Eads Bridge

Theory of the Ribbed Arch

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An appendix to James Eads' 1868 Report of the Engineer-in-Chief contains two articles describing the method used to analyze the arches. The first of these, The Theory of the Ribbed Arch,[1] introduces the equations used for the analysis. The second article, Numerical Results of the Preceding Formulae,[2] uses the equations to estimate forces, moments, and displacements at various locations on the arch. The appendix concludes with an impressive array of charts in which these data are overlayed to reveal the worst-case stress for each component of the arch.

The articles were written by Eads' assistant engineer, Charles Pfeifer, who was responsible for the calculations. A revised version, incorporating changes made after publication of the 1868 report, appears as chapters XXVI and XXVII of Woodward’s A History of the St Louis Bridge. Woodward’s chapters are the subject of the following discussion.

Pfeifer’s articles were written for an audience of professional engineers. He offers a step-by-step account of his procedure but assumes that his audience is familiar with the engineering principals involved. This and the next section of this website provide some of the background that Pfeiffer omitted.   Appendix 1 of the website offers a detailed commentary on Woodward's Chapter XXVI.

Configuration of the Arch

The arches are comprised of four parallel ribs spaced across the width of the bridge. The calculations consider the behavior of a single rib in the center span.

Each of the ribs is similar to a plane truss with curved upper and lower chords. The chords have thicker walls near the ends of the span. These reinforced sections deflect less than the unreinforced center part of the arch under the influence of the same load.

Individual members of the arch rib are subject to axial loads only; their slenderness and flexibility relative to that of the overall structure prevents individual members from experiencing significant bending loads.[4]

Derivation of the Equations

Scope of the Analysis

The use of fixed-end arches is a defining feature of Eads Bridge. The fixed ends stiffen the arches. This enabled them to be more-lightly constructed and permitted them to be erected with less shoring than would have been needed if the arches had hinged ends. The price of these advantages is that the fixed ends render the arches statically-indeterminate and difficult to analyze.

Charles Pfeifer’s calculations investigate the arch’s response to gravity loads and thermal expansion. He also considers the effect of wind and the stability of the piers but takes care to address these topics in ways that do not complicate the already forbidding problem of the indeterminate arch.

Wind loads are evaluated in a separate set of calculations and not added to the stresses in the chords until after the arch analysis is complete. The influence of wind on the arches is reduced by imposing the condition that the maximum wind load and maximum traffic load are not concurrent (Traffic will be suspended during a tornado).[5]

The bearing capacity and lateral stability of the masonry piers are checked. These investigations is kept rigorously separate from the analysis of the arches by the assumption that the piers do not deflect. Immobile support is a precondition for the elastic analysis of the arches.

Fundamentals

Arch and Catenary

Eads and his contemporaries often described suspension bridges as "inverted arches". This emphasized the similarity of the curve followed by the thrust-line of an arch and the curve assumed by a hanging cable or chain.  Both curves are the resultant of the forces acting on the span, only the sign is reversed; tension in the catenary vs compression in an arch.[11] [12]

Because it is flexible, a suspended chain naturally follows the path of the tensile forces acting on it. Unlike a catenary, a rigid arch can’t self-adjust. When designing an arch it is necessary to anticipate the load path so that the arch can be tailored to it. For centuries this was done empirically, by reproducing the proportions of previous successful arches. In 1675, Robert Hooke published his observations about the thrust-line and the catenary curve, initiating the development of analytical methods for designing arches.[13]

Masonry Arches

Masonry is considered to have no tensile strength and only sufficient shear strength to prevent individual bricks or stones from slipping out of position. With no tensile and limited shear strength, masonry arches are supported primarily by compressive force acting along the curve of the arch.

In a symmetrical arch carrying a uniformly-distributed load, the compressive force is horizontal at the crown and bends progressively downward toward to the ends of the arch. 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.  "Safely contained" is subject to interpritation and various definitions have been proposed. The "middle third rule", illustrated below, is representative. 

