Numerical Results
The appendix to the 1868 Report of the Engineer-in-Chief of the Illinois and St. Louis Bridge Company includes two articles by Eads’ chief assistant engineer, Charles Pfeifer, describing the calculations used in design of the arches. The second of these, “Numerical Results of the Preceding Formulae”, uses the equations derived in the first article to predict the stress in components of the arch. A revised version of “Numerical Results” is presented as chapter XXVII of Woodward’s A History of the St Louis Bridge.Chapter XXVII describes a monumental clerical task. Each equation had to be solved repeatedly to account for the permutations of load caused by railroad traffic and by changing temperature. After the full cycle of equations was complete for one location, it had to be repeated until results were obtained for each of 17 stations equally-spaced along the arch. The solutions to all of these calculations were presented in tabular form.
The tables list moments and forces for every load configuration, but the undifferentiated mass of data offers no insight into which are the critical combinations that should guide the design of individual members of the arch. To extract this information, Pfeifer resorts to graphical analysis employing a series of elegant diagrams which sift the data, revealing the governing load in each component of the arch.
Today, the calculations can be performed and the diagrams generated by a computer spreadsheet. The results obtained correlate well with those reported by Woodward. Discrepancies, where they occur, are small and can be attributed to the spreadsheet’s rounding algorithm and to the interpolation of values by Pfeifer’s “computers” as they consulted tables of logarithms and trig functions. The recreated charts are presented below with commentary. For comparison, the original diagrams can be consulted on Plate VI, V, and VI of A History of the St Louis Bridge.
Signs
In the Eads Bridge calculations, Charles Pfeifer employs the following sign conventions.
Force: Compression = positive Tension = negative
Moment: “Hogging” = positive Sagging = negative
Shear: Down at left = positive Up at left = negative
These are the opposite of modern practice. For consistency with the historical documents, Pfeiffer’s convention will be used in the following discussion.
Arch Action and Bending
In the First Report of the Engineer in Chief, in the chapter titled “Suspension and Upright Arch Bridges”, James Eads develops an analogy between suspension and arch bridges. The stiffening truss common in suspension bridges and the bracing incorporated into his proposed arches serve the same function – to prevent deformation of the arch or catenary under the influence of concentrated or asymmetrical loads.[1]Eads' discussion of the bracing recognizes the simultaneous participation of two distinct load-carrying mechanisms. Arch action generates compressive force S’ which acts along the arch’s neutral axis. Bending generates shear normal to the neutral axis and positive and negative forces in the top and bottom chords. Although not explicitly stated in Pfeifer’s articles, the distinction between arch action and bending is helpful when interpreting his analysis.
From Eads “First Report of the Engineer in Chief”
The bracing of the arch-rib and the stiffening truss on the suspension span perform the same function.
Deflection Calculations
Pfeifer’s “strength” calculations predict the forces that will be imposed on individual members of the arch so that they can be sized appropriately. The “deflection” calculations predict how much the arch will deform under the load it carries.The graphical analysis discussed on the following pages summarizes the strength calculations. The deflection calculations will be the subject of a future article. For the time being, the reader is directed to the discussion on Woodward page 353.
Plate IV, Diagram 1
This chart shows bending moments as a train advances onto the arch from the left end, extending progressively 1/8, 2/8, up to 8/8 of the way across. A uniform “moving load” of 0.8 tons per foot of deck is assigned to the part of the span covered by the train. The “moving load” is added to a “permanent load” of 1 ton per foot of deck, extending all of the way across the bridge, representing the weight of the bridge itself. The moments generated by the combined load are calculated for 17 equally-spaced stations along the arch. The resulting data points are plotted with a separate color-coded line for each load combination.
The curve labeled “DL” (dead-load only) and the curve labeled “8/8” (uniform moving load applied to entire span) display very small moments at any location along the arch. These symmetrical loads conform closely to the ideal weight distribution for pure arch action and are kept in equilibrium almost exclusively by the compressive force S’ acting along the neutral axis.
Plate IV – Diagram 2
Arch-action generates compressive force S’ acting along the neutral axis of the arch rib. Half of this force is carried by the top and half by the bottom chord. Each chord’s share of S’ is added to the positive and negative bending forces described in Diagram 1. The sum is plotted separately for each chord and each condition of loading as a train moves across the bridge. These curves are then plotted a second time, reversed left for right, to show the effect of a train entering from the other end of the arch. The maximum and minimum chord force at any position along the arch, generated by any load condition, is described by enveloping lines connecting the extremes of all of the curves. The enveloping lines are labeled Max T, Min T, Max B, Min B in the diagram.
