Appendix 3
Method of Estimating Distributed Load
As described under “The Thrust Line”, above, the load is allocated across the length of an arch by slicing the arch into segments and estimating the load segment by segment.
When analyzing a constructed arch, for which the boundaries of the arch ring and the angles of the voussoir joints (reference planes) are known in advance, the usual practice is to configure the slices so that the divisions fall along the reference planes 1 . This ensures that the volume of the slices above any reference plane corresponds with the tributary area generating the loads imposed on that plane. (Figure 20b)
This approach is not applicable to a natural arch because the boundaries of the effective arch ring are concealed within an undifferentiated mass of stone. Prior to establishing the boundaries, there is no way to define reference planes.
An alternative method is provided by Frank Kidder’s The Architect’s and Builder’s Pocket-Book 2 , a technical handbook published in 1913. Kidder suggests dividing the arch into vertical slices first, then locating the reference planes relative to these slices (Figure 20a). Originally proposed as a labor-saving strategy, this approach enables the slices to be defined and the funicular polygon constructed even though the reference planes are not yet known. With the funicular polygon in hand, the boundaries of the arch and the reference planes can be located later, as described under “The Elevation Drawing”, in Appendix 1.
Results obtained using vertical slices are approximate. As can be seen in Figure 20c, the cumulative area of the slices above reference plane “x” (grey in figure 20c) does not exactly match the tributary area that actually bears on plane “x” (hatched in the figure). This results in a slight disagreement between thrust lines based on radial verses vertical slices.
Thrust lines based on vertical and radial slices are superimposed in Figure 20c. The curves agree at the summit and base of the arch but diverge slightly in the lower quarter, near reference plane “x” in the drawing. The thrust-line plotted using vertical slices (black) drops below the line based on radial slices (red). At the scale of this illustration, the discrepancy is only about a line-thickness, too small to have a significant influence the outcome of the Rainbow Bridge analysis.
Footnotes