Masonry arches are 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. The "Safe Theorem" states that if, for a given distribution of loads,  any safe trajectory can be found (ie entirely within the middle third); then the arch will work for that load.[14]

For someone brought up in the modern construction industry, where structural efficiency is paramount, arches of this type have a through the looking-glass quality.  The material (brick or stone) has a relatively low cost per cubic foot and can be used in lavish quantities.  The proportions of the arch can be so generous that only a small portion of the ultimate strength of the material is used.  Instead of painstakingly predicting the location and magnitude  of the "true" thrust line, the arch is simply bulked-up until it can contain any likely trajectory.   If there is concern that concentrated live-loads, such as the weight of a locomotive, might perturb the location of the thrust-line; the dead-load is jacked up still more, until it overwhelms the moving load, by adding a layer of rubble or concrete over the length of the arch or by adding mass at strategic locations.  With superabundant strength, only the stability of the arch is at issue. This is checked by a relatively simple exercise in vector arithmetic, often accomplished by graphic means (see figure 2).

Metal Arches

All of this changed in the 19th century.  Iron and steel unlocked the door to unprecedented spans, but their cost per volume was orders of magnitude greater than that of brick or stone.  It was no longer feasible to simply throw material at the problem, now it was necessary to precisely model the location and magnitude of forces and to pare back the arch to the minimum skeleton that would sustain those forces.  This was possible because the new materials could resist tension.  This enabled moment-resisting designs, such as Eads Bridge,  in which the force-couple in the top and bottom chord  is used to compel the thrust line (now referred to as compressive force S') to conform to the neutral axis regardless of the configuration of the load.

Determinate and Indeterminate

The action of a two-dimensional structure such as the rib of an arch, in which all of the loads and reactions are contained within a single plane, is summarized by the three equations of static equilibrium: ∑ Fx = 0, ∑ Fy= 0 , and ∑ M = 0.

Determinate structures have a single path by which the loads are transmitted to the supports. For these structures the static equations on their own, without consideration of the elastic properties of the material, provide accurate predictions of forces and moments in the individual members of the structure.

Indeterminate structures have more than one load-path or mechanism of support. The load is allocated between pathways in proportion to the relative stiffness of the paths. The allocation cannot be determined by static analysis alone, it is necessary to consider the elastic properties of the structure. Because they are part of the same construction, the parallel paths are subject to identical deflection at points where they coincide. If the deflection can be predicted at these points, it can be used to determine the corresponding forces and moments in each pathway.

The arches at Eads Bridge are rigidly connected to their supports. There are three mechanisms of support. Part of the load is carried by direct compression along the curve of the arch, part by beam action as the arch resists deflection away from its original curvature, and part by cantilever action, as the fixed ends of the arch resist the rotation caused by deflection. At the ends of the arches these mechanisms contribute to a vertical and a horizontal force and to a bending moment; six unknown reactions in all. Values for three of the reactions are found by considering the elastic deformation of the arch. After these values are known, the equations for static equilibrium provide the remaining three reactions.

Stress and Strain

In the following discussion the term “stress” refers to force per unit of the cross-sectional area of a structural member. “Strain” refers to the change in the dimensions of a material caused by a stress, measured relative to the original dimensions of the material. (This is at variance with the use of the term “strain” by Pfeifer and Woodward; see sidebar)

The elastic analysis begins with the observation that, for wrought iron and steel, the relationship between stress and strain is linear over a wide range. Within this elastic range, the material deforms proportionally as stress is increased and returns to its original dimensions when the stress is removed. The elastic range has a clearly defined upper bound, the “elastic limit,” beyond which the linear relationship breaks down and permanent deformation occurs.

For a given force, the stress in a structural member can be adjusted by changing the member’s cross-sectional area. The designer’s task is to select member sizes that keep the stresses safely within the elastic range.

Neutral Axis

Concentrated or asymmetrical loads tend to flex the arch away from its original, unloaded, curvature. This bending generates tensile and compressive stresses within the rib, with maximum positive and negative values near the top and bottom of the section dropping to zero near the rib's centerline.

The line of zero bending stress, separating the zones of positive and negative, is referred to as the rib's "neutral axis". Because the rib’s cross-section is symmetrical, its neutral axis is centered approximately half-way between the chords.[3] (see figure 1)

Modulus of Elasticity

The relationship of stress to strain is a characteristic of the material out of which a structural member is made and is denoted by the material’s “modulus of elasticity”. “Hooke’s law” defines the modulus as the force per unit of area divided by the corresponding deformation per unit of length. For the steel used in the arches, Charles Pfeifer employs a modulus of elasticity of 1.944 x 10⁶ tons / ft² (27,000 ksi).[6]

Moment of Inertia

A truss or beam’s resistance to bending is determined not only by the modulus of elasticity of the material but also by the way the material is distributed over the cross-section of the truss or beam. This is quantified as the member’s “moment of inertia,” a geometric property based on the dimensions of the cross-section. For the truss-like arch ribs at Eads Bridge, in which the cross sectional area is concentrated in compact chords located far from the neutral axis, Pfeifer approximates the moment of inertia as:[7]

Load

The gravity loads carried by the arch are assumed to be uniformly distributed. To set up the equations, two load conditions are investigated: a symmetrical case in which the uniform load extends all of the way across the arch and an asymmetrical case in which the load extends across one of the reinforced end-pieces and part way across the unreinforced center section.

