Graphical Analysis of a Sandstone Arch

Introduction

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This website explores the traditional graphical approach to designing masonry arches. Rainbow Bridge, a natural sandstone arch in San Juan County, Utah, is used as a test case. The first part of the website summarizes the stability approach to arch design. In the second part, Rainbow Bridge is evaluated using graphical methods based on stability theory.

Part 1 - Stability Analysis

For centuries, masonry arches and vaults were designed by reproducing the proportions of previous successful structures. Knowledge of the critical dimensions and ratios was jealously guarded and handed down by generations of master masons. Innovation progressed slowly, as the rules were adjusted to account for the observed behavior of completed work.

In 1675, Robert Hooke published his observations about the thrust-line and the catenary curve, initiating the development of analytical methods for designing arches. These new methods, referred to as "stability theory" enabled the rapid evolution of new kinds of arched structures to create the masonry infrastructure of the industrial revolution.

Stability analysis reached its apotheosis in the early years of the 20th century in the work of Antonio Gaudi[2], then virtually disappeared as masonry construction fell out of favor. After about 1920, steel and reinforced concrete became the default materials for long spans and large masonry arches were no longer constructed.

The final development of stability theory is documented in textbooks and technical handbooks published in the early 1900s, when graphical methods were widely used to solve engineering problems and while masonry arches and vaults were still being built. The authors of the time acknowledge that stability theory was incomplete – key elements of the underlying mathematics were not in place until well into the 20th century – but they were confident that stability analysis was reliable and that the low precision of its conclusions is appropriate given the irregular character of masonry construction.

The confidence of the fin de siècle theoreticians is vindicated, first, by the success of their structures and, second, when rigorous proof of their ideas was achieved, in the second half of the 20th century, in the form of limit analysis and the “safe theorem”, concepts borrowed from plasticity design of structural steel[3].

Today, stability theory is relegated to the back shelves of engineering libraries but there has been a resurgence of interest, especially in Europe, among engineers tasked with the maintenance and repair of historic masonry structures[4].

Characteristics of Masonry

The term “masonry” denotes an assemblage of discrete elements, usually stone or brick, held together primarily by gravity and friction. The individual pieces might or might not be cemented together with mortar, but the coherence of the mortar is so unreliable that it must be considered to be a filler not an adhesive. Even if the individual units (stones or bricks) are strong, the assembly as a whole is assumed incapable of resisting tension.

The shear strength of masonry is also determined by the joints between the stones. Disregarding the adhesive strength of any mortar which might be present, the shear resistance of masonry is solely due to friction of the stones sliding over each other. A compressive force must be present, acting normal to the shear-plane, to generate the friction needed to lock the stones together.

Masonry has good compressive strength, and because masonry structures tend to be relatively massive and stocky, they are less subject to buckling than the elongated struts typical of wood or steel construction.

The dimensions of most masonry structures are generous relative to the loads they support, resulting in very small internal stress, typically an order of magnitude less than the ultimate strength of the material. The critical issue is stability, not absolute strength. This has profound implications for analysis because strength design requires intimate knowledge of the mechanical properties of the material from which an arch is made, information that may not be reliable where the internal composition is concealed. Stability design focuses on the geometry of the structure and is largely unaffected by its material composition.

The Thrust Line

The primary forces acting on a free-standing arch such as Rainbow Bridge are vertical loads imposed by gravity and the horizontal force generated by the two halves of the arch leaning against each other. With no tensile and limited shear strength, masonry arches resist these vertical and horizontal forces by resolving them into a compressive force which acts along the curve of the arch.

In a symmetrical arch carrying a symmetrical load, the compressive force is horizontal at the crown and bends progressively downward toward to the ends of the span. The horizontal component of the force is constant over the length of the arch. The vertical component at any point is the sum of the gravity loads acting on the arch between that point and the summit. If the trajectory of this vector, known as the "thrust-" or "reaction-line", is safely contained within the body of the arch, the arch will be stable.

To determine the location of a thrust line, the distribution along the arch of the arch’s weight and of any superimposed loads, must be determined. This is done by slicing the arch conceptually, into a series of discrete segments and estimating the load at each slice. The loads are represented by vectors passing through each slice’s center of gravity.

The resultant of these vectors added to the horizontal force follows a faceted path known as the “funicular polygon”. The funicular approximates but does not exactly match the thrust line. It becomes a progressively better approximation if the facets are made smaller by dividing the arch into more segments. At the limit, as the slices become infinitesimal, the funicular polygon converges into a smooth curve. This curve is the thrust line[5].

