Appendix 1
Graphic Analysis, Step-By-Step
For this case-study, thrust lines are plotted using graphical techniques that were developed during the 19th century to aid the design of brick and stone arches. Computer programs are available now which automate this investigation but the hands-on procedure has the advantage of making explicit the logic of the analysis.
For the most part, the method used here follows the suggestions of Jerome Sondericker’s 1907 book Graphic Statics[1]. The reference planes used to locate the arch’s centerline and cross-sections are positioned as recommended in The Architect’s and Builder’s Pocket-Book[2], a technical handbook published in 1913.
The Model
The .stl file from the University of Utah’s Geohazards website was edited using Trimble Sketchup Pro. In Sketchup, the scale was adjusted to match the 273.3 ft span reported by annotation on the NPS’s optical scan. The arch was then sliced into 20 segments of equal width. A larger slice at each end of the span represents the abutments. The center of gravity of each segment was marked in the model. The weight of each slice was estimated by multiplying its volume, reported by Sketchup’s “Entity Info” pallet, by the assumed density of sandstone (150 lbs per cubic foot).
After these manipulations, a 2d elevational view of the model was transferred from Sketchup into TurboCAD Designer 2017, a 2d CAD drawing program. Turbocad was used to prepare the diagrams needed for stability analysis.
Idealized Elliptical Cross-Section
Rainbow Bridge has an irregular cross-section and this section varies at every location along the arch. To simplify the analysis, the cross-sections are idealized as elliptical. These ellipses are made as wide as possible within the confines of each slice of the arch, subject to keeping them aligned with the CG of the actual section. The height of each ellipse is adjusted so that it extends from the interior to the exterior curve of the active portion of the arch.
The ellipse that defines the base of the east (left) abutment extends the full width of the arch's abutment. The ellipse at the base of the west (right) abutment is restricted to the wider part of the abutment. The narrow end of the bridge, extending to the west of the ellipse, is excluded from the abutment and not included in the analysis. (The ellipses and cross-sections appear in Figure 9 on the home page of this website.)
The portions of the arch that fall outside of the ellipses are treated as non-structural. Their weight is included in the load assigned to each slice but only the area of the ellipses is considered when computing stresses. This is conservative, stresses computed in this way will be higher than in the actual structure.
For more about the ellipses, see the article Middle Fourth Rule, on the home page.
Elevation of the Arch
A built arch is distinguished by the radial pattern of the joints between the wedge-shaped “voussoirs” that comprise the arch ring. Masonry above the ring is treated as a dead load supported by the arch.
Because Rainbow Bridge is monolithic this convenient demarcation between structural and non-structural is not available. For the case study, limits of the arch are defined by the following considerations.
The effective area of the arch is the region stressed by the compressive force acting along the curve of the arch.
The bottom of the structurally-engaged region is assumed to coincide with the bottom surface (soffit) of the bridge. If this is not the case and the structurally-active portion of the arch is elevated above the bottom of the bridge, the part of the bridge extending below will not be subject to the clamping action of the arch’s compression. Fragments could easily drop out of this unloaded region, causing the soffit to migrate upward until it reaches the loaded portion of the arch-ring. In this way, the soffit spontaneously conforms itself to the structural shape of the arch.
The upper curve of the arch is not subject to the same spontaneous expression as the bottom. Stone extending above the structurally-active region simply rests on the arch and will stay put indefinitely, or at least until it succumbs to erosion. Since the upper boundary of the structurally-engaged region isn’t visible, it is assumed to follow the highest and widest pressure line that can be made to fit within the confines of the bridge. This line intersects the outer limit of the idealized elliptical section at each abutment and touches the top surface of the bridge near the arch’s summit. This curve does not represent a viable thrust line. Rather, it should be thought of as the approximate path of the stress in the extreme fiber of the active portion of the arch.
Effective Profile
The following illustrations show the steps followed to define the effective profile of the arch:
Digital Model: North elevation of the Sketchup model prior to export of 2-d information. Slices have been cut and centers of gravity are marked. This image provided the information needed to prepare elevation drawing used for stability analysis.