External and Internal Forces and Moments

External forces and moments include loads such as the weight of traffic and of the bridge itself and also the opposing forces (reactions) at the piers or abutments that keep the arch, considered as a whole, in equilibrium. Internal moments and forces are the stresses in the individual members of the arch by which the structure transfers the external loads to the supports.

Polar Coordinates

Pfeifer based his analysis on elastic beam theory but modifies the beam equations to use polar coordinates instead of the Cartesian (x, y) reference frame usually employed. The polar coordinate system, in which location is defined by angles and arc-lengths, provides compact descriptions of positions along the curved arch.

Method of Calculation

Constraints

Two constraints are required as preconditions for the calculations.

1. The length of the neutral axis is not significantly changed by bending. If part of the neutral axis is deflected to a shorter path inside the original curve of the arch, another part must bulge above the original curve, taking a longer path so that the overall length remains the same.
2. The ends of the arch are fixed in position. There is zero deflection at either end of the arch. To satisfy this condition, the sum of the elemental deflections over the full length of the arch must also be zero. (See figure 3)

Arch Action and Bending

As noted above, arch action and two bending mechanisms are active in the arches. By focusing first on bending, Pfeifer is able to derive equations for the three reactions at each end of the arch. With the reactions in hand, a second set of computations provides the compressive force generated by arch action. A final step evaluates the influence of the compressive force on the internal moments and the end reactions.

Normal Section

At any location along the span, behavior of the arch is evaluated relative to a section cut at right angles (normal) to the neutral axis. External forces and moments acting at the normal section must be in equilibrium with the internal forces in the members of the arch intersected by the section.

The loads and reactions between one end of the arch and the normal section can be resolved into a force S and a moment M acting at the point where the neutral axis intersects the section. S can be further resolved into a force S’ acting along the arch, tangent to the neutral axis and a shear force S” acting along the plane of the normal section, toward or away from the center of the arch. (See figure 4)

The influence of moment M on bending is considered first. Because force S and its components S’ and S” act through the point where the neutral axis intersects the section, they do not directly influence the moment and bending calculations. The effects of S’ and S” are considered at a later stage in the calculations.

Rotation of the Normal Section

The external moment M at the normal section is resisted by a force-couple comprised of the axial stress in the top and bottom chords of the arch. These forces slightly stretch one chord and compress the other, resulting in a minute change to the curvature of the neutral axis at the point where M is applied. Because the section remains normal to the neutral axis before and after deflection, the change of curvature can be quantified as a change of the angle of the normal section relative to vertical, an “angular deflection.” The rotation of the normal section is proportional to the moment at each point on the neutral axis.

Bending

Steps in the bending analysis:

1. An equation is written describing moment as a function of position along the arch. Terms in the equation include the loads on the bridge and the (as yet unknown) reactions at the ends.
2. Equations are derived for the angular, vertical, and horizontal deflection of any point on the neutral axis, stated as a function of moment. Constants in the equations describe the mechanical properties of the arch.
3. The moment equation is substituted for the value of M in the deflection equations, producing new equations which state vertical, horizontal,and angular deflection as a function of position along the arch and of the three unknown reactions. Constants in the equations describe the loads on the bridge and the mechanical and dimensional properties of the arch.
4. At the fixed ends of the arch, the value of all three deflections is, by definition, zero.

   Replacing the position variable with its value at the end of the arch provides equations for the deflection at the end, stated in terms of the three reactions. In this form, the equations are amenible to simultaneous solution: they share a common value (zero), they have the same variables (the unknown reactions), and there are at least as many equations as variables. By substitution and elimination, terms are consolidated and three new equations derived, giving the vertical, horizontal, and moment reactions as functions of the (known) loads.

5. With values for the reactions, moments can be computed anywhere along the arch and the associated stress in the chords can be determined.