Although frequently glossed-over, the distinction between funicular polygon and thrust line is important, especially if the slices are fairly large, because the apexes of the funicular polygon lie above the actual line of thrust. This could lead to errors if not taken into consideration when evaluating the thrust line’s conformance with criteria such as the “middle third rule” (See Figure 2).

Indeterminacy

Masonry arches are statically indeterminate.  For a given load, varying horizontal forces produce flatter or steeper trajectories for the thrust line.  Any of these alternative curves that fit within the boundaries of the arch represents a possible solution to the span but the “true” thrust line cannot be identified without an appeal to elastic theory. This is problematic. The characteristics of masonry, described above – discrete elements, no tensile strength, etc – are the opposite of an ideal elastic material which is homogeneous, isotropic, and has reliable tensile and shear strength. Fortunately, the serviceability of an arch can be evaluated without finding the location of the “true” line.

Geometric Factor of Safety

Although the position of the “true” thrust line is unknown, useful information about the behavior of an arch can still be obtained.

For the arch to be stable, the thrust line must lie within the body of the masonry. This limits the range over which potential thrust lines can vary. The extreme cases are lines that touch but do not cross the inside or outside edge of the arch (intrados or extrados).

Although theoretically stable, a thrust line at the edge of the masonry would cause a stress concentration at the edge that might exceed to crushing strength of the stone. Even if the masonry were able to support this, the arch would be teetering on the brink of failure, the slightest perturbance might shift the thrust line outside of the boundary. These conditions are avoided if the thrust line is restricted to a region set back comfortably from the edges of the masonry. The “middle third rule”, commonly cited in engineering texts, is an example of such a restriction.

In his book The Stone Skeleton[6], Jacques Heyman proposes a “geometric factor of safety” which he defines as the ratio between the width of the zone to which the thrust line is restricted and the overall width of the arch. An arch complying with the middle third rule would be assigned a factor of three. Heyman suggests that the minimum practical safety factor is about two. This factor ensures that the thrust will remain safely within the body of the masonry even if it is slightly displaced by unforeseen loading or by settlement or other minor changes to the arch’s shape.

The Safe Theorem

By applying the principal of “least work”, 19th century theoreticians concluded that the true thrust line of an arch will be the one, among the infinite possibilities, that results in the least internal work (least strain energy stored in the arch). This is usually the line lying closest to the arch’s center-line. If any viable thrust line can be identified, the designer can be assured that the arch will spontaneously also find that line or a better (less work) one[7].

These conclusions are confirmed by the “safe theorem”, a tenant of plasticity theory (limit analysis) developed between 1936 and 1960. The safe theorem states that if any thrust line can be found that complies with the criteria for safety (I.e. falls within the middle third) then the actual thrust line certainly also complies. There is no need to locate of the “true” line. In any event, pursuit of the “true” thrust line is futile, the line shifts continuously as the arch responds to temperature changes and other environmental inputs[8].

The Reference Plane

At each location along the arch, the compressive force represented by the thrust line is equal to the internal stress in the masonry summed over the cross-section of the arch. Ideally, the cross-section is normal to the thrust line. Because the location of the thrust line is ambiguous (and completely unknown at the beginning of analysis), for calculating stress, the section is assumed normal to the arch’s centerline. The discrepancy between the slope of the centerline and the thrust line is too small to cause a significant error in the calculation.

The section used in arch design should not be confused with the “normal section” familiar from elastic beam theory. The section used in beam analysis is normal to the neutral axis, in arch theory it is normal to the thrust line (or the arch’s centerline).

Middle Third Rule

The criteria most often cited for location of the thrust line is the “middle third rule”: The thrust line should be contained within the middle third of the thickness of the arch. The justification for this is that, for arches with a rectangular cross-section, location within the middle third ensures that the entire cross-section is subject to compression.

If the thrust line moves outside of the middle third, the part of the arch farthest from the thrust line will be unloaded. The joints between masonry units will tend to open in this region and cracks might form, extending into the arch until they reach the zone still subject to compression. The arch can function in this mode and might be perfectly safe so long as the thrust line still lies within the masonry, but the unloaded region will be subject to damage if water enters the cracks or if bricks or stones in this zone, relieved of the clamping force that holds them in position, shift or drop out of the arch.