Overall Profile: The profile of arch and the boundaries of the slices are traced. The weights of the slices are listed at top of the drawing. The line of action of each weight is denoted by a light dotted line passing through the slice’s center of gravity.
To reduce the influence of surface irregularities, the outer two feet of the arch are excluded from the thrust line analysis. The dotted lines paralleling the perimeter of the elevation represent this two-foot setback.
Upper Boundary: To define the upper boundary of the structurally-active portion of the arch, a preliminary pressure-line is plotted.
See “Elevation of the Arch”, above, for explanation of the significance of the upper boundary.
The Elevation Drawing
The upper boundary of the arch, defined above, is added to the elevation drawing.
The approximate location of the arch’s centerline is established by graphical means. (see Appendix 2).
After the centerline is established, reference planes, are drawn normal to the centerline. These planes are located at the intersections of the centerline with the boundaries between slices 1 through 20 (See Appendix 3 for discussion of vertical slices and reference planes). The reference planes define the position of the idealized elliptical cross-sectons of the arch.
Two curves are plotted through points located 1/8 of the width of the arch, measured along the reference planes, on either side of the centerline. These are the boundaries of the “middle fourth” of the arch which will be used to establish the viability of potential thrust lines.
With this preparation, the elevation drawing is ready to serve as the background for stability analysis.
1 Test Polygon – Least Thrust
The least-thust line is the thrust line representing the smallest admissible horizontal force. It will be the narrowest, tallest, curve that can fit within the center quarter of the arch.
The process of plotting a thrust line can commence at either the right or left end of the arch. Because Rainbow Bridge is slightly asymmetrical, a plot originating at the left is less likely to fall below the center quarter immediately above the plot’s origin.
The method suggested by Sondericker[3] for plotting a funicular polygon through three given points is used to create a test line. Two points will be the ends of the inside edge of the center quarter of the arch. A third point is selected near the summit of the upper boundary of the middle quarter. The trick of projecting the ordinates of the polygon is used to locate this point. (See Appendix 4)
The test polygon (red) does not fit within the middle quarter of the arch. The greatest discrepancy is marked by the arrows labeled “X” on Figure 13.
2 Least-Thrust Line
A slightly wider curve is needed to fit within the center quarter. The shape of the test polygon suggests that the narrowest, tallest curve that will fit, will touch the inside of the center quarter at the left base, the outside near the summit, and will be tangent to the inside of the center quarter near the point “X” located during the previous pass. The left base and point "X" are selected as two of the points defining the new polygon.
The method of projecting ordinates is used again to find the point where the summit of the proposed funicular will touch the top of the center quarter. This establishes the third of the three points that define the funicular polygon. All of this polygon falls within the middle quarter.
Having constructed the funicular polygon (red), the thrust line is plotted (black). The thrust line is a smooth curve plotted through the points where the polygon intersects the reference planes at slices 1 through 20 and also at the left and right bases of the arch.
3 Test Polygon – Maximum Thrust
A preliminary attempt is made to plot the thrust line corresponding to the greatest horizontal force. This will follow the widest, flattest trajectory that will fit within the center quarter. To accomplish this, base points are selected at the extreme ends of the outside curve of the center quarter. An apex point is selected near the summit of the inside curve. The plot of a funicular polygon that will pass through these points is started but discontinued when it is clear that the curve of this thrust line will rise above the center quarter.
Inspection of the plot suggests that an acceptable thrust line would be tangent with the outside of the center quarter in the vicinity of point "Y".
4 Maximum Thrust Line
Taking "Y" as one of the bases, a new thrust line is plotted. This line falls within the middle quarter. This is the maximum-thrust line, corresponding with the greatest horizontal force that satisfies the middle-quarter criteria.
5 Limit Curves
The minimum and maximum thrust curves plotted in the steps above define the range which must contain the "true" thrust line. For discussion, see the Limit Curves article on the home page of this website.
Footnotes
- Sondericker, p.131-133 ^
- Kidder, p.258 ^
- Sondericker, p.25 - 27 ^