Complicating factors

The preceding is an idealized description. If different parts of the arch are subject to different conditions of loading, separate moment equations are required to describe each load condition. If different parts of the arch have different mechanical properties, they will require their own deflection equations.

The arch is divided into regions with differing moments of inertia and further subdivided according to the configuration of the load it carries.

Asymmetrical Loading

A uniform deck load extends from one end of the arch, across one reinforced end-piece and part way across the unreinforced center part of the arch. (The “Loaded Area” in figure 5).

For a given set of loads and reactions, two equations can be written describing M as a function of location along the neutral axis; one for locations within the loaded part of the arch and another for locations outside of the loaded area. Two equations can be written stating angular deflection as a function of M, one for locations within the reinforced end-piece of the arch and the other for locations in the unreinforced part of the arch . Substituting the moment equations for the value of M in the deflection equations produces four equations describing the angular deflection (the change of curvature) of an element of the neutral axis, as a function of the element’s position in the loaded, unloaded, reinforced, and unreinforced sections of the arch. (See figure 5 above. Equations VII, 7, 8a, and 8b in commentary)

Total Angular Deflection

The magnitude of M varies along the arch, generating a different angular deflection in each infinitesimal element of the arch. The absolute rotation of the normal plane at any location (the total angular deflection) is the sum of the incremental deflections of all of the elements between that location and either of the fixed ends of the arch. (See figures 6, 7)

Integrating the equations for elemental deflection produces a new set of equations stating total angular deflection for the given loads and the (as yet unknown) reactions, as a function of position within each of the four parts of the arch. (Equations 9a, IXa, IXb, 9b, in the commentary)

Horizontal and Vertical Deflection

The total angular deflection of each element alters the element’s vertical and horizontal dimensions. Equations are written stating these linear deflections in terms of the total angular deflection. Substituting the angular deflection equation for its value restates these equations directly in terms of loads, reactions, and constants. As with angular deflection, the total vertical and horizontal deflection at any point is the sum of the elemental x and y deflections between that point and one of the fixed ends of the arch. Equations for total deflections are obtained by integrating the equations for elemental linear deflections. (Equations 13 and XIII, 14 and XIV, in commentary)

Piecewise Solution

Angular, vertical, and horizontal deflections are functions of position along the arch. Because the arch is not uniform, the deflection functions are defined “piecewise”, by separate equations representing the different characteristics of each section of the arch (loaded, unloaded, reinforced, unreinforced). To assemble the function for the full length of the arch, these equations are stitched together by constants of integration that align their values at the points of transition.

At the ends of the arch, the angular, vertical, and horizontal deflections are constrained to be zero. At the transitions between regions within the arch, the deflections of adjacent sections are known to be identical. At these locations, the equations can be set equal to each other or to zero, enabling terms to be consolidated or eliminated. By successive substitutions and subtractions, constants of integration are restated in terms of loads and reactions and the deflection equations are reformulated to state the three end reactions in terms of loads and other constants.

With the reactions at one end of the arch defined, simple statics provides the reactions at the other end. Given the end reactions and loads, the value of the moment can be computed for any location along the arch and with that information, the chord forces can be determined.

Symmetrical Loading

When a uniform load extends all the way across the arch, there are only two conditions of loading: Loaded end- and loaded center-piece. Because the load is symmetrical, the vertical reactions at the ends of the arch are identical and are equal to half of the gravity load. Also because the load and the construction of the arch are symmetrical, the deflections and consequently the moment reaction at both ends of the span are the same. Only half of the arch, including one center and one end-piece, needs to be considered.

With these simplifications, analysis of the symmetrical loading condition follows generally the same procedure as the asymmetrical case.

Shear and Force Along the Arch

After the end reactions are known, the “vertical force” (vertical shear), V, can be calculated for any location. The horizontal reaction, Q, is constant over the length of the arch. At any location along the arch, V and Q are resolved into S’ acting along the arch, tangent to the neutral axis, and S” acting at a right-angle (normal) to the neutral axis. (See figure 8)

At each location, S’, the force acting along the neutral axis, is borne equally by the top and bottom chord, where it is added to the forces generated by bending. The normal shear, S”, produces the stress which will be carried by the diagonal bracing.

figure 8 – S’ and S”

Temperature Change

Temperature changes cause the arch to expand and contract uniformly in all directions. The abutments resist the horizontal component of thermal expansion, imposing a horizontal force on the arch. The magnitude of this force is computed from the dimensions of the arch, the degrees of temperature change, and the coefficient of expansion of the material from which the arch is made. Following a procedure similar to that used for symmetrical loading, equations are developed for the moment and the angular, vertical, and horizontal deflections generated by this force, all stated as functions of location along the arch.