Middle Fourth Rule

The middle third rule is frequently cited in engineering textbooks because most constructed arches have a rectangular cross-section. The same cannot be said for natural rock arches. Rainbow Bridge has an irregular cross-section and this section varies at every location along the arch. To simplify the analysis, the cross-section is idealized as elliptical. An ellipse has the advantages of better fitting into the actual profile of the bridge than a rectangle could and of having easily-calculated geometric properties.

The equivalent of the middle third rule for an eliptical section is a middle fourth rule (see figure 3). In the analysis of Rainbow Bridge, the thrust line will be restricted to the middle fourth of the depth of the section.

Middle Fourth Rule for an Ellipse

P = Compressive force (kips)

a = h/2 (Half height of section)

b = w/2 (Half width of section)

y = dist from centerline to fiber under consideration
(At extreme fiber y = a)

e = eccentricity of point of application of P
relative to center of section

M = moment = P∙e

A = area of ellipse = π∙a∙b

I = moment of inertia of ellipse = π∙a³∙b / 4

σ = stress (psi) = M∙y / I

At extreme fiber, y = a:

Maximum stress σ_max = σ_d + σ_b Stress at extreme fiber on same side of centerline as load

Minimum stress σ_min = σ_d – σ_b Stress at extreme fiber on opposite side of centerline from load

For no tension anywhere in section, σ_min must be ≥ 0.

Eccentric Load and Middle Quarter of an Elliptical Section

When compressive force P is centered on the cross-section, the internal stress is uniformly distributed across the section.

If P is off-center but still within the middle fourth of the section, the internal stress varies across the section with the minimum value (σ min) located at the edge of the section farthest from the point of application of P.

If P is on the edge of the center one-fourth, the internal stress varies across the section and equals zero at the edge of the section farthest from P.

If P is outside of the center fourth, the Internal stress varies across the section with σ min = 0 falling within the depth of the section. The part of the arch between the zero value and the edge of the section is unloaded and the effective depth of the section is reduced. The unloaded region is indicated by vertical hatch marks in figure 4.

Part 2 - Rainbow Bridge

Rainbow Bridge is a 273 foot wide[9] sandstone arch located near the south shore of Lake Powell, Utah.

In 2015, Rainbow Bridge was the subject of research investigating the arch’s response to ambient vibration. This study is described in the 2016 Research Letter Anthropogenic Sources Stimulate Resonance of a Natural Rock Bridge,[10] on the website of the American Geophysical Union.

The study included completion of an optical scan and preparation of a 3d computer model of the arch. The optical scan can be viewed online at Rainbow Bridge National Monument - 3D model by National Park Service Geologic Resources Division.

The 3d model, in .stl file format,is available for download from the University of Utah’s Geohazards website, Dynamics of rock arches - Rainbow Bridge (utah.edu).

This unique resource provides the information needed for the present project: stability analysis of the arch using traditional graphical methods.

Observations about the Bridge

When performing stability analysis, an arch is assumed to be a 2-dimensional system in which all loads and reactions lie in a single plane. Tunnel-like arches, which might be subject to different loads at different points across their width (such as a railroad overpass supporting two lines of track) are approximated by dividing their width into strips and considering the strips as separate, parallel, arches.

Rainbow Bridge, being relatively slender in plan, can be treated as a single strip. Except for a tapered area at the west abutment, where the bridge bears against an adjacent rock buttress, the two sides of the arch are nearly parallel. The cross sections at successive locations along the arch, though irregular, are roughly symmetrical with their centers of gravity falling near the arch’s centerline.

Figure 7: Plan and Elevations

Buttress at West End

The configuration of the west end of the arch introduces some out-of-plane forces that need to be reconciled with the “all in one plane” stipulation.

As can be seen in figures 5 and 8, the south side of the abutment bears against a rock buttress. The boundary between bridge and buttress is marked by a deep recess that extends all the way around the intersection, suggesting that a crack completely severs the bridge from the buttress. This separation is reflected in the 3d model, which renders the arch as a free-standing structure.

Figure 8: Lateral Support by Buttress

According to the model, the crack is slightly inclined and partially undercuts the west end of the arch. Fortunately, the lateral support provided by the buttress creates an exception to the requirement that all forces lie in the same plane.