Plate V – Diagram 1
Thermal expansion changes the length of the arch but the abutments do not move. The restrained ends exert a horizontal force, compressing or stretching the span. The arch flexes and the crown rises and falls as needed to accommodate the changed length of the material.The horizontal force exerted on the arch by the abutments generates a force S’ acting along the neutral axis of the arch, shear S” acting in a plane normal to the neutral axis, and moment M which varies along the arch in proportion to the height of the neutral axis above the ends of the span. M imposes positive and negative axial forces on the chords of the arch.
Plate V - Diagram 1 describes the chord forces caused by an 80 degree temperature change above or below an assumed 60 degree “neutral temperature”. This is the sum of the axial forces due to M plus each chord’s share of S’ due to temperature.
T- and B- are the top and bottom chord forces due to a decrease of 80 degrees.
The horizontal force imposed by thermal expansion causes a parabolic moment distribution. The secondary moments generated by the fixed ends of the arch have the effect of shifting the curve vertically, reducing the maximum and increasing the minimum mid span moments and chord forces.
Plate V – Diagram 2
At any location along the arch, the chords must accommodate the greatest force generated by temperature change plus the maximum force due to any distribution of gravity loads.In Plate V, Diagram 2 the light lines labeled Max T, Max B, etc are copies of the enveloping lines from Plate IV, Diagram 2. They represent chord forces due to gravity. The “Total Max” lines are derived from these by adding the maximum and minimum force due to temperature change from Plate V, Diagram 1.
The required cross sectional area of the chords can be read from the scale at the right edge of the chart. The areas are based on an allowable compressive stress of 15 and tensile stress of 10 tons per square inch. In the center part of the span a sectional area of 65 square inches is adequate. At the ends, the required area goes up to about 106 square inches.
The maximum compressive force (Total Max T, Total Max B) determines the required cross sectional area of the chords at every location. The minimum force (Total Min T, Total Min B) is significant near the ends of the arch, where it has a negative value, denoting regions where the chords are sometimes loaded in tension. Although the cross sections established to resist compression are more than adequate for the tension forces, the grooved couplings that connect the chord tubes and the anchor bolts that secure the arches to the piers must be designed to resist this tension.
Axial Shortening
As noted under "Axial Shortening" in Part 1, the areas of the chords should be increased slightly to accommodate the chord forces generated by shortening of the arch under the influence of force S’.According to the as-built dimensions of the chords furnished by Woodward,[2] the first chord tube at the ends of the arch has an area of 116.66 square inches. The mid-span tubes have a cross section of 74.84 square inches. In both cases, about 10 square inches have been added to the area recommended by Diagram 2.
Plate VI Diagram 1, Shear due to Moving plus Permanent Load
This diagram looks at the normal shear force, S”. Shear is plotted for the empty span (DL only) and for each load condition as a train advances from the left 1/8, 2/8 etc of the way across. The enveloping lines, labeled “L Max” and “L Min” describe the extreme magnitude of shear at each point along the span. The enveloping lines are reversed left for right and top for bottom and plotted a second time to represent the effect of a train entering from the right end of the bridge (“R Max and “R Min”).The maximum positive shear at any location along the span occurs on the “R Max” line at the ends and the “L Max” line at the center of the span. Maximum negative shear falls on the “L Min” line at the ends and and “R Min” near the center of the arch.
Plate VI, Diagram 3
Plate V, Diagram 1 depicts the chord forces generated by temperature change. This diagram displays the normal shear associated with those chord forces. The magnitude of normal shear is the derivative of the moment at any point along the arch. The graph of moment due to temperature is a parabola. The graph of Its derivative, normal shear, is a straight line.
Plate VI, Diagram 4
The diagonal bracing is sized to resist the combined shear due to gravity loads and temperature change. The light lines labeled “Max” and “Min” are derived from the maxima and minima lines from Plate VI, Diagram 1 and show the shear generated by the combined live and dead gravity loads. At each point along the curve, the corresponding shear due to temperature change, from Plate VI, Diagram 3, is added to produce the heavy lines.
The pairs of eyebars that comprise each diagonal are reinforced against buckling so that they can function equally well as compression or tension members.
Detail, Woodward Plate XXXI
Footnotes