S" and S', the shear and force along the arch produced by temperature change, are computed as for gravity loads except that, because the arch is free to expand vertically, the value of the “V” term in the equations is zero and the term can be omitted. S’ and S” due to temperature change are added to the shear and compressive force previously calculated for gravity loads.

Axial Shortening

When predicting the deflections caused by bending, it is assumed that the arc-length of the neutral axis is the same before and after flexure. This assumption does not apply to the force acting along the length of the arch. Axial force S’ compresses the arch, shortening the neutral axis. This produces moments and stresses in the arch similar to those produced by reduced temperature. The effect is mitigated by over-sizing arch components by the amount they will be shortened by the load they are expected to carry. Before assembly, the arch will be slightly too long for the span and will need to be “sprung” into position. When the arch is loaded, force S’ compresses the arch to the dimensions anticipated in the calculations. The compressed arch slumps into its designed profile and the moments and forces generated by this deflection neutralize those applied during assembly to squeeze-in the over-sized pieces.

The “extra” length to be added to arch components is established for an S’ half-way between that caused by the span with no traffic on it (dead load only) and the span carrying the maximum allowed live load. Forces generated when S’ departs from this neutral value are estimated using the equations for thermal expansion.

According to Woodward, the difference between “neutral” and maximum or minimum loading is equivalent to a temperature change of about eleven degrees Fahrenheit. The corresponding change to S’ is about 70 tons per chord. This is accommodated by sightly increasing the cross-sectional area of the chords above the area required for gravity and temperature absent the effect of axial shortening.

The elaborate shoring scheme with its hydraulically actuated towers and adjustable closure pieces was necessary, in part, to pre-load the partially completed arch, compressing components to their in-service dimensions so that they could be assembled. For a description of the shoring scheme, see the “Erection of the Arches” article on the “Design and Construction” page of this website.

Compromises

The elastic analysis is logical in summary but proves unwieldy in practice. Integrating the deflections is cumbersome because of the many terms in the equations. The needed labor is compounded because the deflection equations must consider the permutations of loaded and unloaded, reinforced and unreinforced. To keep the calculations manageable, a number of simplifying assumptions are applied.

Permanent Load

The dead load of the structure (permanent load) is treated as a uniform 1 ton per lineal foot of deck. This ignores the varying height of the spandrel uprights which must result in greater weight near the piers.

Moving Load

The live load (moving load) is also represented as a uniform load (0.8 tons per foot of deck in the area covered by a train). Concentrated loads are not considered. (See sidebar “Deck Loading”).

Traffic on the upper deck (pedestrians and horse-drawn vehicles) is not addressed. Possibly these loads are subsumed within the 0.8 ton per foot “moving load”.

Reinforced Sections

The last five segments of the chord tubes at each end of the arch are reinforced by progressively increasing their wall thickness closer to the abutment. This is represented in simplified form in the calculations, as a single stage of reinforcement in the final 1/12 of the span.[8]

Sensitivity

Elastic design depends on a precise description of the deflection of the arch. This is problematic because, since the deflections are very small relative to the dimensions of the bridge, it is necessary to carry some of the calculations out to 7 or 8 decimal places. This renders the analysis very sensitive to small departures from the assumptions underlying the equations and to discrepancies caused by imperfect construction. Foremost among potential issues is the condition that the ends of the arch must be immobile. It is axiometric that the piers and abutments deflect when loaded and this deflection was observed during the proof loading of the bridge. This would appear to undermine the credibility of the calculations.

Charles Pfeifer was aware of this problem and addressed it in his article “Calculation of the Stresses in Arch Bridges” in the June 1876 edition of Van Nostrand’s Eclectic Engineering Magazine.[9]

The great accuracy in the construction of the abutments with regard to their distance and their angle of inclination, which is required of the correctness of the strains, might seem difficult to attain, and even, if attained may be disturbed again by a slight settling of the abutments. The consequences of such an event, however, would not be serious. Whenever, on account of a slight deviation from the dimensions assumed in the calculations some part of the arch is strained beyond the elastic limit, the permanent set thus caused must necessarily be a step towards restoring the conditions upon which the calculation has been based; hence the arch has a tendency to accommodate itself in the course of time to its abutments.

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Footnotes

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