Disregarding the effect of friction, the thrust line of the arch can be resolved into a component parallel to the plane of the crack and a propping force normal to the crack. Because the angle is slight, the component parallel to the crack carries most of the weight of the arch and can be treated as the continuation of the thrust line. The slightly out-of-plane deflection of the thrust line has negligible effect on the trajectory of the thrust line as viewed from the north, in the elevation used for the stability study (see Figure 8).

Analysis of Rainbow Bridge

Stability analysis checks that the shape of an arch is suitable for the configuration of the loads it is expected to support. The procedure focuses exclusively on geometry. Other than exceeding a basic threshold for compressive strength and having a uniform density throughout, the material properties of the arch are not a factor.

Assumptions

1. The sandstone of which Rainbow Arch is composed has a uniform density throughout the arch. With this assumption, the weight of individual slices of the arch can be obtained by measuring the volume of the slices.

2. The body of the arch contains no cracks that fall at acute angles to the compressive stress. Flaws that run across the arch, normal to the thrust line, are not critical; their effect would be similar to the joints between voussoirs in a constructed arch.

3. The working stresses in the arch are extremely low relative to the ultimate strength of the stone. This provides a generous margin of safety to account for inconsistencies in the sandstone.

The Limit Curves

The steepest and shallowest thrust lines that can fit within the center quarter are plotted. These represent the upper and lower bound of the range of acceptable lines. Although the location of the “true” thrust line cannot be determined, it must lie between these extremes.

Figure 9: Limit Curves

Internal Stress

The forces, cross-sectional areas, and eccentricity of the thrust line relative to the centerline of the arch are tabulated for each of the five sections labeled A through E. Using this information, the maximum stress is calculated for each of the sections, assuming that the effective cross-section is the ellipse used to establish the middle quarter. This overstates the stress, the actual cross-sections are larger. Even with this overstatement, none of the stresses exceeds 10% of the nominal crushing strength of sandstone, consistent with the observation that the stress in masonry arches is usually an order of magnitude lower than the ultimate strength of the material.

The true thrust line probably lies closer to the center-line of the arch than either of the limit curves. With less eccentricity, the actual stress will be less than calculated for the limits.

Method

For the most part, the method used here follows the suggestions of Jerome Sondericker’s 1907 book Graphic Statics[11]. 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[12], a technical handbook published in 1913. George Fillmore Swain’s Structural Engineering, 1927[13], provided valuable insights into interpreting the results, especially by pointing out the significance of the principle of least work. All of these books date from the early 20th century and represent the final state of the art, summarizing techniques that evolved over the preceding century.

For a step-by-step account of the procedure, see Appendix 1.

Conclusions

The product of stability analysis is a visual representation of the thrust line, a summary of the stress-field otherwise hidden within the stone. With this information, one can understand how the arch works and predict how it might evolve.

Natural arches sculpt themselves. At a gross level, as discussed in Appendix 1," Elevation of the Arch”; chunks might simply drop out of unstressed portions of the arch, adjusting the arch’s form to conform with its structurally-engaged core. At a more subtle level, recently-published research suggests that compressive stress has a strengthening effect on sandstone[14] [15]. The rock becomes harder at exactly its most-stressed points. Unloaded portions of the arch weather away, revealing a resistant core which is the physical manifestation of the stress-field generated by arch action.

A natural arch is balanced in equilibrium between the tendency for the geometry of the thrust line to assert itself and the preexisting order of strata and bedding. Rainbow Bridge represents one pole of a spectrum. The uniformity of its Navajo sandstone allows a pure expression of the thrust line. Delicate Arch, near Moab, Utah, exemplifies the other extreme, where the geometry of the arch can be seen struggling to emerge from the strata.

Figure 10: Delicate Arch

Arches National Park, Utah

Photo: Thomas Schoch / Wikimedia Commons

Copyright © , David Aynardi

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Footnotes

1. Huerta, 2001, p.49 ^
2. Huerta, 2006, p.324-339 ^
3. Heyman, 1999, p.99-102 ^
4. Huerta, 2001, p.64-68 ^
5. Baker, p.444 ^
6. Heyman, 1995, p.20 ^
7. Heyman, 1999, p.87-88 ^
8. Heyman, 1995, p.22 ^
9. National Park Service Geologic Resources Division, 3D model ^

10. Moore et al. ^
11. Sondericker, p.131-133 ^
12. Kidder, p.258 ^
13. Swain, p.411-412^
14. Thompson ^
15. Bruthans ^

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