots is the baseline of the funicular polygon associated with this thrust line. Somewhere between these points, the line will be tangent with the upper boundary of the center quarter of the arch. Point b’ is a tentative candidate for the point of tangency.

The thrust line is coincident with the funicular polygon at the points where the funicular intersects the reference planes.  b' is defined as a point on the thrust line but because it lies on a reference plane, it also fixes a point on the funicular polygon. With this point defined, the adjacant portion of the polygon can be constructed and the fit of the thrust line can be evaluated.

A preliminary funicular polygon and a first ordinate pair, a-b and a'-b', are constructed.   a-b is an ordinate of the preliminary polygon, a'-b' is the corresponding distance from the baseline of the proposed funicular to point b’. Two additional ordinate pairs, having the same ratio of length as a-b to a’-b’, are constructed, centered below the upper boundary / reference plane intersection at either side of b'.

Ordinates - Detail
Figure 22: Detail, Summit of Ordinate Pairs near Point b'

Figure 22 shows ordinate a'-b' at the upper boundary of the center quarter. Close inspection reveals that the adjacent ordinates fall short of the boundary, indicating that the polygon and thrust line drop away in both directions. This confirms b’ is a good approximation of the point of tangency. (Tangency is actually slightly to the right of b’, but the discrepancy is insignificant)    b' is accepted as the third point defining the funicular polygon. The polygon and thrust line are constructed accordingly.

If b’ had not been the high-point and successive ordinates trended higher above the center quarter in one direction or the other, additional ordinate pairs would have been constructed until the high-point was located. In this case, the intersection of the ordinate extending the farthest above the upper boundary, with the boundary, would be the approximate point of tangency and would be used instead of b’ to construct the funicular polygon.

Between the reference planes, the polygon (red) lies above the thrust line. Apexes of the polygon can be seen in Figure 22, on either side of point b’.



Copyright © , David Aynardi
Footnotes
  1. Sondericker, p.22-23, 26^


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